American House-Carpenter, 1873

Page 1

THE AMERICAN HOUSE-CARPENTER. A TREATISE ON THE ART OF BUILDING, AND THE STRENGTH OF MATERIALS. BY R. G. HATFIELD, Arcnirecr, MEM. AM. INST. OF ARCHITECTS. BEVENTH EDITION, REVISED AND ENLARGED WITH ADDITIONAL ILLUSTRATIONS, NEW YORK: JOHN WILEY & SON, 15 ASTOR PLACE. 1873. r

Page 2

Á Entered sccording to Act of Congress, in the year 1857, by R. G. HATFIELD, Ia the Clerk's Office of the District Court of the United States, for the Southern District of New York.

Page 3

PREFACE. Tuts book is intended for carpenters—for masters, journeymen and apprentices. It has long been the com- plaint of this class that architectural books, intended for their instruction, are of a price so high as to be placed beyond their reach. This is owing, in a great measure, to the costliness of the plates with which they are illus- trated: an unnecessary expense, as illustrations upon wood, printed on good paper, answer every useful pur- pose. Wood engravings, too, can be distributed among the letter-press ; an advantage which plates but partially possess, and one of great importance to the reader. Considerations of this kind induced the author to undertake the preparation of this volume. The subject matter has been gleaned from works of the first autho- rity, and subjected to the most careful examination. The explanations have all been written out from the figures themselves, and not taken from any other work ; and the figures have all been drawn expressly for this book. In doing this, the utmost care has been taken to make everything as plain as the nature of the case would admit. The attention of the reader is particularly directed to the following new inventions, viz; an easy method of describing the curves of mouldings through three given —

Page 4

Vv PREFACE. points; a rule to determine the projection of eave cor nices; a new method of proportioning a cornice to a larger given one; a way to determine the lengths and bevils of rafters for hip-roofs; a way to proportion the rise to the tread i stairs; to determine the true position of butt-joints in hand-rails ; to find the bevils for splayed- work; a general rule for scrolls, &c. Many problems in geometry, also, have been simplified, and new ones intro- duced. Much labour has been bestowed upon the sec- tion on stairs, in which the subject of hand-railing is presented, in many respects, in a new, and it is hoped, more practical form than in previous treatises on that subject. The author has endeavoured to present a fund of use- ful information to the American house-carpenter that would enable him to excel in his vocation; how far he has been successful in that object, the book itself must determine, yew York, Oct. 15, 1844. FIFTH EDITION. Sryce the first edition of this work was published, I have received numerous testimonials of its excellent practical value, from the very best sources, viz. from the workmen themselves who have used it, and who have profited by it. As a convenient manual for reference in respect to every question relating either to the simpler. operations of Carpentry or the more intricate and

Page 5

PREFACE. abstruse problems of Geometry, those who have tried it assure me that they have been greatly assisted in using it. And, indeed, to the true workman, there is, in the - study of the subjects of which this volume treats, a con- tinual source of profitable and plzsurable interest. Gentlemen, in numerous instances, have placed it in the hands of their sons, who have manifested a taste for practical studies; and have also procured it for the use of the workmen upon their estates, as a guide in their mechanical operations. I was not, then, mistaken in my impressions, that a work of this kind was wanted; and this evidence of its usefulness rewards me in a measure for the pains taken in its preparation. New York, Oct. 1, 1852. SEVENTH EDITION. Ir is now thirteen years since the first edition of the American House Carpenter was published. The attempt to furnish the recipients of this book with a fund of useful information in a compact and accessible form, has been so far successful that the sixth edition was exhausted nearly a year ago. At that time it was determined, before issuing another edition, to make a thorough revision of the work. The time occupied in this labour _has been unexpectedly prolonged by at least six months, and this has resulted from various causes, but more especially from the absorbing nature of my professional duties. A large portion of the work has been rewritten,

Page 6

yi PREFACE. about 130 pages of new matter intr.duced and many new cuts inserted. The most important additions to the work will be found in the section on Framing or Construction. Here will be found, now first published, the results of experi- ments on such building materials as are in common use in this country, and an extended series of rules for the application of this experimental knowledge to the prac- tical purposes of building. Some of the rules are new, while others heretofore in use have been simplified. This section has been much improved, and it is hoped that it will be of service, not only to the house carpenter but also to the architect and civil engineer. In preparing the criginal work, a desire to state the eubjects treated of in terms suited to the comprehension of all classes of workmen, precluded the use of algebra- ical symbols and formule. In this edition, however, it has been deemed best to introduce them wherever they would contribute to the clearer elucidation of the sub. ject; but care has been taken to state them in a siraple form at first, and so to explain the symbols as they are introduced that those heretofore uninstructed in regard to them, may comprehend what little is here exhibited, and at the same time be induced to pursue the study more fully in works more strictly mathematical. But for those who may not succeed in comprehending the algebraical formule, it may be stated that all the practical deductions derived from them are written out in words at length, o as to be fully understood without their assistance. Kew York, Sept. 1, 1857. R. G. HL

Page 7

TABLE OF CONTENTS. InTROpUCTION.—Directions for Drawing . . . à Q SECTION L—PRACTICAL GEOMETRY Definitions . Ge . e . e . . Problems on Lines and Angles . è e « ¢ « Problems on the Circie e . ° . e e . Problems on Polygons . œ . . . ° e Problems on Proportions . . ° e ° e ° Problems on the Conie Sections . e . 5o o e Demonstrations.—Definitions, Axioms, &. . . . . Demonstrations.—Propositions and Corollaries . . ° SECTION IL—ARCHITECTURE. History 5 . . o : : o ° ° i . Styles.—Origin, Definitions, Proportions . . . ° Grecian Orders.—Doric, Ionie and Corinthian : . . Roman Orders.—Doric, Tonic, Corinthian and Composite . Beypiai Stylo . e. e e e e`% o >e œ Buildings generally . o o : ° ° . Plans and Elevation fora City Dwelling . : c o Principles of Architecture.—Requisites in a Building . : Principles of Construction.—The Foundations, Column . 3 Principles of Construction.—The Wall, Lintel, Arch . : Principles of Construction.—The Vault, Dome, Roof . e arta, 1-14 15-70 71-80 81-92 93-106 107-110 111-128 130-139 140-167 168-181 182-196 197-211 212-215 216, 217 218-222 223, 224 225-229 230-233 233-235 236-238

Page 8

Vili TABLE OF CONTENTS. BECTION IIL—MOULDINGS, CORNICEB, eo. AERTS Mouldings — Elements, Examples . . . . e 239-256 Cornices.—Designs . z ; . 5 . : . é 3 251 Cornices— Problems e ry . . . . a ° e . 2 5 2-2 5 6 SECTION IV.—FRAMING, OR CONSTRUCTION. First Principles.—Laws of Pressure . > + + + + « 257-282 Resistance of Materials.—Strength, Stifness ; ‘ 6 4 . 283-286 Resistance to Compression.— Various kinds . ; : : : . 287-290 Results of Experiments on American Materials, Tables I, IL . . 291-293 Practical Rules and Examples . c 5 : 3 : : . 294-305 Resistance to Tension . . . : . 306 Results of Experiments on American Materials, Table IIL - « « 807, 308 Practical Rules and Examples . : ; 0 . : ò - 309-316 Resistance to Cross Strains.—Strength, Stiffness . . c i . 3817-319 Resistance to Deflection.—Stiffness, Formula . 5 z - 3820-322 Practical Rules and Examples . é : o : : : . 823-326 Table 1V.— Weight on Beams, Formule . : ó : : : 326 Practical Rules and Examples . . . . . . . . 827-329 Table V.—Dimensions of Beams, Formula . " x : ; , 329 Resistance to Ruptnre —Strength : . . 3 ‘ . 331 Results of Experiments on American Materials, Table VE: : ; 331 Table VII.—Safe Weight on Beams, Formule . A e 9 E 333 Practical Rules and Examples . ; : 5 . : : : 333 Table VIII.—Dimensions of Beams, Formulæ z 7 : `: 334 Practical Rules and Fxamples . a 5 A : : : a 334 Systems of Framing, Simplicity of Designs . C : 6 : 335 Floors.—V arious, Cross-furring, Reduction of Formulæ ; 5 - 836, 337 Practical Rules and Examples . 4 o 4 : . o . 3838-344 Bridging-strips, Girders, Precautions Partitions. —Examples, Load on Partitions, &c. Roofs.—Stability, Inclination T . . - 354, 355 Load. —Roofing, Truss, Ceiling, Wind,- fade é ¢ é ò . 3856-358 Btrains.— Vertical, Oblique, Horizontal A : F è : . 359-368 Resistance of the Material in Rafter and Tie-beam Dimensions.—Rafter, Braces, Tie-beam, Iron Rods Practizal Rules and Examples Table IX.—Weight of Roofs, per Foot . . ‘ . « 845-349 ° ° ` ° e 350-353 . ° . 368 . . . . 870-374 . 7 e 375-383 . ° . . 37€

Page 9

TABLE OF CONTENTS. Examples of Roofs: . . Problems for Hip-rafter Domes.—Examples, Area of Ribs Problems in Domes . . Bridges—Examples . ° Rules for Dimensions . |- Abutments and Piers . . Stone Bridges, Centreing . Joints in Timberwork . c tron Work.—Pins, Nails, Bolts, Straps Tron-Girders.—Cast Girder, Bow-string, Brick Arch Practical Rules and Examples SECTION V.—DOORS, WINDOWS, &a. Doors.—Dimensions, Proportions, Examples Windows.—Form, Size, Arrangement, Problems . Principles, Pitch Board : Platform Stairs, Cylinders, Rail, Face Mould Winding Stairs, Falling Mould, Face Mould, Joints Elucidation of Butt Joint . SECTION VL—STAIRS, Quarter-circle Stairs—Falling Mould, Face-Mould Face Movld.—Elucidation . Face Moulds.—Applied to Plank, Bevils, &c. Face Moulds.—Another methad . Scrolls, Rule, Falling and Face Moulds, Newel Cap SECTION VIL—SHADOWS. Shadows on Mouldings, Curves, Inclinations, &. . ° Shadows.—Reflected Light . Algebraical Signs aT a Trigonometrical Terms . APPENDIX. + 1X ARTA 384-326 ` 387-388 389-391 392-398 399-401 401-405 406, 407 408—417 418—427 428 429-435 431-435 436-441 442-448 449-456 457-468 469-476 417 478-480 481 482-48. 485-488 489-498 499-572 523

Page 10

x TABLE OF CONTENTS. PAGE Glossary of Architectural Terms. . . œ o . 7 Tables of Squares, Cubes and Roots . . a oo 6 ¢ Pa 18 Rules for Reduction of Decimals . ` . ° ° e . 27 Table of Areas and Circumferences of Circles . . oo o 29 Table of Capacity of Wells, Cisternas, 40. . e e « «© -« 33 Table of Areas of Polygons, dc . . ° ` ‘ . . 34 Table of Weights of Materials . « e ° . . ° . 35

Page 11

6 AMERICAN HOUSE-CARPENTER. inch taking for every foot one of those divisions, and for every inch one of the subdivisions into twelfths ; and proceed in like manner in proportioning a drawing to any of the other divisions of the scale. An instrument in the form of a semi-circle, called a protractor, and used for laying down and measuring angles, is of much service to surveyors, but not much to carpenters. 11.—In drawing parallel lines, when they are to be paraliel to either side of the board, use the T-square; but when it is required to draw lines parallel to a line which is drawn in a direction oblique to either side of the board, the set-square must be used. Let a b, (Fig. 2,) be a line, parallel to which it is Fig 2. desired to draw one or more lines. Place any edge, as c d, ot the set-square even with said line; then place the ruler, © h, against one of the other sides, as c e, and hold it firmly; slide the set-square along the edge of the ruler as far as it is desired, as at f; and a line drawn by the edge, i f, will be parallel to a b. 12.—To draw a line, as X% 4, (Fig. 3,) perpendicular to another, as a b, set the shortest edge of the set-square at the line, a b; place the ruler against the longest side, (the hypothenuse of the right-angled triangle ;) hold the ruler firrnly, and slide the set- square along until the side, e d touches the point, Æ; then the line, 2, drawn by it, will be perpendicular to a b. In like

Page 12

INTRODUCTION. 7 manner, the drawing of other problems may be facilitated, as wil) be discovered in using the instruments. Fig. 3. 13.—In drawing a problem, proceed, with the pencil sharpened to a point, to lay down the several lines until the whole figure is completed; observing to let the lines cross each other at the several angles, instead of merely meeting. By this, the length of every line will be clearly defined. With a drop or two of vater, rub one end of the cake of ink upon a plate or saucer, until a sufficiency adheres to it. Be careful to dry the cake of ink; because if it is left wet, it will crack and crumble in pieces. With an inferior camel’s-hair pencil, add a little water to the ink that was rubbed on the plate, and mix it well. It should be diluted sufficiently to flow freely from the pen, and yet be thick enough to make a black line. With the hair pencil, place a little of the ink between the nibs of the drawing-pen, and screw the nibs together until the pen makes a fine line. Beginning with the curved lines, proceed to ink all the lines of the figure ; being careful now to make every line of its requisite length. If they are a trifle too short or too long, the drawing will have a ragged appearance; and this is opposed to that neatness and accuracy which is indispensable to a good drawing. When the ink is dry, efface the pencil-marks with the india-rubber. If

Page 13

8 AMERICAN TIOUSE-CARPENTER. the pencil is used lightly, they will all rub off, leaving those lines only that were inked. 14.—In problems, all auxiliary lines are drawn light ; while the lines given and those songht, in order to be distinguished at a glance, are made much heavier. The heavy lines are made so, by passing over them a second time, having the nibs of the pen scparated far enough to make the lines as heavy as desired. If the heavy lines are made before the drawing is cleaned with the rubber, they will not appear so black and neat; because the india-rubber takes away part of the ink. If the drawing is a ground-plan or elevation of a house, the shade-lines, as they are termed, should not be put in until the drawing is shaded; as there is danger of the heavy lines spreading, when the brush, in shading or coloring, passes over them. If the lines are inked with common writing-ink, they will, however fine they may be made, be subject to the same evil; for which reason, india-ink is the only kind to be used.

Page 14

THE AMERICAN HOUSE-CARPENTER. SECTION IL—PRACTICAL GEOMETRY. DEFINITIONS. 15. — Geometry treats of the properties of magnitudes, 16.—A point has neither length, breadth, nor thickness, 17. —A line has length only. 18.—Superficies has length and breadth only. 19.—A plane is a surface, perfectly straight and even in every direction ; as the face of a panel when not warped nor winding. 20.—<A solid has length, breadth and thickness. 21.—A right, or straight, line is the shortest that can be drawn between two points. , 22.—Parallel lines are equi-distant throughout their length. 23.—An angle is the inclination of two lines towards one another. (Fig. 4.) Fig. 4. Fig. 5 Fig. 6

Page 15

10 AMERICAN HOUSE-CARPZENTER. 21.—A right angle has one line perpendicular to the othe:. (Fig. 5.) 25.—An oblique angle is either greater or less than a right angle. (fig. 4 and 6.) 26.—An acute angle is less than a right angle. (Fig. 4.) 27.—An obtuse angle is greater than a right angle. (Fig. 6.) When an angle is denoted by three letters, the middle one, in the order they” stand, denotes the angular point, and the other two the sides containing the angle ; thus, let a b c, (Fig. 4,) be the angle, then 6 will be the angular point, and a b and b c will be the two sides containing that angle. 28.—A triangle is a superficies having three sides and angles, (Fig. 7, 8, 9 and 10.) as /\ Fig. 7. Fig, & j 2?9—An equi-lateral triangle has its three sides equal. (Er. 7.) 30.—An isosceles triangle has only two sides equal. (Fig. 8.) 31.—A scalene triangle has all its sides unequal. (Fig. 9) ae Fig. iv, 32.--A right-angled triangle has one right angle. (Fig. 10.) 33.—An acute-angled triangle has all its angles acute. (Fig. 7 and 8.) 34.—An obtuse-angled triangle has one obt.se angle, (Fig. 9.) 35.— A quadrangle has four sides and four ann (Fig. 11 to 16.)

Page 16

PRACTICAL GEOMETRY. Il Ea Fig. 11. Fig. 12. 36.—A parallelogram is a quadrangle having its opposite sides parailel. (Fig. 11 to 14.) 37.—A rectangle is a parallelogram, its angles being right angles. (Fig. 11 and 12.) 38.—A square is a rectangle having equal sides, (Fig. 11.) _ 39.—A rhombus is an equi-lateral parallelogram having ob- lique angles. (Fig. 13.) Fig. 13 Fig. 14. A0.—A rhomboid is a parallelogram having cnn angles. (Fig. 14.) 'Al.—A trapezoid is a quadrangle having only two of its sides parallel. (Fig. 15.) Fig. 15. * Fig. 16. 42.—A trapezium is a quadrangle which has no two of its sides parallel. (Fig. 16.) 43.—A polygon is a figure bounded by right lines. 7 44.—A regular polygon has its sides and angies equal. 45.—An irregular polygon has its sides and angles unequal. 46.—A trigon is a polygon of three sides, (Fig. 7 to 10,) a tetragon has four sides, (Fig. 11 to 16;) a pentagon has

Page 17

12 AMERICAN HOUSE-CARPENTER. five, (Fig. 17 ;) a heragon six, (Fig. 18;) a heptagon seven, (Fig. 19;) an octagon eight, (Fig. 20;) a nonagon nine; a decagon ten; an undecagon eleven; and a dodecagon twelve sides. Fig. 17. Fig. 18. Fig. 19. Se A7.—A circle is a figure bounded by a curved line, called the circumference ; which is every where equi-distant from a cer- tain point within, called its centre. The circumference is also called the periphery, and sometimes the circle. 48.—The radius of a cirele is a right line drawn from the centre to any point in the circumference. (a b, Fig. 21.) All the radii of a circle are equal. Fig. 21. 49.—The diameter is a right line passing through the centre, and terminating at two opposite points in the circumference. Hence it is twice the length of the radius. (cd, Fig. 21.) 50.—An arc of a circle is a part of the circumference. (c b or bed, Fig. 21.) 51.—A chord is a right line joining the extremities of an arc. (bd, Fig. 21.)

Page 18

X PRACTICAL GEOMETRY. 13 52.—A segment is any part of a circle bounded by an arc and its chord. (A, Fig. 21.) 53.—A sector is any part of a circle bounded by an arc and two radii, drawn to its extremities. (B, Fig. 21.) 54.—A quadrant, or quarter of a circle, is a sector having a quarter of the circumference for its are. (C, Fig. 21.) 55.—A tangent is a right line, which in passing a curve, touches, without cutting it. (fg, Fig. 21.) 56.—A cone is a solid figure standing upon a circular base diminishing in straight lines to a point at the top, called its vertex. (Fig. 22.) d h Fig. 22. Fig. 23. 57.—The axis of a cone is a right line passing through it, from the vertex to the centre of the circle at the base. 58.—An ellipsis is described if a cone be cut by a plane, not parallel to its base, passing quite through the curved surface. (a b, Fig. 23.) 59.—A parabola is described i a cone be cut by a plane, parallel to a plane touching the curved surface. (e d, Fig. 23— c d being parallel to f g.) - 60.—An hyperbola is described if a cone — cut by a plane, parallel to any plane within the cone that passes through its vertex. (eh, Fig. 23.) 61.—Foci are the points at which the pins are placed in de- scribing an ellipse. (See Art. 115, and f, f, Fig. 24.)

Page 19

14 AMERICAN HOUSE-CARPENTER. - Fig. 24. 62.—The transverse axis is the longest diameter of the ellipsis. (a b, Fig. 24.) 63.—The conjugate axis is the shortest diameter of the ellipsis ; and is, therefore, at right angles to the transverse axis. (ed, Fig. 24.) 64.—The parameter is a right line passing through the focus of an ellipsis, at right angles to the transverse axis, and, termina- ted by the curve. (g hand g t, Fig. 24.) 65.—A diameter of an ellipsis is any right line passing through the centre, and terminated by the curve. (A‘l, or m n, Fie, 24.) 66.—A diameter is conjugate to another when it is parallel to a tangent drawn at the extremity of that other—thus, the diame- ter, m n, (Fig. 24,) being parallel to the tangent, o p, is therefore conjugate to the diameter, & l. g 67.—A double ordinate is any right line, crossing a diameter of an ellipsis, and drawn parallel to a tangent at the extremity of that diameter. (i ¢, Fig. 24.) 68.—A cylinder is a solid generated by the revolution of a right-angled parallelogram, or rectangle, about one of its sides ; and consequently the ends of the cylinder are equal circles. (Fig. 25.)

Page 20

PRACTICAL GEOMETRY. 15 NR Fig. 25. Fig. 26. 69.—The aris of a cylinder is a right line passing through it, from the centres of the. two circles which form the ends. 70.—A segment of a cylinder is comprehended under three planes, and the curved surface of the cylinder. Two of these are segments of circles: the other plane is a parallelogram, called by way of distinction, the plane of the segment. ‘The circular segments are called, the ends of the cylinder. (Fig. 26.) NV. B.—For Algebraical Signs, Trigonometrical Terms, &e., see Appendix.

Page 21

PROBLEMS. RIGHT UINES AND ANGLES. 71.— To bisect a line. Upon the ends of the line, a b, (Fig. 27,) as centres, with any distance for radius greater than halt e d Fig. 27. a b, describe arcs cutting each other in c and d; draw the line, c d, and the point, e, where it cuts a b, will be the middle of the line, a b. In practice, a line is generally divided with the compasses, or dividers ; but this problem is useful where it is desired to. draw, at the middle of another line, one at right angles to it. (See Art. 85.) d ` c ` a b Fig. 28. 72.— To crect a perpendicular. From the point, a, (Fig. 28,)

Page 22

& PRACTICAL GEOMETRY. 17 set off any distance, as a b, and the same distance from a toc, upon c, as a centre, with any distance for radius greater than c a, describe an arc at d; upon b, with the same radius, describe another at d ; join d and a, and the line, d a, will be the per- pendicular required. This, and the three following problems, are more easily per- formed by the use of the set-square—(see Art. 12.)' Yet they are useful when the operation is so large that a set-square cannot be used. a Fig. 29. 73.—To let fall a perpendicular. Let a, (Fig. 29,) be the point, above the line, b c, from which the perpendicular is re- quired to fall. Upon a, with any radius greater than a d, de- scribe an arc, cutting b c at e and f; upon the points, e and f with any radius greater than e d, describe arcs, cutting each other at g ; join a and g,and the line, a d, will be the perpen- dicular required. Fig. 30. 74.— To erect a perpendicular at the end of a line. Let a, (Fig. 30,) at the end of the line, c a, be the point at which the- perpendicular is to bé erected. Take any point, as b, above the line, c a, and with the radius, b a, describe the arc, d a e. through d and b, draw the line, d e ; join e and a, then e a will be the perpendicular required.

Page 23

15 AMERICAN HOUSE-CARPENTER. T'he principle here made use of, is a very important one ; and is applied in many other cases—(see Art. 81, b, and Ari. 84. Vor proof of its correctness, see Art. 156.) f a c b Fig. 31. 74, a.—A second method. Let b, (Fig. 31,) at the end of the line, a b, be the point at which it is required to erect a perpen- dicular. Upon b, with any radius less than b a, describe the are, ced; upon c, with the same radius, describe the small are at e, and upon e, another at d ; upon e and d, with the same or any other radius greater than half e d, describe arcs intersecting at f ; join f and b, and the line, f b, will be the perpendicular required. This method of ereeting a perpendicular and that of the fol- lowing article, depend for accuracy upon the fact that the side of a hexagon is equal to the radius of the circumscribing circle. Fig. 32. T4, h—A third method. Let b, (Fig. 32,) be the given point at which it is required to erect a perpendicular. Upon b, with any radius less than b a, describe the quadrant, d e f; upon d, with the same radius, deseribe an are at e, and upon e, another at c,

Page 24

PRACTICAL GEROMETRY. 19 throigh d and e, draw d c, cutting the arc in ce; jcin c and 6, then cb will be the perpendicular required. This problem can be solved by the sir, eight and ten rule as it is called; which is founded upon the same principle as the problems at Art. 103, 104; and is applied as follows. Let a d, (Fig. 30,) equal eight, and a e, six ; then, if d e equals ten, the angle, e a d, is a right angle. Because the square of six and that of eight, added together, equal the square of ten, thus: 6 x6 = 36, and 8 x 8 = 64; 36 + 64 = 100, and 10 x 10 = 100. Any sizes, taken in the same proportion, as six, eight and ten, will produce the same effect: as 3, 4 and 5, or 12, 16 and 20. (See Art. 103.) By the process shown at Fig. 30, the end of a board may be squared without a carpenters’-square. All that is necessary is a pair of compasses and a ruler. Let c a be the edge of the board, and a the point at which it is required to be squared. Take the point, b, as near as possible at an angle of forty-five degrees, or on a mitre-line, from a, and at about the middle of the board. This is not necessary to the working of the problem, nor does it affect its accuracy, but the result is more easily obtained. Stretch the compasses from b to a, and then bring the leg at a around to d; draw a line from d, through b, out indefinitely ; take the dis- tance, d b, and place it from b to e; join e anda; then e a will be at right angles toe a. In squaring the foundation of a build- ing, or laying-out a garden, a rod and chalk-line may be used instead of compasses and ruler. 75.—To let fall a perpendicular near the end of a line. Let e, (Fig. 30,) be the point above the line, c a, from which the perpendicular is required to fall. From e, draw any line, as e d, obliquely to the line, c a; bisect e d at b; upon b, with the radius, b e, describe the arc, e a d ; join e and a; then e a will be the perpendicular required. c S \ Fig. 33. 76.—To make an angle, (as e d f, Fig. 33,) equal to a given angle, (as b ac.) From the angular point, a, with any radius describe the arc, b c ; and with.the same radius, on the line, d e,

Page 25

20 AMERICAN HOUSE-CARPENTER. and from the point, d, describe the are, fg; take the distance, b c, and upon g, describe the small are at f; join fand d ; and the angle, e d f, will be equal to the angle, b a c. If the given line upon which the angle is to be made, is situa- ted parallel to the similar line of the given angle, this may be performed more readily with the set-square. (See Art. 11.) Fig. 34. 77.— To bisect an angle. Let a b c, (Fig. 34,) be the angle to be bisected. Upon b, with any radius, describe the arc, a c; upon a and c, with a radius greater than half a c, describe arcs cutting each other at d ; join b and d; and b d will bisect the angle, a b c, as was required. This problem is frequently made use of in solving other pro- blems ; it should therefore be well impressed upon the memory. b c Fig. 35. 78.— To trisect a right ang’e. Upon a, (Fig. 35,) with any radius, describe the arc, bc; upon b and c, with the same radius, describe arcs cutting the arc, b c, at d and e ; from d and e, draw lines to a, and they will trisect the angle as was required. The truth of this is made evident by the following operation. Divide a circle into quadrants: also, take the radius in the divi- ders, and space off the circumference. This will divide the circumference into just six parts. A semi-circumference, there-

Page 26

FRACTICAL GEOMETRY. 21 tore, ïs equai to three, and a quadrant to one anà a half of those parts. The radius, therefore, is equal to % of a quadrant; and this is equal to a right angle. e | a = ~ A a Fig. 36. 79.— Through a given point, to draw a line parallel toa given line. Let a, (Fig. 36,) be the given point, and b c the given line. Upon any point, as d, in the line, b c, with the radius, d a, describe the arc, a c; upon a, with the same radius, describe the arc, d e ; make d'e equal to ac; through e and a draw the line, e a ; which will be the line required. This is upon the same principle as Art. 76. í Fig. 37 j 80.— To divide a given line into any number of equal parts. Let a b, (Fig. 37,) be the given line, and 5 the number of parts. Draw a c, at any angle to ab; on ac, from a, set off 5 equal parts of any length, as at 1, 2, 3, 4 and c ; join c and b; through the points, 1, 2, 3 and 4, draw 1 e, 2 f, 3 g and 4 h, parallel te c b; which will divide the line, a b, as was required. The lines, a 6 and ac, are divided in the same proportion. (See Art. 109.) THE CIRCLE. 81.— To find the centre of a circle. Draw any chord, as a b

Page 27

y he AMERICAN HOUSE-CARPENTER. e A E; e d Fig. 38. (Fig. 38,) and bisect it with the perpendicular, ¢ d; tiscct ¢ a with the line, e f, as at g ; then g is the centre as was required. d Fig. 39. 81, a—A second method. Upon any two points in the e1- cumference nearly opposite, as a and b. (Fig. 39,) describe arcs cutting each other at cand d: take any other two points, as e and f, and describe ares intersecting as at g and h ; join g and A, and ¢ and d ; the intersection, o, is the centre. This is upon the same principle as Art. 85. Fig. 40, 81, b.—A third method. Draw any chord, as a b. (Fig. 40,)

Page 28

PRACTICAL GEOMETRY. 23 and from the point, a, draw a c, at right angles to a 6 ; join c and b ; bisect c b at d—which will be the centre of the circle. If a circle be not too large for the purpose, its centre may very readily be ascertained by the help of a carpenters’-square, thus : app'y the corner of the square to any point in the circumference, as ata; by the edges of the square, (which the lines, a b and a c, represent,) draw lines cuttirig the circle, as at b and c ; join band c=; then if b c is bisected, as at d, the point, d, will be the centre. (See Art. 156.) Fig. 41. 82.—At a given point in a circle, to draw a tangent thereto, Let a, (Fig. 41,) be the given point, and b the centre of the cir- cle. Join a and b ; through the point, a, and at right angles to a b; draw c d ; then cd is the tangent required. ZI i aa D G Fig. 42 $3.— The same, without making use of the centre of the circle. Let a, (Fig. 42,) be the given point. From a, set off any distance to b, and the same from b to c; join a and c, upon a, with a b for radius, describe the arc, d b e; make d b * equal to be; through a and d, draw a line; this will be the tangent required. The correctness of this method depends upon the fact that the angle formed by a chord and tangent is equal to any

Page 29

24 AMERICAN HOUSE-CARPENTER. inscribed angle in the opposite segment of the circle, (Art. 163;) a b being the chord, and b ¢ a the angle in the opposite segment of the circle. Now, the angles dab and bca are equal, because the angles d a 6 and b a c are, by construction, equal; and the angles ba cand b ca are equal, because the triangle æ bc is an isosceles triangle, having its two sides, æ } and 6 c, by construction equal; therefore the angles d a b and bcaare equal. Fig. 43. 84.—A circle and a tangent given, to find the point of con- tact. From any point, as a, (Fig. 43,) in the tangent, % c, draw a line to the centre d ; bisect a d at e y upon e, with the radius, e a, describe the arc, a f d ; f is the point of contact required, Iff and d were joined, the line would form right angles wit the taugent, de. (See Art. 156.) j AG Fig. 44, 85.—Through any three points not in a straight line, to draw ee Let a, b and c, (Fig. 44,) be the three given points. Upon a and 6, with any radius greater than half a b, describe

Page 30

iso PRACTICAL GEOMETRY. 5 arcs intersecting a: d and e; upon db and c, wita any radius greater than half 6 c, describe arcs intersecting at f and g; through d and e, draw a right line, also another through f and g; upon the intersection, 4, with the radius, A a, describe the circle, a b c, and it will be the one required. Fig. 45, 86.—Three points not in a straight line being given, to find . a fourth that shall, with the three, lie in the circumference of a circle. Let a b c, (Fig. 45,) be the given points. Connect them with right lines, forming the triangle, æ ¢ b; bisect the angle, c b a, (Art..77,) with the line 6d; also bisect c a in e, and erect e d, perpendicular to æ c, cutting b d in d; then dis the fourth point required. A fifth point may be found, as at f, by assuming a, d and 3, as the three given points, and proceeding as before. So, also, any number of points may be found; simply by using any three already found. This problem will be serviceable in obtaining short pieces of very flat sweeps. (See Art. 397.) The proof of the correctness of this method is found in the fact that equal chords subtend equal angles, (Art. 162.) Join d and c; then since a e and e c are, by construction, equal, therefore the chords a d and d c are equal; hence the angles they subtend, d 6 a and d bc, are equal. So likewise chords drawn from a to f, and from f to d, are equal, and subtend the equal angles, d bf and fb a. Additional points, beyond æ or b, may be obtained on the same principle. To obtain a point beyond a, on b, as a centre, describe with any radius the are io n, make o n equal to 07; through b and n draw b g; on was 4

Page 31

26 AMERICAN HOUSE-CARPENTER. centre and with af for radius, describe the arc, cutting g b at g, then g is the point sought. d e Fig. 46. f 7.—To describe a segment of a circle by a set-triangle. Let a b, (Fig. 46,) be the chord, and ¢ d the height of the segment. Secure two straight-edges, or rulers, in the position, ¢ e and cf, by nailing them together at c, and aflixing a brace from e to f; putin pins at a and b; move the angular point, c, in the direction, æ ¢ b; keeping the edges of the triangle hard against the pins, æ and b ; a pencil held at ¢ will describe the are, a ¢ b. A curve described by this process is accurately circular, and is not a mere approximation to a circular are, as some may suppose. This method produces a circular curve, because all inscribed angles on one side of a chord line are equal. (Art. 161.) To obtain the radius from a chord and its versed sine, see Art. 165. If the angle formed by the rulers at ¢ be a right angle, the segment described will be a semi-circle. This problem is use ful in describing centres for brick arches, when they are re- quired to be rather flat. Also, for the head hanging-stile of a window-frame, where a brick arch, instead of a stone lintel, is to be placed over it. 87 a.—To find the radius of an are of a circle when the chord and versed sine are given. The radius is equal to the sum of the squares of half the chord and of the versed sine, divided by twice the versed sine. This is expressed, algebraic- (s+0" : ally, thus—r= EE, where 7 is the radius, ¢ the chord, and v the versed sine. (Art. 165.) Erample.—In a given are of a circle, a chord of 12 feet has

Page 32

PRACTICAL GEOMETRY. 27 tne rise at the middle, or the versed sine, equal to 2 feet, what is the radius ? Half the chord equals 6, the square of 6 is, 6 x 6 = 36 The square of the versed sine is, 2 x 2 = 4 Their sum equals, 40 Twice the versed sine equals 4, and 40 divided by 4 equals 10 Therefore the radius, in this case, is 10 feet. This result ic shown in less space and more neatly by using the above alge- braical formula. For the letters, substituting their value, the +e CPER a 2v Ta” the arithmetical operations here indicated equals CT BFE 40 4 = —— eS 4 4 87 b.—To find the versed sine of an are of a circle when the radius and chord are given. The versed sine is equal to the radius, less the square root of the difference of the squares of the radius and half chord : expressed algebraically thus—v = r formula 7 = becomes 7 = d performing — yr —(Sy, where r is the radius, v the versed sine, and ¢ the chord. (Art. 158.) Example.—In an are of a circle whose radius is 75 feet, what is the versed sine to a chord of 120 feet? By the table in the Appendix it will be seen that— The square of the radius, 75, equals, : . 5625 The square of half the chord, 60, equals, - 8600 The difference is, 2025 The square root of this is, ou 2 : - 45 This deducted from the radius, . ; : 75 The remainder is the versed sine, = ` 30 This is expressed by the formula thus— v=75 — V 75 — GY =75 — v 5625 — 3600 = 75 — 45 = 3C csal P 3 te eB we 1k YF a R2 3 d 3 2 1 ò

Page 33

9S AMERICAN IOUSE-CAKPENTER. 88.—7» deseribe the segment of a circle by intersection of lines. Let a b, (Fig. 47,) be the chord, and ¢ d the height of the segment. Through ¢, draw e f, parallel to a b; draw b f at right angles to cġ; make c e equal to c f; draw ag and bh, at right angles to ab; divide c e, cf, da, d b, a g, and D h, each into a like number of equal parts, as four ; draw the lines, 1 1, 2 2, &e., and from the points, 0, 0 and 9, draw lines to c; at the intersection of these lines, trace the curve, ac b, which will be the segment required. In very large work, or in laying out ornamented gardens, &e., this will be found useful; and where the centre of the sroposed are of a circle is inaccessible it will be invaluable. (To trace the curve, see note at Art. 117.) The lines e a, ed and f ù, would, were they extended, meet in a point, and that point would be in the opposite side of the circumference of the circle of which ac b is a segment. The lines 1 1, 2 2, 8 8, would likewise, if extended, meet in the same point. The line, ¢ d, if extended to the opposite side of the circle, would become a diameter. The line, f b, forms, by construction, a right angle with 5 c, and hence the extension of fb would also form a right angle with 3 c, on the opposite side of bc; and this right angle would be the inscribed angle in the semicircle; and since this is required to be a right angle, (Art. 156,) therefore the construction thus far is correct, and it will be found likewise that at each point in the curve formed by the intersection of the radiating lines, these intersecting lines are at right angles. Fig. 47 a. 88 a.—Points in the circumference of a circle may be ob- tained arithinetically, and positively accurate, by the calcula- tion of ordinates, or the parallel lines, 01, 0 2, 0 3, 0 4. (Fig.

Page 34

PRACTICAL GEOMETRY. -> 29 47 a.) These ordinates are drawn at right angles to the chord line, @ b, and they may be drawn at any distance apart, either equally distant or unequally, and there may be as many of them as is desirable; the more there are the more points in the curve will be obtained. If they are located in pairs, equally distant from the versed sine, ¢ d, calculation need be made only for those on one side of ¢ d, as those on the opposite side will be of equal lengths, respectively ; for example, 0 1, on the left-hand side of ¢ d, is equal to 0 1 on the right-hand side, 0 2 on the right equals 0 2 on the left, and in like manner for the others. The ‘length of any ordinate is equal to the square root of the difference of the squares of the radius and abscissa, less the difference between the radius and versed sine. (Art. 166.) The abscissa being the distance from the foot of the versed sine to the foot of the ordinate. Algebraically, y= Vr—a? — (x —v), where y is put to represent the ordinate; w, the ab- scissa ; v, the versed sine; and 7, the radius. Example.—An are of a circle has its chord, a b, (Fig. 47 a,) 100 feet long, and its versed sine, ¢ d, 5 feet. It is required to ascertain the length of ordinates for a sufficient number of points through which to describe the curve. To this end it is requisite, first, to ascertain the radius. This is readily done in E ee a 2 2x 5 259°5 = radius. Having the radius, the curve might at once be described without the ordinate points, but for the impracti- eability that usually occurs, in large, flat segments of the circle, of getting a location for the centre; the centre usually being inaccessible. The ordinates are, therefore, to be calculated. In Fig. 47 a the ordinates are located equidistant, and are 10 feet apart. It will only be requisite, therefore, to calculate those on one side of the versed sine, c d. Forsthe first ordinate, 01, the formula, y = Vr — x° — (r — v) becomes y = V 252:5 — 10° — (252:5 — 5). v 6375625 — 100 — 2475. 2523019 — 2475. 4:8019 = the first ordinate, 0 1. accordance with Art. 87a. For, Heal W

Page 35

o0 AMERICAN HOUSH -CARPENTER v For tne second— y = 2525 — 20? —(252°5 — 5). = ¥%51°7066 — 25. = 42066 = the second ordinate, 02. For the third— =a y = V25 — 30! — 247°5. = 2507115 — 247-5. = 3:2115 = the third ordinate, 03. For the fourth— y = y 2525 — 40° — 2475. = 2493115 — 2475. = 1:8115 = the fourth ordinate, 04. The results here obtained are in feet and decimals of a foot. To reduce these to feet, inches, and eighths of an inch, proceed as at Reduction of Decimals in the Appendix. If the two-feet rule, used by carpenters and others, were decimally divided, there would be no necessity of this reduction, aud it is to be hoped that the rule will yet be thus divided, as such a reform would much lessen the labor of computations, and insure more accurate measurements. Versed sine, ¢ d, = ft. 5:0 = ft. 5°0 inches. -Ordinates, O1,= 48019 = 4°93 inches nearly. s 2,= 42066 = 4°23 inches nearly. s 03,= 32115 = 324 inches nearly. i 04,= 18115 = 19ł inches nearly. b 1 2 3 & £ c Fig. 48. 89.—In a given angle, to describe a tanged curve. Let ab Gs (Fig. 48,) be the given angle, and 1 in the line, æ b, and 5 in the line, 6 c, the termination of the curve. Divide 1} and $ 5 into a like number of equal parts, as at 1, 2, 3, 4 and 5; join 1 and 1, 2 and 2, 3 and 3, &c.; and a regular curve will be formed that will be tangical to the line, a 3, at the point, 1, and to dc at 5. This is of much use ir. stair-building, in easing the angles formed between the wall-string and the base of the hall, alse

Page 36

PRACTICAL GEOMETRY. l 31 between the front string and level facia, and in many other ‘instances. The curve is not circular, but of the form of the parabola, (Fig. 935) yet in large angles the difference is not perceptible. This problem can be applied to describing the e a c ò Fig. 49. curve for door heads, window-heads, &c., to rather better ad- vantage than Art. 87. For instance, let æ b, (Wig. 49,) be the width of the opening, and ¢ d the height.of the arc. Extend ¢ d, and make d e equal to e d; join a and e, also e and b; and proceed as directed above. a b ig. 50 90.—To describe a circle within any gwen triangle, so that the sides of the triangle shall be tangical. Let ab c, (Fig. 50,) be the given triangle. Bisect the angles æ and b, according to Art. TT; upon d, the point of intersection of the bisecting lines, with the radius, d e, describe the required circle.

Page 37

32 AMERICAN HOUSE-CARPENTER. ad 91.—About a given cirele, to describe an equi-lateral tri- angle. Leta d b c, Ly. 51) be the given a Draw the diuneter, e d; upon d, with the radius of the given circle, describe the are,aeb; join a and b; draw fg, at right angles tode; make f c and ¢ g, each equal to æ b; from f, through a, draw f h, also from g, through b, draw gh; then fg A will be the triangle required. 92.—To find a right line nearly equal to the circumference of a circle. Let abe d, (Fig. 52,) be the given circle. Draw the diameter, æ e p on this erect an equi-lateral triangle, @ e c, according to Art. 93; draw g f, parallel toa e; extend e c to f, also e a tog; then g f will be nearly the length of the semi-circle, adc; and twice g f will nearly equal the circum- ference of the cirele, a b e d, as was reqnired. Lines drawn from e, through any points in the circle, as 0, 6 and o, to p, p and p, will divide g f in the same way as the semi-circle, a d ¢, is divided. So, any portion of a cirele may be transferred to a straight line. This is a very useful pro- blem, and should be well studied ; as it is frequently used to solve problems on stairs, domes, Ce. 92, a.—Another method. Let ab f c, (Fig. 53,) be the given circle. Draw the diameter, ac ; from d, the centre, and at right angles to æ c, draw d b; join band c; bisect b cate; from d, through e, draw d f; then e f added to three times the diameter, will equal the cirenmference of the circle sufficiently near for

Page 38

PRACTICAL GEOMETRY. 33 b ~J fi a C = Fig. 53, many uses. The result is a trifle too large, If the circumfer- erence found by this rule, be divided by 64822, the quotient will be the excess. Deduct this excess, and the remainder will be the true circumference. This problem is rather more curious than useful, as it is less labor to perform the operation arithmetically: simply multiplying the given diameter by 3:1416, or where a great degree of accuracy is needed by 371415926. POLYGONS, &0. Fig. 54, 93.— Upon a given line to construct an equi-lateral triangve. Let a b, (Fig. 54,) be the given line. Upon a and b, with a b for radius, describe arcs, intersecting at cy join æ and c, also ¢ and 6; then ac 6 will be the triangle required. * 94.—To describe an equi-lateral rectangle, or square. Let a b, (Fig. 55,) be the length of a side of the proposed square. Upon a and 6, with a b for radius, describe the arcs a d and bc; bisect the arc, ae, in f; upon e, with e f for radius, de- 5

Page 39

34 AMERICAN HOUSE-CARFENTER. yo d a uv Fig. 55, scribe the are, cfd; join a and o, ¢ and d, d and b; then ao db will be the square required. 95.— Within a given circle, to inseribe an equilateral iri- angle, hexagon or dodecagon. Let a b c d, (Fig. 56,) be the given circle. Draw the diameter, 6 d; upon 6, with the radius of the given circle, describe the are, a e c ; join a and c, also a and d, and ¢ and d—and the triangle is completed. For the hexagon: from a, also from c, through e, draw the lines, a f and ¢ g ; join a and b, b and c, c and f, &., and the hexagon is completed. The dodecagon may be formed by bisecting the sides of the hexagon. Fach side of a regular hexagon is exactly equal to the radius of the circle that cireumseribes the figure. For the radius is equal to a chord of an are of 60 degrees; and, as every circle is supposed to be divided into 360 degrees, there is just 6 times 60, or 6 ares of 60 degrees, in the whole cireum- ference. A line drawn from each angle of the hexagon to the centre, (as in the figure,) divides it into six equal, equi-lateral triangles. 96.— Within a square to inscribe an octagon. Let ab cd,

Page 40

PRACTICAL GEOMETRY. 35 (Fig. 57,) be the given square. Draw the diagonals, a d and bc; upon a, b, c and d, with a e for radius, describe ares cut- ` ting the sides of the square at 1, 2, 3, 4, 5, 6, 7 and 8; join 1 and 2, 3 and 4, 5 and 6, &c., and the figure is completed. In order to eight-square a hand-rail, or any piece that is to be afterwards rounded, draw the diagonals, a d and b ce, upon the end of it, after it has been squared-up. Set a gauge to the distance, æ e, and run it upon the whole length of the stuff, from each corner both ways. This will show how much is to be chamfered off, in order to make the piece octagonal. (Art. 159.) Fig. 58. Fig. 59. Fig. 60. 97.— Within a given circle to inscribe any regular polygon. Let a b c 2, (Fig. 58, 59 and 60,) be given circles. Draw the diameter, æ c; upon this, erect an equi-lateral triangle, a e ¢, according to Art. 93; divide æ ¢ into as many equal parts as the. polygon is to have sides, as at 1, 2, 3, 4, &c.; from e

Page 41

36 AMERICAN HOUSE-CARPENTER. through each even number, as 2, 4, 6, &e., draw lines cutting the cirele in the points, 2, 4, &e.; from these points and at right angles to ac, draw lines to the opposite part of the circle; this will give the remaining points for the polygon, as 6, f, ce. In forming a hexagon, the sides of the triangle erected upon ac, (as at Fig. 59,). mark the points b and f. This method of locating the angles of a polygon is an approximation suffi- ciently near for many purposes; it is based upon the like prin- ciple with the method of obtaining a right line nearly equal to a circle. (Art. 92.) The method shown at Art. 98 is accurate. Fig? 61. Fig. 62. Fig. 63. 98.—Upon a given line to describe any regular polygon. Let a b, (Fig. 61, 62 and 63,) be given lines, equal to a side of the required figure. From b, draw b c, at right angles to a b ; upon a and J, with a ọ for radius, describe the ares, @ ¢ d and feb, divide a ¢ into as many equal parts as the polygon is to have sides, and extend those divisions from ¢ towards d; from the second point of division counting from ¢ towards a, as 3, (Zig. 61,) 4, (Fig. 62,) and 5, (Fig. 68,) draw a line to 6; take the distance from said point of division to a, and set it from 6 toe, joine anda; upon the intersection, o, with the radius, o a, describe the circle a f d b; then radiating lines, drawn from 6 through the even numbers on the are, a d, will cut the circle at the several angles of the required figure. In the hexagon, (Fig. 62,) the divisions on the are, a d, are not necessary ; for the point, o, is at the intersection of the arcs, ad and 7 0, the points, 7 and d, are determined by the intersection of those ares with the circle, and the points above, g and A, ean be found by drawing lines from a and b, through the centre, o. In polygons of a greater number of sides than the hexagon, the in- tersectiou, 0, comes above the ares; in such case, therefore, the

Page 42

PRACTICAL GEOMETRY. 3a lines ae and $ 5, (Fig. 63,) have to be extended before they will intersect. This method of describing polygons is founded on correct principles, and is therefore accurate. In the circle equal arcs subtend equal angles, (Arts. 86 and 162.) Although this method is accurate, yet polygons may be described as accu- rately and more simply in the following manner. It will be observed that much of the process in this method is for the pur pose of ascertaining the centre of a circle that will circumscribe the proposed polygon. By reference to the Table of Polygons in the Appendix it will be seen how this centre may be obtained arithmetically. This is the /?wle.—Multiply the given side by the tabular radius for polygons of a like number of sides with the proposed figure, and the product will be the radius of the required circumscribing circle. Divide this circle into as many equal parts as.the polygon is to have sides, connect the points of division by straight lines, and the figure is complete. For exam- ple: It is desired to describe a polygon of 7 sides, and 20 inches aside. The tabular radius is 1°1523824. This multiplied by 20, the product, 23°047648 is the required radius in inches. The Rules for the Reduction of Decimals, also in the Appendix, show how to change decimals to the fractions of a foot or an inch. From this, 23-047648 is equal to 23:5 inches nearly. It is not needed to take all the decimals in the table, three or four of them will give a result suilicrently near for all ordinary practice. 99.—To construct a mae ohio sides shall be severally equal to three given lines. Let a, b and c, (Fig. 64,) be the given lines. Draw the line, de, and make it equal toc; upon e, with 6 for radius, describe an are at f; upon d, with a for radius, describe an arc intersecting the other at f; join d and f, also Jand e; then df e will be the triangle required. b h —. a A Fig. 65. Fig. 66.

Page 43

os PRACTICAL GEOMETRY. 100.--To construct a figure equal to a g.ven, right-lined figure. Let ab c d, (Fig. 65,) be the given figure. Make ef, (Fig. 66,) equal to c d ; upon f, with d a for radins, describe an arc at g; upon e, with ¢ a for radius, describe an are intersecting the other at g ; join g and e ; upon fand g, with d b anda b for radius, describe ares intersecting at A; join g and A, also A and f; then Fig. 66 will every way equal Fig. 65. So, right-lined figures of any number of sides may be copied, py first dividing them into triangles, and then proceeding as above. 'The shape of the floor of any room, or of any piece of land, &c., may be accurately laid out by this problem, at a scale upon paper; and the contents in square feet be ascertained by the next. a d Fig. 67. 101.— To make a parallelogram equal to a given triangle. Let a b c, (Fig. 67,) be the given triangle. From a, draw a d, at right angles to bc; bisect a d in e; through. e, draw f g, parallel to b c; from b and c, draw b Jand c g, parallel to de; then b f g c will be a parallelogram containing a surface exactly equal to that of the triangle, a b c. Unless the parallelogram is required to be a rectangle, the lines, b f and c g, need not be drawn parallel to d e. If a rhomboid is desired, they may be drawn at an oblique angle, provided they be parallel to one another. To ascertain the area of a triangle, multiply the base, b c, by half the perpendicular height, d a. In doing this, it matters not which side is taken for base.

Page 44

AMERICAN HOUSE-CARPENTER. ‘89 102.—A parallelogram being given, to construct another equal to it, and having a side equal to a givenline. Let A (Fig. 68,) be the given parallelogram, and B the given line Produce the sides of the parallelogram, as at a, b, c and d ; make e d equal to B ; through d, draw c f, parallel to g b; through e, draw the diagonal, c a-; from a, draw a f, parallel toe d then C will be equal to A. (See Art. 144.) Fig 69. 103.— To make a square equal to two or more given squares. Let A and B, (Fig. 69,) be two given squares. Place them sc as to form a right angle, as at a ; join b and c ; then the square, C, formed upon the line, b c, will be equal in extent to the squares, A and B, added together. Again: if a b, (Fig. 70,) be equal to Fig. 70. the side of a given square, ¢ a, placed at right angles to a b, be the side of another given square, and c d, placed at right angles to

Page 45

+0 PRACTICAL GEOMETRY. c b, be the side of a third given square; then the square, A, formed upon the line, d b, will be equal to the three given squares. (See Art. 157.) The usefulness and importance of this problem are proverbial. ‘To ascertain the length of braces and of rafters in framing, the length of stair-strings, &c., are some of the purposes to which it may be applied in carpentry. (See note to Art. 74, b.) If the length of any two sides of a right-angled triangle is known, that of the third can be ascertained. Because the square of the hypothenuse is equal to the united squares of the two sides that contain the right angle. (1.)—The two sides containing the right angle being known, to find ihe hypothenuse. /tu/e.—Square each given side, add the squares together, and from the product extract the square- root: this will be the answer. For instance, suppose it were required to find the length of a rafter for a house, 34 feet wide,— the ridge of the roof to be 9 feet high, above the level of the wall-plates. Then 17 feet, half of the span, is one, and 9 feet, the height, is the other of the sides that contain the right angle Proceed as directed by the rule: 17 9 17 g 119 81 = square of 9. 17 289 = square of 17. od ——— 289 = square of 17. 370 Product. 1) 370 ( 19:235 + = square-root of 370; equal 19 feet, 2: in. eel nearly : which would be the required -— — length of the rafter. 29 ) 270 2 2b 382) °° 900 2 764 3843 ) 13600 3 11529 88465): 207100 (By reference to the table of sqnare-roots 192325 in the Appendix, the root of almost any number may be found ready calculated ; also, to change the decimals of a foot to inches and parts, see Rules tor the Reduction of Decimals in the Appendix.)

Page 46

AMERICAN HOUSE-CARPENTER. 4} Agai : suppose it be required, in a frame building, to tind the length of a brace, having a run of three feet each way from the point of the right angle. The length of the sides containing the right angle will be each 3 feet: then, as before— | wes 9 = square of one side. 3 times 3 = 9 = square of the other side. 18 Product: the square-root of which is 4:2426 + ft., or 4 feet, 2 inches and ths. full. (2.)}—The hypothenuse and one side being: known, to find the other side. Rude.—Subtract the square of the given side from the square of the hypothenuse, and the square-root of the product will be the answer. . Suppose it were required to ascertain the greatest perpendicular height a roof of a given span may have, when pieces of timber of a given length are to be used as rafters. Let the span be 20 feet, and the rafters of 3x4 hemlock joist. These come about 13 feet long. The known hypothenuse, then, is 13 feet, and the known side, 10 feet—that being half the span of the building. 13 13 39 13 169 = square of hypothenuse. 10 times 10 = 100 = square of the given side. l 69 Product: the square-root of which is 8 -3066 + feet, or 8 feet, 3 inches and eths. full. This will be the greatest perpendicular height, as required. Again: suppose that in a story of 8 feet, from floor to floor, a step-ladder is re- - quired, the strings of which are to be of plank, 12 feet long ; and it is desirable to know the greatest run such a length of string will afford. In this case, the two given sides are—hypothenuse 12, perpendicular 8 feet. “12 times 12 = 144 = square of hypothenuse. 8 times 8 = 64 = square of perpendicular. 80 Product : the square-root of which is 8:9442 + feet. or 8 feet, ‘1 inches and ,ths.—the answer, as required. 6

Page 47

49 PRACTICAL GEOMETRY. Many other cases might be adduced to show the utility of this problem. A practical and ready method of ascertaining the length of braces, rafters, &c., when not of a great leugth, is to apply a rule across the carpenters’-square. Suppose, for the length of a rafter, the base be 12 feet and the height 7. Apply the rule diagonally on the square, so that it touches 12 inches from the corner on one side, and 7 inches from the corner on the other. The number of inches on the rule, which are intercepted by the sides of the square, 134 nearly, will be the length of the rafter in feet; viz, 13 feet and 3ths of a foot. .If the dimensions are large, as 30 feet and 20, take the haif of each on the sides of the square, viz, 15 and 10 inches; then the length in inches across, will be one-half the number of feet the rafter is long. This method is just as accurate as the preceding ; but when the length of a very long rafter is sought, it requires great care and precision to ascertain the fractions. For the least variation on the square, or in the length taken on the rule, would make perhaps several inches difference in the length of the rafter. For shorter dimensions, however, the result will be true enough. ~ Cc Fig. 7. 101.— To make a circle equal to two given circles. Let A and B, (Fig. 71,) be the given circles. In the right-angled tri- angle, a b c, make a b equal to the diameter of the circle, B, and ¢ b equal to the diameter of the circle, A ; then the hypothenuse, SÀ Fig. 72.

Page 48

AMERICAN HOUSE-CARPENTER. 43 a c, Will be the diameter of a circle, C, which will be equal in area to the two circles, A and B, added together. Any polygonal figure, as A, (Fig. 72,) formed on the hypo- thenuse of a right-angled triangle, will be equal to two similar figures,* as B and C, formed on the two legs of the triangle. Fig. 73 105.— To construct a square equal to a given rectangle. Let A, (Fig. 73,) be the given rectangle. Extend the side, a b, and make 6 c equal to b e ; bisect a c in f, and upon f, with the radius, f a, describe the semi-circle, a g c; extend e b, till it cuts the curve in g ; then a square, b g h d, formed on the line, b g, will be equal in area to the rectangle, A. =. Fig. 74. 105, a.—Another. method. Let A, (Fig. 74,) be the given rectangle. Extend the side, a b, and make a d equal toac, * Similar figures are such as have their several angles respectively equal, and their als respectively proportionate.

Page 49

44 PRACTICAL GEOMETRY. bisect a d in e; upon e, with the radius, e a, describe the sem circle, a f d ; extend g b till it euts the curve in f; join a and f; then the square, B, formed on the line, a f, will be equal in area to the rectangle, A. (See Art. 156 and 157.) 106.—To form a square equal to a given triangle. Let ab, (Fig. 73,) equal the base of the given triangle, and b e equal half its perpendicular height, (see Fig. 67;) then proeeed ag directed at Art. 105. Fig. 75. 107.— Two right lines being given, to find a third propor- tional thereto. Let A and B, (Fig. 75,) be the given lines. Make a b equal to A; from a, draw a c, at any angle with a b ; make a c and a d each equal to B; join cand b ; from d, draw d e, parallel to c b ; then a e will be the third pruportional re- quired. That is, a e bears the same proportion to B, as B does to A. A C ¢é Fig. 76. 108.— Three right lines being given, to find a fourth pro portional thereto. Let A, Band C, (Fig. 76,) be the given lines. Make a b equal to A ; from a, draw a c, at any angle with a b; make a c equal to B, and a e equal to C; join c and b ; from e, draw e f, parallel to c b ; thena f will be the fourth proportional required. That is, a f bears the same proportion to C, as B does to A.

Page 50

AMERICAN HOUSE-CARPENTER. 43 ‘i'o apply this problem, suppose the two axes of a given ellipsis and the longer axis of a proposed ellipsis are given. Then, by this problem, the length of the shorter axis to the proposed ellip- sis, can be found ; so that it will bear the same proportion to the longer axis, as the shorter of the given ellipsis does to its longer. {See also, Art. 126.) Fig. 77. + 109.—A line with certain divisions being given, to divide another, longer or shorter, given line in the same proportion. Let A, (Fig. 77,) be the line to be divided,-and B the line with its divisions. Make a b equal to B, with all its divisions, as at 1,2, 3, &c.; from a, draw a c, at any angle with a b; make ac equal to A ; join c and b ; from the points, 1, 2, 3, &c., draw lines, parallel to c b; then these will divide the line, a c, in the same proportion as B is divided—as was required. ‘This problem will be found useful in proportioning the mem- bers of a proposed cornice, in the same proportion as those of a given cornice of another size. (See Art. 253 and 254.) So of a pilaster, architrave, &c. A B & e b c Fig. 78. 110.— Between two given right lines, to find a mean pro- portional. Let A and B, (Fig. 78,) be the given lines. On the line, a c, make a b equa! to A, and b c equal to B; bisect a cine; upon e, with e a for radius, describe the semi-circle, a d

Page 51

46 AMERICAN HOUSE-CARPENTER. c; at b, erect bd, at right angles to ac; then 0 g will be the mean proportional between Aand B. That is, ab is tob d as distobe This is usaally stated thus—a b:bb:bd: be, and since the product of the means equale the product of the extremes, therefore, ab x b e = d. This is shown geometri- cally at Art. 105. CONIO SECTIONS. 111,—If a cone, standing upon a base that is at right angles with its axis, be cut by a plane, perpendicular to its base and passing through its axis, the section will be an isosceles triangle ; n Fig. 79. ‘as a b c, Fig. 79;) and the base will be a semi-circle. Ifa cone be cut by a plane in the direction, e f, the section will be an ellepsis ; if in the direction, m } the section will be a parabola ; and if in the direction, r o, an hyperbola. (See Art. 56 to 60.) If the cutting planes be at right angles with the plane, æ b c, then— 112.— T9 find the axes of the ellipsis, bisect e f, (Fig. 79,) in g; througn g, draw / îi, parallel to a b; bisect A 7 ing; upon j, with J A for radius, describe the semi-circle, A4 & 7; from g, draw gk, at right angles to hz; then twice g & will be the conjugate axis, and e f the transverse.

Page 52

AMERICAN HOUSE-CARPENTER. 47 113.— T% find the axis and base of the parabola. Let ml, (Fig. 79,) parallel to a c, be the direction of the cutting plane From m, draw m d, at right angles to a b ; then 1 m will be the axis and height, and m d an ordinate and half the base; asat Pig, 92093. 114.—To find the height, base and transverse axis of an hyperbola. Leto r, (Fig. 79,) be the direction of the cutting plane. Extend or and ac till they meet at n ; from o, draw o p, at right angles to a b; then ro will be the height, n r the transverse axis, and o p half the base; asat Pig. 94. Fig. 80. 115.— The azes being given, to find the foci, and to describe an ellipsis with a string. Let a b, (Fig. 80,) and c d, be the given axes. Upon c, with ae orb e for radius, describe the arc, ff; then f and f, the points at which the arc cuts the transverse axis, will be the foci. At f and f place two pins, and another at c ; tiea string about the three pins, so as to form the triangle, f fc ; remove the pin from c, and place a pencil in its stead ; keeping the string taut, move the pencil in the direction, c g a; it will then describe the required ellipsis. 'The lines, f g and g f, show the position of the string when the pencil arrives at g. This method, when performed correctly, is perfectly accurate; but the string is liable to stretch, and is, therefore, not so good to Jise as the trammel. In making an ellipse by a string or twine, that kind should be used which has the least tendency to elasticity, For this reason, a cotton cord, such as chalk-lines are commonly made of, is not proper for the purpose: a linen, cr flaxen cord is mush better.

Page 53

4S PRACTICAL GEOMETRY. a Fig. 8l 116.— The ares being given, to describe an ellipsis with a trammel. Jet a band c d, (Fig. 81,) be the given axes. Place the trammel so that a line passing through the centre of the grooves, would coincide with the axes ; make the distance from the pencil, e, to the nut, f, equal to half cd; also, from the pen- cil, e, to the nut, g, equal to half a b; letting the pins under the nuts slide in the grooves, move the trammel, e g, in the direction, cb d; then the pencil at e will describe the required ellipse. A trammel may be constructed thus: take two straight strips ot board, and make a groove on their face, in the centre of their width ; join them together, in the middle of their length, at right angles to one another; as is seenat Fig. 81. A rod is then to be prepared, having two moveable nuts made of wood, with a mor- tice through them of the size of the rod, and pins under them large enough to fill the grooves. Make a hole at one end of the rod, in which to place a pencil. In the absence of a regular tram- mel, a temporary one may be made, which, for any short job, will answer every purpose. Fasten two straight-edges at right angles to oneanother. Lay them so as to coineide with the axes of the proposed ellipse, having the angular point at the centre. Then, ina rod having a hole for the pencil at one end, place two brad-awls at the distances described at Art. 116. While the pencil is moved in the direction of the curve, keep the brad-awls- hard against the straight-edges, as directed for using the tram- mel-rod, and one-quarter of the ellipse will be drawn. Then. by shifting the straight-edges, the other three quarters in succes- sion may be drawn. If the required ellipse be not too large, a carpenters’-square may be made use of, in place of the straight edges. An improved method of constructing the trammel, is as fol lows : make the sides of the grooves bevilling from the face ot the stuff, or dove-tailing instead of square. Prepare two slips ot wood, each about two inches long, which shall be of a shape to ‘ust fill the groove when slipped in at the end. 'These, instead of

Page 54

AMERICAN HOUSE-CARPENTER. 49 pins, are to be attached one to each of the moveable nuts with a screw, loose enough for the nut to move freely about the screw as an axis. The advantage of this contrivance is, in preventing the nuts from slipping out of their places, during the operation of describing the curve. Fig. 82. 117.— To describe an ellipsis by ordinates. Leta b and cq, (Fig. 82,) be given axes. With c e ore d for radius, de- scribe the quadrant, f g h; divide 7 h, a e and e b, each into a like number of equal parts, as at 1, 2 and 3; through these points, draw ordinates, parallel to c d and f œ ; take the distance, 1%, and place it at 1 Z, transfer 2 j to 2 m, and3 k to3 n; through the points, a, n, m, l and ¢, trace a curve, and the ellipsis will be completed. The greater the number of divisions on a e, &c., in this-and the following problem, the more points in the curve can be found, and the more accurate the curve can be traced. If pins are placed in the points, n, m, l, &c., and a thin slip of wood bent around by them, the curve can be made quite correct. This method is mostly used in’ tracing face-moulds for stair hand- railing. d Fig 83. 118.— To describe an ellipsis by intersection of lines. Let a

Page 55

5O PRACTICAL GEOMETRY. a b and c d, (Fig. 83,) be given axes. Through c, draw fg, parallel tc a b; from a and b, draw a fand b g, at right angles to ab; divide f a, g b, ae and eb, each into a like number of equal parts, as at 1, 2, 3 and o, 0, o ; from 1, 2 and 3, draw lines toc; through o, o and o, draw lines from d, intersecting those drawn to c; then a curve, traced through the points, ¿, 2, 2, will ye that of an ellipsis. Where neither trammel nor string is at hand, this, perhaps, is the mest ready method of drawing an ellipsis. The divisions should be small, where accuracy is desirable. By this method, an ellipsis may be traced without the axes, provided that a diame- ter and its conjugate be given. Thus, a b and c d, (ig. 84,) are conjugate diameters: f g 1s drawn parallel to a b, instead of being at right angles to c d ; also, f aand g b are drawn para'lel to c d, instead of being at right angles to a b. d Fig. 85. 119.— To describe an ellipsis by intersecting arcs. Let ab

Page 56

AMERICAN HOUSE-CARPENTER. < 5I ana cd, (Fig. 85,) be given axes. Between one of the foci, f aid f, and the centre, e, mark any number of points, at random, as 1,2 and 3; upan fand f, with b 1 for radius, describe arcs at £,£,g and g ; upon fand f, with a 1 for radius, describe ares inter- secting the others at g, g, g and g ; then these points of intersection will bein the curve of the ellipsis. 'The other points, Aand 2, are found in like manner, viz: A is found by taking b 2 for one radius, and a 2 for the other; tis found by taking b 3 for one radius, and a 3 for the other, always using the foci for centres. Thén by tracing a curve through the points, c, g, h, i, b, &c., the ellipse will be completed. This problem is founded upon the same principle as that of the string. This is obvious, when we reflect that the length of the string is equal to the transverse axis, added to the distance between the foci. See Fig. 80; in which c f equals a e, the half of the transverse axis. i i g Fig. 86. 120.— To describe a figure nearly in the shape of an ellip- sis, by a pair of compasses. Let a b and c d, (Fig. 86,) be given axes. Frome, draw c e, parallel to a b; from a, draw a e, parallel to c d; join e and d ; bisect e ain f ; join fand c, inter- secting e d inz; bisect i cin o ; from o, draw og, at right angles ‘to i c; meeting c d extended to g ; join ¿ and g, cutting the trans- verse axis in r; make h j equal toh g,and h k equal to Ar, from j, through r and k, draw j mand j n ; also, from g, through k, araw gl; upon g and j, with g c for radius, describe the

Page 57

59 PRACTICAL GEOMETRY. arcs, i l and m n; upon ratl k, with r a for radius, describe the arcs, m iand l n, this will compiete the figure. When the axes are proportioned to one another as 2 to 3, the extremities, c and d, of the shortest axis, will be the centres for describing the arcs, i Land m n; and the intersection of e d with the transverse axis, will be the centre for describing the arc, m 17, &e. As the elliptic curve is continually changing its course from that of a circle, a true ellipsis cannot be described with a pair ot compasses. The above, therefore, is only an approximation. d Fig. 87. 121.— To draw an oval in the proportion, seven by nine. Let cd, (Fig. 87,) be the given conjugate axis. Bisect c d ino, and through o, draw a b, at right angles toed ; bisectc oine, upon o, with o e for radius, describe the circle,e f g h; from e, through A and f, draw e jand e i; also, from g, through hand f, draw gkand gl; upon g, with g c for radius, describe the arc, kl; upon e, with e d for radius, describe the arc, j i ; upon A and f, with h k for radius, describe the arcs, j kand li; this will complete the figure. This is an approximation to an ellipsis; and perhaps no method can be found, by which a well-shaped oval can be drawn with oe facility. By a little variation in the process, ovals of different proportions may be obtained. If quarter cf the trans- verse axis is taken for the radius of the circle, e f g h, one will be drawn in the proportion, five by seven.

Page 58

AMERICAN HOUSE-CARPENTYER. 5a E e Ze yy: e b Fig. 88. 122.— To draw a tangent to an ellipsis. Leta bcd, (Fig. 88,) be the given ellipsis, and d the point of contact. Find the foci, (Art. 115,) f and f, and from them, through d, draw fe and f d; bisect the angle, (Art. 77,) e d o, with the line, sr; then s r will be the tangent required. | c Fig 89 123.—An ellipsis with a tangent given, to detect the point of contact. Letagb f, (Fig. 89,) be the given ellipsis and tan- gent. Through the centre, e, draw a b, parallel to the tangent; any where between e and f, draw c d, parallel to a b ; bisecte din o; through o and e, draw f g ; then g will be the point of con- tact required. 124.—A diameter of anellipsis given, to find its conjugate. “ Let a b, (Fig. 89,) be the given diameter. Find the line, f g, by the last problem; then f g will be th2 diameter required.

Page 59

PRACTICAL GEOMETRY. Fig. 90. 125.—Any diameter and its conjugate being given, to as- certain the two axes, and thence to describe the ellipsis. wet aband cd, (Fig. 90,) be the given diameters, conjugate to one another. Through c, draw e f, parallel toa b; from c, draw c g, at right angles toe f; make c g equal to a h or hb; join g and h; upon g, with æ c for radius, describe the arc, i & c ae upon A, with the same radius, describe the arc, l n ; through the intersections, 2 and n, draw n o, cutting the tangent, e f, ino; upon 0, with o g for radins, describe the semi-circle, ei g f ; join e and g, also g and f, cutting the arc, i c j, ink and t; from e, through h, draw e m, also from f, through h, draw f p; from k and ¢t, draw kr and ¢ s, parallel to g h, cutting e m in r, and f p in s; make h m equal to hr, and h p equal to hs; then r m and s p will be the axes required, by, which the ellipsis may be drawn in the usual way. 126.— To describe an ellipsis, whose axes shall be propor- tionate to the axes of a larger o- smaller given one. Leta cbd, (Fig. 91,) be the given ellipsis and axes, and i j the trans- verse axis of a proposed smaller one. Join a and c; from i lraw ie, parallel to a c; make o f equal to oe ; then e f will be

Page 60

AMERICAN HOUSE-CARPENTER:. 55 c LE QE r} Fig. 91. the conjugate axis required, and will bear the same proportior. to “J asc ddoestoab. (See Art. 108.) Pa al 3 2f 1 127.— To describe a parabola by intersection of lines. Ler m l, (Fig. 92,) be the axis and height, (see Fig. 79,) and d d, a double ordinate and base of the proposed parabola. Through J, draw a a, parallel todd; through d and d, draw d a and d a, parallel to m l; divide a d and d m, each into a like number of equal parts; from each point of division in d m, draw the lines, 1 1, 22, &c., parallel to ml; from each point of division in d a, draw lines to J; then a curve traced through the points ot intersection, o, o and o, will be that of a parabola. 127, a.—Another method. Let m l, (Fig. 93,) be the axis and height, and d d the base. Extend m l, and make / a equal to m “¢; join @ and d, and a and d; divide a d and ad, each into a lixe number of equal parts, as at 1, 2, 3, &c.; join 1 and 1, 2 and 2, &c., and the parabola will be completed

Page 61

56 PRACTICAL GEOMETRY. m Fig. 93. 128.—To describe an hyperbola by intersection of lines. Let r o, (Fig. 94,) be the height, p p the base, and n r the trans- verse axis. (See Fig. 79.) Through r, draw a a, parallel to p p; trom p, draw a p, parallel to r o; divide a p and p o, each into a like number of equal parts ; from each of the points of di- Visions in the base, draw lines to n ; from each of the points of division in a p, draw lines to r ; then a curve traced through the points of intersection, o0, 0, &c., will be that of an hyperbola. The parabola anà hyperbola afford handsome curves for various moldings.

Page 62

DEMONSTRATIONS. 129.—T'o impress more deeply upon the mind of the learner some of the more important of the preceding problems, and to indulge a very common and praiseworthy curiosity to discover the cause of things,’are some of the reasons why the following exercises are introduced. In all reasoning, definitions are ne- cessary; in order to insure, in the minds of the proponent and respondent, identity. of ideas. A corollary is an inference deduced from a previous course of reasoning. An axiom is a proposition evident at first sight. In the following demonstrations, there are many axioms taken for granted; (such as, things equal to the same thing are equal to one another, &c.;) these it was thought « not necessary to introduce in form. € 5m i Fig. 95. 130.— Definition. If a straight line, as a b, (Fig. 95,) stand upon another straight line, as ¢ &, so that the two angles made at

Page 63

5S PRACTICAL GEOMETRY. the point, b, are equal—a b c to a b d, (see note to Art. 27,) then each of the two angles is called a right angle. 131.—Definition. The circumference of every circle is sup- posed to be divided: into 360 equal parts, called degrees ; hence a semi-circle contains 180 degrees, a quadrant 90, &c. e Cc b d Fig. 96. 132.— Definition. The measure of an angle is the number of degrees contained between its two sides, using the angular point as a centre upon which to describe the arc. Thus the arc, c e, (Fig. 96,) is the measure of the angle, c b e; e a, of the angle, eb a; and ad, of theangle, a b d. 133.—Corollary. As the two angles at b, (Fig. 95,) are right angles, and as the semi-circle, c a d, contains 180 degrees, (Art. 131,) the measure of two right angles, therefore, is 180 degrees ; of one right angle, 90 degrees; of half a right angle, 45; of one-third of a right angle, 30, &c. 134.— Definition. In measuring an angle, (Art. 132,) no re- gard is to be had to the length of its sides, but only to the degree of their inclination. Hence equal angles are such as have the same degree of inclination, without regard to the length of their sides. « Fig. 97. 135.—Aziom. If two straight lines, parallel to one another, >

Page 64

AMERICAN HOUSE-CARPENTER. 59 as a b and c d, (Fig. 97,) stand upon another straight line, as e f, the angles, a b fand c d f, are equal; and the angle, a b e, 1s equal to the angle, c d e. 136.—Definition. If a straight line, as a b, (Fig. 96,) stand obliquely upon another straight line, as c d, then one of the an- gles, as ab c, is called an obtuse angle, and the other, as a b d, an acute angle. 137.—Axiom. The two angles, a b d and a b c, (Fig. 96,) are together equal to two right angles, (Art. 130, 133;) also, the three angles, a b d,e b a and c be, are together equal to two right angles. 138.—Corollary. Hence all the angles that can be made upon one side of a line, meeting in a point in that line, are together equal to two right angles. 139.—Corollary. Hence all the angles that can be made on both sides of a line, at a pointin that line, or all the angles that can be made about a point, are together equal to four right angles. 6 Fig 98. 140.—Proposition. If to each of two equal angles a third angle be added, their sums will be equal. Leta b cand d ef, (Fig. 98,) be equal angles, and the angle, i j k, the one to be added. Make the angles, g b a and A e d, each equal to the given ‘angle, i j k ; then the angle, g b ¢, will be equal to the angle, h e f; for, if ab cand defbe angles of 90 degrees, and ¿ j k, 30, then the angles, g bc and h e f, will te each equal to 90 and 30 added, viz: 120 degrees. .

Page 65

60 PRACTICAL GEOMETRY. a d, b c e Í Fig. 99. 141.—-Proposition. 'Triangles that have two of their sides and the angle contained between them respectively equal, have also their third sides and the two remaining angles equal; and consequently one triangle will every way equal the other. Let a b c, (Fig. 99,) and d e f be two given triangles, having the angle at a equal to the angle at d, the side, a b, equal to the side, d e, and the side, a c, equal to the side, d f; then the third side of one, b c, is equal to the third side of the other, e f; the angle at b is equal to the angle at e, and the angle at c is equal to the angle atf. For, if one triangle be applied to the other, the three points, b, a, c, coinciding with the three points, e, d, f, the line, b c, must coincide with the line, e f; the angle atb with the angle at e; the angle at c with the angle at f ; and the triangle, b a c, be every way eoual tothe triangle, e d f. 6 d c Fig. 100. 142.—Proposition. The two angles at the base of an isoceles triangle are equal. Let ab c, (Fig. 100,) be an isoceles triangle, of which the sides, a b and ac, are equal. Bisect the angle, (Art.

Page 66

AMERICAN HOUSE-CARPENTER. 61 77,) bac, by the line, ad. 'Then the line, b a, being equal te the line, a c ; the line, a d, of the triangle, A, being equal to the line, a d, of the triangle, B, being common to each; the angle, b a d, being equal to the angle, d ac; the line, b d, must, accord- ing to Art. 141, be equal to the line, dc ; and the angle at b must be equal to the angle at c. a b b . a c ad d a Fig. 101. 143.—Proposition. A diagonal crossing a parallelogram di. vides it into two equal triangles. Leta bcd, (Fig. 101,) bea given parallelogram, and b c,a line crossing it diagonally. Then, as a cis equal to b d, anda b to cd, the angle at a to the angle at d, the triangle, A, must, according to Art. 141, be equal to the triangle, B. 144.—Proposition. Leta b c d, (Fig. 102,) be a given pa- rallelogram, and b ca diagonal. At any distance between a b and c d, draw e f, parallel to a b ; through the point, g, the intersection of the lines, b c and e f, draw A i, parallel to b d. In every paral- lelogram thus divided, the parallelogram, A, is equal to the paral- lelogram, B. According to Ari. 143, the triangle, a bc, is “equal to the triangle, bc d; the triangle, C, to the triangle, D ; and E to F'; this being the case, take D and F from the triangle, bcd, and C and E from the triangle, a b c, and what remains

Page 67

62 PRACTICAL GEOMETRY. in one must be equal to what remains in the other; therefore, the parallelogram, A, is equal to the parallelogram, B. a b e f Fig. 103. 145.—Proposition. Parallelograms standing upon the same base and between the same parallels, are equal. Leta b cd and e f c d, (Fig. 103,) be given parallelograms, standing upon the same base, c d, and between the same parallels, a f and c d. Then, ab and e f being equal to c d, are equal to one another; b e being added to both a b and e f, a e equals b f; the line, ac, being equal to b d, and a eto b f, and the angle, v a e, being equal, (Art. 135,) to the angle, d b f, the triangle, a e c, must be _ equal, (Art. 141,) to the triangle, b f d ; these two triangles being equal, take the same amount, the triangle, be g, from each, and what remains in one, a b g c, must be equal to what remains in the other, e fd g; these two quadrangles being equal, add the same amount, the triangle, c æ d, to each, and they must still be equal; therefore, the parallelogram, a b c d, is equal to the paral- lelogram, e f c d. 146.— Corollary. Hence, if a parallelogram and triangle stand upon the same base and between the same parallels, the parallelo- gram will be equal to double the triangle. Thus, the paral- lelogram, a d, (Fig. 103,) is double, (Art. 143,) the triangle, ced. 147.— Proposition. Let a b c d, (Fig. 104,) be a given quad- rangle with the diagonal, a d. From b, draw ù e, parallel toa d, extend c dto e ; join a ande ; then the triangle, a ec, will be equal in area to the quadrangle, a bc d. Since the triangles, ad b and a d e, stand upon the same base, a d, and between the same paral-

Page 68

AMERICAN HOUSE-CARPENTER,. 63 d Fig. 104. lels, a d and b e, they are therefore equal, (Art. 145, 146 ;) and since the triangle, C, is common to both, the remaining triangles, A: and B, are therefore equal; then B being equal to A, the triangle, a e c, is equal to the quadrangle, a b c d. b Fig. 105. 148.—Proposition. If two straight lines cut each other, aś a band cd, (Fig. 105,) the vertical, or opposite angles, A and C, are equal. Thus, a e, standing upon c d, forms the angles, Band C, which together amount, (Art. 137,) to two right angles ; in the same manner, the angles, A and B, form two right angles ; since the angles, A and B, are equal to B and C, take the same amount, the angle, B, from each pair, and what remains of one pair is equal to what remains of the other; therefore, the an- gle, A, is equal to the angle, C. The same can be proved of the opposite angles, B and D. , 149.—Proposition. The three angles of any triangle are equal to two rightangles. Leta b c, (Fig. 106,} be a given tri- angle, with its sides extended to f, e, and d, and the line, €g,

Page 69

64 PRACTICAL GEOMETRY. J me e Fig. 106. drawn parallel tobe. As g cis parallel to e b, the angle, gc &, is, equal, (Ar. 135,) to the angle, e bd; as the lines, f c and b e, cut one another at a, the opposite angles, fa e and b a c, are equal, (Art. 148 ;) as the angle, f a e, is equal, (Art. 135,) to the angle, a c g, the angle, ac g, is equal to the angle, b ac; there- fore, the three angles meeting at c, are equal to the three angles of the triangle, a b c; and since the three angles at c are equal, (Art. 137,) to two right angles, the three angles of the triangle, a b c, must likewise be equal to two right angles. Any triangle can be subjected to the same proof. 150.— Corollary. Hence, if one angle of a triangle be a right angle, the other two angles amount to just one right angle. 151.—Corollary. If one angle of a triangle be a right angle, and the two remaining angles «re equal to one another, these are each equal to half a right angle. 152.— Corollary. If any two angles of a triangle amount to a right angle, the remaining angle is a right angle. 153.—Corollary. If any two angles of a triangle are together equal to the remaining angle. that remaining angle is a right angle. 154.—Corollary. If any two angles of a triangle are each equal to two-thirds of a right angle, the remaining angie is alsc equal to two-thirds of a right angle. 155.— Corollary. Hence, the angles of an equi-lateral trian- gle. are each equal to two-thirds of a right angle.

Page 70

AMERICAN HOUSE-CARPENTER, 65 é e e Fig. W. 156.—Proposition. If from tle extremities of the diameter of a sem-circle, two straight lines be drawn to any point in tne cir- cumferencé, the angle formed by them af that point wih be a rigntangle. Let abc, (Fig. 107,) be a given semi-circle , and ab and b c lines drawn from the extremities of the diameter, a c, to the given point, b; the angle formed at that point by these lines, is a right angle. Join the point, b, and the centre, d ; the lines, d a, d b and d c, being radii of the same circle, are equal ; the angle at a is therefore equal, (Art. 142.) to the angie, a b d, also, the angle at c is, for the same reason, equal to the angie, d Ł c; the angle, a b c, being equal to the angles at a and c taken together, must therefore, (Arf. 153,) be a right angle. Fig. 103. .57.—Proposittion. The square of the hypothenuse of a right-angled triangle, is equal to the squares of the two remaining sides. Leta b c, (Fig. 108.) be a given right-angled triangle. having a square formed on each of its sides: then, the square, 6 e, is equal to the squares, Ac and Fi taken together. 'Thıscan be

Page 71

66 PRACTICAL GEOMETRY. proved by showing that the parallelogram, b l is equal to the square, g b ; and thatthe parallelogram, c J, is equal to the square, A e. The angle, c b d, is a right angle, and the angle; a b f, is aright angle ; add to each of these the angle, a b c ; then the angle, f bc, will evi- dently be equal, (Art. 140,) to the angle, a b d ; the triangle, f b c, and the square, g b, being both upon the same base, f b, and between the sume parallels, f b and g c, the square, g b, is equal, (Art. 146,) to twice the triangle, f b c; the triangle, a b d, and the parallelo- gram, b l, being both’ upon the same base, b d, and between the same parallels, b d and al, the parallelogram, b l, is equal to twice the triangle, a b d; the triangles, f b c and a b d, being equal to one another, (Art. 141,) the square, æ b, is equal to the parallelo- gram, b l, either being equal to twice the triangle, fb c ora bd. The method of proving h c equal to c / is exactly similar—thus proving the square, b e, equal to the squares, h c and g b, taken together. This problem, which is the 47th of the First Book of Euclid is said to have been demonstrated first b} Pythagoras. It is sta. ted, (but the story is of doubtful authority,) that as a thank-offer ing for its discovery he sacrificed a hundred oxen to the gods. From this circumstance, it is sometimes called the hecatomb pro- blem. It is of great value in the exact sciences, more especially in Mensuration and Astionomy, in which many otherwise intri- eate calculations are by it made easy of solution. 158.—Proposttion. In a segment of a circle, the versed sine equals the radius, less the square root of the difference of the squares of the radius and half-chord. That is, the versed sine, ac, (Fig. 109,) equals a b, less cb. Now ab is radius, hence the radius, minus ¢ b, equals æ c, the versed sine. To find the value of ¢ b, it will be observed that cb is the side of the square, cf, whie the radius b d is the side of the square, b A, and the half-chord, c d, is the side of the square, c e; also, that these three squares are made upon the three sides of the right angled

Page 72

AMERICAN HOUSE-CARPENTER. 37 e a AA i Fig 109. triangle, 6 cd, and the square, b A, is therefore equal to the two squares, ¢ e and c f, (Art. 157;) therefore, the square, ¢ f, is equal to the square, b A, minus the square, ¢ e ;—or, is equal to the difference of the squares on ġ d and cd. Consequently the square root of cf is equal to the square root of the difference of the squares on 6 d and c d ; and since c b is the square root of cf, therefore c b equals the square root of the difference of the squares on bd and c d—or, equals the square root of the difference of the squares of the radius and the half-chord. Having found an expression for the value of ¢ 6, it remains merely to deduct this value from the radius, and the residue equals the versed sine ; for, as before stated, the versed sine, a c, equals the radius, æ b, minus cb; therefore, the versed sine equals the radius, minus the square root of the difference of the squares on the radius and half-chord. The rule expressed algebraically is v=r—v/7*—a’, where v is the versed sine, 7 the radius, and-a the half-chord. It is read, v equals 7, minus the square root of the difference of the squares of r and a. ~ 159.—Proposition. In an equilateral octagon the semi- - diagonal of a circumscribed square, having its sides coincident with four of the s'des of the octagon, equals the distance along

Page 73

PRACTICAL GEOMETRY Fig. 110. a side of the square from its corner to the more remote arigle of the octagon occurring on that side of the square. To prove this, it need only to be shown that the triangle, a o d, (Fig. 110,) is an isosceles triangle having its sides æ o and a d, equal. The octagon being equi-lateral, it is also equi-angular, therefore the angles, bc 0, e c 0, a d 0, &c., are all equal. Of the right- angled triangle, fe c, fe and fe being equal, the two angles, fec and f c eare equal, (Art. 142,) and are therefore, (Art. 151,) each equal to half a right angle. In like manner it may be shown that fab and fb a are also each equal to half a right angle. And since fee and fab are equal angles, therefore the lines e ¢ and ab are parallel, (Art. 135,) and hence the angles, e c o and ao d, are equal. These being equal, and the angles e c o and a d o being, by construction, equal, as before shown, therefore the angles a o d and a d o are equal, and con- sequently the lines a o and ad are equal. (Art. 142.) 160.—Proposition. An angle at the circumference of a circle is measured by half the are that subtends it: that is, the angle abc, (Fig. 111,) is equal to half the angle adc. Through the centre, d, draw the diameter, b ¢ The triangle a b d is an isosceles triangle, a d and b d being radii, and there- fore equal; hence the two angles, d a b and d b a, are equal.

Page 74

AMERICAN HOUSE-CARPENTER. 3 e Fig. 111. (Art. 142,) and the sum of these two angles is equal to the angle a de, (Art. 149,) and therefore one of them, ab d, is equal to the half of a de. The anglesa d e and a b d (or a6) are both subtended by the arcae. Now, since the angle, a d e, is measured by the are a e, which subtends it, therefore the half of the angle, æ d e, would be measured by the half of the arcae; and since a b d is equal to the half of ade, therefore a b d, or a b e, is measured by the half of the arc a e. It may be shown in like manner that the angle e b ¢ is mea- sured by half the are ec, and hence it follows that the angle, a b c, is measured by half the arc, a c, that subtends it. _ 161.—Proposition. In a circle, all the inscribed angles, a b c, (Fig. 112,) which stand upon the same side of the chord de, are equal. For each angle is measured by half the are df e, (Art. 169,) hence the angles are all equal. + 162.—Q@orollary. Equal chords, in the same circle, subtend equal angles. l 163.—Proposition. The angle formed by a chord and tan- gent is equal to any inscribed angle in the opposite segment

Page 75

C PRACTICAL GEOMETRY. Fig. 112. e Fig. 118 of the circle; that is, the angle D, (Fig. 113,) equals the angle A. Leto f be the chord, and a b the tangent; draw the dia- meter, dc; then d cb is a right angle, also d f ¢ is a right angle. (Art. 156.) The angles A and B together equal a right angle, (Art. 150 ;) also the angles B and D together equal a right angle, (equal the angle d c b ;) therefore the sum of A and B equals the sum of B and D. From each of these two equals, ¿aking the like quantity B, the remainders, A and

Page 76

AMERICAN HOUSE-CARPENTER. V1 D, are equal. Thus, it is proved for the angle at d; ıt is also true for any other angle; for, since all other inscribed angles on that side of the chord line, c f, equal the angle A, (Art. 161,) therefore the angle formed by a chord and tangent equals any angle in the opposite segment of the circle. This being proved for the acute angle, D, it is also true for the obtuse angle, a e f; for, from any point, n, (Fig. 114,) in the are c n f, draw lines to d, f and c; now, if it can be proved that the angle a c f equals the angle f n c, the entire proposition is proved, for the angle f n c equals any of all the inscribed angles that can be drawn on that side of the chord. (Art. 161.) To prove, then, that æ e f equals en f: the angle a ef equals the sum, of the angles A and B; also the angle e n f equals the sum of the angles Cand D. The angles B and D, being inscribed angles on the same chord, d f, are equal. The angles C and A being right angles, (Art. 156,) are likewise equal. Now, since A equals C, and B equals D, therefore the sum of A and B equals the sum of C and D—or the angle a c f equals the angle cn f. . 164.—Proposition. Two chords, a b and c d, (Fig. 115,) intersecting, the parallelogram, or rectangle formed by the two parts of one is equal to the rectangle formed by the two parts of the other. That is, c e mı ltiplied by e d, the product is

Page 77

72 PRACTICAL GEOMETRY. Fig. 115. equal to the product of a e multiplied by e b. The triangle A is similar to the triangle B, because it has corresponding an- gles. The angle ¿ equals the angle e, (Art. 148 ;) the angle at c equals the angle at æ because they stand upon the same chord, d b, (Art. 161) for the same reason the angle 6 equals the anyle d, for each stands upon the same chord, a e. There- fore, the triangle A having the same angles as the triangle B, the length of the sides of one are in like proportion as the length of the sides in the other. So, ed:aeneb:oce Hence, æ e multiplied by e 0 is equal to e d mutiplied by ¢ e— or the product of the means equals the product of the ex- tremes. 165.—Proposition. In any circle, when a segment is given, the radius is equal to the sum of the squares of half the chord and of the versed sine, divided by twice the versed sine. Let a b, (Fig. 116,) be the chord line, and v the versed sine of the segment. By the preceding article the triangle A is shown to be like the triangle B, having equal angles and proportionate

Page 78

AMERICAN HOUSE-CARPENTER. 3 length of sides. Therefore, v : nism : 4, or ~ =1; that is, @ is equal to the square of n (or n x n) divided by v. This result being added to v equals the diameter o v, which may be indi cated by the letter d ; thus, = +v=t1+v=4d; and the half ny 7 Te 2 sion by multiplying the numerator and denominator each by the of this, or =< = r = the radius. Reducing this expres- v+v like quantity, viz. v, there results, ae? and where c yee 20 as stated above, the radius is equal to the sum of the squares + of half the chord and of the versed sine, divided by twice the versed sine. 166.—Proposition. Any ordinate, m n, (Fig. 117,) in the segment of a circle, is equal to the square root of the difference 10 represents the chord, the expression is, =r: that is,

Page 79

44 PRACTICAL GEOMETRY. e o Fig. 117. of the squares of the radius and abscissa, (d n,) less the differ- ence of the radius and versed sine. So, if the chord a b, and the versed sine ¢ d, be given, the length of any number of ordinates may be found by which to describe the are. Find the radius, ¢ e, by the preceding Article. It will be observed that e m is also radius. Then, to find the length of the ordi- nate, m n, make e o equal to dn: now, according to Article 157, the square of e o taken from the square of e m, the residue equals the square of o m, and the square root of this residue will be the length of the line o m. Then from o m take o n equal to e d, and the result will be the length of m n. That is, the ordinate is equal to the square root of the difference of the squares of the radius and abscissa, less the difference of the radius and versed sine. This may be expressed algebraically thus: y= yr —a* — (r — v), where y is the ordinate, 7 the radius, w the abscissa, and v the versed sine ;—d n being the abscissa of the ordinate n m, d g the abscissa of the ordinate

Page 80

AMERICAN HOUSE-CARPENTER. 75 gf, &c.: the abscissa being in each case the distance from the foot of the versed sine, c d, to the foot of the ordinate whose length is sought. C mm Fig. 118, 167.—Proposition. The sides of any quadrangle being bisected, and lines drawn joining the points of bisection in the adjacent sides, these lines will form a parallelogram. Draw the diagonals, a b and c d, (Fig. 118.) It will here be per- ceived that the two triangles, æ e o and ac d, are homologous, having like angles and proportionate sides. Two of the sides of one triangle lie coincident with the two corresponding sides of the other triangle, therefore the contained angles between these sides in each triangle are identical. By construction, these corresponding sides are proportionate; æ ¢ being equal to twice a e, and ad being equal to twice ao; therefore the remaining sides are proportionate, c d being equal to twice e 0, hence the remaining corresponding angles are equal. Since, then, the angles a e o and a c d are equal, therefore the line e 0 is parallel with the diagonal c d—so, likewise, the line m n is parallel to the same diagonal, c œ. If, therefore, these two lines, e o and m n, are parallel to the same line, ¢ d, they must be parallel to each other. In the same manner the lineson and e m are proved parallel to the disgonal, a b, and to each

Page 81

76 PRACTICAL GEOMETRY. other; therefore the inscribed figure, m eo n, is a paralleto- gram. It may be remarked also, that the parallelogram so formed will contair just one-half the area of the circumscribing quadrangle. These demonstrations, which relate mostly to the problems previously given, are introduced to satisfy the learner in regard to their mathematical accuracy. By studying and thoroughly i understanding them, he will soonest arrive at a knowledge of their importance, and be likely the longer to retain them in memory. Should he have a relish for such exercises, and wish to continue them farther, he may consult Euclid’s Elements, in which the whole subject of theoretical geometry is treated of in a manner sufliciently intelligible to be understood by the young mechanic. The house-carpenter, especially, needs infor- mation of this kind, and were he thoroughly acquainted with the principles of geometry, he would be much less liable to commit mistakes, and be better qualified to excel in the execu- tion of his often difficult undertakings.

Page 82

SECTION JL—ARCHITECTURE. HISTORY OF ARCHITECTURE. 168.— Architecture has been defined to be—“ the art of build ing ;” but, in its common acceptation, it is—“ the art of designing and constructing buildings, in accordance with such principles as constitute stability, utility and beauty.” The literal signification of the Greek word archi-tecton, from which the word architect is derived, is chief-carpenter; but the architect has always been known as the chief designer rather than the chief builder. Of the three classes into which architecture has been divided—viz., Civil, Military, and Naval, the first is that which refers to the construction of edifices known as dwellings, churches and other public buildings, bridges, &c., for the accommodation of civilized man—and is the subject of the remarks which follow. 169.—This is one of the most ancient of the arts: the scrip- tures inform us of its existence at a very early period. Cain, the son of Adam,—‘ builded: a city, and called the name of the city after the name of his son, Enoch”—but of the peculiar styla or manner of building we are not informed. It is presumed that it was not remarkable for beauty, but that utility and perhaps sta- bility were its characteristics. Soon after the deluge—that me

Page 83

TS AMERICAN HOUSE-CARPENTER. morable event, whick removed from existence all traces of the works of man—the Tower of Babel was commenced. This was a work of such magnitude that the gathering of the materials, according to some writers, occupied three years; the period from its commencement until the work was abandoned, was twenty- two years; and the bricks were like blocks of stone, being twenty feet long, fifteen broad and seven thick. Learned men have given it as their opinion, that the tower in the temple of Belus at Baby lon was the same as that which in the scriptures is called the Tower of Babel. The tower of the temnle of Belns was square at its base, eacn side measuring one turlong, and consequently half a mile in circumference. Its form was that of a pyramid and its height was 660 feet. It hada winding passage on the outside from the base to the summit, which was wide enough for two carriages. 170.—Historical accounts of ancient cities, of which there are aow but few remains—such as Babylon, Palmyra and Ninevah of the Assyrians; Sidon, Tyre, Aradus and Serepta of the Phæ- nicians ; and Jerusalem, with its splendid temple, of the Israelites —show that architecture among them had made great advances, Ancient monuments of the art are found also among other nations; the subterraneous temples of the Hindoos upon the islands, Ele- phanta and Salsetta; the ruins of Persepolis in Persia ; pyramids, obelisks, temples, palaces and sepulchres in Egypt—all prove that the architects of those early times were possessed of skill and judgment highly cultivated. The principal characteristics of their works, are gigantic dimensions, immoveable solidity, and, in some instances, harmonious splendour. The extraordinary size of some is illustrated in the pyramids of Egypt. The largest of these stands not far from the city of Cairo: its base, which is Square, covers about 114 acres, and its height is nearly 500 feet The stones of which it is built are immense—the smallest being full thirty feet long. 171.- -Among the Greeks, architecture was cultivated as a fine

Page 84

` ARCHITECTURE. 79 art, and rapidly advanced towards perfection. Dignity anà grace were added to stability and magnificence. In the Doric order, their first style of building, this is fully exemplified. Phidias, _ Ictinus and Callicrates, are spoken of as masters in the art at this period: the encouragement and support of Pericles stimulated them to a noble emulation. The beautiful temple of Minerva, erected upon the acropolis of Athens, the Propyleum, the Odeum and others, were lasting monuments of their success. The Ionic and Corinthian orders were added to the Doric, and many mag- nificent edifices arose. ‘These exemplified, in their chaste propor- tions, the elegant refinement of Grecian taste. Improvement in Grecian architecture continued to advance, until perfection seems to have been attained. The specimens which have been partially preserved, exhibit a combination of elegant proportion, dignified simplicity and majestic grandeur. Architecture among the Greeks was at the height of its glory at the period immediately preceding the Peloponnesian war: after which the art declined. An excess of enrichment succeeded its former simple grandeur; yet a strict regularity was maintained amid the profusion of orna- ment. After the death of Alexander, 323 B. C., a love of gaudy splendour increased: the consequent decline of the art was visible, and the Greeks afterwards paid but little attention to the science. 172.—While the Greeks were masters in architecture, which they applied mostly to their temples and other public buildings, the Romans gave their attention to the science in the construction of the many aqueducts and sewers with which Rome abounded; building no such splendid edifices as adorned Athens, Corinth and Ephesus, until.about 200 years B. C., when their intercourse with the Greeks became more extended. Grecian architecture was introduced into Rome by Sylla ; by whom, as also by Marius and Cæsar, many large edifices were erected in various cities of Italy. But under Cæsar Augustus, at about the beginning of the christian era, the art arose to the greatest perfection it ever at-

Page 85

80 AMERICAN HOUSE-CARPENTER. tained in Italy. Under his patronage, Grecian artists were en- couraged, and many emigrated to Rome. It was at about this time that Solomon’s temple at Jerusalem was rebuilt by Herod— a Roman. This was 46 years in the erection, and was most pro- bably of the Grecian style of buildmg—perhaps of the Corin- thian order. Some of the stones of which it was built were 46 feet long, 21 feet high and 14 thick; and others were of the astonishing length of 82 feet. 'The porch rose toa great height ; the whole being built of white marble exquisitely polished. This is the building concerning which it was remarked—-“ Master, see what manner of stones, and what buildings are here.” For the construction of private habitations also, finished artists were em- ployed by the Romans: their dwellings being often built with the finest marble, and their villas splendidly adomed. After Augus- tus, his successors continued td beautify the city, until the reign of Constantine; who, having removed the imperial residence to Constantinople, neglected to add to the splendour of Rome; and the art, in consequence, soon fell from its high excellence. Thus we find that Rome was indebted to Greece for what she possessed of architecture—not only for the knowledge of its prin- ciples, but also for many of the best buildings themselves; these having been originally erected in Greece, and stolen by the un- principled conquerors—taken down and removed to Rome. Greece was thus robbed of her best monuments of architecture. ‘Touched by the Romans, Grecian architecture lost much of its elegance and dignity. The Romans, though justly celebrated for their scientific knowledge as displayed in the construction of their various edifices, were not capable of appreciating the simple grandeur, the refined elegance of the Grecian style; but sought to improve upon it by the addition of luxurious enrichment, and thus deprived it of true elegance. In the days of Nero, whose palace of gold is so celebrated, buildings were lavishly adorned. Adrian did much to encourage the art; but not satisfied with the simplicity of the Grecian style, the artists of his time aimed at

Page 86

N ARCHITECTURE. 81 inventing new ones, and added to the already reduidant embel- lishments of the previous age. Hence the origin of the pedestal, the great variety of intricate ornaments, the convex frieze, the round and the open pediments, &c. The rage for luxury continued until Alexander Severus, who made some impr>ve- ment; but very soon after his reign, the art began rapidly to decline, as particularly evidenced in the mean and trifling charac- ter of the ornaments. 173.—The Goths and Vandals, when they overran the coun- tries of Italy, Greece, Asia and Africa, destroyed most of the works of ancient architecture. Cultivating no art but that of war, these savage hordes could not be expected to take any interest in the beautiful forms and proportions of their habitations. From this time, architecture assumed an entirely different aspect. 'The celebrated styles of Greece were unappreciated and forgotten; and modern architecture took its first step on the platform of existence. ‘The Goths, in their conquering invasions, gradually extended it over Italy, France, Spain, Portugal and Germany, into England. From the reign of Gallienus may be reckoned the total extinction of the arts among the Romans. From his time until the 6th or 7th century, architecture was almost entirely neglected. The buildings which were erected during this suspension of the arts, were very rude. Being constructed of the fragments of the edi- fices which had been demolished by the Visigoths in their unre- strained fury, and the builders being destitute of a proper know- ledge of architecture, many sad blunders and extensive patch- work might have been seen in their construction—entablatures imrverted, columns standing on their wrong ends, and other ridi- culous arrangements characterized their clumsy work. The vast number of columns which the ruins around them afforded, thev ‘used as piers in the construction of arcades—which by some is thought, after having passed through various changes, to have been the origin of the plan of the Gothic cathedral. Buildings generally, which ar2 not of the classical styles, and which were mm

Page 87

s2 AMERICAN HOUSE-CARPENTER. erected after the fall of the Roman empire, have by some been indiscriminately included under the term Gothic. But the changes which architecture underwent during the dark ages, show that there were several distinct modes of building. 174:—Theodoric, king of the Ostrogoths, a friend of the arts, who reigned in Italy from A. D. 493 to 525, endeavoured to re- store and preserve some of the ancient buildings ; and erected others, the ruins of which are still seen at Verona and Ravenna. Simplicity and strength are the characteristics of the structures erected by him; they are, however, devoid of grandeur and ele- gance, or fine proportions. These are properly of the Goruic style; by some called the old Gothic to distinguish it from the pointed style, which is generally called modern Gothic. 175.—The Lombards, who ruled in Italy from A. D. 568, had no taste for architecture nor respect for antiquities. Accordingly, ` they pulled down the splendid monuments of ¢lassic architecture which they found standing, and erected in their stead huge build- ings of stone which were greatly destitute of proportion, elegance or utility—their characteristics being scarcely any thing more than stability and immensity combined with ornaments ofa pucrile cha- racter.. ‘Their churches were disfigured with rows of smal! columns along the cornice of the pediment, small doors and windows with circular heads, roofs supported by. arches having arched buttresses to resist their thrust, anda lavish display of incongruous orna- ments. This kind of architecture is called, the LomBarp style, and was employed in the 7th century in Pavia, the chief city of the Lombards; at which city, as also at many other places, a great many edifices were erected in accordance with its inelegant forms. 176.—-The Byzantine architects, from Byzantium, Constantino- ple, erected many spacious edifices; among which are included the cathedra.s of Bamberg, Worms and Mentz, and the most an cient part of the minster at Strasburg ; in all of these they com- bined the Roman-Ionic order with the Gothic of the Lombards.

Page 88

ARCHITECTURE. 83 This style is called the LomBparp-Byzantine. 'To the last style there were afterwards added cupolas similar to those used in the east, together with numerous sleuder pillars with tasteless capi- tals, and the many minarets which are the characteristics of the proper Byzantine, or Oriental style. 177:—In the eighth century, when the Arabs and Moors de- stroyed the kingdom of the Goths, the arts and. sciences were mostly in possession of the Musselmen-conquerors; at which time there were three kinds of architecture practised; viz: the Arabian, the Moorish and the modern-Gothic. The ARABIAN style was formed from Greek models, having circular arches | added, and towers which terminated with globes and minarets. ‘The Moortsn is very similar to the Arabian, being distinguished from it by arches in the form of a horse-shoe. It originated in Spain in the erection of buildings with the ruins of Roman archi- tecture, and is seei in all its splendour in the ancient palace of the Mohammedan monarchs at Grenada, called the Alhambra, or red- house. The Mopern-Gotuic was originated by the Visigoths in Spain by’a combination of the Arabian and Moorish styles; and introduced by Charlemagne into Germany. On account of the changes and improvements it there underwent, it was, at about the 13th or 14th century, termed the German, or romantic style. It is exhibited in great perfection in the towers of the minster of Strasburgh, the cathedral of Cologne and other edifices. The most remarkable features of this lofty and aspiring style, are the Jancet or pointed arch, clustered pillars, lofty towers and flying buttresses. It was principally employed in ecclesiastical archi- tecture, and in this capacity introduced into France, Italy, Spain, and England. 178.—The Gothic architecture of England is divided into the * Norman, the Early-English, the Decorated, and the Perpen- dicular styles. 'The Norman is principally distinguished by the character of its ornaments—the chevron, or zigzag, being the most common. Buildings in this style were erected in the 12th

Page 89

S4 AMERICAN HOUSE-CARPENTER. century. The Early-English is celebrated for the beauty of its edifices, the chaste simplicity and purity of design which they display, and the peculiarly graceful character of its foliage. This - style is of the 12th century. The Decorated style, as its name implies, is characterized by a great profusion of enrichment, which consists principally of the crocket, or feathered-ornament, and ball-flower. It was mostly in use in the 14th century. ‘The Perpendicular style, which dates from the 15th century, is distin- guished by its high towers, and parapets surmounted with spires similar in number and grouping to oriental minarets. 179.—Thus these several styles,-which have been erroneously termed Gothic, were distinguished by peculiar characteristics as well as by different names. The first symptoms of a desire to return toa pure style in architecture, after the ruin caused by the Goths, was manifested in the character of the art as displayed in the church of St. Sophia at Constantinople, which was erected by Justinian in the 6th century. The church of St. Mark at Venice, which arose in the 10th or 11th century, was the work of Grecian archi- tects, and resembles in magnificence the forms of ancient archi- tecture. ‘The cathedral at Pisa, a wonderful structure for the age, was erected by a Grecian architect in 1016. The marble with which the walls of this building were faced, and of which the four rows of columns that support the roof are composed, is said to be of an excellent character. The Campanile, or leaning-tower as it is usually called, was erected near the cathedral in the 12th cen- tury. Its inclination is generally supposed to have arisen from a poor foundation ; although by some it is said to have been thus constructed originally, in order to inspire in the minds of the beholder sensations of sublimity and awe. In the 13th century, the science in Italy was slowly progressing ; many fine churches were erected, the style of which displayed a decided advance in the progress towards pure classical architecture. In other parts ot Europe, the Gothic, or pointed style, was prevalent. The cathedral at Strasburg, designed ` y Irwin Steinbeck, was erected

Page 90

ARCHITECTURE. 85 in the 18th and 14th centuries. In France and England dur ing the 14th century, many very superior edifices were erected ’ in this style. 180.—In the 14th and 15th centuries, and particularly in the latter, architecture in Italy was greatly revived. The masters began to study the remains of ancient Roman edifices; and many splendid buildings were erected, which displayed a purer taste in the science. Among others, St. Peter’s of Rome, which was built about this time, is a lasting monument of the architectural skill of the age. Giocondo, Michael Angelo, Palladio, Vignola, and other celebrated architects, each in their turn, did much to restore the art to its former excellence. In the edifices which were erected under their direction, however, it is plainly to be seen that they studied not from the pure models of Greece, but from the remains of the deteriorated architecture of Rome. The high pedestal, the coupled columns, the rounded pediment, the many curved-and-twisted enrichments, and the convex frieze, were unknown to pure Grecian architecture. Yet their efforts were serviceable in correcting, to a good degree, the very impure taste that had prevailed since the overthrow of the Ro- man empire. 181.—At about this time, the Italian masters and numerous artists who had visited Italy for the purpose, spread the Roman style over various countries of Europe; which was gradually re- ceived into favor in place of the modern-Gothic. This fell into disuse ; although it has of late years been again cultivated. It requires a building of great magnitude and complexity for a per- fect display of its beauties. In America, the pure Grecian style was at first more or less studied; and perhaps the simplicity of its principles would be better adapted to a republican country, “than the intricacy and extent of those of the Gothic; but at the present time the latter style is being introduced, especially for ecclesiastical structures.

Page 91

86 AMERICAN HOUSE-CARPENTER. STYLES OF ARCHITECTURE. 152.—It is generally acknowledged that the various styles in architecture, were originated in accordance with the different pursuits of the early inhabitants of the earth ; and were brought by their descendants to their present state of perfection, through the propensity for imitation and desire of emulation which are found more or less among all nations. Those that followed agricultural pursuits, from being employed constantly upon the same piece of land, needed a permanent residence, and the wooden Aut was the offspring of their wants; while the shep- herd, who followed his flocks and was compelled to traverse large tracts of country for pasture, found the tent to be the most portable habitation; again, the man devoted to hunting and fishing—an idle and vagabond way of living—is naturally supposed to have been content with the cavern as a place of shelter. The latter is said to have been the origin of the Egyptian style; while the curved roof of Chinese structures gives a strong indication of their having had the tent for their model; and the simplicity of the original style of the Greeks, (the Doric,) shows quite conclusively, as is generally conceded, that its original was of wood. The modern-Gothic, or pointed style, which was most generally confined to ecclesiastica. structures, is said by some to have originated in an attempt to imitate the bower, or grove of trees, in which the ancients per- formed their idol-worship. 183.—There are numerous styles, or orders, in architecture ; and a knowledge of the peculiarities of each is important to the student in the art. An Orper, in architecture, is composed of three principal parts, viz: the Stylobate, the Column and the Entablature. 184.—The Sryzosate is the substructure, or basement, upon which the columns of an order are arranged. In Roman archi- tecture—especially in the interior of an edifice—it frequently occursythat cach column has a separate substructure; this is

Page 92

ARCHITECTURE. 87 called a pedestal. If possible, the pedestal should be avoided in all cases; because it gives to the column, the appearance of having been originally designed for a small building, and after- wards pieced-out to make it long enough for a larger one. 185.—The Corum is composed of the base, shaft and capital. 186.—The Eyrasrarunn, above and supported by the co- Iumng, is horizontal; and is composed of the architrave, frieze and-cornice. These principal parts are again divided into vavious members and mouldings. (See Sect. IIL.) 187.—The Bass of a column is so called from basis, a found- ation, or footing. . #188.—The Smarr, the upright part of a column standing upon the base and crowned with the capital, is from shafto, to dig—in the manner of a well, whose inside is not unlike the form of a column. 189.—The Caprrat, from kephale or caput, he head, is the uppermost and crowning part of the column. 190.—The Axcurrrave, from archi, chief or principal, and trahs, a beam, is that part of the entablature which lies in immediate connection with the column. ` 191.—The Frieze, from fibron, a fringe or border, is that part of the entablature which is immediately above the archi- trave and beneath the cornice. It was called by some of the ancients, zophorus, because it was usually enriched with sculp- tured animals. 192.—The Cornicr, from corona, a crown, is the upper and projecting part of the entablature—being also the uppermost and crowning part of the whole order. 193.—The Prpment, above the entablature, is the tr ey clare portion which is formed by the inclined edges of the roof at - the end of the building. In Gothic architecture, the pediment is called, a gable. 194.—The Tympanva is the perpendicular triangular surface which is enclosed by the ce rnice of the pediment. e

Page 93

88 AMERICAN HOUSE-CARPENTER. 195.—The Arric is a small order, consisting of pilasters and entablature, raised above a larger order, instead of a pediment. An attic story is the upper story, its windows being usually square. 196.—An order, in architecture, has its several parts and members proportioned to one another by a scale of 60 equal parts, which are called minutes. If the height of buildings were always the same, the scale of equal parts would be a fixed quantity—an exact number of feet and inches. But as buildings are erected of different heights, the column and its accompaniments are required to be of different dimensions. To ascertain the scale of equal parts, it is necessary to know the height to which the whole order is to be erected. This must be divided by the number of diameters which is directed for the order under consideration. Then the quotient obtained by such division, is the length of the scale of equal parts—and is, also, the diameter of the column next above the base. For instance, in the Grecian Doric order the whole height, includ- ing column and entablature, is 8 diameters. Suppose now it were desirable to construct an example of this order, forty feet high. Then 40 feet divided by 8, gives 5 feet for the length of the seale; and this being divided by 60, the scale is com- pleted. The upright- columns of figures, marked H and P, by the side of the drawings illustrating the orders, designate the height and the projection of the members. The projection of each member is reckoned from a line passing through the axis of the column, and extending above it to the top of the enta- blature. The figures represent minutes, or 60ths, of the major diameter of the shaft of the column. 197.—Grecian Sryzes. The original method of building among the Greeks, was in what is called the Doric order: to this were afterwards added the Jonic and the Corinthian. These three were the only styles known among them. Each is distingt ish- d from the other two, by not only a peculiarity of

Page 94

ARCHITECTURE. ; 89 some one or more of its principal parts, but also by a particular destination. The character of the Doric is robust, manly and Herculean-like; that of the Ionic is more delicate, feminine, matronly ; while that of the Corinthian is extremely delicate, youthful and virgin-like. However they may differ in their general character, they are alike famous for grace and dignity, elegance and grandeur, to a high degree of perfection. 198.—The Doric Oxper, (Fig. 120,) is so ancient that its origin is unknown—although some have pretended to have discovered it. But the most general opinion is, that it is an improvement upon the original wooden buildings of the Grecians. These no doubt were very rude, and perhaps not unlike the following figure. | The trunks of trees, set perpendicularly to support the roof, may be taken for columns; the tree laid upon the tops of the perpendicular ones, the architrave ; the ends of the cross-beams which rest upon the architrave, the triglyphs; the tree laid on the cross-beams as a support for the ends of the rafters, the bed-moulding of the cornice; the ends of the rafters which project beyond. the bed-moulding, the mutules; and perhaps the projection of the roof in front, to screen the entrance from the weather, gave origin to the portico. i The peculiarities of the Doric order are the triglyphs—those parts of the frieze which a Tair channels cut m

Page 95

90 DORIC ORDER. I estas — = cg lee ia aia _ —_ a ——= 54 diameters. 2 Fig. 120

Page 96

ARCHITECTURE. 91 their surface; the absence of a base to the column—as alio of fillets between the flutings of the column, and the plainness of the capital. The triglyphs are to be so disposed that the width of the metopes—the spaces between the triglyphs—shall be equal to their height. f .199.—The intercolumniation, or space between the columns, is regulated by placing the centres of the columns under the centres of the triglyphs—except at the angle of the building; where, as may be seen in Fig. 120, one edge of the triglyph must be over the centre of the column.* Where the columns are so disposed that one of them stands beneath every other tri- glyph, the arrangement is called, mono-triglyph, and is most common. When a column is placed beneath every third tri- - glyph, the arrangement is called diastyle ; and when beneath every fourth, ar@ostyle. This last style is the worst, and is sel- dom adopted. 200.—The Doric prided! is suitable for buildings that are des- tined for national purposes, for banking-houses, &e. Its ap- pearance, though massive and grand, is nevertheless rich and graceful. The Patent Office at Washington,eand the Custom- House at New York, are good specimens of this order. *Grecian Doric ORDER., When the width to be occuped by the whole front is limited; to deter: mine the diameter of the column, The relation between the parts may be expressed thus: =. _ 6a . ~~ d(b-++e) + (60 — c) Where a equals the width in feet occupied by the columns, and their intercolumniations taken collectively, measured at the base; b equals the width of the metope, in minutes; ¢ equals the width of the triglyphs in minutes; d equals the number of metopes, and z equals the diameter in feet. Example.—A front of six columns—hexastyle—61 feet wide; the frieze having one triglyph over each intercolumniation, or mono-triglyph. In this case, there being five intercolumniations and two metopes over each, therefore there are 5 X 2 = 10 metopes. Let the metope equal 42 minutes and - the triglyph equal 3, Then a= 61; b= 42; c=28; and d= 10; and the formula above becomes, 60 x 6f 60 x 61 3660 = DeF Fa Foa ox 0438 pS 5 feet = the diameter required, Exampie.—An octastyle front, 8 columns, 184 feet wide, three metopes over each intercolumnia Yon, 21 in all, and the metope and triglyph 42 anc 28, as before. Then, 60 x 184 11040 = ee auia Šī = 91 a2 + 2) + (GO — 8) > 1302 = 138 z feet =the diameter required.

Page 97

IONIC ORDER, ©) Lo i OY Yumy O i aie) Fe | 221 a a &

Page 98

ARCHITECTURE. 93 201.—The Ionic Orvrer. (Fig. 121.) The Doric was for some time the only order in use among the Greeks. They gave their attention to the cultivation of it, until perfection seems to have been attained. Their temples were the principal objects upon which their skill in the art was displayed; and as the Doric order seems to have been well fitted, by its massive pro- portions, to represent the character of their male deities rather than the female, there seems to have been a necessity for an- other style which should be emblematical of feminine graces, and with which they might decorate such temples as were de- dicated to the goddesses. Hence the origin of the Ionic order. This was invented, according to historians, by Hermogenes of Alabanda; and he being a native of Caria, then in the posses- sion of the Ionians, the order was called, the Ionic. 202.—The distinguishing features of this order are the vo lutes, or spirals of the capital; and the dentils among the bed- mouldings of the cornice: although in some instances, dentils are wanting. The volutes are said to have been designed as a representation of curls of hair on the head of a matron, of whom the whole column is taken as a semblance. 208.—The intercolumniation of this and the other orders— both Roman and Grecian, with the exception of the Doric— are distinguished as follows. When the interval is one and a half diameters, it is called, pycnostyle, or columns thick-set ; when two diameters, systyle ; when two and a quarter diame- ters, eustyle ; when three diameters, diastyle ; and when more than three diameters, arwostyle, or columns thin-set. In all the orders, when there are four columns in one row, the arrange- ment is called, ¢etrastyle ; when there are six in a row, hexa- style ; and when eight, octastyle. 204.—The Ionic order is appropriate for churches, colleges, seminaries, libraries, all edifices dedicated to literature and the arts, and all places of peace and tranquillity. The front of the

Page 99

94. ARCHITECTURE. Merchants’ Exchange, New York city, is a good specimen of this order. 205.— To describe the Ionic volute. Draw a perpendicular from a to s, (Fig. 122,) and make a s equal to 20 min. or to + of the whole height, ac; draw s 0, at right angles to s a, and equal to 14 min.; upon o, with 24 min. for radius, describe the eye of the volute; about o, the centre of the eye, draw the square, 7r ¿1 2, with sides equal to half the diameter of the eye, viz. 24 min., and divide it into 144 equal parts, as shown Cc Fig. 122.

Page 100

AMERICAN HOUSE-CARPENTER. 95 Fig. 123. at Fig. 123. The several centres in rotation are at the angles — formed by the heavy lines, as figured, 1, 2, 3, 4, 5, 6, &e. The position of these angles is determined by commencing at the point, 1, and making each heavy line one part less in length than the preceding one. No. 1 is the centre for the arc, a b, (Fig. 122;) 2 is the centre for the arc, c; and so on to the last. The inside spiral line is to be described from the centres, x, x, w, &e., (Fig. 123,) being the centre of the first small square towards the middle of the eye from the centre for the outside arc. The breadth of the fillet at æ 7, is to be made equal to 2:3 min. This is for a spiral of three revolutions; but one of any number of revolutions, as 4 or 6, may be drawn, by dividing o f, (Fig. 123,) into a corresponding number of equal parts. Then divide the part nearest the centre, 0, into two »parts, as at A; join o and 1, also o and 2; draw A 3, parallel to o 1, and Ž 4, parallel to o 2; then the lines, o 1, o 2, 4 3, A 4, will determine the length of the heavy lines, and the place of the centres. (See Art. 489.) l

Page 101

96 AMERICAN HOUSE-CARPENTER. 206.—The Corman ORDER, (Fig. 125,) is in general like the Ionic, though the proportions are lighter. The Corinthian displays a more airy elegance, a richer appearance ; but its distinguishing feature is its beautiful capital. This is gene- rally supposed to have had its origin in the capitals of the columns of Egyptian temples ; which, thongh not approaching it in elegance, have yet a similarity of form with the Corin- thian. The oft-repeated story of its origin which is told by Vitrnvins—an architect who flourished in Rome, in the days of Augustus Ceesar—though pretty generally considered to be fabulous, is nevertheless worthy of being again recited. It is this: a young lady of Corinth was sick, and finally died. Her nurse gathered into a deep basket, such trinkets and keepsakes as the lady had been fond of when alive, and placed them upon her grave; covering the basket with a flat stone or tile, that its contents might not be disturbed. The basket was placed accidentally upon the stem of an acanthus plant, which, shooting forth, enclosed the basket with its foli- age ; some of which, reaching the tile, turned gracefully over in the form of a volute. a A celebrated sculptor, Calima- chus, saw the basket thus deco- rated, and from the hint which it suggested, conceived and con- structed a`ʻcapital for a column. ` This was called Corinthian from the fact that it was invented and first made use of at Corinth. 207.—The Corinthian being the gayest, the richest, and most lovely of all the orders, it is appropriate for edifices which are dedicated to amusement, banqueting and festiv- ity—for all places where delicacy, gayety and splendour are desirable. , 208.—In addition to the three regular orders of architecture,

Page 102

“4 @ ei ARCHITECTURE. | eo is Ve af "eg WAY ANA | D er o A E a CORINTHIAN ORDER.—Fig. 125.

Page 103

9S AMERICAN HOUSE-CARPENTER. it was sometimes customary among the Greeks—and after. wards among other nations—to employ representations of the human form, instead of columns, to support entablatures; these were called Persians and Caryatides. 209.—Perrsians are statues of men, and are so called in commemoration of a victory gained over the Persians by Pau- sanias. The Persian prisoners were brought to Athens and condemned to abject slavery; and in order to represent them in the lowest state of servitude and degradation, the statues were loaded with the heaviest entablature, the Doric. 210.—Caryatipes are statues of women dressed in long robes after the Asiatic manner. Their origin is as follows. In a war between the Greeks and the Caryans, the latter were totally vanquished, their male population extinguished, and their females carried to Athens. To perpetuate the memory of this event, statues of females, having the form and dress of the Caryans, were erected, and crowned with the Ionic or Co- riuthian entablature. The caryatides were generally formed of about the human size, but the persians much larger; in order to produce the greater awe and astonishment in the beholder. The entablatures were proportioned to a statue in like manner as to a column of the same height. l 211.—These semblances of slavery have been in frequent use among moderns as well as ancients; and as a relief from the stateliness and formality of the regular orders, are capable of forming a thousand varieties; yet in a land of liberty such marks of human degradation ought not to be perpetuated. 212.—Rtoman Sryres. Strictly speaking, Rome had no. architecture of her own—all she possessed was borrowed from other nations. Before the Romans exchanged intercourse with the Greeks, they possessed some edifices of considerable extent and merit, which were erected by architects from Etru- ria; but Romë was principally indebted to Greece for what she acquired of the art. Although there is no such thing as

Page 104

ARCHITECTURE. 99 spore OBAN OF IT TPO) aa a 1 Tate HET a [A i RO a UNN A (5 CC a Ce D a een ee 94 +" ` i HL | i ae a i a ~ > ia 09 oli

Page 105

JO AMERICAN HOUSE-CARPENTER. an architecture of Roman invention, yet no nation, perhaps, ever was so devoted to the cultivation of the art as the Ro- man. Whether we consider the number and extent of their structures, or the lavish richness and splendour with which they were adorned, we are compelled to yield to them our admiration and praise. At one time, under the consuls and emperors, Rome employed 400 architects. The public works —such as theatres, circuses, baths, aqueducts, &c.—were, in extent and grandeur, beyond any thing attempted in modern times. Aqueduets were built to convey water from a distance of 60 miles or more. In the prosecution of this work, rocks and mountains were tunnelled, and valleys bridged. Some of the latter descended 200 feet below the level of the water; and in passing them the canals were supported by an arcade, or succession of arches. Public baths are spoken of as large as cities; being fitted up with numerous conveniences for exercise and amusement. Their decorations were most splen- did; indeed, the exuberance of the ornaments alone was offen- sive to good taste. So overloaded with enrichments were the baths of Diocletian, that on an occasion of public festivity, great quantities of sculpture fell from the ceilings and entabla- tures, killing many of the people. 213.—The three orders of Greece were introduced into Rome in all the richness and elegance of their perfection. But the luxurious Romans, not satisfied with the simple ele- gance of their refined proportions, sought to improve upon them by lavish displays of ornament. They transformed in many instances, the true elegance of the Grecian art into a gaudy splendour, better suited to their less refined taste. The Romans remodelled each of the orders: the Doric, (Fig. 126,) was modified by increasing the heignt of the column to 8 dia- meters; by changing the echinus of the capital for an ovolo, or quarter round, and adding an astragal and neck below it; by placing the centre, instead of one edge, of the first triglyph

Page 106

ARCHITECTURE. 101 i ——————————— Sd BECO ECC ETE EVE ELBOW SLEUTH MRL CE A REO EL ECE ELT t UA jj CAN IRS NNR RR SUNT SEN X partt Raa A N mig Xg TIRER ENEN J A) 2 E ETN aaa Room Fig 127.

Page 107

102 AMERICAN IOUSE-CARPENTER. od over the centre of the column; and introducing horizontal instead of inclined mutules in the cornice, and in some instan ces dispensing with them altogether. The Ionie was modified by diminishing the size of the volutes, and, in some specimens, introducing a new capital in which the volutes were diago nally arranged, (Fig. 127.) This new capital has becn termed modern Ionic. The favorite order at Rome and her colonies was the Corinthian, (Fig. 128.) But this order, the Roman artists in their search for novelty, subjected to many altera- tions—especially in the foliage of its capital. Into the upper part of this, they introduced the modified Ionic capital; thus combining the two in one. This change was dignified with the importance of an order, and received the appellation, Composirr, or oman: the best specimen of which is found in the Arch of Titus, (Zig. 129.) This style was not much used among the Romans themselves, and is but slightly appreciated now. 214.—The Tuscan Orper is said to have been introduced to the Romans by the Etruscan architects, and to have been the only style used in Italy before the introduction of the Grecian orders. However this may be, its similarity to the Doric order gives strong indications of its having been a rude imita- tion of that style: this is very probable, since history informs us that the Etruscans held intercourse with the Greeks at a remote period. The rudeness of this order prevented its ex- tensive use in Italy. All that is known concerning it is from Vitruvius—no remains of buildings in this style being found among ancient ruins. 215.—For mills, factories, markets, barns, stables, &c., where utility and strength are of more importance than beauty, the improved modification of this order, called the modern Tuscan, (Fig. 180,) will be useful; and its simplicity recommends it where economy is desirable. 216.—Eeyptian Sry. The architecture of the ancient

Page 108

ARCHITECTURE. m eo RTE NE, alee ae. Beas BD bar wii 327. Tye ¢ Wo a7 Sow | (ONO ZO) = WE KR AV Re Ve PD nen orn TAV TA) S ca g Cae Zn vines ny $ mf | are? fj. WEIN O teeth oa to AtA a S

Page 109

104 AMERICAN HOUSE-CARPENTER. Pe Il. ; J 33 E SEG E A EEE A R NE EA AA ANI g ye Be EATERS RUN TS AE ARAT AY EER A: PEAN ROSCA UE ARS A AMAA ES REAM AS RR ERANS Y 2 KE repe w29 EN w Sug eE x hs = OOO RO ROBO ROO HOLE FP RARBA SERN TARTAN TR INET LUNN hg A pr 7 AO ESE NE EGY EGR DTG 54 ISSN RS RE fie CR A Fe PERE 35 YIP SSE NAUDE NE N/URE ARO UNERNA SECS RA RE LW RB AANA US AUN UO SE © Fig. 129,

Page 110

TUSCAN ORDER. 105 | 30 6 Dia. pe g | © Leal | = | igis | l

Page 111

196 ARCHITECTURE. Egyptians -to which that of the ancient Hindoos bears some re- semiblance—is characterized by boldness of outline, solidity and grandeur. ‘The amazing labyrinths and extensive artificial lakes, the splendid palaces and gloomy cemeteries, the gigantic pyramids and towering obelisks, of the Egyptians, were works of immen- sity and durability; and their extensive remains are enduring proofs of the enlightened skill of this once-powerful, but iong since extinct nation, The principal features of the Egyptian Style of architecture are—uniformity of plan, never deviating from right lines and angles; thick walls, having the outer surface slightly deviating inwardly from the perpendicular; the whole building low; roof flat, composed of stones reaching in one piece from pier to pier, these being supported by enormous columns, very stout in proportion to their height; the shaft sometimes polygonal, having no base but with a great variety of handsome capitals, the foliage of these being of the palm, lotus and other leaves; entablatures having simply an architrave, crowned with a huge cavetto orna- mented with sculpture; and the intercolumniation very narrow, usually 13 diameters and seldom exceeding 24. In the remains of a temple, the walls were found to be 24 feet thick ; and at the gates of Thebes, the walls at the foundation were 50 feet thick and perfectly solid. The immense stones of which these, as well as Egyptian walls generally, were built, had both their inside and outside surfaces faced, and the joints throughout the body of the wall as perfectly close as upon the outer surface. For this reason, as well as that the buildings generally partake of the pyramidal form, arise their great solidity and durability. The dimensions and extent of the buildings may be judged from the temple ot Jupiter at Thebes, which was 1400 feet long and 300 feet wide— exclusive of the porticos, of which there was a great number. It is estimated by Mr. Gliddon, U. S. consul in Egypt, that not less than 25,000,000 tons of hewn stone were employed in the erection of the Pyramids of Memphis alone,—or enough to con- struct 3,000 Bunker-Hill monuments. Some of the blocs are 40

Page 112

EGYPTIAN STYLE.

Page 113

128$ . ARCHITECTURE. E: feet long, and polished with emery to a surprising degree. Itis conjectured that the stone for these pyramids was brought, by rafts and canals, from a distance of 6 or 7 hundred miles. 217.—The general appearance of the Egyptian style of archi- tecture is that of solemn grandeur—amounting sometimes to sepulchral gloom. For this reason it is appropriate for cemete- ries, prisons, &c.; and being adopted for these purposes, it is gradually gaining favour. A great dissimilarity exists in the proportion, form and general features of Egyptian columns. In some instances, there is no uniformity even in those of the same building, each differing from the others either in its shaft or capital. For practical use in this country, Fig. 131 may be taken as a standard of this style. The Halls of Justice in Centre-street, New-York city, is a building in general accordance with the principles of Egyptian architecture. Buildings in General. 218.—That style of architecture is to be preferred in whicn utility, stability and regularity, are gracefully blended with gran- deur and elegance. But as an arrangement designed for a warm country would be inappropriate for a colder climate, it would seem that the style of building ought to be modified to suit the wants of the people for whom it is designed. High roofs to resist the pressure of heavy snows, and arrangements for artificial heat, are indispensable in northern climes; while they would be regarded as entirely out of place in buildings at the equator. 219.—Among the Greeks, architecture was employed chiefly upon their temples and other large buildings; and the proportions of the orders, as determined by them, when executed to such large dimensions, have the happiest effect. But when used for small buildings,porticos, porches, &c., especially in country-places, they are rather heavy and clumsy; in such cases, more slender proportions will be found to produce a better effect. The

Page 114

AMERICAN HOUSE-Cé RPENTER. 10% English cottage-style is rather more appropriate, and is becom- ing extensively practised for small buildings in the country. 220.—Every building should bear an expression suited to its destination. If it be intended for national purposes, it should be magnificent—grand; for a private residence, neat and modest ; for a banqueting-house, gay and splendid; for a monument or cemetery, gloomy—melancholy ; or, if forachurch, majestic and graceful. By some it has been said—‘‘somewhat dark and gloomy, as being favourable to a devotional state of feeling ;” but such impressions can only result from a misapprehension of the nature of true devotion. “Her ways are ways of pleasantness, and all her paths are peace.” ‘The church should rather be a type of that brighter world to which it leads. 221.—However happily the several parts of an edifice may be disposed, and however pleasing it may appear as a whole, yet much depends upon its size, as also upon the character and style of the structures in its immediate vicinity, and the degree of cul- tivation of the adjacentcountry. A splendid country-seat should have the out-houses and fences in the same style with itself, the trees and shrubbery neatly trimmed, and the grounds well cul- tivated. 292, Europeans express surprise that so many houses in this country are built of wood. And yet, in a new country, where wood is plenty, that this should be so is no cause for wonder. Still, the practice should not be encouraged. Buildings erected with brick or stone are far preferable to those of wood; they are more durable; not so liable to injury by fire, nor to need repairs : and will be found in the end quite as economical. A wooden house is suitable for a temporary residence only; and those who would bequeath a dwelling to their children, will endeavour to buil& with a more durable material. Wooden cornices and gut- ters, attached to brick houses, are objectionable—not only on 2e- count of their frail nature, but also because they render the build- ing liable to destruction by fire.

Page 115

110 AMERICAN HOUSE-CARPENTER. Fig. 132 Fig. 133. Onder-Cellar. Basement,

Page 116

» ARCHITECTURE. 111 223.—Dwelling houses are built of various dimensions and styles, according to their destination ; and to give designs and directions for their erection, it is necessary to know their situa- tion and object. A dwelling intended for a gardener, would require very different dimensions and arrangements from one intended for a retired gentleman—with his servants, horses, &¢.; nor would a house designed for the city be appropriate for the country. For city houses, arrangements that would be convenient for one family might be very inconvenient for twc or more. Fig. 182, 183, 184, 135, 136, and 137, represent the ichnographical projection, or ground-plan, of the floors of an ordinary city house, designed to be occupied by one family only. Fig. 189 is an elevation, or front-view, of the same house: all these plans are drawn at the same scale—which is tbat at the bottom of Fig. 139. Fig. 132 is a Plan of the Under-Cellar. a, is the coal-vault, 6 by 10 feet. b, is the furnace for heating the house. c, d, are front and rear areas. Fig. 133 is a Plan of the Basement. a, is the library, or ordinary dining-room, 15 by 20 feet. b, is the kitchen, 15 by 22 feet. | c, is the store-room, 6 by 9 feet. d, is the pantry, 4 by 7 feet. e, is the china closet, 4 by 7 feet. J; is the servants’ water-closet. 9, is a closet. h, is a closet with a dumb-waiter to the first story above, 2, is an ash closet under the front stoop. j, is the kitchen-range. k, is the sink for washing and drawing water l, are wash trays.

Page 117

AMERICAN HOUSE-CARPENTER. 112 Fig. 135. Second Story. Fig. 134. First Story

Page 118

ARCHITECTURE. 118 Fig. 134 is a Plan of the First Story. a, is the parlor, 15 by 34 feet. l b, is the dining-room, 16 by 23 feet. ¢, is the vestibule. e, is the closet containing the dumb-waiter from the basement. J; is the closet containing butler’s sink. g, 9, are closets. h, is a closet for hats and cloaks. t, j, are front and rear balconies. Fig. 135 is the Second Story. a, a, are chambers, 15 by 19 feet. b, is a bed-room, 7% by 18 feet. c, is the bath-room, 74 by 13 feet. d, d, are dressing-rooms, 6 by 78 feet. e, €, are closets. J, J, are wardrobes. 9, J, are cupboards. Fig. 136 is the Third Story. a, a, are chambers, 15 by 19 feet. b, b, are bed-rooms, 74 by 13 feet. c, c, are closets. d, is a linen closet, 5 by 7 feet. e, e, are dressing-closets. SF, are wardrobes. 9, 9, are cupboards. Fig. 137 is the Fourth Story. „ a, æ, are chambers, 14 by 17 feet. b, b, are bed-rooms, 84 by 17 feet. C, C, ©, are closets. d, is the step-ladder to the root. 15

Page 119

1i4 AMERICAN HOUSE-CARPENTER. Fig. 136. Fig. 137. Third Story. Fourth Story.

Page 120

ARCHITECTURE. 115 Fig. 188 is the Section of the House showing the heights of the several stories. s Fig. 139 is the Front Elevation. The size of the house is 25 feet front by 55 feet deep; this is about the average depth, although some are extended to 60 and 65 feet in depth. These are introduced to give some general ideas of the prin- ciples to be followed in designing city houses. In placing the chimneys in the parlours, set the chimney-breasts equi-distant from the ends of the room. The basement chimney-breasts may be placed nearly in the middle of the side of the room, as -there is but one flue to pass through the chimney-breast above ; but in the second story, as there are two flues, one from the basement and one from the parlour, the breast will have to be placed nearly perpendicular over the parlour breast, so as to receive the flues within the jambs of the fire-place. As it is desirable to have the chimney-breast as near the middle of the room as possible, it may be placed a few inches towards that point from over the breast below. So in arranging thosé of the stories above, always make provision for the flues from below. 224.—In placing the stairs, there should be at least as much room in the passage at the side of the stairs, as upon them; and in regard to the length of the passage in the second story, there must be room for the doors which open from each of the principal rooms into the hall, and more if the stairs require it. Having assigned a position for the stairs of the second story, now generally placed in the centre of the depth of the house, let the winders of the other stories be placed perpendicularly “over and under them; and be careful to provide for head- room. ‘To ascertain this, when it is doubtful, it is well to draw a vertical section of the whole stairs; but in ordinary cases, this is not necessary. To dispose the windows properly, the

Page 121

116 AMERICAN HOUSE-CARPENTER. Sa NOL@ Kez 0 (a)! EOE: ae Fig. 138. Fig. 139, Section. 4levaticn. ae.

Page 122

ARCHITECTURE. 117 middle window of each story should be exactly in the middle of the front; but the pier between the two windows which light the parlour, should be in the centre of that rooin; be- cause when chandeliers or any similar ornaments, hang from the centre-pieces of the parlour ceilings, it is important, in order to give the better effect, that the pier-glasses at the front and rear, be in a range with them. If both these ob- jects cannot be attained, an approximation to each must be attempted. The piers should in no case be less in width than the window openings, else the blinds or shutters when thrown open will interfere with one another ; in general practice, it is well to make the ontside piers 3 of the width of one of the middle piers. When this is desirable, deduct the amount of the three openings from the width of the front, and the re- mainder will be the amount of the width of all the piers; divide this by 10, and the product will be 4 of a middle pier ; and then, if the parlour arrangements do not interfere, give twice this amount to each corner pier, and three times the same amount to each of the middle piers. PRINCIPLES OF ARCHITECTURE. 225.—In the construction of the first habitations of men, frail and rude as they must have been, the first and principal object was, doubtless, utility—a mere shelter from sun and rain. But as successive storms shattered the poor tenement, man was taught by experience the necessity of building with an idea to durability. And when in his walks abroad, the symmetry, proportion and beauty of nature met his admiring gaze, contrasting so strangely with the misshapen and dispro- > portioned work of his own hands, he was led to make gradual changes; till his abode was rendered not only commodious and durable, but pleasant in its appearance; and building became a fine art, having utility for its basis.

Page 123

118 AMERICAN HOUSE-CARPENTER. 996,—In all designs for buildings of importance, utility, durability and beauty, the first great principles of architec- ture, should be pre-eminent. In order that the edifice be useful, commodions and comfortable, the arrangement of the apartments should be such as to fit them for their several des- tinations; for public assemblies, oratory, state, visitors, retir- ing, eating, reading, sleeping, bathing, dressing, &c.—these should each have its own peculiar form and situation. To accomplish this, and at the same time to make their relative situation agreeable and pleasant, producing regularity and harmony, require in some instances much skill and sound judgment. Convenience and regularity are very important, aud each should have due attention; yet when both cannot be obtained, the latter should in most cases give place to the former. A building that is neither convenient nor regular, whatever other good qualities it may possess, will be sure of disapprobation. 227.—The utmost importance should be attached to such arrangements as are calculated to promote health: among these, ventilation is by no means the least. For this purpose, the ceilings of the apartments should have a respectable height; and the sky-light, or any part of the roof that can be made moveable, should be arranged with cord and pullies, so as to be easily raised and lowered. Small openings near the ceiling, that may be closed at pleasure, should be made in the partitions that separate the rooms from the passages—espe- cially for those rooms which are used for sleeping apartments. All the apartments should be so arranged as to secure their being easily kept dry and clean. In dwellings, suitable apart- ments shonld be fitted up for bathing with all the necessary apparatus for conveying the water. 228.—To i sure stability in an edifice, it should be designed upon well-known geometrical principles: such as science has de- monstrated to be necessary and sufficient fcr firmness and dura

Page 124

AMERICAN HOUSE-CARPENTER. 119 bility. It is well, also, that it have the appearance of stability as well as the reality ; for should it seem tottering and unsafe, the sensation of fear, rather than those of admiration and pleasure, will be excited in the beholder. ‘To secure certainty and accu- racy in the application of those principles, a knowledge of the strength and other properties of the materials used, is indispensa- ble; and in order that the whole design be so made as to be capable of execution, a practical knowledge of the requisite mechanical operations is quite important. 229.—-T he elegance of an architectural design, although chiefly depending upon a just proportion and harmony of the parts, will be promoted by the introduction of ornaments—provided this be judiciously performed. -For enrichments should not only be of a proper character to suit the style of the building, but should also have their true position, and be bestowed in proper quantity. The most common fault, and one which is prominent in Roman archi- tecture, is an excess of enrichment: an error which is carefully to be guarded against. ` But those who take the Grecian models for their standard, will not be liable to go to that extreme. In ornamenting a cornice, or any other assemblage of mouldings, at least every alternate member should be left plain; and those that are near the eye should be more finished than those which are dis- tant. Although the characteristics of good architecture are utili- ty and elegance, in connection with durability, yet some buildings are designed expressly for use, and others again for ornament : in the former, utility, and in the latter, beauty, should be the gov- erning principle. 23().—The builder should be intimately acquainted with the principles upon which the essential, elementary parts of a build- ing are founded. A scientific knowledge of these will insure > certainty and security, and enable the mechanic toerect the most extensive and lofty edifices with confidence. The more important parts are the foundation, the column, the wall, the lintel, the arch, the vault, the dome and the roof. A separate description of the

Page 125

120 ARCHITECTURE. oJ peculiarities of each, would seem to be necessary; and cannot perhaps be better expressed than in the following language of a modern writer on this subject. 231.—“In laying the Founparion of any building, it is ne- cessary to dig to a certain depth in the earth, to secure a solid basis, below the reach of frost and common accidents. The most solid basis is rock, or gravel which has not been moved. Next to these are clay and sand, provided no other excavations have been made in the immediate neighbourhood. From this basis a stone wall is carried up to the surface of the ground, and constitutes the foundation. Where it is intended that the super- structure shall press unequally, as at its piers, chimneys, or columns, it is sometimes of use to occupy the space between the points of pressure by an inverted arch. This distributes the pressure equally, and prevents the foundation from springing be- tween the different points. In loose or muddy situations, it is always unsafe to build, unless we can reach the solid bottom below. In salt marshes and flats, this is done by depositing tim- bers, or driving wooden piles into the earth, and raising walls upon them. The preservative quality of the salt will keep these timbers unimpaired for a great length of time, and makes the foundation equally secure with one of brick or stone. 232.—The simplest member in any building, though by no means an essential one to all, is the Coum, or pillar. This is a perpendicular part, commonly of equal breadth and thickness, not intended for the purpose of enclosure, but simply for the sup- port of some part of the superstructure. 'The principal force which a column has to resist, is that of perpendicular pressure. In its shape, the shaft of a column should not be exactly cylin- drical, but, since the lower part must support the weight of the superior part, in addition to the weight which presses equally on the whole column, the thickness should gradually decrease from bottom to top. The outline of columns should bea little curved, 80 as to represent a portion of a very long spheroid, or paraboloid,

Page 126

AMERICAN HOUSE-CARPENTER. “191 rather than of acone. This figure is the joint result of two cal- culations, independent of beauty of appearance. One of these is, that the form best adapted for stability of base is that of a cone; the other is, that the figure, which would be of equal strength throughout for supporting a superincumbent weight, would be generated by the revolution of two parabolas round the axis of the column, the vertices of the curves being at its ex- tremities. ‘The swell of the shafts of columns was called the e%- tasis by the ancients. It has been lately found, that the columns of the Parthenon, at Athens, which have been commonly sup. posed straight, deviate about an inch from a straight line,.and that their greatest swell is at about one third of their height. Columns in the antique orders are usually made to diminish one sixth or one seventh of their diameter, and sometimes even one fourth. The Gothic pillar is commonly of equal thickness throughout. 233.—The Watt, another elementary part of a building, may be considered as the lateral continuation of the column, answer- ing the purpose both of enclosure and support. A wall must diminish as it rises, for the same reasons, and in the same propor- tion, as the column. It must diminish still more rapidly if it ex- tends through several stories, supporting weights at different heights. A wall, to possess the greatest strength, must also con- sist of pieces, the upper and lower surfaces of which are horizon- tal and regular, not rounded nor oblique. The walls of most of the ancient structures which have stood to the present time, are constructed in this manner, and frequently have their stones bound together with bolts and cramps of iron; The same method is adopted in such modern structuresas are intended to possess great strength and durability, and, insome cases, the stones are even “dove-tailed together, as in the light-houses at Eddystone and Bell Rock. But many of our modern stone walls, for the sake ot cheapness, have only one face of the stones squared, the inner half of the wall being completed with brick; so that they can, 16

Page 127

122 ARCHITECTURE. in reality, be considered only as brick walls faced with stone Such walls are said to be liable to become convex outwardly, from the difference in the shrinking of the cement. ubble walls are made of rough, irregular stones, laid in mortar. ‘The stones should be broken, if possible, so as to produce horizontal surfaces The coffer walls of the ancient Romans were made by enclosing successive portions of the intended wall in a box, and filling it with stones, sand, and mortar, promiscuously. This kind of structure must have been extremely insecure. 'The Pantheon, and various other Roman buildings, are surrounded with a double brick wall, having its vacancy filled up with loose bricks and cement. The whole has gradually consolidated into a mass of great firmness. ' The reticulated walls of the Romans, having bricks with oblique surfaces, would, at the present day, be thought highly unphilosophical. Indeed, they could not long have stood, had it not been for the great strength of their cement. Modern brick walls are laid with’ great precision, and depend for firmness more upon their position than upon the strength of their cement. The -bricks being laid in horizontal courses, and continually overlaying each other, or breaking joints, the whole mass is strongly inter- ` woven, and bound together. Wooden walls, composed of timbers covered with boards, are a common, but more perishable kind. They require to be constantly covered with a coating of a foreign substance, as paint or plaster, to preserve them from spontaneous decomposition. In some parts of France, and elsewhere, a kind of wallis made of earth, rendered compact by ramming it in moulds or cases. This method ‘is called building in pisé, and is much more durable than the nature of the material would lead us to suppose. Walls of all kinds are greatly strengthened by angles and curves, also by projections, such as pilasters, chimneys and buttresses. ‘These projections serve to increase the breadth of the foundation, and are always to be made use of in large buildings, and in walls of considerable length.

Page 128

AMERICAN HOUSE-CARPENTER. 123 234.—- The LıNTEL, or beam, extends in a right line over a vacant space, from one column or wall to another. 'The strength of the lintel will be greater in proportion as its transverse vertica] diameter exceeds the horizontal, the strength being always as the square of the depth. The floor is the lateral continuation or connection of beams by means of a covering of boards. 235.—The Arc is a transverse member of a building, an- swering the same purpose as the lintel, but vastly exceeding it in strength. The arch, unlike the lintel, may consist of any num- ber of constituent pieces, without impairing its strength. It is, however, necessary that all the pieces should possess a uniform shape,—the shape of a portion of a wedge,—and that the joints, formed by the contact of their surfaces, should point towards a common centre. In this case, no one portion of the arch can be displaced or forced inward; and the arch cannot be broken by any force which is not sufficient to crush the materials of which itis made. In arches made of common bricks, the sides of which are parallel, any one of the bricks might be forced inward, were it not for the adhesion of the cement. Any two of the bricks, however, by the disposition of their mortar, cannot colJective-~ ly be forced inward. An arch of the proper form, when com- plete, is rendered stronger, instead of weaker, by the pressure of a considerable weight, provided this pressure be uniform. While building, however, it requires to be supported by a centring of the shape of its internal surface, until itis complete. ‘The upper stone of an arch is called the key-stone, but is not more essential than any other. In regard to the shape of the arch, its most simple form is that of the semi-circle. It is, however, very fre- quently a smaller arc of a circle, and, still more frequently, a por- tion of an ellipse. The simplest theory of an arch supporting itself only, is that of Dr. Hooke. The arch, when it’ has only its own ‘weight to bear, may be considered as the inversion of a chain, suspended at each end. The chain hangs in such a form, that the weight of each link or portion is held in equilibrium by

Page 129

124 ARCHITECTURE. the result of two forces acting at its extremities ; and these forces, or tensions, are produced, the one by the weight of the portion of the chain below the link, the other by the same weight increased by that of the link itself, both of them acting originally in a ver- tical direction. Now, supposing the chain inverted, so as to con- stitute an arch of the same form and weight, the relative situa- tions of the forces will be the same, only they will act in contrary directions, so that they are compounded in a similar manner, and balance each other on the same conditions. The arch thus formed is denominated a catenary arch. In common cases, it differs but little from a circular arch of the extent of about one third of a whole circle, and rising from the abut- ments with an obliquity of about 30 degrees from a perpendicu- lar. But though the catenary arch is the best form for support- ing its own weight, and also all additional weight which presses in a vertical direction, it is not the best form to resist latera! pressure, or pressure like that of fluids, acting equally in all direc- tions. 'Thus the arches of bridges and similar structures, when covered with loose stones and earth, are pressed sideways, as well as vertically, in the same manner as if they supported a weight of fluid. In this case, it is necessary that the arch should arise more perpendicularly from the abutment, and that its general figure should be that of the longitudinal segment of an ellipse. In small arches, in common buildings, where the disturbing force is not great, it is of little consequence what is the shape of the curve. The outlines may even be perfectly straight, as in the tier of bricks which we frequently see over a window. This is, strictly speaking, a real arch, provided the surfaces of the bricks tend towards a common centre. It is the weakest kind of arch, and a part of itis necessarily superfluous, since no greater portion can act in supporting a weight above it, than can be included be- tween two curved or arched lines. Besides the arches already mentioned, various others are in use The acute or lancet arch, much used in Gothic architecture, is

Page 130

AMERICAN HOUSE-CARPENTER. 198 “described usually from two centres outside the arch. It is ə strong arch for supporting vertical pressure. ‘The rampant arch is one in which the two ends spring from unequal heights. The horse-shoe or Moorisharch is described from one or more centres placed above the base line. In this arch, the lower parts are in danger of being forced inward. The ogee arch is concavo-con- vex, and therefore fit only for ornament. In describing arches, the upper surface is called the extrados, and the inner, the in- trados. 'The springing lines are those where the intrados meets the abutments, or supporting walls. The span is the distance from one springing line to the other. The wedge-shaped stones, which form an arch, are sometimes called vowssoirs, the upper- most being the key-stone. The part of a pier from which an arch springs is called the impost, and the curve formed by the upper side of the voussoirs, the archivolt. Itis necessary that the walls, abutments and piers,on which arches are supported, should be so firm as to resist the lateral thrust, as well as vertical pressure, of the arch. It will at once be seen, that the lateral or sideway pressure of an arch is very considerable, when we recol- lect that every stone, or portion of the arch, is a wedge, a part of whose force acts to separate the abutments. For want of atten- tion tothis circumstance, important mistakes have been committed, the strength of buildings materially impaired, and their ruin ac- celerated. In some cases,.the want of lateral firmness in the walls is compensated by a bar of iron stretched across the span ot the arch, and connecting the abutments, like the tie-beam of a roof. ‘This is the case in the cathedral of Milan and some other Gothic buildings. In an arcade, or continuation of arches, it is only necessary that the outer supports of the terminal arches should bestrong enough ‘to resist horizontal pressure. In the intermediate arches, the lat- eral force of each arch is counteracted by the opposing lateral force of the one contiguous to it. In bridges, however, where individual arches are liable to be destroyed by accident, it is desi

Page 131

126 ARCHITECTURE. rable that each of the piers should possess sufficient horizontal strength to resist the lateral pressure of the adjoining arches. 236.—The VauttT is the lateral continuation of an arch, serving to cover an area or passage, and bearing the same relation to the arch that the wall does to the column. A simple vault is con- structed on the principles of the arch, and distributes its pressure equally along the walls or abutments. A complex or groined vault is made by two vaults intersecting each other, in which case the pressure is thrown upon springing points, and is greatly increased at those points. The groined vault is common in Gothie architecture. 237.—The Dome, sometimes called cupola, is a concave cover- ing to a building, or part of it, and may be either a segment of a sphere, of a spheroid, or of any similar figure. When built of stone, it is a very strong kind of structure, even more so than the arch, since the tendency of each part to fall is counteracted, not only by those above and below it, but also by those on each side. It is only necessary that the constituent pieces should have a common form, and that this form should be somewhat like the frustum of a pyramid, so that, when placed in its situation, its four angles may point toward the centre, or axis, of the dome. During the erection of a dome, it is not necessary that it should be supported by a centring, until complete, as is done in the arch. Each circle of stones, when laid, is gapable of supporting itself without aid from those above it. It follows that the dome may be left open at top, without a key-stone, and yet be perfectly secure in this respect, being the reverse of the arch. The dome of the Pantheon, at Rome, has been always open at top, and yet has stood unimpaired for nearly 2000 years. The upper circle of stones, though apparently the weakest, is nevertheless often made to support the additional weight of a lantern or tower above it. In several of the largest cathedrals, there are two domes, one within. the other, which contribute their joint support to the lan- tern, which rests upon the top. In these buildings, the dome

Page 132

AMERICAN HOUSE-CARPENTER. 127 rests upon a circular wall, -which is supported, in its turn, by arches upon massive pillars or piers. This construction is called building upon pendentives, and gives open space and room for passage beneath the dome. The remarks which have been made in regard to the abutments of the arch, apply equally to the walls immediately supporting a dome. They must be of sufficient thickness and solidity to resist the lateral pressure of the dome, which is very great. The walls of the Roman Pantheon are of great depth and solidity. In order that a dome in itself should be perfectly secure, its lower parts must not be too nearly vertical, since, in this case, they partake of the nature of perpendicular walls, and are acted upon by the spreading force of the parts above them. The dome of St. Paul’s church, in London, and some others of similar construction, are bound with chains or hoops of iron, to prevent them from spreading at bottom. Domes which are made of wood depend, in part, for their strength, on their in- ternal carpentry. ‘The Halle du Bled, in Paris, had originally a wooden dome more than 200 feet in diameter, and only one foot in thickness. This has since been replaced by a dome of iron (See Aré. 389.) 238.—The Roor is the most common and cheap method of covering buildings, to protect them from rain and other effects of the weather. It is sometimes flat, but more frequently oblique, in its shape. The flat or platform-roof is the least advantageous for shedding rain, and is seldom used in northern countries. The pent roof, consisting of two oblique sides meeting at top, is the most common form. These roofs are made steepest in cold cli- mates, where they are liable to be loaded with snow. Where the four sides of the roof are all oblique, it is denominated a hipped roof, and where there are two portions to the roof, of different ob- ` liquity, it is a curb, or mansard roof. In modern-times, roofs are made almost exclusively of wood, though frequently covered with incombustible materials. 'The internal structure or carpen- try of roofs is a subject of considerable mechanical contrivance, L

Page 133

128 ARCHITECTURE. The roof is supported by rafters, which abut on the walls on each side, like the extremities of anarch. If no other timbers existed, except the rafters, they would exert a strong lateral pres- sure on the walls, tending to separate and overthrow them. To counteract this lateral force, a fie-beam, as it is called, extends across, receiving the ends of the rafters, and protecting the wall from their horizontal thrust. To prevent the tie-beam from sagging, or bending downward with its own weight, a king- post is erected from this beam, to the upper angle of the rafters, serving to connect the whole, and to suspend the weight of the beam. This is called trussing. Queen-posts are sometimes added, parallel to the king-post, in large roofs ; also various other connecting timbers. In Gothic buildings, where the vaults do not admit of the use of a tie-beam, the rafters are prevented from spreading, as in an arch, by the strength of the buttresses. In comparing the lateral pressure of a high roof with that of a low one, the length of the tie-beam being the same, it will be seen that a high roof, from its containing most materials, may produce the greatest pressure, as far as weight is concerned. On the other hand, if the weight of both be equal, then the low roof will exert the greater pressure ; and this will increase in propor- tion to the distance of the point at which perpendiculars, drawn from the end of each rafter, would meet. In roofs, as well as in wooden domes and bridges, the materials are subjected to an in- ternal strain, to resist which, the cohesive strength of the material is relied on. On this account, beams should, when possible, be of one piece. Where this cannot be effected, two or more beams are connected together by splicing. Spliced beams are never so strong as whole ones, yet they may be made toapproach the same strength, by affixing lateral pieces, or by making the ends overlay each other, and connecting them with bolts and straps of iron. ‘The tendency to separate is also resisted, by letting the two pieces into each ether by the process called scarfing. Mortices, in-

Page 134

AMERICAN HOUSE-CARPENTER. 129 tended to russ or suspend one piece by another, should be formed upon similar principles. Roofs in the United States, after being boarded, receive a se- condary covering of shingles. When intended to be incombustible, they are covered with slates or earthern tiles, or with sheets of lead, copper or tinned iron. Slates are preferable to tiles, being lighter, and absorbing less moisture. Metallic sheets are chiefly used for flat roofs, wooden domes, and curved and angular surfaces, which require a flexible material to cover them, or have not a sufficient pitch to shed the rain from slates or shingles. Various artificial compositions are occasionally used to cover roofs, the most com- mon of which are mixtures of tar with lime, and sometimes with sand and gravel.”—Eincy, Am. (See Art. 354.) 17

Page 135

SECTION IIIL—MOULDINGS, CORNICES, &«. MOULDINGS. 239.—A moulding is so called, because of its being of the same determinate shape ¿loug its whole length, as though the whole of it had been cast in the same mould or form. The regular mouldings, as found in remains of ancient architecture, are eight in number ; and are known by the following names: Annulet, band, cincture, fillet, listel or square. Fig. 140, as Astragal or bead. Fig. 141. E ars o1 ture. Fig. 142. Be Scotia, trochilus or mouth, Fig. 143. Ovolo, quarter-round or echinus. Fig. 144,

Page 136

AMERICAN HOUSE-CARPENTER. 131 | Cavetto, cove or hollow. Fig. 145, Cymatium, or cyma-recta. Fig. 146. Ogee. Inverted cymatium, or cymoa-reversa Fig. 147, Some of the terms are derived thus: fillet, from the French word fil; thread. Astragal, from astragalos, a bone of the heel —or the curvature of the heel. Bead, because this moulding, when properly carved, resembles a string of beads. Torus, or tore, the Greek for rope, which it resembles, when on the base of acolumn. Scotia, from shotia, darkness, because of the strong shadow which its depth produces, and which is increased by the projection of the torus above it. Ovolo, from ovum, an egg, which this member resembles, when carved, as in the Ionic capi- tal. Cavetto, from cavus, hollow. Cymatium, from kumaton a wave. 240.—Neither of these mouldings is peculiar to any one of the orders of architecture, but each one is common to all; and al- though each has its appropriate use, yet it is by no means con- fined to any certain position in an assemblage of mouldings The use of the fillet is to bind the parts, as also that of the astra- gal and torus, which resemble ropes. The ovolo and cyma-re- versa are strong at their upper extremities, and are therefore used to support projecting parts above them. The cyma-recta and cavetto, being weak at their upper extremities, are not used as supporters, but are placed uppermost to cover and shelter the other parts. The scotia is introduced in the base of a column, tu

Page 137

132 MOULDINGS, CORNICES, &C. separate the upper and lower torus, and to produce a pleasing variety aud relief. The form of the bead, and that of the toras, is the ane the reasons for giving distinct names to them are, that the torus, in every order, is always considerably larger than the bead, and is placed among the base mouldings, whereas the bead is never placed there, but on the capital or entablature; the torus, also, is seldom carved, whereas the bead is; and while the torus among the Greeks is frequently elliptical in its form, the bead retains its circular shape. While the scotia is the reverse of the torus, the cavetto is the reverse of the ovolo, and the eyma- recta and cyma-reversa are combinations of the ovolo and cavetto. 241.—'The curves of mouldings, in Roman architecture, were most generally composed of parts of circles; while those of the Greeks were almost always elliptical, or of some one of the conic sections, but rarely circular, except in the case of the bead, which was always, among both Greeks and Romans, of the form of a semi-circle. Sections of the cone afford a greater variety ot forms than those of the sphere; and perhaps this is one reason why the Grecian architecture so much excels the Roman. 'The quick turnings of the ovolo and cyma-reversa, in particular, when exposed to a bright sun, cause those narrow, well-defined streaks of light, which give life and splendour to the whole. 242.—A profile is an assemblage of essential parts and mould- ings. hat profile produces the happiest effect which is com- posed of but few members, varied in form and size, and arranged so that the plane and the curved surfaces succeed each other al- ternately. * 243.— To describe the Grecian torus and scotia. Join the extremities, a and b, (Fig. 1483) and from f, the given projection of the moulding, draw f o, at right angles to the fillets; from b, draw b A, at right angles to a b; bisect a bin c; join f «d c, and upon c, with the radius, ¢ f, describe the arc, f h, cutting b A in A; through c, draw d e, parallel with the fillets ; make dc and ¢ e, each equal to bh ; then d e and a b will be conjugate diame-

Page 138

AMERICAN HOUSE-CARPENTER. 133 ters of the required ellipse. To describe the curve by intersec- tion of lines, proceed as directed at Aré. 118 and note; by a trammel, see Art. 116; and to find the foci, in order to describe it with a string, see Art. 115. Fig. 150. 244.--Fig. 149 to 156 exhibit various modifications of the Grecian ovolo, sometimes called echinus. Fig. 149 to 153 are

Page 139

Toa MOULDINGS, CORNICES, &C. Fig, 151. Fig. 152. Fig. 156. elliptical, a b and b ¢ being given tangents to the curve; parallel to which, the semi-conjugate diameters, a d and d c, are drawn. In Fig. 149 and 150, the lines, a d and d c, are semi-axes, the tangents, a b and b c, being at right angles to each other. To draw the curve, see Aré. 118. In Fig. 153, the curve is para- bolical, and is drawn according to Art. 127. In Fig. 155 and 156, the curve is hyperbolical, being described according to Art. 128. The length of the transverse axis, a b, being taken at pleasure in order to flatten the curve, a b should be made short in propor- tion to ac.

Page 140

AMERICAN HOUSE-CARPENTER. 125 KA Fig. 157. j 245.— To describe the Grecian cavetto, (Fig. 157 and 158,) having the height and projection given, see Art. 118. Fig. 159, Fig. 160. 246.— To describe the Grecian cyma-recta. When the pro- jection is more than the height, as at Fig. 159, make a b equal to the height, and divide g b c d into 4 equal parallelograms; then proceed as directed in note to Art. 118. When the projec- tion is less than the height, draw d a, (Fig. 160,) at right angles to a b; complete the rectangle, a b c d ; divide this into 4 equal rectangles, and proceed according to Art. 118. Fig. 161. d Fig. 162. 247.—To describe the Grecian cyma-reversa. When the

Page 141

136 MOULDINGS, CORNICES, &C. projection is more than the height, as at Fig. 161, proceed as di rected for the last figure; the curve being the same as that, the position only being changed. When the projection is less than the height, draw a d, (Fig. 162,) at right angles to the fillet , make a d equal to the projection of the moulding : then proceed as directed for Fig. 159. i 248.—Roman mouldings are composed of parts of circles, and have, therefore, less beauty of form than the Grecian. The bead and torus are of the form of the semi-circle, and the scotia, also, in some instances; but the latter is often composed of two quad- rants, having different radii, as at Fig. 163 and 164, which re- semble the elliptical curve. The ovolo and cavetto are generally a quadrant, but often less. When they are less, as at Fig. 167, the centre is found thus: join the extremities, a and b, and bisect abin c; from c, andat right angles to a b, draw c d, cutting a level line drawn from a in d ; then d will be the centre. This moulding projects less than its height. When the projection is more than the height, as at Fig. 169, extend the line from c until Fig. 163. Fig. 164, Fig. 165. Fig. 166.

Page 142

AMERICAN HOUSE-CARPENTER, 137 a [eee | | Fig. 167. : ig. 168. Fig. 169. Fig. 170. Fig. 17L Fig. 172. . Fig. 173, Fig, 174, 18

Page 143

138 MOULDINGS, CORNICES, &C, Fig 175. Fig. 176, b Fig. 177. it cuts a perpendicular drawn from a, as at d ; and that will be the centre of the curve. In a similar manner, the centres are found for the mouldings at Fig. 164, 168, 170, 173, 174, 175, and 176. The centres for the curves at Fig. 177 and 178, are found thus: bisect the line, œ b, at c; upon a, ¢ and b, suc- cessively, with æ c or c b for radius, describe arcs intersecting at d and d, then those intersections will be the centres. 249.—/ig. 179 to 186 represent mouldings of modern inven- tion. They have been quite extensively and successfully used in inside finishing. Zig. 179 is appropriate for a bed-moulding under a low projecting shelf, and is frequently used under man- tle-shelves. The tangent, ¿ A, is found thus: bisect the line, ad, atc,anddcatd; from d, draw d e, at right angles toed; from b, draw bf, parallel to ed; upon b, with b d for radius, describe thearc, d f; divide this arc into 7 equal parts, and set one of the parts from s, the limit of the projection, too; make o A equal to ve; from A, through c, draw the tangent, Ai; divide bd, ie, ce and č q, each into a like number of equal parts, and draw the in-

Page 144

Page 145

MOULDINGS, CORNICES, &C. a pes Fig 184 Fig. 185. j Fig. 186, tersecting lines as directed at Art. 89. If a bolder form is desired, draw the tangent, ¢ A, nearer horizontal, and describe an elliptic curve as shown in /%g. 148 and 181. Fig. 180 is much used on base, or skirting of rooms, and in deep panelling. The curve is found in the same manner as that of Fig. 179. In this ease, however, where the moulding has so little projection

Page 146

AMERICAN HOUSE-CARPENTER. 141 in comparison with its height, the point, e, being found as in the last figure, h s may be made equal to s e, instead of o e as in tne last figure. Fig. 181 is appropriate for a crown moulding of a cornice. In this figure the height and projection are given; the direction of the diameter, a 6, drawn through the middle cf the diagonal, e f,is taken at pleasure; and dc is parallel toa e. To find the length of d c, draw b A, at right angles to a b; upon o, with o f for radius, describe the arc, f h, cutting b h in h ; then make o cand o d, each equal tobh.* To draw the curve, see note to Art. 118. Fig. 182 to 186 are peculiarly distinct from ancient mouldings, being composed principally of straight lines ; the few curves they possess are quite short and quick Fig. 187. o Fig. 138. 250.—Fig. 187 and 188 are designs for ante caps. The .*The manner of ascertaining the length of the conjugate diameter, d c, in this figure, and also in Fig, 148, 198 and 199 is new, and is important in this application. It ia founded upon well-known mathematical principles, viz: All the parallelograms that may be circumscribed about an ellipsis are equal to one another, and conzequently any one is equal to the rectangle of the two axes. And again: the sum of the squares of every pair of conjugate diameters is equal to the sum of the squares of the two axes.

Page 147

v 142 AMERICAN HOUSE-CARPENTER. diameter of the antæ is divided into 20 equal parts, and the height and projection of the members. are regulated in accord- ance with those parts, as denoted under Æ and P, height and projection. The projection is measured from the middle of the ante. These will be found appropriate for porticos, door- ways, mantel-pieces, door and window trimmings, &c. The height of the ante: for. mantel-pieces, should be from 5 to 6 diameters, having an entablature of from 2 to 24 diameters. This is a good proportion, it being similar to the Dorie order. But for a portico these proportions are much too heavy; an ante, 15 diameters high, and an entablature of 3 diameters, will have a better appearance. CORNICES. 251.—/ig. 189 to 197 are designs for cave cornices, and Fig. 198 and 199 are for stucco cornices for the inside finish of rooms. In some of these the projection of the uppermost member from the facia, is divided into twenty equal parts, Fig 189.

Page 148

MOULDINGS, CORNICES, &C. 143 and the various members are proportioned according to those parts, as figured under Z and P. isip] JLA LLY OR sa ele C DADANA AA AEA ji Hi L SL

Page 149

AMERICAN HOUSE-CARPENTER. 144 Fig. 192.

Page 150

MOULDINGS, CORNICES, &0. 145 Fig. 194. 8/16: apeceees seecemecorentsecnng, , se., eed A Go ea - - oy Fig. 195. 19

Page 151

146 AMERICAN HOUSE-CARPENTER. Fig. 196. Fig, 197,

Page 152

Page 153

148 AMERICAN HOUSE-CARPENTER. 252.— To proportion an eave cornice in accordance with the height of the building. Draw the line, a c, (Fig. 200,) and make b c and b a, each equal to 36 inches; from b, draw 6 d, at right angles to a ¢, and equal in length to 3 of a c; bisect b d in e, and from a, through e, draw a f; upon a, with a e for radius, describe the arc, c f, and upon e, with e f for radius, describe the arc, f d; divide the curve, df c, into 7 equal parts, as at 10, 20, 30, &c., and from these points of division, draw lines to b ¢, pa- rallel tod b ; then the distance, b 1, is the projection of a cornice for a building 10 feet high ; b 2, the projection at 20 feet high; b 3, the projection at 30 feet, &c. If the projection of a cornice for a building 34 feet high, is required, divide the arc between 30 and 40 into 10 equal parts, and from the fourth point from 30, draw a line to the base, b c, parallel with b d ; then the distance of the point, at which that line cuts the base, from b, will be the projec- tion required. So proceed for a cornice of any height within 70 feet. The above is based on the supposition that 36 inches is the proper projection for a cornice 70 feet high. This, for general purposes, will be found correct; still, the length of the line, b c, may be varied to suit the judgment of those who think differ- ently. Having obtained the projection of a cornice, divide it into 20 equal parts, and apportion the several members according to its destination—as is shown at Fig. 195, 196, and 197.

Page 154

MOULDINGS, CORNICES, &C. 149 Fig. 201. 253.-— To proportion a cornice according to a smaller given eve. Let the cornice at Fig. 201 be the given one. Upon any point in the lowest line of the lowest member, as at a, with the height of the required cornice for radius, describe an intersecting arc across the uppermost line, as at b ; join a and b; then b 1 will oe the perpendicular height of the upper fillet for the proposed cor- nice, 1 2 the height of the crown moulding—and so of all the members requiring to be enlarged to the sizes indicated on this line. For the projection of the proposed cornice, draw a d, at right angles to a b, and c d, at right angles to bc; parallel with cd, draw lines from each projection of the given cornice to the line, ad; then ed will be the required projection for the proposed cornice, and the perpendicular lines falling upon e d will indicate the proper projection for the members. 254.— To proportion a cornice according toa larger given one. Let A, (Fig. 202,) be the given cornice. Extend a o to b, and draw c d, at right angles to a b; extend the horizontal lines of the cornice, A, until they touch o d; place the height of the proposed cornice from o to e, and join f and e ; upon o, with the projection of the given cornice, o a, for radius, describe the quad- rant, ad; from d, draw d b, parallel to fe; upon o, with o b for radius, describe the quadrant, b c ; then o ¢ will be the proper pro- jection for the proposed cornice. Join a and c ; draw lines from the

Page 155

150 AMERICAN HOUSE-CARPENTER. Fig. 202, d projection of the different members of the given cornice to ao, parallel to o d ; from these divisions on the line, a o, draw lines to the line, o c, parallel to a c ; from the divisions on the line, o f, draw lines to the line, o e, parallel to the line, fe; then the di- visions on the lines, o e and o c, will indicate the proper height and projection for the different members of the proposed cornice. In this process, we have assumed the height, o e, of the proposed cornice to be given; but if the projection, o c, alone be given, we can obtain the same result bya different process. Thus: upon o. with o c for radius, describe the quadrant, c b ; upon o, with o a for radius, describe the quadrant, ad ; join d and 6 ; from f, draw f e, parallel to d b ; then e will be the proper height for the pro- posed cornice, and the height and projection of the different mem- bers can be obtained by the above directions. By this problem, a cornice can be proportioned according to a smaller given one as well as to alarger ; but the method described in the previous article is much more simple for that purpose. 255.— To find the angle-bracket for a cornice. Let A, (Fig. 203,) be the wall of the building, and B the given bracket, which, for the present purpose, is turned down horizontally. The angle- bracket, C,is obtained thus: through the extremity, a, and paral-

Page 156

Fig. 204. lel with the wall, fd, draw the line,ab; make e c equal a f, and through c, draw c b, parallel with e d ; join d and b, and from the several angular points in B, draw ordinates to cut d b in 1, 2 and 3; at those points erect lines perpendicular to d b; from A, draw h g, parallel tof a; take the ordinates, 1 0, 2 o, &c., at B, and transfér them to C, and the angle-bracket, C, will be defined. In the same manner, the angle-bracket for an internal cornice, or theangle-rib of a coved ceiling, or of groins, as at Fig. 204, can be found. 256.—A level crown moulding being given, to find theraking moulding and a level return at the top. Let A, (Fig. 205,) be the given moulding, and A b the rake of the roof. Divide the curve of the given moulding into any number of parts, equal or unequal, as at 1, 2, and 3; from these points, draw horizonta: -lines to a perpendicular erected from c; at any convenient place on the rake, as at B, draw ac, at right angles to A b ; also, from b, draw the horizontal line, b a; place the thickness, d a, of the moulding at A, from b to a, and from a, draw the perpendicular "line, a e ; from the points, 1, 2, 3, at A, draw lines to C, parallel to Ab; make a1, a 2 and a 3, at B and at C, equal to a 1, &c., at A; through the points, 1, 2 and 3, at B, trace the curve—this will be the proper form for the raking moulding. From 1, 2 and

Page 157

R 152 AMERICAN HOUSE-CARPENTER. Fig 205. 3, at C, drov perpendiculars to the corresponding ordinates from 1,2 and 3, at A; through the poirts of intersection, trace the curve—this will be the proper form for the return at the top.

Page 158

» SECTION IV.—FRAMING. 257.—This subject is, to the carpenter, of the highest impor- tance; and deserves more attention and a larger place in a volume of this kind, than is generally allotted to it. Something, indeed, has: been said upon the geometrical principles, by which the seve- ral lines for the Joints and the lengths of timber, may be ascer- tained; yet, besides this, there is much to be learned. For how- ever precise or workmanlike the joints may be made, what will it avail, should the system of framing, from an erroneous position of its timbers, &c., change its form, or become incapable of sus- taining even its own weight? Hence the necessity for a know- ledge of the laws of pressure and the strength of timber. These being once understood, we can with confidence determine the best position and dimensions for the several timbers which compose a floor or a roof, a partition or a bridge. As systems of framing are more or less exposed to heavy weights and strains, and, in case of failure, cause not only a loss of labour and material, but frequently that of life itself, itis very important that the materials ' employed be of the proper quantity and quality to serve their des tination. And, on the otherhand, any superfluous material is not only useless, but a positive injury, it being an unnecessary load upon the points of support. It is necessary, therefore, to know 20

Page 159

154 AMERICAN HOUSE-CARPENTER. the least quantity of timber that will suffice for strength. The greatest fault in framing is that of using an excess of material. Economy, at least, would seem to require that this evil be abated. Before proceeding to conside! the principles upon which a sys- tem of framing should be constructed, let us attend to a few of the elementary laws in Mechanics, which will be found to be of great value in determining those principles. 258.—Laws or Pressure. (1.) A heavy body always exerts a pressure equal to its own weight in a vertical direction. Example: Suppose an iron ball, weighing 100 lbs., be supported upon the top of a perpendicular post, (Fig. 220;) then the pressure exerted upon that post will be equal to the weight of the vall; viz., 100 lbs. (2.) But if two inclined posts, (Fig. 206,} be substituted for the perpendicular support, the united pressures upon these posts will be more than equal to the weight, and will be in proportion to their position. 'The farther apart their feet are spread the greater will be the pressure, an@ vice versa. Hence tremendous strains may be exerted by a comparatively small weight. And it follows, therefore, that a piece of timber intend- ed for a strut or post, should be so placed that its axis may coin- cide, as near as possible, with the direction of the pressure. 'The direction of the pressure of the weight, IV, (Fig. 206,) is in the vertical line, b d; and the weight, W, would fall in that line, if the two posts were removed, hence the best position for a support Fig. 206,

Page 160

FRAMING. 155 for the weight would be in that line. But, as it rarely occurs in systems of framing that weights can be supported by any single resistance, they requiring generally two or more sup- ports, (as in the case of a roof supported by its rafters,) it be comes important, therefore, to know the exact amount of pres- sure any certain weight is capable of exerting upon oblique supports. Now it has been ascertained that the three lines of a triangle, drawn parallel with the direction of three concur- ring forces in equilibrium, are in proportion respectively to these forces. For example, in Fig. 206, we have a represen- tation of three forces concurring in a point, which forces are in equilibrium and at rest; thus, the weight, W, is one force, and the resistance exerted by the two pieces of timber are the other two forces. The direction in which the first force acts is vertical—downwards ; the direction of the two other forces is in the axis of each piece of timber respectively. These three forces all tend towards the point, b. Draw the axes, æ b and b c, of the two supports; make b d vertical, and from d draw d e and d J parallel with the axes, b c and ò a, respectively. Then the triangle, 6 d e, has its lines parallel respectively with the direction of the three forces; thus, 6 d is in the direction of the weight, W, d e parallel with the axis of the timber 0 c, and e b is in the direction of the timber æ b. In accordance with the principle above stated, the lengths of the sides of the triangle, 5 d e, are in proportion respectively to the three forces aforesaid ; thus— As the length of the line, b d, Is to the number of pounds in the weight, W, So is the length of the line, b e, To the number of pounds’ pressure resisted by the timber, ab. Again— As the length of the line, 6 d, Is to the number of pounds in the weight, W,

Page 161

156 AMERICAN HOUSE-CARPENTER. So is the length of the line, d e, To the number of pounds’ pressure resisted by the timber, be. And again— As the length of the line, b e, Is to the pounds’ pressure resisted by a b, So is the length of the line, d e, To the pounds’ pressure resisted by 0 c. These proportions are more briefly stated thus— Ist. bd: VW::be: P, P being used as a symbol to represent the number of younds’ pressure resisted by the timber, a 0. 2nd. bad BW ide: Q; Q representing the number of pounds’ pressure resisted by the timber, 0 c. 3d. be: Pat de: O: 259.—This relation between lines and pressures is important, and is of extensive application in ascertaining the pressures induced by known weights throughout any system of framing. The parallelogram, b e d f, is called the Paralelogram of Forces ; the two lines, b e and b f, being called the compo- ments, and the line 6 d the resultant. Where it is required to find the components from a given resultant, (Fig. 206,) it is not needed to draw the fourth line, d f, for the triangle, 6 d e, gives the desired result. But when the resultant is to be ascertained from given components, (Fig. 212,) it is more con- venient to draw the fourth line. 260.—The Resolution of Forces is the finding of two or more forces, which, acting in different directions, shall exactly balance the pressure of any given single force. To make a practical application of this, let it be required to ascertain the oblique pressure in Fig. 206. In this Fig. the line 6 g measures half an inch, (0°5 inch,) and the line ġ e three. tenths of an inch, (0°3 inch.) Now if the weight, IW, be sup-

Page 162

FRAMING. 157 posed to be 1200 pounds, then the first stated proportion above, | bd: W::be:P, becomes -e 0°5 $ 1200 :: 0°33 P. And since the product of the means divided by one of the extremes gives the other extreme, this proportion may be put in the form of an equation, thus— 1200 x 0:3 0:5 Performing the arithmetical operation here indicated, that is, multiplying together the two quantities above the line, and dividing the product by the quantity under the line, the quo- tient will be equal to the quantity represented by P, viz., the pressure resisted by the timber, æ b. Thus— 1200 0'3 =F. 0°5)360-0 2 = er. The strain upon the timber, a b, is, therefore, equal to 720 pounds; and the strain upon the other timber, 6 c , is also 720 pounds; for in this case, the two timbers being inclined equally from the vertical, the line e d is therefore equal to the line 6 e. j a Fig. 207.

Page 163

158 AMERICAN HOUSE-CARPENTER. 961.—In Fig. 207, the two supports are inclined at different angles, and the pressures are proportionately unequal. The supports are also unequal in length. The length of the sup- ports does not alter the amount of pressure from the concen- trated load supported; but generally long timbers are not so capable of resistance as shorter ones. They yield more readily laterally, as they are not so stiff, and shorten more, as the com- pression is in proportion to the length. To ascertain the pres- sures in Fig. 207, let the weight suspended from b d be equal to two and three-quarter tons, (2°75 tons.) The line 6 d mea- sures five and a half tenths of an inch, (0°55 inch,) and the line b e half an inch, (0°5 inch.) Therefore, the proportion bd : W:: be: P, becomes 055 : %75 3; 05 : P, Zip x 0T P ee = a 2°75 0°5 0:55)1:375(2°5 = P. 110 275 275 The strain upon the timber, 0 e, is, therefore, equal to two and a half tons. Again, the line e d measures four-tenths of an inch, (0:4 inch ;) therefore, the proportion bd: W :: ed: Q, becomes 0°55 : 2°75 :: 0-4: Q, 2°75 x O-4 and ~ O55 T 2°75 0:4 0°55)1°100(2 = Q. 110

Page 164

FRAMING. 15S The strain upon the timber, b f, is, therefore, equal to two tons. 262.—Thus it is seen that the united pressures exerted by a weight upon two inclined supports always exceed the weight. In the last case 23 tons exerts a pressure of 24 and two tons, equal together to 44 tons;‘and in the former case, 1200 pounds exerts a pressure of twice 720 pounds, equal to 1440 pounds. The smaller the angle of inclination to the horizon- tal, the greater will be the pressure upon the supports. So, in the frame of a roof, the strain upon the rafters decreases gra- dually with the increase of the angle of inclination to the horizon, the length of the rafter remaining the same. 263.—This is true in comparing systems of framing with each other; but in a system where the concentrated weight to be supported is not in the middle, (see Fig. 207,) and, in consequence, the supports are not inclined equally, the strain will be greatest upon the support that has the greatest inclina- ston to the horizon. 264.—In ordinary cases, in roofs for example, the load is uot concentrated but is that of the framing itself. Here the amount of the load will be in proportion to the length of the rafter, and the rafter increases in length with the increase of the angle of inclination, the span remaining the same. So it is seen that in enlarging the angle of inclination to the horizon in order to lessen the oblique thrust, the load is increased in consequence of the elongation of the rafter, thus increasing the oblique thrust. Hence there is a limit to the angle of inclina- tion. A rafter will have the least oblique thrust when its angle of inclination to the horizon is 35° 16’ nearly. This angle is attained very nearly when the rafter rises 84 inches -per foot; or, when the height, B C, (Fig. 216,) is to the base, A 0, as 88 is to 12, or as 0°7071 is to 1°0. . 265.—Correct ideas of the comparative pressures exerted upon timbers, according to their position, will be readily

Page 165

160 AMERICAN HOUSE-CARPENTER. formed by drawing various designs of framing, and estimating the several strains in accordance with the parallelogram ot forces, always drawing the triangle, b d e, so that the three lines shall be parallel with the three forces, or pressures, re- spectively. The length of the lines forming this triangle is unimportant, but it will be found more convenient if the line drawn parallel with the known force is made to contain as many inches as the known force contains pounds, or as many tenths of an inch as pounds, or as many inches as tons, or tenths of an inch as tons: or, in general, as many divisions of any convenient scale as there are units of weight or pressure in the known force. If drawn in this manner, then the num- ber of divisions of the same scale found in the other two lines of the triangle will equal the units of pressure or weight of the other two forces respectively, and the pressures sought will be ascertained simply by applying the scale to the lines of the triangle. For example, in Fig. 207, the vertical line, b d, of the tri- angle, measures fifty-five hundredths of an inch, (0°55 inch ;) the line, 4 e, fifty-hundredths, (0°50 inch ;) and the line, e d, forty, (0°40 inch.) Now, if it be supposed that the vertical pres- sure, or the weight suspended below b d, is equal to 55 pounds, then the pressure on b e will equal 50 pounds, and that on e d will equal 40 pounds; for, by the proportion above stated, ba V: bes 2, bbs Op 22 00+ 303 and so of the other pressure. 266.—If a scale cannot be had of equal proportions with the forces, the arithmetical process will be shortened somewhat by making the line of the triangle that represents the known weight equal to unity of a decimally divided scale, then the other lines will be measured in tenths or hundredths; and in the numerical statement of the proportions between the lines and forces, the first term being unity, the fourth term will be

Page 166

FRAMING. 161 ascertained simply by multiplying the second and third terms together. For example, if the three lines are 1, 0°7 and 1:3, and the known weight is 6 tons, then bd: W::be: P, becomes 1:6::07: P= 42, equals four and two-tenths tons. Again— bd: W:: ed: Q, becomes 1:6 3198 78, equals seven and eight-tenths tons. Fig. 208, 267.—In Fig. 208 the weight, W, exerts a pressure on the struts in the direction of their length; their feet, n n, have, therefore, a tendency to move‘in the direction n o, and would so move, were they not opposed by a sufficient resistance from the blocks, A and A. If a piece of each block be cut off at the horizontal line, æ n, the feet of the struts would slide away from each other along that line, in the direction, n a; but if, instead of these, two pieces were cut off at the vertical line, n.b, then the struts would descend vertically. To estimate the horizontal and the vertical pressures exerted by the struts, let n o be made equal (upon any scale of equal parts) to the num- i 21

Page 167

162 AMERICAN IOUSE-CARPENTER. ber of tons with which the strut is pressed; construct tlie parallelogram of forces by drawing o e parallel to an, and o f parallel tod; then 7 f, (by the same scale,) shows the num- ber of tons pressure that is exerted by the strut in the direc- tion n a, and n e shows the amount exerted in the direction n b. By constructing designs similar to this, giving various and dissimilar positions to the struts, and then estimating the pressures, it will be found in every case that the horizontal pressure of one strut is exactly equal to that of the other, how- ever much one strut may be inclined more than the other ; and also, that the united vertical pressure of the two struts is exactly equal to the weight, W. (In this calculation the weight of the timbers has not been taken into consideration, simply to avoid complication to the learner. In practice it is requisite to include the weight of the framing with the load upon the framing.) Fig. 209. 268.—Suppose that the two struts, Æ and B, (Fig. 208.) were rafters of a roof, and that instead of the blocks, A and A; the walls of a building were the supports: then, to prevent the walls from being thrown over by the thrust of B and B, it would be desirable to remove the horizontal pressure. This

Page 168

FRAMING, 163 may be done by uniting the feet of the rafters with a rope, iron rod, or piece of timber, as in Fig. 209. This figure is similar to the truss of a roof. The horizontal strains on the tie-beam, tending to pull it asunder in the direction of its length, may be measured at the foot of-the rafter, as was shown at Fig. 208; but it can be more readily and as accu- rately measured, by drawing from f and e horizontal lines to the vertical line, b d, meeting it in o and o; then f o will be the horizontal thrust at B, and eo at A ; these will be found to equal one another. When the rafters of a roof are thus connected, all tendency to thrust the walls horizontally is removed, the only pressure on them is in a vertical direction, being equal to the weight of the roof and whatever it has to support. This pressure is beneficial rather than otherwise, as a roof having trusses thus formed, and the trusses well braced to each other, tends to steady the walls. l Fig. 210, Fig. 211. 269.—Fig. 210 and 211 exhibit methods of framing for sup- porting the equal weights, Wand W. Suppose it be required

Page 169

164 AMERICAN HOUSE-CARPENTER. to measure and compare the strains produced on the pieces, A Band AC. Construct the parallelogram of forces, e b f d, according to Art. 258. Then b will show the strain on A B, and 6 e the strain on A C. By comparing the figures, b d be- ing equal in each, it will be seen that the strains in Pig. 210 are about three times as great as those in Zig. 211: the posi- tion of the pieces, A Æ and A C, in Fig. 211, is therefore far preferable. C Fig. 212, 270.—The Composition of Forces consists in ascertaining the direction and amount of one force, which shall be just capable of balancing two or more given forces, acting in different directions. This is only the reverse of the resolution of forces, and the two are founded on one and the same principle, and may be solved in the same manner. For example, let A and B, (Fig. 212,) be two pieces of timber, pressed in the direction of their length towards b—A by a force equal to 6 tons weight, and B equal to 9. To find the direction and dmount of pres- sure they would unitedly exert, draw the lines, 6 e and b f, in a line with the axes of the timbers, and make 6 e equal to the pressure exerted by B, viz., 9; also make b f equal to the pressure on A, viz., 6, and complete the parallelogram of forces, e b f d; then bd, the diagonal of the parallelogram, will be the direction, and its length, 9:25, will be the amount,

Page 170

FRAMING. 165 of the united pressures of A and of B. The line, b d, is termed the resultant of the two forces, bf and be. IfA and B are to be supported by one post, O, the best position for that post will be in the direction of the diagonal, 6d; and it will require to be sufficiently strong to support the united pressures of A and of B, which are equal to 9°25 or 9} tons. Fig. 218. 271.—Another example: let /%g. 213 represent a piece of framing commonly called a crane, which is used for hoisting heavy weights by means of the rope, B b f, which passes over a pulley at db. This is similar to Fig. 210 and 211, yet it is materially different. In those figures, the strain is in one direction only, viz., from 6 to d; but in this there are two strains, from A to B and from A to W. The strain in the direction A B is evidently equal to that in the direction A W. “To ascertain the best position for the strut, A C, make ġ e equal to b f, and complete the parallelogram of forces, e bf d; _ then draw the diagonal, b d, and it will be the position re- quired. Should the foot, C, of the strut be placed either higher or lower, the strain on A Ọ would be increased. In constructing cranes, it is advisable, in order that the piece, B A, may be under a gentle pressure, to place the foot of the °

Page 171

166 AMERICAN HOUSE-CARPENTER, strut a trifle lower than where the diagonal, b d, would indi cate, but never higher. Fig. 214, wW 272.—Ties and Struts. Timbers in a state of tension are called ties, while such as are in a state of compression are termed struts. This subject can be illustrated in the following manner: . Let A and B, (Fig. 214,) represent bears of timber support- ing the weights, W, W and W; A having but one support, which is in the middle of its length, and B two, one at each end. To show the nature of the strains, let each beam be sawed in the middle from a tod. The effects are obvious: the cut in the beam, A, will open, whereas that in B will close. If the weights are heavy enough, the beam, A, will break at b; while the cut in B will be closed perfectly tight at a,and the beam be very little injured by it. But if, on the other hand, the cuts be made in the bottom edge of the tim- bers, from o to b, B will be seriously injured, while A will scarcely be affected. By this it appears evident that, in a piece of timber subject to a pressure across the direction of its length, the fibres are exposed to contrary strains. If the tim- ber is supported at both ends, as at B, those from the top edge down to the middle are compressed in the direction of their length, while those from the middle to the bottom edge are in a state of tension; but if the beam is supported as at A, the contrary effect is produced; while the fibres at the middle of either beam are not at all strained. The strains in a framed

Page 172

FRAMING. 167 truss are of the same nature as those in a single beam. The truss for a roof, being supported at each end, has its tie-beam in a state of tension, while its rafters are compressed in the direction of their length. By this, it appears highly important that pieces in a state of tension should be distinguished from such as are compressed, in order that the former may be pre- served continuous. A strut may be constructed of two or more pieces; yet, where there are many joints, it will not resist compression so well. 278.—To distinguish ties from struts. This may be done by the following rule. In Fig. 206, the timbers, a b and b o, are the sustaining forces, and the weight, W, is the straining force; and, if the support be removed, the straining force | would move from thè point of support, b, towards d. Let it be required to ascertain whether the sustaining forces are stretched or pressed by the straining force. ule: upon the direction of the straining force, b d, as a diagonal, construct a parallelo- gram, e b f d, whose sides shail be parallel with the direction of the sustaining forces, a b and ¢ d; through the point, b, draw a line, parallel to the diagonal, e fy this may then be called the dividing line between ties and struts. Because all those supports which are on that side of the dividing line, which the straining force would oceupy if unresisted, are com- pressed, while those on the other side of the dividing line are. stretched. In Fig. 206, the supports are both compressed, being on that side of the dividing line which the straining force would occupy if unresisted. In (ig. 210 and 211, in which A B and A Č are the sustaining forces, A C is compressed, whereas „A Bisin a state of tension; A C being on that side of the line, A 7, which the straining force would occupy if unresisted, and A B on the opposite side. The place of the latter might be supplied by a chain or rope. In Fig. 209, the foot of the rafter at A is sustained by two forces, the wall and the tie

Page 173

168 AMERICAN HOUSE-CARPENTER. beam, one perpendicular and the other horizontal: the direc tion of the straining force is indicated by the line, da. The dividing line, A 2, ascertained by the rule, shows that the wall is pressed and the tie-beam stretched. Fig. 215. 274.—Another example: let FA B F, (Fig. 215,) represent a gate, supported by hinges at A and Æ. In this case, the straining force is the weight of the materials, and the direction of course vertical. Aseertain the dividing line at the several points, G, B, Z, J, H and F. It will then appear that the force at G is sustained by A G and G Æ, and the dividing line shows that the former is stretched and the latter com- pressed. The force at M is supported by A H and H A—the former stretched and the latter compressed. The force at B is opposed by M B and A B, one pressed, the other stretched. The force at F is sustained by @ F and F E, G F being stretched and F’ # pressed. By this it appears that A B is in a state of tension, and Æ F, of compression ; also, that A H and G F are stretched, while B H and G Æ are compressed : which shows the necessity of having A H and G F, each in one whole length, while B H and G@ # may be, as they are shown, each in two pieces. The force at J is sustained by G J and J II, the former stretched and the latter compressed.

Page 174

FRAMING. 169 The piece, O D, is neither stretched nor pressed, and coul l be dispensed with if the joinings at J and J could be made as effectually without it. In case A B should fail, then O D would be in a state of tension. : 275.—The centre of gravity. The ai of gravity of a uniform prism or cylinder, is in its axis, at the middle of its length ; that of a triangle, is in a line drawn from one angle to the middle of the opposite side and at one-third of the length of the line from that side; that of a right-angled triangle, at a point distant from the perpendicular equal to one-third of the base, and distant from the base equal to one-third of the per- pendicular ; that of a pyramid or cone, in the axis and at one- quarter of the height from the base. 276.—The centre of gravity of a trapezoid, (a four-sided figure having only two of its sides parallel,) is in a line joining the centres of the, two parallel sides, and at a distance from the longest of the parallel sides equal to the product of the length into the sum of twice the shorter added to the longer of the parallel sides, divided by three times the sum of the two parallel sides. Algebraically thus— _1@a+d) ~ 8(@+5) where d equals the distance froin the longest of the parallel sides, 2 the length of the line joining the two parallel sides, } and @ the shorter and 6 the longer of the parallel sides. Example.—A. rafter, 25 feet long, has the larger end 14 inches wide, and the smaller end 10 inches wide, how far from the larger end is the centre of gravity located ? Here, 2 = 25, a= 4%, and d= 14 l 1 = ra - hence cet b) 25(2 X 44 4-14) _ 25 x 34 eF 6) a 868+ H) 8x o 95x 84 850 sor T = 11:8 = 11 feet 98 inches nearly. In irregular bodies with plain sides, the centre of gravity 22

Page 175

er 170 AMERICAN HOUSE-CARPENTER. may be found by balancing them upon the edge of a prism— upon the edge of a table—in two positions, making a line each time upon the body in a line with the edge of the prism, and the intersection of those lines will indicate the point required. Or suspend the article by a cord or thread attached to one corner or edge; also, from the same point of suspension, hang a plumb-line, and mark its position on the face of the article ; again, suspend the article from another corner or side, (nearly at right angles to its former position,) and mark the position of the plumb-line upon its face; then the intersection of the two lines will be the centre of gravity. Fig. 216, 277.—The effect of the weight of inclined beams. An in- clined post or strut, supporting some heavy pressure applied at its upper end, as at Fig. 209, exerts a pressure at its foot in the direction of its length, or nearly so. But when such a beam is loaded uniformly over its whole length, as the rafter of a roof, the pressure at its foot varies considerably from the direction of its length. For example, let A B, (Fig. 216,) be a beam leaning against the wall, B c, and supported at its foot by the abutment, A, in the beam, A c, and let o be the centre of gravity of the beam. Through o, draw the vertical line, 6 d, and from B, draw the horizontal line, B b, cutting bdind; join b and A, and b A will be the direction of the thrust. To prevent the beam from loosing its footing, the joint at A should be made at right angles tod A. The amount of pressure will be found thus: let b d, (by any scale of equal

Page 176

FRAMING. 17] parts,) equal the number of tons upon the beam, A B; draw d e, parallel to B b; then d e, (by the same scale,) equals the pressure in the direction, 6 A; and e d, the pressure against the wall at B—and also the horizontal thrust at A, as these are always equal in a construction of this kind. -278.—The horizontal thrust of an inclined beam, (Fig. 216,) —the effect of its own weight—may be calculated thus : Rule.—Maultiply the weight of the beam in pounds by its base, A 0, in feet, and by the distance in feet of its centre of gravity, o, (see Art. 275 and 276,) from the lower end, at A ; and divide this product by the product of the length, A B, into the height, B C, and the quotient will be the horizontal thrust in pounds. This may be stated thus: H P where d equals the distance of the centre of gravity, o, from the lower end; 6 equals the base, A C; w equals the weight of the beam; / equals the height, B C; lequals the length of the beam; and H equals the horizontal thrust. Example.—A beam, 20 feet long, weighs 300 pounds; its centre of gravity is at 9 feet from its lower end; it is so inclined that its base is 16 feet and its height 12 feet; what is the horizontal thrust ? l Here Z2 £ becomes ZX 16 x 300 _9 x 4 x 25 yo o 12 x 20 5 = 180 = H = the horizontal thrust. This rule is for cases where the centre of gravity does not occur at the middle of the length of the beam, although it is applicable when it does occur at the middle; yet a shorter rule will suffice in this case,—and it is thus :— _ Rule—Multiply the weight of the rafter in pounds by the base, A C, (Fig. 216,) in feet, and divide the product by twice the height, B C, in feet; and the quotient will be the horizon tal thrust, when the cer tre of gravity occurs at the middle of the beam. =9x4xK5 ©

Page 177

172 AMERICAN HOUSE-CARPENTER. If the inclined beam is loaded with an equally distributed load, add this load to the weight of the beam, and use this total weight in the rule instead of the weight of the beam. And generally, if the centre of gravity of the combined weights of the beara and load does not occur at the centre of the length of the beam then the former rule is to be used. B Fig. 217, 279.—In Fig. 217, two equal beams are supported at their feet by the abutments in the tie-beam. This case is similar to the last; for it is obvious that each beam is in precisely the position of the beam in Fig. 216. The horizontal pressures at B, being equal and opposite, balance one another; and their horizontal thrusts at the tie-beam are also equal. (See Art. 268—/ig. 209.) When the height of a roof, (Fig. 217,) is one-fourth of the span, or of a shed, (Fig. 216,) is one-half the span, the horizontal thrust of a rafter, whose centre of gravity is at the middle of its length, is exactly equal to the weight distributed uniformly over its surface. Fig 218,

Page 178

FRAMING. 173 280.—In shed, or lean-to roofs, as Fig. 216, tte horizontal pressure will be entirely removed, if the bearings of the raft- ers, as A B, (Fig. 218,) are made horizontal—provided, how- ever, that the rafters and other framing do not bend between the points of support. If a beam or rafter have a natural curve, the convex or rounding edge should be laid uppermost. 281.—A beam laid horizontally, supported at each end and aniformly loaded, is subject to the greatest strain at the mid- dle of its length. Hence mortices, large knots and other de- fects, should be kept as far as possible from that point; and, in resting a load upon a beam, as a partition upon a floor beam, the weight should be so adjusted, if possible, that it will bear at or near the ends. Twice the weight that will break a beam, acting at the centre of its length, is required to break it when equally dis- tributed over its length; and precisely the same deflection or sag will be produced on a beam by a load equally distributed, that five-eighths of the load will produce if acting at the centre of its length. 282.—When a beam, supported at each end on horizontal bearings, (the beam itself being either horizontal or inclined,) has its load equally distributed, the amount of pressure caused by the load on each point of support is equal to one half the ° load ; and this is also the case, when the load is concentrated at the middle of the beam, or has its centre of gravity at the middle of the beam; but, when the load is unequally distri- buted or concentrated, so that its centre of gravity occurs at some other point than the middle of the beam, then the amount of pressure caused by the load on one of the points of support is unequal to that on the other. The precise amount on each may be ascertained by the following rule. Rule.—Multiply the weight w, (Fig. 219,) by its distance, CB, from its nearest point of support, B, and divide the product by the length, A B, of the beam, and the quotient wil be the

Page 179

174 AMERICAN HOUSE-CARPENTER. Fig. 219. amount of pressure on the remote point of support, A. Again, deduct this amount from the weight, w, and the remainder will be the amount of pressure on the near point of support, B; or, multiply the weight, w, by its distance, A C, from the remote point of support, A, and divide the product by the length, A B, and the quotient will be the amount of pressure on the near point of support, B. When 7 equals the length, A B; a=AC; b=CB, and = the load, then b = = A =the amount of pressure at A, and wa 7T =B= the amount of pressure at B. nae beam, 20 feet long between the bearings, has a load of 100 pounds concentrated at 3 feet from one of the bearings, what is the portion of this weight sustained by each bearing ? Here w = 100; a, 17; 6, 3; and Z, 20. wb 100 x 3 Hence A =—— 7 = 99. = 15 wa 100x17 And B=G-= =—99 = = 85 Loadon A = 15 pounds. Loadon B = 85 pounds. Total weight = 1(0 pounds.

Page 180

FRAMING. 175 RESISTANCE OF MATERIALS. 283.—Before a roof truss, or other piece cf framing, can be properly designed, two things are required to be known. The one is, the effect of gravity acting upon the various parts of the intended structure ; the other, the power of resistance possessed by the materials of which the framing is to be con- structed. In the preceding pages, the former subject having been treated of, it remains now to call attention to the latter. 284.—Materials used in construction are constituted in their structure either of fibres (threads) or of grains, and are termed, the former fibrous, the latter granular. All woods and wrought metals are fibrous, while cast iron, stone, glass, &c., are gra- nular. The strength of a granular material lies in the power of attraction, acting among the grains of matter of which the material is composed, by which it resists any attempt to sepa- rate its grains or particles of matter. A fibre of wood or of wrought metal has a strength by which it resists being com- pressed or shortened, and finally crushed; also a strength by which it resists being extended or made longer, and finally sundered. There is another kind of strength in a fibrous mate- rial ; it is the adhesion of one fibre to another along their -j or the lateral adhesion of the fibres. 285.—In the strain applied to a piece of timber, as a post supporting a weight imposed upon it, (Fig. .220,) we have an instance of an attempt to shorten the fibres of which the tim- ber is composed. The strength of the timber in this case is termed the resistance to compression. In the strain on a piece of timber like a king-post or suspending piece, (A, Fig. 221,) we have an instance of an attempt to-extend or lengthen the fibres of the material. The strength here exhibited is termed the resistance to tension. When a piece of timber is strained like a floor beam, or any horizontal piece carrying a load, (Fig. 222,) we have an instance.in which the two strains of

Page 181

176 AMERICAN HOUSE-CARPENTER. Fig. 222. compression and tension are brought into action; the fibres of the upper portion of the beam being compressed, and those of the under part being stretched. This kind of strength of tim ber is termed resistance to cross strains. In each of these three kinds of strain to which timber is subjected, the power of resistance is in a measure due to the lateral adhesion of the fibres, not so much perhaps in the simple tensile strain, yet to a considerable degree in the compressive and cross strains. But the power of timber, by which it resists a pressure acting compressively in the direction of the length of the fibres, tend- ing to separate the timber by splitting off a part, as in the case of the end of a tie beam, against which the foot of the rafter presses—is wholly due to the lateral adhesion of the fibres. 286.—The strength of materials is that power by which they resist fracture, while the stiffness of materials is that quality which enables them to resist deflection or sagging. A know- ledge of their strength is useful, in order to determine their

Page 182

FRAMING. 177 limits of size to sustain given weights safely ; but a knowledge of their stifness is more important, as in almost all construe- tions it is desirable not only that the load be safely sustained, but that no appearance of weakness be manifested by any sen- ahe deflection or sagging. I.—RESISTANCE TO COMPRESSION. 287.—The resistance of materials to the force SE m _ may be considered in four several ways, viz. 1st. When the- pressure is pit to me fibres id nally, and on short pieces. 2d. When the pressure is applied to the fibres longitudi- nally, and on long pieces. 3d. When the pressure is applied to the fibres longitudi- nally, and so as to split off. the part pressed against, causing the fibres to separate by sliding. 4th. When the pressure is applied to the fibres trans- versely. Posts having their height less than ten times their least side will crush before bending; these belong to the first case: while posts, whose height is ten times their least side, or more than ten times, will bend before crushing ; these belong to the second case. 288.—In the above first and fourth: cases of compression, experiment has shown that the resistance is in proportion to the number of fibres pressed, that is, in proportion to the area. For example, if 5,000 pounds is required to crush a prism with a base 1 inch square, it will require 20,000 pounds to crush a prism having a base of 2 by 2 inches, equal to 4 inches area; because 4 times 5,000 equals 20,000. Experiment has also shown that, in the third case, the resistance is in proportion’ to the area of the surface separated without regard to the form of the surface. 289.—In the second case of compression, the resistance is in 23

Page 183

178 AMERICAN HOUSE-CARPENTER,. proportion to the area of the cross section of the piece, multi plied by the square of its thickness, and inversely in propor: tion to the square of the length, multiplied by the weight. Whien the piece is square, it will bend and break in the direc- tion of its diagonal ; here, the resistance is in proportion to the square of the diagonal multiplied by the square of the dia- gonal, and inversely proportional to the square of the length multiplied by the weight. Ifthe piece is round or cylindrical, its resistance will be in accordance with the square of the dia- meter multiplied by the square of the diameter, and inversely proportional to the square of the length, multiplied by the weight. 290.—These relations between the dimensions of the piece strained and its resistance, have resulted from the discussion of the snbject by various authors, and rules based upon these relations are in general use, yet their accuracy is not fully established. Some experiments, especially those by Prof. Hodgkinson, have shown that the resistance is in proportion to a less power of the diameter, and inversely to a less power of the height; yet the variance is not great, and inasmuch as the material is restricted in the rules to a strain decidedly within its limits of resistance, no serious error can be made in the use of rules based on the aforesaid relations. 291.—Kxperiments. In the investigation of the laws appli- cable to the resistance of materials, only such of the relations of the parts have been considered as apply alike to wood and metal, stone and glass, or other material, leaving to experi- ment the task of ascertaining the compactness and cohesion of particles, and the tenacity and adhesion of fibres; those quali- ` ties upon which depend the superiority of one kind of material over another, and which is represented in the rules by a constant number, each specific kind of material having its own special constant, obtained by experimenting on specimens of that peculiar material.

Page 184

FRAMING. 179 292.—The following table exhibits the results of experiments on such woods as are in most common use in this country for the purpose of construction. The resistance of timber’ of the TABLE I.—COMPRESSION. ‘ p EN E © £ Z © Es EE S JAA e |= ae Kind of Material. a | 25 | 8a |355| 2 |B [aes é [ei sean te. foe 2 aa Sa piu] sa | ee Sao ° Ez| Za) ef jpg] of | BES] ef $ | sm | #3 | suk | 58 | S83 | 2a lll aa a a , i Pounds Pounds Pounds per in. per in. per in. White wood, . . | °397 | 2482 | 600 600 | 390 Mahogany (Baywood), . | ‘439 | 3527 | 880 1300 | 650 Ash, : . | ‘517 | 4175 | 1040 2300 | 1150 Spruce, 5 a : . | -369 | 4199 | 1050 | 470 160 | 500 | 250 Chestnut, 5 White pine, : 5 388 | 4806 | 1200 | 490 160 600 300 Ohio pine, 6 “686 | 4809 | 1200 388 130 | 1250 625 Oak, . e . . z 612 | 5316 | 1330 780 260 | 1900 950 Hemlock, . e o - | "423 | 5400 | 1350 540 180 600 300 Black walnut, o : 421 | 5594 | 1400 1600 800 Maple, o : : 574 | 6061 | 1515 2050 | 1025 Cherry, . . e s 494 | 6477 | 1620 1900 950 White oak, : 6 . | 774 | 6660 | 1665 2000 | 1000 Georgia pine, . . f 613 | 6767 | 1700 | 510] 170 | 1700 | 850 Locust, . : e ` «| 762 | 7652 | 1910 | 1180 | 400 | 2100 | 1050 Live oak, - | °916 | 7936 | 1980 5100 | 2550 Mahogany, (St. Domingo), . | 837 | 8280 | 2070 4300 | 2150 Lignum vite, . « | 1282 | 8650 } 2160 5800 | 2900 Hickory, . : . e | °877 | 9817 | 2450 3100 | 1550 same name varies much ; depending as it obviously must on the soil in which it grew, on its age before and after cutting, on the time of year when cut, and on the manner in which it has been kept since it was cut. And of wood from the same tree, much depends upon its location, whether at the butt or towards the limbs, and whether at the heart or at the sap, or at a point midway from the centre to the circumference of the tree. The pieces submitted to experiment were of ordinary good quality, such as would be deemed proper to be used in framing. The prisms crushed were 2 inches long, and from 1 inch to 14 inches square ; some were wider one way than the

Page 185

180 AMERICAN IIOUSE-CARPENTER. other, but all containing in area of eross section froin 1 to 3 inches. There were generally three speeimens of each. kind. The weight given in the table is the average crushing weight per superficial inch. . In the preceding table the first column contains the specific gravity of the several kinds of wood, showing their eompara- tive density. The weight in pounds of a enbic foot of any kind of wood or other material, is equal to its specific gravity multiplied by 62-5 ; this number being the weight in pounds of a cubic foot of water. The second column contains the weight in pounds required to crush a prism having a base of one inch square; the pressure applied to the fibres longitudi- nally. The third column contains the value of Cin the rules; C being equal to one-fourth of the crushing weight in the preceding column. The fourth column contains the weight in pounds, which, applied to the fibres longitudinally, is required to force off a part of the piece, causing the fibres to separate by sliding, the surface separated being one inch square. The fifth column contains the value of /Z in the rules, ZZ being equal to one third of the weight in the preced- ing column. The sixth column contains the weight in pounds required to erush the piece when the pressure is applied to the fibres transversely, the piece being one ineh thick, and the surface crushed being one ineh square, and depressed one twentieth of an inch deep. The seventh column contains the value of Pin the rules; P being the weight in pounds applied to the fibres transversely, which is required to make a sensible impression one inch square on the side of the piece, this being the greatest weight that would ‘be proper for a post to be loaded with per inch surface of bearing, resting on the’side of the kind of wood set opposite in the table. A greater weight would, in proportion to the excess, erush the side of the wood under the post, and proportionably derange the framing, if not eause a total failure. It will be observed that the measure of

Page 186

‘FRAMING. 181 tnis resistance is useful in limiting the load on a post accord- ing to the kind of material contained, not in the post, but in the timber upon which the post presses. 293.—In Table II. are the results of experiments made tc test the resistance of materials to flexure: first, the flexure produced by compression, the force acting on the ends of the fibres longitudinally; secondly, the flexure arising from the effects of a cross strain, the force acting on the side of the fibres transversely, the beams being laid on chairs or rests. Of white oak, No. 1, there were eight specimens, of 2 by 4 inches, and 34 feet long, seasoned more than a year after they were prepared for experiment., Of the other kinds of wood there were from three to five specimens of each, of 14 by 24 inches, and from 12 to 23 feet long. Of the cast iron there "were six specimens, of 1 inch square and 1 foot long; and of the wrought iron there were five specimens of American, three of 2 by 2 inches, and two of 14 inches square, and three specimens of common English, 4 by 2 inches; the eight speci- mens being each 19 inches long, clear bearing. In each case the result is the average of the stiffness of the several specimens. The numbers contained in the second column are the weights producing the first degree of -flexure in a post or strut, where the post or strut is one foot long and one inch square; so, likewise, the numbers in the fifth column, and which are represented in the rules by Æ, are the weights required to deflect a beam one’ inch, where the beam is one foot long, clear bearing, and one inch square—(See remarks upon this, Art. (821.) The numbers in the third column are equal to one-half of those in the second. The numbers con- tained in the fourth column, and represented by n in the ‘rules, show the greatest rate of deflection that the material may be subjected to without injury. This rate multiplied by the length in feet, equals the total deflection within the limite of elasticity.

Page 187

182 AMERICAN HOUSE-CARPENTER. TABLE IJ.-—-FLEXURE. Under ) Under Compression. Cross Strain, Specific pas ag Gravity.) Pounds pro- | Value of | Value of | Value of ducing the Bin nin T first degree | the Rules. | the Rules, the Rules of flexure. Hemlock, : ` ; . | 0:402 2640 1320 |008794 1240 Spruce, . F 6 . . "432 4190 2095 | 009197 1550 White pine, . è : =| $4.07 2350 1175 =| 071022 1750 Ohio yellow pine, . . - | 586 6000 3000 | 0°049 1970 Chestnut, . 3 : : =] o2 4720 8860 | 0:07541 2350 White oak, No. 1, . =| 82 009152 2520 White oak, No. 2, . : . | 7805 6950 3475 | 0°0567 2590 Georgia pine, . : ; 2i a65 9660 4830 | 007723 2970 Locust, . . . z . 863 10920 5460 |00686615 8280 Cast iron, : 6 5 s |7042 0:0148 80500 Wrought iron, common English, | 7°576 : 008717 | 45500 Wrought iron, American, . e |1576 004038 | 51400 PRACTICAL RULES FOR COMPRESSION. First Case. 294.—To find the weight that can be safely sustained by a post, when the height of the post is less than ten times the diameter if round, or ten times the thickness if rectangular, and the direction of the pressure coinciding with the axis. Lule 1.—Multiply the area of the cross-section of the post, in inches, by the value of C in Table I, the product will be the required weight in pounds. Example.—A Georgia pine post is 6 feet high, and in cross- section, 8 x 12 inches, what weight will it safely sustain ? The area = 8 x 12 = 96 inches; this multiplied by 1700, the value of C, in the table, set opposite Georgia pine, the result, 163,200, is the weight in pounds required. It will be observed that the weight would be the same for a Georgia pine post of any height less than 10 times 8 inches = 80 inches = 6 feet 8

Page 188

FRAMING. 183 inches, provided its breadth and thickness remain the same, 12 and 8 inches. 295.—To find the area of the cross-section of a post to sus- tain a given’weight safely, the height of the post being less than ten times the diameter if round, or ten times the least side if rectangular; the pressure coinciding with the axis. Rule U.—Divide the given weight in pounds by the value of C, in Table I., and the product will be the required area in inches w g~ A. (2.) l Example.—A weight of 38,400 pounds is to be sustained by a white pine post 4 feet high, what must be its area of section in order to sustain the weight safely? Here, 38,400 divided by 1200, the value of C, in Table I., set opposite white pine, gives a quotient of 32; this, therefore, is the required area, and such a post may be 5 x 6'4 inches. To find the least side, so that it shall not be less than one-tenth of the height, divide the height, reduced to inches, by 10, and make the least side to exceed this quotient. The area, divided by the least side so determined, will give the wide side. If, however, by this process, the first side found should prove to be the greatest, then the size of the post is to be found by Rule VIT., VIIL, or IN. 296.—If the post is to be round, by reference to the Table of Circles in the Appendix, the diameter will be found in the column of diameters, set opposite to the area of the post found in the column of areas, or opposite to the next nearest area. - For example, suppose the required area, as just found by the example under Rule II., is 32; by reference to the column of areas, 83°183 is the nearest to 32, and the diameter set opposite “is 65. The post may, therefore, be 64 inches diameter. Second Case. 297.- -To ascertain the weight that can be sustained safely

Page 189

184 AMERICAN HOUSE-CARPENTER. by a post whose height is, at least, ten times its least side if rectangular, or ten times its diameter if round, the direction of the pressure coinciding with the axis. Rule WL— When the post is round the weight may be found by this rule: Multiply the square of the diameter in inches by the square of the diameter in inches, and multiply the product by 0°589 times the value of B, in Table II., divide this product by the square of the height in feet, and the quo- tient will be the required weight in pounds. 0559A DPD 05595 p: w = oo a = =g (3.) Example-—What weight will a Georgia pine post sustain safely, whose diameter is 10 inches and height 10 feet? The square of the diameter is 100; 100 x 100 = 10,000. And 10,000 by 0°589 times 4830, the value of Æ, Table IL, set opposite Georgia pine, = 28,448,700, and this divided by 100, the square of the height, equals 284,487, the weight required, in pounds, Lule 1V.—If the post be rectangular the weight is found by this rule: Multiply the area of the cross-section of the post by the square of the thickness, both in inches, and by the value of B, Table II. Divide the product by the square of the height in feet, and the quotient will be the required weight in pounds. APERO E C= = P T` a (£) Exvample—What weight will a white pine post sustain safely, whose height is 12 feet, and sides 8 and 12 inches re- spectively? The area = 8 x 12 = 96 inches; the square of the thickness, 8, = 64. The area by the square ofthe thick- ness, 96 x 64, = 61443 and this by 1175, the valne of B, for white pine, equals 7,219,200. This, divided by 144, the square of the he'ght = 50,1331, the reqiired weight in pounds,

Page 190

FRAMING. 185 ` Rule V.—Lf the post be square, the weight is found by this rule: Multiply the value of B, Table II., by the square of the area of the post in inches, and divide the product by the square of the height in feet, and the quotient will be the required weight in pounds. © AUB DIA T z — (5.) Fxample—What weight will a white oak post sustain safely, whose height is 9 feet, and sides each 6 inches? The. value of B, set opposite white oak, is 3475 ; this, by (86 x 36 =) 1296, the square of the area, equals 4,503,600. This pro- duct, divided by 81, the square of the height, gives for quo- tient, 55,600, the required weight in pounds. 298.—To ascertain the size of a post to sustain safely a given weight when the height of the post is at least ten times the least side or diameter. kule V1.— When the post is to be round or cylindrical, the size may be obtained by this rule: Divide the weight in pounds by 0°589 times the value of B, Table II., and extract the square root of the product; multiply the square root by the height in feet, and the square root of this pr oduct will be the diameter of the post in inches. bai in| = Jas O 589 B Lxample.—What must be the diameter of a locust post, 10 feet high, to sustain safely 40,000 pounds? Here 0:589 times 5,460, the value of B for locust, Table II., equals 3215-9, The weight, 40,000, divided by 3215:9, equals 12:438. The _ square root of this, 3°5268, multiplied by 10, the height, equals 35268, and the square root of this is 5°9386 or 514 inches, the required diameter of the post. Rule vS the post is to be rectangular, the size may be ' obtained by this rule: Multiply the square of the height in 24

Page 191

186 AMERICAN HOUSE-CARPENTER. feet by the weight in pounds, and divide the product by the value of B, Table II. Now, if the breadth is known, divide the quotient by the breadth in inches, and the cube root of this quotient will be the thickness in inches. But if the thick- » ness is known, and the breadth desired, divide, instead, by the cube of the thickness in inches, and the quotient will be the breadth in inches. s/h? w LEN a (7.) Vw —. — 8. BË o) Example.—What thickness must a hemlock post have, whose breadth is 4 inches and height 12 feet, to sustain safely 1,000 pounds? The square of the height equals 144; this, by 1,000, the weight, equals 144,000. This, divided by 1,320, the value of B for hemlock, Table II., equals 109-091. This, divided by 4, the breadth, equals 27:273, and the cube root of this is 3°01, a trifle over 3 inches, and this is the thickness required. l Another Example.—W hat breadth must a spruce post have, whose thickness is 4 inches and height 10 feet, to sustain safely 10,000 pounds? The square of the height, 100, by 10,000, the weight, equals 1,000,000. This, divided by 2095, the value of B, Table IL, for spruce, equals 477:09 ; and this, divided by 64, the cube of the thickness, equals 7:45, nearly 74 inches, the breadth required. Rule VII.—Jf the post is to be square, the size may be obtained by this rule. Divide the weight in pounds by the value of B, Table II., and multiply the square root of the pro- duct by the height in feet, and the square root of this product will be the dimension of a side of the post in inches. Vive J Example.—What dimension must the side of a square post

Page 192

FRAMING. 187 have, whose height is 15 feet, the post being of Georgia pine, to sustain safely 50,000 pounds? The weight 50,000, divided by 4830, the value of B, Table II., for Georgia pine, equals 10°352. The square root of this, 3-2175, multiplied by 15, the height, equals 48°362, and the square root of this is 6:9472, nearly 7 inches, the size of a side of the required post. 299.—A square post is not the stiffest that can be made from a given amount of material. The stiffest rectangular post is that whose sides are in proportion as 6 is to 10. When this proportion is desired it may be obtained by the following rule. Fule [X.—Divide six-tenths of the weight in pounds by the value of B, Table IL, and extract the square root of the quo- tient; multiply the square root by the height in feet, and then the square root of this product will be the thickness in inches. The breadth is equal to the thickness divided by 0°6. a J h /06w 4/06 h* w (10.) i B TRN (14.) Example.—What must be the breadth and thickness of a white pine post, 10 feet high, to sustain safely 25,000 pounds. Here f; of 25,000, the weight, divided by 1175, the value of B, Table IL, for white pine, equals 12-766. The square root of this, 3:5729, multiplied by 10, the height, equals 35-729, and the square root of this is 5-977, nearly 6 inches, the thickness required. Th’s, divided by 0-6, equals 10, equals the breadth in inches required. i 300.—The sides of a post may be obtained in any desirable proportion by Rule IX., simply by changing the decimal 0'6 to such decimal as will be in proportion to unity as one side is to be to the other. For example, if it be desired to have the sides in proportion as 10 is to 9, then 0'9 is the required decimal; if as 10 is to 8, then 0'8 is the decimal; if as 10

Page 193

158 AMERICAN HOUSE-CARPENTER. is to 7, then 0-7 is the decimal to be used in p.ace of 0-6 in the rule. And generally let 6 equal the broad side and z the narrow side, or let these letters represent respectively the numbers that the sides are to be in proportion to; then, where v equals the decimal sought, d:¢::1ia= z- = the required decimal, or fraction. For a fraction may be used in place of the decimal, where it would be more convenient, as is the case when the sides are desired to be in proportion as 3 to 2. Here 3:2::1:@=2. This fraction should be used in the rule in place of the decimal 0°6—rather than its equivalent decimal ; simply because the decimal contains many figures, and there- fore would not be convenient. The decimal equivalent to % is 0:666666 +. Third Case. 301.—To ascertain what weight may be sustained safely by the resistance of a given area of surface, when the weight tends to split off the part pressed against by causing one sur- face to slide on the other, in case of fracture. Rule X.—Multiply the area of the surface by the value of LI, in Table I., and the prodnet will be the weight required in pounds. . AH=w. | (12.) Lxvample.—tThe foot of a rafter is framed into the end of its tie-beam, so that the uncut substance of the tie-beam is 15 inches long from the end of the tie-beam to the joint of the rafter; the tie-beam is of white pine, and is six inches thick ; what amount of horizontal thrust will this end of the tie-beam sustain, without danger of having the end of the tie-beain split off? Tere the area of surface that sustains the pressure is 6 by 15 inches, equal to 90 inches. This, multiplied by 160, the value of ZI, set opposite to white pine, Table I., gives a product of 14,400, and this is the required weight ın pounds.

Page 194

FRAMING. 189 302.—To ascertain the area of surface that is required to sustain a given weight safely, when the weight tends to split off the part pressed against, by causing one surface to slide on the other, in case of fracture. Rule X1.—Divide the given weight in pounds by the value of H, Table I., and the quotient will be the required area in inches. 4 z i (13.) Example.—The load on a rafter causes a horizontal thrust at its foot of 40,000 pounds, tending to split off the end of the tie- beam, what must be the length of the tie-beam beyond the line, where the foot of the rafter is framed into it, the tie-beam being of Georgia pine, and nine inches thick? The weight, or horizontal thrust, 40,000, divided. by 170, the value of M, Table I., set opposite Georgia pine, gives a quotient of 235-3. This, the area of surface in inches, divided by 9, the breadth of the surface strained, (equal to the thickness of the tie-beam,) the quotient, 2671, is the length in inches from the end of the tie-beam to the rafter joint, say 26 inches. 303.—A knowledge of this kind of resistance of materials is useful, also, in ascertaining the length of framed tenons, so as to prevent the pin, or key, with which they are fastened from tearing out; and, also, in cases where tie-beams, or other timber under a tensile strain, are spliced, this rule gives the length of the joggle on each end of the splice. | Fourth Case. 304.—To ascertain what weight a post may be loaded with, so as not to crush the surface against which it presses. Kule X11.—Maultiply the area of the post in inches by the value of P, Table I., and the product is the weight ere in pounds, ae? (14.)

Page 195

190 AMERICAN HOQUSE-CARPENTER. Ecample.—A post, 8 by 10 inches, stands upon a white pine girder; the area equals 8 x 10 = 80 inches. This, by 300, the value of P, Table I., set opposite white pine, the product, 24,000, is the required weight in pounds. 305.—To ascertain what area a post must have in order to prevent the post, loaded with a given weight, from crushing the surface against which it presses. Rule XU1.—Divide the given weight in pounds by the value of P, Table I., and the quotient will be the area required in inches. A == (15.) Erample.—A post standing on a Georgia pine girder is loaded with 100,000 pounds, what must be its area? The weight, 100,000, divided by 850, the value of P, Table I., set opposite Georgia pine, the quotient, 117°65, is the reqnired area in inches. The post may be 10 by 113, or 10 x 12 inches, or, if square, each side will be 10°84 inches, or 12} inches diameter, if round. II.——RESISTANCE TO TENSION. 306.—The resistance of materials to the force of stretching, as exemplified in the case of a rope from which a weight is suspended, is termed the resistance to tension. In fibrous materials, this force will be different in the same specimen, in accordance with the direction in which the force acts, whether in the direction of the length of the fibres, or at right angles to the direction of their length. It has been found that, in hard woods, the resistance in the former direction is about 8 to 10 times what it is in the latter; and in soft woods, straight grained, such as white pine, the resistance is from 16 to 20 times. A knowledge of the resistance in the direction of the fibres is the most useful in practice. 307.--In the following table, the experiments recorded were

Page 196

FRAMING. 193 to test this resistance in sncli woods, also iron, as are in common use. Each specimen was turned cylindrical, and about 2 inches diameter, and then the middle part for 10 inches in - length reduced to ths of an inch diameter, at the middle of the reduced part, and gradually increased toward each end, where it was about an eighth of an inch larger at its junction with the enlarged end. l TABLE II.——TENSION. Specific Weight produc- Value of f b ing fi Kind of Material Gravity. g zene e 1 Gina Pounds, Hickory, 5 : P : 7 0751 20,700 3,450 Locust, z : : : ; 794 15,900 2,650 Maple, . 5 0 : "694 15,400 2,567 White pine, . 6 A : A "458, . 14,200 2,367 Ash, : : 9 o 6 5 608 - 11,700 1,950 Oak, k 5 0 . 5 728 10,000 1,667 White oak, . 5 d a o 774 17,000 2,833 Georgia pine, . : 5 : 5 *650 17,000 2,883 cast Tn from . 0 z % 7-200 17,000 2,833 ’ to o 5 : a 7-600 80,000 5,000 American wrought iron, 2 in. diam., 7-000 80,000 5,000 Do. do. and} do, 7-800 55,000 9,166 Do. do. wire, No. 3,. : 102,000 17,000 Do. do. do. No. 0,. : 74,500 12,416 Do. annealed do. No. 0,. : 53,000 8,5833 308.—The value of Tin the rules, as contained in the last column of the above table, is one-sixth of tke weight pro-. ducing fracture per square inch of cross section, as recorded in the preceding column. This proportion of the breaking weight is deemed the proper one, from the fact that in prac- tice, through defects in workmanship, the attachments may be so made as to cause the strain to act along one side of the „ piece, instead of through its axis; and as in this case % has been found, that fracture will be produced with 4 of the strain that can be sustained through the axis, therefore one half of this reduced strain, (equal to } of the strain through the axis), is the largest that a due regard to security will permit to be

Page 197

192 AMERICAN HOUSE-CARPENTER. used. And in some cases it may be deemed advisable to load the material with even a still smaller strain. 309.—To ascertain the weight or pressure that may be safely applied to a beam as a tensile strain. Rule XTV.—Multiply the area of the cross section of the beam in inches by the value of 7, Table IIL, and the product will be the required weight in pounds. The cross section here intended is that taken at the smallest part of the beam or rod. A beam is usually cut with mortices in framing; the area will probably be smallest at the severest cutting: the area used in the rule must be only of the uncut fibres. Aly =o. (16.) Example.—A tie-beam of a roof truss is of white pine, and 6 x 10 inches; the cutting for the foot of the rafter reduces the uncut area to 40 inches: what amount of horizontal thrust from the foot of the rafter will this tie-beam safely sustain ? Here 40 times 2,367, the value of T, equals 94,680, the required weight in pounds. 310.—To ascertain the sectional area of a beam or rod that will sustain a given weight safely, when applied as a tensile strain. Mule XV.—Divide the given weight in pounds by the value of T, Table III., and the quotient will be the area required in inches: this will be the smallest area of uncut fibres. If the piece is to be cut for mortices, or for any other purpose, then a sufficient addition is to be made to the result found by the rule. w Ro A (1T) Erample.—A rafter produces a thrust horizontally of 80,000 pounds ; the tie-beam is to be of oak: what must the area of the cross section of the tie-beam be, in order to sustain the rafter safely? The given weight, 80,000, divided by 1,667, the value of 7; the quotient, 48, is the area of uncut

Page 198

FRAMING. 193 fibres. This should have usually one-half of its amount added to it as an allowance for cutting; therefore 48 + 24 = 72. The tie-beam may be 6 x 12 inches. 311.—In these rules nothing has been said of an allowance for the weight of the beam itself, in cases where the beam is piaced vertically, and the weight suspended from the end. Usually, in timber, this is small in comparison with the load, and may be neglected; although in very long timbers, and where accuracy is decidedly essential, it may form a part of the rule. 312.—Taking the effect of the weight of the beam into account, the relation existing between the weights and parts of the beam, may be stated algebraically thus :— AT=w+k (18.) Where A equals the area of the section of uncut fibres, 7 equals the tabular constant in the rules, which is equal to the load that may be safely trusted on a rod of like material with the beam and one inch square; w equals the load, and & equals the weight of the beam. Now, the weight of the beam equals its cubical contents in feet, multiplied by the weight of a cubic foot of like material; and a cubic foot of the material equals 62-5 times its specific gravity, while the cubical contents of the beam in feet equals <a l, where Æ equals the sectional area in inches, and Z equals the length in feet. Hence— k 0:5 f rand, (19.) where f equals the specific gravity. It will be observed that A equals the sectional area of the uncut fibres, while Z? equals the seotional area of the entire beam; and, where the excess of over A may be stated as a proportional part of 4, or when A+” A = K, (n being a decimal in proportion to unity, as the excess of Zè over A is to -A,) or R—A -Jj =n. Then, [from (18) ] — 25

Page 199

194 AMERICAN HOUSE-CARPENTER. At] ww ke poe es a 2-5 saat and w = A T — 0°434 (n+ 1) A fl, w= A (T—0°4384(n+1)f1;) — (20.) Ww . and A a a A (21.) When A is found, to find X, we have from R=-A+nA, eA 1) (22.) As the excess of Z over A decreases, n also decreases, until finally, when Æ = A, n becomes zero. For— iA n = —) and when A = X, then RB P When n equals zero, it disappears from the rules, and (20) becomes w= A (T—0-484,f 1 (23.) and (21) becomes Ww A= T 0IANT, (24.) and (22) becomes R= A, (25.) 313.—These rules stated in words at length are as fol- 10WS :— To ascertain the weight that may be suspended safely from a vertical beam, when the weight of the beam itself is to be taken into account, and when a portion of the fibres are cut, in framing. Rule XVI.—From the sectional area of the beam, deduct

Page 200

FRAMING. 195 the sectional area of uncut fibres, and divide the remainder by the sectional area of the uncut fibres, and to the quotient add unity; multiply this sum by 0°434 times the specific gravity of the beam, and by its length in feet; substract this product from the value of T, Table III., and the remainder, multiplied by the sectional area of the uncut fibres, will be the required weight in pounds. : w= A (T— 0434 (n+ 1) f2.) (20.) Example.—A. white pine beam, set vertically, 5 x 9 inches and 30 feet long, is so cut by mortices as to have remaining only 5 x 6 inches sectional area of uncut fibres: what weight will such a beam sustain safely, as a tensile strain? The uncut fibres, 5 x 6 = 30, deducted from the area of the beam, 5 x 9 = 45, there remains 15. This remainder, divided by 30, the area of the uncut fibres, the quotient is 0°5. This added to unity, the sum is1°5. This, by 0:434 times 0°458, the specific gravity set opposite white pine in Table III, and by 30, the length of the beam in feet, the product is 8-95. This product, deducted from 2,367, the value of T set opposite white pine in Table IIL, the remainder is 2,358°05. This remainder multi- plied by 80, the sectional arca of the uncut fibres, the product, 70,741°5, is the required weight in pounds. 314.—When the beam is uncut for mortices or other pur- poses, the former part of the rule is not needed; the weight will then be found by the following rule. Rule XVIl—Deduct 0°484 times the specific gravity of the beam, multiplied by its length in feet, from the value of T, Table IIL; the remainder, multiplied by the sectional area of the beam in inches, will be the required weight in * pounds. l w= A (T — 0-434 f I). . (23.) Example.—A Georgia, pine beam, set vertically, is 25 feet long and 7 x 9 inches in sectional aréa: what weight will it sustain safely, as a tensile strain? By the rule, 0-434 times

Page 201

196 AMERICAN HOUSE-CARPENTER. 0°65, the specific gravity of Georgia pine, as in Table II., mul tiplied by 25, the length in feet, the product is 7:05. This product, deducted from 2,833, the value of 7, Table IIL, set opposite Georgia pine, and the remainder, 2,825-95, multiplied by 63, the sectional area, the product, 178,034°85, is the reqnired weight in pounds. 815.—To ascertain the sectional area of a vertical beam that will safely sustain a given tensile strain, where the weight of the beam itself is to be considered. Rule XVIIL.—Where the beam is cut for mortices or other purposes, let the relative proportion of the uncut fibres to those that are cut, be as 1 is to n, (n being a decimal to be fixed on at pleasure.) Then to the value of n add unity, and multiply ing the sum by 0:434 times the specific gravity in Table IIL, and by the length in feet. Ieduct this product from the value of 7, Table IH., divide the given weight in pounds by this remainder, and the quotient will be the area of the uneut fibres in inches. Add unity to the value of n, as above, and multiply the sum by the area of the uncut fibres; the product will be the required area of the beam in inches. uw A P R (n+ IFL o R=A(n+1), (22.) Example.—A vertical beam of white oak, 30 feet long, is required to resist effectually a tensile strain of 80,000 pounds : what must be its sectional area? The relative proportion of the uncut fibres is to be to those that are cut as 1 is to Ov4. To 0:4, the value of n, add 1-5 the sum is 1-4. This, by 0-434 times ‘774, the specific gravity of white oak in Table III., and by 30, the length, the product is 14'109. This, deducted from 2,833, the value of 7 for white oak in Table III., the remainder is 2,818°891. The given weight, 80,000, divided by 2,818-891, the remainder, as above, the quotient, 28°38, is the area of the uncut fibres. This multiplied by the sum of 0°4 and 1, (or

Page 202

FRAMING. 197 the value of n and unity = 1:4,) the prcduct, 39°732, is the required area of the beam in inches. 316.—When the fibres are uncut, then their sectional area equals the area of tle beam, and may be found by the follow- ing rule. Kule XTX.—Deduct 0:434 times the specific gravity in Table HI., multiplied by the length in feet, from the value of T, Table III., and divide the weight in pounds by the remain- ` der. The quotient will be the required area in inches, w l A= BTFT. oe) Example—aA vertical beam of locust, 15 feet long, fibres all uncut, is required to sustain a tensile strain equal to 25,000 pounds: what must be its area? Here 0-434 times -794, the specific gravity for locust in Table III., multiplied by 15, the length in feet, is 5-17. This, from 2,650, the value of T for — locust, Table IH., the remainder is 2,644°83. The given weight, 25,000, divided by 2,644°83, the remainder, as above, the quotient, 9:45, will be the required area in inches. JI.—RESISTANCE TO CROSS-STRAINS. 317.—A load placed upon a beam, laid horizontal or in-. clined, tends to bend it, and if the weight be proportionally large, to break it. The power in the material that resists this - bending or breaking, is termed the resistance to cross-strains, or transverse strains. While in posts or struts the material is compressed or shortened, and in ties and suspending-pieces it is extended or lengthened; in beams subjected to cross-strains ` the material-is both compressed and extended. (See Art -254.) When the beam is bent, the fibres on the concave side are compressed, while those on the convex side are extended. The line where these two portions of the beam meet—that is, the portion compressed and the portion extended—the hori- zontal line of juncture, is termed the neutral line or plane. It a

Page 203

198 AMERICAN HOUSE-CARPENTER. is so called because at this line the fibres are neither com pressed nor extended, and hence are under no strain whatever. The location of this line or plane is not far from the middle of the depth of the beam, when the strain is not sufficient to injure the elasticity of the material; but it removes towards the concave or convex side of the beam as the strain is increased, until, at the period of rupture, its distance from the top of the beam is in proportion to its distance from the bot- tom of the beam as the tensile strength of the material is to its compressive strength. 318.—In order that the strength of a beam be injured as little as possible by the cutting required in framing, all mor- tices should be located at or near the middle of the depth. There is a prevalent idea among some, who are aware that the upper fibres of a beam are compressed when subject to cross-strains, that it is not injurious to cut these top fibres, provided that the cutting be for the insertion of another piece of timber——as in the case of gaining the ends of beams into the side of a girder. They suppose that the piece filled in will as effectually resist the compression as the part removed would have done, had it not been taken out. Now, besides the effect of shrinkage, which of itself is quite sufficient to prevent the proper resistance to the strain, there is the mecha- nical difficulty of fitting the joints perfectly throughout ; and, also, a great loss in the power of resistance, as the material is so much less capable of resistance when pressed at right angles to the direction of the fibres, than when directly with them, as the results of the experiments in the tables show. 319.—In treating upon the resistance to cross-strains, the subject is divided naturally into two parts, viz. stiffness and strength: the former being the power to resist deflection or bending, and the latter the resistance to rupture. 320.—Resistance to Deflection. When a load is placed upon a beam supported at each end, the beam bends more or

Page 204

FRAMING. 199 less ; the distance that the beam descends wider the operatior of the load, measured at the middle of its length, is termed its deflection. In an investigation of the laws of deflection it has been demonstrated, and experiments have confirmed it, that while the elasticity of the material remains uninjured by the pressure, or is injured in but a small degree, the amount of deflection is directly in proportion to the weight producing it, and is as the cube of the length; and, in pieces of rectangular sections, it is inversely proportional to the breadth and the cube of the depth: or, inversely proportional to the fourth power of the side of a square beam or of the diameter of a cylindrical one. Or, when ¿ equals the length between the supports, w the weight or pressure, 6 the breadth, d the depth, and p the deflection ; then— 3 fas = uh, (26.) equals a constant quantity for beams of all dimensions made from a like material. Also, D w — s equals a side of a square beam ; and Ew osem p= o 8) where D equals the diameter of a cylindrical beam. The constant here is less than in the case of the square and of the rectangular beams. It is as much less as the, circular beam is less stiff than a square beam whose side is equal to the diame- ter of the cylindrical one. The constant, fyi is therefore mul- tiplied by the decimal 0-589. _ . 821.—It may be observed that Zin (26) and (27) would be equal to w, in case the dimensions of the beam and the amount of deflection were each made equal to unity; and in (28) equal to w divided by 0:589. That is, when in (26) the length is 1, the breadth 1, and the depth 1, then Z would be

Page 205

200 AMERICAN IIOUSE-CARPENTER. equal to the weight that would depress the beam from its ori ginal line equal to 1. Thus— the dimensions all taken in inches except the length, and this taken in feet. This is an extreme state of the case, for in most kinds of material this amount of depression would exceed the limits of elasticity ; and hence the rule would here fail to give the correct relation among the dimensions and pressure. For the law of deflection as above stated, (the deflection being equal for equal weights,) is true only while the depressions are small in comparison with the length. Nothing useful is, therefore, derived from this position of the question, except to give an idea of the nature of the quantity represented by the constant, Æ; it being in reality a measure of the stiffness of the kind of material used in comparing one material with another. Whatever may be the dimensions of the beam, Æ, calculated by (26,) will always be the same quantity for the same material; but when various materials are used, Æ will vary according to the flexibility or stiffness of each particular material. For example, Æ will be much greater for iron than for wood ; and again, among the various kinds of wood, it will be larger for the stiff woods than for those that are flexible. 322.—If the amount of deflection that would be proper in beams used in framing generally, (such as floor beams, girders and rafters,) were agreed upon, the rules would be shortened, and the labor of calculation abridged. Tredgold proposed to make the deflection in proportion to the length of the beam, and the amount at the rate of onefortieth of an inch (= 00253 inch) for every foot of length. He was undoubtedly right in the manner and probably so in the rate; yet, as this is a mat- ter of opinion, it were better perhaps to leave the rate of de- flection open for the decision of those who use the rules, and then it may be varied to suit the peculiarities of each case

Page 206

FRAMING. 201 that may arise. Any deflection within the limits of the elas- ticity of the material, may be given to beams used for some purposes, while others require to be restricted to that amount of deflection that shall not be perceptible to a casual observer. "Let 2 represent, in the decimal of an inch, the rate of deflec- tion per foot of the length of the beam ; then the product of n, multiplied by the number of feet contained in the length of the beam, will equal the total deflection, =n J. Now,ifnd be substituted for p in the formulas, (26,) (27) and (28,) they will be rendered more available for general use. For example, let this substitution be made in (26,) and there results— Ç w 2 w Engin iFa Pk) where / is in feet, and 6, d and n in inches; and for (27)— Cw Pw = eee 0) also for (28)— P w ? w E= ysg Dnt oss Dw Ct) where the notation is as before, with also s and D in inches. In these formulas, w represents the weight in pounds concen- trated at the middle of the length of the beam. If the weight, instead thereof, is equally distributed over the length of the beam, then, since $ of it concentrated at the middle will de- flect a beam to “he same depth that the whole eos when equally distributed, (Arz. 281,) therefore— J= b Ew (32.) we r (33.) gw f= 5559 Dw (34) where w equals the whole of the equally distributed load, „Again, if the load is borne by more beams than one, laid parallel to each other—as, for example, a series or tier of floor 26

Page 207

202 AMERICAN IOUSE-CARPENTER. beams—and the load is equally distributed over the supported surface or floor; then, if f represents the number of pounds ot the load contained on each square foot of the floor, or the pounds’ weight per foot superficial, and ¢ represents the dis- tance in feet between each two beams, or rather the distance from their centres, and ¿ the length of the beam in feet, in the clear, between the supports at the ends; then ¢ Z will equal the area of surface supported by one of the beams, and fel will represent the load borne by it, equally distributed over its length. Now, if this representation of the load be substituted for w in (32,) (33) and (84) there results— 2 s E= an = Yaw 63) pele. ofer Lo en (36.) i FELU fel’ L 0539 Dn = 0589 D’ nm Gr Practical Rules and Examples. 323.—To ascertain the weight, placed upon the middle of a beam, tliat will canse a given deflection. Rule XX.—Multiply the area of the cross-section of the beam by the square of the depth and by the rate of the deflec- tion, all in inches ¿multiply the product by the value of Æ, Table IT., and divide this product by the square of the length in feet, and the quotient will be the weight in pounds required. Example—What weight can be supported upon the middle of a Georgia pine girder, ten feet long, eight inches broad, and ten inches deep, the deflection limited to three-tenths of an inch, or at the rate of 0-03 of an inch per foot of the length ? Here the area equals 8 x 10 = 80; the square of the depth equals 10 x 10 = 100: 80 x 100 = 8,000; this by 0:03, the rate of deflection, the product is 240; and this by 2970, the valne of Æ for Georgia pine, Table I., equals 712,800. This

Page 208

FRAMING. 203 product, divided by 100, the square of the length, the quotient, 7,128, is the weight required in pounds. Rule XXI.—Where the beam is square the weight may be found by the preceding rule or by this :—Multiply the square of the area of the cross-section by the rate of deflection, both in inches, and the product by the value of Æ, Table II., and divide this product by the square of the length in feet, and the quotient will be the weight required in pounds. Example.—W hat weight placed on the middle of a spruce beam will deflect it seven-tenths of an inch, the beam being 20 feet long, 6 inches broad, and 6 inches deep? Here the area is 6 x 6 = 36, and its square is 36 x 36 = 1296; the rate of deflection is equal to the total deflection divided by the _o ~ 20 this by 1550, the value of Æ for spruce, Table II., equals 70,308. This, divided by 400, the square of the length, equals 175°77, the required weight in pounds. Kule XX1I.—When the beam is round find the weight by this rule :—Multiply the square of the diameter of the cross- section by the square of the diameter, and the product by the ' rate of deflection, all in inches, and this product by 0°589 times the value of Æ, Table II. This last product, divided by the square of the length in feet, will give the required weight length, = 0-0353 ; therefore, 1296 x 0:035 = 45°36, and in pounds. Example.—What weight on the middle of a. round white pine beam will cause a deflection of 0-028 of an inch per foot, the beam being 10 inches diameter and 20 feet long? The square of the diameter equals 10 x 10=100; 100 x 100 = 10,000; this by the rate, 0-028, = 280, and this by 0-589 x 1750, the value of Z, Table IJI., for white pine, equals 288,610. This last product, divided by 400, the square of the length, equals 721°5, the required weight in pounds, 324.—To ascertain the weight that will produce a given de

Page 209

20+ AMERICAN HOUSE-CARPENTER. flection, when the weight is equally distributed over the length of the beam. Rule XXIIL.-—The rules for this are the same as the three preeeding rules, with this modification, viz., instead of the square of the length, divide by five-eighths of the square of the length. 325.—In a series or tier of beams, to ascertain the weight per foot, equally distributed over the supported surface, that will canse a given deflection in the beam. Rule XXIV.—The rules for this are the same as Rules XX., XXI., and XXIL, with this modification, viz., instead of the square of the length, divide by the product of the distance apart in feet between each two beams, (measured from the centres of their breadths,) multiplied by five-eighths of the cube of the length, and the quotient will be the required weight in pounds that may be placed upon each superticial foot of the floor or other surface supported by the beams. In this and all the other rules, the weight of the material com- posing the beams, floor, and other parts of the constructions is understood to be a part of the load. Therefore from the ascer- tained weight deduct the weight of the framing, floor, plaster- ing, or other parts of the construction, and the remainder will be the neat load required. Excample.—tin a tier of white pine beams, 4 x 12 inches, 20 feet long, placed 16 inches or 14 feet from centres, what weight per foot superficial may be equally distribnted over the floor covering said beams—the rate of deflection to be not more than 0-025 of an inch per foot of the length of the beams. Proceeding by Rule XX. as above modified, the area of the cross-section, 4 x 12, eqnals 48; this by 144, the square of the depth, eqnals 6912, and this by 0-025, the rate of deflection, equals 172°8. Then this product, multiplied by 1750, the value of Æ, Table II., for white pine, eqnals 302,400. The distance between the centres of the beams is 14 feet, the eube

Page 210

FRAMING. 208 of the length is 8,000, and $ by & of 8,000 equals 6,6663. The above 302,400, divided by 6,6662, the quotient, 45°36, equals the required weight in pounds per foot superficial. The weight of beams, floor plank, cross-furring, and plastering oc- - curring under every square foot of the surface of the floor, is now to be ascertained. Of the timber in every 16 inches by 12 inches, there occurs 4 x 12 inches, one -foot long; this equals one-third of a cubic foot. Now, by proportion, if 16 inches in width contains $ of a cubic foot, what will 12 inches n : 1x12 12 : ? OCO =~ = 2 = e in width contain ? ig “3x16 3 = 4 of p cubic foot The floor plank (Georgia pine) is 12 x 12 inches, and 14 inches thick, equal to z of a cubic foot, equals i equals y. Of the furring strips, 1 x 2 inches, placed 12 inches from centres, there will occur one of a foot long in every superficial foot. Now, since in a cubic foot there is 144 rods, one inch square and one foot long, therefore, this furring strip, 1 x 2 x 12 inches, equals ;2; = 7 of a cubic foot. The weight of the timber and furring strips, being of white pine, may be esti- mated together: } + 73 = 44 + qz =13 of a cubic foot. White pine varies from 23 to 30 pounds. If it be taken at 30 pounds, the beam and furring together will weigh 30 x 48 pounds, equals 7-92 pounds. Georgia pine may be taken at 50 pounds per cubic foot ;* the weight of the floor plank, then, is 50 x ` ap = 5°21 pounds. A superficial foot of lath and plastering ` will weigh about 10 Ibs. Thus, the white pine, 7°92, Georgia v pine, 5:21, and the plastering, 10, together equal 23°13 pounds ; this from 45°36, as before ascertained, leaves 22°23, say 224 pounds, the neat weight per foot superficial that may be equally distributed over the floor as its load. * To get the weight of wood or any other material, multiply its specific gravity by 62°5, For Spe cific Gravriies see Tables I., II., and TIT. and the Appendix for Weight of Materials.

Page 211

206 AMERICAN HOUSE-CARPENTER. 826.—To ascertain the weight when the beam is laid not horizontal, but inclined. Rule XXV.—In each of the foregoing rules, multiply the result there obtained by the length in feet, and divide the pro- duet by the horizontal distanee between the supports in feet, and the quotient will be the required weight in pounds. The foregoing Rules, stated algebraically, are placed in the following table :— r TABLE IV.—STIFFNESS OF BEAMS ; WEIGHT. When the beam is When the z pea Eled When the weight is a acilen Square. Round. (88.) (39.) (40.) TIorizontal {Concentrated at middle, w, in pounds, equals Bee = ee (41.) (42.) (43.) Enba E Horizontal Equally distributed, w, in pounds, equals = Re Í- Lei, ce 78 (44) (45.) (46.) Enba: | Ens | 9424 En Di Horizontal |By the foot superficial, 4 in pounds, equals Ser | ser a (47.) (48.) | (49.) Enbæ | Ena 599 En DA Inclining |Concentrated at middle, 2, in pounds, equals T Th E 0.) | 6L) (52.) Enbd: | Ens! | 9494 En Dt Inclining (Equally distributed, w, iu pounds, equals |) Sih gih th 63) | (4) (65.) Enba: | Ens | 9424 Bn Di Inclining By the foot superficial, f in pounds, equals ohh | Rel meh In the above table, b equals the breadth, and d equals the depth of cross-section of beam 3's equals the breadth of a side of a square beam, and D equals the diameter of a round beam ; 7 equals the rate of deflection per foot of the length ;

Page 212

FRAMING. 207 D, s, b, d and v, all in inches; Z equals the length, ¢ equals the distance between two parallel beams medsured from the centres of their breadth; % equals the horizontal distance between the supports of an inclined beam; Z, ¢ and A in feet ; w equals the weight in pounds on the beam; f equals the weight upon each superficial foot of a floor or roof supported by two or more beams laid parallel and at equal distances apart; Fis a constant, the value of which is found in Table II.; r is any decimal, chosen at pleasure, in proportion to unity, as b is to d, from which proportion 6 equals d r. 327.—To ascertain the dimensions of the cross-section of a beam to support the required weight with a given deflection. Rule XXVI.—Preliminary. When the weight is concen- trated at the middle of the length. Multiply the weight in pounds by the square of the length in feet, and divide the pro- duct by the product of the rate of deflection multiplied by the . value of Æ., Table II., and the quotient equals a quantity which mmay be represented by M—referred to in succeeding rules. w 2 i M. (56.) Rule XXVII.. Preliminary. When the weight is equally distributed over the length. Multiply five-eighths of the weight in pounds by the square of the length in feet, and divide the product by the rate of deflection multiplied by the value of EZ, Table II., and the quotient equals a quantity which may be represented by V—referred to in succeeding rules. “Ba ~T (57.) Rule XXVIUI.—Preliminary. When the weight is given per foot superficial and supported by two or more beams. Multiply the distance apart between two of the beams, (mea- sured from the centres of their breadth,} by the cube of the length, both in feet, and muitiply the product by five-eighths of the weight per foot superficial; divide this product by the

Page 213

208 AMERICAN HOUSE-CARPENTER. product of the rate of deflection, multiplied by the value of # Table II., and the quotient equals a quantity which may be represented by U—referred to in succeeding rules. ie = U. (58.) Rule XXIX.—Preliminary. When the beam is laid not horizontal, but inclining. In Rules XXVI. and XXVIL, instead of the square of the length multiply by the length, and by the horizontal distance between the supports, in feet. And in Rule XXVIII., instead of the cube of the length, multiply by the square of the length, and by the horizontal distance between the supports, in feet. From (56) ue = M,. (59.) From (57) wlhh 3 En = N, e (60.) From (58) sf eee Sle (61.) Lule XXX.— When the beam is rectangular to find the dimensions of the cross-section. Divide the quantity repre- sented by M, N or U, (in preceding preliminary rules,) by the breadth in inches, and the cube root of the quotient will equal the required depth in inches. Or, divide the quantity represented by M, V or U, by the cube of the depth in inches, and the quotient will equal the required breadth in inches. Or, again, if it be desired to have the breadth and depth in proportion, as 7 is to unity, (where 7 equals any required deci- mal,) divide the quantity represented by M, W or U, by the value of 7, and extract the square root of the quotient: and the square root extracted the second time, will equal the depth in inches. Multiply the depth thus found by the value of 7, and the product will equal the breadth in inches.

Page 214

FRAMING. 209 Example.—To find the depth. A beam of spruce, laid on supports with a clear bearing of 20 feet, is required to support a load of 1674 pounds at the middle, and the deflection not to exceed 0°05 of an inch per foot; what must be the depth when the breadth is 5 inchés. By Rule XXVI. for load at middle: the product of 1674, the weight, by 400, the square of the length, equals 669,600. The product of 0:05, the rate of de- flection, multiplied by 1550, the value of Æ, from Table Il., set opposite spruce, is 77°5. The aforesaid product, 669,600, divided by 77°5, equals 8640, the value of M. Then by Rule XXX., 8640, the value of Jf, divided by 5, the breadth, the quotient is 1728, and 12, the cube root of this, found in the table of the Appendix, equals the required depth in inches. Ecample.—To find the breadth. Suppose that in the last example it were required to have the depth 13 inches; in that case what must be the breadth? The value of M, 8640, as just found, divided by 2197, the cube of the depth, equals 3:9326, the required breadth—nearly 4 inches. Example.—To-find both breadth and depth, and in a certain proportion. Suppose, in the above example, that neither the breadth nor the depth are given, but that they are desired to be in proportion as 0°5 is to 1°0. Now, having ascertained the value of M, by Rule XXVI., to be 8640, as above, then, by Rule XXX., 8640, divided by 0°5, the ratio, gives for quotient 17,280. The-square root of this (by the table in the Appen- dix,) is 131-45, and the square root of this square root is 11-465, the required depth. The breadth equals 11:465 x 0°5, which ' equals 5°7325. The depth and breadth may be 11! by 52 inches. In cases where the load is equally distributed over the length of the beam, the process is precisely the same as set forth in the three preceding examples, except that five-cightiis of the weight is to be used in place of the whole weight; and hence it would be a useless repetition to give examples to illustrate such cases, 27

Page 215

210 AMERICAN IOUSE-CARPENTER. Example.— When the weight is per foot superficial to find the depth. A floor is to be constructed to support 500 pounds on every superficial foot of its surface. The beams to be ot white pine, 16 feet long in the clear of the supports or walls, placed 16 inches apart, from centres, to be 4 inches thick, and the amount of deflection not objectionable provided it be within the limits of elasticity. Proceeding by Rule XXVIIL, the produet of 14 feet, (equal to 16 inches,) multiplied by 4096, the cube of the length, equals 54614. This, multiplied by 312-5, (equal to # of the weight,) equals 1,706,666. The largest rate of deflection within the limits of the elasticity of white pine is 0:1022, as per Table II. This, multiplied by 1750, the value of F for white pine, Table IL, equals 178°85. The former product, 1,706,666, divided by the latter, 178°85, equals 9,542°5, the valne of U. Now, by Rule XXX., this value of U, 9,549°5, divided by 4, the breadth, equals 2885-6, the cube root of which, 18:362, is the required depth—nearly 13% inches. Example.—To find the breadth. Suppose, in the last exam- ple, that the depth is known but not the breadth, and that the depth is to be 13 inches. Having found the value of JU, as before, to be 9542°5, then by Rule XXX., dividing 9542°5, the value of U, by 2197, the cube of the depth, gives a quotient of 43434 and this equals the breadth—nearly 4% inches. Example.—To find the depth and breadth in a given propor- tion. Suppose, in the above example, that the breadth and depth are both unknown, and that it is desired to have them ih proportion as 0°7 is to 1:0. Having found the value of U, as before, to be 9542-5, then by Rule XXX., dividing 9542°5, the value of U, by 0-7, the quotient is 13,632, the square root of which is 116-75, and the square root of this is 10°805, the depth in inches. Then 10-805, multiplied by 0*7, the product, 75635, is the breadth in inches. The size may be Ts by 10:3 inches. 828.-- Example—In the case of inclined berms to find the

Page 216

FRAMING. 4 tepth. A beam of white pine, 10 feet long in the clear of the bearing, and laid at such an inclination that the horizontal distance between the supports is 9 feet, is required to support 12,000 pounds at the centre of its length, with the greatest allowable deflection within the limits of elasticity ; what must be its depth when its breadth is fixed at 6 inches? By refer- ence to Table II. it is seen that the greatest value of. n, Within the limits of elasticity, is 0°1022. By Rule XAVLI, for con- centrated load, and Rule XXIX., for inclined beams, 12,000, the weight, multiplied by 10, the length, and by 9, the hori- zontal distance, equals 1,080,000. The product of 0:1022, the greatest rate of deflection, by 1750, the value of Æ, Table II., for white pine, equals 178°85. “Dividing 1,080,000 by 178-85, the quotient is 6038-58, the value of M. Now, by Rule XXX, for rectangular beams, 6038°58, the value of J, divid- ed by 6, the breadth, the quotient is 1006-43. The cube root of this, 10-02, a trifle over 10 inches, is the depth required. Example.—In case of inclined beams to find the breadth. In the last example suppose the depth fixed at 12 inches; then by Rule XXX., 6038°58, the value of M, as above found, divided by 1728, the cube of the depth, equals 3:4945, or nearly 34 inches—tie breadth required. i Example.—Again, in case the breadth and depth are to be in a certain proportion; as, for example, as 0-4 is to mnity. Then by Rule XXX., 6038:58, the value of M, found as above, divided by 0°4, equals 15,096°45, the square root of which is 122-87, and the square root of this square root is 11-0843, a trifle over 11 inches—the depth required. Again, 11 multi-. plied by the decimal 0-4, (as above,) equals 4-4, a little over 4% inches—the breadth required. In the three preceding examples, the weight is ar to be concentrated at the middle. If, however, the weight had been equally distributed, the same process would have been used to obtain the dimensions of the cross-section, with

Page 217

212 AMERICAN HOUSE-CARPENTER. only one exception ; viz. % of the weight instead of the whole weight would have been nsed. (See Rule XXVIL) Fxample.—In case of inclined beams ; the weight per foot superficial, and borne by two or more beams. A tier of spruce beams, laid with a clear bearing of 10 feet, and at 20 inches apart from centres, and laid so inclining that the horizontal distance between bearings is 8 feet, are required to sustain 40 pounds per superficial foot, with a deflection not to exceed 0-02 inch per foot of the length; what must be the depth when the breadth is 3 inches? Proceeding by Rule XXIX. for inclined beams, and by Rule XXVIII., 1%, (= 20 inches,) the distance from centres, multiplied by 100, the square of the length, and by 8, the horizontal distance between bearings, equals 1,38384; this, by § x 40, five-eighths of the weight, equals 33,3334. This, divided by 0:02 x 1550, the rate of deflection, by the value of Æ, Table H., for spruce, equal to 31, equals 1075-27, the value of U. Now by Rule XXX. for rectangular beams, 107527, divided by 3, the breadth, equals 358°42, the cube root of which, 7-1, is the required depth in inches. Example.—The same as the preceding; but to find the breadth, when the depth is fixed at 8 inches. By Rule XXX., 1075-27, the value of U, divided by 512, the cube of the depth, equals 2°1—the breadth required in inches. Lxample.—The same as the neat but one preceding ; but to Jind the breadth and depth in the proportion of 0°8 to 1:0, or 8 to 10. By Rule XXX., 1075-27, the value of U, divided by 0-3, the value of r, equals 3584-28. The square root of this is 59°869, and the square root of this square root is 7°737—the depth required in inches. This 7:737, multiplied by 0°3, the value of 7, equals 2°3211—the required breadth in inches. The dimensions may, therefore, be 23; by 72 inches. Rule XXXI.— When the. beam is square to find the side. Extract the square root of the quantity represented by Jf, WV

Page 218

Franch 218 or U, in preliminary Rules XXVI, XXVII. and XXVIL, and the square root of this square root will equal the side required. ' Example.—aA beam of chestnut, having a clear bearing of 8 feet, is required to sustain at the middle a load of 1500 pounds; what must be the size of its sides in order that the deflection shall not exceed 0-03 inch per foot of its length? By Rule XXVI., 1500, the load, multiplied by 64, the square of the length, equals 96,000. This product divided by 0:03 times 2330, the value of Æ, Table I., for chestnut, gives a quotient of 1373-4, the quantity represented by M. Now by Rule. XXXI., the square root of 1373-4 is ‘87-05, and the square root of this is 6°087. The beam must, therefore, be 6 inches ‘square. In this example, had the load, instead of being con- centrated at the middle, been equally distributed over its length, the side would have been equal to the side just found, multiplied by the fourth root of 2 or of 0°625, equal to 6°087 x 0°889 = 5-4 inches. (See Rules XXVI. and XXX1.) Example.—In the case where the weight is per foot superfi- cial and borne by two or more beams. A floor, the beams of which are of oak, and placed 20 inches or 12 feet apart from centres, and which have a clear bearing of 20 feet, is required to sustain 200 pounds per superficial foot, the deflection not to exceed 0°025 inch per foot of the length, and the beam to be square. By Rule XXVIIL, 1%, the distance from centres, multiplied by 8000, the cube of the length, equals 18,3338 ; and this by 125, (being & of 200 pounds,) equals 1,666,666. Dividing this by 0-025 times 2520, the value of Æ, Table it. for oak, the quotient is 26,455—a number represented by J. Now by Rule XXXI., the square root of this number is 162°65. and the square root of this square root is 12°753—the required side. The beam may be 122 inches square. Example.—Inclined square beams, load at middle. A bar of cast-iron, 6 feet long in the clear of bearings, and laid

Page 219

O14 AMERICAN HOUSE-CARPENTER. inclining so that the horizontal distance between the bearings is 5 feet, is required to sustain at the middle 8000 pounds, and the deflection not to exceed 0:01 inch per foot of its length ; what must be the size of its sides? By Rule XXVI. for load at middle, modified by Rule XXIX. for inclined beams; 3000, the weight, multiplied by 6, the length, and by 5, the horizontal distance between bear- ings, equals 90,000. The rate of deflection, 0-01, by 30,500, the value of Æ, Table II., for cast-iron, equals 305 ; and 9000 divided by 305, equals 295-082, the value of M. Now by Rule XXXI. for square beams, the square root of 295-082 is 17-18, the square root of which is 4:145—the size of the side required ; a trifle over 4%, the bar may, therefore, be 44 inches square. Example.—Same as preceding, but the weeghé equally distri- buted. By Rule XX VII. 3 of the weight is to be used instead of the weight; therefore 295-082, the value of M, as above, multiplied by 8, will equal 184°426, the value of W. By Rule XXXI. the square root of 184°426 is 13°58, the square root of which is 3°685—the size of the side required ; equal to nearly 312 inches square. Example.—Same as preceding case, but the weight per foot superficial, and sustained by 2 or more bars, placed 2 feet from centres, the load being 250 pounds per foot superficial. By Rule XXVIII., modified by Rule XXIX., the distance from centres, 2, multiplied by 86, the square of the length, and by 5, the horizontal distance, equals 860. This by 156-25, five-eighths of the weight, equals 56,250. The rate of deflec- tion, 0°01, by 30,500, the value of Æ, Table II., for cast-iron, equals 805. The above 56,250, divided by 305, equals 184:426, the value of Ọ. Now by Rule XXXI. the square root of 184426, the value of U, is 13:58, the square root of which is 3°685—the size of the side required. It will be observed that this result is precisely like that in the last example. This is as it should be, for each beam has to sustain the weight on 2 x €

Page 220

Ld FRAMING. 215 =: 12 superficial feet, equal to 12 x 250, equal 3(00 pounds; and all tle other conditions are parallel. Rule XXXIL— When the beam is round to find the diame- ter. Divide the value of M, W or U, found by Rules XXVI., XXVII. or XXVIIL, by the decimal 0:589, and extract the square root: and the square root of this square root will be the diameter required. ° Example.—In the case of a concentrated load at middle. A round bar of American iron, of 5 feet clear bearing, is required to sustain 800 pounds at the middle, with a deflection not to exceed 0°02 inch per foot; what must be its diameter? By Rule XXVI. for load at middle, 800, the weight, multiplied by 25, the square of the length, equals 20,000. The rate of deflection, 0°02, by 51,400, the value of Æ, Table II., for Ame- rican wrought iron, equals 1028. The above 20,000, divided by 1028, equals 19:4552, the value of M. Now, by Rule XXXII., 19°4552, the valme of M, divided by 0-589 equals 33°03, the square root of which is 5-747, and the square root of this is 2°397, nearly 2*4, the diameter required in inches, equal to 28 large. Example.—Same case as the preceding, but the load equally distributed. By Rule XXVII, five-eighths of the weight is to be used instead of the whole weight; therefore the above 33°03, multiplied by 8, equals 20°64375, the square root of which is 4°544, and the square root of this square root is 2:182, the diameter required in inches, 24 inches large. Example.— When the weight ts per foot superficial, and sus- tained by two or more bars or beams. The conditions being the same as in the preceding examples, but the weight, 100 pounds per foot, is to be sustained on a series of round rods, placed 18 inches apart from centres, equal 1°5 feet. By Rule XXVIII., for weight per foot superficial, 1:5, the distance from centres, multiplied by 125, the cube of the length, and by 62:5, five-eighths of the weight, equals 11,718°75. This

Page 221

216 AMERICAN HOUSE-CARPENTER. divided by 1028, the product of the rate of deflection by the value of Æ, as found in the preceding example, equals 11:4, the value of U. Now by Rule XXVII., 11-4, the value of U, divided by 0:589, equals 19°42, the square root of which is 4-407, and the square root of this square root is 2°099, the diameter required—very nearly 2;'; inches. Example.— When the beam is round and laid inclining, the weight concentrated at the middle. A round beam of white pine, 20 feet long between bearings, and laid inclining so that the horizontal distance between bearings is 18 feet, is required to support 1250 pounds at the middle, with a deflection not to exceed 0°05 inch per foot; what must be its diameter? By Rule XXVI. for load at middle, modified by Rule XXIX. for inclined beams, 1250, the weight, multiplied by 20, the length, and by 18, the horizontal distance, equals 450,000. The rate of deflection, 0:05, multiplied by 1750, the value of Æ, Table IIL., for White pine, equals 87:5. The above 450,000 divided by 87:5, equals 5142°86, the value of M. Now by Rule XXXII. for round beams, 5142:86, the value of M, divided by 0:589, equals 8731°5, the square root of which is 93-44, and the square root of this square root is 9:667, the diameter re- quired—equal to 92 inches. Example.—Same as in preceding example, but the weight equally distributed. By Rule XXVIL., five-eighths of the weight is to be used instead of the whole weight, therefore 8731-5, the result in the last example just previous to taking the square root, multiplied by #, equals 5457-2, the square root of which is 73°87, and the square root of this square root is 8:59, the diameter required—nearly 8% inches. Example.—Same as in the next but one preceding example, but the weight per foot superficial, and supported by two or more beams. A series of round hemlock poles or beams, 10 fect long clear bearing, laid inclining so as that the horizontal distance between the supports equals 7 feet, and laid 2 feet

Page 222

FRAMING. 217 and 6 inches apart from centres, are required to support 20 pounds per superficial foot without regard to the amount o deflection, provided that the elasticity of the material be not injured; what must be their diameter? By Rule XXVUL for weight per foot superficial, modified by Rule XXIX. for inclined beams, 2°5, the distance from centres, multiplied by 100, the square of the length, and by 7, the horizontal distance between bearings, and by five-eighths of the weight, 12°5, equals 21,875. The greatest value of n, Table II., for hem- . lock, 0°08794, multiplied by 1240, the value-of Æ, Table II., for hemlock, equals 109-0456. The above 21,875, divided ~ 109-0456, equals 200:6, the value of U. Now by Rule XXXII, the above 200:6, divided by 0:589, equals 340°6, the square root of which is 18°46, and the square root of this square root is 4'296, the diameter required—equal to 4,5 inches nearly. 329.—The greater the depth of a beam in geopuntion to the thickness, the greater the strength. But when the difference between the depth and the breadth is great, the beam must be stayed, (as at Fig. 228,) to prevent its falling over and break- ing sideways. Their shrinking is another objection to deep beams; but where these evils can be remedied, the advantage of increasing the depth is considerable. The following rule is, to find the strongest form for a beam out of a given quantity of timber. Liule.—Multiply the length in feet by the decimal, 0°6, and divide the given area in inches by the product; and the square of the quotient will give the depth in inches. Example.—What is the strongest form for a beam whose - given area of section is 48 inches, and length of bearing 20 feet? The length in.feet, 20, multiplied by the decimal, 0°6, gives 12; the given area in inches, 48, divided by 12, gives a quotient of 4, the square of which is 16—this is the depth in inches; and the breadth must be 3 inches. A beam 16 inches 28 ‘

Page 223

218 AMERICAN HOUSE-CARPENTER. by 3 would bear twice as much as a square beam of the same drea of section; which shows how important it is to make beams deep and thin. In many old buildings, and even in - new ones, in country places, the very reverse of this has beer practised; the principal beams being oftener laid on the broad side than on the narrower one. The foregoing rules, stated algebraically, are plaved in the following table. TABLE V.—STIFFNESS OF BEAMS, DIMENSIONS. eee Rectangular. Square. Round. phe | When the weight is |——__ ae ees Se pees Is laid Valne of Value of |When?=dr,| Value ofa |Valne of the depth. breadth. jvalne of d. side. diameter | Se SS ee) ee ee es eee (62.) (63.) (64.) (65.) (66.) yur wl 7 wl- i fw tt $/ wht Concentrated at middle BAT End ve an ve 59 En 3 7 : : = (67.) (68 ! (69.) , (70.) : (71.) N 6 36 wl! ss wk aJ wl 1/4 wh 47 wht E Equally distributed ae End V EnF = V 0124 En 72.) (73) (74.) (75.) (76.) 2/56 fe li 5s febs 4/756 fel | 275 feB Sc By the foot eapertitat | Vr | aS | VEE | VEE Vara (77.) (78.) (79.) (80.) (81.) Sfwih wth s7oth velh | wth Concentrated at middle End End Enr En 539 En 2 (82.) (83.) (84.) (85.) (86.) 2 ywlh 56 wlh 5 wth | s/s wlh jy wth 8 Equally distributed Vie: A E tae IT Te 0494 Ea t (87.) (88.) (89.) (90.) (91.) oft elth Bfekh| 4/fetth Jch By the foot superficial Vžrel is fer nd o = ve En V 9494 En In the above table, b equals the breadth, and d the depth of eross-section of beam; 2 equals the rate of deflection per foot of the length ; ġ, d and n, all in inches. Also, Z equals tlie length, e the distance between two parallel beams measured from the

Page 224

FRAMING. 219 centres of their breadth, and A equals the horizontal distance between the supports of an inclined beam; Z, c and A, all in feet. Again, w equals the weight on the beam, f equals the weight upon each superficial foot of a floor or roof, supported by two or more beams laid parallel and at equal distances apart; w and f in pounds. And 7 is any decimal, chosen at pleasure, in proportion to unity, as 6 is to d—from which pro- portion b=d r. Lis a constant the value of which is found in Table II. . 330.—To ascertain the scantling of the stiffest beam that can be cut from a cylinder. Let d a c b, (Fig. 223,) be the section, and e the centre, of a given cylinder. Draw’the diameter, ab; upon a and b, with the radius of the section, describe the ares, d e and ec; join d and a, a and c, c and b, and b and d; then the rectangle, d a c b, will be a section of the beam required. ò b Fig. 223. 3881.—Resistance to Rupture.—The resistance to deflection having been treated of in the preceding articles, it now re- mains to speak of the other branch of resistance to cross strains, namely, the resistance to rupture. When a beam is laid horizontally and supported at each end, its strength to resist a cross strain, caused by a weight or vertical pressure at the middle of its length, is directly as the breadth and square of the depth and inversely as the length. If the beam is square, or the depth equal to the breadth, then the strength is directly

Page 225

220 AMERICAN JIOUSE-CARPENTER. as the cube of a side of the beam and inversely as the length, and if the beam is round the strength is directly as the cube of the diameter and inversely as the length. When the weight is concentrated at any point in the length, the strength of the beam is directly as the length, breadth, and square of the depth, and inversely as the product of the two parts into which the length is divided by the point at which the weight is located. When the beam is laid not horizontal but inclining, the strength is the same as in each case above stated, and also in proportion, inversely as the cosine of the angle of inclination with the horizon, or, which is the same thing, directly as the length and inversely as the horizontal distance between the points of support. When the weight is equally diffused over the length of a beam, it will sustain just twice the weight that could be sus- tained at the middle of its length. A beam secured at one end only, will sustain at the other end just one-quarter of the weight that could be sustained at its middle were the beam supported at each end. These relations between the strain and the strength exist in all materials. For any particular kind of material, 1 Pr = 8; (92.) S, representing a constant quantity for all materials of like strength. The superior strength of one kind of material over another is ascertained by experiment; the value of S being ascertained by a substitution of the dimensions of the piece tried for the symbols in the above formula. Having thus obtained the value of ©, the formula, by proper inversion, becomes use- ful in ascertaining the dimensions of a beam that will require a certain weight to break it; or to ascertain the weight that will be required to break a certain beam. It will be observed in the preceding formula, that if each of the dimensions of the

Page 226

FRAMING. l 221 beam equal unity, then w = ©. Hence, S is equal to the weight required to break a beam one inch square and one foot long. The values of S, for various materials, have been ascertained from experiment, and are here recorded :— TABLE VI.—STRENGTH. Number of Materials. Value of J. Experiments. Green plate-glass . . e o «© àe 178 4 ruce i RS o e 845 5 emlock . . 863 T Soft white pine 890 9 Hard white pine . 1 Ohio yellow pine 2 Chestnut . . . . 9 e e 508 2 Georsiapino o eE E on 510 7 Oak (ar ee a es 574 2 BSN og a N a e, , 742 2 Cast-iron (from 1550 to 2280). e . « 1926 29 The specimens broken were of various dimensions, from one foot long to three feet, and from one inch square to one by three inches. The cast-iron specimens were of the various kinds of iron used in this country in the mechanic arts. S may be taken at 2,000 for a good quality of cast-iron. It is usual in determining the dimensions of a beam to suppose it capable of sustaining safely one-third of the breaking weight, and yet Tredgold asserts that one-fifth of the breaking weight will in time injure the beam so as to give it a permanent set or bend, and Hodgkinson says that cast-iron is injured by any weight however small, or, in other words, that it has no elastic power. However this may be, experience has proved cast- iron quite reliable in sustaining safely immense weights for a long period. Practice has shown that beams will sustain safely from one-third to one-sixth of their breaking weight. If the load is leid on quietly, and is to remain where laid, at rest, beams may be trusted with one-third of their breaking weight, but if the load is moveable, or subject to vibration,

Page 227

299 AMERICAN HOUSE-CARPENTER @ one-quarter, one-fifth, or even, in some cases, one-sixth is quite a suflicient proportion of the breaking load. 332.—The dimensions of beams should be ascertained only by means of the rules for the stiffness of materials, (Arts. 320. 323, et scq.,) as these rules show more accurately the amount of pressure the material is capable of sustaining without injury. Yet owing to the fact that the rules for the strength of mate- rials are somewhat shorter, they are more frequently used than those for the stifness of materials. In order that the proportion of the breaking weight may be adjusted to suit cir- cumstances it is well to introduce into the formula a symbol to represent it. The proportion represented by the symbol may then be varied at discretion. Let this symbol be a; a decimal in proportion to unity as the safe load is to the break- ing load, then S a will equal the safe load. Hence, Sa bE L - (93) for a safe load at middle on a horizontal beam supported at both ends; and 2 Sabl w = —— l for a safe load eqnally diffused over the length of the beam: and (94.) Do R (95) for the load, per superficial foot, that can be sustained safely upon a floor supported by two or more beams, ¢ being the dis- tance in feet from centres between each two beams, and f the load in pounds per superficial foot of the floor. Generally, in (93,) (94,) and (95,) w equals the load in pounds; S, a constant, the value of which is found in Table VI.; æ a decimal, in pro- portion to unity as the safe load is to the breaking load; 2 the length in feet between the bearings; and b and d the breadth and depth in inches.

Page 228

FRAMING. 233 TO FIND THE WEIGE Tf. 333.—The formulas for ascertaining the weight in the seve- ral cases are arranged in the following table, where c, f, w, S, a, l b and d represent as above; and also s equals a side of a square beam; D equals the diameter of a cylindrical beam ; m and n equal respectively the two parts into which the length is divided by the point at which the weight is located ; and A equals the horizontal distance between the supports of an inclined beam. TABLE VII. STRENGTH OF BEAMS; SAFE WEIGHT. When the beam is When the beim (aia When the weight is : Rectangular, Square. Ronnd, Concentrated at middle, 2, i pi au 4 oncentrated at middle, 2, in p g , e equals a W, sa ses 589 D? Sa (99.) (100.) (101,) Equally distributed, w, in) 9saba@ 2 Sae 1178 D8 Sa 4 pounds, equals == i ee l 8 = ed 5 (102.) (108.) (104) By the foot superficial, j in 2 Sabda 2 Sag 1178 D3 Sa pounds, equals eR cl: ce (105.) (106.) (107.) Concentrated at any point in Sabdi Sals 147 D3 Sat the length, 2, in pounds, equals “imn e D ae (108.) (109.) (110.) Concentrated at middie, w, in Sab@ Sas 589 DS Sa pounds, equals >t = =I m j ail.) (112.) (113.) Equally distributed, w, in 28aba 2S ae? 1178 D: Sa Ei pounds, equals - - >; a oe 3 ae rr o l M a * (14) (115.) (116.) By the foot snperficial, Z in) 2Sabd? 25a % 1178 D3 Sa ponnds, equals — ohl Tehi che (117.) (118.) (119.) Concentrated at any point in| Babe P ,. Sate 147 D Sark j length, w, in ponnds, “ye ~4hmn- 4hmn hms

Page 229

224 AMERICAN HOUSE-CARPENTER. Practical Rules and Examples. Rule XXXIII.—To find the weight that may be supportea safely at the middle of a beam laid horizontally. Multiply the value of S, Table VI., by a decimal that is in proportion to unity as the safe weight is to the breaking weight, and divide the product by the length in feet. Then, if the beam is rectangular, multiply this quotient by the breadth and by the square of the depth, and the product will be the required weight in pounds; or, if the beam is square, multiply the said quotient, instead, by the cube of a side of the beam and the product will be the required weight in pounds; but, if the beam is round, multiply the aforesaid quotient, instead, by 589 times the cube of the diameter, and the product will be the required weight in pounds. Feample.—What weight will a rectangular white pine beam, 20 feet long, and 3 by 10 inches, sustain safely at the middle, the portion of the breaking weight allowable being 0:3? By the above rule, 390, the value of S for white pine, Table VI., multiplied by 0'3, the decimal referred to, equals 117, and this divided by 20, the length, the quotient is 5'85. Now the beam being rectangular, this quotient multiplied by 3 and by 100, the breadth and the square of the depth, the product, 1755, is the desired weight in pounds. Example.—lf the above beam had been square, and 6 by 6 inches, then the quotient, 5°85, multiplied by 216, the cube of 6, a side, the product, 1263°6, is the weight required in pounds. Example.—If the above beam had been round, and 6 inches diameter, then the above quotient, 5°85, multiplied by -589 times 216, the cube of the diameter, the product, 744°26, would be the required weight in pounds. Rule XXXIV.—To find the weight that may be supported safely when equally distributed over the length of a beam, lait horizontally. Multiply the result obtained, by Rule

Page 230

FRAMING. 995 XXXIIL, by 2, and the Gak will ie the required weight in Baile Example.—In the example, under Rale XXXII, the safo weight at middle of rectangular beam is found to be 1755 pounds. This multiplied by 2, the product, 3510, is the weight the beam will bear safely if equally distributed over its length. Example.—So in the case of the sguare beam, 2527:2 pounds is the weight, equally distributed, that may be safely sus- tained. Example.—And for ae round beam 1488-52 is the required weight. Rule XXXV.—To ascertain the weight per superficial foot that may be safely sustained on a floor resting on two or more ` beams laid horizontally and parallel. Multiply twice the value of S, Table VI., by the decimal that is in proportion to unity, as the safe weight is to the breaking weight, and divide tl e product by the square of the length, in feet, multiplied by the distance apart, in feet, between the beams measured from their centres. Now, if the beams are rectangular, multiply this quotient by the breadth and by the square of the depth, both in inches, and the product will be the required weight in pounds; or if the beams are square, multiply said quotient, instead, by the cube of a side of a beam and the product will be the required weight in pounds. Bnt if the beams are round, multiply the aforesaid quotient, instead, by ‘589 times ` the cube of the diameter, and the product will be the weight required in pounds. Example.—What weight may be safely sustained on each foot superficial of a floor resting on spruce beams, 10 feet long, 3°by 9 inches, placed 16 inches, or 14 feet, from centres: the portion of the breaking weight allowable being 0:25? By the Rule, 690, twice the value of spruce, Table VI., multiplied by 0-25, the decimal aforesaid, equals 172-5. This product divided by 100, the square of the length, multiplied by 14, the distance 29

Page 231

226 AMERICAN HOUSE-CARPENTER. from centres, equals 1:294. Now this quotient multiplied by 3, the breadth, and by 81, the square of the depth, the product 31444 is the required weight in pounds. Had these beams been square, and 6 by 6 inches, the re- quired weight would be 279°5 pounds. Or, if round, and 6 inches diameter, 16463 pounds. Rule XXXVI.—To ascertain the weight that may be sus- tained safely on a beam when concentrated at any point of its length. Multiply the value of S, Table VI., by the decimal in proportion to unity, as the safe weight is to the breaking weight, and by the length in feet, and divide the product by four times the product of the two parts, in feet, into which the ` length is divided, by the point at which the weight is concen- . trated. Then, if the beam is rectangular, multiply this quo- tient by the breadth and by the cube of the depth, both in inches, and the product will be the required weight in pounds. Or, if the beam is square, multiply the said quotient, instead, by the cube of a side of the beam, and the product will be the required weight in pounds. But if the beam is round, multi- ply the aforesaid quotient by °589 times the cube of the dia- meter, and the product will be the weight required. ° Excample-—What weight may be safely supported on a Georgia pine beam, 5 by 12 inches, and 20 feet long; the weight placed at 5 feet from one end, and the proportion of the break- ing weight allowable being 0-2? By the rule, 510, the value of S for Georgia pine, Table VI., multiplied by 0-2, the decimal referred to, equals 102; this by 20, the length, equals 2040.; this divided by 800, (= 4 x 5 x 15,) or 4 times the product of the two parts into which the length is divided by the point at which the weight is located, equals 6:8. The beam being rect- angular, this quotient multiplied by 5, the breadth, and by 144, the square of the depth, equals 4896, the required weight. A beam, 8 inches square, other conditions being the same as in the preceding case, would sustain safely 3481-6 pounds.

Page 232

FRAMING. 227 And a round beam, 8 inches diameter, will sustain safely, under like conditions, 2050-66 pounds. Rule XXXVIL—To find the weight that may be safely sustained on inclined beams. Multiply the result found for horizontal beams in preceding rules, by the length, in feet, and divide the product by the horizontal distance between the supports, in feet, and the quotient will be the required weight. Ezample.—What weight may be safely sustained at the middle of an oak beam, 6 x 10 inches, and 10 feet long, (set inclining, so that the horizontal distance between the supports is 8 feet,) the portion of the breaking weight allowable being 0:3? The result for a horizontal beam, by Rule X XXIII, is 10,332 pounds. This, multiplied by 10, the length, and divid- ed by 8, the horizontal a ie equals 12,915 pounds, the required weight. TO FIND THE DIMENSIONS. 334.—The following table exhibits, algebraically, rules for ascertaining the dimensions of beams required to support given weights; where b equals the breadth, and g the depth of a rectangular beam, in inches; Z the length between sup- ports; A the horizontal distance between the supports of an inclined beam, and ¢ the distance apart of two parallel beams, measured from the centres of their breadth, Z,.4, and c, in feet; w equals the weight on a beam; f the weight on each super- ficial foot of a floor resting on two or more parallel beams; A equals a load on a beam, and m and n the distances, respect- ively, at which’ 7 is located from the two supports; also P is a weight, and g and & the distances, respectively, at which P is located from the two supports; alsom+n=l=g+k; w, J, R, and P, all in pounds; m, n, g, and &, in feet. S is a constant, ae value of which is found in Table VI.; aisa decimal in proportion to unity as the safe load is to the break

Page 233

228 AMERICAN IOUSE-CARPENTER. ing load; 7 is a decimal in proportion to unity as b is to d; from which b =dr; a and r to be chosen at discretion. TABLE VIII.—STRENGTH Rectangular. When the When the peam is laid weight is Value of depth. Value of breadth. (120.) (121.) Concentrated st y wl wl middle gab Bad (125.) (126.) Equally distribut- y_wl wl ed 28ab ¥ 8a dz F : ne (181 ) x By the foot super- ck Se $ ficial Visat sad =a) Co a k (135.) (186.) neentrate a AwWmn 4wmn ee ee Sabot Saat (140.) (141.) at he Ia the! V4(Rmn+ Pok + ke.) 4(Rmn+ Pgk+ dc.) ength Sabl Sad? (145.) (146.) Concentrated at vy wh wh middle Sab Sad (150.) (151.) Equally distribut- y wh wh ed 3ga 35a de a (158.) (156.) E By the foot super- yfehl fehl G ficial 28að 2 Sad bh Concentrated at : C Gei any point in the yihwmn 4hwmn length Sabte Varho At two or more (165.) 4 6e.) ints in the) yih(Rmn+Pgk+gc)| 4h(Rmn+Pgk+ e.) length Sabk 1G nae 2 ge — oy

Page 234

FRAMING. ; 229 OF BEAMS , DIMENSIONS. Square. Bound. When 6 = d r. value of d. Value of a side. Value of the diameter. (122.) az.) a24) 3/ wl 3/wl 3/ wl wr ar Vor E Sa (127.) (129.) 3 wt V wl 2S8ar 1178 Sa (182.) (184) /fek a7 for 2Sar 1178 8a (187.) (139.) s/homn umn Sart “147 Sat (142.) (148.) (144) S/s(Rmn+ Pgk + ge) S/4(Rmn + Pgk+ ga) Vii mn+Pgk + ge.) Sart Sat E set 2 (147.) (148.) (149.) 3/ wh 3 Joh 8/7 wh gar Vor 589 Sa 152.) a58.) (154) 3/ wh 3/ wh 3/ wh 3Sar 28a 1178 Sa (157) (158) (159.) Vieht 3/foht 3/ fehl 28ar 28a Vims ae) © (164) æ S/thomn s/f homn Sart: 147 Sak (167.) j (168.) (169.) /iAh(Rmn+Pok+ go.) S/4h(Rmn+Pok+ do) Ssh(Rmn+Pgk + go) Sart: -E pag 7 a ser

Page 235

230 AMERICAN HOUSE-CARPENTER. Practical Rules and Examples. Rule XXXVUI—Preliminary, When the weight is con centrated at the middle. Multiply the weight, in pounds, by the length, in feet, and divide the product by the value of 8, Table VI., multiplied by a decimal that is in proportion to unity as the safe weight is to the breaking weight, and the quotient is a quantity which may be represented by J, referred to in succeeding rules. wb Fa” J (170.) Rule XXXIX.—Preliminary. When the weight is equally distributed. One-half of the quotient obtained by the preced- ing rule is a quantity which may be represented by K, referred to in succeeding rules. U S (171.) Rule XL.—Preliminary. When the weight is per foot su- perficial. Multiply the weight per foot superficial, in pounds, by the square of the length, in feet, and by the distance apart from centres between two parallel beams, and divide the pro- duct by twice the value of S, Table VI., multiplied by a deci- mal in proportion to unity as the safe weight is to the break- ing weight, and the quotient is a quantity which may be re- presented by Z, referred to in succeeding rules. e? ss eo Rule XLI.— Preliminary. When the weight is concentrated at any point in the length. Multiply the distance, in feet, from the loaded point to one support, by the distance, in feet, from the same point to the other support, and by four times the weight in pounds, and divide the product by the value of S, Table VI., multiplied by a decimal in proportion to unity as the safe weight is to the breaking weight, and by the length,

Page 236

FRAMING. 231 i in feet; and the quotient is a quantity which may be repre. sented by Q, referred to in the rules. soar =¢ (73) ule XLIL.— Preliminary. When two or more weights are concentrated at any points in the length of the beams. Multi- ply each weight by each of, the two parts, in feet, into which the length is divided by the point at which the weight is located, and divide four times the sum of these products by the value of S, Table VI., multiplied by a decimal in propor- tion to unity as the safe weight is to the breaking weight, and by the length, in feet, and the quotient is a quantity which may be represented by V, referred to in the rules. 4(Rmn+Pgk+ ce. ee tL y (74) Rule XLII.—Preliminary. When the beam is not laid horizontal, but inclining. In the five preceding preliminary rules, multiply the result there obtained by the horizontal dis- tance between the supports, in feet, and divide the product by the length, in feet, and the quotient in each case is to be used for beams when inclined, as referred to in succeéding rules. TO FIND THE DIMENSIONS. Rule XLIV.—When the beam is rectangular. To find the depth. Divide the quantity represented in preceding rules by J, K, L, Q, or V, by the breadth, in inches, and the square root of the quotient will be the depth required in inches. To find the breadth. Divide the quantity represented by J, K, L, Q, or V, by the square of the depth, and the quotient will be the required breadth, in inches. To find both breadth and depth, when they are to be ina given proportion. Divide the quantity represented by J, K, L, Q, or V, by a decimal in proportion to unity as the breadth

Page 237

932 AMERICAN HOUSE-CARPENTER. is to be to the depth, and the cube root of the quotient will be the depth in inéhes. Multiply the depth by the aforesaid de- cimal and the quotient will be the dread¢?. in inches. Evanwle.—A locust beam, 10 feet long in the clear of the supports, is required to sustain safely 8,000 pounds at the middle of its length, the portion of the breaking weight allow- able being 0°3; what is the required breadth and depth? Proceeding by the rule for weight eoneentrated at middle, (Rule XX XVIII.) 3,000, the weight, by 10, the length, equals 30,000. The value of S, Table VI., for locust, is 742: this by 0'3 the deeimal, as above, equals 222-6 ; the 30,000 aforesaid divided by this 222°6 equals 184-77, equals the quantity repre- sented by J. Now to find the depth when the breadth is 4 inches, 18477 divided by 4, the breadth, as above required, the quotient is 33°69, and the square root of this, 5°8, is the required depth in inehes. But to find the breadth, when the depth is known, let the depth be 6 inehes, then 18477 divided by 86, the square of: the depth, equals 3°74, the breadth required in inches. Again, to find both breadth and depth in a given proportion, say, as 0°6 is to 1:0. Here 134-77 divided by 0°6 equals 224-617, the eube root of whieh is 6°08, the re- quired depth in inches, and 6:08 by 0°6 equals 3: eae the re- quired breadth in inches. Thus it is seen, in this example, that a piece of locust tim- ber, 10 feet long, having 3,000 pounds concentrated at the middle of its length, as 3; of its breaking load, is required to be 4 by 513 inches, or 82 by 6- inches, or 38 by 64 inches. If this load were equally diffused over the length, the dimensions required would be found to be 4 by 4:1, or 1:87 by 6, or 2-895 by 4-825 inches, in the three cases respectively. Example—A tier of ehestnut beams, 20 feet long, placed cne foot apart from centres, is required to sustain 100 pounds per superficial foot upon the floor laid upon them: this load to be 0'2 of the breaking weight; what is the required dimen-

Page 238

FRAMING. 233 sions of the cross-section? By Rule XL., the rule for a load per foot superficial, 100 by 20 x 20 and by 1 equals 40,000. Twice 508, the value of S for chestnut, Table VI., by 0°2 equals 201:2. The above 40,000 divided by 201:2 equals 198°8, the value of Z. Now if the breadth is known, and is 3 inches, 198°8 divided by 3 equals 66-27, the square root of which is 8-14, the réquired depth. But if the depth is known and is 9 inches, 198°8 divided by (9 x 9 =) 81 equals 2°454 inches, the required breadth. Again, when the breadth and depth are required in the proportion of 0°25 to 1:0, then 198°8 divided by 0°25 equals 795-2, the cube root of which is 9:265, the required depth in inches, and 9:265 by 0°25 equals 2°316, the required breadth in inches. Example.—A. cast-iron bar, 10 feet long, is required to sus- tain safely 5,000 pounds placed at 3 feet from one end, and consequently at 7 feet from the other end, the, portion of the breaking load allowable being 0'3 ; what must be the size of the cross section? By Rule XLI., the rule for a concentrated load at any point in the length of the beam, 3 x 7 x 4 x 5000 = 420,000. And 1926, the value of S for cast-iron, Table VI., by 0:3 and by 10 equals 5778. The aforesaid 420,000 divided by this, 5778, equals 72°689, the value of Q. Now if the breadth is fixed at 1:5 then 72°689 divided by 1°5 equals 48-459, the square root of which is 6:96, the required depth in inches. But if the depth is fixed at 6 inches, then 72°689, the value of Q, divided by 36, the square of 6, equals 2-019, the required breadth in inches. Again, if the breadth and depth are required in the proportion 0'2 to 1:0; then Q, 72°689, divided by 0-2, equals’ 363-445, the cube root of which is 7-136, “the required depth in inches; and T136 by 0:2 n 1:427 the required breadth in inches. Jule XLV.—When the beam is square to find the breadth of aside. The cube root of the quantity represented by J, K, L, Q, or V, in preceding rules, is the breadth of the side required 30

Page 239

234 AMERICAN HOUSE-CARPENTER. Example.—A. Georgia pine beam, 10 feet long, is required to sustain, as 0°3 of the breaking load, a weight of 30,000 pounds equally distributed over its length, and the beam to be square, what must be the breadth of the side of such a beam ? By the rule for an equally distributed load, (Rule XXXIX.) 30,000 x 10 = 300,000, and 510 (the value of S, for Georgia pine, Table VI.) x 0°3 = 153. 300,000 divided by 153 equals 1960-784, and one-half of this equals 980-392, the value of A. Now the cube root of this is 9°934 inches, or 914, the required side. Had the weight been concentrated at the middle 1960-784 would be the value of J, and the cube root of this 12515, or 124 inches, would be the size of a side of the beam. Fxample.—A. square oak beam, 20 feet long, is required to sustain, as 0°25 of the breaking strength, three loads, one of 8,000 pounds at 5 feet from one end, one of 7,000 pounds at 14 feet, and one of 5,000 pounds at 8 feet from one end, what must be the breadth of a side of the beam? The value of S, for oak, Table VI, is 574. By the rule for this case, (Rule XLII.) 8000 x 5 x 15 equals 600,000; and 7000 x 14 x 6 equals 588,000 ; and 5000 x 8 x 12 equals 480,000. The sum of these products is 1,668,000; this by 4 equals 6,672,000. Now 574 x 0°25 x 20 equals 2870, and the 6,672,000 divided by the 2870 equals 2325, the number represented by V ; the cube root of which is 13°25, the required size of a side of the beam, 134 inches. This is for a horizontal beam. Now it this beam be laid inclining, so that the horizontal distance between the bearings is 15 feet, then to find the size by the rule for this case, viz. XLII., the abové number V, equal to 2325, multiplied by 15, the “horizontal distance, equals 34,875, and this divided by 20, the length, equals 1743:75. Now by Rule XLV., the cube root of this is 12°04, the required size of a side—12 inches full. Rule XLVI.—When the beam is round. Divide the quan-

Page 240

FRAMING. 235 tity represented by J, K, Z, Q, or V by the decimal 0°589, and the cube root of the quotient will be the required diameter. Example.—A white pine beam or pole, 10 feet long, is re- quired te sustain, as the 0°2 of the breaking strength, a load of 5,000 pounds concentrated at the middle, what must be the diameter? The value of S, for white pine, Table VI., is 390. Now by the rule for load at middle, (XXXVIII, 5000 x 10 = 50,000; and 390 x 02 =78; and 50,000 +78 = 641 = J. By this rule, 641 + 0°589 = 1088-28, the cube root of which, 10°28, is the required diameter. If this beam be inclined, so that the horizontal distance between the supports is 7 feet, then to find the diameter, by Rule XLII., value of J as above, 641 multiplied by 7 and divided by 10 equals 448-7. Now by this rule, 448°7 + 0-589 = 761-796, the cube root of which, 9°133, is the required diameter. Example.—A spruce pole, 10 feet long, is required to sus- tain, as the 0-334 or 4 of the breaking weight, a load of 1,000 pounds at 3 feet from one end, what must be the diameter ? The value of S for spruce, Table VI, is 345. By the rule for this case, (Rule XLL,) 3 x 7 x 4 x 1000 = 84,000; and 345 x $x 10 = 1150; and 84,000 + 1150 = 93-04, the value of Q. Now by this rule, 73°04 + 589 = 124, the cube root of which, 4-9866, is the required diameter in inches. 335.—Systems of Framing. In the various parts of framing known as floors, partitions, roofs, bridges, &c., each has a spe- cific object; and, in all designs for such constructions, this — object should be kept clearly in view ; the various parts being so disposed as to serve the design with the least quantity of material. The simplest form is the best, not only because it is the most economical, but for many other reasons. The great number of joints, in a complex design, render the construction liable to derangement by multiplied compressions, shrinkage, and, in consequence, highly increased oblique strains; by which its stability and durability are greatly lessened.

Page 241

236 AMERICAN HOUSE-CARPENTER. FLOORS. 336.—Floors are most generally constructed single, that is simply a series of parallel beams, each spanning the width of Fig 224. the floor, as seen at Fig. 224. Occasionally floors are con Fig. 225. structed double, as at Fig. 225; and sometimes framed, as at

Page 242

FRAMING. ' 937 Fig. 226; but these methods are seldom practised, inasmuch as either of these require more timber than the single floor. ‘Where lathing and plastering is attached to the floor -beams tc form a ceiling below, the springing of the beams, by custo- mary use, is liable to crack the plastering. To obviate this in good dwellings, the double and framed floors have been resorted to, but more in former times than now, as the cross- Surring (a series of narrow strips of board or plank, nailed transversely to the underside of the beams to receive the lath- ing for the plastering,) serves a like purpose very nearly as well. 337.—In single floors the dimensions of the beams are to be ascertained by the preceding rules for the stifness of materials. These rules give the required dimensions for the various kinds of material in common use. The rules may be somewhat abridged for ordinary use, if some of the quantities repre- sented in the formula be made constant within certain limits. For example, if the load per foot superficial, and the rate of deflection, be fixed, then these, together with the š, and the

Page 243

238 AMERICAN HOUSE-CARPENTER. constant represented by Æ, may be reduced to one ccnstant. For dwellings, the load per foot may be taken at 66 pounds, as this is the weight, that has been ascertained by experiment, to arise from a crowd of people on their feet. To this add 20 for the weight of the material of which the floor is composed, and the sum, 86, is the valne of f, or the weight per foot superficial for dwellings. The rate of deflection allowable for this load may be fixed at 0:03 inch per foot of the length. Then (44) transposed, Dole sIn ask becomes 5 x 86 cP Bx 003 a which, reduced, is 1800 f 180 = for five of the most common woods, and there Reducing results, rejecting small decimals, and putting LP =a, x equal, for " Georgia pine a o ee a a ò : 0'6 ORT se E E ey oe | White pime ree ele Se eee) 10 Spruce : : . . ‘ . : e 115 Hemlock 3 ; 5 . . o . «e 145 Therefore, the rule is reduced to vel =b d. And for white pine, the wood most used for floor bearns, æ = 1:0, and therefore disappears from the formula, rendering it still more simple, thus, . cP =b d (176.) The dimensions of beams for stores, for all ordinary business, may also be calculated by this modified rule, (175,) for it will require about 8} times the weight used in this rule, or about

Page 244

FRAMING. 239 300 pounds, to increase the deflection to the limit of elasticity in white pine, and nearly that in the other woods. But for warehouses, taking the rate of deflection at its limit, and fixing the weight per foot at 500 pounds, including the weight of the material of which the floor is constructed, and letting y repre- sent the constant, then gel=bdg (177.) and y equals, for Georgia pine _—- ». 8 1°85 Oak . ¢ o . . e : . 1°35 Whitepimat 2mm Cl lll a Spruce o . . . . . . . 2°2 Hemlock . o . : o o . . 2°85 338.—Hence to find the dimensions of floor beams for dwell- ings when the rate of deflection is 0°03 inch per foot, or for ordinary stores when the load is about 300 pounds per foot, and the deflection caused by this weight is within the limits of the elasticity of the material, we have the following rule: _ Rule XLVIL—Maultiply the cube of the length by the dis- tance apart between the beams, (from centres,) both in feet, and multiply the product by the value of x, (Art. 337.) Now to find the breadth, divide this product by the cube of the depth in inches, and the quotient will be the breadth in inches. But if the depth is sought, divide the said product by the breadth in inches, and the cube root of the quotient will be the depth in inches; or, if the breadth and depth are to be in proportion, as 7 is to unity, 7 representing any required deci- mal, then divide the aforesaid product by the value of 7, and extract the square root of the quotient, and the square root of this square root will be the depth required in inches, and the depth multiplied by the value of r will be the breadth in inches. i Fsample.—To find the breadth. In a dwelling or ordinary store what must be the breadth of the beams, when placed 15

Page 245

240 AMERICAN HOUSE-CARPENTER. inches from centres, to support a floor covering a span of 16 feet, the depth being 11 inches, the beams of oak? By the rule, 4096, the cube of the length, by 14, the distance from centres, and by 0°7, the value of æ, for oak, equals 3584. This divided by 1331, the cube of the depth, equals 2°69 inches, or 211 inches, the required breadth. F Erample.—To find ihe depth. The conditions being the same as in the last example, what must be the depth when the breadth is 3 inches. The product, 3584, as above, divided by 3, the breadth, equals 1194% ; the cube root of this is 10°61, or 102 inches nearly. Example.—To find the breadth and depth in proportion, say, as 0'3 to 1:0. The aforesaid product, 3584, divided by 0:3, the value of 7, equals 11,9462, the square root of which is 109:3, and the square root of this is 10°45, the required depth. This multiplied by 0°3, the value of 7, equals 3°135, the re- quired breadth, the beam is therefore to be 34 by 104 inches. 339.—And to find the breadth and depth of the beams for a floor of a warehouse sufficient to sustain 500 pounds per foot superficial, (including weight of the material in the floor,) with a deflection not exceeding the limits of the elasticity of the material, we have the following rule: lule XLVUI.—The same as XLVIL, with the exception that the value of y (Art. 387) is to be used instead of the value of x. Lxample.—To find the breadth. The beams of a warehouse floor are to be of Georgia pine, with a clear bearing between the walls of 15 feet, and placed 14 inches from centres, what must be the breadth when the depth is 11 inches? By the ‘rule, 8875, the cube of the length, and 1}, the distance from centres, and 1°35, the value of y, for Georgia pine, all multi- plied together, equals 5315-625; and this product divided by 1331, the cube of the depth, equals 3:994, the required depth, or 4 inches.

Page 246

FRAMING. 241 Example.—To find the depth. The conditions remaining, as in last example, what must be the depth when the breadth is 8 inches? 5315-625, the said product, divided by 3, the breadth, equals 1771°875, and the cube root of this, 12-1, or 12 inches, is the depth required. Example.—To find the breadth and depth in a given propor- tion, say, 0°35 to 1:0. 5315/625 aforesaid, divided by 0°35, the value of 7, equals 15187-5, the square root of which is 121°8, and the square rovt of this square root is 11-04, or 11 inches, the required depth. And 11-04 multiplied by 0°35, the value of r, equals 3:864, the required breadth—3$ inches. ' 3840.—It is sometimes desirable, when the breadth and depth of the beams are fixed, or when the beams have been sawed and are now ready for use, to know the distance from centres at which such beams should be placed, in order that the floor be sufficiently stiff. In this case, (175,) transposed, and put- ting «= -a there results À MET oy This in words, at length, is, as follows : Rule XLIX.—Multiply the cube of the depth by the breadth, both in inches, and divide the product by the cube of the length, in feet, multiplied by the value of æ, for dwell- (178.) ings, and for ordinary stores, or by y for warehouses; and the ` quotient will be the distance apart from centres in feet. Example.—A span of 17 feet, in a dwelling, is to be covered by white pine beams, 3 x 12 inches, at what distance apart from centres must they be placed? By the rule, 1728, the cube of the depth, multiplied by 3, the breadth, equals 5184. The cube of 17 is 4913, this by 1-0, the value of æ, ror white © pine, equals 4913. The aforesaid 5184, divided by this, 4913, equals 1-055 feet, or 1 foot and $ of an inch. 841.—Where chimneys, flues, stairs, etc., occur to interrupt RI

Page 247

242 AMERICAN HOUSE-CARPENTER. the bearing, the beams are framed into a piece, b, (Fig. 227,) called a header. The beams, a a, into which the header is framed, are called trimmers or carriage-beams. These framed beams require to be made thicker than the common beams. The header must be strong enough to sustain one-half of the weight that is sustained upon the ¢az beams, ¢ c, (the wall at the opposite end or another header there snstaining the other half) and the trimmers must each sustain one-half of the weight sustained by the header in addition to the weight it supports as a common beam. It is usual in practice to make these framed beams one inch thicker than the common beams for dwellings, and two inches thicker for heavy stores. This practice in ordinary cases answers very well, but in extreme eases these dimensions are not proper. Rules applicable gene- rally must be deduced from the conditions of the case—the load to be sustained and the strength of the material. 342.—For the header, formula (68,) Table V., is applicable. The weight, represented by w, is equal to the snperficial area of the floor supported by the header, multiphed by the load on every superficial foot of the floor. This is equal to the length of the header multiplied by half the length of the tail beams, and by 86 pounds for dwellings and ordinary stores, or

Page 248

FRAMING. 243 by 500 pounds for warehouses. Calling the length of the tail . beams, in feet, g, formula (68,) becomes E ig? ee a Then if f equals 86, and n equals 0°03, there results 900 g I = 179. (= Re ep This in words, is, as follows: Rule L.—Multiply 900 times the length of the tail beams by the cube of the length of the header, both in feet. The product, divided by the cube of the depth, multiplied by the value of Æ, Table lI., will equal the breadth, in inches, for dwellings or ordinary stores. Example.—A header of white pine, for a dwelling, is 10 feet long, and sustains tail beams 20 feet long, its depth is 12 inches, what must be its breadth? By the rule, 900 x 20 x 10° = 18,000,000. This, divided by (12° x 1750 =) 3,024,000, equals 5-95, say 6 inches, the required breadth. For heavy warehouses the rule is the same as the above, only using 1550 in the place of the 900. This constant may be varied, at discretion, to anything between 900 and 5000, in accordance with the use to which the floor is to be put. 343.—In regard to the trimmer or carriage beam, formula (136,) Table VIIL, is applicable. The load thrown upon the trimmer, in addition to its load as a common beam, is equal to one-half of the load on the header, and therefore, as has been seen in last article, is equal to one-half of the superticial area of the floor, supported by the tail beams, multiplied by the weight per superficial foot of the load upon the floor; “therefore, when the length of the header, in feet, is represented by 7, and the length of the tail beam by n, w equals Jy 5 XA = equals $ fj n, and therefore (136,) of Table VIII., becomes pim w — Sad

Page 249

9244 AMERICAN HOUSE-CARPENTER. equals the additional thickness to be given to a common beam when used as a trimmer, and for dwellings when f equals 86 and a equals 0°3, this part of the formula reduces to 2862, or, for simplicity, call it 300, which would be the same as fixing f at 90 instead of 86. Then we have 300 j m n? Sail This, in words, is as follows : Rule LI.— For dwellings. Multiply 300 times the length of the header by the square of the length of the tail beams, and by the difference in length of the trimmer and tail beams, all in feet. Divide this product by the square of the depth in inches, multiplied by the length of the trimmer in feet, and by the value of S, Table VI., and the quotient added to the thick- ness of a common beam of the floor, will equal the required b= (180.) thickness of the trimmer beam. Example.—A tier of 3 x 12 ineh beams of white pine, hav- ing a clear bearing of 20 feet, has a framed well-hole at one side, of 5 by 12 feet, the header being 12 feet long, what must be the thickness of the trimmer beams? Ly the rule, 800 x 12 x 15° x 5, divided by the product of 12° x 20 x 390, equals 3°6, and this added to 3, the thickness of one of the common beams, equals 6°6, the breadth required, 64 inches. For stores and warehouses the rule is the same as the above, only the constant, 300, must be enlarged in proportion to the load intended for the floor, making it as high as 1600 for heavy warehouses, 344.—When a framed opening occurs at any point removed from the wall, requiring two headers, then the load from the headers rest at two points on the carriage beam, and here for- mula (141,) Table YIII., is applicable. In this special case this formula reduces to __ 300,j (m* n+ g) |

Page 250

FRAMING. 945 where 6 equals the additional thickness, in inches, to be given to the carriage beam over the thickness of the common beams ; J, the length of the header, in feet; m and & the length, in feet, respectively, of the two sets of tail beams, and m + n = k +g=l. The constant in the above, (181,) is for dwellings ; if the floor is to be loaded more than dwelling floors, then it must be increased in proportion to the increase of load up to as high as 1600 for warehouses. Rule LIL—Trimmer beams for framed openings occurring so as to require two headers. Multiply the square of the length of each tail beam by the difference of length of the tail beam and trimmer, all in feet, and add the products; multiply their sum by 300 times the length, in feet, of the header, and divide this product by the product of the square of the depth, in inches, by the length, in feet, and by the value of S, Table VI; and the quotient, added to the thickness of a common beam of the tier, will equal the thickness of the trimmer beams. . _ a Example.—A. tier of white pine beams, 4 x 14 inches, 20 feet long, is to have an opening of 5 x 10 feet, framed so that the length of one series of tail beams is 7 feet, the other 8 feet, what must be the breadth of the trimmers? Here, (T x 18) + (8° x 12) equals 1405. This by 300 x 10 equals 4,215,000. This divided by 1,528,800 (= 14’ + 20 x 390) equals 2°75, and this added to 4, the breadth, equals 6°75, or 6%, the breadth required, in inches. 345.—Additional stiffness is given to a floor by the insertion of bridging strips, or struts, as at a a, (Fig. 228.) These pre- vent the turning or twisting of the beams, and when a weight is placed npon the floor, concentrated over one beam, they prevent this beam from descending below the adjoining beams to the injury of the plastering upon the underside. It is usual to insert a course of hridging at every 5 to 8 feet of the length

Page 251

246 AMERICAN HOUSE-CARPENTER. Fig. 228, af the beam. Strips of board or plank nailed to the underside of the floor beams to receive the lathing, are termed crass- JSurring, and should not be over 2 inches wide, and placed 12 inches from centres. It is desirable that all furring be narrow, in order that the clinch of the mortar be interrupted but little. When it is desirable to prevent the passage of sound, the openings between the beams, at about 3 inches from the upper edge, are closed by short pieces of boards, which rest on cleets, nailed to the beam along its whole length. This forms a floor, on which mortar is laid from 1 to 2 inches deep. This is called deafening. 346.—When the distance between the walls of a building is great, it becomes requisite to introduce girders, as an addi- tional support, beneath the beams. The dimensions of girders may be ascertained by the general rules for stiffness. For- mulas (72,) (78,) and (74,) Table V., are applicable, taking f, at 86, fur dwellings and ordinary stores, and increased in pro- portion to the load, up to 500, for heavy warehouses. When but one girder occurs, in the length of the beam, the distance from centres, c, is equal to one-half the length of the beam. 347.—When the breadth of a girder is more than about 12 inches, it is recommended to divide it by sawing from end to end, vertically through the middle, and then to bolt it to

Page 252

FRAMING. 247 gether with the sawn sides outwards. This is not tv strength- en the girder, as some have supposed, but to reduce the size of the timber, in order that it may dry sooner. The opera- tion affords also an opportunity to examine the heart of the stick—a necessary precaution; as large trees are frequently in a state of decay at the heart, although outwardly they are seemingly sound. When the halves are bolted together, thin slips of wood should be inserted between them at the severab points at which they are bolted, in order to leave sufficient space for the air to circulate between. This tends to prevent decay; which will be found first at such parts as are not exactly tight, nor yet far enough apart to permit the escape of moisture. 348.—When girders are required for a long bearing, it is usual to truss them; that is, to insert between the halves two pieces of oak which are inclined towards each other, and which meet at the centre of the length of the girder, like the rafters of a roof-truss, though nearly if not quite concealed within the girder. This, and many similar methods, though _ extensively practised, are generally worse than useless; since it has been ascertained that, in nearly all such cases, the ope- ration has positively weakened the girder. A girder may be strengthened by mechanical contrivance, when its depth is required to be greater than any one piece ot Fig. 229.

Page 253

248 AMERICAN JIOUSE-CARPENTER, timber will allow. J/g. 229 shows a very simple yet invalu able method of doing this. The two pieces of which the gir- der is composed are bolted, or pinned together, having keys inserted between to prevent the pieces from sliding. The keys should be of hard wood, well seasoned. The two pieces should be about eqnal in depth, in order that the joint be- tween them may be in the neutral line. (See Art. 317.) The thickness of the keys should be about half their breadth, and the amount of their united thicknesses should be equal to a trifle over the depth and one-third of the depth of the girder. Instead of bolts or pins, iron hoops are sometimes used; and when they can be procured, they are far preferable. In this ease, the girder is diminished at the ends, and the hoops driven from each end towards the middle. 349.—DBeams may be spliced, if none of a sufficient length van be obtained, though not at or near the middle, if it can be avoided. (See Art. 281.) Girders should rest from 9 to 12 inches on the wall, and a space should be left for the air to circulate around the ends, that the dampness may evapo- rate. Floor-timbers are supported at their ends by walls of considerable height. They should not be permitted to rest upon intervening partitions, which are not likely to settle as much as the walls; otherwise the unequal settlements will derange the level of the floor. As all floors, however well- constructed, settle in some degree, it is advisable to frame the beams a little higher at the middle of the room than at its sides,—as also the ceiling-joists aud cross-furring, when either are used. In single floors, for the same reason, the rounded edge of the stick, if it have one, should be placed uppermost. If the floor-plank are laid down temporarily at first, and left to season a few months before they are finally driven together and secured, the joints will remain much closer. But if the edges of the plank are planed after the first laying, they will

Page 254

FRAMING. 249 shrink again ; as it is the nature of wood to shrink after every planing however dry it may have been before. PARTITIONS. 350.—Too little attention has been given to the construction of this part of the frame-work of a house. The settling of floors and the cracking of ceilings and walls, which disfigure to so great an extent the apartments of even our most costly houses, may be attributed almost solely to this negligence. A square of partitioning weighs nearly a ton, a greater weight, when added to its eustomary load, such as furniture, storage, &c., than any ordinary floor is calculated to sustain. Hence the timbers bend, the ceilings and eornices erack, and the whole interior part of the honse settles; showing the necessity for providing adequate supports independent of the floor- timbers. A partition should, if practicable, be supported by the walls with which it is connected, in order, if the walls set- tle, that it may settle with them. This would prevent the separation of the plastering at the angles of rooms. For the same reason, a firm connection with the eeiling is an im- portant object in the eonstruetion of a partition. 351.—The joists in a partition shonld be so placed as to dis- charge the weight upon the points of support. All oblique pieces in a partition, that tend not to this objeet, are much better omitted. Jig. 230 represents a partition having a door in the middle. Its construction is simple but effective. Fig. 231 shows the manner of constructing a partition having doors near the ends. The truss is formed above the door-heads, “and the lower parts are suspended from it. The posts, @ and b, are halved, and nailed to the tie, ¢ d, and the sill, e f. The braees in a trussed partition should be placed so as to form, as near as possible, an angle of 40 degrees with the horizon. In partitions that are intended to support only their own weight, 32

Page 255

IRJ AMERICAN HOUSE-CARPENTER. Fig. 230. we’ (DEE iu [Tm d Fig. 231. the principal timbers may be 3 x 4 inches for a 20 feet span, 34 x 5 for 30 feet, and 4x6 for 40. The thickness of the filling-in stuff may be regulated according to what is said at Art. 345, in regard to the width of furring for plastering. The filling-in pieces should be stiffened at about every three feet by short struts between. All superfluous timber, besides being an unnecessary load upon the points of support, tends to injnre the stability of the plastering ; for, as the strength of the plastering depends, in a great measure, upon its clinch, formed by pressing the mortar

Page 256

r FRAMING. 251 through the space between the laths, the narrower the surface, therefore, upon which the laths are nailed, the less will be the quantity of plastering unclinched, and hence its greater secu- rity from fractures. For this reason, the principal timbers of the partition should have their edges reduced, by chamfering off the corners. Fig. 282. 352.—When the principal timbers of a partition require to: be large for the purpose of greater strength, it is a good plan to omit the’ upright filling-in pieces, and in their stead, to place a few horizontal pieces; in order, upon these and the principal timbers, to nail upright battens at the proper dis- tances for lathing, as in Fig. 232. A partition thus con- structed requires a little more space than others; but it has the advantage of insuring greater stability to the plastering, and also of preventing to a good degree the conversation of one room from being heard in the other. When a. partition is required to support, in addition to its own weight, that of a floor or some other burden resting upon it, the dimensions of the timbers may be ascertained, by applying the principles which regulate the laws of pressure and those of the resistance of timber, as explained at the first part of this section. The following data, however, may assist in calculating the amount of pressure upon partitions :

Page 257

252 AMERICAN HOUSE-CARPENTER. Wh.te pine timber weighs from 22 to 32 pounds per cubic foot, varying in accordance with the amount of seasoning it has had. Assuming it to weigh 30 pounds, the weight of the beams and floor plank in every superficial foot of the flooring will be, when the beams are 8 x 8inches, and placed 20 inches from centres, 6 pounds. 8 x 10 é “é é 18 ae «sé t ét veo ae 3 x 12 a “cc éi 1 6 a ae a 9 “ 3 x 13 a é ae 1 $ a é “ 1 1 4 4 x 1 9 i “u as 1 9 s ét (ii 1] 3 sc 4 x 14 é: a cs 14 “u its é: 13 i In addition to the beams and plank, there is generally the plastering of the eeiling of the apartments beneath, and some- times the deafening. Plastering may be assumed to weigh 9 pounds per superficial foot, and deafening 11 pounds. Hemlock weighs about the same as white pine. <A parti- tion of 3 x £ joists of hemlock, set 12 inches from centres. therefore, will weigh about 24 pounds per foot superficial, and when plastered on both sides, 20$ pounds. 3538.—When floor beams are supported at the extremities, and by a partition or girder at any point between the extre- mnities, one-half of the weight of the whole floor will then be supported by the partition or girder. As the settling of parti- tions and floors, which is so disastrous to plastering, is fre- quently owing to the shrinking of the timber and to ill-made joints, it is very important that the timber be seasoned and the work well execnted. Where practicable, the joists of a partition ought to extend down between the floor beams tc the plate of the partition beneath, to avoid the settlement con- sequent upon the shrinkage of the floor beams. ROOFS.* 354.—In ancient buildings, the Norman and the Gothie, the * See also Art. 238,

Page 258

FRAMING. 253 walls and buttresses were erected so massive and firm, that it was customary to construct their roofs without a tie-beam; the walls being abundantly capable of resisting the lateral pressure exerted by the ratters. But in modern buildings, the walls are so slightly built as to be incapable of resisting scarcely any oblique pressure; and hence the necessity of constructing the roof so that all oblique and lateral strains may be removed; as, also, that instead of having a tendency to separate the walls, the roof may contribute to bind and steady them. j 355.—In estimating the pressures upon any certain roof, for the purpose of ascertaining the proper sizes for the timbers, calculation must be made for the pressure exerted by the wind, and, if in a cold climate, for the weight of snow, in addition te the weight of the materials of which the roof is composea. The weight of snow will be of course according to the depth it acquires. Snow weighs 8 lbs. per cnbic foot, and more when saturated with water. In a severe climate, roofs onght to be constructed steeper than in a milder one, in order that the snow may have a tendency to slide off before it becomes of sufficient weight to endanger the safety of the roof. The inclination should be regulated in accordance with the qualities of the material with which the roof is to be covered. The following table may be useful in determining the smallest in- clination, and in estimating the weight of the various kinds of covering Material. Inclination. heug ong a Tin =. § to 14 Iba Copper a, 1 tol} “ Lea s s . . « . 4 to 7 e Zino 3 4 š lłtoa “ Short pine shingles Me, , lto? © Long cypress — K . A 8 tos * Slate . . —. , 5 tog “

Page 259

254 AMERICAN HO {SE-CARPENTER. The weight of the covering, as above estimated, is that of the material only, with the weight of whatever is used to fix it to the roof, such as nails, &e. What the material is laid . on, such as plank, boards or lath, is not included. The weight of plank is about 8 pounds per foot superficial; of boards, 2 pounds; and lath, about a half pound. 356.—The weights and pressures on a roof arise from the roofing, the truss, the ceiling, wind and snow, and may be stated as follows: First, the Ioofing—On each foot superficial of the inclined surface, Slating. . . . . . ~~ will weigh about 7 Ibs. Roof plank, 14 inches thick . : É uy EO nE Roof beams or jack rafters. : i eo ae Total, 12 Ibs. This is the weight per foot on the inclined surface; but it is desirable to know how mnch per foot, measured horizontally, this is equal to. The horizontal measure of one foot of the inclined surface is equal to the cosine of the angle of inclina- tion. Therefore, cos. ; 1 +P ere where p represents the pressure on a foot of the inclined sur- face, and w the weight of the roof per foot, measured horizon- tally. The cosine of an angle is equal to the base of the right- angled triangle divided by the hypothenuse, which in this case would be half the span divided by the length of the rafter, or = where s is the span, and 7 the length -f the rafter. Hence, . Ee a ote cos. 2, s? or, twice the pressure per foot of inclined surface, multiplied

Page 260

v bead FRAMING. 25 by the length of the rafter, and diviaed by the span, will give the weight per foot measured horizontally ; or, ou! = w (182.) equals the weight per foot, measured horizontally, of the roof beams, plank, and covering for a slate roof. Sevond, the Truss.—The weight of the framed truss is nearly in proportien to the length of the truss, and to the distance apart at which the trusses are placed. w= 52c8 (183.) equals the weight, in pounds, of a white pine truss with iron suspension rods and a horizontal tie beam, near enough for the requirements of our present purpose; where s equals the length or span of the truss, and ethe distance apart at which the trusses are placed, both in feet. It is desirable to know how much this is equal to per foot of the area over which the truss is to sustain a covering. This is found by dividing the weight of the truss by the span, and by the distance apart from centres at which the trusses are placed; or, 9 . r = (184) equals the weight in pounds per foot to be allowed for the truss. Third, the Ceiling.—The weight supported by the tie beams, viz.: that of the ceiling beams, furring and plastering, is about 9 pounds per superficial foot. Fourth, the Wind.—The force of wind has been known as high as 50 pounds per superficial foot against a vertical sur- face. The effect of a horizontal force on an inclined surface is in proportion to the sine of the angle of inclination, the ef- fect produéed being in the direction at right angles to the in- clined surface. The force thus acting may be resolved into forces acting in two directions—the one horizontal, the other vertical; the former tending, in the case of a roof, to thrust +

Page 261

256 AMERICAN HOUSE-CARPENTER. aside the walls on which the roof rests, and the latter acting directly on the materials of which the roof is constructed— this latter force being in proportion to the sine of the angle of inclination multiplied by the cosine. This is the vertical effect of the wind upon a roof, without regard to the surface it acts upon. The wind, acting horizontally through one foot snperficial of vertical section, acts on an area of inclined sur- face equal to the reciprocal of the sine of inclination, and the horizontal measurement of this inclined surface is equal to the cosine of the angle of inclination divided by the sine. This is the horizontal measurement of the inclined surface, and the vertical force acting on this surface is, as above stated, in pro- portion to the sine multiplied by the cosine. Combining these, it is found that the vertical power of the wind is in proportion to the square of the sine of the angle of inclination. There- fore, if the power of wind against a vertical surface be taken at 50 pounds per superficial foot, then the vertical effect on a roof is equal to ; w = 50 sin.” = sot for each piece of the inclined surface, the horizontal measure- ment of which eqnals one foot; where / equals the length of the rafter, and / the height of the roof. Fifth, Snow.—-The weight of show will be in proportion to the depth it acquires, and this will be in proportion to the rigour of the climate of the place at which the building is to be erected. Upon roofs of most of the usual inclinations, snow, if deposited in the absence of wind, will uot slide off. When it has acquired some depth, and not till then, it will have a tendency, in proportion to the angle of inclination, to slide off in a body. The weight of snow may be taken, there- fore, at its weight per cubic foot, 8 pounds, multiplied by the depth it is usual for it to acquire. This, in the latitude of New York, may be stated at about 24 fcet. Its weight would, (185.) w

Page 262

FRAMING. 257 therefore, be 20 pounds per foot superficial, measured horizon- tally. i 357.—There is one other cause of strain upon a roof; namely, the load that may be deposited ın the roof when used as a room for storage, or for dormitories. But this seldom occurs. When a case of this kind does occur, allowance is to be made for it as shown in the article on floors. But in the general rule, now under consideration, it may be omitted. 358.—The following, therefore, comprehends all the pressures or weights that occur on roofs generally, per foot superficial ; l For roof beams, plank, and slate (182) 3 s o ` 24 : Ibs. “ the truss (184) 5 > 5 e e . . 52 “ “ ceiling : o . dl . . > ° : 9 s i p ` 20 “4 Having found the weight per foot, the total weight for any part of the roof is found by multiplying the weight per foot by the area of that part. This process will give the weight sup- ported by braces and suspension rods, and also that supported by the rafters and tie beam. But in these last two, only half of the pressure of the wind is to be taken, for the wind will act only on one side of the roof at the same time. The vertical pressure on the head of a brace, then, equals (186.) W= 4on(6- 4 8:55 + 195% 8 d And Wz=cpn, where p equals (65 + 8°55 + 12°5 , equals the weight per foot. “ And the aggregate load of the roof on each truss equals i Ay aa W = 408(6-+ 855 + 625 7) (187.) l h And W = cgs, where g = 4(6- + 8:55 + 6°25 a) equals the 33

Page 263

958 AMERICAN HOUSE-CARPENTER. weight per foot ; where ¢ equals the distance apart frorn centres at which the trusses are placed; n the distance horizontally between the heads of the braces, or, if these are not located at equal distances, then » is the distance horizontally from a point half-way to the next brace on one side to a point half-way to the next brace on the other side; 2 the length of the rafter; s the span, and / the height—all in feet. 859.—By the parallelogram of forces, the weight of the roof is in proportion to the oblique thrust or pressure in the axis of the rafter, as twice the height of the roof is to the length of the rafter; or, W:H3: 2h: lor Wl . 2h? where Æ equals the pressure in the axis of the rafter. And the weight of the roof is in proportion to the horizontal thrust in the tie beam, as twice the height of the roof is to half the Diet es Te S (188.) span; or, WoT 2h $e, OF a Pa (189.) an where Z7 equals the horizontal thrust in the tie beam; the valne of W in (188) and (189) being shown at (187), and (187) being compounded as explained in Art. 356. The weight is that for a slate roof. If other material is used for covering, ee ry = this Wi t= or should there be other conditions modifying the weight in any particular case, an examination of Art. 356 will show how to modify the formula accordingly. 360.—The pressures may be. obtained geometrically, as shown in Fig. 233, where A B represents the axis of the tie beam, A C the axis of the rafter, D Z and FB the axes of the braces, and D G, F E, and C B, the axes of the suspension rods, In this design for a truss, the distance A B is divided into three

Page 264

FRAMING. 259 equal parts, and the rods located at the two points of division, Gand Æ. By this arrangement the rafter AC is supported at equi-distant points, D and F. The point D supports the rafter for a distance extending half-way to A and half-way to F, and the point / sustains half-way to D and half-way to ©. Also, the point C sustains half-way to F and, on the other rafter, half-way to the corresponding point to F. And because these points of support are located at equal distances apart, there- fore the load on each is the same in amount. On DG make D a equal to 100 of any decimally divided scale, and let Da represent the load on D, and draw the parallelogram ab De. Then, by the same scale, Db represents (Art. 258) the pressure in the axis of the rafter by thé load at D ; also, Dc the pressure in the brace D Æ. Draw ¢ d horizontal; then D d is the ver- tical pressure exerted by the brace DE at Æ. The point F sustains, besides the common load represented by 100 of the scale, also the vertical pressure exerted by the brace D £; therefore, since a represents the common load on D, F, or C, make Fe equal to the sum of Da and D d, and draw the parallelogram /’gef. Then /’g, measured by the scale, is the pressure in the axis of the rafter caused by the load at F, and Ff is the load in the axis of the brace FB. Draw fA horizontal; then ZA is the vertical pressure exerted by the brace / B at B. The point O, besides the common load re- presented by D a, sustains the vertical pressure FA caused by the brace FB, and a like amount from the corresponding brace on the opposite side. Therefore, make Cj equal to the sum of Da and twice FA, and draw 7k parallel to the opposite rafter. Then C% is the pressure in the axis of the rafter at ©. This is not the only pressure in the rafter, althongh it is the total pressure at its head C. At the point Z, besides the pressure C k, there is Fg. At the point D, besides these two pressures, there is the pressure Db. At the foot, at A, there is still au additional pressure: while the point D sustains the load half-

Page 265

260 AMERICAN HOUSE-CARPENTER. way to F' and halfway to A, the point A sustains the load half-way to D. This load is, in this case, just half the load at D. Therefore draw A m vertieal, and equal to 50 of the seale, or half of Da. Extend CA tol; draw mZ horizontal. Then AJ is the pressure in the rafter at A caused by the weight of the roof from A half-way to D. Now the total of the pressures in the rafter is equal to the sum of AZ + Db + Fg added to Ck. Therefore make kn equal to the sum of AZ + Db + Fg, and draw no parallel with the opposite rafter, and nj hori- zontal. Then Co, measured by the same seale, will be found equal to the total weight of the roof on both sides of C B. If Da =100 represent the portion of the weight borne by the point D, then Co, representing the whole weight of the roof. should equal 600, (as it does by the seale,) for D supports just one-sixth of the whole load. As Cn is the total oblique thrust in the axis of the rafter at its foot, therefore n j is the horizon- tal thrust in the tie beam. 361.—In stating the amount of pressures in the above ag being equal to certain lines, it was so stated with the under- standing that the lines were simply in proportion to the weights. To obtain the weight represented by a line, multiply its length (measured by the scale used) by the load resting at D, (or at F or C, as these are all equal in this example,) and divide the produet by 100, and the quotient will be the weight required. For, as 100 of the scale is to the load it represents, so is any other dimension on the same scale to the load é represents. 362.— Example. Let A B (Fig. 233) equal 26 feet, CB 12 feet, and A C 29 fect, and A G, GZ, and EB, each 8? feet. Let the trusses be placed 10 feet apart. Then the weight on D, for the use of a braces and Ei is, per (186), equal to son (6: +855 +1955) = 4 x 10 x 83(6 x Z + 8:55 + 124 x 34)

Page 266

FRAMING. 263. = 346% x 14:398 = 4991°3. This is the common load at the points D, F, and C, and each of the lines denoting pressures multiplied by it and divided by | 49913 100 duct will be the weight required. 49-918 may be called 50, for simplicity ; therefore the pressure in the brace D Z equals 112 x 50 = 5600 pounds, and in the brace F B, 140 x 50 = 7000 pounds, and in like manner for any other strain. For the rafters and tie beam the total weight, as per (187), equals 4os (62+ 8:55 + 64 S 100, or multiplied by the quotient of = 49:913, the pro- =4 x 10 x 52(6 x SD + 8:55 + GF x a) = 2080 x 13°148 = 27343-683 pounds. This is the total weight of the roof supported by one truss. The oblique thrust in the rafter A C is, per (188), equal to LW _ 29 x 27343-68 2h 2 x 13 = 3049872 pounds. To obtain this oblique thrust geometrically : Co (Fig. 233) represents the weight of the roof, and measures 600 by the scale ; and the line On, representing the oblique thrust, measures 670. By the proportion, 600 : 670 3: 27343-68 : 30533-8, = the oblique thrust. The result here found is a few pounds more than the other. This is owing to the fact that the line O'n is not exactly 670, nor is the length of the rafter precisely 29 feet. Were the exact dimensions used in each case the results would be identical; but the result in either case is near enough for the purpose. The horizontal strain is, per (189), equal to

Page 267

262 AMERICAN HOUSE-CARPENTER. Ws 27843°68 x 52 Gh &xiB = 27348°68 pounds, The result gives the horizontal thrust precisely equal to the weight. This is as it should be in all cases where the height of the roof is equal to one-fourth of the span, but not other- wise; for the result depends (189) upon this relation of the height to the span. Geometrically, the result is the same, for Co and nj (Fig. 233,) representing the weight and horizontal thrust, are precisely equal by measurement. 363.—The weight at the head of a brace is sustained partly at the foot of the brace and partly at the foot of the rafter. The sum of the vertical effects at these two points is just equal to the weight at the head of the brace. The portion of the weight sustained at either point is in proportion, inversely, to the horizontal distance of that point from the weight; there- fore, V= ws (190.) where V equals the vertical effect at the foot of the brace; W, the weight at the head of the brace; g, the horizontal distance from the foot of the rafter to the head of the brace; and a, the distance from the same point to the foot of the brace. 364.—For the oblique thrust in the brace: from the triangle Ef h (Fig. 233, ) Pee A Sain, 2 rad. Sines Pade dS therefore, T=—=V, (191.) where Z equals the oblique thrust in the brace; V, the verti- cal pressure caused by 7’ at the foot of the brace (190); a Zand À the length and height respectively of the brace. 365.—Lcample. Brace D E, Fig. 233. In this ease, equals the product of the weight per superficial foot, m

Page 268

FRAMING. j 263 plied by the area supported at the point D, equals 5000 pounds, (Art. 862.) The length g equals 84 feet, and @ equals 174 feet. Therefore (190), = Y= wi= 5000 x d- 2500 pounds equals the vertical pressure at Æ caused by the brace D Æ. Ther for the oblique thrust, ¿ equals 9°6 feet, and A equals 4'3 feet. Therefore, froin (191), T= ve = 2500 x = = 5581-4 pounds equals the oblique thrust in the brace D Æ. In Art. 362 it was found to be 5600. The discrepancy is owing to like causes of want of accuracy in the case of the rafter, as explained in Art. 362. Another Example.—Brace FB, Fig. 233. In this case, W equals the product of the weight per superficial foot, multiplied by the area supported by the point F, added to the vertical strain caused by the brace DE. From Art. 362 the weight of roof on F equals 5000 pounds, and the vertical strain from brace D F is, as just ascertained, = 2500, total 7500, equals W. The length, g, equals two-thirds of 26 feet, equals 174, and a equals 26 feet. Therefore, from (190), V = W2= 7500 x 13 = zo00 a 26 equals the vertical effect at B caused by the brace FD. Then, for the oblique thrust in the brace, Z equals 12-2, and A equals 83. Therefore, from (191) f= Vy = 5000 x SE = 7038°5 equals the oblique thrust or strain in the axis of the brace. It was 7000 by the geometrical process, (Art. 362.) 366.—-The strain upon the first rod, D G, equals simply the weight of the ceiling supported by it, added to the part of the tie beam it sustains. The weight of the tie beam will equal

Page 269

264 AMERICAN HOUSE-CARPENTER.

Page 270

b FRAMING. 265 about one pound per superficial foot of the ceiling. The weight per foot for the ceiling is stated (see Art. 356 third, and 358, at 9 pounds. To this add 1 pound for the tie beam, and the sum is 10. Then NV =10en. (192.) The strain on the second tie rod equals the weight of ceiling supported, = JV, added to the vertical effect of the strain in the brace it sustains, [see (190)] or.equal to O=10en+ V. (193.) The strain on the third rod is equal to JV, added to the ver- tical effect of the strain in the brace it sustains, and this is the strain on any rod. The first rod has no brace to sustain, and the middle rod sustains two braces. In this case the strain equals U= 10en+ 20. (194.) It may be observed that V represents the vertical strain caused by that brace that is sustained by the rod under consi- deration ; and, as the vertical strain caused by any one brace is more than that caused by any other brace nearer the foot of the rafter, therefore the V of (193) is not equal to the V of (194): Hence a necessity for care lest the two be confounded and thus cause error. 367.—Examples. The rod D G (Fig. 233) has a strain which equals (192) N = 10 cn = 10 x 10 x 8Ẹ = 867 pounds. The strain on rod FE equals (193) O = 10 en + V = 867 + 2500 = 3367 pounds. The strain on rod C B, the middle rod, equals (194) U =10cn + 2V = 867 + 2 x 5000 = 10867 pounds. 368.—The load, and the strains caused thereby, having been discussed, it remains to speak of the resistance of the ma- terials. Í First, of the Rafter. —Generally this piece of timber is so pinioned by the roof beams or purlins as to prevent any late- 34

Page 271

266 AMERICAN NOUSE-CARPENTER. ral movement, and the braces keep it from deflection; there fore it is not liable to yield by flexure. Hence the manner of its yielding, when overloaded, will be by crushing at the ends, or it will crush the tie beam against which it presses. The fibres of timber yield much more readily when pressed toge- ther by a force acting at right angles to the direction of their length, than when it acts in a dene with their length. The value of timber subjected to pressure in these two ways is shown in Art. 292, Table I., the value per square inch of the first stated resistance being expressed by P, and that of the other by ©. Timber pressed in an oblique direction yields with a force exceeding that expressed by ŽP, and less than that by ©. When the angle of inclination at which the force acts is just 45°, then the force will be an average between P and C. And for any angle of inclination, the force will vary in- versely as the angle; approaching P as the angle is enlarged, and approaching C as the angle is diminished. It will be equal to C when the angle becomes zero, and equal P when the angle becomes 90°. The resistance of timber per square inch to an oblique force is therefore expressed by M=P+H(0-P, (195.) where 4° equals the complement of the angle of inclination. In a roof, A° is the acute angle formed by the rafter with a vertical line. If no convenient instrument be at hand to measure the angle, describe an arc upon the plan of the truss—thus: with CB (Fig. 233) for radius, describe the are B g, and get the length of this are by stepping it off with a pair of dividers. Then AE a 90 = 0-633 5 where @ equals the length of the arc, and 4 equals B C, the height of the roof. Therefore,

Page 272

r FRAMING. 267 M = P +063} 70- P) (196.) equals the value of timber per square inch in a tie beam, C and P being obtained from Table L, Art. 292. When C for the kind of wood in the tie beam exceeds C’ set opposite -the kiud of wood in the rafter, then the latter is to be used in the rules instead of the former. 369.—Having obtained the strain to which the material is subjected in a roof, and the capability of the material to resist that strain, it only remains now to state the rules for determin- ing the dimensions of the material. | : 370.—To obtain the dimensions of the rafter:—It has been shown that the strain in the axis of the rafter equals (188), l k= Ws; This is the strain in pounds. Timber is capable of resisting effectually, in every square inch of the surface pressed (196), P + 063% (C — P) pounds. And when the strain and resistance are equal, R =bd[P + 0-6385(C — P), where 6 and d are respectively the breadth and depth of the rafter. Hence bd= SS acer P + 06335 (C— P) _Example.—(Fig. 233.) The strain in the axis of the rafter in this example, ascertained in Art. 362, is 30498°72 pounds. If the timber used be white pine, then P = 800 and C = 1200. The length of the are Bg is 144 feet, and A=13. Therefore Pe 30498:72 an 300 + (0:634 x 42 x 900) This is the area of the abutting surface at the tie beam— say 6 by 54 inches. At least half this amount should be added

Page 273

268 AMERICAN HOUSE-CARPENTER. to allow for the shoulder, and for cutting at the joints fo. braces, &c. The rafter may therefore be 6 by 9 inches. l The above method is based upon the supposition that the rafter is effectually secured from flexure by the braces and roof beams. Should this not be tte case, then the dimensions of the rafter are to be obtained by rules in Art. 298, for posts. Nevertheless, the abutting surface in the joint is to be deter. mined by the above formula (197). . 371.—To obtain the dimensions of the braces :—Usually, braces are so slender as to require their dimensions to be ob- tained by rules in Árt. 298; the strain in the axis of the brace having been obtained by formula (191), or geometrically as in Art. 360. The abutting surface of the joint of the brace is to be ob- tained, as in the case of the rafter, by formula (195); 4° be- ing the number of degrees contained in the acute angle formed by the brace and a vertical line, for the joint at the tie beam ; but for the joint at the rafter, A° is the number of degrees contained in the acute angle formed by the brace and a line perpendicular to the rafter, or it is 90, dimninished by the num- ber of degrees contained in the acute angle formed by the rafter and brace. Example.—Fig. 233, Brace D E, of white pine. In this brace the strain was found (Art. 362) to be 5600 pounds, the length of the brace is 9°6 feet. By Art. 298, the brace is therefore required to be 4°18 x 6 inches. For the abutting surface at the joints, for white pine, / equals 800 and, C 1200. The angle D EF equals 68° 26’. By (197) and (195), pd- T 5600 B ea a ee One an P+35(C—P) 300+ [7 x (1200 - 800)] 5600 : =r 6 inches. e This is the area of the abuttir g surface of the joint at the tie

Page 274

FRAMING. 269 beam. To obtain the joint at the rafter, the angle FD Æ equals 53° 8’, and hence _ : 5600 bd= = ee ae TH 2G — P) 300 +[ = x (1200 — 300)) = _ ON BoT anche 300 + A x 900) This is the area of the abutting surface of the joint at the rafter. Another Ecample.—Brace FB, Fig. 233, of white pine, 12:2 feet long. The strain in its axis is (Art. 862) 7000 pounds. By Art. 298, the brace is required to be 54 x 6 inches. For the abutting surface of the joints, P equals 300, C equals 1200, and the angle F B C equals 45° ; therefore, piety ODO naa = 94 inches. 800-+[g x (1200 — 300)] This is the area of the abutting surface at the tie beam. For the surface at the rafter, the angle CFB equals 71°, and 90 — 71 = 19, equals the angle to be used in the formula; therefore, = san = 14°3 inches, nearly. 300+ [8 x (1200 — 300)] This is the area of the abutting surface of the joint at the rafter. 372.—To obtain the dimensions of the tie beam :—A tie beam must be of such dimensions as will enable it to resist effectually the tensile strain caused by the horizontal thrust of the rafter and the cross strains arising from the weight of the ceiling, and from any load that may be Eipcod upon it in the roof. From (17), Art. 310, w H 2 oar er where Æ equals the horizontal thrust, and from (189),

Page 275

bo -l © AMERICAN HOUSE-CARPENTER. therefore, where W equals the weight of the roof in pounds, as shown at (187); s, the span; A, the height, both in feet; and T, a constant set opposite the kind of material, in Table III. ; and A equals the area of unent fibres in the tie beam. About one-half of this should be added to allow for the requisite eut- ting at the joints; or, the area of the cross section of the tie beam should be equal to at least 3 of the area of uncut fibres ; or, when }d equals the area of the tie beam, then E a (198). Example.—The weight on the truss at Fig. 233 is shown to be (Art. 362) 27343°68 pounds, say 27500 pounds; the span is 52 feet, the height 13, and the value of 7 for white pine is (Table III.) 2367, therefore bd = ee =x ae = 174 inches equals the area of cross section of the tie beam requisite to resist the tensile strain. This is smaller, as will be shown, than what is required to resist the cross strains, and this will be found to be the ease generally. The weight of the ceiling is 9 pounds per superficial foot; the length of the longest unsup- ported part of the tie beam is 8% feet; then, if the defleetion per lineal foot be allowed at 0-015 ineh, the depth of the tie beam will be required ((72), Table V.) to be 6-14 inches. But in order effectually to resist the strains tie beans are subjected to at the hands of the workmen, in the process of framing and elevating, the area of cross seetion in inehes should be at least equal to the length in feet. Were it possible to guard against this cause of strain, the size ascertained by the rule, 6

Page 276

FRAMING. orl by 6°14, would be sufficient; but to resist this strain, the size should be 6 by 9. There is yet one other dimension for the tie beam required, and that is, the distance at which the joint for the rafter must be located from the end of the tie beam, in order that the thrust of the rafter may not split off the part against which it presses. This may be ascertained by Rule XI., Art. 302, for all cases where no iron strap or bolt is used to secure the joint; but where these fastenings are used the abutment may be of any convenient length. And in using irons here, care should be exercised to have the surface of pressure against the iron of sufficient area to prevent indentation. 373.—To obtain the dimensions of the iron suspension rods. By Art. 310, (17), and 7’ varies (Table IIJ.) from 5000 to 17000, according to the diameter inversely ; for the smaller rods are stronger in pro- ‘portion than the larger ones. Example.—Taking T equal 5000, then the area of the rod D G (Fig. 233) requires (Art. 367) to be equal to 867 : A = B00 0°173 inch, corresponding to 0°469 inch diameter. This rod may be half inch diameter. | Another Example.—The rod FE (Fig. 233) is loaded with (Art. 3867) 8367 pounds, therefore ; ba 330 see A = zgo = 0613 inch equals the area of the rod, the corresponding diameter of which is 0925. This rod may be one inch diameter. Again, a third example; the rod CB. This rod is loaded with (Art. 367) 10867 pounds, therefore

Page 277

272 AMERICAN HOUSE-CARPENTER. = el = 2178 inch equals the required area of the rod, the diameter correspond- ing to which is 1°66. This rod may therefore be 12 inches diameter. 374.—While discussing the principles of strains in roofs and deducing rules therefrom, the truss indicated in (7g. 233 has been examined thronghout. The result is as follows: rafter, 6 x 9; tie beam, (6 x 6, or) 6 x 9; the first brace from the wall, 44 x 6 inches, with an abutting surface at the lower end of 6 inches, and at the upper end of 8% inches; the other brace, 54 x 6 inches, with an abutting surface at the lower end of 94 inches, and at the upper end of 14,3, inches; the shortest rod, 4 inch diameter; the next, 1 inch diameter ; and the middle rod, 14 inches diameter. PRACTICAL RULES AND EXAMPLES. For Roofs Loaded as per Art. 356. 375.—ule LIII. To obtain the dimensions of the rafter _ Multiply the value of X (Table IX., Art. 376) by the span of the roof, by the length of the rafter, and by the distance apart from centres at which the roof trusses are placed, all in feet, and divide the product by the sum of twice the height of the roof multiplied by the value of P, Table I., set opposite the kind of wood used in the tie beam, added to the difference of the values of C and P in said table multiplied by 12 times the length of the arc that measures the acute angle formed between the rafter and a vertical line, the arc having the height of the roof for radius (see are B G, Fig. 233), and the quo tient will be the area of the abutting surface of the joint at the foot of the rafter. To the abutting surface add its half, and the sum will be the area of the cross section of the rafter

Page 278

FRAMING. 273 This rule is upon the presumption that the rafter is secured from flexure by the roof beams and by braces and ties at short intervals, as in /%g. 233. In roofs where the rafter does not extend up to the ridge of the roof but abuts against a horizontal straining beam (e, Fig. 237), in the rule for rafters, take for the length of the rafter the distance from the foot of the rafter to the ridge of the roof; or, a distance equal to what the rafter would be in the absence of a straining beam. The area of cross section of the straining beam should be made equal to that of the rafter, as found by the rule so modified. Example.—Find the dimensions of a rafter for a roof truss whose span is 52 feet, and height 13; the length of the rafter being 29 feet, the trusses placed 10 feet apart from centres, and the arc*measuring the angle at the head of the rafter (having the ‘height of the roof for radius) being 144 feet, white pine being used in the tie beam. The height of this roof being in proportion to the span as 1 to 4, the value of Æ in Table IX. is 52°6; multiplying this, in accordance with the rule, by 52, the span of the roof, and by 29, the length of the rafter, and by 10, the distance between the roof trusses, the product is 793208. The value of P for white pine in Table I. is 300; multiplying this by 2 x 13 = 26, twice the height of the roof, the product is 7800. The value of C for white pine, (Table I.) is 1200, hence the difference of the values of C and P is 1200—300 = 900; this multiplied by 14, and by 144, the length of the arc, the product is 16031; this added to the 7800 aforesaid, the sum is 28831. The aforesaid product of 793208, divided by this 23831, the quotient,.33°3, equals the area in inches of the abutting surface of the joint at the tie beam. To this add 16:7, its half, and the sum, 50, equals the area of cross section of the rafter. This divided by the thick- ness of the rafter, say 6 inches, the quotient, 8%, is the breadth. The rafter is therefore to be 6 x 83 inches. It may be made 6 x 9, avoiding the fractions. , 35

Page 279

274 AMERICAN HOJSE-CARFENTER. 376.—The following table, caleulated upon data in Art. 358, presents the weight per foot for roofs of various inclinations, and covered with slate. TABLE IX. The vertical strain per foot of surface supported, measured horizontally, When height of roof is to span as on rafters = R == on braces = Q = 1 to 8 48 pounds 49°5 pounds, IET 48°6 “ 50:5 = 1 cis 6 49°4 “ 519 “se 1 ce 5 50°6 “ 54: ii 1“ 4 52°6 “ 57°6 “ 1) 563 “ 64" “ 1 66 DA 63°7 a 762 “ faa | Se s 101° “ To get the proportion that the height bears to the span, di- vide the span by the height; then unity will be to the quotient as the height is to the span. In case the quotient is not a whole number, the required value of /? or Q will not be found in the above table, but may be obtained thus: multiply the decimal part of the quotient by the difference of the values of T? set opposite the two proportions, between which the given proportion occurs as an intermediate, and subtract the product from the larger of the two said values of 2; the remainder will be the value of Æ required. The process is the same for the values of Q. Lxample.—A roof whose span is 60 feet, has a height of 2b feet. Then 60 divided by 25 equals 2-4. The proportion, therefore, between the height and span is 1 to 2*4. This pro- portion js an intermediate between 1 to 2 and 1to3. The values of X, opposite these two, are 63°7 and 56°3. The dif- ference between these values is 7-4; this multiplied by 0-4, the decimal portion of the quotient, equals 2°96; this subtracted from 63:7, the larger value of R, the remainder, 60°74, is the required value of A.

Page 280

FRAMING. 275 The values of Æ and Q are those for a roof covered with slate weighing 7 pounds per superficial foot of the roof sur- face.. When the roof covering is either lighter or heavier, sub- tract from or add to the table values, the sifference of weight between 7 pounds and the weight of the covering used, and the remainder, or sum, will be the value of 2 or Q required. 377.—Rule LIV. To obtain the dimensions of braces. Multiply the value of Q (Table IX., Art. 376) by the distance apart in feet at which the roof trusses are placed, and by the horizontal distance in feet from a point half-way to the next point of support of the rafter on one side of the brace, to a corresponding point on the other side. The product will be the weight in pounds sustained at the head of the brace. To this add the vertical strain (Art. 360) on the suspension rod located at the head of the brace, and make a vertical line dropped from the head of the brace, as F'e, Fig. 233, equal, by any convenient scale, to this sum, and draw the parallelo- gram #’feg. Then F'f, measured by the same scale, equals the pressure in the axis of the brace F'B. Multiply this pres- sure in pounds by the square of the length of the brace in feet, and divide the product by the breadth of the brace in ‘inches multiplied by the value of B (Table IL, Art. 293). The cube root of the quotient will be the thickness of the brace in inches. If this cube root should exceed the breadth of the brace, the result is not correct, and the calculation will have to be made anew, taking a larger dimension for the breadth. Example.—tThe brace /’ B (Fig. 233) is of white pine, and is required to sustain a pressure in its axis of 7000 pounds : (Art. 362). The length of the brace is 12 feet and its breadth 6 inches, what must be its thickness? Here 7000, the pressure, multiplied by 144, the square of the length, equals 1008000. The value of B is 1175; this by 6, the breadth of the brace, equals 7050. The product 1008000 divided by the product 7050 equals 143, the cube root of which, 5°23, is the required

Page 281

276 AMERICAN HOUSE-CARPENTER. thickness of the brace in inches. The brace will therefore be 5:23 by 6 inches, or 54 by 6. 378.—Rule LV. To obtain the area of the abutting sur- face of the ends of braces. Divide the number of degrees contained in the complement of the angle of inclination by 90, and multiply the quotient by the difference of the values of C’ and P, set opposite the kind of wood in the tie beam or rafter, in Table I., Art. 292; and to the product add the said value of P, and by the sum divide the pressure in the axis of the brace, and the quotient will be the area of the abutting sur- face. The complement of the angle of inclination referred to is, for the foot of the brace, the acute angle contained between the brace and a vertical line; and for the head of the brace, the acute angle contained between the brace and a line per- pendicular to the rafter. Example.—To find the abutting surface of the ends of the brace 7” B (Fig. 233). The complement of the angle of incli- nation, for the foot of the brace, is that contained between the lines /’ B and F E, and measures by the protractor, 45°. The tie beam is of white pine, and the values of P and C for this wood are 300 and 1200 respectively, and the pressure in the axis of the brace is 7000 pounds. Now by the rule, 45 di- vided by 90 equals 0°5, this by the 900, the difference of the values of C and P, equals 450; to this add 300, the value of P, and the sum is 750. The pressure in the axis of the brace, 7000, divided by this 750, equals 94, the required area of the abutting surface at the foot of the rafter. The complement of the angle of inclination for the head of the brace is that con- tained between the lines B F' and Fp, and measures by the protractor 19°. The rafter being of white pine, the values of P and C are as before. By the rule, 19 divided by 90 equals 0°23, and this multiplied by 900, the difference of the values of P and C, equals 190; to this add 300, the value cf P, and

Page 282

r FRAMING. 27% the sum is 490. The pressure, 7000, divided by this 490, equals 14:3 inches, the required area of the abutting surface at the head of the brace. 379.—To obtain the dimensions of the tie beam. Tie beams are subjected to two kinds of strain—tensile and transverse. Tule LVI.—To guard against the tensile strain, multiply the value of A (Table IX., Art. 376) by three times the distance apart at which the trusses are placed, and by the square of the span of the truss, both in feet. Divide this product by the value of T, (Table HI., Art. 308) set opposite the kind of wood in the tie beam, multiplied by 8 times the height of the roof in feet, and by the breadth of the tie beam in inches. The quotient will be the required depth in inches. The result thus obtained is usually smaller than that re- quired to resist the ‘cross strain to which the tie beam is sub- jected. The dimensions required to resist this strain, where there is simply the weight of the ceiling to support, may be obtained by this rule: Lule LVII.—Maultiply the cube of the longest unsupported part of the tie beam by 400 times the distance apart at which the trusses are placed, both in feet; and divide the product by the breadth of the tie beam in inches, multiplied by the value of Æ, (Table II., Art. 293) set opposite the kind of wood in the tie beam, and the cube root of the quotient will be the re- quired depth of the tie beam in inches. The result thus obtained may not be sufficient, in some cases, to resist the strains to which the tie beam is subjected in the hands of the workmen during the process of framing. kule LVUL—To resist these strains the area of cross sec- ` tion in inches should be at least equal to the length in feet. Example—the tie beam in Fig. 233. For this case we have the value of Æ 52-6, the trusses placed 10 feet from centres, the span 52 feet, the height 13 feet, the breadth 6 inches, and the value of 7 2367. Then by the rule, 52°6 x 3 x 10 x 52"

Page 283

aie, AMERICAN HOUSE-CARPENTER. =: 4266912, and 2367 x 8 x 13 x 6 = 1477008; the former product divided by the latter, the quotient equals 2-9, equals the required depth of the tie beam in inches. The other strains will require the depth to be more. To resist the cross strains, we have the longest unsupported part of the tie beam 83 feet, (this dimension is frequently greater than this,) distance from centres 10 feet, and breadth 6 inches. Then, by the rule, 823 x 400 x 10 = 2603852, and 6 x 1750 = 10500; the former produet divided by the latter, the quotient is 248, the cube root of which, 6°28, equals the required depth in inches. The tie beam therefore is to be 6 by 6°28 inches, or 6 x 7 inches. But if not guarded against severe accidental strains from careless handling this size would be too small. It would, in this case, require to be 52 inches area of cross section, say 6 x 9 inches. 380.—To obtain the diameter of the suspension rods, when made of good wrought iron. Rule LIX.—Divide the weight or vertical strain, in pounds, by 4000. The square root of the quotient will be the required diameter of the rod in inches. Exvample.—A. suspension rod is reqnired to sustain 16000 pounds, what must be its diameter? Dividing by 4000, the quotient is 4; the square root of which, 2, is the required diameter. The vertical strain on any rod is equal to the weight of so much of the ceiling as is supported by the rod, added to the vertical strain caused by each brace that is footed in the tie beam at the rod. The weight of the ceiling supported by a rod, is equal to ten times the distance apart in feet at which the trusses are placed, multiplied by half the distance in feet between the two next points of support, one on either side of the rod. The vertical strain caused by the braces can be as certained geometrically, as in Art. 360. 381.—When the suspension rods are located as in Fig. 233, dividing the span into equal y arts, the diameter of the rods

Page 284

FRAMING. 279 may be obtained without the preliminary calculation cf the strain, as follows :! Lule LX.—For the first rod from the wall. Multiply the distance apart at which the trusses are placed by the distance apart between the suspension rods, and divide the produet by 400. The square root of the quotient will be the required diameter of the rod. Example.—Rod D G, Fig. 233. In this figure the rods are located at 8% feet apart, and the distance between the trusses is 10 feet. Therefore, 10 x 83 = 862; this divided by 400, the quotient is 0:2167, the square root of which, 0°465, is the required diameter. The diameter may be half an inch. Kule LXI.—For the second rod from the wall. To the value of Q (Table IX., Art. 376) add 20, and multiply the sum by the distance apart at which the trusses are placed and by the distance between the rods, both in feet, and divide the pro- duct by 8000. The square root of the quotient will be the re- quired diameter. l ` Hxeample—Rod F E, Fig. 233. The distances apart in this case are as stated in last example. The value of Q is 57-6, and when added to 20 equals 77-6. Therefore, 77-6 x 10 x 83 = 66734; this divided by 8000, the quotient is 0ʻ8341, the square root of which, 0°91, is the required diameter. This rod may be one inch diameter. Rute LXII.—For the centre rod. To the value of Q (Table | IX., Art. 376) add 5, and multiply the sum by the distance apart at which the trusses are placed and by the distance apart between the rods, both in feet, and divide the produet by 2000. The square root of the quotient will equal the required diameter. Example.—Rod CB, Fig. 233. The distances apart as be- fore, and the value of Q the same. To Q add 5, and the sum is 62°6. Then 62°6 x 10 x 8% = 54254; this divided by 2000, the quotient is 2°7126, the square root of which, 1°647, equals the required diameter. This rod may be 12 inches diameter.

Page 285

280 AMERICAN HOUSE-CARPENTER. 382.—For all wrought iron straps and bolts the dimensions may be found by this rule. ; Lule LXHI.— Divide the tensile strain on the piece, in pounds, by 5000, and the quotient will be the area of cross section of the required bar or bolt, in inches. 383.—Roof-beams, jack-rafters, and purlins. All pieces of tinber subject to cross strains will sustain safely much greater strains when extended in one piece over two, three, or more distances between bearings; therefore roof-beams, jack-rafters, and purlins should, if possible, be made in as long lengths as practicable; the roof-beams and purlins laid on, not framed into, the principal rafters, and extended over at least two spaces, the joints alternating on the-trusses; and likewise the jack-rafters laid on the purlins in long lengths. The dimen- sions of these several pieces may be obtained by the following rule: . Rule LXTV.—From the value of Q (Table IX., Art. 376) deduct 10, and multiply the remainder by 33 times the distance from centres in feet at which the pieces are placed, and by the cube of the distance between bearings in feet; divide the pro- duct by the value of # (Table IL, Art. 298) for the kind of wood nsed and extract the square root of the quotient. The square root of this square root will be the required depth in inches. Multiply the depth thus obtained by the decimal 0-6, - and the product wilh be the required breadth in inches. Lxample.—Roof-beams of white pine placed 2 feet from cen- tres, resting on trusses placed 10 feet from centres, the height and the span of the roof being in proportion as 1 to 4. In this case the valne of Q is 57°6. By the rule, 57°6 — 10 = 47°6, and 47°6 x 33 x 2 x 10° = 3141600. This divided by 1750, the quotient is 1795-2, the square root of which is 42°37, and the square root of 42°37 is 6'5, the required depth. This multi- plied by 06 equals 3:9, the required breadth. These roof beams may therefore be 4 by 64 inches.

Page 286

? FRAMING. 281 384.—Five examples of roofs are shown at Figs. 234, 235, 236, 237, and 238. In Fig. 234, æ is an iron suspension rod, bb are braces. In Fig. 235, a, a, and 6 are iron rods, and dd,ec, are braces. In Fig. 236, æ b are iron rods, dd braces, and ¢ the straining beam, In Fig. 237, aa, bb, Fig. 236. are iron rods, ¢e, d d, are braces, and ¢ is a straining beam. In Fig. 238, purlins are located at P P, &c. ; the inclined beam that lies upon them is the jack-rafter; the post at the ridge is the king post, the others are queen posts. In this design the tie beam is increased in height along the middle by a strengthen- ing piece (Art. 348), for the purpose of sustaining additional weight placed in the room formed in the truss. 385.—Fig. 239 shows a method of constructing a truss having a built-rib in the place of principal rafters. The proper form for the curve is that of a parabola, (Art. 127.) This curve, © 86

Page 287

982 AMERICAN HOUSE-CARPENTER. when as flat as is described in the figure, approximates so near to that of the circle, that the latter may be used in its stead. The height, æ b, is just half of ac, the curve to pass through the middle of the rib. The rib is composed of two series of abutting pieces, bolted toge- ther. These pieces should be as long as the dimensions of the tim- ber will admit, in order that there may be but few joints. The sus- pending pieces are in halves, notched and bolted to the tie- beam and rib, and a purlin “is framed upon the upper end of each. <A truss of this construc- tion needs, for ordinary roofs, no diagonal braces between the sus- pending pieces, but if extra strength is required the braces may be added. The best place for the suspending picces is at the joints of the rib. A rib of this kind will be sufficiently strong, if the area of its section contain about one-fourth more timber, than is required for that of a raf- ter for a roof of the same size. The proportion of the depth to the thickness should be about as 10 is to 7. 386—Some writers have given designs for roofs similar to Fig. 240, having the tie-beam omitted for the accommodation of an arch in the ceiling. This and all similar designs are se-

Page 288

FRAMING. 283 riously objectionable, and should always be avoided; as the small height gained by the omission of the tie-beam can never Fig. 240. compensate for the powerful lateral strains, which are exerted by the oblique position of the supports, tending to separate the

Page 289

284 AMERICAN HOUSE-CARPENTER. walls. Where an arch is required in the ceiling, the best plan is to carry up the walls as high as the top of the arch. Then, by using a horizontal tie-beam, the oblique strains will be en- tirely removed. Many a public building, by my own obser- vation, has been all but ruined by the settling of the roof, consequent npon a defective plan in the formation of the truss in this respect. It is very necessary, therefore, that the hori- zontal tie-beam be used, except where the walls are made so strong and firm by buttresses, or other sapport. as to prevent a possibility of their separating. anh a Fig. 241. 387.—Fig. 241 isa method of obtaining the proper lengths and bevils for rafters in a hip-roof: ad and b ¢ are walls at the angle of the building; ġe is the seat of the hip-rafter and gf of a jack or cripple pane Draw e A, at right angles to be, and make it equal to the rise of the roof; join b and A, and 4 6 will be the length of the hip-rafter. Through e, draw dî, at right angles to be; upon J, with the radius, b A, describe the arc, A4, cutting dz in 7, join and į, and extend g f to meet bi ing ; then gj will

Page 290

FRAMING. 285 be the length of the jack-rafter. The length of each jack-rafter is found in the same manner—by extending its seat to cut the line, bi. From f, draw fk, at right angles tof g, also f l, at right angles to be; make f k equal to fl by the arc, l k, or make g k equal to g j by the arc, J k; then the angle at 7 will be the top- bevil of the jack-rafters, and the one at & will be the down-bevil.* 388.— To find the backing of the hip-rafter. At any con- venient place in b e, (Fig. 241,) as o, draw m n, at right angles to be; from o, tangical to b h, describe a semi-circle, cutting b e in s; join m and s and n and s; then these lines will form at s the proper angle for beviling the top of the hip-rafter. DOMES.İ Fig. 248. * The lengths and bevils of rafters for roof-valleys can also be found by ne above process t See also Art. 237.

Page 291

286 AMERICAN HOUSE-CARPENTER. 389.—The most usual form for domes is that of the sphere, the base being cireular. When the interior dome does rot rise toc high, a horizontal tie may be thrown across, by which any de- gree of strength required may be obtained. Jig. 242 shows a section, and Fg. 243 the plan, of a dome of this kind, a b being the tie-beam in both. Two trusses of this kind, (Fig. 242,) pa- rallel to each other, are to be placed one on each side of the open- ing in the top of the dome. Upon these the whole framework is to depend for support, and their strength must be calculated accord- ingly. (See the first part of this section, and Art. 356.) If the dome is large and of importance, two other trusses may be intro- duced at right angles to the foregoing, the tie-beams being pre- served in one continuous length by framing them high enough to pass over the others. Fig. 244. 390.—-When the interior dome rises too high to admit of a leve,

Page 292

FRAMING. ° 287 tie-beam, the framing may be composed of a succession of ribs standing upon a continuous circular curb of timber, as seen at Fg. 244 and 245,—the latter being a plan and the former a sec- tion. This curb must be well secured, as it serves in the place of a tie-beam to resist the lateral thrust of the ribs. In small domes, these ribs may be easily cut from wide plank ; but, where an extensive structure is required, they must be built in two thicknesses so as to break joints, in the same manner as is descri- bed for a roof at Art. 385. They should be placed at about two feet apart at the base, and strutted as at a in Fig. 244. 391.—The scantling of each thickness of the rib may be as follows : For domes of 24 feet diameter, 1x8 inches. cs (3 36 (13 14x10 cc We ‘ 60 (14 2x13 ec cc tr 90 ce 4x13 6c (44 (44 108 74 3x13 (14 392.—Although the outer and the inner surfaces of a dome may be finished to any curve that may be desired, yet the framing should be constructed of such a form, as to insure that the curve of equilibrium will pass through the middle of the depth of the framing. The nature of this curve is such that, if an arch or dome be constructed in accordance with it, no one part of the structure will be less capable than another of resisting the strains and pressures to which the whole fabric may be exposed. 'The curve of equilibrium for an arched vault or a roof, where the load is equally diffused over the whole surface, is that of a parabola, (Art. 127 ;) for a dome, having no lantern, tower or cupola above it, a cubic parabola, (Fig. 246 ;) and for one having a tower, &c., above it, a curveapproaching that of an hyperbola must be adopted, as the greatest strength is required at its upper parts. If the curve of a dome be circular, (as in the vertical section, Fig. 244,) ihe pressure will have a tendency to burst the dome 2utwards at about one-third of its height. ‘Therefore, when this form is used

Page 293

253 AMERICAN HOUSE-CARPENTER. in the construction of an extensive dome, an iron kand should be placed around the framework at that height; and whatever may be the form of the curve, a band or tie of some kind is necessary around or across the base. If the framing be of a form Jess convex than the curve of equilibrium, the weight will have a tendency to crush the ribs in- wards, but this pressure may be effectually overcome by strutting between the ribs ; and hence it is important that the struts be so placed as to form c ntinuous horizontal circles. ajai kge J e : Fig. 246. 393.— To describe a cubic parabola. Let a b, (Fig. 246,) be the base and b c the height. Bisect a bat d, and divide a d into 100 equal parts; of these give d e 26, e f 184, f g 14}, g h 128, h i 103, i j 93, and the balance, 83, to j a; divide b c into 8 equal parts, and, from the points of division, draw lines paralle! to a b, to meet perpendiculars from the several points of division in a b, at the points, o, o, o, &c. Then a curve traced through these points will be the one required. 394.—Small domes to light stairways, &c., are frequently made elliptical in both plan and section; and as no two of the ribs in one quarter of the dome are alike in form, a method for obtaining the curves is necessary. . 395.— Tn find the curves for the ribs of an elliptical dome Let a be d, (Fig. 247,) be the plan of a dome, and e f the seat

Page 294

FRAMING. Fig. 247. ot one of the ribs. Then take e ffor the transverse aris ana ‘twice the rise, o g, of the dome for the conjugate, and describe (according to Art. 115, 116, &c.,) the semi-ellipse, e æ f, which will be the curve required for the rib, eg f. The other ribs are found in the same manner. cal Fig. 248. 396.—To find the shape of the covering for a spherical dome. Let A, (Fig. 248,) be the plan and B the section of a given dome. From a, draw ac, at right angles to a b ; find the _Stretch-out, (Art. 92,) of o b, and make d c equal to it; divide the arc, o b, and the line, d c, each into a like number of equal! parts, 37

Page 295

290 AMERICAN HOUSE-CARPENTER. as 5, (a large number will insure greater accuracy than a small one ;) upon c, through the several points of division in ¢ d, describe the arcs, o d 0, Le 1, 2 f 2, &c.; make d o equal to half the width of one of the boards, and draw o s, parallel to a c; join s and a, and from the points of division in the arc, o b, drop perpendicu- lars, meeting a s in ij k l; from these points, draw 7 4, j 3, &c., parallel to a c; make d o, e 1, &c., on the lower side of a c, equal to d o, e 1, &c., on the upper side; trace a curve through the points, 0, 1, 2, 3,4, c, on each side of d c;, then o co will be the proper shape for the board. By dividing the circumference of the base, A, into equal parts, and making the bottom, o d o, of the board of a size equal to one of those parts, every board may be made of the same size. In the same manner as the above, the shape of the covering for sections of another form may be found, such as an ogee, cove, &c. 4 D Fig. 249. 397.— To find the curve of the boards when laid in horizon- fal courses. Let A B C, (Fig. 249,) be the section of a given dome, and D B its axis. Divide B C into as many parts as there are to be courses of boards, in the points, 1, 2, 3, &e.; through 1 and 2, draw a line to meet the axis extended at @; then @ will be the centre for describing the edges of the board, Æ. Through 3 and 2, draw 36 ; then b will be the centre for describing F. Through 4 and 3, draw 4d; then d will be the centre for G, B is the centre for the arc, 1 o. If this methoc is taker. tè find

Page 296

FRAMING. 991 the centres for the boards at the base of the dome, they would occur so distant as to make it impracticable : the following metl od is preferable for this purpose. G being the last board obtained by the above method, extend the curve of its inner edge until it meets the axis, D B, in e; from 3, through e, draw 3 f, meeting the arc, A B,in f; join f and 4, f and 5 and f and 6, cutting the axis, D B, in s, n and m ; from 4, 5 and 6, draw lines parallel to A Cand cutting the axis in c, p and r; make c4, (Fig. 250,) m oe | 8 4 c .4 Fig. 250. equal to c 4 in the previous figure, and c s equal to ¢ s also in the previous figure; then describe the inner edge of the board, H, according to Art. 87 : the outer edge can be obtained by gauging from the inner edge. In like manner proceed to obtain the next board—taking p 5 for half the chord and p n for the height of the segment. Should the segment be too large to be described easily, reduce; it by finding intermediate points in the curve, as at Art. 86. : H Fig. 251, 398.— To find the shape of the angle-rib for a polygonal deme. Let A G H, (Fig. 251,) be the plan of a given dome, and

Page 297

292 AMERICAN HOUSE-CARPENTER. C Da vertical section taken at the line, e f. From 1, 2,3, &c. in the arc, C D, draw ordinates, parallel to A D, to meet f G, from the points of intersection on f G, draw ordinates at right- angles to f G; make s 1 equal to o 1, s 2 equal to o 2, &c.; then G f B, obtained in this way, will be the angle-rib required. The best position for the sheathing-boards for a dome of this kind is horizontal, but if they are required to be bent from the base tc the vertex, their shape may be found in a similar manner to that shown at Fig. 248. BRIDGES. 399.—Various plans have been adopted for the construction of bridges, of which perhaps the following are the most useful. Fig. 252 shows a method of constructing wooden bridges, where the banks of the river are high enough to permit the use of the tie-beam, a b. The upright pieces, c d, are notched and bolted on in pairs, for the support of the tie-beam. A bridge of this construction exerts no lateral pressure upon the abutments. This method may be employed even where the banks of the river are low, by letting the timbers for the roadway rest immediately upon the tie-beam. In this case, the framework above will serve the purpose of a railing. Fig. 252. 400.—Fiig. 253 exhibits a wooden bridge without a tie-beam. Where staunch buttresses can be obtained, this method may be recommended; but if there is any doubt of their stability, ıt

Page 298

FRAMING. 293 Fig. 253. should not be attempted, as it is evident that such a system of framing is capable of a tremendous lateral thrust. Fig 254. 401.—FYg. 254 represents a wooden bridge in which a built-1 ib, (see Art. 385,) 1s introduced as a chief support. The curve of equilibrium will not differ much from that of a parabola: this, therefore, may be used—especially if the rib is made gradually a little stronger as it anproaches the buttresses. As it is desirable that a bridge be kept low, the following table is given to show the least rise that may be given to the rib. . Span in feet.| Least rise in feet. | Span in feet.| Least rise in feet. Spanin feet.| Least rise in feet. st Ss ee een 8 EE —E EEE | i 30 05 | 120 vi 280 24 40 0:8 140 8 300 23 50 1-4 160 10 320 32 60 2 180 11 350 39 | 70 24 200 12 380 47 80 3 220 14 400 53 90 | 4 | 240 m 100 5 Il 260 20 The rise should never be made less than this, but in all cases

Page 299

294 AMERICAN HOUSE-CARPENTER. greater if practicable; as a small rise requires a greater quantity of timber to make the bridge equally strong. ‘The greatest uni- form weight with which a bridge is likely to be loaded is, prota- bly, that of a dense crowd of people. ‘This may be estimated at 66 pounds per square foot, and the framing and gravelled road- way at 234 pounds more; which amounts to 300 pounds on a square foot. The following rule, based upon this estimate, may be useful in determining the area of the ribs. Mule LXV.— Multiply the width of the bridge by the square of half the span, both in feet; and divide this product by the rise in feet, multi- plied by the number of ribs; the quotient, multiplied by the decimal, 0:0011, will give the area of each rib in feet. When the roadway is only planked, use the decimal, 0:0007, instead of 0-0011. Laample.—What should be the area of the ribs for a bridge of 200 fect span, to rise 15 feet, and be 30 feet wide, with 3 curved ribs? The half of the span is 100 and its square is 10,000; this, multiplied by 30, gives 300,000, and 15, multi- plied by 3, gives 45; then 300,000, divided by 45, gives 66663, which, multiplied by 0:0011, gives 7:333 feet, or 1056 inches for the area of each rib. Such a rib may be 24 inches thick by 44 inches deep, and composed of 6 pieces, 2 in width and 3 in depth. Fig. 255. 402.—The above rule gives the area of a rib, that would be re- qnisite to support the greatest possible uñńiform load. But ir. large bridges, a variable load, such as a heavy wagon, is capable of exerting much greater strains; in such cases, therefore, the rib should be made larger. The greatest concentrated load a

Page 300

FRAMING, 295 bridge will be likely to encounter, may be estimated at from alout 20 to 50 thousand pounds, according to the size of the bridge. This is capable of exerting the greatest strain, when placed at about one-third of the span from one of the abutments, as at ò (Fig. 255.) The weakest point of the segment, b g c, is at g, the most distant point from the chord line. The pressure exerted at b by the above weight, may be considered to be in the direction of the chord lines, b a and b c ; then, by constructing the paral- lelograr. of ferces, e b f d, according to Art. 258, b f will show the vressure in the direction, b e. ‘Fhen the scantling for the rib may be found by the following rule. kule LXV1.—Multiply the pressure in pounds in the direc- tion be, by the distance g A, and by the square of the distance bc, both in feet; and divide the product by the united breadth in inches of the several ribs, multiplied by the value of B (Table II., Art. 293) for the kind of wood used; and the cube root of the quotient will be the required depth of the rib in inches. l Erample.—A bridge is to have three white pine ribs each 20 inches wide; the pressure in the direction b c, (Fig. 255) is-equal to 60,000 pounds, the distance be equals 60 feet, and the distance gh equals 10 feet. What must be the depth of the ribs, the value of B (Table II.) being for white pine 1175? Here, by the rule, 60,000 x 10 x 60° = 2,160,000,000. Then 1175 x 3 x 20 = 70,500. The former product divided by the latter equals 30,638, the cube root of which, 31:29, equals the required depth in inches, The ribs are, therefore, to be 20 by 314 inches. 403.—In constructing these ribs, if the span be not over 50 feet, each rib may be made in two or three thicknesses of timber, (three thicknesses is preferable) of convenient lengths bolted together ; but, in larger spans, where the rib will be such as to render it difficult to procure timber of sufficient breadth, they may be constructed by bending the pieces to the proper curve

Page 301

296 AMERICAN HOUSE-CARPENTER. and bolting them together. In this case, where timber of suffi sient length to span the opening cannot be obtained and scarfing is necessary, such joints must be made as will resist both tension and compression, (see Fig. 264) ‘To ascertain the greatest depth for the pieces which compose the rib, so that the process of bend ing may not injure theirelasticity, multiply the radius of curvature in feet by the decimal, 0°05, and the product wil: se the depth ır. inches. 'zample.—Suppose thecurve of the rib to be described with a radius of 100 feet, then what should be the depth? The radius in feet, 100, multiplied by 0-05, gives a product of 5 inches. White pine or oak timber, 5 inches thick, would freely bend tc the above curve ; and, if the required depth of such a rib be 2t inches, it would have to be composed of at least 4 pieces. Pitch pine is not quite so elastic as white pine or oak—its thickness may be found by using the decimal, 0'046, instead of 0-05. Fig. 256, 404.—When the span is over 250 feet, a framed rib, formed as in Fig. 256, would be preferable to the foregoing. Of this, the upper and the lower edges are formed as just described, by bend- ing the timber to the proper curve. The pieces that tend to the centre of the curve, called radials, are notehed and bolted on in pairs, and the cross-braces are halved together in the middle, and abut end to end between the radials. ‘The distance between the ribs of a bridge should not exceed atout 8 fect. The roadway

Page 302

FRAMING. 297 should be supported by vertical standards bolted to the ribs ai about every 10 to 15 feet. At the place where they rest on the ribs, a double, horizontal tie should be notched and bolted on the back of the ribs, and also another on the under side; and diago- na! braces should be framed between the standards, over the space between the ribs, to prevent lateral motion. The timbers for the roadway may be as light as their situation will admit, as all use- less timber is only an unnecessary load upon the arch. 405.—It is found that if a roadway be 18 feet wide, two car- riages can pass one another without inconvenience. Its width, therefore, should be either 9, 18, 27 or 36 feet, according to the amount of travel. The width of the foot-path should be 2 feet for every person. When a stream of water has a rapid current, as few piers as practicable should be allowed to obstruct its course ; otherwise the bridge will be liable to be swept away by freshets. When the span is not over 300 feet, and the banks of the river are of sufficient height to admit of it, only one arch should be employed. ‘The rise of the arch is limited by the form of the roadway, and by the height of the banks of the river (See Art. 401.) The rise of the roadway shou!d not exceed one in 24 feet, but, as the framing settles about one in 72, the roadway should be framed to rise one in 18, that it may be one in 24 after settling. 'Thecommencement of the arch at the abutments—the spring, as it is termed, should not be below high-water mark : and the bridge should be placed at right angles with the course of the current. 406.—The best material for the abutments and piers of a bridge, is stone ; and, if possible, stone should be procured for the i purpose. The following rule is to determine the extent of the abutments, they being rectangular, and built with stone weighing 120 lbs. to a ecubic-foot. Jule LX VII.—Multiply the square of the height of the abutment by 160, and divide this product by the weight of a square foot of the arch, and by the rise of the arch; add unity to the quotient, and extract the square-root. Diminish the square-root by unity, and m uitiply the root, so diminished, by

Page 303

298 AMERICAN NOUSE-CARPENTER. half the span of the arch, and by the weight of a square-foot of the arch. Divide the last product by 120 times the height of the abutment, and the quotient will be the thickness of the abutment. Evample.—Let the height of the abutment from the base to the springing of the arch be 20 feet, half the span 100 feet, the weight of a square foot of the arch, including the greatest possible load upon it, 300 pounds, and the rise of the arch 18 feet—what should be its thickness? The square of the height of the abutment, 400, multiplied by 160, gives 64,000, and 300 by 18, gives 5400 ; 64,000, divided by 5400, gives a quotient of 11:852, one added to this makes 12:852, the square-root of which is 3°6 ; this, less one, is 2°6; this, multiplied by 100, gives 260, and this again by 300, gives 78,000 ; this, divided by 120 times the height of the abut- ment, 2400, gives 32 feet 6 inches, the thickness required. The dimensions of a pier will be found by the same rule. For, although the thrust of an arch may be balanced by an ad- joining arch, when the bridge is finished, and while it remains uninjured ; yet, during the erection, and in the event of one arch being destroyed, the pier should be capable of sustaining the en- tire thrust of the other. — l 407.—Piers are sometimes constructed of timber, their princi- pal strength depending on piles driven into the earth, but such piers should never be adopted where it is possible to avoid them; for, being alternately wet and dry, they decay much socner than the upper parts of the bridge. Spruce and elm are considered good for ‘piles. Where the height from the bottom of the river to the roadway is great, it is a good plan to cut them off at a little below low-water mark, cap them with a horizontal tie. and upon this erect the posts for the support of the roadway. This method cuts off the part that is continually wet from that which is only occasionally so, and thus affords an opportunity for replacing the upper part. The pieces which are immersed will last a great length of time, especially when of elm; for it isa well-established fact, that timber is less durable when subject te

Page 304

FRAMING. 299 altern: te dryness and moisture, than when it is either continually wet or continually dry. It has been ascertained that the piles ander London bridge, after having been driven about 600 years, were not materially decayed. ‘These piles are chiefly of elm, and vholly immersed. Fig. 257. 408.—Centres for stone bridges. Fig. 257 is a design for a centre for a stone bridge where intermediate supports, as piles driven into the bed of the river, are practicable. Its timbers are so distributed as to sustain the weight of the arch-stones as they are being laid, without destroying the original form of the centre ; and also to prevent its destruction or settlement, should any of the piles be sweptaway. The most usual error in badly-constructed centres is, that the timbers are disposed so as to cause the framing to rise at the crown, during the laying of the arch-stones up the sides. To remedy this evil, some have loaded the crown with heavy stones ; but a centre properly constructed will need no such precaution. Experiments have shown that an arch-stone does not.press apon the centring, until its bed is inclined to the horizon at an angle of from 30 to 45 degrees, according to the hardness of the stone, and whether it is laid in mortar or not. For general pur- poses, the point at which the pressure commences, may be con- sidered to be at that joint which forms an angle of 32 degrees with the horizon. At this puint, the pressure is inconsiderable,

Page 305

300 AMERICAN HOUSE-CARPENTER. but gradually increases towards the crown. The following table gives the portion of the weight of the arch stones that presses upon the framing at the various angles of inclination formed by the bed of the stone with the horizon. The press- ure perpendicular to the curve is equal to the weight of the arch stone multiplied by the decimal 0, when the angle of inclination is 32 degrees. mo pe o a aa aeea qa S mu a oi Gg t oy ge a estes Se ee a I: Pats ies n g o uy 25 (23 (1 “ 6s tty & 44 (1 moan a g a oag S so apt os i as m a ee g (5 e aia ee a e G å a a Moa a a cee ca eos a es ee ee 2 eS St et a a g u ra ag ao Lo a From this it is seen that at the inclination of 44 degrees the pressure equals one-quarter the weight of the stone; at 57 de- grees, half the weight; and when a vertical line, as æ b, (Fig. 258,) passing through the centre of gravity of the arch-stone, does not fall within its bed, cd, the pressure may be considered equal to the whole weight of the Pe stone. This will be the case at about 60 de- grees, when the depth of the stone is double its breadth. The direction of these pressures is considered in a line with the ra- dius of the curve. The weight upon a centre being known, the pressure may be estimated and the timber calculated ac- cordingly. But it must be remembered that the whole weight is never placed upon the framing at once—as seems to have

Page 306

FRAMING. 301 been the idea had in view by the designers of some centres. In building the arch, it chould be commenced at each buttress at the same time, (as is generally the case,) and each side should progress equally towards the crown. In designing the framing, the effect produced by each successive layer of stone should be considered. The pressure of the stones upon one side should, by the arrangement of the struts, be counterpoised by that of the stones upon the other side. 409.—Over a river whose stream is rapid, or where it is ne- cessary to preserve an uninterrupted passage for the purposes of navigation, the centre must be constructed without interme- diate supports, and without a continued horizontal tie at the Fig. 259. base; such a centre is shown at Fig. 259. In laying the stones from the base up to a and c, the pieces, d and 6 d, act as ties to prevent any rising at b After this, while the stones are being laid from a and from ¢ to b, they act as struts: the piece, Jg, is added for additional security. Upon this plan, with some variation to suit circumstances, centres may be con- structed for any span usual in stone-bridge building. 410.—In bridge centres, the principal timbers should abut, and not be intercepted by a suspension or radial piece between. These should be in halves, notched on each side and bolted. The timbers should. intersect as little as possible, for the more

Page 307

302 AMERICAN HOUSE-CARPENTER. joints the greater is the settling; and halving them together is a bad practice, as it destroys nearly one-half the strength of the timber. Tiesshould be introduced across, especially where many timbers mect; and as the centre is to serve but a tem- porary purpose, the whole should be designed with a view to employ the timber afterwards for other uses. For this reason, all unnecessary cutting should be avoided. 411.—Centres should be sufficiently strong to preserve a staunch and steady form during the whole process of building; for any shaking or trembling will have a tendency to prevent the mortar or cement from setting. For this purpose, also, the centre should be lowered a trifle immediately after the key-stone is laid, in order that the stones may take their bear- ing before the mortar is set; otherwise the joints will open on the under side. The trusses, in centring, are placed at the distance of from 4 to 6 feet apart, according to their strength and the weight of the arch. Between every two trusses, diago- nal braces shonld be introduced to prevent lateral motion. 412.—In order that the centre may be easily lowered, the frames, or trusses, shonld be placed upon wedge-formed sills ; as is shown at d, (Fig. 259.) These are contrived so as to ad- mit of the settling of the frame by driving the wedge, d, with, a maul, or, in large centres, with a piece of timber mounted as a battering-ram. The operation of lowering a centre shoul be very slowly performed, in order that the parts of the arch may take their bearing uniformly. The wedge pieces, instead of being placed parallel with the truss, are sometimes made sufficiently long and laid through the arch, in a direction at right angles to that shown at Zig. 259. This method obviates the necessity of stationing men beneath the arch during the process of lowering; and was originally adopted with success soon after the occurrence of an accident, in lowering a centre, by which nine men were killed. 413.—To give some idea of the manner of estimating the pres

Page 308

FRAMING. 303 sures, in order to select timber of the proper scantling, calculate (Ait. 408) the pressure of the arch-stones from ¢ to b, (Fig. 259,) and suppose half this pressure concentrated at a, and acting in the direction af. Then, by the parallelogram of forces, (Art. 258,) the strain in the several pieces composing the frame, bda, may be computed. . Again, calculate the pressure of that portion of the arch included between æ and c, and consider half of it collected at 6, and acting in a vertical direction ; then, by the parallelogram of forces, the pressure on the beams, bd and bd, may be found. Add the pressure of that portion of the arch which is included: between ¢ and b to half the weight of the centre, and consider this amount concentrated at d, and acting in a vertical direction; then, by constructing the parallelogram of forces, the pressure upon dj may be as- certained. 414.—The strains having been obtained, the dimensions of the several pieces in the frames b a d and b c d, may be found by computation, as directed in the case of roof trusses, from Arts. 875 to 880. The tie-beams 6 d, b d, if made of sufficient size to resist the compressive strain acting upon them from the load at b, will be more than large enough to resist the tensile strain upon them during the laying of the first part of the arch-stones below @ and e. ` 415.—In the construction of arches, the voussoirs, or arch- stones, are so shaped that the joints between them are perpen- dicular to the curve-of the arch, or to its tangent at the point at which the joint intersects the curve. Ina circular arch, the joints tend toward the centré of the circle: in an elliptical arch, the joints may be found by the following process : Fig. 260,

Page 309

304 AMERICAN HOUSE-CARPENTER. 416.— To find the direction of the joints for an elliptical arch, A joint being wanted at a, (Fig. 260,) draw lines from that point to the foci, f and f; bisect the angle, fa f, with the line, a6, then ab will be the direction of the joint. Fig. 261, hg 417.—To find the direction of the joints for a parabolic arch A joint being wanted at a, (Fig. 261,) draw ae, at right angles to the axis, eg; make cg equal to ce, and join a and g; draw ah, at right angles to ag; then a A will be the direction of the joint. The direction of the joint from 6 is found in the same manner. The lines, ag and b f, are tangents to the curve at those points respectively; and any number of joints in the curve may be obtained, by first ascertaining the tangents, and then drawing lines at right angles to them, JOINTS. Cae Fig. 262. 418.—Fig. 262 shows a simple and quite strong method of lengthening a tie-beam; but the strength consists wholly in the bolts, and in the friction of the parts produced by screwing the pieces firmly together. Should the timber shrink to even a small degree, the strength would depend altogether on the bolts. It would be made much stronger by indenting the pieces together; as at the upper edge of the tie-beam in Z'ig. 263; or by placing keys in the joints, as at the lower edge in

Page 310

FRAMING. 305 the same figure. This process, however, weakens the beam in proportion to the depth of the indents. Fig. 263. 419.—/ig. 264 shows a method of scarfing, or splicing, a tue-beam without bolts. The keys are to be of well-seasoned, Fig, 264. nard wood, and, if possible, very cross-grained. The addition of bolts would make this a very strong splice, or even white- oak pins would add materially to its strength. Aen Fig. 265. 490.—Fig. 265 shows about as strong a splice, perhaps, as can well be made. Itis to be recommended for its simplicity ; as, on account of there being no oblique joints in it, it can be readily and accurately executed. A complicated joint is the worst that can be adopted; still, some have proposed joints that seem to have little else besides complication to recom mend them. ` 421.—In proportioning the parts of these scarfs, ‘the depths “ of all the indents taken together should be equal to one-third of the depth of the beam, In oak, ash or elm, the whole length of the scarf should be six times the depth, or thickness, of the beam, when there are no bolts; but, if bolts instead of indents are used, then three times the breadth ; and, when both methods are combined, twice the depth of the beam. The 39

Page 311

306 AMEKICAN HOUSE-CARPENTER. . length of the scarf in pine and similar soft woods, depending wholly on indents, should be about 12 times the thickness, or depth, of the beam; when depending wholly on bolts, 6 times the breadth; and, when both methods are combined, 4 times the depth. Fig. 266. 422.—Sometimes beams have to be pieced that are required to resist cross strains—such as a girder, or the tie-beam of a roof when supporting the ceiling. In such beams, the fibres of the wood in the upper part are compressed; and therefore a simple butt joint at that place, (as in 7%. 266,) is far prefer- able to any other. In such case, an oblique joint is tlie very worst. The under side of the beam being in a state of tension, it must be indented or bolted, or both; and an iron plate un- der the heads of the bolts, gives a great addition of strength. Searfing requires accuracy and care, as all the indents should bear equally ; otherwise, one being strained more than another, there would be a tendency to splinter off the parts. Hence the simplest form that will attain the object, is by far the best. In all beams that are compressed endwise, abutting joints, formed at right angles to the direction of their length, are at once the simplest and the best. For a temporary purpose, Fig. 262 would do very well; it would be improved, however, by having a piece bolted on all four sides. Zig. 263, and indeed each of the others, since they have no oblique joints, would resist compression well. 423.—In framing one beam into another for bearing pur poses, such as a floor-beam into a trimmer, the best place to make the mortice in the trimmer is in the neutral line, (Arts 317, 818,) which is in the middle of its depth. Some have thought that, as the fibres of the npper edge are compressed, a

Page 312

FRAMING. 307 mortice might be made there, and the tenon driver. in tight enough to make the parts as capable of resisting the compres- sion, as they would be without it; and they have therefore concluded that plan to be the best. This could not be the case, even if the tenon would not shrink; for a joint between two pieces cannot possibly be made to resist compression, so well as a solid piece without joints. The proper place, therefore, for the mortice, is at the middle of the depth of the beam; but the best place for the tenon, in the floor-beam, is at its bottom edge. For the nearer this is placed to the upper edge, the greater is the liability for it to splinter off; if the joint is Fig. 261. formed, therefore, as at Fig. 267, it will combine all the ad- vantages that can be obtained. Double tenons are objection- able, because the piece framed into is needlessly weakened, and the tenons are seldom so accurately made as to bear equally. For this reason, unless the tusk at æ in the figure fits exactly, so as to bear equally with the tenon, it had better be omitted. And in sawing the shoulders, care’should be taken not to saw into the tenon in the least, as it would wound the beam in the place least able to bear it. 424.—Thus it will be seen that framing weakens both pieces, imore or less. It should, therefore, be avoided as much as pos- sible; and where it is practicable one piece should rest upon the other, rather than be framed into it. This remark applies to the bearing of floor-beams on a girder, to the purlins and jack-rafters of a roof, &c. 425.—In a framed truss for a roof, bridge, partition, &c., the joints should be so constructed as to direct the pressures

Page 313

308 AMERICAN HOUSE-CARPENTER. through the axes of the several pieces, and also to avoid every tendency of the parts to slide. To attain this object, the abut- Fig. 268, Fig. 269. Fig. 270. ting surface on the end of a strut should be at right angles to the direction of the pressure; as at the joint shown in Fig. 268 for the foot of a rafter, (see Art. 277,) in Fig. 269 for the head of a rafter, and in Fig. 270 for the foot of a strut or brace. The joint at Fig. 268 is not cut completely across the tie-beam, but a narrow lip is left standing in the middle, and a corresponding indent is made in the rafter, to prevent the parts from separating sideways. The abutting surface should be made as large as the attainment of other necessary objects will admit. The iron strap is added to prevent the rafter slid- ing out, should the end of the tie-beam, by decay or otherwise, splinter off. In making the joint shown at Fig. 269, it should be left a little open at a, so as to bring the parts to a fair bear- ing at the settling of the truss, which must necessarily take place from the shrinking of the king-post and other parts. If the joint is made fair at first, when the truss settles it will cause it to open at the under side of the rafter, thus throwing the whole. pressure upon the sharp edge at a. This will cause an indentation in the king-post, by which the truss will be made to settle further; and, this pressure not being in the axis of the rafter, it will be greatly increased, thereby rendering the rafter liable to split and break. 426.—If the rafters and struts were made to abut end to end, as in Figs. 271, 272 and 273, and the king or queen post notched on in halves and bolted, the ill effects of shrinking

Page 314

FRAMING. 309 wonld be avoided. This method has Deen practised with suc- cess, in some of the most celebrated bridges and roofs in Eu- Fig. 271. Fig. 272. Fig. 278. rope; and, were its use adopted in this country, the unseemly sight of a hogged ridge would seldom be met with. A plate of cast iron between the abutting surfaces will equalize the pressure. — Fig. 274 Fig. 275. 427.— Fig. 274 is a proper joint for a collar-beam in a small roof: the principle shown here should characterize all tie- joints. The dovetail joint, although extensively practised in the above and similar cases, is the very worst that can be em- ployed. The shrinking of the timber, if only to a small de- „gree, permits the tie to withdraw—as is shown at Fig. 275. The dotted line shows the position of the tie after it has shrunk. ` 428.—Locust and white-oak pins are great additions to the strength of a joint. In many cases they would supply the place of iron bolts; and, on account of their small cost, they should be nsed in preference wherever the strength of iron is

Page 315

310 AMERICAN HOUSE-CARPENTER. not requisite. In small framing, good cut’ nails are of great service at the joints; but they should not be trusted to bear any considerable pressure, as they are apt to be brittle. Iron straps are seldom necessary, as all the joinings in carpentry may be made without them. They can be used to advantage, , however, at the foot of suspending-pieces, and for the rafter at the end of the tie-beam. In roofs for ordinary purposes, the iron straps for suspending-pieces may be as follows: When the longest unsupported part of the tie-beam is 10 feet, the strap may be 1 inch wide by ,3, thick. 15 (Z4 (29 19 14 (as (2s ł (<9 90 (<4 (29 (a3 9, (<3 (23 4 « In fastening a strap, its hold on the suspending-piece will be much increased by turning its ends into the wood. Iron straps should be protected from rust; for thin plates of iron decay very soon, especially when exposed to dampness. For this purpose, as soon as the strap is made, let it be heated to about a blue heat, and, while it is hot, pour over its entire surface raw linseed oil, or rub it with beeswax. Either of these*will give it a coating which dampness will not penetrate. IRON GIRDERS. Fig. 976 Fig, 977. 429,— Fig. 276 represents the front view, and Fig. 277 the cross section at middle, of a cast iron girder of proper form for sustaining a weight equally diffused over its length. The curve is that of a parabola: generally an arc of a circle is

Page 316

FRAMING. i 311 used, and is near enough. Beams of this form are much used to sustain brick walls of buildings; the brickwork resting upon the bottom flange, and laid, not arching, but horizontal. In the cross section, the bottom flange is made to contain in area four times as much as the top flange. The strength will be in proportion to the area of the bottom flange, and to the height or depth. Hence, to obtain the greatest strength from a given amount of material, it is requisite to make the upright part, or ` the blade, rather thin ; yet, in order to prevent injurious strains in the casting while it is cooling, the parts should be nearly equal in thickness. The thickness of the three parts—blade, top flange and bottom flange, may be made in proportion.as 5, 6 and 8. For a beam of this form, the weight equally dif- fused over it equals w= gone”, (199.) l The depth equals | Lw The area of the bottom flange equals lw = Joor (201.) where w equals the weight in pounds equally diffused over the length; d, the depth, or height in inches of the cross section at middle; a, the area of the bottom flange in inches; J, the length of the beam in feet, in the clear between the bearings ; and ¢, a decimal in proportion to unity as the safe weight is to the breaking weight. This is usually from 0-2 to 0°3, or from one-fifth to one-third, at discretion. 430.—Beams of this form, laid in series, are much used in sustaining brick arches turned over vaults and other fire-proof rooms, forming a roof to the vault or room, and a floor above; the arches springing from the flanges, one on either side of the beam, as in Fig. 278.

Page 317

312 AMERICAN HOUSE-CARPENTER. For this use the depth of cross section at middle equals eS aay d — 9000ta (202.) The area of the bottom flange equals es oa (203.) where the symbols signify as before, and ¢ equals the distance apart from centres in feet at which the beams are placed, and J the weight per superficial foot, in pounds, including the weight of the material of which the floor is constructed. Practical Rules and Examples. 431.—For a single girder the dimensions may be found by the following rule, ((200) and (201):) Rule LX VUI.—Divide the weight in pounds equally dif- fused over the length of the girder by a decimal in proportion to unity as the safe weight is to the breaking weight, multiply the quotient by the length in feet, and divide the product by 9000. Then this quotient, divided by the depth of the beam at middle, will give the area of the bottom flange; or, if di- vided by the area of the bottom flange, will give the depth— the area and depth both in inches. Example—Let the weight equally diffused over a girder equal 60000 pounds ; the decimal that is in proportion to unity as the safe weight is to be to the breaking weight, equal 0°3

Page 318

FRAMING. 813 the length in the clear of the bearings equal 20 feet. Then 60000 divided by 0'3 equals 200000, and this by 20 equals 4000000; this divided by 9000 equals 4444. Now if the depth is fixed, say at 20 inches, then 4444, divided by 20, equals 223, equals the area of the bottom flange in inches. But if the area is given, say 24 inches, then to find the depth, divide 4444 by 24, and the quotient, 18°5, equals the depth in inches; and such a girder may be made with a bottom flange of 2 by 12 inches, top flange, (equal to 4 of bottom flange,) 14 by 4 inches, and the blade 14 inches thick. 432.—For a series of girders or iron beams, the dimensions may be found by the following rule: (202) and (208). Rule LXIX.—Divide the weight per superficial foot, in pounds, by a decimal in proportion to unity as the safe weight is to the breaking weight, and inultiply the quotient by the square of the length of the beams and by the distance apart at > which the beams are placed from centres, both in feet, and di- vide the product by 9000. Then this quotient, divided by the depth of the beams at middle, will give the area of the bottom flange; or, if divided by the area of the bottom flange, will give the depth of the beam—the depth and area both in inches. Example.—Let the weight per superficial foot resting upon an arched floor be 200 pounds, and the weight of the arches, concrete, &c., equal 100 pounds, total 300 pounds per superfi- cial foot. Let the proportion of the breaking weight to be trusted on the beams equal 0:3, the length of the beams in the ` clear of the bearings equal 12 feet, and the distance apart from centres at which they are placed equal 4 feet. Then 300 di- vided by 0°3 equals 1000; this multiplied by 144 (the square of 12), equals 144000, and this by 4, equals 576000; this di- vided by 9000, equals 64. Now if the depth is fixed, and at 8 inches, then 64 divided by 8 equals 8, equals the area of the bottom flange. But if the area of the bottom flange is fixed, and at 6 inches, then 64, divided by 6, equals 10%, the depth 40 ;

Page 319

alt AMERICAN HOUSE-CARPENTER. required. Such a beam may be made with the bottom flange 1 by 6 inches, the top flange, (equal to one-quarter of the bot- tom flange,) ¢ by 2 inches, and the blade $ inch thick. 433.—The kind of girder shown at Fig. 280, (a cast iron arch with a wrought iron tie rod,) is extensively used as a sup- port upon which to build brick walls where the space below is required to be free. The objections to its use are, the dispro- portion between the material and the strains, and the enhanced cost over the cast iron girder formed as in Figs. 276 and 277. The material in the cast arch, (Fig. 280,) is greatly in excess over the amount needed to resist effectually the compressive strains induced by the load through the axis of the arch, while the wrought metal in the tie is usually much less than is re- quired to resist the horizontal thrust of the arch; absolute fail- ure being prevented, partly by the weight of the walls resting on the haunches, and partly by the presence of adjoining buildings, their walls acting as buttresses to the arch. Some instances have occurred where the tie has parted. Where this arched girder is used it is customary to lay the first courses of brick in the form of an arch. This brick arch of itself is quite sufficient to sustain the compressive strain, and, were there proper resistance to the horizontal thrust pro- vided, the brick arch would entirely supersede the necessity for the girder. Indeed, the instances are not rare where con- structions of this nature have proved quite satisfactory, the horizontal thrust of the arch being sustained by a tie rod secured to a pair of cast iron heel plates, as in Fig. 219. The

Page 320

FRAMING. 81b brick arch, in this case, being built upon a wooden centre, which was afterwards removed. The diameter of the rod required for an arch of this kind is equal to ws D= z000 F (204.) where w equals the weight in pounds equally diffused over the arch; s, the length of the rod, clear of the heel plates, in feet ; and A, the height at the middle, or rise, of the arc, in inches; D, the diameter, being also in inches. When the diameter found by formula (204) is impracticably large, this difficulty may be overcome by dividing the metal into two rods. In the bow-string girder, (Fig. 280,) two rods cannot be used with advantage, because of the difficulty in adjusting their lengths so as to ensure to each an equal amount of the strain. But in the case of the brick arch, the two heel plates being disconnected, any discrepancy of length in the rods is adjusted simply by the pressure of the arch acting on the plates. When there are to be two rods, the diameter of each rod equals WS D=\/— (205.) Practical Rule and Example. 434.—To obtain the diameter of wrought iron tie-rods for heel plates, asin Fig. 279, proceed by this rule. - Kule LXX.—Multiply the weight in pounds equally distri- buted over the arch by the length of the tierod in feet, clear of the heel plates, and divide the product by the height of the arc in inches, (that is, the height at the middle, from the axis of the tie-rod to the centre of the depth of the brick arch,) then, if there is to be but one tie-rod, divide the quotient by 3000 ;

Page 321

316 AMERICAN HOUSE-CARPENTER. but if two, then divide by 6000, and the square root of the quotient, in either case, will be the required diameter. Example.—The weight to be supported on a brick arch, equally distributed, is 24000 pounds; the length of the tie-rod, clear of the heel plates, is 10 feet; and the height, or rise, of the arc is 10 inches. Now by the rule, 24000 x 10 = 240000. This divided by 10, equals 24000. Upon the presumption that one tie-rod only will be needed, divide by 3000, and the quo- tient is 8, the square root of which is 2°82 inches. This is rather large, therefore there had better be two rods. In this case the quotient, 24000, divided by 6000, equals 4, the square root of which is 2, the diameter required. The arch should, therefore, have two rods of 2 inches diameter. Two rods are preferable to one. The iron is stronger per inch in small rods than in large ones, and the rules require no more metal in the two rods than in the one. Fig. 280. 435.—The Bow-string Girder, as per Fig. 280, has little to recommend it, (see Art. 433,) yet because it has by some been much used, it is well to show the rules that govern its strength, if only for the benefit of those who are willing to be governed by reason rather than precedent. To resist the horizontal thrust of the cast arch, the diameter of the rod must equal (204) u Veron 3000 A’ But the cast iron arch has a certa’n amount of strength to re

Page 322

FRAMING. alr ast cross strains: this strength must be considered. Upon the presumption that the cross section of the cast arch at the mid- dle is of the most favorable form, as in Fig. 277, or at least that it have a bottom flange, (although the most of those cast -are without it), the strength of the cast arch to resist cross strains is shown by formula (199), when J, its length, is changed to s, its span. The weight in pounds equally diffused over the arch will then equal 9000 tad a te. This is the weight borne by the cast arch acting simply as a beam. Deducting this weight from the whole weight, the re- mainder is the weight to be sustained by the rod. Calling the whole weight w, then 9000t¢ad ws — 9000 tad = W 8 8 Therefore, from (204), thè diameter equals Wes 8000 A = (es 8000 A ws—9000tad 7 Va 30004 ’ a where D equals the diameter of the rod in inches; w, the weight in pounds equally diffused over the arch; s, the span of the arch in feet; A, the rise or height of the arc at middle, in inches; g, the height or depth of the cross section of the cast arch in inches; a, the area of the bottom flange of the cross section of the cast arch in inches; and ¢, a decimal in proportion to unity as the safe weight is to be to the breaking weight. The rule in words at length, is Rule LXXI.—Multiply the decimal in proportion to unity

Page 323

318 AMERICAN HOUSE-CARPENTER. as the safe weight is to be to the breaking weight, by 9000 times the depth of the cross section of the cast arch at middle, and by the area of the bottom flange of said section, both in inches, and deduct the product from the weight in pounds equally diffused over the arch multiplied by the span in feet, and divide the remainder by 3000 times the height of the arc in inches, measured from the axis of the tie-rod to the centre of the depth of.the cast arch at middle, and the square root of the quotient will be the diameter of the rod in inches. Example.—tThe rear wall of a building is of brick, and is 40 feet high, and 21 feet wide in the clear between the piers of the story below. Allowing for the voids for windows, this wall will weigh about 63000 pounds; and it is proposed to support it by a bow-string girder, of which the cross section at middle of the cast arch is 8 inches deep, and has a bottom flange containing 12 inches area. The rise of the curve or arc is 24 inches. What must be the diameter of the rod, the por- tion of the breaking weight of the cast arch, considered safe to trust, being three-tenths or 0°3% By the rule, 0-3 x 9000 x 8 x 12 = 259200; then 63000 x 21 — 259200 = 1063800. This remainder divided by (3000 x 24 =) 72000, the quotient equals 14°775 ; the square root of which, 3°84, or nearly 3% inches, is the required diameter. This size, though impracticably large, is as small as a due regard for safety will permit; yet it is not unusual to find the rods in girders intended for as heavy a load as in this exam- ple, only 24 and 24 inches! Were it possible to attach the rod so as not to injure its strength in the process of shrinking it in—putting it to its place hot, and depending on the con- traction of the metal in cooling to bring it to a proper bearing —and were it possible to have the bearings so true as to induce the strain through the axis of the rod, and not along its side, (Art. 308,) then a less diameter than that given by the rule would suffice. But while these contingencies remain, the rule

Page 324

FRAMING. 319 cannot safely be reduced, for, in the rule, the value of T, for wruught iron, (Table ILL, Art. 308,) is taken at nearly 6006 pounds, a point rather high in consideration of the size of the rod and the injuries, before stated, to which it is subjected. In eases where a girder wholly of cast iron (Fig. 276) is not preferred, it were better to build a brick areh resting on heel plates, (Zig. 279,) in which the metal required to resist the thrust may be divided into two rods instead of one, thus render- ing the size more practical, and at the same time avoiding the injuries to which rods in areh girders are subjected. The heel- plate arch is also to be preferred to the cast arch on the score of economy; inasmuch as the brick which is substituted for the east areh will cost less than iron. For example, suppose the eross section of the iron arch to be thus: the blade or up- right part 8 by 14 inches, the top flange 12 by 1} inches, and the bottom flange 6 by 12 inches. At these dimensions, the area of the cross section will equal 12 + 15 + 104 = 374 inches. A bar of cast iron, one foot long and one inch square, will weigh 3:2 pounds; therefore, 374 x 3'2 = 120 pounds, equals the weight of the east arch per lineal foot. The price of castings per pound, as also the price of brickwork per. cubic foot, of course will depend upon the locality and the state of the market at the time, but for a comparison they may be stated, the one at three and a half cents-per pound, and the other at thirty cents per cubie foot. At these prices the cast arch will cost 120 x 34 = $4 20 per lineal foot; while the brick arch—12 inches high and 12 inches thick—will cost 30 „cents per lineal foot. The difference is $3 90. This amount is not all to’ be credited to the aceount of the brick arch. Proper allowance is to be made for the cost of the heel plates, and of the wooden centre; also for the cost of a small addi- tion to the size of the tie rods, which is required to sustain the strain otherwise borne by the cast arch in its resistance to a cross strain (Art. 435). Deducting the cost of these items, 4

Page 325

320 AMERICAN HOUSE-CARPENTER. the difference in favor of the brick arch will be about $3 pet foot. This,on a girder 25 feet long, amounts to $75. The difference in all cases will not equal this, but will be sufficiontly great to be worth saving.

Page 326

SECTION V.—DOORS, WINDOWS, &c. ` DOORS. 436.—Among the several architectural arrangements of an edi- fice, the door is by no means the least in importance ; and, if pro- perly constructed, it is not only an article of use, but also of or- nament, adding materially to the regularity and elegance of the apartments. Thediniensions and style of finish of a door, should be in accordance with the size and style of the building, or the apartment for which it is designed. ‘As regards the utility of doors, the principal door to a public building should be of suffi- cient width to admit of a free passage for a crowd of people; while that of a private apartment will be wide enough, if it per- mit one person to pass without being incommoded. Experience has determined that the least width allowable for this is 2 feet 8 “inches ; although doors leading to inferior and unimportant rooms may, if circumstances require it, be as narrow as 2 feet 6 inches ; and doors for closets, where an entrance is seldom required, may be but 2 feet wide. The width of the principal door to a public building may be from 6 to 12 feet, according to the size of the building ; and the width of doors for a dwelling may be from 2 feet 8 inches, to 3 feet 6 inches. If the importance of an apart- ment in a dwelling be such as to require a door of greater width 3 A=

Page 327

S22 AMERICAN HOUSE-CARPENTER. than 3 feet 6 inches, the opening should be closed with two doors, or a door in two folds; generally, in such cases, where the opening is from 5 to8 feet, folding or sliding doors are adopted. As to the height of a door, it should in no case be less than about 6 feet 3 inches; and generally not less than 6 feet 8 inches. 437.—The proportion between the width and height of single doors, for a dwelling, should be as 2 is to5; and, for entrance- doors to public buildings, as 1 is to 2. If the width is given and the height required of a door for a dwelling, multiply the width by 5, and divide the product vy 2; but, if the height ts given and the width required, divide by 5, and multiply by 2. Where two or more doors of different widths show in the same room, it is well to proportion the dimensions of the more important by the above rule, and make the narrower doors of the same height as the wider ones; as all the doors in a suit of apartments, except the folding or sliding doors, have the best appearance when of one height. The proportions for folding or sliding doors should be such that the width may be equal to 4 of the height; yet this rule needs some qualification: for, if the width of the opening be greater than one-half the width of the room, there will not be a sufficient space left for opening the doors; also, the height should be about one-tenth greater than that of the adjacent single doors. 438.—Where doors have but two panels in width, let the stiles and muntins be each + of the width; or, whatever number of panels there may be, let the united widths of the stiles and the muntins, or the whole width of the solid, be equal to # of the width of the door. Thus: ina door, 35 inches wide, containing two panels in width, the stiles should be 5 inches wide ; and in a door, 3 feet 6 inches wide, the stiles should be 6 inches. If a door, 3 feet 6 inches wide, is to have 3 panels in width, the stiles and muntins should be each 44 inches wide, each panel being 8 inches. The bottom rail and the lock rail ought to be each equal in width to ṣẹ of the height of the door; and the top rail, and all

Page 328

DOORS, WINDOWS, &c. 323 others, of the same width as the stiles. The moulding on the panel should be equal in width to 4 of the width of the stile Fig. 281. 439,— Fig. 281 shows an approved method of trimming doors : a is the door stud ; b, the lath aud plaster; c, the ground ; d, the jamb; e, the stop ; f and g, architrave casings ; and A, the door stile. It is customary.in ordinary work to form the stop for the © door by rebating the jamb. But, when the door is thick and heavy, a better plan is to nail on a piece asat e in the figure. This piece can be fitted to the door, and put on after the door is hung; so, should the door be a trifle winding, this will correct - the evil, and the door be made to shut solid. 440.— Fig. 282 is an elevation of a door and trimmings suita- ble for the best rooms of a dwelling. (For trimmings generally, see Sect. III.) ‘The number of panels into which a door should be divided, is adjusted at pleasure; yet the present style of finish- ing requires, that the number be as small as a proper regard for strength will admit. In some of our best dwellings, doors have been made having only two upright panels. A few years expe- : -ience, however, has proved that the omission of the lock rail is at the expense of the strength and durability of the door; ə four-panel door, therefore, is the best that can be made. 441.—The doors of a dwelling should all be hung so as to open into the principal rooms ; and, in general, no door should be hung to open into the hall, or passage. As to the proper edge of the door on which to affix the hinges, no general rule can be assigned

Page 329

324 AMERICAN HOUSE-CARPENTER. Oe es Q OAX OAN TN VAX VAN SOZSO ESOS ROZSO Fig. 282. WINDOWS. 449A window should be of such dimensions, and in such a position, as to admit a sufficiency of light to that part of the apartment for which it is designed. No definite rule for the size

Page 330

DOORS, WINDOWS, &C. 325 can well be given, that will answer in all cases; yet, as an ap- proximation, the following has been used for general purposes, Multiply together the length and the breadth in feet of the apart- ment to be lighted, and the product by the height in feet; then the square-root of this product will show the required number of square feet of glass. 443.—To ascertain the dimensions of window frames, add 44 inches to the width of the glass for their width, and 6+ inches to the height of the glass for their height. ‘These give the dimen- sions, in the clear, of ordinary.frames for 12-light windows; the height being taken at the inside edge of the sill. Ina brick wall, the width of the opening is 8 inches more than the width of the glass—44 for the stiles of the sash, and 34 for hanging stiles— aud the height between the stone sill and lintel is about 104 inches more than the height of the glass, it being varied according to thek thickness of the sill of the frame. 444.—In hanging inside shutters to fold into bores, it is ne cessary to have the box shutter about one inch wider than the flap, in order that the flap may not interfere when both are folded into the box. ‘The usual margin shown between the face of the shutter when folded into the box and the quirk of the stop bead, or edge of the casing, is half an inch; and, in the usual method of letting the whole of the thickness of the butt hinge into the edge of the box shutter, it is necessary to make allowance for the throw of the hinge. This may, in general, be estimated at } of an inch at each hinging; which being added to the margin, the ~ entire width of the shutters will be 14 inches more than the width of the frame in the clear. Then, to ascertain .the width of the box shutter, add 14 inches to the width of the frame in the clear, between the pulley stiles ; divide this product by 4, and add half an inch to the quotient; and the last product will be the re- quired width. For example, suppose the window to have 3 lights in width, 11 inches each. Then, 3 times 11 is 33, and 4% added for the wood of the sash, gives 37} 372 and 1} is 39

Page 331

826 AMERICAN HOUSE CARPENTER. and 39, divided by 4, gives 93; to which add half an inch, and the result will be 104 inches, the width required for the box shutter. 445.—In disposing and proportioning windows for the walls of a building, the rules of architectural taste require that they be of different heights in different stories, but of the same width. The windows of the upper stories should all range perpendicularly over those of the first, or principal, story; and they should be disposed so as to exhibit a balance of parts throughout the front of the building. ‘To aid in this, it is always proper to place the front door in the middle of the front of the building ; and, where the size of the house will admit of it, this plan should be adopted. (See the latter part of Art. 224.) The proportion that the height should bear to the width, may be, in accordance with general usage, as follows : The height of basement windows, 1} of the width. > e c principal-story “ 2} be se $ second-story “ 4} $ X E third-story “ R . 4 fourth-story lo. ae A ss attic-story “ the same as the width. But, in determining the height of the windows for the several stories, it is necessary to take into consideration the height of the story in which the window is to be placed. For, in addition to the height from the floor, which is generally required to be from 28 to 30 inches, room is wanted above the head of the window for the window-trimming and the cornice of the room, besides some respectable space which there ought to be between these. 446.—Doors and windows are usually sguare-headed, or termi- nate in a horizontal line at top. These require no special direc- tions for their trimmings. But eircular-headed doors and win- dows are more difficult of execution, and require some attention. If the jambs of a door or window be placed at right angles to the face of the wall, the edges of the soffit, or surface of the head, would be straight, and its length be found by getting the

Page 332

DOORS, WINDOWS, &C. 327 stretch-out of the circle, (Ari. 92;) but, when the jaisbs are placed obliquely to the face of the wall, occasioned by the de- mand for light in an oblique direction, the form of the soffit will be obtained by the following article: and, when the face of the wall is circular, as in the succeeding one. Fig. 283. 447.—To find the form of the soffit for circular window heads, when the light is received in an oblique direction. Let a bcd, (Fig. 283,) be the ground-plan of a given window, ande f a, a vertical section taken at right angles to the face of the jambs. From a, through e, draw ag, at right angles to a b ; obtain the stretch-out of e f a,and make e g equal to it; divide e æ and e f a, each into a like number of equal parts, and drop perpen- diculars from the points of division in each ; from the points of intersection, 1, 2, 3, &c., in the line, a d, draw horizontal lines to “meet corresponding perpendiculars from e g ; then those points of intersection will give the curve line, d g, which will be the one required for the edge of the soffit. The other edge, ¢ h, is found in the same manner. 448.— To find the form of the soffit for circular window- heads, when the face of the wall is curved. Letabed, (Fig. 284,) be the ground-plan of a given window, and e f a, a vertical section of the head taken at right angles to the face of the jambs.

Page 333

328 AMERICAN HOUSE-CARPENTER. f Proceed as in the foregoing article to obtain the line, d g; ther that will be the curve required for the edge of the soffit; the other edge being found in the same manner. If the given vertical section be taken in a line with the face of the wall, instead of at right angles to the face of the jambs, place it upon the line, ¢ b, (Fig. 283 ;) and, having drawn ordinates at right angles to c b, transfer them to e Ja; inthis way, a section at right angles to the jambs can be obtained,

Page 334

SECTION VI—STAIRS. 449.—The srarrs is that mechanical arrangement in a build: ing by which access is obtained from one story to another. Theit position, form and finish, when determined with discriminating taste, add greatly to the comfort and elegance of a structure. As regards their position, the first object should be to have them near the middle of the building, in order that an equally easy access may be obtained from all the rooms and passages. Next in im- portance is light; to obtain which they would seem to be best situated near an outer wall, in which windows might be construc- ted for the purpose ; yet a sky-light, or opening in the roof, would -not only provide light, and so secure a central position for the stairs, but may be made, also, to assist materially as an ornament to the building, and, what is of more importance, afford an op- portunity for better ventilation. * 450.—It would seem that the length of the raking side of the pitch-board, or the distance from the top of one riser to the top ot the next, should be about the same in all cases ; for, whether stairs be intended for large buildings or for small, for public or for pri- vate, the accommodation of men of the same stature is to be con- sulted in every instance. But it is evident that, with the same effort, a longer step can be taken on level than on rising ground 49

Page 335

339 AMERICAN HOUSE-CARPENTER. and that, although the tread and rise cannot be proportioned merely in accordance with the style and importance of the build- ing, yet this may be done according to the angle at which the flight rises. If itis required to ascend gradually and easy, the length from the top of one rise to that of another, or the hypothe nuse of the pitch-board, may be long; but, if the flight is steep the length must be shorter. Upon this data the following piobler is constructed. ee 28 451.— To proportion the rise and tread to one another, Make the line, a b, (Fig. 285,) equal to 24 inches ; from b, erect ‘bc, at right angles to a b, and make b ¢ equal to 12 inches; join a and c, and the triangle, a b c, will form a scale upon which te graduate the sides of the pitch-board. For example, suppose a very easy stairs is required, and the tread is fixed at 14 inches. Place it from b to f, and from f, draw f g, at right angles to a b ; then the length of f g will be found to be 5 inches, which is a proper rise for 14 inches tread, and the angle, f b g, will show the degree of inclination at which the flight will ascend. But, in a majority of instances, the height of a story is fixed, while the length of tread, or the space that the stairs occupy on the lower floor, is optional. The height of a story being determined, the height of each rise will of course depend upon the number intc which the whole height is divided ; the angle of ascent being more easy if the’number be great, than if it be smaller. By dividing

Page 336

STAIRS. 331 the whole height of a story into a certain number of rises, sup- pose tlie length of each is found to be 6 inches. Place this length from b to A, and draw A i, parallel to a b; then A i, or b j will be the proper tread for that rise, and j b i will show the angle of as- cent. On the other hand, if the angle of-ascent be given, as a b l, (b l being 104 inches, the proper length of run for a step- ladder,) drop the perpendicular, l k, from Ito k; then ¿kb will be the proper proportion for the sides of a pitch-board for that run. 452.—The angle of ascent will vary according to circum- stances. The following treads will determine about the right in- clination for the different classes of buildings specified. In public edifices, tread about 14 inches. In first-class dwellings 1 124 “ In second-class “‘ ie s In third-class “ and cottages “ on" Step-ladders to ascend to scuttles, &c., should have from 10 to l1 inches run on the rake of the string. (See notes at Art. 103.” 453.—The length of the steps is regulated according to the ex- tent and importance of the building in which they are placed, -varying from 3 to 12 feet, and sometimes longer. Where two per- sons are expected to pass each other conveniently, the shortest length that will admit of it is 3 feet ; still, in crowded cities where land is so valuable, the space allowed for passages me very small, they are frequently executed at 24 feet. 454.— To find the dimensions of the pitch-board. ‘The first „thing in commencing to build a stairs, is to make the pitch-board ; this is done in the following manner. Obtain very accurately, in feet and inches, the perpendicular height of the story in which the stairs are to be placed. This must be taken from the top ot the floor in the lower story to the top of the floorin the upper story. ‘Then, to obtain the number of rises, the height in inches thus obtained must be divided by 5, 6, 7, 8, or 9, according to the quality and style of the building in which the stairs are to be

Page 337

332 AMERICAN HOUSE-CARPENTER. built. For instance, suppose the building to be a first-class dwelling, and the height ascertained is 13 feet 4 inches, or 166 inches. The proper rise for a stairs in a house of this class is about 6inches. Then, 160 divided by 6, gives 26% inches. 'This being nearer 27 than 26, the number of risers, should be 27. ° Then divide the height, 160 inches, by 27, and the quotient will give the height of one rise. On performing this operation, the quotient will be found to be 5 inches, 4 and ṣẹ of an inch. Then, if the space for the extension of the stairs is not limited, the tread can be found asat Art.451. But, if the contrary is the case, the whole distance given for the treads must be divided by the number of treads required. On account of the upper floor forming a step for the last riser, the number of treads is always one less than the number of risers. Having obtained this rise and tread, the pitch-board may be made in the follow- ing manner. Upon a piece of well-seasoned board about 3 of an inch thick, having one edge jointed straight and square, lay the | corner of a carpenters’-square, as shown at Fig. 286. Make ab ve Fig 286. equal to the rise, and b c equal to the tread; mark along those edges with a knife, and cut it out by the marks, making the edges perfectly square. The grain of the wood must run in the direction indicated in the figure, because, if it shrinks a trifle, the rise and the tread will be equally affected by it. When a pitch-board is first made, the dimensions of the rise and tread should be pre- served in figures, in order that, should the first shrink, a second could be made. 455.— To lay out the string. The space required for timber

Page 338

STAIRS. 333 € d e if Fig. 257, and plastering under the steps, is about 5 inches for ordinary stairs; set a gauge, therefore, at 5 inches, and run it on the lower edge of the plank, as a b, (Fig. 287.) Commencing at one end, lay the longest side of the pitch-board against the gauge-mark, a b, es atc, and draw by the edges the lines for the first rise and tread; then place it successively as at d, e and f, until the re- quired number of risers shall be laid down. Fig. 268. j 456.—F'ig. 288 represents a section of a step and riser, joined after the most approved method. In this, a represents the end of a block about 2 inches long, two of which are glued in the corner in the length of the step. The cove at b is planed up square, glued in, and stuck after the glue is set. . PLATFORM STAIRS. 457.—A platform stairs ascends from one story to another in two or more flights, having platforms between for resting and to cnange their direction. This kindof stairs is the most easily constructed, and is therefore the most common. The cylin-

Page 339

334 AMERICAN HOUSE-CARPENTER, d Fig. 289. £ der is generally of small diameter, in most cases about 6 ixiches. It may be worked out of one solid piece, but a better way is to glue together three pieces, as in Fig. 289; in which the pieces, a, b and c, compose the cylinder, and d and e represent parts of the strings. The strings, after being glued to the cylinder, are secured with screws. ‘The joining ato and ois the most proper for that kind of joint. 458.— To obtain the form of the lower edge of the cylinder. Find the stretch-out, d e, (Fig. 290,) of the face of the cylinder a b c, according to Art. 92; from d and e, draw d f and e g, at right angles to d e ; draw A g, parallel to d e, and make k f and g i, each equal to one rise; from ¿į and f, draw ¿i j and f k, paral- lel toh g ; place the tread of the pitch-board at these last lines, and draw by the lower edge the lines, & h andi l; parallel to titese, draw m n and o p, at the requisite distance for the dimen- sions of the string; from s, the centre of the plan, draw s q, parallel to d f; divide h qand q g, each into 2 equal parts, as at vand w; from v and w, draw vn and w o, parallel to f d; joinn and o, cutting q sinr; then the angles, u n r and r o é, being eased off according to Art. 89, will give the proper curve for the bottom edge of the cylinder. A centre may be found upon which to describe these curves thus: from u, draw uz, at right angles to mi; iromr, drawr z, at right angles to no; then x will be the centre for the curve, ù r. . The centre for the surve, r £, is found in the same manner.

Page 340

STAIRS. BRS 459.— To find the position for the balusters. ‘Place the centre of the first baluster, (b. Fig. 291,) 4 its diameter from the face of the riser, c d, and 4 its diameter from the end of the step, e d; and place the centre of the other baluster, a, half the tread from the centre of the first. The centre of the rail must be placed over the centre of the balusters. Their usual length is 2 feet - 5 inches, and 2 feet 9 inches, for the short and the long balusters respectively. Fig, 291.

Page 341

300 AMERICAN HOUSE-CARPENTER.» Fig. 292. c 460.— To find the face-mould for a round hand-rail to plat- form stairs. Čase 1.— When the cylinder is small. In Fig. 292, j and e represent a vertical section of the last two steps of the first flight, and d and i the first two steps of the second flight, of a platform stairs, the line, e f, being the platform; and a b c is the plan of a line passing through the centre of the rail around the cylinder. Through i and d, draw i X, and through j and e, draw jk; from k, draw kl, parallel to fe ; from b,draw b m, parallel tog d; from Z, draw Zr, parallel to kj ; from n, draw n é at right angles to 7 k; on the line, o b, make o ¢ equal to n t; join c and ¢: on the line, j c, (Fig. 298) make e c equal to e nat Fig. 292; from c, draw ec ¢, at right angles to j c, and make c ¢

Page 342

STAIRS. SO Fig. 298. vqual to ¢ ¢ at Fig. 292; through ¢, draw p J, parallel to 7 ¢, and make ¢ / equal toż? at Lg. 292; join 7 and c, and complete the parallelogram, e c ls; find the points, o, o, o, according to Art. 118; upon e, 0, 0, 0, and l, successively, with a radius equal to half the width of the rail, describe the circles shown in the figure ; then a curve traced on both sides of these circles and just touch- ing them, will give the proper form for the mould. The joint at lis drawn at right angles to ¢ l. 461,—Hlucidation of the foregoing method. 'This excellent plan for obtaining the face-moulds for the hand-rail of a platform stairs, has never before been published. It was communicated to me by an eminent stair-builder of this city: and having seen rails put up from it, I am enabled to give it my unqualified re- commendation. In order to have it fully understood, I have in- troduced Fig. 294; in which the cylinder, for this purpose, is made rectangular instead of circular. The figure gives a per- spective view of a part of the upper and of the lower flights, and a part of the platform about the cylinder. 'The heavy lines, i m, m c and e j, show the direction of the rail, and are supposed ‘to pass through the centre of it. When the rake of the second flight is the same as that of the first, whicl. is here and is gene- rally the case, the face-mould for the lower twist will, when re- versed, do for the upper flight: that part of the rail, therefore, which passes from e to c and from c to J, is all that will need ex- planation. Suppose, then, that the parallelogram, e a o c, represent a plane lying perpendicularly over e a b f, being inclined in the direction, e c, and level in the direction, c o; suppose this plane, e a o c, 43

Page 343

838 AMERICAN HUUSE-CARPENTER., Fig. 294, be revolved on e c as an axis, in the manner indicated by the arcs, o nand a z, until it coincides with the plane, e r ¢ c; the line, a o, will then be represented by the line, v n ; then add the paral- lelogram, z r é n,and the triangle, c ¿ l, deducting the triangle, er s; and the edges of the plane, e s l c, inclined in the direction, ec, and also in the direction, c 7, will lie perpendicularly over the plane, e abf. From this we gather that the line, co, being at 1ight angles ta

Page 344

STAIRS. ? 339 e c, must, in order to reach the point, Z, be lengthened the distance, n t, and the right angle, e c ¢, be made obtuse by the addition to it of the angle, c l. By reference to Fig. 292, it will be seen that this lengthening is performed by forming the right-angled triangle, c o t, corresponding to the triangle, co ¢, in Fig. 294. The line, c £, is then transferred to Fig. 293, and placed at right angles to ec; this angle, e c ¢, being increased by adding the an- gle, £ c l, corresponding to ¢ c l, Fig. 294, the point, J, is reached, and the proper position and length of the lines, e c and e J ob- tained. To obtain the face-mould for a rail over a cylindrica’ well-hole, the same process is necessary to be followed until the the length and position of these lines are found ; then, by forming the parallelogram, e c ls, and describing a quarter of an ellipse therein, the proper form will be given. i f Fig 295, 462.—0asE 2 — When the cylinder is large. Fig. 295 re

Page 345

840 AMERICAN HOUSE-CARPENTER. presents a plan and a vertical section of a line passing through the centre of the rail as before. From b, draw b k, parallel tocd ; ex-: | tend the lines, i d and j e, until they meet kb in k and f ; from n, draw n l, parallel to o b ; through J, draw / ¢, parallel to 7k, from k, draw k t, at right angles to 7 k ; on the line, o b, make o ¢ equal to kt. Makee c, (Fig. 296.) equal toe kat Fig. 295; from c, p 8 it Fig. 296, draw c t,at right angles to e c, and equal toc tat Fig. 295° from t, draw ¢ p, parallel toc e, and make ¢/ equal to ¢/ at Fig. 295; complete the parallelogram, e c ls, and find the points, 0, 0, 0, as before; then describe the circles and complete the mould as in Fig. 293 The difference between this and Case 1 is, that the line, c t, instead of being raised and thrown out, is lowered and drawn in. (See note at page 381.) Fig. 297, c 463.—UaseE 3.— Where the rake meets the level. In Fig

Page 346

r STAIRS. : 341 297, a b c isthe plan of a line passing through the centre tf tne rail around the cylinder as before, and j and e is a vertical section of two steps starting from the floor, A g. Bisect e hind, and through d, draw d f, paraliel to h g ; bisect f n in J, and trom J, draw Z ¢, parallel to n j; from n, draw n ż, atrightangles toj n, on the line, o b, make o ¢ equal to n ¢. Then, to obtain a mould for the twist going up the flight, proceed as at Fig. 293 ; making e cin that figure equal to e n in Fig. 297, and the other lines of. a length and position such as is indicated by the letters of reference in each figure. 'To obtain the mould for the level rail, extend 4 o, (Fig. 297,) toi; make oì equal to f L and joiniandc; make. c i, (Fig. 298,) equal to ciat Fig. 297; through c, draw c 4, at o i d c Fig. 298. right angles to ci; make d c equal to d f at Fig. 297, and com plete the parallelogram, o d c i; then proceed as in the previons cases to find the mould. | 464.—All the moulds obtained by the preceding examples have been for round rails. For these, the mould may be applied to .a plank of the same thickness as the rail is intended to be, and the plank sawed square through, the joints being cut square from the face of the plank. A twist thus cut and truly rounded will hang in a proper position over the plan, and present a perfect and graceful wreath. : 465.— To bore for the balusters of a round rail before ro:nd- ing it. Make the angle, o c ¢, (Fig. 299,) equal to the angle, o ct, at Fig. 292; upon c, describe a circle with a radius equal to half the thickness of the rail; draw the tangent, b d, parallel to t c, and complete the rectangle, e b d f, having sides tangica! to the circle; from c, draw c a, at right angles to oc; then, bd being the bottom of the ra‘], set a gauge from b to a, and run it the whole length of the stuff; in boring, place the centre of the

Page 347

342 AMERICAN HOUSE-CARPENTER. vo Fig. 269. bit in the gauge-mark at a, and bore in the direction, ac. To de this easily, make chucks as represented in the figure, the bottom edge, g h, being parallel to o c, and having a place sawed out, as e f, to receive the rail. These being nailed to the bench, the rail will be held steadily in its proper place for boring vertically, The distance apart that the balusters require to be, on the under side of the rail, is one-half the length of the rake-side of the pitch-board. i ' l | l i l 1 | ' I l 1 b I

Page 348

STAIRS. 342 466.—To obtain, by the foregoing principles, the face-mould for the twists of a moulded rail upon platform stairs In Fig. 300, abc is the plan of a line passing through the centre of the rail around the cylinder as before, and the lines above it are a vertical section of steps, risers and platform, with the lines for the rail obtained as in Fig. 292. Set half the width of the rail from b to f and from b to r, and from f and r, draw f e andr d, parallel toca At Fig. 301, the centre lines of the rai., k c and c n, are obtained as in the previous examples. Make ciandc J, each equal to c i at-Fig. 300, and draw the lines, 7 m and j g, parallel toc k; make n e and n d equal to ne and n d at Fig. 800, and draw do and e l, parallel to n c; also, through k, draw s g, parallel to n c; then, in the parallelograms, m s d o and g sel, find the elliptic curves, d m and e g, according to Art.- 118, and they will define the curves. The.line, d e, being drawn through n parallel to k c, defines the joint, which is to be cut through the plank vertieally. If the rail crosses the platform rather steep, a butt joint will be preferable, to obtain which see Art. 498.

Page 349

344 AMERICAN HOUSE-CARPENTER. 467.—To apply the mould to the plank. 'The mould obtained according to the last article must be applied to both sides of the plank, as shown at Fig. 802. Before applying the mould, the edge, e f, must be bevilled according to the angle, ¢ t x, at Lig 500; if the rail is to be eanted up, the edge must be bevilled at an obtuse angle with the upper faee; but if it 1s to be canted down, the angle that the edge makes with the upper faee mnst be acute. From the spring of the curve, a, and the end, c, draw vertical lines across the edge of the plank by applying the pitch- board, a b c; then, in applying the mould to the other side, place the points, a and c, at band f ; and, after marking around it, saw the rail out vertically. After the rail is sawed out, the bottom and the top surfaces must be squared from the sides. 468.— To ascertain the thickness of stuff required for the twists. The thickness of stuff required for the twists of a round rail, as before observed, is the same as that for the straight; but for a moulded rail, the stuff for the twists niust be thicker than that for the straight. In Fig. 300, draw a section of the rail be- tween the lines, d r and ¢ f, and as close to the line, d e, as possi- ble ; at the lower corner of the section, draw g h, parallel to d e; then the distance that these lines are apart, will be the thickness required for the twists of a moulded rail. The foregoing method of finding moulds for rails is applicable to all stairs which have continued rails around cylinders, and are without winders. WINDING STAIRS. 469. than at the other. In some stairs, there are steps of parallel width Winding stairs have steps tapering narrower at one er.d incorporated with tapering steps ; the former are then called flyers and the latter sofders. 470.— To descrite a regular geometrical winding stairs. In Fig. 303, ab c d represents the inner surface of the wall en. closing the space allotted to the stairs, a e the length of the steps, and e f gh the cylinder, or face of the front string. The line,

Page 350

STAIRS. 345 Fig. 308 a e, iS given as the face of the first riser, and the point, 7, for the limi: of the last. Make e i equal to 18 inches, and upon o, with o i for radius, describe the arc, ij; obtain the number of risers and of treads required to ascend to the floor at j, according to Art. 454, and divide the arc, ¢ j, into the same number of equal parts as there are to be treads; through the points of division, 1, 2, 3, &c., and from the wall-string to the front-string, draw lines tend- ing to the centre, o ; then these lines will represent the face ot each riser, and determine the form and width of thesteps. Allow the necessary projection for the nosing beyond a e, which should be equal tothe thickness of the step, and then ael k will be the dimensions for each step. Make a pitch-board for the wall-string having a k for the tread, and the rise as previously ascertained ; with this, lay out on a thicknessed plank the several risers and treads, as at Fig. 287, gauging from the upper edge of the strirg for the line at which to set the pitch-board. Upon the back of the string, with a 1} inch dado plane, make

Page 351

346 AMERICAN HOUSE-CARPENTER. a succession of grooves 1} inches apart, and parallel with the lines for the risers on the face. ‘These grooves must be tut along the whole length of the plank, and deep enough to admit of the plank’s bending around the curve, a bcd. ‘Then construct a drum, or cylinder, of any common kind of stuff, and made to fit a curve having a radius the thickness of the string less than oa ; upon this the string must be bent, and the grooves filled with strips of wood, called keys, which must be very nicely fitted and glued in. After it has dried, a board thin enough to bend around on the outside of the string, must be glued on from one end to the other and nailed with clout nails. In doing this, be careful not to nail into any place where a riser or step is to enter on the face. After the string has been on the drum a sufficient time for the glue to set, take it off, and cut the mortices for the steps and risers on the face at the lines previously made; which may be done by boring with acentre-bit half through the string, and nicely chiseling to the line. The drum need not be made so large as the whole space occupied by the stairs, but merely large enough to receive one piece of the wall-string at once—for it is evident that more than one will be required. The front string may be constructed in the same manner; taking e J instead of a k for the tread of the pitch-board, dadoing it with a smaller dado plane, and bending it on a drum of the proper size. Fig. 304 471.—-To find the shape and position of the timbers neces- sary to support a winding stairs. The dotted lines in Fig. 308 show the proper position of the timbers as regards the plan : the shape of each is obtained as follows. In Fig. 304, the line, 1 a, is equal to ariser, less the thickness of the floor, and the ines, 2 m, 3 n, 4 0,5 p and 6 q, are each equal to one riser. The

Page 352

STAIRS. 347 line, a 2, is equal to a m in Fig. 303, the line, m 3 to m n in that figure, &c. In drawing this figure, commence at a,.and make the lines, a 1 and a 2, of the length above specified, and draw them at right angles to each other; draw 2 m, at right angles to a 2, and m 3, at right angles to m 2, and make 2 m and m 3 of the lengths as above specified ; and so proceed to the end. Then, through the points, 1, 2, 3, 4, 5 and 6, trace the line, ib; upon the points, 1, 2, 3, 4, &c., with the size of the timber for radius, describe arcs as shown in the figure, and by these the lower line may be traced parallel to the upper. This will give the proper shape for the timber, a b, in Fig. 303 ; and that of the others may be found in the same manner. In ordinary cases, the shape of one face of the timber will be sufficient, for a good workman can easily hew it to its proper level by that; but where great accuracy is desirable, a pattern for the other side may be found in the same manner as for the first. | ` 412.—To find the falling-mould for the rail of a winding stairs. In Fig. 305, a cb represents the plan of a rail around half the cylinder, A the cap of the newel, and 1, 2, 3, &c., the face of the risers in the order they ascend. Find the stretch-out, e f, of a.c b, according to Art. 92; from o, through the point of the mitre at the newel-cap, draw o s ; obtain on the tangent, e d, the position of the points, sand h’,* as at t and f° ; from e t f° and f, draw e z, t u, f° g* and f h, all at right angles to e d; make e g equal to one rise and f° g° equal to 12, as this line is drawn ` from the 12th riser; from g, through 2°, drawg i; make g r equal to about three-fourths of a rise, (the top of the newel, v, should be 34 feet from the floor ;) draw x u, at right angles to ez, and ease off the angle at u; at a distance equal to the thickness of * In the above, the references, a2, 62, &c., are introduced for the first time. During the time taken to refer to the figure, the memory of the form of these may pass from the mind, While that of the sound alone remains; they mey then be mistaken for a 2, b 2, &c. This an be avoided in reading by giving them a sovnd corresponding to their meaning, which la seconda second b, &c. or a second, b second.

Page 353

348 AMERICAN HOUSE-CARPENTER. the rail, draw v w y, parallel to x u i; from the centre of the plan, o, draw o J, at right angles to e d ; bisect hn in p, and through p, at right angles to œ i, draw a line for the joint; in the same manner, draw the jointat k ; then x y will be the falling-mould for that part of the rail which extends from s to b on the plan. 473.— To find tle face-mould for the rail of awinding-stairs. From the extremities of the joints in the falling-mould, as ¥, z and y, (Fig. 305,) draw k a’, z b and y d, at rightangles toe d ; make b e’ equal to f d. Then, to obtain the direction of the joint, a’ c’, or b? d’, proceed as at Fig 306, at which the parts are

Page 354

STAIRS, 349 l \. T | | i e Ji Fig. 306. shown at half their full size. A is the plan of the rail, and B is the falling-mould ; in which k z is the direction of the butt-joint. From k, draw k b, parallel to Z o, and & e, at right angles tok b: from b, draw b f, tending to the centre of the plan, and from f, draw f e, parallel to b k; from J, through e draw / i, and from i, draw 4 d, parallel to e f ; join d.and b, and d b will be the proper direction

Page 355

250 AMERICAN HOUSE-CARPENTER. for the joint or. the plan. The direction of the joint on the other side, a c, can be found by transferring the distances, z b and o d tozaandoc. (See Art. 477.) Fig. 807. Having obtained the direction of the joint, make s r d b, (Fig. 207,) equal to s r d’ b’ in Fig. 805; through r and d, draw ta, through s and from d, draw ¢ u and d e, at right angles tota; make ¢ u and d e equal to £ u and b? m, respectively, in Fis. 805 ; from u, through e, draw u o ; through b, from 7, and from as many other points in the line, ¢ a, as is thought necessary, as f, A and 7 draw the ordinates, r c, f g, h i j k and ao ; from u, c, g, i k, e and o, draw the ordinates, 2 1, c 2, g 3, i 4, k 5, e 6 ando 7, at right angles to uo; make u 1 equal to ¢ s, c 2 equal tor 2, g 3 equal to f 3, &c., and trace the curve, 1 7, through the points thus found ; find the curve, c e, in the same manner, by transfer- ring the distances between the line, a, and the arc, rd; join 1 and c, also e and 7; then, 1 c e 7 will be the face-mould required for that part of the rail which is denoted by the letters, s r d? $°, on the plan at Fig. 305. To ascertain the mould for the next quarter, make a c je, (Fig

Page 356

STAIRS. Fig. 808, 308,) equal to a? è j è at Fig. 305; atany convenient height on the line, d i, in that figure, draw q 7’, parallel to e d; through c andj, (Fig. 308,) draw b d ; through a, and from j, draw b k and j o, at right angles to bd; make b k and j o equal tow k and q i, respectively, in Fig. 305; from k, through o, draw kf; and proceed as in the last figure to obtain the face-mould, A. 474.—To ascertain the requisite thickness of stuff. Case 1.— When the falling-mould is straight. Make o h and k m, (Fig. 308,) equal to į y at Fig. 8305; draw A ¿ and m n, parallel tob d; through the corner farthest from k f, as n or i, draw ni, parallel to Æ f; then the distance between k f and ni will give “ the thickness required. 475.—Case 2.— When the falling-mould is curved. In Fig. 309, sr d bis equal to s r Ë b’ in Fig. 305. Make a c equal to the stretch-out of the arc, s b, according to Art. 92, and divide a ¢ and s b,each iuto a like number of equal parts ; from a and c, and from each point of division in the line, ac, draw a k, e 1, &c., at right an- gles to ac, make a kequal tot u in Fig-805,andc jequalto b’ m

Page 357

392 AMERICAN HOUSE-CARPENTER. in that figure, and complete the talling-mould, / j, every way equal to u m in Fig. 3805; fromthe points of division in the are, sb, draw lines radiating towards the centre of the.circle, dividing the are. r d, in the same proportion as s b is divided; from d and b, draw dt and b u, at right angles to a d, and from jand v, draw j u and v w, at right angles toj c; then ¢ ww will bea vertical projection of the joint, d b. Supposing every radiating line across s r d b— corresponding to the vertical lines across & 7—to represent a joint, find their vertical projection, as at 1, 2, 3, 4, 5 and 6 ; through the corners of those parallelograms, trace the curve lines shown in the figure ; then 6 u will bea kelinet, or vertical projection, of s r d b. 'To find the thickness of plank necessary to get out this part of the rail, draw the line, z ¢, touching the upper side of the helinet in two places: through the corner farthest projecting from that line, as w, draw y w, parallel to z ¢; then the distance between those lines will be the proper thickness of stuff for this part of the rail. The same process is necessary to find the thickness of stuff in all cases in which the falling-mould is in any way curved. 476.—To apply the face-mould to the plank. In Fig. 810, A represents the plank with its best side and edge in view, and B the same plank turned up so as to bring in view the other side

Page 358

STAIRS. ° 353 and the same edge, this being square from the face. Apply the tips of the mould at the edge of the plank, as at «a and o, (A,) and mark out the shape of the twist; from a and o, draw the lines, a b and o ¢, across the edge of the plank, the angles, e a b and e o c, corresponding with k fdat Fig. 308; turning the plank upas at B, apply the tips of the mould at b and c, and mark it out as shown in the figure. In sawing out the twist, the saw must be be moved in the direction, a b; which direction will be perpen- dicular when the twist is held up in its proper position. In sawing by the face-mould, the sides of the rail are obtained ; the top and bottom, or the upper and the lower surfaces, are ob- tained by squaring from the sides, after having bent the falling- “mould around the outer, or convex side, and marked by its edges. Marking across by. the ends of the falling-mould will give the position of the butt-joint. 477.—EHlucidation of the process by which the direction of - the butt-joint is obtained in Art. 478. Mr. Nicholson, in his Carpenters Guide, has given the joint a different direction to that here shown; he radiates it towards the centre of the cylin- der. This is erroneous—as can be shown by the following operation : In Fig. 311, a r j is the plan of a part of the rail about the joint, s u is the stretch-out of a i, and g p is the helinet, or ver- tical projection of the plan, a r j i, obtained according to Art.

Page 359

out * AMERICAN HOUSE-CARPENTER. r Fig. 811. j 475, Bisectr ż, part of an ordinate from the centre of the plan, and through the middle, draw c b, at right angles to æ v; from b and c, draw c d and b e, at right angles to s u; from d and e, draw lines radiating towards the centre of the plan: then d o and e m will be the direction of the joint on the plan, according to Nicholson, and ¢ b its direction on the falling-mould. It will be admitted that all the lines on the upper or the lower side of the rail which radiate towards the centre of the cylinder, as d o, e m or tj, are level; for instance, the level line, w v, on the top of the

Page 360

STAIRS. 355 rail in the helinet, is a true representation of the radiating line, 7 i, on the plan. The line, b A, therefore, on the top of the rail in the helinet, is a true representation of e m on the plan, and & c on the bottom of the rail truly represents d o. From k, draw k l, parallel to c b, and from A, draw hf, parallel to b c; join ¿and b, also c and f ; then c k lb will be a true representation of the end of the lower piece, B, and c fh b of the end of the upper piece, A ; and f k or h l will show how much the joint is open on the inner, og concave side of the rail.

Page 361

356 AMERICAN HOUSE-CARPENTER. To show that the process followed in Art. 473 is correct, let do and e m, (Fig. 312,) be the direction of the butt-joint found as at Fig. 306. Now, to project, on the top of the rail in the helinet, a line that does not radiate towards the centre of the cylinder, as j k, draw vertical lines from j and & to wand A, and join wand h ; tnen it will be evident that w A isa true representation in the helinet of j k on the plan, it being in the same plane as 7 k, and also in the same winding surface asw v. The liue, l n, also, is a true repre- sentation on the bottom of the helinet of the line, 7 k, in the plan. The line of the joint, e m, therefore, is projected in the same way and truly by i b on the top of the helinet; and the line, d o, by c aon the bottom. Join a andi, and then it will be seen that the lines, c a, ai andi b, exactly coincide with c b, the line of he joint on the convex side of the rail; thus proving the lower tnd of the upper piece, A, and the upper end of the lower piece, B, to be in one and the same plane, and that the direction of the joint on the plan is the true one. By reference to Fig. 306 it will be seen that the line, Z i, corresponds to v i in Fig. 3812; and that e k in that figure is a representation of f b, andi k of d b. Fiz. 818. In getting out the twists, the joints, before the falling-mould 1s

Page 362

STAIRS 357 applied, are cut perpendicularly, the face moti being long enough to include the overplus necessary for a butt-joint. he face-mould for A, therefore, would have to extend to the line, i b ; and that for B, to the line, yz. Being sawed vertically at first, a section of the joint at the end of the face-mould for A, would be represented in the helinet by bi fg. To obtain the position of the line, b i, on the end of the twist, draw 7 s, (Fig. 318,) at right angles to i f, and make i s equal to me at Fig. 312; through s. draw s g, pa rallel to i f, and make s b equal to s b at Fig. 312; join b and è} make i f equal toż fat Fig. 312, and from f, draw f g, parallel to + b ; then i b æ f will be a perpendicular section of the rail over the line, e m, on the plan at Fig. 312, corresponding to ib g f in the helinet at that figure ; and when the rail is squared, the top, or back, must be trimmed off to the line, i b, and the bottom to the line, f g. 478.—To grade the front string of a stairs, having winders in a quarter-circle at the top of the flight connected with flyers at the bottom. In Fig. 314, a b represents the line of the facia along the floor of the upper story, b e c the face of the cylinder, and c d the face of the front string. Make g b equal to } of ‘the diameter of the baluster, and draw the centre-line of the rail, f g, g hiand ij, parallel to ab, bec andc d; make gkandgl each equal to half the width of the rail, and through & and J, draw lines for the convex and the concave sides of the rail, parallel tothe centre-line ; tangical to the convex side of the rail, and parallel to k m, draw n o; obtain the stretch-out, q 7, of the semi-circle, & p m, according to Art. 92; extend a b to £, and k mto s; make c s equal to the length of the steps, and ¿u equal to 18 inches, and de- scribe the arcs, s tand u 6, parallel to sn p ; from t, draw ¢ w, tend- ing to thecentre of the cylinder ; from 6, and on the line, 6 wz, run off the regular tread, as at 5, 4, 3,2, 1 and v; make u x equal to half the arc, u 6, and make the point of division nearest to v, as v, the limit of the parallel steps, or flyers ; make r o equal to m z ; from o, draw o a°. at right angles ton o, and equal to one rise;

Page 363

358 AMERICAN HOUSE-CARPENTER. Fig. 814, from a’, draw a’ s, parallel to n 0, and equal to one tread ; from s through o, draw s b°. Then from w, draw w c’, at right angles to n o, and set up, on the line, w œ, the same number of risers that the floor, A, is above the first winder, B, as at 1, 2, 3, 4, 5 and 6; through 5, (on the arc, 6 u,) draw d’ e, tending to the centre of the cylinder; from &, draw e f’, at right angles tomo, and through 5, (on the line,

Page 364

STAIRS. 859 w &,) draw g° f’, parallel tomo; through 6, (on the line, w c) and f*, draw the line, h’ b; make 6 c’ equal to half a rise, and from ¢ and 6, draw œ 7’ and6 f, parallel ton o ; make h? 7 equal to h? f°; from 7’, draw 7%, at right angles to 7? h’, and from f’, draw f° k’, at right angles to f° h’; upon X, with # f° for radius, describe the arc, f° 7; make b F equal to b’ f°, and ease off the angle at b? by the curve, f? F. In the figure, the curve is de- scribed from a centre, but in afull-size plan, this would be imprac- ticable ; the best way to ease the angle, therefore, would be with a tanged curve, according to Art. 89. Then from 1, 2, 3 and 4, (on the line, w c’,) draw lines parallel to z 0, meeting the curve in m°, n’, ° and p°; from these points, draw lines at right angles to no, and meeting it in 2°, 7’, $ and £; from 2° and 7°, draw lines tending to «x, and meeting the convex side of the rail in 4° and z’; make m v’ equal to r s*, and m w? equal to r ê; from y’, z“, x°, and w’, through 4, 3, 2 and 1, draw lines meeting the line of the wall-string in a’, 6°, c’ and d’ ; from e, where the centre-line of the rail crosses the line of the floor, draw e° f*, at right angles to u o, and from f°, through 6, draw f° g’ ; then the heavy lines, f° 2’, ed, a, zd’, vc, w d’, and z y, will be the lines for the risers, which, being extended to the'line of the front string, b e c d, will give the dimensions of the winders, and the grading of the front string, as was required. 479.— To obtain the falling-mould for the twists of the last- mentioned stairs. Make? g* and 7 h’, (Fig. 314,) each equal to half the thickness of the rail; through /° and o°, draw h? ¢ and g° 7°, parallel to 7? s ; assuming & 4° and m m’ on the plan as the amount of straight to be got out with the twists, make n q equal to & k3, and r F equal to mm’; from n and &, draw lines at right angles to m o, meeting the top of the falling-mould in 7’ and o° ; from o*, draw a line crossing the falling-mould at right angles to a chord of the curve, f? l; through the centre of the cylinder, draw x° 8, at right angles to z o ; through 8, draw 7 9, tending tc k? ; then n° 7 will be the falling-mould for the upper twist, and 7 o” the falling-mould for the lower twist.

Page 365

860 AMERICAN HOUSE-CABRPENTER. 480.— To obtain the face-moulds. The moulds for the twists of this stairs may be obtained as at Art. 478; but, as the falling- mould in its course departs considerably from a straight line, it would, according to that method, require a very thick plank for the rail, and consequently cause a great waste of stuff. In order, therefore, to economize the material, the following method is to be preferred—in which it will be seen that the heights are taken in three places instead of two only, as is done in the previous method. | Case 1.— When the middle height is abové a line joining the other two. Having found at Fig. 314 the direction of the joint, w $ and p e, according to Art. 473, make i p e a, (Fig. 315,) equal to A? přep in Fig.314; join b and c, and from o, draw o h, at right angles to b ¢ ; obtain the stretch-out of d e, as d f, andat Fig. 314, place it from the axis of the cylinder, p, to ¢ ; from ¢ in that figure, draw q° r°, at right angles to n o ; also, at a convenient height on the line, n 7°, in that figure, and at right angles to that line, draw uv’ v*; from b and c, in Fig. 315,

Page 366

STAIRS. 36% draw b jand c l, at right angles to b c ; make b 7 equal to u’ wir Fig. 314, i h equal to w° r°in that figure, and c l equal to v° 9; from J, through 7, draw 1m ; from h, draw h n, parallel to c b; from n, draw n r, at right angles to b c, and join r and s ; through the lowest corner of the plan, as p, draw v e, parallel tob c; from a, e, u, p, k, t, and from as many other points as is thought ne- cessary, draw ordinates to the base-line, v e, parallel to rs; through k, draw w v, at right angles to m l; upon n, with r s for radius, describe an intersecting arc at z, and join n and z; from the points at which the ordinates from the plan meet the base- line, v e, draw ordinates to meet the line, m l, at right angles to v e; and from the points of intersection on m l, draw correspond- ‘ing ordinates, parallel to n x; make the ordinates which are pa- rallel to n x of a length corresponding to those which are parallel to r s,and through the points thus found, trace the face-mould as required. Case 2.— When the middle height is below a line joining the other two. The lower twist in Fig. 314 is of this nature. The face-mould for this is found at Fig. 316 in a manner similar to that at Fig. 315. The heights are all taken from the top of the falling-mould at Fig. 314; b j being equal to w 6 in Fig. 314, ih equal to 2° y° in that figure, and clto fo. Draw a line through 7 and J, and from A, draw h n, parallel to bc; from n, draw nr, at right angles to b c, and join r and s ; then r s will be the bevil for the lower ordinates. From h, draw h x, at right an- gles to j 1; upon n, with r s for radius, describe an intersecting are at z,and join mand x; then n x will be the bevil for the upper ordinates, upon which the face-mould is found as in Case 1. 481.—Elucidation of the foregoing method.—This method of finding the face-moulds for the handrailing of winding stairs, being founded on principles which govern cylindric sections, may be illustrated by the following figures. Fig. 317 and 318 repre- sent solid blocks, or prisms, standing upright on a level base, b d ; the upner surface, j a forming oblique angles with the face, b l— 46

Page 367

262 AMERICAN HOUSE-CARPENTER. Fig. 816. in Fig. 317 obtuse, and in Fig. 318 acute. Upon the base, de scribe the semi-circle, b s c ; from the centre, 7, draw 7s, at right angles to bc; from s, draw s 2, at right angles to e d, and from i draw 7 A, at right angles tobe; make A equal to s 2, and join hand z ; then, A and v being of the same height, the line, h z, joining them, is a level line. From h, draw h n, parallel to b c, and from n, draw n r, at right angles to b c; join r and s,also n

Page 368

* STAIRS. 263 and r; then, n and z being of the same height, n ris a 'eve! line; and this line lying perpendicularly over r s, n x and r s must be of the same length. So, all lines on the top, drawn parallel to 2 x, and perpendicularly over corresponding lines drawn’parallel to r son the base, must be equal to those lines on the base; and by drawing a number of these on the semi-circle at the base and others of the same length at the top, it is evident that a curve, j z l, maybe traced through the ends of those on the top, which shall lie perpendicularly over the semi-circle at the base. It is upon this principle that the process at Fig. 315 and 316 is founded. The pian of the rail at the bottom of those figures is supposed to lie perpendicularly under the face-mould at the top ; and each ordinate at the top over a corresponding one at the base. The ordinates, n x and r s, in those figures, correspond to n's and r s in Fig. 317 and 318. In Fig. 319, the top, e a, forms a right angle with the face, d c ; all that is necessary, therefore, in this figure, is to find a line ` corresponding to % x in the last two figures, and that will lie level and in the upper surface; so that all ordinates at right angles to d r on the base, will correspond to those that are at right angles

Page 369

364 AMERICAN HOUSE CARPENTER. Fig 319 r tueconthetop. This elucidates Fig. 307; at waich the lines, n 9 and i 8. correspond to Å 9 and 7 8 in this figue. p Fig. 820. 482.— To find the bevil for the edge of the plank. The plank, before the face-mould is applied, must be bevilled accord- ing to the angle which the top of the imaginary block, or prism, in the previous figures, makes with the face. This angle is de- termined in the following manner: draw w i, (Fig. 820,) at right angles tozs, and equal to what Fig. 615; make zs equal to ¿i s in that figure, and join wands ; then sw p will be the bevil required in order to apply the face-mould at Fig. 815. In Fig. 316, the middle height being below the line joining the other two, the bevil is therefore acute. To determine this, draw i s, (Fig. 321,) at

Page 370

STAIRS. 365 i —3 { w P Fig. 821. right angles toi p, a dequal to ¿ s in Fig. 316; make s w equal toh win Fig. 316, andjoin w andi; then w i p will be the bevil required in ordər to apply the face-mould at Fig. 316. Al though the falling-mould in these cases is curved, yet, as the plank is sprung, or bevilled on its edge, the thickuess necessary to get out the twist may be ascertained according to Art. 474— taking the vertical distance across the falling-mould at the joints, and placing it down from the two outside heights in Fig. 315 or 316. After bevilling the plank, the moulds are applied as at Art. 476—applying the pitch-board on the bevilled instead of a square edge, and placing the tips of the mould so that they will bear the same relation to the edge of the plank, as they do to the line, 7 l, in Fig. 315 or 316. i Fig. 322. 483.—To apply the moulds without bevilling the plank. Make w p, (Fig. 322,) equal to w p at Fig. 320, and the angle, b c d, equal to b j lin Fig. 315; make pa equal to the thick- ness of the plank, as w a ir. Fig. 320, and froma draw ao, pa- rallel to w d ; frome, draw c e, at right angles to w d, and join e

Page 371

3866 AMERICAN HOUSE-CARPENTER, and b ; then the angle, b e o, on a square edge of the plank, hav ing a line on the upper face at the distance, p a, in Fig. 820, at which to apply the tips of the mould—will answer the same pur pose as bevilling the edge. If the bevilled edge of the plank, which reaches from p to w, is supposed to be in the plane of the paper, and the point, a, te be above the plane of the paper as much as a, in Fig. 320, is dis- tant from the line, w p; and the plank to be revolved on p b as an axis until the line, p w, falls below the plane of the paper, and the line, p a, arrives in it; then, it is evident that the point, c, will fall, in the line, c e, until it lies directly behind the point, e, and the line, b c, will lie directly behind b e. Fig. 823. 484.— To find the bevils for splayed work. 'The principle employed in the last figure is one that will serve to find the bevils for splayed work—such as hoppers, bread-trays, &c.—and a way of applying it to that purpose had better, perhaps, be introducea in this connection. In Fig. 323, a b c is the angle at which the work is splayed, and b d, on the upper edge of the board, is at right angles toa b; make the angle, f g 7, equal to a bc, and from f, draw f h, parallel toe a; from b, draw b o, at right an gles toa b; through o, draw i e, parallel to c b, and join e and d ; then the angle, a e d, will be the proper bevil for the ends from the inside, or Æ d efrom the outside. If a mitre-joint is re-

Page 372

X É) ~ STAIRS. 267 quired, set f g, the thickness of the stuff on tke level, from e to m, and join m and d ; then k d m will be the proper bevil for a mitre-joint. lf the upper edges of the splayed work is to be bevilled, so as to be horizontal when the work is placed in its proper position, S £j, being the same as a b c, will be the proper bevil for that purpose. Suppose, therefore, that a piece indicated by the lines, kg,g f andfh, were taken off; then a line drawn upon the bevilled surface from d, at right angles to k d, would show the true position of the joint, because it would be in the direction of the board for the other side; but a line so drawn would pass through the point, oọ—thus proving the principle correct. So, if a line were drawn upon the bevilled surface from d, at an angle of 45 degrees to & d, it would pass through the point, n. 485.—Another method for face-moulds. It will be seen by reference to Art. 481, that the principal object had in view in the preparatory process of finding a face-mould, is to ascertain upon it the direction of a horizontal line. 'This can be found by a method different from any previously proposed ; and as it requires fewer lines, and admits of less complication, it is probably to be preferred. Tt can be best introduced, perhaps, by the following explanation. In Fig. 324, j d represents a prisin standing upon a level base, b d, its upper surface forming an acute angle with the face, bi,as at Fig. 318. Extend the base line, bc, and the raking line, j J, to meet at f; also, extend ed and g a, to meet at k; from f, through k, draw fm.. If we suppose the prism to stand upon a level floor, o f m, and the plane, j g a l, to be extended to meet that floor, then it -will be obvious that the intersection between that plane and the plane of the floor would be in the line, f k; and the line, f k, being in the plane of the floor, and also in the inclined plane, j g k f, any line made in the plane, 7 g bi, parallel to f k, must be a level line. By finding the position of a perpendicular plane, at right angles to the raking plane, j f k g, we shall greatly shorten the process for obtaining ordinates.

Page 373

oOo S į : 368 AMERICAN HOUSE-CARPENTER. This may be done thus: from f, draw f o, at right angles to fm ; extend e b to o, and g j, tot; from o, draw o ¢, at right angles to o f, and join¢ and f ; then to f will bea perpendicular plane, at right angles to the inclined plane, ¢ œ kf; because the base of the former, o f, is at right angles to the base of the latter, f k, both these lines being in the same plane. From b, draw b p, at right angles to o f, or parallel tof m ; from p, draw p q, at right angles to o f, and from q, draw a line on the upper plane, parallel to f m, or at right angles to ¿f ; then this line will obviously be drawn to the point, 7, and the line, q 7, be equal to p b. Proceed, in the same way, from the points, s and c, to find x and J. Now, to apply the principle here explained, let the curve, b s c, (Fig. 325,) be the base of a cylindric segment, and let it be re- quired to find the shape of a section of this segment, cut by a plane passing through three given points in its curved surface: one perpendicularly over b, at the height, b j; one perpendicu- larly over s, at the height, s x ; and the other over c, at the height, c /—these lines being drawn at right angles to the chord of the base, bc. From j, through l, draw a line to meet the chord line extended to f ; from s, draw s k, parallel to b f, and from 2, draw x k, parallel to jf; from f, through k, draw fm ; then f m will be the intersecting line of the plane of the section with ths

Page 374

STAIRS. 269 Fig. 825. plane of thebase. This line can be proved to be the intersection of these planes in another way ; from b, through s, and from j, through v, draw lines meeting at m ; then the point, m, will be in the intersecting line, as is shown in the figure, and also at Fig. 324. From f, draw f p, at right angles to f m; from b and c, and fromas many other points as is thought necessary, draw ordinates, parallel to f m; make p q equal to b j, and join q and f; from the points at which the ordinates meet the line, q f, draw others at right angles to q f; make each ordinate at A equal to its cor- responding ordinate at C, and trace the curve, gn i, through the points thus found. Now jt may be observed that A is the plane of the section, B the plane of the segment, corresponding to the plane, q p f, of Fig. 324, and C is the plane of the base. To give these planes their proyer position, let A be turned ong f as an axis until it

Page 375

evn AMERICAN HOUSE-CARPENTER. stands perpendicularly over the line, q f, and at right angles tc the plane, B ; then, while A and B are fixed at right angles, let B be turned on the line, p J, as an axis until it stands perpendicu- larly over p f, and at right angles to the plane, C; then the plane, A, will lie over the plane, C, with the several lines on one corres- ponding to those on the other ; the point, 7, resting at J, the point, n, at x, and g at j ; and the curve, g n i, lying perpendicularly over b s c—as was required. If we suppose the cylinder to be cut by a level plane passing through the point, J, (as is done in finding a face-mould,) it will be obvious that lines corresponding to q fand p f would meet in Z; and the plane of the section, A, the plane of the segment, B, and the plane of the base, C, would all meet in that point. 486.— To find the face-mould for a hand-rail according to the principles explained in the previous article. In Fig. 326, aec/f is the plan of a hand-rail over a quarter of a cylinder; and in Fig.327, a b ed is the falling-mould; f e being equal to the stretch-out of a df in Fig. 326. From c, draw c h, parallel to ef; bisect À ¢ in i, and find a point, as b, in the are, d f, (Fig. 326,) corresponding to 7 in the line, A c; from i, (Fig. 327,) to the top of the falling-mould, draw 7 j,at right angles to A c; at Fig. 326, from c, through b, draw c g, and from b and c, draw b j and ck, at right angles tog c; makec k equal toh g at Fig. 327, and b j equal to ¿j at that figure; from Æ, through j, draw k g, and from g, through a, draw œ p ; then gp will be the intersecting line, corresponding to f m in Fig. 324 and 325; through e, draw p 6, at right angles to g p, and from c, draw c q, parallel to g p ; make r q equal to A g at Fig. 827; joinp and q, and proceed as in the previous examples to find the face-mould, A. The joint of the face-mould, « v, will be more accurately determined by tinding the projection of the centre of the plan, o, as at w; joining s and w, and drawing u v, parallel to s w. It may be noticed that c & and b jare not of a length corres- ponding to the above directions: they are but 4 the length given.

Page 376

Page 377

T Po AMERICAN HOUSE ARPENTER. Fig. 827, The object of drawing these lines is to find the point, g, and that can be done by taking any proportional parts of the lines given, as well as by taking the whole lines. For instance, supposing ¢ k and b j to be the full length of the given lines, bisect one ini and the other in m; then a line drawn from m, through i, will give the point, æ, as was required. The point, æ, may also be

Page 378

STAIRS. 373 obtained thus: at Fig. 327, make A l equal to c b in Fig. 326 irom J, draw Jk, at right angles toh c; from j, draw j k, parallel to h c; from g, trough k,draw œ n; at Fig. 326, make b g equal to Znin Fig. 327; then g will be the point required. The reason why the points, a, b and c, inthe plan of the rail at Fig. 326, are taken for resting points instead of e, 7 and f, is this: the topof the rail being level, it is evident that the points, a and e, in the section a e, are of the same height ; also that the point, 2, is of the same height as b, and cas f. Now, if a is taken for a point in the inclined plane rising from the line g p, e must be below that plane ; if b is taken for a point in that plane, 2 must be below it; and if cis in the plane, f must be below it. The rule, then, for taking these points, is to take in each section the one that is nearest to the line, g p. Sometimes the line of intersection, g p, happens to come almost in the direction of the line,e r : in such case, after finding the line, see if the points from which the heights were taken agree with the above rule; if the heights were taken at the wrong points, take them according to the rule above, and then find the true line of intersection, which will not vary much from the one already found. Fig. 828. 487.— To apply the face-mould thus found to the plank. The face-mould, when obtained by this method, is to be applied to a square-edged plank, as dirested at Art. 476, with this differ- ence: instead of applying both tips of the mould to the edge of

Page 379

3874 AMERICAN HOUSE-CARPENTER. the plank, one of them is to be set as far from the edge of the plank, as z, in Fig. 326, is from the chord of the section p q—as is shown at Fig. 328. A, in this figure, is the mould applied on the upper side of the plank, B, the edge of the plank, and C, the mould applied on the under side; a b and c d being made equal to q z in Fig. 326, and the angle, e a c, on the edge, equal to the angle, p q r, at Fig. 826. In order to avoid a waste of stuff, it would be advisable to apply the tips of the mould, e and b, im- mediately at the edge of the plank. To do this, suppose the moulds to be applied as shown in the figure; then let A be re- volved upon e until the point, b, arrives at g, causing the line, e 6, to coincide with e g: the mould upon the under side of the plank must now be revolved upon a point that is perpendicularly beneath e, as f ; from f, draw f h, parallel to ¿i d, and from d, draw dh, at right angles to i d ; then revolve the mould, C, upon `f, until the point, h, arrives at j, causing the line, f h, to coincide with f j, and the line, ¿i d, to coincide with k Z; then the tips of the mould will be at k and l. The rule for doing this, then, will be as follows: make the an- gle, i f k, equal to the angle q v x, at Fig. 326; make f k equal to fz, and through k, draw k l, parallel to 77; then apply the corner of the mould, i, at k, and the other corner d, at the line, k J. The thickness of stuff is found as at Art. 474. 488.—To regulate the application of the falling-mould. Obtain, on the line, k c, (Fig. 327,) the several points, r, q, p, l and m, corresponding to the points, 6’, a’, z, Y, &c., at Fig. 326 , from r q p, &c., draw the lines, r ¢, q u, p v, &c., at right angles tohe; makehs,r t, q u, &c., respectively equal to 6c’, r q, 5 d’, &c., at Fig. 326; through the points thus found, trace the curve, sw c. Then get out the piece, œ s c, attached to the fall- ing-mould at several places alorg its length, as at z, z, z, &c. In applying the falling-mould with this strip thus attached, the edge, s wc, will coincide with the upper surface of the rail piece wre

Page 380

a STAIRS. 8T before it is squared ; and thus show the proper position of the fall. ing-mould along its whole length. (See Art. 496.) SCROLLS FOR HAND-RAILS. 459.— General rule for finding the size and position of the regulating square. The breadth which the scroll 1s to occupy, the number of its revolutions, and the relative size of the regula ting square to the eye of the scroll, being given, multiply the number of revolutions by 4, and to the product add the number of times a side‘of the square*is contained in the diameter of the eye, and the sum will be the number of eqnal parts into which the breadth is to be divided. Make a side of the regulating square equal to one of these parts. To the breadth of the scroll add one of the parts thus found, and half the sum will be the length of the longest ordinate. Fig. 829 490.— To find the proper centres in the regulating square. Let a 2 1 b, (Fig. 329,) be the size of a regulating square, found according to the previous rule, the required number of revolu- tions being 13. Divide two adjacent sides, asa 2 and 21, into ‘as many equal parts as there are quarters in the number of revo- lutions, as seven ; from those points of division, draw lines across the square, at right angles to the lines divided ; then, 1 being the first centre, 2, 3, 4, 5, 6 and 7, are the centres for the other quar ters, and 8 is the centre for the eye; the heavy lines that deter-

Page 381

376 AMERICAN HOUSE-CARPENTER mine these centres being each one part less in length than its pre ceding line. ho c Fig. 830. 491.— To describe the scroll for a hand-rail over a curtail step. Leta b, (Fig. 330,) be the given breadth, 1% the given number of revolutions, and let the relative size of the regulating square to the eye be 4 of the diameter of the eye. Then, by the rule, 1% multiplied by 4 gives 7, and 3, the number of times a side of the square is contained in the eye, being added, the sum is 10. Divide a b, therefore, into 10 equal parts, and set one from b toc; bisectac ine; then a e will be the length of the longest ordinate, (1 d or 1 e.) From a, draw a d, from e, draw e 1, and from b, draw b f, all at right angles to a b ; make e 1 equal to e a, and through 1, draw 1 d, parallel to a b ; set b c from 1 to 2, and upon 1 2, complete the regulating square; divide this square asat Fig. 329; then describe the arcs that compose the scroll, as follows: upon 1, describe d e; upon 2, describe e f; upon 3, describe f g ; upon 4 describe g h, &c.; make d lequal to the

Page 382

STAIRS., 8i7 width of the rail, and upon 1, describe l m`; upon 2, aescribe m n, &c.; describe the eye upon 8, and the scroll is completed. 492.—To describe the scroll for a curtail step. Bisect d l (Fig. 330,) ino, and make o v equal to $ of the diameter of a baluster; make v w equal to the projection of the nosing, and e z equal to w l; upon 1, describe w y, and upon 2, describe y z ; also upon 2, describe x i; upon 3, describe ij, and so around to 2; and the scroll for the step will be completed. 493.— To determine the position of the balusters under the scroll. Bisect di, (Fig. 330,)in o, and upon 1, with 1 o for ra- dius, describe the circle, o r u; set the baluster at p fair with the face of the second riser, c’, and from p, with half the tread in the dividers, space off as at 0, q, r, $, t, u, &c., as far as q°; upon 2, 3, 4 and 5, describe the centre-line of the rail around to the eye of the scroll; from the points of division in the circle, o r u, draw lines to the centre-line of the rail, tending to the centre of the eye, 8; then, the intersection of these radiating lines with the centre-line of the rail, will determine the position of the balusters, as shown in the figure. Fig. 381, 494.— To obtain the falling-mould for the raking part of the scroll. Tangicalto the rail at h, (Fig. 330,) draw h k, parallel to d a; then k a? will be the joint between the twist and the other part of the scroll. Make d e equal.to the stretch-out of de,and upon d 48

Page 383

378 AMERICAN HOUSE-CARPENTER. e’, find the position of the point, k, asat? ; at Fig. 331, make ed equal toe’ d in Fig. 330, and d c equal to d œ in that figure; from c, draw ca, at right angles to e c, and equal to one rise; make c b equal to one tread, and from b, through a, draw bj, bisect a ¢ in J, and through /, draw m q, paralleltoeh; m q is the height of the level part of ascroll, which should always be about 34 feet from the floor; ease off the angle, m fj, according to Art. 89, and draw g w n, parallel to m z j, and at a distance equal to the thickness of the rail; at a convenient place for the joint, as 7, draw in, at right angles to b j; through n, draw j h, at right angles to e h ; make d k equal tod /* in Fig. 330, and from ķ, draw ko, at right angles to eh; at Fig. 330, make d h° equal to d h in Fig. 331, and draw A’ 5’, at right angles to d h; then k a and h? ï will be the position of the joints on the plan, and at Fig. 331, o p and i n, their position on the falling- mould; and p o in, (Fig. 331,) will be the falling-mould re quired. Fig. 882. 495.— To describe the face-mould. At Fig. 330, fromk, draw kr’, at right angles tor? d; at Fig. 331, make A r equal to h? r° in Fig. 330, and from r, draw r s, at right angles to rh ; from the intersection of r s with the level line, m q, through 7, draw s t; at Fig. 330, make h? b? equal to q ¢ in Fig. 331, and join b’ and r?; from a’, and from as many other points in the arcs, a° l and k d, as is thought necessary, draw ordinates to 7’ d, at right angles to the latter; make r b, (Fig. 382,) equal in its length and in its divisions to the line; 7° b, in Fig. 320; fromr, n, 0, p,q

Page 384

STAIRS. 879 and J, draw the lines, r k, n d, o a, p e, q fand le, at right an- gles to r b, and equal to r° k, d's’, f? a’, &., in Fig. 330; through the points thus found, trace the curves, & land a c, and complete the face-mould, as shown in the figure. This mould is to be applied to a square-edged plank, with the edge, J b, parallel to the edge of the plank. The rake lines upon the edge of the plank are to be made to correspond to the angle, s ¢ h, in Fig. 331. The thickness of stuff required for this mould is shown at Fig. 381, between the lines s ¢ and u v—u v being drawn pa- rallel to s ¢. 496.—All the previous examples given for finding face-moulds over winders, are intended for moulded rails. For round rails, the same process is to be followed with this difference: instead of working from the sides of the rail, work from a centre-line. After finding the projection of that line upon the upper plane, describe circles upon it, as at Fig. 293, and trace the sides of the moulds by the points so found. The thickness of stuff for the twists of a round rail, is the same as for the straight; and the twists are to be sawed square through.

Page 385

880 AMERICAN HOUSE-CARPENTER 497.—To ascertain the form of the newel-cap from a section of the rail. Draw-a b, (Fig. 333,) through the widest part of the given section, and parallel to c d; bisect ab in e, and through a, e and b, draw à i, f g, and k j, at right angles toa b; at a con- venient place on the line, fg, as o, with a radius equal to’ half the width of the cap, describe the circle, 2 7 g; make r l equal toe borea; join / and j, also land i; from the curve, f b, to the line, 7 7, draw as many ordinates as is thought necessary, parallel to f g; from the points at which these ordinates meet the line, Z 7, and upon the centre, o, describe arcs in continuation to meet op; from n, t, x, &c., draw ns, tu, &c., parallel to f g; make n s, t u, &c., equal to e f, w v, &c.; make æ y, &c., equal to z d, &c.; make o2, 03, &c., equal to on, o t, &c.; make 2 4 equal to n s, and in this way find the length of the lines crossing om; through the points thus found, describe the section of the newel-cap, as shown in the figure. Fig 834. 498.— To find the true position of a butt joint for the twists of a moulded rail over platform stairs. Obtain the shape of the mould according to Art. 466, and make the line a b, Fig. 384, equal to a c, Fig. 300; from b, draw b c, at right angles to a b, and equal in length to n m, Fig. 300: join ø and c, and bisect ac in o; through o draw e f, at right angles to a c, and d k, parallel to cb; make od and o k cach equal to half e h at Fig. 300; through e and f, draw h i and g j, parallel to a c. At Fig. 301, make n a equal to e &, Fig. 334, and through a, draw r p, at right angles to n c; then r p will be the true position on the face-mould for a butt joint, as was required. The sides must be sawn verti .

Page 386

STAIRS. 381 cally as described at Art. 467, but the joint is to be sawn square through the plank. The moulds obtained for round rails, (Art. 464,) give the line for the joint, when apyilied to either side of the plank ; but here, for moulded rails, ths line’ for the joint can be obtained from only one side. Whe, the rail is canted up, the joint is taken from the mould laid on the upper side of the lower twist, and on the under side of the upper twist; but when it is canted down, a course just the reverse of this is to be pursued. When the rail is not canted, either up or dows. the vertical joint, obtained as at Art. 466, will be a butt joint, and therefore, in such a case, the process described in this article will be unnecessary, NOTE TO ARTICLE 462, Platform stairs wih a large cylinder. Instead of ‘placing the platform-risers at the spring of the cyl- inder, a more easy and graceful appearance may be given to the rail, and the necessity of canting either o? the twists entirely obviated, by fixing the place of the above risers at a certain distance within the cyl- inder, as shown in the annexed cut—the lines indi- cating the face of the risers cutting the cylinder at & and Z, instead of at p and q, the spring of the cylin- der. To ascertain the position of the risers, let ad c be the pitch-board of the lower flight, and c d e that of the upper flight, these being placed so that b c and c d shall form a right line. Extend a c to cut dein f; draw f g parallel to d b, and of indefinite length: draw g o at right angles to f g, and equal ‘in. length to the radius of the circle formed by the centre of the rail in passing around the cylinder; on o as centre describe the semicircle j g i; make o h equal to the radius of the cylinder, and describe on o the face of the cylinder ph q; then extend d b across the cylinder, cutting it in l and A—giving the position of the face of the risers, as required. To find the face-mould for the twists is simple and ob- vious: it being merely a quarter of an ellipse, hav- ing o j for semi-minor axis, and the distance on the rake corresponding to o g, on the plan, for the semi-major axis, found thus,—extend ¢ j to meet a f, then from this point of meeting to f is the semi-major axis.

Page 387

SECTION VII.—SHADOWS. 499.—The art of drawing consists in representing solids upou a plane surface: so that a curious and nice adjustment of lines is nade to present the same appearance to the eye, as does the human figure, a tree, or a house. It is by the effects of light, in its reflection, shade, and shadow, that the presence of an object is made known to us; so, upon paper, it is necessary, in order that the delineation may appear real, to represent fully all the shades and shadows that would be seen upon the object itself. In this section I propose to illustrate, by a few plain examples, the simple ` elementary principles upon which shading, in architectural sub- jects, is based. The necessary knowledge of drawing, prelim- inary to this subject, is treated of in the Introduction, from Art. 1 to 14. 500.— The inclination of the line of shadow. This is always, in architectural drawing, 45 degrees, both on the elevation and the plan; and the sun is supposed to be behind the spectator, and over his left shoulder. This can be illustrated by reference to Fig. 335, in which A represents a horizontal plane, and B and C two vertical planes placed at right angles to each other. A rep- resents the plan, C the elevation, and B a vertical projection from the elevation. In finding the shadow of th: plane, B, the

Page 388

SHADOWS, 383 Fig. 825. _hne, a b, is drawn at an angle of 45 degrees with the horizon, and the line cb, at the same angle with the vertical plane, B. The plane, B, being a rectangle, this makes the true direction of the sun’s rays to be in a course parallel to d b; which direction has been proved to be at an angle of 35 degrees and 16 minutes with the horizon. It is convenient, in shading, to have a set-square with the two sides that contain the right angle of equal length ; this will make the two acute angles each 45 degrees; and will give .the requisite bevil when worked upon the edge of the T- square. One reason why this angle is chosen in preference to another, is, that when shadows are properly made upon the draw- ing by it, the depth of every recess is more readily known, since the breadth of shadow and the depth of the recess will be equal. To distinguish between the terms shade and shadow, it will be understood that all such parts of a body as are not exposed to the direct action of the sun’s rays, are in shade; while those parts which are deprived of light by the interposition of other bodies, are in shadow. _

Page 389

384 AMERICAN HOUSE-CARPENTER. A eam Fig. 336. Fig. 887. Fig. 839. 501.—To find the line of shadow on mouldings and other ho- rizontally straight projections. Fig. 336, 337, 338, and 339, represent various mouldings in elevation, returned at the left, in the usual manner of mitreing around a projection. A mere in- spection of the figures is sufficient to see how the line of shadow is obtained ; bearing in mind that the ray, @ b, is drawn from the projections at an angle of 45 degrees. Where there is no return at the end, it is necessary to draw a section, at any place in the length of the mouldings, and find the line of shadow from that. 502.— To find the line of shadow cast by a shelf. In Fig. 340, A is the plan, and B is the elevation of a shelf attached to a wall. From a and c, draw a b and c d, according to the angle previously directed ; from b, erect a perpendicular intersecting c dat d; from d, draw d e, parallel to the shelf; then the lines, c d and d e, will define the shadow cast by the shelf. There is another method of finding the shadow, without the plan, A. Extend the lower line o tne shelf to f, and make c f equal to the projection of the sheif

Page 390

SHADOWS. 885 from the wall; from f, draw f g, at the customary angle, and from £, drop the vertical line, c g, intersecting f g at g; from g, draw g e, parallel to the shelf, and from c, draw c d, at the usual angle ; then the lines, c d and d e, will determine the extent of the shadow as before. 503.—To find the shadow cast by a shelf, which ts wider at one end than at the other. In Fig. 341, A is the plan, and B the elevation. Find the point, d, as in the previous example, and from any other point in the front of the shelf, as a,-erect the perpen- dicular, a e; from a and e, draw a b and e c, at the proper angle, and from b, erect the perpendicular, b c, intersecting ec in c:

Page 391

386 AMERICAN HOUSE-CARPENTER. from d, through c, draw d o ; then the lines, i d and d o, will give the limit of the shadow cast by the shelf. Fig. 842. 504.—To find the shadow of a shelf having one ena acute or obtuse angled. Fig. 342 shows the plan and elevation of an acute-angled shelf. Find the line, e g, as before; from a, erect the perpendicular, a b; join b and e; then b e and e g will define the boundary of shadow. ine ars: ji a Ze 505.—To find the shadow cast by an inclined shelf. In Fig. 343, the plan and elevation of such a shelf is shown, having also one end wider than the other. Proceed as directed for finding the shadows of Fig. 341, and find the points, d and c; then ad anl d c will be the shadow required. If the shelf had been

Page 392

SHADOWS. 38’ parallel in width on the plan, then the line, d c, would have bee: parallel with the shelf, a b. 506.—To find the shadow cast by a shelf inclined in its ver tical section either upward or downward. From a, (Fig. 34 and 345,) draw a b, at the usual angle, and from b, draw b ¢ parallel with the shelf; obtain the point, e, by drawing a lin from d, at the usual angle. In Fig. 344, join e andi; then i and e c will define the shadow. In Fig. 345, from o, draw o i parallel with the shelf; join č and e; then ż e and e c will be th shadow required. The projections in these several examples are bounded b straight lines; but the shadows of curved lines may be found i: the same manner, by projecting shadows from several points i the curved line, and tracing the curve of shadow through thes points. Thus— Fig. 846.

Page 393

388 AMERICAN HOUSE CARPENTER. 507.—To find the shadow of a shelf having its front edge, or end, curved on the plan. In Fig. 346 and 347, A and A show an example of each kind. From several points, as a, a, in the plan, and from the corresponding points, o, 0, in the elevation, draw rays and perpendiculars intersecting at e, e, &c.; through these points of intersection trace the curve, and it will define the shadow. 508.—To find the shadow of a shelf curved in the elevation. In Fig. 348, find the points of intersection, e, e and e, as in the last examples, and a curve traced through them will define the shadow. i The preceding examples show how to find shadows when cast upon a vertical plane; shadows thrown upon curved surfaces are ascertained in a similar manner. ‘Thus—

Page 394

SHADOWS. 359 509.—To find the shadow cast upon a cylindrical wall by a vrojection of any kind. By an inspection of Fig. 349, it will be seen that the only difference between this and the last examples, is, that the rays in the plan die against the circle, a b, instead of a straight line. Fig. 850. 510.—To find the shadow cast by a shelf upon an inclined wall. Cast the ray, a b, (Fig. 350,) from the end of the shelf to the face of the wall, and from b, draw b c, parallel to the shelf; cast the ray, d e, from the end of the shelf; then the lines, d e and e c, will define the shadow. These examples might be multiplied, but enough has been given to illustrate the general principle, by which shadows in all instances are found. Let us attend now to the application of this principle to such familiar objects as are likely to occur in practice.

Page 395

390 AMERICAN HOUSE-CARPENTER. 511.—To find the shadow of a projecting horizontal beam From the points, a, a, &c., (Fig. 351,) cast rays upon the wall . the intersections, e, e, e, of those rays with the perpendiculars drawn from the plan, will define the shadow. If the beara be ım- clined, either on the plan or elevation, at any angle other than a right angle, the difference in the manner of proceeding can be seen by reference to the preceding examples of inclined shelves &c. HEE UGHHG tH | c Fig. 852. 512.—To find the shadow in a recess. From the point, a, (Fag. 352,) in the plan, and b in the clevation, draw tk: rays, a c and be; from c, erect the perpendicular, c e, and frum e, draw the horizontal line, e d; then the lines, c e and e d, will show the extent of the shadow. This applies only where the back of the recess is paralle] with the face of the wall. 513.—To find the shadow in a recess, when the face of the wall is inclined, and the back of the recess is vertical. In Fig. 353, A shows the section and B the eleration of a recess of this

Page 396

¢ SHADOWS. 391 kind. From b, and from any other point in the line, be asa draw the rays,bcandae; from c, a, and e, draw the horizonta’ lines, c g, a f, ande h; from d and f, cast the rays, d i and fh: from 7, through 4, draw 7s; then sz and ¿g will define the shadow. Fig. 854. 514.—To find the shadow in a fireplace. From a and b, (Fig. 354,) cast the rays, a c and be, and from c, erect the per- pendicular, ce; from e, draw the horizontal line, e o, and join e and d ; then c e, e o, and o d, will give the extent of the shadow. Fig. 855. 515.—To find the shadow of a moulded window-lintel. Cast rays from the projections, a, ©, & :., in the plan, (Fig. 355,) and d, e, &c., in the elevation, anc. draw the usual perpendiculars in- tersecting the rays at 1,7, and 2; these intersections connected

Page 397

392 AMERICAN HOUSE-CARPENTER. and horizontal lines drawn from them, will define the shadow The shadow on the face of the lintel is found by casting a ray back frem z to s, and drawing the horizontal line, s n. 516.—To find the shadow cast by the nosing of a step. From a, (Fig. 356,) and its corresponding point, c, cast the rays, a b and c d, and from b, erect the perpendicular, b d; tangical to the curve at e, cast the ray, e f,.and from e, drop the perpendicular, e o, meeting the mitre-line, a g, ino; cast a ray from o to 2, and from 2, erect the perpendicular, 7 f; from h, draw the ray, hk; from f to d and from d to k, trace the curve as shown in the figure ; from k and A, draw the horizontal lines, k n and h s ;' then the limit of the shadow will be completed. 517.—To find the shadow thrown by a pedestal upon steps. From a, (Fig. 357,) in the plan, and from c in the elevation, draw the rays, a b and c e; then ao will show the extent of the shadow on the first riser, as at A; f g will determine the shadow on the second riser, as at B; cd gives the amount of shadow on the first tread, as at C, and ki that on the second tread, as at D; which completes the shadow of the left-hand pedestal,. both on the plar and elevation. A mere inspection of the figure will be suf-

Page 398

SIIADOWS, 3892 Fig. 857, ficient to show how the shadow of the right-hand pedestal is obtained. ht gu : N Fig. 858. Fig. 859. 518.—To find the shadow thrown on a column by a square abacus. From aand b, (Fig. 858,) draw the rays, a c and be, and from c, erect the perpendicular, c e ; tangical to the curve at d, draw the ray, d f, and from h, corresponding to f in the plan, draw the ray, k 0; take any point between a and f, as i, and from this, as also from a corresponding point, n, draw the rays, 7 r and ns; from r, and from d, erect the perpendiculars, r s and do; through the points, e, s, and o, trace the curve as shown. in the figure ; then the extent of the shadow will be defined. 519.—To find the shadow thrown on a column by a circular abacus. This is so near like the .ast example, that no explanation ~ will be necessary farther than a reference to the preceding article 50

Page 399

394 AMERICAN HOUSE-CARPENTER. — SY Fig. 360. 520.—To find the shadows on the capital of a column. This may be done according to the principles explained in the examples already given; a quicker way of doing it, however, is as follows If we take into consideration one ray of light in connection with all those perpendicularly under and over it, it is evident that these several rays would form a vertical plane, standing at an angle of 45 degrees with the face of the elevation. Now, we mav sup pose the column to be sliced, so to speak, with planes of this

Page 400

SIIADOWS. 395 Fig. 861. nature—cutting it in the lines, a b, c d, &c., (Fig. 360,) and, in the elevation, find, by squaring up from the plan, the lines of sec- tion which these planes would make thereupon. For instance : in finding upon the elevation the line of section, a b, the plane cuts the ovolo at e, and therefore f will be the corresponding point upon the elevation; A corresponds with g, 7 with 7, o with s, and l with b. Now, to find the shadows upon this line of section, cast from m, the ray, m n, from A, the ray, h o, &c.; then that part of the section indicated by the letters, m f i n, and that part also be- tween h and o, will be under shadow. By an inspection of the figure, it will be seen that the same process is applied to each line of section, and in that way the points, put, t, U, U, WP owasenae Il ; 2,3, &c., are successively found, and the lines of shadow traced through them. Fig. 861 is an example of the same capital with all the shadows finished in accordance with the lines obtained on Fig. 360. ia = To find the shadow thrown on a vertical wall by a column and entablature standing ‘n advance of said wall. Cast

Page 401

3896 AMERICAN HOUSE-CARPENTER Fig. 862. . rays from a and b, (Fig. 362,) and find the point, c, as in the previous examples; from d, draw the ray, d e, and from e, the horizontal line, e f; tangical to the curve at g and h, draw the rays, g j and hz, and from and j, erect the perpendiculars, i / andj k; from m and n, draw the rays, m f and n k, and trace the curve between k and f; cast a ray from o to p, a vertical line from p to s, and through s, draw the horizontal line, s £; the shadow as required will then be completed.

Page 402

SHADOWS. 897 Fig. 363 is an example of the same kind as the last, with all the shadows filled in, according to the lines obtained in the pre- ceding figure. Fig. 864. 522 —Fig. 364 and 365 are examples of the Tuscan cornice. The manner of obtainirg the shadows is evident.

Page 403

398 AMERICAN HOUSE-CARPENTER. Fig. 365. 523.—Reflected light.. In shading, the finish and life of an object depend much on reflected light This is seen to advantage in Fig. 361 and on the column in Fig. 363. Reflected rays are thrown in a direction exactly the reverse of direct rays; therefore. on that part of an object which is subject to reflected light, the shadows are reversed. The fillet of the ovolo in Fig. 361 is an example of this. On the right-hand side of the column, the face of the fillet is much darker than the cove directly under it. ‘The reason of this is, the face of the fillet is deprived both of direct and reflected light, whereas the cove is subject to the latter. Other instances of the effect of reflected light will te seen in the other examples.

Page 404

APPENDIE.

Page 405

Page 406

ALGEBRAICAL SIGNS. +, plus, signifies addition, and that the two quantities between which it stands are to be added together; as a + b, read a added to b. —, minus, signifies subtraction, or that of the two quantities between which it occurs, the latter is to be subtracted from the former; as a — b, read a minus 6. x, multiplied by, or the sign of multiplication. It denotes that the two quantities between which it occurs are to be multiplied together ; as a x b, read a multiplied by b, or a times b This sign is usually omitted between symbols or letters, and is then understood, as ab, This has the same meaning as a x b. It is never omitted between arithmetical numbers; as 9 x 5, read nine times five. - =, divided by, or the sign of division, and denotes that of the two quanti- ties between which it occurs, the former is to be divided by the latter ; as a + b, read a divided by b Division is also represented thus : a, in the form of a fraction. This signifies that a is to be divided by b b When more than one symbol occurs above or below the line, or both, as ma, it denotes that the product of the symbols above the line is to be divided by the product of those below the line. =, is equal to, or sign of equality, and denotes that the quantity or quantities on. its left are equal to those on its right; as a — b = 0, read @ minus b is equal to c, or equals c; or, 9 — 5 = 4, read nine minus five equals four. This sign, together with the symbols on each side of it, when spoken of as a whole, is called an eguatzon. a? denotes a squared, or a multiplied by a, or the second power of a, and a’ denotes a cubed, or a multiplied by a and again multiplied by a, or the third power of a. The small figure, 2, 3, or 4, &e., is termed the index or exponent of the power. It indicates how many times the symbol is to be taken. Thus, a? —aa,a’ =aaa, a —aaaa. J is the radical sign, and denotes that the square root of the quantity e following it is to be extracted, and

Page 407

4 APPENDIX. V denotes that the cube root of the quantity following it is to be ex. tracted. Thus, V9 = 3, and ¥27=3. The extraction of roots is also denoted by a fractional index or exponent, thus a% denotes the square root of a, a% denotes the cube root of a, až% denotes the cube root of the square of a, &e.

Page 408

TRIGONOMETRICAL TERMS. Fig. 366. In Fig. 366, where AB is the radius of the circle B C H, draw a line 4 F, from A, through any point, C, of the are B G. From C draw C D perpendicular to A B ; from B draw B E perpendicular to AB; and from G draw G F perpendicular to A G. Then, for the angle FA B, when the radius A C equals unity, CD 1s the sine ; AD the cosine ; D B the versed sine ; BE the tangent, G F the cotangent; AE the secant ; and A F the cosecant.

Page 409

6 APPENDIX. But if the angle be larger than one right angle, yet less than two right angles, as B A H, extend H A to K and EB to K, and from H draw H J perpendicular to A J. Then, for the angle BAH, when the radius A H equals unity, HJ is the sine ; AJ the cosine ; BJ the versed sine ; BK the tangent; and AK the secant. : When the number of degrees contained in a given angle is known, then the value of the sine, cosine, œc., corresponding to that angle, may be found in a table of Natural Sines, Cosines, &c. In the absence of such a table, and when the degrees contained in the given angle are unknown, the values of the sine, cosine, &c., may be found by computation, as follows:—Let B A C, (Fig. 367,) be the C A B Fig. 867, given angle. At any distance from A, draw c perpendicular to A B. By any scale of equal parts obtain the length of each of the three lines a, b, c. Then for the angle at A we have, by proportion, a:e: LO: Sn = a b a: beT i cos, = o. a ° ae bici 10: tan =>. c ` D R a a N S TER a clas 1:0 > coset = =. Or, in any right angled triangle, for the angle contained between the base and hypothenuse— Wher perp. divided by hyp., the quotient equals the sine. “ 6 6 h (23 6 6c & base : YP- cosine. G3 perp. “u “ base, “ “ “ (33 tangent. « base 3 “ perp., “ 6 “ “ cotangent. “ hyp. 6 “ base, “ “ “ “ secant. “ hyp. (73 73 perp. “ 6“ (13 (3 cosecant,

Page 410

GLOSSARY. Terms not found here can be found in the lists of definitions in other parts of this book, or in common dictionaries. Abacus.—The uppermost member of a capital. Abbatoir.—A slaughter-house. Abbey.—The residence of an abbot or abbess. Abutment.—That part of a pier from which the arch springs. Acanthus.—A plant called in English, bear’s-breech. Its leaves are employed for decorating the Corinthian and the Composite capitals. Acropolis.—The highest part of a city; generally the citadel. Acroteria.—The small pedestals placed on the extremities and apex oï a pediment, originally intended as a base for sculpture. Aisle.—Passage to and from the pews of a church. In Gothic ar- chitecture, the lean-to wings on the sides of the nave. Alcove.—Part of a chamber separated by an estrade, or partition of columns. Recess with seats, &c., in gerdens. " Altar.—A pedestal whereon sacrifice was offered. In modern churches, the area within the railing in front of the pulpit. Alto-relievo.—High relief; sculpture projecting from a surface so as to appear nearly isolated. Amphitieatre —a double theatre, employed by the ancients for the exhibition of giadiatorial fights and other shows. Ancones.— Trusses employed as an apparent support to a cornice upon the flanks o” the architrave. g Annulet.—A small square. moulding used to separate others; the fillets in the Doric capital under the ovolo, and those which separate the flutings of columns, are known by this term. Aute.—A pilaster attached to a wall. Apiary.—A place for keeping beehives. Arabesque.—A building after the Arabian style. ; Areostyle.—An intercolumniation of from four to five diameters. Arcade—A series of arches. n Arch.—An arrangement of stones or other material in a curvilinear form, so as to perform the office of a lintel and carry superincumbent weights. : Architrave-—That part of the entablature which rests upon the capital of a column, and is beneath the frieze. The casing and mouldings about a door or window.

Page 411

APPENDIX. Archivolt.—The ceiling of a vault: the under surface of an arch. Area.—Superficial measurement. An open space, below the leve: of the ground, in front of basement windows. Arsenal.—A public establishment for the deposition of arms and warlike stores. Astragal.—A small moulding consisting of a half-round with a fillet on each side. Attic.—A low story erected over an order of architecture. A low additional story immediately under the roof of a building. -Aviary.—A place for keeping and breeding birds. Balcony.—An open gallery projecting from the front of a building. Baluster.—A small pillar or pilaster supporting a rail. Balustrade.—A series of balusters connected by a rail. Barge-course.—That part of the covering which projects over the gable of a building. Base.—The lowest part of a wall, column, &c. Basement-story—That which is immediately under the principal story, and included within the foundation of the building. Basso-relievo.—Low relief; sculptured figures projecting from a surface one-half their thickness or less. See Alto-redievo. Battering.—See Talus. Battlement.—Indentations on the top of a wall or parapet. Bay-window.—A window projecting in two or more planes, and not forming the segment of a circle. Bazaar.—A species of mart or exchange for the sale of various ar- ticles of merchandise. Bead.—A circular moulding. Bed-mouldings.—Those mouldings which are between the corona and the frieze. Beifry.—That part of a steeple in which the bells are hung: an- ciently called campanile. Belvedere.—An ornamental turret or observatory commanding a pleasant prospect. Bow-window.—A window projecting in curved lines. Bressummer.—Abeam or iron tie supporting a wall over a gateway or other opening. 4 Prick-nogging.—The brickwork between studs of partitions. * Buttress-—A projection from a wall to give additional strength. Cable.—A cylindrical moulding placed in flutes at the lower part of the column. Camber.—To give a convexity to the upper surface of a beam. Campanile.—-A tower for the reception of bells, usually, in Italy, separated from the church. Canopy.—An ornamental covering over a seat of state. Cantalivers.—The, ends of rafters under a projecting roof. Pieces of wood or stone supporting the eaves. Capital_—The uppermost part of a column included between the shaft and the architrave.

Page 412

APPENDIX, 9 Caravansera.—lIn the East, a large public building for tue reception of travellers by caravans in the desert. Carpentry.—(From the Latin, carpentum, carved wood.) That de- partment of science and art which treats of the disposition, the con. struction and the relative strength of timber. Tho first is called de- scriptive, the second constructive, and the last mechanical carpentry. _ Caryatides.—Figures of women used instead of columns to support an entablature. Casino.—A small country-house. Castellated.—Built with battlements and turrets in imitation of an- cient castles. ; Castle—A building fortified for military defence. A house with towers, usually encompassed with walls and moats, and having a don- jon, or keep, in the centre. Catacombs.—Subterraneous places for burying the dead. Cathedral_—The principal church of a province or diocese, wherein the throne of the archbishop or bishop is placed. Cavetto.—A concave moulding comprising the quadrant of a circle. Cemetery.—An edifice or area where the dead are interred. Cenotaph.—A monument erected to the memory of a person buried in another place. Centring.—The temporary woodwork, or framing, whereon any vaulted work is constructed. ; Cesspool.—A well under a drain or pavement to receive the waste- water and sediment. Chamfer.—The bevilled edge of any thing originally right-angled. Chancel.— That part of a Gothic church in which the altar is placed. Chaniry.—A little chapel in ancient churches, with an endowment for one or more priests to say mass for the relief of souls out of purga- tory. : : Chen building for religious worship, erected separately from a church, and served by a chaplain. Chaplet.—A moulding carved into beads, olives, &c. Cincture.—The ring, listel, or fillet, at the top and bottom of aco- lumn, which divides the shaft of the column from its capital and base. Circus.—A straight, long, narrow building used by the Romans for the exhibition of public spectacles and chariot races. At the present day, a building enclosing an arena for the exhibition of feats of horse. manship. g Clere-story.—The upper part of the nave of a church above the roofs of the aisles. ` Cloister.—The square space attached to a regular monastery o1 large church, having a peristyle or ambulatory around it, covered with a range of buildings. f Coffer-dam.—A case of piling, water-tight, fixed in the bed of a river, for the purpose of excluding the water while any work, such as a wharf, wall, or the pier of a bridge, is carried up. — Collar-beam.—A horizontal beam fra ned between two principal rafters above the tie-beam. Cullenade.—.A range of columns. Columbarium.—aA_ pigeon-house.

Page 413

10 APPENDIX. Column.—A vertical, cylindrical support under the entablature of an order. Common-rafters.—The same as jack-rafters, which see Conduit.—A long, narrow, walled passage underground, for secret communication between different apartments. A canal or pipe for the conveyance of water. Conservatory.—A building for preserving curious and rare exotic plants. Consoles.--The same as ancones, which see. Tontour.—The external lines which bound and terminate a figure. Convent.—A building for the reception of a society of religious per- sons. Coping.—Stones laid on the top of a wall to defend it from the weather. Corbels.—Stones or timbers fixed in a wall to sustain the timbers of a floor or roof. Cornice.—Any moulded projection which crowns or finishes the part to which it is affixed. Corona.—That part of a cornice which is between the crown- moulding and the bed-mouldings. Cornucopia.—The horn of plenty. Corridor.—An open gallery or communication to the different apart- ments of a house. Cove.—A concave moulding. Cripple-rafters—The short rafters which are spiked to the hip-rafter of a roof. Crockets.—In Gothic architecture, the ornaments placed along the angles of pediments, pinnacles, &c. Crosettes.—The same as ancones, which see. Crypt.—The under or hidden part of a building. Culvert.—An arched channel of masonry or brickwork, built be- neath the bed of a canal for the purpose of conducting water under it. Any arched channel for water underground. Cupola.—A small building on the top of a dome. Curtail-step.—A step with a spiral end, usually the first of the flight. Cusps.—The pendents of a pointed arch. Cyma.—An ogee. There are two kinds; the cyma-recta, having the upper part concave and the lower convex, and the cyma-reversa, swith the upper part convex and the lower concave. Dado.—The die, or part between the base and cornice of a pedestal. Dairy.—An apartment or building for the preservation of milk, and the manufacture of it into butter, cheese, dc. Dead-shoar.—A piece of timber or stone stood vertically in brick- work, to support a superincumbent weight until the brickwork which is to carry it has set or become hard. Decastyle.—A building having ten columns in front. Dentils—({F rom the Latin, dentes, teeth.) Small rectangular blocks used in the bed-mouldings of some of the orders. Diastyle.—An intercolumniation of three, or, as some say, four diameters.

Page 414

APPENDIX. 11 Ine.-~That part of a pedestal included between the base and the sornice ; it is also called a dado. i Dodecastyle.—A building having twelve columns in front. Donjon.—A massive tower within ancient castles to which the gar rison might retreat in case of necessity. Dooks.— A Scotch term given to wooden bricks. ` Dormer.—-A window placed on the roof of a house, the frame being p.aced vertically on the rafters. Dormitory.—A sleeping-room. Dovecote.—A building for keeping tame pigeons. A columbarium. Echinus.—The Grecian ovolo. Elevation.—A geometrical projection drawn on a plane at right an- gles to the horizon. Entablature.-—That part of an order which is supported by the co- lumns ; consisting of the architrave, frieze, and cornice. Eustyle.—An intercolumniation of two and a quarter diameters. Exchange.—A building in which merchants and brokers meet to transact business. Extrados.—The exterior curve of an arch. Facade.—The principal front of any building. Face-mould—The pattern for marking the plank, out of which hand. railing is to be cut for stairs, &c. Facia, or Fascia.—A flat member like a band or broad fillet. Falling-mould.—The mould applied to the convex, vertical surface of the rail-piece, in order to form the back and under surface of the rail, and finish the squaring. Festoon.—An ornament representing a wreath of flowers and leaves. Fillet.—A narrow flat band, listel, or annulet, used for the separa- ion of one moulding from another, and to give breadth and firmness io the edges of mouldings. Flutes.—Upright channels on the shafts of columns. Flyers.—Steps in a flight of stairs that are parallel to each other. Forum.—Jn ancient architecture, a public market; also, a place where the common courts were held, and law pleadings carried on. Foundry.—A building in which various metals are cast into moulde or shapes. A Frieze.—That part of an entablature included between the archi- trave and the cornice. Gable.—The vertical, triangular piece of wall at the end of a root, from the level of the eaves to the summit. Gain.—A recess made to receive a tenon or tusk. ‘Gallery.—4 common passage to several rooms in an upper story. A long room for the reception of pictures. A platform raised on co- lumne, pilasters, or piers. ) N Girder.—The principal beam in a floor for supporting the binding and other joists, whereby the bearing or length is lessened. > Glyph.—A vertical, sunken channel. From their number, those in the Doric order are called triglyphs. ®

Page 415

12 APPENDIX. Granary.—A building for storing grain, especially that intended te be kept for a considerable time. Groin.—The line formed by the intersection of two arches, which cross each other at any angle. Guite.—The small cylindrical pendent ornaments, otherwise called drops, used in the Doric order under the triglyphs, and also pendent from the mutuli of the cornice. Gymnasium.—Orig aally, a space measured out and covered with sand for the exercise « f athletic games: afterwards, spacious buildings devoted to the mental as well as corporeal instruction of youth. Hall.—The first large apartment on entering a house. The public room of a corporate body. A manor-house. Ham.—A house or dwelling-place. A street or village: hence Nottingham, Buckingham, &c. Hamlet, the diminutive of ham, is a small street er village. Helix.—The small volute, or twist, under the abacus in the Corin- thian capital. Hem.—The projecting spiral fillet of the Ionic capital. Hexastyle.—A building having six columns in front. Hip-rafter.—<A piece of timber placed at the angle made by two ad- jacent inclined roofs. Homestall.—A mansion-house, or seat in the country. Hotel, or Hostel.—A large inn or place of public entertainment. A large house or palace. Hot-house.—A glass building used in gardening. Hovel.—An open shed. ' Hut.—A small cottage or hovel generally constructed of earthy materials, as strong loamy clay, &c. Inpost.—The capital of a pier or pilaster which supports an arch. Intaglio.—Sculpture in which the subject is hollowed out, so that the impression from it presents the appearance of a bas-relief. Intercolumniation.—The distance between two columns. Intrados.—The interior and lower curve of an arch. Jack-rafters.—Rafters that fill in between the principal rafters of a roof; called also common-rafters. Jail.—A place of legal confinement. Jambs.—The vertical sides of an aperture. Joggle-piece.—A post to receive struts. Joisis.—The timbers to which the boards of a floor or the laths of a ceiling are nailed. Keep.—The same as donjon, which see. Key-stone.—The highest central stone of an arch. Kiln.—A building for the accumulation and retention of heat, in c1. der to dry or burn certain materials deposited within it. King-post.—The centre-post in a trussed roof. Knee.—A convex bend in the back of a hand-rail. See Ramp.

Page 416

APPENDIX. 13 Lactarium.—The same as dairy, which see. Lantern.—A cupola having windows in the sides for lighting an apartment beneath. Larmier.—The same as corona, which see. Lattice. —A reticulated window for the admission of air, rather than light, as in dairies and cellars. Lever-boards.—Blind-slats: a set of boards so fastened that they may be turned at any angle to admit more or less light, or to lap upon zach other so as to exclude all air or light through apertures. Lintel.—A piece of timber or stone placed horizontally over a door, window, or other opening. Listel—The same as fillet, which see. Lobby.—An enclosed space, or passage, communicating with the principal room or rooms of a house. Lodge.—A small house near and subordinate to the mansion. A cottage placed at the gate of the road leading to a mansion. Loop.—A small narrow window. Loophole is a term applied to the vertical series of doors in a warehouse, through which goods are de- livered by means of a crane. Luffer-boarding.—The same as lever-boqrds, which see. Luthern.—The same as dormer, which see. Mausoleum.—A_ sepulchral building—so called from a very cele- brated one erected to the memory of Mausolus, king of Caria, by his wife Artemisia. i Metopa.—The square space in the frieze between the triglyphs of the Doric order. Mezzanine.—A story of small height introduced between two of greater height. Minaret.—A slender, lofty turret having projecting balconies, com- mon in Mohammedan countries. Minster.—A church to which an ecclesiastical fraternity has been or is attached. Moat.—An excavated reservoir of water, surrounding a house, cas tle or town. Modillion.—A projection under the corona of the richer orders, re sembling a bracket. : Module.—The semi-diameter of a column, used by the architect as a measure by which to proportion the parts of an order. ; Monastery —A building or buildings appropriated to the reception of monks. Monopteron.—A circular collonade supporting a dome without an enclosing wall. i ; Mosaic.—A mode of representing objects by the inlaying of small cubes of glass, stone, marble, shells, &c. À Mosque.—A Mohammedan temple, or place of worship. _ , Mullions.—The upright posts or bars, which divide the lights in a Gothic window. ; Muniment-house.—A strong, fire-proof apartment for the keeping and preservation of evidences, charters, seals, &c., called muniments

Page 417

14 APPENDIX. Museum.—A repository of natural, scientific and literary, curiosities or of works of art. Mutule.—A projecting ornament of the Doric cornice supposed tc .epresent the ends of rafters. Nave.—The main body cf a Gothic church. Newel.—A post at the starting o~ landing of a flight of stairs. Niche.—A cavity or hollow place in a wali for the reception of a statue, vase, &c. Nogs.-—W ooden bricks. Nesing.—The sounded and projecting edge of a step in stairs. Nunnery.—A building or buildings appropriated for the reception of nuns. i Obelisk.—A lofty pillar of a rectangular form. Octastyle.—A building with eight columns in front. Odeum.—Among the Greeks, a species of theatre wherein the poets and musicians rehearsed their compositions previous to the public pro- duction of them. Ogee.—See Cyma. a Orangery.—A gallery or building in a garden or parterre fronting the south. Oriel-window.—A large bay or recessed window in a hall, chapel, or other apartment. Ovolo.—A convex projecting moulding whose profile is the quad- rant of a circle. Pagoda.—A temple or place of worship in India. Palisade.—A fence of pales or stakes driven into the ground. Parapet.—A small wall of any material for protection on the sides of bridges, quays, or high buildings. Pavilion.—A turret or small building generally insulated and com- prised under a single roof. Pedestal. —A square foundation used to elevate and sustain a co- lumn, statue, &c. Pediment.—The triangular crowning part of a portico or aperture which terminates vertically the sloping parts of the roof: this, in Gothic architecture, is called a gable. Penitentiary.—A prison for the confinement of criminals whose crimes are not of a very heinous nature. Piazza.—A square, open space surrounded by buildings. This term is often improperly used to denote a portico. Prer.—A rectangular pillar without any regular base or capital. The upright, narrow portions of walls between doors and windows are known by this term. Pilaster.—A square pillar, sometimes insulated, but more common ly engaged in a wall, and projecting only a part of its thickness. Piles.—Large timbers driven into the ground to make a secure foundation in marshy pla es, or in the bed of a river. Pillar.—A column of irregular form, always disengaged, and al-

Page 418

APPEND’ X. 15 ways deviating from the proportions of the orders ; whence the distin tion between a pillar and a column. Pinunacle.—A small spire used to ornament Gothic buildings. Planceer.——The same as soffit, which see. Phata. —The lower square member of the base of a column, pedes- tal, or wall. Porck.—-An exterior appendage to a building, forming a covered apploacn to one of its principal doorways. Portal.— The arch over a door or gate ; the framework of the gate ; the lesser gate, when there are two of different dimensions at oue en- trance. Portcullis.—A strong timber gate to old castles, made to slide up and down vertically. Portico.—A colonnade supporting a shelter over a walk, or ambu- latory. i Priory.—A building similar in its constitution to a monastery or abbey, the head whereof was called a prior or prioress. Prism.—A solid bounded on the sides by parallelograms, and on the ends by polygonal figures in parallel planes. Prostyle.—A building with columns in front only. Purlines.—Those pieces of timber which lie under and at right an- gles to the rafters to prevent them from sinking. Pycnostyle.—An intercolumniation of one and a half diameters. Pyramid.—A solid body standing on a square, triangular or poly- gonal basis, and terminating in a point at the top. Quarry.—A place whence stones and slates are procured. Quay.—(Pronounced, key.) A bank formed towards the sea or on the side of a river for free passage, or for the purpose of unloading merchandise. Quoin.—An external angle. See Rustic quoins. Rabbet, or Rebate.-—A groove or channel in the edge of a board. Ramp.—A concave bend in the back of a hand-rail. Rampant arch.—One having abutments of different heights. Regula.—The band below the tenia in the Doric order. Riser.—-In stairs, the v>rtical board forming the front of a step. Rostrum.—An elevated platform from which a speaker addresses an audience. Rotunda.—A circular building. Rubble-wall.—A wall built of unhewn stone. Rudenture.—The same as cable, which see. Rustic quoins.—The stones placed on the external angle of a build- ing, projecting beyond the face of the wall, and having their edges bevilled. Rustic-work.—A mode of building masonry wherein the faces of the stones are left rough, the sides only being wrought smooth where the union of the stones takes place.

Page 419

16 APPENDIX. Salon, or Saloon.—A lofty and spacious apartment comprehending the height of two stories with two tiers of windows. Sarcophagus.—A tomb or coffin made of one stone. Scaniling.—The measu.e to which a piece of timber is to be or has been cut. Scarfing.—The joining of two pieces of timber by bolting or nailing transversely together, so that the two appear but one. Scotia. —The hollow moulding in the base of a column, between the fillets of the tori. Scro/l.—A carved curvilinear ornament, somewhat resembling in profile the turnings of a ram’s horn. Sepulchre.-—A grave, tomb, or place of interment. Sewer.—A drain or conduit for carrying off soil or water from any place. Shaft.—The cylindrical part between the base and the capital of a column. Shoar.—A piece of timber placed in an oblique direction to support a building or wall. Siil.—The horizontal piece of timber at the bottom of framing ; the timber or stone at the bottom of doors and windows. Sofit—The underside of an architrave, corona, &c. The underside of the heads of doors, windows, &c. Summer.—The lintel of a door or window ; a beam tenoned into a’ girder to support the ends of joists on both sides of it. Systyle.— An intercolumniation of two diameters. Tenia.—The fillet which separates the Doric frieze from the archi- trave. Tulus.—The slope or inclination of a wall, among workmen called battering. : Terrace.—An area raised before a building, above the level of the ground, to serve as a walk. Tesselated pavement.—A curious pavement of Mosaic work, com- posed of small square stones. Tetrastyle.—A building having four columns in front. Thatch.—A covering of straw or reeds used on the roofs of cottages, barns, &e. Theatre.—A building appropriated to the representation of drama..e spectacles. Tile.—A thin piece or plate of baked ciay or other material used for the external covering of a rcof. Tomb.—A grave, or place for the interment of a human body, in- cluding also any commemorative monument raised over such a piace. Torus.—A moulding of semi-circular profile used in the bases of columns. Tower.—A lofty building of severai stories, round or polygona!. Transept.—The transverse portion -f a cruciform church. Transom.—The beam across a double-lighted window ; if the win dow have no transom, it is called a clere-story window.

Page 420

APPENDIX. 17 Tread.—Tha; part of a step which is included between the face of ite riser and that of the riser above. Trellis—A reticulated framing made of thin bars of wood for screens, windows, &c. : Triglyph.—The vertical tablets in the Doric frieze, chamfered on :he two vertical edges, and having two channels in the middle. Tripod.--A table or seat with three legs. Trochilus.—--The same as scotia, which see. Truss.—An &:~angement of timbers for increasing the resistance to cross-strains, consisting of a tie, two struts and a suspending-piece. Turret—A small tower, often crowning the angle of a wall, &c. Tusk—A short projection under a tenon to increase its strength. , Tympanum.—The naked face of a pediment, included between the evel and the raking mouldings. Underpinning.—The wall under the ground-sills of a building. University. An assemblage of colleges under the supervision of a senate, &c. Vault.—A concave arched ceiling resting upon two opposite paral- lel walls. Venetian-door.—A door having side-lights. Venetian-window.—A window having three separate apertures. Veranda.—An awning. An open portico under the extended roof of a building. Vestibule.—An apartment which serves as the medium of commu- nication to another room or series of rooms. Vestry.—An apartment in a church, or attached to-it, for the pre- servation of the sacred vestments and utensils. Villa.—A country-house for the residence of an opulent person. Vinery.—A house for the cultivation of vines. Volute-——A spiral scroll, which forms the principal feature of the lonic and the Composite capitals. Voussoirs.—Arch-stones ` Wainscoting.—W ooden lining of walls, generally in panels. Water-table.—The stone covering to the projecting foundation or other walls of a building. Well.—The space occupied by a flight of stairs. The space lefi beyond the ends of the steps is called the well-hoce. Wicket.—A small door made in a gate. Winders.—In stairs, steps not parallel to each other. Lophorus.—The same as frieze, which see. , Zystos—Among the ancients, a portico of unusual length, common ly appropriated to gymnastic exercises.

Page 421

TABLE OF SQUARES, CUBES, AND ROOTS. No, | Square. | Cube. | 1 1 1 2 4 8 3 9 27 4 16 61 5 25 125 6 36 216 T 49 313 8 61 512 9 81 729 10 100 1040 11 121 1331 12 14 liz 13 169 2197 lf 196 2744 15 225 3375 16 256 4036 17 239 4913 18 324 5332 19 361 6353 20 400 8000 21 441 9261 22 484 10643 23 529 12167 24 576 13324 25 625 15625 26 676 17576 27 729 19683 : 21952 29 841 24389 30 900 97000 31 961 29791 32| 1024 32768 33 | 10939 35937 34 | 1156 39304 35 | 1225 42875 36 | 1296 46636 37 | 1369 50653 "33 1] 1444 54372 33 | 1521 59319 40 | 1600 64090 41 | 1631 63921 s2 | 1764 74088 43 | 1819 79507 44 | 1936 85184 45 | 2025 91125 46 | 2116 97336 47 | 2209 103323 48 | 2304 110592 49 2401 117649 50 | 2500 125000 51} 2601 132551 52 | 2704 140603 53 | 2309 143377 54| 2916 157464 55 | 3025 166375 50 | 3136 175616 57| 3219 135193 ās | 3364 195112 593| 3431 W537 bu 3600 216000 61} 3721 226931 62| 3411 233328 63 | 396d 250047 4096 262144 4229 274025 4356 237436 (From Hutton’s Mathematics.) 1-0000000' 1:090000 1 4142136, 1259921, 17320503 1442250 20000000) 1-537401! 22360680, 1:709976. 2°4491897| 1317121 26457513) 1912931 2:8234271| 2-0000U0, 3-0200000' 203098-4; 3°1622777) 2154435; 33165243" 2223330 34641016 2233429'| 3:6 55513 2351335) 37416574, 2410142 38729833 2466212 40030000; 2519342 41231056 2571232 42426407. 2-620741 43583939 2-663402 | 44721350} 2714018, 45325757) 2758028 46901153 2302033 47959315! 2343367! 4893735] 2°38 1499 50000000) 2-924018 50990195; 2962496 5-1961524: 3009009! 5 2915925) 3-036539 53951649, 3072317 54772256) 3-107232 5 5677644; 3-141351 56568542, 3-174802 57415626 3-297531, 5:8399519 3-233612 59160798 3-271066; 6:0000000 3 391927) 60327625 3332222, G6 IGEEL40) 3-351975' 62419980! 3-391211 6°3245553 3.419952, 64031242 3448217 64807407 3-476027 6:5574335 3-503399 66332198! 3-539318 67082033 3:356893 67823300! 3583913 68556546, 3603326 6-9232032 3-631241 7-000000, 3-659336 70710678 71414253) 72111026 3732511 72301099; 3-756286 73181692 3-779703 74161985 3-302952! 74833143! 3825862.. 7-519334 1 3-843591 76157731 77459667! 3911883, 78102497) 3-936437 73740979, 3-957391! 7-9372539, 3-979057.. R-D000I0. 4-099000 | 80622577, U29726 $-1240334) 4041240; 3-634031 3-703433) 3370377), 76311457, 3-332996)| | i i | 1 | | f { | | Sq. Root. loubeRoots| No. | Square. “110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 123 129 130 131 132 133 134 pt | H33] 300763 | 815353523 4061513 134) 17956 Cube. 46%) 314432 4761| 328509 4900} 313000 5041, 357911 5184| 373248 5329| 389917 5476| 405224 5625) 421875 5776| 433976 5929, 456533 6034| 474552 6211| 493039 6100} 512000 6561) 531441 6724| 551368 68339! 571787 7058| 592704 7225) 614125 7396 636055 7569| 653503 7744) 68H72 7921| 704969 8100) 729900 821! 753571 8161| 773683 8643) 804357 8836! 839534 9025) 857375 9216| 831736 9409) 912673 9604, 941192 9301} 970299 10000; 1000000 10201) 1032301 10404) 1061208 10609) 1092727 10816) 1124861 11025!) 1157625 11236) 1191016 11449, 1225043 11664| 1259712 11831) 1295029 12100} 1331000 12321) 1367631 12514| 1404928 12769! 1442897 12996; 1481514 13225! 1520875 13456) 1560896 13689) 1601613 13924) 1643032 14161) 1685159 14400) 1723000 14641! 1771561 14884) 18158183 15129, 1860867 15376 1906624 15625, 1953125 15376 2000376 16129 2043333 16334 2097152 16641, 2146639 16300, 2197000, 17161 2248091; 17424) 2299968 17689. 2352637 2496104! TURTHRB6al 5 117230 Sq. Root. |CubeRoot, 82462113, 83066239 8:3666003 8-426 1498, 8-4852314 85440037) 86023253 86602540) 8:7177979 87749644 8:8317609) 88881944 89442719, 9-0000000! 9055335 1 91104336) 9:1651514, 92195445: 9-2736 185, 9-327379 1 9-3308315! 9-4339311 9436833) 9:5393)2)) 9-5916630 9 643653 9-6953597 9°7467943) 9-7979590 98183573, 98991949, 99498744) 4:6 10-0000000 10-0498755, 10-0995019: 10:14339:5! 10-1939330 10-24695.3' 102055301. + 10:3 140304, 10°3923)44 10-4403955" 10-4830335 10:535653%1 10-5330053! 10-6301459} 106770733. 10 7238033; 10-7703225! 108166533 10°8627805 10-9037124 10-95445 11 110000008 11-04533 1c 11-0905355) 111355237; 11:180330¥| : 112219722 11-2694277] 11-31370835 11:3573 167 11-4017543) 11455231 114831253 115325626! 4-08165 4 101566 4:179333, 4:198336 4272659 4-290540) 4°203+69) 4-3-6719) 4344481 4:36207] 4:379519 4°396830) 4562903 4-573357 4:594701 4:876999 4890973 4994863 4-918685 4946987 5026526 5-039681 5052774 5-065797 5078753

Page 422

APPENDIX. 19 Cube. Sq, Root. ICubeR oot. 8242408! 142126704! 536746 H, 8365427; 14-2478068| 5877131 8189664] 14-2323569 5836765 8615125] 14-3178211] 5396365 8741816] 14:3527001| 5905341 8869743) 143374946) 5-91548 8998912) 11-4222051] 5-9249% 9129329) 14-4568323) 5-93447 9261000} 14-4913767| 5-943 322 9393931] 14-5258390| 5:953342 9528123] 145602193] 5902732 9663597] 14-5945195! 5-972093 5°129328)) 3 5-142563) 5155137] 5-167649 5-180101 5192494 5-204828 5217103; 5-229321! 5241183 5253538 5265637, 2571353) 11°7046999 2628072) 11-7473401 2685619] 11-7898261 2744000) 11-8321596 2303221) 118743424 2863283| 11-9163753 2924207] 11-9592607 120000000 3048625] 12-0415946 3112136] 12-08304 147| 21609! = 3176523) 12°1243557| 5277632] 4 9300344; 14-6287333! 5°93142; 148; 21904) 3241792} 12°1655251) 5°289572) < 9933375] 14-6623783; 5 950725 149) 22201) 3307949] 12-2065555; 5-391459 10077696) 14-6969335| 6-OU0U0, 150; 22500) 3375000) 12-2474487) 5°313293, 10218313! 147309199] 6-003215 10360232| 14:7648231| 6013462 10503159) 147985486| 6-027650 10613000) 14-8323970| 6036311 10793361) 148660687} 6-035943 10941048) 14-8936644| 605504 11033567| 14-9331845; 606412 11239424) 14-9656295, 6-073173 11399625| 15-0000090! 6-US2: 11543176, 15-0332961| 609119 11697083) 15-0665192 6-109317 11352352| 15-0996639} 6-199115 12008939| 15-1327460; 6-113033 12167000, 15-1657509 612632 12326391| 15:1936342| 6-135732 12487169| 15-2315462| 6-144634 12649337| 15-2643375 6-15344: 12812904| 15-2970585) 6-1622 4i 12977875} 15-3297097 6-171005 13144256| 15°3622915) 6-179747 13312053, 15-3349013 613346 13191272} 15:4272486| 6-197154 13651919| 15-4596249, 6205322 3442951| 12-2332057| 5'325074|| 3 3511303| 12-3238280] 3-336303 3531577| 123693169) 5318481] 3652264] 12-4096736) 5-360103| « 3723875| 12449399: 5-371635, 3796416] 12-4399960| 5-333213! | 3869393| 12-5299641| 5-394691 153, 24964| 3944312) 12-5698051] 5-406120 | 25281] 4019679) 126095202] 5-417501 160, 25600} — 4096000| 12-6491106; 5-428835 181| 25921] 4173281) 12-6335775! 5-440122 162; 26244] 4251528) 12-7279221| 5-451362! 163! 26569) 4330747] 12°7671453, 5462556 Ha) 5506) 441094) 128062435), 5473704 27225] 4492125! 128452326) 5-434897 165! ¥7556) 4574296| 12°8840987) 5-495365 167! 27839} 4657463, 12 9228480) 5506873)! 163, 23224| 4741632} 129614814) 5-517349 169, 28561] 4826809) 13- 5523775 170; 28300 4913000] 130334043) 5-539658 171| 29241} 5000211} 130766963) 5-559499] 2 : 13-1143770] 5561293, 173) 29929; 5177717) 13-1529464) 5:57205 13324000) 15-4919334! 6-214465}. 174, 30276) 5268024) 13-19 5 532770 13947521! 15:5241747| 6223031]. 175| 30625) 5359375! 13-2287566 5593415, 14172433 15-5563192 6:23163 | 176| 30976) 5151776, 13-2664992 5-604079, 14318307) 15:5334573) 62402 177| 31329 9915233) 13°3041347| 5°614672,, 14526784) 156204994) 624330 : 178, 31684; 5639752, 13°3416641| 5-625226 14706125) 15-6524753| 6°257325 , 179) 32041] 5735339! 13-3730892) 5-635741 14336936; 15°6343371) 6°255327 ' 1830| 32400) 5832000) 13-4164079) 5-646216 15069223) 15-7162336; 6274305 181) 32761| 5929741) 134536240 5:656653 15252992) 157489157; 6232761 182! 33124; 6028568) 13-4907376! 5-667051 15433249| 15-7797335; 6291195 183| 33499) 6128487| 135277493 5-677411 15625000| 15-8113333; 6-299605 181| 33356) 6229504) 13-5645600 5-637734 15313251) 15-8429795) 6'3JTIH 185) 34225; 6331625| 13:6014705) 5-693019 16003008) 15:8715079; 6316369 185} 34596) 6434856) 13°6331817 5-708267) P 16194277) 15:9059737) 6'32:7 187} 34969] 6539203) 13°6747943 5-718479) 16387064! 15-9373775) 633392 16531375| 15 9637194| 6311325 16777216| 16 0039090 6-319564 16974593| 16-6312195| 6357361 17173512| 16-0623734! 6:36%0J7 17373979} 16-0934769| 6:3743: i 17576000) 16-1215155, 6339259: 17779531} 16:1554344| 63396 183| 35344) 6644672) 137113092; 5-723554, 189) 35721; 6751269) 13-7477271 5°738794 | 190; 36100; 6859000) 13:781 5743397, 191| 36431} 6967871| 13-8202750 5-753965)) 253 192! 36864; 7077888, 13-9564065 5-763995) 259 193, 37249) 7189057) 13-8924440) 5°773996, 269% 67 194} 37636) 7301384} 13-9283883) 5-783 | 261 195} 38025; 7414875) 13-9542400; 5°793390)| 262 17984723| 13°1864141) 6°39332 196; 38416] 7529536) 14-0000000; 5°303736)| 26 18191447 162172747) 6-196353 197] 38809} 7645373| 14-0376633; 5:313643) 264 13399744 16-2130763; 6-415969 193) 39204| 7762392; 14-07124173) 5823177; 265 18699625| 16-2738236. 6:42315 199} 39601) 7890599! 14-1067360] 5-833272! 266 13821096) 163035064! 643122- 200 300| 14°1421356 ee 26 19034163) 16°3101346 6439277 201| 40401) 8120601! 14°1774469 5-857766| 2638, 71824 19248832] 16-3707055] 64473 : 53

Page 423

20 | No! Square. | Cube. we har} oe © 100489 , 101124 . 101761 | 103041 72381 72300 73141 73984 74529 75076 75625 76176 76729 77231 T7841 78400 7396) 79524 80089 80656 81225 81796 82369 82944 83521 81100 84681 85264 85349 86 136 87025 87616 83299 88804 89401 90000 90691 91204 91809 92416 93025 93636 94249 94864 95431 96100 96721 97344 973469 98596 99225 99336 102400 22: 103584 101329 101976 105625 106276 106929 107581 108241 108900 109551 110224 110839 111556 112225 a 19465109 19633000 19902511 20123648 20346417 20570824 20796975 21024576 21253933 21481952 21717639 21952000 22183041 22425763 22665187 22906304! 23149125 23393656 23639903 23337872 24137569 24389900 24642171 24897083 25153757 25412184 25672375 25931336 25198073 26163592 _ 26739899 27000000 27270901, 275413603 27818127 28094464 28372625 23652616 28934443 29218112 29593629 29791000 30030231 39871323 3.664297 30959144 31255875 31554496 31855013 32157432 32461759 32763000! 33076161 33336218 33693267 31012224 34323125 31645976 31965783 35287552 35611239 35937000 36264691 36591363 36926037 37259704" 37595375, APPENDIX 16-4012195| 6455315 16-4316767| 6463304! 16:4620776| 6-471274 16-4924225! 6-479224 16-5227116 6437151 16-5529454! 6495965 16-5331240 6-5 16-6132477 16-6433170 16-6733320 16-7032931 16-7332005 16-7630546 16-7923556 16-8226939 16-8522995 16-8819430 16 9115315 16-9410743 16-9705627 17-0000009 17-0293364 17-0597221] 17-0330075 17-1172423 17-1464232 17-1755610 172046505 17-2336379 17-2626765 17-2916165 17:3205981 17-3193516 17-3731472 17-4063952 17-4355958 17-4642492 17-4929557 17-5214155 17-5199238 17-5783953 17-6063169 17-6351921 17-6635217 17-6918060 17-7200151 177482333 177763333 17-8044933 17-8325515 17-8605711 17-8835 133 17-9164729 17-94 13334 17-972208 18-0000000 18-0277564 18-03541701 180331413 18-1107703 18-1333571 18-1659921 18-19314054! 19-2208572 18-2482376! 6-931391 18-2756669| 6-933232 13-3030052. 6945150) 6-519912 6-557672 6-565414 6:573139 6530344 | 6-533532) 6:5965202 | 6603354; 6-611439 6-619106. 6-626705 6-634237 | 6-641852 6-649400 6-656930 6-664444 6-671949 667912) 6°635883 6694323) 6-709173 6°716570) 6°723951 6°731316 6°733664 6°745997 6°753313; 6°7606 14 6°767899 6°775169) 6782123 6-789661 6-796831! 6811235) 6813162! 6°825624) 6839904 | 6-847021 6-854124 6-861212 6863235: 6-875344] 6832339 6-889419|| 6-896435 6-903136 6:910423|! 6-917336]! 6-924356 6701759, : 6-904092); < 6832771) : 336 337 333 339 400 401 | 126736 | 151321 Sq. Root. (CubeRoot | No, | Square. 112896 113569 114244 114921 115500 116281 116964 117649 118336 119025 119716 120409 121104 12189] 122509 123291 123904 124699 125316 126025 127449 128164 129881 129600) 130321 131044 131769 132494 133225) 133956 134639 135124 136161 136999 137641 133384 139129 139876 140625 111376 142129 142884 14364] 144400 145161 145924 146639 147456 148225 148996 149769 1505144 152100: 159381 153664 154449) 155236 156025 153816 157609! 158404 159291 160000 160301 402| 161604| Cube, 37933056 38272753 33614472 3808219 3926 1000 39651821 40001683 40353607) 18- 40707584 41063525 41421736 41781923 42144192 42508549 42875900 43243531 43614203 4398 5977 4426 1864 44733875 45118016 45199293 45382712 46263279 46655000 470538) 47437928 47832147 48223544 43627125 49027395 49130863 49835032 5243409 50653900 51064811 51478848 51895117 52313624 52734375 53157376 53582633 51010152 54439939 54872000 55306311! 55742968 56181887 56623104 57066625 57512456 57960603 53111072 5 3363369. 59319000 59776471! 69236233 60693457 61162934 61629875! 62099133 62570773. 63044792 63521199 64000000: 64431201 64964308 j Sq Root. \CubeRoot. 18-3303928! 6-952053 18:4390839| 6979532 184661853) 6-986363 18-6910752, 7-020349 18-6279360) 7-027 106 18-6547581| 7-033350 18-761653 "| 7-060697 187332942 7-067377 18-8148977: 7-074044 18-8414437, 7-080699 13-8679623 19-8914136, 7-093 371 18-92 '8379 13-9472953 18-9736669 19-0000000 19-0262976 19-0525539 19-0737810 19-1049732 19-1311265 191572441) 7-159599 19-2973915 19-3132979 19-3390796 19-3649167) 7211248 19-3907194 19°4164878| 722404 194422221 19-4679223) 7° 19-4935887 19-5192213 19:5443203| 72555 19-5703353, 7-262167 19-5959179; 7263432 19-6214169 7-274786 19-6469327| 7-231079 19:6723156| 7-297362 19-6977155; 7293633 19°7230829) 7-299994 19-7434177| 7306144 19-7737199 7-312383 19°7939899| 7-318611 19-8242276! 732482 19-8494332) 7 331037 19-8746059| 7-337231 19:8397437| 7-343420 19-9249583) 7349597 20 1499377| 7-330323

Page 424

162409 163216 164025 164836 165 49 j 166464! 167281 168100 168921 169744 170569 171396 172225 173056 #} 173889 174724 175561 176100 177241 173684 | 173929 181476 7| 182329 423) 183184 429| 181011 430| 184900 431 | 204304! 205203 | 206116: 453, 207025] _ 207936] 457! 208349 458. 209764 459° 210681 460 211600 464; 212521 462213444 463 21436) 464 215296 465 216225 466 217156) 467, 218189) 468, 219024) 469| 219961) £5939264; 20-0997512 €6433125| 20-1246118 66923116] 20-1494417] 67419143) 20-1742410) 67917312. 20-1990099 68117929] 292237434 68921090] 202484567 694267 31) 23-2731349 69934! 23| 20-2977331 70444997, 20-3224014 70357944| 20-3463399 71473375| 20-3715483 71991296, 20-3960731 72511713, 29-4205773 73034632, 20-4150183 73560059, 20-46)4395 74033000} 20-4939015 74618161) 20-5182345 75151448) 29-5126336 75636967) 20-5663638 76225024, 20-591260; 76765625 296155281 77303776; 20°6397674 77354493 236633733 73402752) 20-6831609 78953539, 20-7123152 72507000, 20-7364414 80062991; 20-7605395 80621568. 20-7846997 81182737) 20-8086523 81746504! 208326667 82312875 208566525 82881856 20-8806130 83453453 20904545! 84027672! 299284495 20-9523268 20-9761770] 21-0000000) 21-0237960 21-0475652 21-0713975) 21-0950231 21-1187121 21-1423745j 21-1660105 21-1336291 21-2132034: 212367606 21-2502916 2128379671 21-3072759) 21-3307290 21-3541595, 21:3775533; 214009346] 214212353; 21-4176106 21-4709106 214941553, 215174348 21-5406592 215633537! 21-5370331 21-6101523 21-6333977 21-656 1078) 84601519 85131000 85765121 86350388 86938307, 87528334 89121125 88716536, 89314623 89915392 90518349 91125000) 91733851) 92315103 92959677, 93576664 94196375 94818816 95413993 96071912 702579 7336000 97972181 93511128 93252347 9J897344 100514625 101194696 101817563 102503232 103161709 APPENDIX, 7-336437 7-392542, 7-398636! 7404721 7110795 7416859 TIRIH 7-428959 7434994 7441019 7447034 7-453040 7-459035 7465022 7-470999 7-476966 7-482924 T488972 7434311 750741 7-536661 7512571 7518473! 75 24365) 7530243 7535122) 7541987 TSA), 7-553639 7559526, 7-555355" 7:571174|| 7-576985" 7-532786) 7:583579|! 7-534363" 7600133. 7-605905 7611663 7617412 7-623152; 7623381 7634607; 7 61032 7646027, 7651725" 7657414 7663994 7663766" 7674430! 7630086: 7635733 7691372 7637002 7702625, 7708239. 7713345 7719443 7725932 TTG. TT36183 7741753 7747311} T 7528611 1753402, 7763-36. 7-769462 Sq. Root. CubeRoot | No. i Square. 65450527! 20-0748599 470 471 472 473 474 475 476 4 478 479 430 431) 432: 433 434; 435 486 437 438 439) 430) 491 432 493 494. 435! 498i 497 493 49 590 591 502 503) 504! 505! 506! 597| 508) 509} 510! 511 512 513 514 515l 516: 517 5] 51 5: 521 522 523) 524 525 526 527 23 529 530 531 532 533 534 535 536 223900! 221841 2297 223729 224676 225625 226576 227529 223481 229441 230499 231361 232324 233289 234256! 235225 236156 237169 238144] 239121 210100 241031! 212064 213049! 241033 245025 2460 lb 247009 243004) 249001) 250000 251001 252004 253009 254016 255025 256036 257043| 253061) 259031 260100 261121 262144 263169) 264196 265295 266256 267289 263324 269351| 270400; 271441) 272434 273529) 271576' 275625 276676) 277723! 273734 279341 280990! 281961 233024 281039) 285156 286225 287206 ——. Cube. 21-6794334 21702553H 21-7255610 217435632 217715411 21-7944947 218171242 21-8403297 218632111 21-8360686 21-9083023 21-9317122 21:95 14934 219772610 ZL'VUOVOIU 220227155 22 0351077 220689765 Z2 UI0722Y 29 11334144 22 1359135 2271535133 22 1810739 222033033 222291103 222485355 2227105 (5! 222534963 22°3159135 22°3333U79! 22-36U5798 22333293. 22-4053565" 2242766 15| 224499413 22-4722051 22-4944433 22-51665u5) 225339953! 22 5d]Uz83) 225331793 2200531 226274170 22-6195033 226715531! 2269301 l4; 22 7156334 227376340: 227530131] 22-7315715 103323000 104187111 105154048 105823817 106496124 107171875 107850176; 108531333) 109215352) 109902239} 1105920.0 111281641 111930163 112678537 113379904 114082125! 114791256! 115301303 116214272 116930169 117649000 11837077 1 119035133; 119823457 120553734 121237375} 122623336! 122763473) 123505992 124251499 125000000 125751501 126506008! 127263527) 128024064 128787625 129554216 130323343 1310505 tz! 131372223 13255 1000 133432331 134217723 135005697 135796744! 136530375 137383096) 19518415) | 138991832 13979335y! 1406U8v0u0! 14142u761 142236645 143055667 13877321 144703125 145531576 146363183 147197952 143935333 1483770 149721291 150553763 151419407] 152273304) 153130375; 153990656 ' 223173193 223091933 228910463 223128185 22:9316 93! 22953436 229732500 23 VUJOJUU L3 UIT 230434372 230651252! 230307920 23-1034400 23-1340670 22°3u35085! 223254044 23-1516738, Sq. Root. |CubeRout. 797911: T93A3 H 7989570 7-99473 8-025957 8-031129% 8-036029: 5441451 8-092677 509775 8-102337 d: 107917 S112385 8&-11s0il 8-1239.

Page 425

288369 239444 293521 291600 292681 293764! 294349! 295936 297025 298116 299209 300304 301401 302590 3933801 394704 335309 306916 208025 309136 310249 311334 539} 312431 313600 314721 315344 316969 318096 5 319225 5| 323356 321489 322624] 323761 324909 326041] 327184 323323 | 329476 330625! | 331776] 332929 | 334034 335241 335400 337561 | 333724 | 339339 311056 342225 313396 344569 345744 346921| 343100, 349281 350464! 351649 352836 354025 355216, 356409" 357604) 358801, 360000 361201 362404; 363609) EN Cube. 154854153 231732695! 81253145.) 155720872. 156590819 157464050: 158340421 159220088) 160103007 160939184 161873525, 162771336, 163667323, 161565592 165469149 166373000 167234151 163196603 169112377 170031464 170353375, 171379616 1723086933 173741112 174670573. 175516000 176553481] 1775 34328 173453547) 179410144 130362125 131321496. 182231263 133250432 184220009 185193000 136169411 187149248 183132517 189119224 190109375 191102976 192100033 193100532 194104539 195112000 196122941 197137368 198155287 199176704 200201625 201230055 202282003 208297472 204336469 235379000 206425071 207474683 208527857 209584584 210644875 21170873€ 212776173 213347192 214921799 216000000 217081801 218167208 219956227 Sq. Root. 23-1948 270 23-2163735) 23-2379001] 237591067] 23-2503935 23-3023604] 23-3233076. 23-3152351 23-3665429 233330311 23-41093998 23-4307490 23-4520738 234733392 23-4946802 235159520 23-5372046 235534380] 23-5796522 23-G003174 23 6220236 23-643 1808. 23-6643191 236354336 237063392 237276210 IZTAIBESA2 237697285 237907515 235117613! £ 23° 83275.. Gh 23°8537 209) 238746728 8 23 8356063) 23-9165215! 23-9374 184 239532971 23-9731576" 24-0000000 210208243 210116396 240624188 240531991 241033416 24-1246762 24-1453929 24-1660919 24-1867732 242974369. 24-2230829 2i 24187113 24269322% 212999156 24-3104916 213310501. 243515913" 243121152 243926218; 244131112 244335334. 21-4510385) 24.4744165 24-4943974 245153013 245356333! 245560533! APPENDIX. CuheRoot- i} 8: 133187) 8-133223! 8-143253 8-143276! 8:153294: 8:153305 8163310 8-168339) 8-173302 8-178299! 8:183269|. 8-188244 8:193213 8198175 8203132) 8205032 8213027 8217966 8222893, 8232746), 8-237661 || 8-212571! 8217474. 8-252371 6 8°257263 8-462149 8267029]! 8271904 8:276773|! 528163 8256493! 8291344); 8-246190] 8:39103)) 8:3J5965, 3310624) 8: ee 832085 8325147 8-329951 | 8-331755)! 8-339551 8344341! 8-349126 8-353905" 8: Sean) 8363447] 8363209; | 8 372963) toh 377719 8 332465, 8: 337206) 8: 391942!! 8°396673): 8-401393), 8-406118.. & 410333, 841554 8-420246] 8424945], 8429633); 8434327); 8-439010; 8: 443683), 8-418360] j Na. 6J; 605! 606 607 608 609 669, 670 Square. j 395641 | 403225 i| 405769 | 412164 | 417316 3, 416025 | 418609 | 429025 | 432964 | 434281 | 435600 | 436921 ` 438244 _ 449896 ` 442225) 294079625 | 444839! 298740963 ` 446224 298077632 447561 299418309 Cube. 220343354 221445125! 222315016 293643543! 224755712! 225866529 22698 LUGO 228099131 229220928 230346397 231475544 23260837 233744895 234385113 235029032 237176659 233328000 239433061 240641848 241304367 242970624 244146625 215314376 246491833 249673152 248853189 250047 Ov0) 251239591 252435958; 253636137 254840104 256047375 257259456) 258 174853 259694072 260917119 262144000 263374721 264609283 265847707 267089984 263336125 269586136 270340023 272097792! 273359449 274625000 275894451) 277 167808 273445077 279726264 281011375 282300416! 283593393: 234890312. 286191179, 7496000. 2 288804781 290117528: 291434247 292734944 3.4816 3-6025 357236 368149 269664! 37088 1! 37210% 373321 371554 373769 376996 375225 3794156 330689 331924 333161 331400 335641 386334 33 129 333376 390525 39187 393129 3941334 3I69UC 333161 339424 490683 491956 404196 107044 408321 4096.10 410831 413419 414736 419904 421201 422500 23801 425104 426409 427716 43.336 431649 439569) 443956 295408296 448900; 300763000) 245764115, Sq. Rezt. 21596747 8 24-61 70673. 24-63737001 8- 24-6576560, 246779254 24 6981731 247184142; 21-7386339; 2475898359 JOuteRuot ~ 8 453096 8: "457691 ‘8-480926 8435559 8 490185 849499 25 1592913; 24-7790234 24°7991935 218193473 248394347} 8- 248596058 8-517340 218797106 243997992 219198716 249399278 219594673 24:97 9992U 250000000 25019992) 25-0399681 25-0599232 250792724 2F099SUUs, è 2511971341 k 25°1395102] 8-591581 8536205 25-1793566. 8-590724 254992063] 8595233 252190104 859974 25°2334539) 8604252 252586619] 8-6753 25 2731493 8-613243 25.2932213 8-617739) 253179778) 8622225 25°3377139| 8-626706 253574447 253771551 %5 3968502 8631183 8635655 8.610123 257416539} 5644585 25 4751734] 8° 25°495097 a 25°5147016 8-666331 255342907! 8-671266 255533647] 8 675697 255734237! 8-689124 255929678 8-684546 256124969, 8-63896 256320112! 8:69337 25°65 15107; 8-697731 = 6709953 870218 56904652. 870653 23° 7099203) 8-7108 25°7293607| 8-715373 25°7487854) 8-719760 25°7681975, 8724141 257875939 8-728515, 25°8069758) 8-722992 25°8263431| 8-737260 258156960! 8-741625 25-8650343 874598 25-8842582, 8°75034u)

Page 426

Cube. l 302111711 303464448 304321217 306182024 307546375 450241 451534 452929 454276 455625 456976 453329 459684 461041 462400 | 463761 465124 466489 467356 469225 470596 T| 471969 473344 474721) 476100 310288733 311665752 313046839 314432000 315821241 317214568 318611987 320013504 321419125 322828356 324242703 325660672 327082769 323509U00 329939371 331373588 332312557 334255334 335702375 337153536 338608373 340063392 341532099 343000000 344472101 345948408 347428927 318913664 350402625 351895316 353393243 354894912 356400829 357911 359425431 360944128 362467097 363994344 363525875 367061696 368601813 370146232 371694959 373248000 ER or] 377933067 379503424 331078125 382657 176 384240583 385828352 387420489 389017000 390617391 342223168 393832837 395440944 337065375 395683256 4003 15553! 491401 492504 494209 495616 5| 497025 493436 499849) : 501264 5U263 1 904100 505521 712| 506944 508369 509796 511225 p! 512656 514089 515524 516961 518400 519841 | 521234! 922729 ! 524176 525625! 5 527076 | 528529 23; 529984 531441; , 532900 534361) 32: 935524: 33! J729 z | $3576 » 54u25 | 5HY 37; 343109 25-9036677 308915776] 26 APPENDIX, Sq. Root. 259229628 25-9422435 25-9615100 259807621 "0000000 26-0192237 26-0381331 26.0576281 26-0763096 26-0959767 26-1151297 26-1342687 26-1533937 26-1725047 26-1916017 26-21068418 26-2297541 25-2483095 262678511 26-2868789 26-3U53929 26-3248932 26343877) 8°853598 26-3628527 263318119 26-4007576 26-4196896 26-4386081 26-4575131 264761046 26-4952826 26-5141472 26-5329983 26-5518361 26-5706605 265894716 26-6082694 266270539 266458252 26-6645833 26:6833281 26-7020598 26° 7207784 $*°7394839 26° Peas 8°78934 8°793659 | 8-797968 88022721 8- -806572 8 bet Se 267763557 267955220 268141754 268328157 268514432 268700577 26'8886593 269072481 26-9258240 26-9443872 269629375. 26-9814751| 8-995883 270000000) 9-000U0} 270185122, 9-004113) 10370117] 9-008223 27-0554985. 9-012329 970739727! 9016431 270924344! 9-020529 27.1108534: 9023624 27-1293199. 9-023715 271477430! 9-u323U2\' ieee 747 , td! 749 750 346 iel 547600 549031 550564 552049 953536; 555025 556516 558009 559504 551001 562500 Square. | Cube. | 445949744 D| 504353336 644| 401047272 27-1661554! 403583419! 405224000 406369021 408518488 410172407, 411830734 413193625] 4151609361 416332723 418508992 420189749 421875000 423561751 425259003 426957777 423661064 430369875 432081216 43379009 435519512 437245479 438976000 240711081 442450728 444194947 447697125 449455056 451217663 452984832 454756609 456533000 458314011 460099648 461889917 463684824 465434375 467288576 469097433 470910952 472729139 474552000 476379541 470211768 439303872 491169069 493039000 494913671 496793088 498677257 500566184 5024593875 506261573 513922401 515849608 517781627) 23 519718464 |s Sq. Roct. j| 230356915 23 |CubeRoot. 27:1845544| 9 272029410 272213152 27° 2396769 27-294688 1 27°31300C6 27-3313007 27°3195337 27°3678644 27-3861279 274043792 27-4225134 27-4403455 w7: 4590694 274772633 27-4351542 z7 513933. 275947335 27-5449940 27°36809735 2733802204 27-604 3475 276224546 276405493 27-6580334 27-6767U5U) 27°6917648 277128129 IT TIu 277485739 277663865 27°7043880 278025775 27°8208595 LT d355218 27°8567700 27-°8747197 27°39265 14) 27-9105715 BTU 279403772 279642629 274821372 238-UUUU000U 23 0173515 88 SS oe SR EF YB UI352W3 280713377 g3 U5 1435 23° 1069386 Bs" 4247222 23° 142440 23° 1602557 28° 1750050 25° 1957444 252134720 23-231 lood V8 ‘243895 23-2665881 25238427 le 25°30 1434 gs Ilt’ -5372510 929410. “BSS1SYIS 9207

Page 427

24 805 806 807 Bog 809 Slu 811 812 813 814 815 816 817 818 819 820 82] 822) 823 824 825 26 $27 828 829 830 331 832) 833 S34 835, 036, 837) 833, 839, S40! 841) 842 813, Bit BAS) 816 847 843) 849) 850 851 852! 853 854 835, 856 857 858 859) 860, 861, 362! 363) 864 865 366; 867: 868, 869, 70: 371| 643025! 649636 651219 652864 6541181 656100 657721 659314 660969 662596 6642235 663356 667489 669124 70761 672400 674041 675684 677329 678976 680625 632276 633529 685584 687241 6839 -U 690561 69.224 693839 645559 697225 693896 700569) FueL244 qJu3yz21 705690 GU7<281 708: 64) 710649 712336 714025 715716; 717409 719104! 720801 722500 724201 725904 T2T60») 729316 7310:5 732736) 734449 736164 73733) 733600) 741321] 743044 744769 746196 743.25 749956 75168v 753424) 755161 75690u 75864 i No. Square. | Cube. | Sq. Root. 521669125 5236u6616 525557343. 527514112) 529475129! 531441000 533411731) 53538732 537367797 539353144 511343375 543338496 515333513! 517313432 549353259) 551363000; 553337661 555412248 557441767 559476224 561515625 5563559976 565609283 567663552 569722789 571737000 573856191 75930368 578009537 530093704 582189875) 584277056 586376253 5353480472 590539719 532704000 594823321 596947688 599077107 601211584 603351125 6054195736, 607645123, 609300192) 611960049, 614125000 6162. 5051 618470203; 620659477] 622835864) 625926375; 627222016! 629422793 631623712 633839779, 626356000 634277331) 610503923) 642735647 6449725 44 647214625 649161395 651714363: 6533572032 655234909) 655503000; 660776311. APPENDIX. CubeRoot. t 28 3725219 28-3901391 234077454 28 4253403 234429253 284604989 23-4780617 23-4956137 235131549 235306352 23-5482048 23-5657137 28-5832119 28-6006993 23-6181760 28-6356421 28-6530976 28-6705424 236379766 237054002 287228132 28-7402157 287576077 28774991 287923601 28-8097206 | 28°8270706! 9401569) 288414102, 97405339) 28°8617394; 9409105; 238790582, 9°4 12869; 238963666! 9-416630 29:9136546| 9-420337 289309523) 9424142! 28-9482297| 9427834, 28:9654967/ 9431642, 289827535! 9435398; 29-0000000! 9439131 29-0172363) 9442370 29:03414623 9446607 290516781! 9450341, 2906833837! 9451072 29-0860791, 9-457800 291032644, 9461525 29:1234396| 9465217 29-1376046) 9463966, 29-1547595) 9472682 29-1719043" 9476336 29-1890330. 94301u6 29-2061637) 9483314 292232784, 9437518; 292403839" 9491229; 29-2574777, 9494919, 29-2745623, 9498615" 29-2916370! 950238 293087018. 9505993 29-3257565, 9509985 43°3423015 9513370" 29°3598365 9517051 29-3768616. 9520730 293933763 9°524406. z¥41U3ssz3 1528079 294273779 9531750 294448637 9535417 294615397 953908% 23-4738059. 9512744; wyedyaT624 95416403 295127021 9554059 9-306323. 9-310175; 9314019 9-317860, 9-321697 9325532 9-325363) 9333192 9337017 9-340839 9-344657 9°3:8173 9-352236 9-355095" 9-359902 9-363705; 9 367505 9371302 9-375096 9:373387 9332675 9-336460! 9-390212 9-394021 93.7796) No. | 873 874 873| 876 877 878 879 830 910 911 712 913 914 91d 916 91; yls 91y WEAR 92; 925 929 §3u 931 932 $33 934 935! 92:6 937 93s! — 9302477 | 872| 760384 i 783225 | 784997 5| 820836, Cute, 663954848 665338617 667627624 669921875 672221376 74526133 676836 152 679151439 681472000 683797341 686128968 688165387! 690807101 693154125 695505456 697864103 700227072 702595369 701969000 707347971 709732288 712121957 714516934 716917375 719323136 721734273 724150792 726572693 729900000 731432701 73387080! 736314327) 30-0499584 733763264! 30-0665923 741217625) 30-0332179 747677416] 30°099833) 716142643) 30°1164407 7486 13312| 30-1330383 751089429) 30-1496269 753571000. 30-16562963 756053031) 33-1827765 758550528) 30°1993377 761018197) 302153399 76255 1944| 30°2324329 766060875! 30-2489669 763575296! 30-9654919 771095213| 30-2820079 773620632) 39-2385148 776151559| 30-3150128 773638000} 30-3315018 781229961| 30-3479818 783777448! 30-3644529 786330167 3 Sota Square. 29-5296461 29-5465734 29-5634910 29-5303989 29-5972972 29-6141858 29-6310648 29-6479342 29-6647939 29-6816442 23.6984843 23-7153159| 29-7321375 29-7189496 29-7657521] 29-7825452 29-7993239 29-8161030 29-8323678, 29-8496231. 29-8663690 239831056 29-8998328. 23-9165506, 2% 5332591 23-9499533 29-9666491 29°9833237, 30-0000000 30-0166620 762129 763376 765625 76737 769129 770884 712641 774400 776161 771924 779639 731456 786769 738544 790321 792100 793881) 795664 797449 799236 801025 802816 804609: 806404 808201 810000 811801 813604; 815109 817216 819025 822649 824464 826231 828100 829921 831744 833569 835396 837225 839056 840889) 812724 841561] 846400. 848241 850084 551929 853776 855625! 857476 859329 861184 863041 864900. 866764 S634. BT) 7838889024) 30-39373633 791453125 794022776 796597983 799179752 801765039 8C4357000 Su605 1491 809557568 30-4138 12 33-4392491 309-4466747 30-4630924 30-4795013! 30-4959014! 30-5122926 30-5286750 812166237] 30-5450497 872356" 814736504] 30-5614136 874225 817400375) 39-5777697] 876096 820025356) 305241171) 877969, 822656153, 3)°61U455 7 879814! 825293672| 3v-6267857, Sq. Reat. \CubeRoot. 30°0333148) 9: 9-55371 DO1ITYI 9-611539: 9-619002 9-622693 9-626202 9-623797 9-633291 9-636381 9-64056. 9-654415: 9-647737 9-672740 9-676392 907-860 9-690521 9-694006: 9-764437 9-767992 9-771184 9-774974 9-775462 9 731947 9-735429 9-7359U9

Page 428

APPENDIX. 25 Sq. Root. |CubeRoot.'| No, | Square. Cube. 881721, 827936019) 30-6431069 940900) 912673000 883600 830584000) 30-6594 194 942841) 915498611 885481) 833237621) 30-6757233 944784) 918330048 837364 835896888; 30-6920185 946729} 921167317 889249 838561807) 30-7083051 948676] 924010424 891136) 841232334) 30-7245830 950625) 926859375 893025, 843908625 952576] 929714176 894916) 846590536) 307571130 954529) 932574833 SRA 849278123, 30-7733651 956484| 935441352 898704! 851971392) 307896086 958441) 938313739 900601| 854670349 308058436 960400! 941192000 902500! 857375000) 30-8220700 962361) 944076141 904401) 860085351! 30:8382879 964324) 946966168 906304) 862801408) 30-8544972 966289} 949962087 908209 865523177) 30°870698 1 968256; 952763904 910116] 868250664 30-8868904} 9°844254 | 985| 970225) 955671625 912025) 870983875; 30-9030743! 9°847632 | 986| 972196) 958585256 913936) 873722816 30-9192497| 9:851128 | 987| 974169) 961504803 915849) 876467493) 30-9354166) 9°854562 | 288) 976144) 964430272 917764] 879217912, 30-9515751! 9°857993 | 989] 978121) 96736166 919681} 881974079 30-9677251| 9-86 1422 | 990] 980100] 970299000 921600; 884736000) 30°9938668) 9-864848 | 991) 982081) 973242271 923521! 887503681, 31-0000000! 9-868272 | 992; 981064) 976191488 925444! £90277128 31-0161248! 9:871694| 993) 986049) 979146657 927369; 823056347, 31-0322413| 9°875113| 994) 988036] 982107784 929206, 895841344) 31-0483494) 9°878530 | 995; 990025) 985071875 931225) 898632125 31-0644491} 9°881945 | 996| 992016] 988047936 933156) 901428696: 21-0805405) 9-885357.) 997; 994009) 991026973 935089} 904231063 31.1126081 9-883767, 2) 996004! 994011992 £ z 31-4960315| 9-973262 315119025) 9-976651! 315277655) 9-979960 315436206 9983305 315594677) 9-98664y 315753068! 9-989990 315911330| 9-99332: 31-6069613! 9-996666 31-6227766|10-0000V0) ' 968| 937024; 907039232 31:1126984| 9:892175| 999; 998001| 997002999 969; 938961| 909853209 31-1287648; 9-895580 | 1000/1000000 1000000000 The following rules are for finding the squares, cubes and roots, of numbers exceeding 1,000. To find the square of any number divisible without a remainder. Rule.—Divide the given number by such a number, from the forego- ing table, as will divide it without a remainder ; then the square of the quotient, multiplied by the square of the number found in the table, will give the answer. : Example — -What is the square of 2,000? 2,000, divided by 1,000, a number round in the table, gives a quotient of 2, the square of which is 4, and the square of 1,000 is 1,000,000, therefore : 4 X 1,000,000 = 4,000,000: the Ans. Another example.—What is the square of 1,230? 1,230, being dt vided by 123, the quotient will be 10, the square of which is 100, and the square of 123 is 15,129, therefore : 100 X 15,129 == 1,512,900: the Ans. To find the square of any number not divisible without a remainder. Rule.—Add together the squares of such two adjoining numbers, from the table, as shall together equal the given number, and multiply the sum by 2; then this product, less 1, will be the answer. Example.—What is the square of 1,487? ‘The adjoining numbers, 743 and ‘744, added,together, equal the given number, 1,487, and the square of 743 = 552,049, the square of 744 = 553,536, and these added, = 1;105,585, therefore : 1,105,585 X 2 = 2,211,170 — 1 == 2,211,169: the Ans. To find the cube of any number uwisiHe without a remainder. Rule.-—Divide the given number by sven a number, from the forego

Page 429

26 APPENDIX. ing table, as will divide it without a remainder ; then, the cube of toe quotient, multiplied by the cube of the number found in the table, will give the answer. Example.—W hat is the cube of 2,700? 2,700, being divided by 900, the quotient is 3, the cube of which is 27, and the cube’ of 900 is 729,000,000, ibere : 27 X 729,000,000 = 19,683,000,000: the Ans. To find the square or cube root of numbers higher than is found in the table. Rule.—Select, in the column of squares or cubes, as the case may require, that number which is nearest the given number ; then the answer, when decimals are not of importance, will be found di- rectly opposite in the column of numbers. Example.—What is the square-root of 87,620? In the column of squares, 87,616 is nearest to the given number; therefore, 296, im- mediately opposite in the column of numbers, is the answer, nearly. Another example.—What is the cube-root of 110,591? Jn the co- lumn of cubes, 110,592 is found to be nearest to the given number ; therefore, 48, the number Opposite, is the answer, nearly. To find the cube-root more accurately. Rule.—Select, from the co- lumn of cubes, that number which is nearest the given number, and add twice the number so selected to the given number; also, add twice the given number to the number selected from the table. Then, as the former product is to the latter, so is the root of the numbar selected to the root of the number given. Example.—What is the cube-root of 9,200? The nearest number in the column of cubes is 9,261, the root of which is 21, therefore : 9261 9200 2 s 18522 18400 9200 9261 As 27,722 is to 27,661, so is 21 to 20-953 ++ the Ans. Thus, 27661 x 21 = 580881, and this divided Ly 27722 = 20:953 + To find the square or cube root of a whole number with decimals, Rule—Subtract the root of the whole number from the root of the next higher number, and multiply the remainder by the given decimal; then the product, added to the root of the given whole number, will give the answer correctly to three places of decimals in the square root, and to seven in the cube root. FHrample.— What is the square-root of 11:14? The square-root of 11 is 3°3166, and the square-root of the next higher number, 12, is 3:4641; the former from the latter, the remainder is 0°1475, and this by 0°14 equals 002065. This added to 3°3166, the sum, 333725, is the square root of 11°14. To find the roots of decimals by ‘the use of the table. Pule—Seek for the given decimal in the column of numbers, and opposite in the col- umns of roots will be found the answer, correct as to the figures, but re- quiring the decimal point to be shifted. The transposition of the deci- mal point is to be performed thus: For every place the decimal point is removed in the root, remove it in the number two places for the square root and three places ‘for the cube root.

Page 430

APPENDIX. 27 _ Hxamples.—By the table the square root of 86:0 is 9:2736, conse- quently, by the rule the sqiare root of 0°86 is 092736. The square root of 9° is 3°, hence the square root of 0'09 is 0'3. For the square root of 0°0657 we have .0°25632; found opposite No. 657. So, also, the square root of 0°000927 is 0°030446, found opposite No. 927. And the square root of 8°73 (whole number with decimals) is 2°9546, found opposite No. 873. The cube root of 0°8 is 0-928, found at No.-800; the cube root of 0:08 is 0°4308, found opposite No. 80, and the cube root of 0°008 is 0'2, as 2°0 is the cube root of 8°0. So also the cube root of 0:047 is 0°36088, found opposite No. 47. RULES FOR THE REDUCTION OF DECIMAIS. To reduce a fraction to its equivalent decimal. Rule—Divide the numerator by the denominator, annexing cyphers as required. Example—W hat is the decimal of a foot equivalent to 3 inches? 3 inches is ;} of a foot, therefore: ° Fe +. - 12) 3-00 25 Ans. Another examjite—W hat is the equivalent decimal of 7 of an inch? Į... 2) 7008 ‘875 Ans. ; To reduce a compound fraction to its equivalent decimal. Hule——TIn accordance with the preceding rule, reduce each fraction, commencing at the lowest, to the decimal ot the next higher denomination, to which add the numerator of the next higher fraction, and reduce the sum to the decimal of the next higher denomination, and so procecd to the last ; and the final product will be the answer. a Example—What is the decimal of a foot equivalent to 5 inches, 3 and ,, of an inch. The fractions in this case are, } of an eighth, 2 of an inch, and +f; of a foot, therefore: : >. ere 5 3° eighths, $...... 8) S0000 "4375 5: inches, Pg e» e e o 12) 5437500 ‘453125 Ans

Page 431

235 APPENDIX. The process may be condensed, thus; write the numerators of the given fractions, from the least to the greatest, under each other, and place each denominator to the left of its numerator, thus: ae 2| 10 ==> Y RES, 8 | 3:5000 ARTE 12 | 5437500 453125 Ans. ; To reduce a decimal to its equivalent in terms of lower denominations. Rule—Multiply the given decimal by the number of parts in the next less denomination, and point off from the product as many figures to the right hand, as there are in the given decimal; then multiply the figures pointed off, by the number of parts in the next lower denomina- tion, and point off as before, and so proceed to the end; then the seve- ral figures pointed off to the left will be the answer. Hxample—W hat is the expression in inches of 0°390625 fect? Feet 0°390625 12 inches in a foot, Inches 4°687500 8 eighths in an inch. Eighths 5:5000 2 sixteenths in an eighth. Sixteenth 1°0 Ans., 4 inches, 3 and 4. Another example-—What is the expression, in fractions of an inch, of (6875 inches? Inches 0°6875 8 eighths in an inch. . Eighths 55000 2 sixteenths in an eighth, ' Sixteenth 1:0 Ans, § and 44

Page 432

TABLE OF CIRCLES. (From Gregory’s Mathematics.) from this table may be found by inspection the area or circum‘e cence of a ciicle of any diameter, and the side of a square equal to the area of any given circle from 1 to 100 inches, feet, yards, miles, &c. If the given diameter is in inches, the area, circumference, &c., set opposite, will be inches ; if in feet, then feet, &c. Side of Sıde of Diam. Area. | Circum, equal sq. || Diam. Area. Circum. equal s 25 1903 -78539) -22135||_-75|_—«90-76257| 3377212 952693 5 19635) 157079) = -44311/] 11- 9503317) 3155751) 974319 75 ‘44178' 235619) -66467/| 25! 9940195) 3534291) 9-97005 r 73533, 3-14159| 88622] 5 | 10386890) 35-12331| 10-19160 25 122718; 3-92699] 1:10778|| -75! 108-434093! 3691371) 10-41316 5 1-76714| 4-71238, 1°32934)} 12- 113-03733| 3769911} 10-6372 ‘73 2-405293) 5-49773/ 15508ə|| -25| 117:853831| 33-43451) 1085627 2 314159) 623318) 177245] 5| 12271846) 3926990; 1107733 25i 97607) 7068358) 199401 "75| 127°67628! 4005530) 1129933 5 4-20973| 7-835393) 2-21556)| 13- 13273228) 40-84070| 11:52095 -15 593357, 863937] 243712|| -25| 137-836416) 4162610| 1174259 3 706853 942477) 265363) -5| 14313331) 42-41150! 11-96406 25 82.576, 10-21017; 283023|| -75| 143-495934] 43-19689) 12-1356z 5 9-62112| - 10-99557] 3-10179|| 14 15393804 43-98229) 12-4071 “5 1104466) 1178037| 3°32335)] -25| 15948491) 44-76769) 1262373 4 12°56637| 12-56637 351490 ‘5 | 165°12996 . 43-55309! 12-8502 25 1418625; 13-35176| 3°76646/| -75| 170:87318| 46-33349| 13-07131 5 15:99431| 1413716) 398802] 15- 17671458) 47-12333; 13-2934) 73 1772954! 1492256. 420957 -25| 18265416| 9 47-90923, 1351496 5 1963195] 1570796} 443113] -5| 19369190) 43:69%463) 13:7365] 25) 21-64753| 1649336) 4°65269|] -75| 19482783! 4949003, 1395307 S| 2375929; 1727875| 487424) 16- | 201-06192 50:2051 4 1796 eo e Bens! samal 3 | sissi] | 518gse7! 1462278 25| 3067961) 1963495; 553391 75. 220-353271 5262167) 14-3130: 5 33-18307| 2942035) 576V47i; 17 22693006) 53-40707! 15-06535 -5| 3578170) 2120575) 5-98203|! -25! 23370504) 5419247! 15-2374] T 33:484156| 21-99114) 620359i) -5 | 24052813] 5497737) 1550897 25| 4128219) 2277654) 642514) 75| 24744950) 55 76326 15:73052…, 5 4117864) 2356194) 661670: 13- | 26146900] 56-54866 15-9520 DO ka Sum Heil 3°) msa Simi lege 8- 50:2654 -132 T : “8025 ; 3991 E E E DE E- Eo 35! cami arise i 775413 95 291-03910 6947565; 17-0596 Sale Paia Sana Tial a oan) Orosi 1 S0 95 D ! § -05 7 . we . © J 7k ; 5 7083218| 2984513) 841915]| 20 31415926 6283185] iie ‘5) 7466191) 30°63052; 864071|| -25| 32206233, 636172 won 10- 7853931, 31-41592, 8862%)| 5 330-06357| 64410264 [a 25| 8251539) 3220132! 903382|| -75| 3389-16299 65:19301) 1 ~ ‘5 | 8659014) 32986721 930533. 21- | 346-360593) 6597344; 19-6107

Page 433

an w 2h “The RoR a | 31 Area. 35465635 36305030 37154241 3580-13271 38882117 397-60782 406-49263 41547562 42455679 43373613 443-01365 452-38934 46 1:85320 47143524 48110546 49087385 50074041 51070515 520-768306 530:92915 54118842 55154586 562-00147 57255526 58320722 59395736 60780567 61575216 626-796832 637-93965| 64913066 660-51985 67195721 683:49275 6995-12646 706°85834 718 68840 130-6 1664 742 643U5 751767631 7866-99039 77931132 79173043 80424771 81686317 §29°57631 81233561 85529859 86830675 8531-41308 89461759 90792027 92132113 93482016 94841736 962°] 1275 97590630 98979803 1003:78794 101787601 1632-06227 104634670 1060°72930 1073-21003 1089-78903 1104-46616 1119-2414" Circum. 66°75884 6754424 68:32964 69-11503' 69-90043 70-68583 71-47123 7225663 73-04202 7382742 7461282 75.39822. 76-18362' 76-96902' 77.75441] 78-5398] | 79-32521 80-1106] 80-8960] 81-68140 82-46680 83-25220, 84-03760 84-82300 85-60839 86:39379 | 87-17919 87-96459 88-74999) 89-53539 90-32078) 91-10613 91-39153, 92-67698 93-46233! 9424777 95-03317 9581857 96-60397 97-38937 98-1747 98:96016. 99°71556: 100-53096 101-31636! 102-10176 102-88715 103-67255 104-45795 105-24335 10602875 106°81415 10759954 10833494 109-17034 10995574 11074114 11152653 112-31193 113-09733 113-88273 11466313 115-453353 116-2382 11702432 11780972 11859572 APPENDIX. Side of Circum. e ee a e ee a ee 18:83232 | 38: 19:05387 1927543 19°49699 | 1971854 | 39 1994010 2016166 20°38321 20°60477 | 20-82633' 2104788. 21-26944, 2149100: 2171255. 293411! 2215567, 22:37722 22-59878, 22:82034 | 23-04190: 23-26345' 23-48501! 2370657! 23-92812' 24-14968 2437124, 24-56279, 2481135 2503591: 2525746" 25:47902! 2570053 2592213, 2614369 26365251 2653680" 2680836, 27-02992' 2725147 2747303" 2769459; 2791614; 2813770: 28:35926. 2958081; 2810237! 29 0233) 29-21548 29°46704! 29-68860) 29-91015. 30°131711| 30°35327' 30°57482 30°79638' 31-01794 |: 3123949 || 31:46105/ 31:6826 1 | 31-904 16 32:12572 | 32°31728} 32°56883} 32-7903) 33-01195 33°23350 33°45506 an " ache On & È v or wn oN — won m3 O09 R ESE Ka Oral Aa RSEN ana a a > œo th A āo —_ w nw ow no on an Py . . .° arn TR an . are (eR o n "75 52: 2 5 A an 53 1134-11494) 1149-08660 1164-15642 1179-32442 1194-59060 1209-95495 1225-41748 1210-97818 1256-63704 127239411 1288-24933 1304-20272 1320-25431 1336-40406 1352-65198 1368-99808 1385-44236 1401-98480 1418-62543 1435-36423 1452-20120 1469-13635 1486-16967 1503-30117 1520-53084 1537-85869 1556-28471 1572-80840 1590-43128 1608-15182 1625-97054 164385744 1661-90251 1680-01575 1698-22717 1716-53677 1734-94454 175345048 177205460 1790-75639 1809 55736 1828-45601 1847-45232 1866-54782! 188574099 190583233 1924-42184 1943-90954 1963-49540 198317944 2002-96166 292284205 204282062 206289735 2083-07227 210331536 2123-71663 2141-18607 2164-75268 2185-41947 2206°18344 2227-04557 2248-00589 2269-06433 2290-22104 2311:4758 2332-82889) 120°16591 120-95131 121-73671 122-52211 123-30751 124-09290 124:87830! 12566370 126-44910 127-23450 128-01990 128-80529 129-59069) 130°37609! 131161149) 13194689! 13273228; 133-51768: 131-30308' 1235-08348! 135:87383; 13665923! 13744467, 13823007; 129-01547 13980087 140°58627 14137166 142-15706 142-9 1246! 14372786! 144 51326) 14529866 146084105; 14686945! 14765455, 148-4425 1419-22565) 150-01104 15979644 15158184 15236724 15315264 15393304 15472343! 155503983; 15629423 15707963; 1537:9653 153-65012 159°43582 160°22122 161-01 662 161-79202 162°57741 163°36281 16414821 16493351 16571901 166-593441] 16729980, 16807520. 168-856060! 1693-64600 170°43140 171-21679 Side of equal sq. 3367662 33:89817 8411973 3434129 3456285 3178440 35:00596 3522752 35°44907 35:67063 35-89219 36°11374 36-33530 36-55686 36-7841 36-99997 37-22153 3744308] 37.66464 37-38620 33:10775 33-3293] 38:55087 3377242 38-9938 3921554 39-43709} 39 65365 39-3302} 40-16176 40°32332 40:54488 40°766-13} 40-9879 4120955 41431101 41-65245| 41 87422; 42095771 42317323 42-53889) 4276044 4298200 4320356 43 42511 4364667 4386323 4403978 44-31134 4453290 4175445 4497601 4519757 4541912 4564068 4585224 4o vego 46:30535 46 52691 46°74847 4697002 4719158 47-41314 47-63469 4785625 43:07781 4829936

Page 434

Diam. ios) Vs Ge LSS al "<2 ch oo a a i) SR Mh w . . pa ASIE X] nE U e 5 a 23542303, Arca, | Circum. 2375-82244! 2337-47693; 2419-22269; 2141:06657| 2463-00861 2185-01387! 2507-18723. 2520-42337 | 2551-75363) 2574-19156: 2596-72267; 2619-35196! 2642-07942 2664 90505 . 2687-82836, 2710-85084. 2733-97100, 2757-18933, 2780-50584! 2803-92053. 2827-43333. 2851-04442. 2874-75362) 2893-56100) 2922-46656. 2946-47029 2970-57220 2994-77228) ° 3019-07054 3043 40697 3067-96157 3092-55435: 3117-21531 3142-03144! 3166-92174) 3191-90722) 3216-99087] 3242-17270) 3267-45270) 3292-83088) 3318-30721! 3313-88176] 3309-53447) 3395-32534 3421-19439 3417-16162! 3173-22702) 3199-39060 3525-65235 3552-01223 3578-47033 3605-02665 3631-68110 3658-43373 368528453 3712-23350 3739-23065 3766-42597 79366947 3521-01115 3931-35959 3959-19214 3987-12286 172-00219 172-73759 17357299 174-3583) 175°14379 175-92918 17671453 177-49998 178-28538 179-07078 179°85617 180-64 157 181-42697 182 21237 182-9977 183-78317 181-5685 185-3536 186: 13936 1836-92476 1867-71016 183°49555 189-28095 19085175 19085175 191-63715 152-42255 193-20794 1938-99334 194°77874 195-56-414 196°34954 197° 13493 197-92033 198°70573 1989-49113 200°27653 2601-06192 201°84732 202 63272 20341812 20429352 2941-98892 205:77431 206°55571 207-345 11 208°13051 208-91591 209°70130 21048670 211:27210 21205750 21284290 213-625330 21441369 215-19909 215-98449 216-76989 2217-55529 2183-34068 219° 12608 21991148 22069683 22143228 222-26'/68 22305307 22383347] | APPENDIX. Side of |; equal sq. | Diam. Area, Circum. 43-52092)| 71:5 | 4015-15176) 22162337 48-75218) 75; 4043-27893] 225-409227 48-96403;) 72- | 4071:50407| 226°19467 49-18559.| 25: 409982750! 226-98006 49-40715.| -5 | 4128-24909) 227-76546 49-62870'| 75; 4156-76886) 228-55086 49-85026 | 73: | 4185-33681) 229-33626 50-07182'| 25; 4214-10293| 23012166 5029337 | 5 | 4242-91721 2930-90706 50-51493| -75) 4271-82969, 231-69245 50-73649 | 74: 4300-84034) 232-4778 50-95804|| 25; 4329-94916 233-26325 51-17960;} 5 | 4359-15615) 2341-04855 51-40116| -75| 4383-46132! 234-83405 5162271 | 75 4417-86466! 235-61944 51-84427,| 25| 4447-36618) 23640484 52-06583,| 5 | 417696548! 237-196024 52-28738 | "75| 4506-66374; 237-97564 52:50894;| 76 4536-45979, 238°76104) 52-73050;| +25) 4566 35400! 23954613 52:95205j| -5 | 452634640 2410-33183! 5317364 -75| 4626-43696, 24111723 5339517 77- | 4656-62571 241-9263 53-61672|| -25| 4696-91262" 242-6483 53-83828| 5 | 4717-29771] 24347343 5405984| 75| 4747-78093, 24425382 54-28139)| 78- 4778-36242 245 04422 5450295, 25] 4809-04204 245-82962 5472451) 5 | 4839-81983 246-6152 54-94606 | -75| 4570-79579! 2217-40042 - §5:16762'| 79° 4901-66992 248-18581 §5°33918| 25| 4932-74225 2348-97121 5561073 | 5 | 4963-91974 2149-75661 55-83229 | 75| 4995-18140 250-34201 56-05385 | 80- 5026-54824 251-32741 56:27540|| 25; 5058-01325 25211281 56-49696)| 5 | 5089-57644 252-8920 56-71952|| °73} 5121-22781! 2533-68360 56-94007'| 81 5152-93735 254-4694) 5716163 |- 25| 5184-25506 255:25449 57-33319i| 5 | 5216-81095. 2356-03380 57:60475i| 75| 5218-86501, 25682579 57-82630)} 82 5281-01725. 2357-61059 58-04786|| -25| 531326766; 2593-39599 5326942| -5 | 5345-41624 25918139 53-49097|| 75] 5378-06301 259-96679 53-7 1253)! 83° 5410-60794 260°75219 58°93409|/ 25| 5443-95105, 261:53758 59°15564/] 5 | 5475-99234 2262-32298 59°37720|| -75| 5508-83180) 263-10833 59-59876|| 84- 5541-76944) 263-89378 59-82031|| 25| 5574-80525. 264-67918 60-04187|| 5 | 5607-93923 265-46457 6026343); °75| 5641-17139’ 2266-24997 60°48498]| 85- 5674-50173. 267-03537 60°706£4|} 25) 5707-93023, 2267-82077 60°92810|| 5 | 5741-45692, 268-60617 61-14965|| 75| 5775-08178, 26939157 61:37121|| 86- 5808-80481) 270-17696 6159277 || -25| 5842-62602, 270-96236 61:81432/| 5 | 5976-54540, 271-74776 62-03538|| -75] 591056296, 272-53316 62-25744|| 87- | 594467869, 273-31956 62°47899|| 25| 5978-89260, 27410335 62°70055|| 5 | 6013-20468, 274188935 62°92211|| °75) 6047-61494) 275-67475 6082'12337| 27646015 63°143661| 88° 7399994 7422150 74443065 7466461 7488617 1510773 75:3292 7555084 7577240 1599395 76-2155] 7643707 7665362 76°88) le 7710174 TTI TT 5A4R5 77°766-41 TI9STH

Page 435

32 APPENDIX. Side of | Side ot Diam. Area. Circum, equal eq. Area. Circum. equal sq. 611672993) 27724555; 7320952|) 94:25, 6976-74097; 29609510 6151-43476! 278:03094| 78:43103 3 | 701880194; 296:88050 613623772! 2738:81634; 78:65263 "75; 705096109] 297-66590 6221-13885; 27960174) 7857419) 95 7083218412, 29845130 6256-13315! 23033714; 7909575 23) 7125-57992) 29923670 6291-23563) 281:17254 7931730 "5 | 7163-02759) 30002209 6326°43129) 28195794; 7953836 : 7200-579441 300°80749 6361:72512| 28274333) 7976042 ; 723322947] 30159239 6397-11712| 233°52373) 7998193 25, 727597767) 3302-37829 6432:60730| 281:31413| 8020353 : 7313°82404| 303°16369) 6463-19556) 2835-09953; 80°12509 $ 735176359] 303-94908 6503:83219| 285-88493) 8964669] 97- 7339:81131| 30473448 6534°66639| 2286-67032) 8086829 : 7427-95221; 3305-51983 6575:51977) 237:45572| 81-03976 : 7466-19129) 39630523 6611-53082) 2058-24112) 8131132 ; 7504-52853) 3U7-09068 6547-61005) 23902652 8153237) í 754296396| 307:87603 6633-73745) 28931192] 8175443 : 758149755] 3038-66147 672906303) 290-59734) 8197529 : 7620°12933) 309-44637) 6756-43678, 29133271) B2-SOTS4 : 7653°85927) 31023227 673299871) 292°16811| 8241910 : 7697-68739, 31101767 6829-47831! 292-95351 2-5 1066) : 7736°61369| 311:80307 6866°14709| 2593-73391 2-862: : 777563316) 31258346 690291354! 29452431 303837 : 7814-76031! 31337336 6939778317] 29530970) 8330533/100- 7353-93163; 31415926; The following rules are for extending the use of the above table. To find the area, circumference, or side of equal square, of a circie having a diameter of more than 100 inches, feet, §c. Rule.—Divide the given diameter by a number that will give a quotient equal to some one of the diameters in the table; then the circumference or side of equal square, opposite that diameter, multiplied by that divisor, or, the area opposite that diameter, multiplied by the square of the aforesaid divisor, will give the answer. Example.—W hat is the circumference of a circle whose diameter is 228 feet ? 228, divided by 3, gives 76, a diameter of the table, the cir- cumference of which is 238-761, therefore : . 238-761 3 : 716-283 feet. Ans. Another example.—What is the area of a circle having a diameter of 150 inches ? 150, divided by 10, gives 15, one of the diameters in the table, the area of which is 176-71458, therefore : 176-71458 100 = 10 x 10 17,671-45800 inches. Ans. To find the area, circumference, or side of equal square, of a circle having an intermediate diameter to those in the table. Rule.—Multiply the given diameter by a number that will give a product equal to some one of the diameters in the table; then the circumference or side of equal square opposite that diameter, divided by that multiplier, or, the area opposite that diameter diviled by the cquare of the aforesaid mul tiplier, will give the answer.

Page 436

APPENDIX. 32 Example.—W hat is the circumference of a circle whose diameter is 6%, or 6-125 inches? 6-125, multiplied by 2, gives 12-25, one of the diameters of the table, whose circumference is 38:484, therefore : 2)58-484 19:242 inches. Ans. Another example. —What is the area of a circle, the diameter of which is 3°2 feet? 3-2, multiplied by 5, gives 16, and the area of 16 is 201-0619, therefore : 5 X 5 = 25)201-0619(8-0424 + feet. Ans. 200 erent 106 100 E 61 50 119 100 19 Note.—The diameter of a circle, multiplied by 3-14159, will give its circumference ; the square of the diameter, multiplied by :78539, will give its area ; and the diameter, multiplied by -88622, will give the side of a square equal to the area of the circle. TABLE SHOWING THE CAPACITY OF WELLS, CISTERNS, &C. The gallon of the State of New York, by an act passed April 11, 1851, is required to conform to the standard gallon of the United States | pare This standard gallon contains 231 cubile Inches. In CoufOriilts with this standard the following table has been computed. One foot in depth of a cistern of 3 feet diameter will contain 52°872 gallons 34 s “ e e e e e 71:965 se 4 Gs “ e e e e Q 93°995 G 4, “ G . . e 5 . 118963 = ð “ = . ° . ° - 146868 “ 54 Cs “ a (J . e e 177-710 oe 6 ce he 5 ° . b . 21490 5 64 Ke “ . ‘ : A . 248207 E 7 E G . . - : 2608C ™ 8 “ 6s A 5 A : e siogsz ™ 9 “ “ k J 2 p . AIDSo2 | 10 “ “ 3 P ; s -SBAT ™ 12 “ E a T Note —To reduce cubic feet to gallons, multiply by 7°48.

Page 437

TABLE OF POLYGONS. ‘From Gregory’s Mathematics.) | Muhipliers for | Radins of cir- | Factors for areas. | cum. circle. sides, 3| Trigon T 0:4330127| 05773503) 1-732051 4 | Tetragon, or Square 1-0000000] 0°7071068) 1:414214 5 | Pentagon - - 1:7204774) 0:8506508| 1:175570 6 | Hexagon - - 2:5980762| 1:0000000} 1-000000 7 Heptagon - - 3:6339124| 1:1523824| 0-867767 & | Octagon - - 4:8284271| 13065628) 0765367 9 | _Nonagon - - 6:1818242| 1:4619022| 0:684040 10 Decagon - - 7°6942088) 1:6180340| 0-618034 11 Undecagon . - 9°3656399| 1:7747324| 0°563465 12 Dodecagon - - 11:1961524| 1:9318517| 0:517638 To find the area of any regular polygon, whose sides do not exceed twelve. Ru/e.—Multiply the square of a side of the given polygon by the number in the column termed Multipliers for areas, standing op posite the name of the given polygon, and the product will be the an- swer. Erample.—What is the area of a regular heptagon, whose sides measure each 2 feet ? 3°6339124 4=2X 2 14-5356496: Ans. To find the radius of a circle which will circumscribe any regular polygon given, whose sides do not exceed twelve. Ruie.—Multiply a side of the given polygon by the number in the column termed Radius of circumscribing circle, standing opposite the name of the given poly- gon, and the product will give the answer. Example.—What is the radius of a circle which will circumscribe a regular pentagon, whose sides measure each 10 feet ? *8506508 10 8°5065080: Ans. To find the side of any regular polygon that may be inscribed within a given circle. Rule.——Multiply the radius of the given circle by the number in the column termed Factors for sides, standing opposite the name of the given polygon, and the product will be the answer. Ez- ample.—What is the side of a regular octagon thet may be inscrived within a circle, whose radius is 5 feet ? °765367 5 3-826935: Ans.

Page 438

WEIGHT OF MATERIALS bs. ina bbs. ina Woods. cubic foot. Metals. cule foot. Apple, - += + + 49| Wire-drawn brass, - 534 Ash, 5 - - 45| Cast brass, ° - - 506 Beach, - - > - 40| Sheet-copper, - - 549 Birch, - : : 45| Pure cast gold, - - 1210 Box, - : : - 60| Bar-iron, - 475 to 487 Cedar, 2 - - 28] Castiron, - - 450 to 475 Virginian red cedar, - 40| Milled lead, - - . 713 Cherry, A : : 38| Cast lead, : : 709 Sweet chestnut, - - 36| Pewter, > - - 453 Horse-chestnut, - - 384| Pure platina, - - 1345 Cork, - 2 > - 15) Pure cast silver, - - 654 , Cypress, - ©- =- 28] Steel, - -= 486 to 490 Ebony, - - = - 83| Tm, =- J: - - 456 Elder, ^ - - - 43} Zinc, z - : 439 Elm, $ > 3 - 84] Stone, Earths, &c. Fir, (white spruce,) : 29| Brick, Phila. stretchers, 105 Hickory, - - - 52| North river common hard Lance-wood, - - 59 brick, - - - 107 Larch, - >o o = on Do. salmon brick, 100 Larch, (whitewood,) - 22| Brickwork, about - 95 Lignum-vitæ, - -~ - 83| Cast Roman cement, - 100 Logwood, - - - . 57| Do. and sand inequal parts, 113 St. Domingo mahogany, - 45| Chalk, - 144 to 166 Honduras, or bay mahogany, 35| Clay, - - - - 116 Maple, - - - 47| Potter’s clay, - 112 to 130 White oak, - - 43to53| Common earth, 95 to 124 Canadian oak, - - 54] Flint, - - - - 163 Red oak, z - - 47] Plate-glass, - 172 Live oak, - - - 76| Crown-glass,- - =- 157 White pine, - - 28to 30} Granite, - - 158 to 187 Yellow pine, 34 to 44| Quincy granite, - - 166 Pitch pine, - - 461058; Gravel, - : - 109 Poplar, : : z 25| Grindstone, - 4 - 184 Sycamore, 2 - - 36 Gypsum, (Plaster-stone,) 142 Walnut, - - 40 Unslaked lime, - 5 52

Page 439

36 APPENDIX. lbs, ma | lbs. ina ' cubic foot. cubic foot, Limestone, - - 118 to 198| Common blue stone, - 160 Marble, - - 161 tol77| Silver-gray flagging, - 185 New mortar, - - - 107| Stonework, about, - 120 Dry mortar, - - 90| Common plain tiles, - 115 Mortar with hair, (Plaster- Sundries. ing,) - > - - 105| Atmospheric air, - 0075 Do. dry, - - 86| Yellow beeswax, - - 60 Do. do. including lath Birch-charcoal, - - 34 and nails, from 7 to 11 Oak-charcoal, 5 - 21 Ibs. per superficial foot. Pine-charcoal, - - 17 Crystallized quartz, - 165; Solid gunpowder, - - 109 Pure quartz-sand, - 171| Shaken gunpowder, - 5S Clean and coarse sand, 100| Honey, - - e- 90 Welsh slate, - = - 180 Milk, z - Š 64 Paving stone, - - 151 Pitch, - - - = l Pumice stone, - - 656 Sea-water, - - 64 Nyack brown stone, - 148 Rain-water, - - - 62:5 Connecticut brown stone, 170 Snow, - A - 8 ‘Tarrytown blue stone, - 171 Wood-ashes, - : 58 THE END.

Page 440

New and Important Works ON THE RESISTANCE OF MATERIALS AND THE CONSTRUCTION OF BRIDGES AND ROOFS. Designed as Text-Books and for Practical Use. BY DE VOLSON WOOD, AM, €E, PROFESSOR OF MATHEMATICS AND MECHANICS IN STEVEN'S INSTITUTE OF TECH- NOLOGY, AND FORMERLY PROFESSOR OF CIVIL ENGINEERING IN THE UNIVERSITY OF MICHIGAN; MEMBER OF THE AMERICAN SOCIETY OF CIVIL ENGINEERS, AND CORRESPONDING MEMBER OF THE AMERICAN SOCIETY OF ARCHITECTS. I. A TREATISE ON THE RESISTANCE OF MATERIALS, with an Appendix on the Preservation of Timber. Illustrated by 112 Wood Engravings. One vol. 8vo, cloth. $2.50. This work gives the results of the experiments which have been made from time to timb upon various materials—mostly Wood, Stone, Wrought-Iron, Cast-Iron, and Steel—and the theo- ries which follow from these experiments. It is intended by the author to show an intimate relation between experiments and the theories which he maintains. It treats of the elastic resistances and the strength of materials, when subjected to strains of tension, compression, transverse and longitudinal shearing, transverse strains, and torsive strains. It also includes the results of experi- ments upon boiler plates, the collapse of boiler flues, and the buckling of tubes. It also treats of beams of uniform resistance, e effect of “shocks” upon the strength, and remarks upon the crystallization of iron and steel. There is an Appendix which gives a summary of the several processes which have been devised for the preservation of wood.

Page 441

II. A TREATISE UPON THE THEORY OF THE CONSTRUCTION OF BRIDGES AND ROOFS. Illustrated by 124 Wood En- gravings. One vol. 8vo, cloth, $3.00. This work is intended to be a logical development of the theory of the strains in trussed girders. It begins with the sim- plest structures, and passes, step by step, to the more complex, including in its consideration an analysis of all the important trussed bridges which are now in common use. The chapter on Roofs is intended to be sufficiently compre- hensive to enable a person who is acquainted with ordinary mathematics to proportion a structure correctly. The last chapter reverses the process of the preceding chapters, and deduces all the consequences from the fundamental equa- tions of statics. The work is supposed to be especially adapted to the wants of those who desire to establish themselves on correct principles in regard to the elements of Bridge Construction. The above-named volumes, and any other of our publications, will be sent prepaid, by mail, on the receipt of the prices affixed. Our new General Classified and Descriptive Scientific Catalogue including our own publications, will be sent gratis, and prepaid by mail. JOHN WILEY & SON, 15 Astor Place, New York, FEBRUARY, 1873.

Page 442

New York, March, 1873. JOHN WILEY & SON’S LIST OF PUBLICATIONS, 1S ASTOR PLAGE Under the Mercantile Library and Trade Svterooms. DOWNING. (33 66 6s t. DOWNING & LINDLEY. DOWNING, - AGRICULTURE. FRUITS AND FRUIT-TREES OF AMERICA; or the Culture, Propagation, and Management in the Garden and Orchard, of Fruit-trees generally, with descriptions of all the finest varieties of Fruit, Native and Foreign, cultivated in this ‘country. By A.J. Downing. Second revision and correction, with large additions. By Chas. Downing. 1 vol. 8vo, over 1100 pages, with several hundred outline engravings, Price, with Supplement for 1872..............ceeceeee .....$5 00 “ As a work of reference it has no equal in this country, and deserves a place in the Library of every Pomolozist in America."—Marshall P. Wilder. ENCYCLOPEDIA CF FRUITS; or, Fruits and Fruit- Trees of America. Part 1.—APPLES. With an Appendix containing many new varieties, and brought down to 1872, By Chas. Downing. With numerous outline engravings. 8vo, fall clothe ecese «sce cs COE is Moe cls: a $2 50 ENCYCLOPEDIA OF FRUITS; or, Fruits and Fruit- Trees of America. Part 2.—CHERRIES, GRAPES, PEACHES, Pears, &C. With an Appendix containing many new varie- ties, and brought down to 1872. By Chas. Downing. With numerous outline engravings. 8vo, full cloth......... $2 50 FRUITS AND FRUIT-TRESS OF AMERICA. By A. J. Downing. First revised edition. By Chas. Downing. 12mo, clothe e. «ere Be COUR» SBM 00000000000% $2 00 SELECTED FRUITS. From Downing’s Fruits and Fruit- Trees of America. With some new varieties, including their Culture, Propagation, and Management in the Garden and Orchard, with a Guide to the selection of Fruits, with refer- ence to the Time of Ripening. By Chas, Downing. Ilus- trated with upwards of four hundred outlines of Apples, Cherries, Grapes, Plums, Pears, Ke, 1 vol., 12mo....§2 50 LOUDON'S GARDENING FOR LADIES, AND COM- PANION TO THE FLOWER-GARDEN. Second American from third London edition. Edited by A. J. Downing. 1 vol.,12mo ......... bia A So c $2 00 THE THEORY OF HORTICULTURE. By J. Lindley. With additions by A. J. Downing. 12mo, cloth....... $2 00 COTTAGS RESIDENCES, A Series of Designs for Rural Cottages and Cottage Villas, with Garden Grounds. By A. J. Downing. Containing a revised List of Trees, Shrubs, and Plants, and the most recent and best selected Fruit, with some account of the newer style of Gardens. By Henry Winthrop Sargent and Charles Downing. With many new designs in Rural Architecture. By George E Harney, Architect. 1 vol. 4t0.....sesesesssrsecsesees eese. ge OG

Page 443

92 JOHN WILEY & SON’S LIST OF PUBLICATIONS. DOWNING & WIGHTWICK. KEMP. LIEBIG. 6é PARSONS. DOWNING. DOWNING & WICHTW CK. HATFIELD. HOLLY. &6 RUSKIN HINTS TO PERSONS ABOUT BUILDING IN THE COUNTRY. By A. J. Downing. And HINTS To YOUNG ARCHITECTS, calculated to facilitate their practical operations. By George Wightwick, Architect. Wood engravings. 8vo, cloths... .6...6 66. cc 0+ vc. oe $2 00 LANDSCAPE GARDENING;; or, How to Lay Out a Gar- den. Intended as a gencral guide in choosing, forming, or improving an estate (from a quarter of an acre to a hun- dred acres in extent), with reference to both design and exe- cution. With numerous fine wood engravings, By Edward Kemp- I yol 12mo cloth sen. cereo os. eres $2 59 CHEMISTRY IN ITS APPLICATION TO AGRICUL- TURE, &c. By Justus Von Liebig. 12mo, cloth....$1 00 LETTERS ON MODERN AGRICULTURE. By Baron Von Liebig. Edited by John Blyth, M.D. With addenda by a practical Agriculturist, embracing valuable suggestions, adapted to the wants of American Farmers. 1 vol. 12me, Cloth oaeee enaa systatoveisioie EEEa sie ove seus e aae $1 00 PRINCIPLES OF AGRICULTURAL CHEMISTRY, with special reference to the late researches made in England. By Justus Von Liebig. 1 vol. 12mo...............0- 75 cents. HISTORY AND CULTURE OF THE ROSE. By S. B. Parsons: pam aVOle ol om Orem tects eee tis te reine coe $1 25 ARCHITECTURE. COTTAGE RESIDENCES; or, a Series of Designs for Rural Cottages and Cottage Villas and their Gardens and Grounds, adapted to North America. By A. J. Downing. Containing a revised List of Trees, Shrubs, Plants, and the most recent and best selected Fruits. With some account of the newer style of Gardens, by Henry Wentworth Sargent and Charles Downing. With many new designs in Rural Architecture by George B. Harney, Architect.. .. seste esee essees ees. pO 00 HINTS TO PERSONS ABOUT BUILDING IN THE COUNTRY. By A. J. Downing. And HINTS TO YOUNG ARCHITECTS, calculated to facilitate their practical operations. By George Wightwick, Architect. With many wood-cuts. 8vo, cloth.......sesessesecee $2 00 THE AMERICAN HOUSE CARPENTER. A Treatise upon Architecture, Cornices, and Mouldings, Framing, Doors, Windows, and Stairs; together with the most important principles of Practical Geometry. New, thoroughly revised, and improved edition, with about 150 additional pages, and numerous additional le By R. G. Hatfield. 1 vol. OVO eaer ee aeee a EE EE ae e T $3 50 NOTICES OF THE WORK. i “The clearest and most thoronghly practical work on the subject.” tt This work is a most excellent one, very comprehensive, and Incidly arranged.” “ This work commends itself by its practical excellence.” “ It is a valuable addition to the brary of the architect, and almost indispensable to every scientific master-mechanic."—A. R. Journal. CARPENTERS’ AND JOINERS’ HAND-BOOK, contain- ing a Treatise on Framing, Roofs, etc., and useful Rules ae Tables. By H. W. Holly. 1 vol. "18mo, cloth. «nee $0 7 THE ART OF SAW-FILING SCIENTIFICALLY TREATED AND EXPLAINED. With Directions f putting in order all kinds of Saws. By H. W. Holly. 18m: clorn a a E Tone nOo E e ee $07 SEVEN LAMPS OF ARCHITECTURE. 1 vol 12mo, cloth pPlites aie een cise A ee E a de $1 75

Page 444

JOHN WILEY & SON’S LIST OF PUBLICATIONS, 93 RUSKIN. LECTURES ON ARCHITECTURE AND PAINTING. 1 vol. 12mo, cloth, plates........, er. cose. G1 50 Ole LECTURE BEFORE SOCIETY OF ARCHITECTS. 0 15 WOOD. A TREATISH ON THE RESISTANCE OF MA- TERIALS, and an Appendix on the Preservation of Timber. By De Volson Wood, Prof. of Engineering, University of Michigan. 1 vol. 8vo, cloth................ slap see $2 50 This work is used as a Text-Book in Iowa University, Iowa icaltural Coll eee oa Me at Scientific a ted Haven, Coone: [nsti ew York, Polytechnic Coll Brooki niversi i and other institutions. 3 i Ua — “ A TREATISE ON BRIDGES. Designed as a Text-book and for Practical Use. By De Volson Wood. 1 vol. 8vo, nu- merous illustrations, cloth (shortiy)..... sae Ps $3 00 ASSAYING-ASTRONOMY., BODEMANN. A TREATISE ON THE ASSAYING OF LEAD, SILVER, COPPER, GOLD, AND MERCURY. By Bodemann and Kerl. Translated by W. A. Goodyear. i vol. 12mo, CLO cris E teiaaus.cicine bet e ss E ee $2 50 MITCHELL. A MANUAL OF PRACTICAL ASSAYING. By: John Mitchell. Third edition, edited by William Crookes. 1 vol. F thick Svo, clothe a aeae eee o Ee ee ee. ee $10 00 NORTON.. A TREATISE ON ASTRONOMY, SPHERICAL AND PHYSICAL, with Astronomical Problems and Solar, Lunar, and other Astronomical Tables for the use of Colleges and Scientific Schools. By William A, Norton. Fourth edition, revised, remodelled, and enlarged. Numerous plates. 8vo, OO ee ee COMMODROER DADS os6 56 $3 50 BIBLES, &c. BACSTER. THE COMMENTARY WHOLLY BIBLICAL. Contents: —The Commentary: an Exposition of the Old and New Tes- taments in the very words of Scripture. 2264 pp. II. An ‘ outline of the Geography and History of the Nations men- tioned in Scripture. III. Tables of Measures, Weights, and Coins, IV. An Itinerary of the Children of Israel from Egypt to the Promised Land. V. A Chronological compara- tive Table of the Kings and Prophets of Israel and Judah. VL A Chart of the World’s History from Adam to the Third Century, A.D. VII. A complete Series of Illustrative Maps. TX. A Chronological Arrangement of the Old and New Tes- taments. X. An Index to Doctrines and Subjects, with numerous Selected Passages, quoted in full. XL An Index to the Names of Persons mentioned in Scripture. XII. An Index to the Names of Places found in Scripture. XIIL The Names, Titles. and Characters of Jesus Christ our Lord, as revealed in the Scriptures, methodically arranged. eavoluntes:4to, Clothwaverereivecucie « ctelelsiays «aire + eek $19 50 2 volumes 4to, half morocco, gilt edges............. 26 00 2 volumes 4to, morocco, gilt edges.....-... saeco cce! 89 3 volumes Ato, Cloth. -usse . asa naan eneee Rec 20 00 3 volumes 4to, half moroce, gilt edges...... Too 33 00 3 volumes 4t», morocco, gilt edges... .......seee-ess 40 00 BLANK-PACED THE HOLY SCRIPTURES OF THE OLD AND NEW BIBLE. TESTAMENTS; with copious references to parallel and ilustrative passages, and the alternate pages ruled for Ms. notes. age . œ e . - . ry ted by tre This edition of the Scriptures contains the Authorized Version, illustra: references of ** Aie Polyglot Bible," and cnriched with accurate maps, nseful tables, and un Index of Subjects. 1 vol. 8vo, half moroccO........ ..cccerreecec reese on 1 vol. 8vo, morocco.......... mono uagodeppoounDade:

Page 445

94 JOHN WILEY & SON’S LIST OF PUBLICATIONS. THE TREASURY Containing the authorized English version of the Holy Scriptures, BIBLE, interleaved with a Treasury of more than 500,000 Parallel Passages from Canne, Brown, Blayney, Scott, and others, With numerous illustrative notes. Tivol half boundi ves cee ecco ies cu neako wale --.9 50 i vol SMOIOCEO. aoe eeen oaas ET esas 10100 COMMON PRAYER, 48mo Size. A (Done in London expressly for us.) COMMON No. 1. Gilt and red edges, imitation morocco........... $0 623 PRAYER, No. 2. Gilt and red edges, rims..............0cceceeee 874 No. 3. Gilt and red edges, best morocco and calf....... 1 2 No. 4. Gilt and red edges, best morocco and calf, rims.. 1 50 BOOK-KEEPING, : JONES. BOOKKEEPING AND ACCOUNTANTSHIP. Elementary and Practical, In two parts, with a Key for Teachers. By Thomas Jones, Accountant and Teacher. 1 volume 8vo, CLOUDY, 5 coc sre ens ean es a ieee ene eet og nine oie $2 50 es BOOKKEEPING AND ACCOUNTANTSHIP. School Edi- tion. By Thomas Jones, 1 vol. 8vo, half roan....... $1 50 s BOOKKEEPING AND ACCOUNTANTSHIP. Set of Blanks. In 6 parts. By Thomas Jones.............. $1 50 ae BOOKKEEPING AND ACCOUNTANTSHIP. Double Entry; Results obtained from Single Entry; Equation of Payments, etc. By Thomas Jones. 1 vol. thin 8vo...$0 75 CHEMISTRY. CRAFTS. A SHORT COURSE IN QUALITATIVE ANALYSIS; with the new notation. By Prof. J. M. Crafts. Second edition: 1 vol 12mo Clone... 2 sce see wea sss ea ee $1 50 JOHNSON’S A MANUAL OF QUALITATIVE CHEMICAL ANALY- FRESENIUS. SIS. By ©. R. Fresenius. Edited by S. W. Johnson, Pro- fessor in Sheffield Scientific School, Yale College. With Chemical Notation and Nomenclature, old and new. 1 vol. OVO pCO ryote ane ce eerie ea ae ones uae en ee $4 50 a A SYSTEM OF INSTRUCTION IN QUANTITATIVE CHEMICAL ANALYSIS. By C. R. Fresenius. From latest editions, edited, with additions, by Prof. S. W. John- son. With Chemical Notation and Nomenclature, ree o KIRKWOOD COLLECTION OF REPORTS (CONDENSED) AND OPINIONS OF CHEMISTS IN REGARD TO THE USE OF LEAD PIPE FOR SERVICE PIPE, in the Distribution of Water for the Supply of Cities. By Jas. P. Kirkwood. 8vo, cloth......... TO cies bass eats $1 50 MILLER. ELEMENTS OF CHEMISTRY, THEORETICAL AND PRACTICAL. By Wm. Allen Miller. 3 vols. 8vo..$18 00 ss Part 1—CHEMICAL PHYSICS. 1 vol. 8vo.......... $4 00 i Part I.—INORGANIC CHEMISTRY. 1 vol. 8vo..... 6 00 st Part I1I.—ORGANIC CHEMISTRY. 1 vol. 8vo....... 19) 00 “Dr. Miller's Chemistry is a work of which the author has every reason to feel proud. It is now by far the largest and most accnrately written Treatise on Chemistry in the Engtish language,” etc.—Dublin Med. Journal. si MAGNETISM AND ELECTRICITY. By Wm. Allen Miller. Tyo SVOM o otee 6 settee cine es Barer cvace E A $2 50

Page 446

JOHN WILEY & SON’s LIST OF PUBLICATIONS, 95 MUSPRATT. CHEMISTRY — THEORETICAL PRACTICAL, AND ANALYTICAL —as applied and relating to the Arts ama Manufactures, By Dr. Sheridan Muspratt. 2 vols. 8vo, sloth, 18100; halfmat... M a eG $24 00 NOAD, A MANUAL OF QUALITATIVE AND QUANTITATIVE CHEMIOAL ANALYSIS. For the use of Students. By H. M. Noad, author of “Manual of Electricity.” 1 voL 12mo. (London.) Complete..... Do Geeta E e -$6 00 6 QUANTITATIVE ANALYSIS. 1 vol. cloth.......... 4 00 PERKINS, AN ELEMENTARY MANUAL OF QUALITATIVE CHEMICAL ANALYSIS. By Maurice Perkins, 12mo, eloth..... CCT ES csc). oe eee $1 00 DRAWING AND PAINTING. BOUVIER HANDBOOK ON OIL PAINTING. Handbook of Young AND OTHERS. Artists and Amateurs in Oil Painting; being chiefly a con- densed compilation from the celebrated Manual of Bouvier, t with additional matter selected from the labors of Merriwell, De Montalbert, and other distinguished Continental writers on the art. In 7 parts. Adapted for a Text-Book in Academies of both sexes, as well as for self-instruction. Appended, a new Explanatory and Critical Vocabulary. By an American Artist. 12mo, cloth.................... $2 00 COE. NEW SERIES OF DRAWING CARDS, by Benj. H. Coe, viz.: OG DRAWING FOR LITTLE FOLES; or, First Lessons for the Nursery. 30 cards. Neat case....... ecto = ee $0 30 ae FIRST STUDIES IN DRAWING. Containing Elementary Exercises, Drawings from Objects, Animals, and Rustic Figures. Complete in three numbers of 18 cards each, in neat Cane. SESCH aise acct e se eM ku a e... $0 30 < COTTAGES. An Introduction to Landscape Drawing. Con» taining 72 Studies, Complete in four numbers of 18 cards each, in neat case. Each..... clodo.ad S CRO E o. $0 30 Æ BASY LESSONS IN LANDSCAPE. Complete in four numbers of 40 Studies each. In neat 8vo case. Eaċh, $0 30 se HEADS, ANIMALS, AND FIGURES. Adapted to Pencil Drawing. Complete in three numbers of 10 Studies each. In neat Svo case. Bache... .eereaerecareesenee. ue $0 30 as COPY BOOK, WITH INSTRUCTIONS............ -$0 87} USKIN. THE ELEMENTS OF DRAWING. In Three Letters to des Beginners. By John Ruskin, 1 vol. 12mo...... o---- Hl 00 e THE ELEMENTS OF PERSPECTIVE. Arranged for the use of Schools. By John Ruskin................0006 $i 00 . A MANUAL OF TOPOGRAPHICAL DRAWING. By s Prof. R. S. Smith. Second edition. 1 vol. 8vo,, cloth, plates. Wes... «shyt... p Oe E cOLOGOND « $2 00 go MANUAL OF LINEAR PERSPECTIVE. Form, Shade, Shadow, and Reflection. By Prof. R. S. Smith. 1 vol, 8vo, plates, cloth...........+. E ooo a EEEN $2 00 UCTIVE GEOMETRY AND INDUSTRIAL —e ow in G. By S. Edward Warren, Professor in the Mas- sachusetts Institute of Technology, Boston:— I. ELEMENTARY WORKS. i 1. ELEMENTARY FREE-HAND GEOMETRICAL DRAWING. "A series of progressive exercises on regular lines and forms, including systematic instruction in lettering; a wacan of the eye and hand for all who are learning to draw. 12mo, Cloth, many cuts. ....eseseseereesesososreseeeerece í Di Vols. 1 and 3, bound in 1 vol...... ees etnies ce eee

Page 447

WARREN JOHN WILEY & SON’S LIST OF PUBLICATIONS, ELEMENTARY WORKS.—Continued. ; 2. PLANE PROBLEMS IN ELEMENTARY GEOMETRY. With numerous wood-cuts. 12mo, cloth................6- $1 25 3. DRAFTING INSTRUMENTS AND OPERATIONS. Con- taining full information about all the instruments and materials used by the draftsmen, with full directions for their use. With plates and wood-cuts. One vol. 12mo, cloth, $1 25 4, ELEMENTARY PROJECTION DRAWING. Revised and en- larged edition. Im five divisions. This and the last volume are favorite text-books, especially valuable to all Mechanical Artisans, and are particularly recommended for the use of all higher public and private schools. New revised and enlarged edition, with numerous wood-cuts and plates. (1872.) 12mo, ClO o E svni esis secs cyetc intense eecra E ota, $1 50 5. ELEMENTARY LINEAR PERSPECTIVE OF FORMS AND SHADOWS. Part 1.—Primitive Methods, with an Introduc- tion. Part U.—Derivative Methods, with Notes on Aerial Perspective. and many Practical Examples. Numerous wood- cuts. 1 vol. 12mo, cloth...... E E oases OO I. HIGHER WORKS. ` These are designed principally for Schools of Engineering and Architecture, and for the members generally of those professions; and the first three are also designed for use in those colleges which provide courses of study adapted to the preliminary general training of candidates for the scientific professions, as well as for those technical schools which undertake that training themselves, 1. GENERAL PROBLEMS OF ORTHOGRAPHIC PROJEC- TIONS. The foundation course for the subsequent theo- retical and practical works, A new cdition of this work will soon appear. 2. GENERAL PROBLEMS OF SHADES AND SHADOWS. A wider range of problems than can elsewhere be found in English, and the principles of shading. 1 vol. 8vo, with numerous plates. Cloth....... E a ester tte ecenessieusrares $3 50 3. HIGHER LINEAR PERSPECTIVE. Distinguished by its con- cise summary of various methods of perspective construction ; a full set of standard problems, and a careful discussion of special higher ones. With numerous large plates 8vo, LOG secon A E cree ears ene ence AE ee ay otc eee $4 00 4. ELEMENTS OF MACHINE CONSTRUCTION AND DRAW- ING; or, Machine Drawings. With some elements of descrip- tive and rational cinematics. A Text-Book for Schools of Civil and Mechanical Engineering, and for the use of Me- chanical Establishments, Artisans, and Inventors, Containing the principles of gearings, screw propellers, valve motions, and governors, and many standard and novel examples, mostly from present American practice. By 5. Edward Warren. 2 vols. 8vo. 1 vol. text and cuts, and 1 vol. large plates......$7 50 A FEW FROM MANY TESTIMONIALS. “Tt seems to me that your Works only need a thorough examination to be intro- duced and permanently used in all the Scientific and Engineering Schools,” —Prof. J. G. FOX, Collegiate and Engineering Institute, New York City. “I have used several of your Elementary Works, and believe them to be better adapted to the purposes of instraction than any others with which I am acquainted."—H. F. WALLING, Prof. of Civil and Topographical Engi- neering, Lafayette College, Easton, Pa. “Yonr Works appear to me to fill a very important gap in the literature of the subjects treated. Any effort to draw Artisans, etc., away from the ‘rule of thumb,’ and give them an insight into principles, is in the right direction, and ineets my heartiest approval. This is the distinguishing feature of your Elementary Works."—Prof. H. L. EUSTIS, Lawrence Scientific School, Cambridge, Mass. The author has happily divided the subjects into two great portions: tle former embracing those processes and problems proper to be tanght to all students in Institutions of Elementary Instruction; the latter, those suited to advanced students preparing for technical purposes. The Elementary Books ought to be nsed in all High Schools and Academies; the Higher ones in Schools of Technology."—WM. W. FOLWELL, President of University of Minnesota,

Page 448

JOHN WILEY & SON’S LIST OF PUBLICATIONS, 7 DYEING, &c. MACFARLANE. A PRACTICAL TREATISE ON DYEING AND CALICO. PRINTING. Including the latest Inventions and Improve- ments. With an Appendix, comprising definitions of chemical terms, with tables of Weights, Measures, &e. By an expe- rienced Dyer. With a supplement, containing the most recent discoveries in color chemistry. By Robert Macfarlane. TENGIROVO..... ahr. sie. aie art E 0 00 COO BIBS ae.. $0 00 REIMANN. A TREATISE ON THE MANUFACTURE OF ANILINE AND ANILINE COLORS. By M. Reimann. To which is added the Report on the Coloring Matters derived from Coal Tar, as shown at the French Exhibition, 1867. By Dr. Hofmann. Edited by Wm. Crookes. 1 vol. 8vo, cloth, $2 50 “Dr, Reimann’s portion of the Treatise, written in concise language. is profonndly practical, giving the minutest details of the processes for obtaining nll the more important colors, with woodcuts of apparatus. Taken in conjunction with Hofmann’s Report, we have now a complete history of Coal Tar Dyes, both theoretical and practical.”—-Chenust and Druggist. ENGINEERING. AUSTIN. A PRACTICAL TREATISE ON THE PREPARATION, COMBINATION, AND APPLICATION OF CALCA- REOUS AND HYDRAULIC LIMES AND CEMENTS. To which is added many useful recipes for various scientific, mercantile, and domestic purposes. By James G. Austin. 1 yol. temo. cee. «© Ba E a See E A $2 00 COLBURN. LOCOMOTIVE ENGINEERING AND THE MECHAN- ISM OF RAILWAYS. A Treatise on the Principles and Construction of the Locomotive Engine, Railway Carriages, and Railway Plant, with examples. Ilustrated by Sixty-four large engravings and two hundred and forty woodcuts. By Zerah Colburn. Complete, 20 parts, or 2 vols. cloth. .$20 00 Or, half morocco, gilt top........- sce. cceeneee 25 00 KNIGHT. THE MBECHANICIAN AND CONSTRUCTOR FOR EN- GINEERS. Comprising Forging, Planing, Lining, Slotting, Shaping, Turning, Screw-cutting, &c. Ilustrated with ninety-six plates. By Cameron Knight. 1 vol. 4to, half JNOTOCCO)... oo e ciao anae aen E ee A + sce E E A $15 00 MAHAN, AN ELEMENTARY COURSE OF CIVIL ENGINEER- ING, for the use of the Cadets of the U. S. Military Academy. By H. P. Mahan. 1 vol. 8vo, with numerous woodcuts. New edition, with large Addenda, &c. Full cloth..... $4 00 oe DESCRIPTIVE GEOME?RY, as applied to the Drawing of Fortifications and Stone-Cu‘ting. For the use of the Cadets of the U. S. Military Academy. By Prof. D. H Mahan. IRvolngyvon Plates cer 6 anesbos 5000.00.00 0 AEE oe oe: $1 50 z INDUSTRIAL DRAWING. Comprising the Description and Uses of Drawing Instruments, the Construction of Plane Figures, the Projections and Sections of Ge metrical Solids, Architectural Elements, Mechanism, and Topographical Drawing. With remarks on the method of Teaching the subject. For the use of Academies and Comm n Schools. By Prof. D. H. Mahan. 1 vol. 8vo, Twenty steel pate. Fall cloth®.....-02 25. i cere on oe r ral 00 & EATISE ON FIELD FORTiF IC - Contain- ; z as instructions on the Methods of Laying Out, Constructing, Defending, and Attacking Entrenchments. With the General Outlines, also, of the Arrangement, the Attack, and Defence of Permanent Fortifications. By Prof. D. H. Mahan. N ew edition, revised and enlarged. 1 vol. 8vo, full Se oD plates ewoesereeteoeeo eee eaoaeaeeataeoae eens e@eeeeeeter ee ERMANENT FORTIFICATIONS. By s nA D. a 1 vol. 8vo, with numerous large naye oe pe a eS $6

Page 449

JOHN WILEY & SON’S LIST OF PUBLICATIONS. MAHAN. MAHAN & MOSELY. MAHAN & BRESSE. WOOD. BACSTER. GREEN. 66 LETTERIS. ADVANCED GUARD, OUT-POST, and Detachment Service of Troops, with the Essential Principles of Strategy and Grand Tactics. For the use of Officers of the Militia and Volunteers. By Prof. D. H. Mahan. New edition, with large additions and 12 plates. 1 vol. 18mo, cloth...... $1 50 MECHANICAL PRINCIPLES OF ENGINEERING AND ARCHITECTURE. By Henry Mosely, M.A., F.R.S. From last London edition, with considerable additions. by Prof. D. H. Mahan, LL.D., of the U. S. Military Academy. 1 vol. 8vo, 700 pages. With numerous cuts. Cloth...$5 06 HYDRAULIC MOTORS. Translated from the French Coura de Mecanique, appliquée par M. Bresse. By Lieut. F. A. Mahan, and revised by Prof. D. H. Mahan. 1 vol. 8vo, platesn Erte n a A a ee a a a a E ee $2 50 A TREATISE ON THE RESISTANCE OF MATE- RIALS, and an Appendix on the Preservation of Timber. By De Volson Wood, Professor of Engineering, University of Michigan. 1 vol. 8vo, cloth....... T eet nets T $2 50 A TREATISE ON BRIDGES. Designed as a Text-book and for Practical Use. By De Volson Wood. 1 vol. 8vo, nume- rous illustrations, cloth (shortly)..............--ceee. $2 00 GREEK. GREEK TESTAMENT, ETC. The Critical Greek and English New Testament in Parallel Columns, consisting of the Greek Text of Scholz, readings of Griesbach, etc., etc. I yol. 18mo, half morocco, 6.0 c..0s6 5 cece senso acies $3 00 do. Full morocco, gilt edges........ T 4 50 - With Lexicon, by T S. Green. MHalf-bound........ 4 50 do. Full morocco, gilt edges.............-..-. 6 00 do. With Concordance and Lexicon. Half mor., 6 00 on Limp Morocco.. aeee ee a cies 7 50 THE ANALYTICAL GREEK LEXICON TO THE NEW TESTAMENT. In which, by an alphabetical arrangement. is found every word in the Greek text in every form in which it appears—that is to say, every occurrent person, number, tense or mood of verbs, every case and number of nouns, pro- nouns, &c.,is placed in its alphabetical order, fully explained by a careful grammatical analysis and referred to its root, so that no uncertainty as to the grammatical structure of any word can perplex the beginner, but, assured of the precise grammatical force of any word he may desire to interpret, he is able immediately to apply his knowledge of the English meaning of the root with accuracy and satisfaction. 1 vol. small ALO Hak BOUnd... sse casecc cv sans os ssassn" $6 50 GREEK-ENGLISH LEXICON TO TESTAMENT. By T. S. Greon. Half morocco: .....c¢ ccc ces esse coos se $1 50 HEBREW. . A GRAMMAR OF THE HEBREW LANGUAGE. With copious Appendixes. By W. H. Green, D.D., Professor in Princeton Theological Seminary, 1 vol. 8vo, cloth... .$3 50 AN ELEMENTARY HEBREW GRAMMAR. With Tables, Reading Exercises, and Vocabulary. By Prof. W. H. Green, D.D. 1 vol. 12m0, cloin.. seses ecirar enere. $1 50 HEBREW CHRESTOMATHY; or, Lessons in Reading and Writing Hebrew. By Prof. W. H. Green, D.D. 1 vol. Svo, A a n E oc, oo ono o oes E E D $2 00 A NEW AND BEAUTIFUL EDITION OF THE HE- BREW BIBLE. Revised and carefully examined by Myer Levi Letteris. 1 vol. 8vo, with key, marble edges... .. $2 50 “This edition has a large and much more legible type than the known one volnme editions, and the print is excellent, while the name of LETTERIS İs a sufficient guarantee for correetness.” --Rev. Dr. J. M. WISE, Editor of the ISRAELITE,

Page 450

BACSTER'S CESENIUS, BACSTER’S BODEMANN. CROOKES. FAIRBAIRN. FRENCH. KIRKWOOD LESLEY. FITZCERALD. HOLLY. pe KNICHT. JOHN WILEY & SON’S LIST OF PUBLICATIONS. 99 * BAGSTHR’8 COMPLETE EDITION OF GESENIUS’ HEBREW AND OHALDEH LEXICON. In large, clear, and perfect type. Translated and edited with addi- tions and corrections, by S. P. Tregelles, LL.D. In this editlon great care has been taken to guard the student from Neologian tendencies by suitable remarks whenever needed. “The careful revisal to which the Lexicon has been subjected by a faithful and Orthodox translator exceedingly enhances the practical value of this edition.” — Edinburgh Ecclesiastical Journal. Small 4to, half bound......... A E O A $7 50 NEW POCKET HEBREW AND ENGLISH LEXICON. The arrangement of this Manual Lexicon combines two things—the etymological order of roots and the alphabetical order of words. This arrangement tends to lead the learner onward; for, as he becomes more at home with roots and derivatives, he learns to turn at once to the root, without first searching for the particular word in its alphabetic order. 1 vol, 18mo, cloth......... R RESIS isi sie lagersvs roots $2 00 * This is the most beautiful, and at the same time the most correct and perfect Manual Hebrew Lexicon we have ever used."—Kelectic Review, IRON, METALLURGY, &c. A TREATISE ON THE ASSAYING OF LEAD, SILVER, COPPER, GOLD, AND MERCURY. By Bodemann & Kerl. Translated by W. A. Goodyear. 1 vol. 12mo, $? 50 A PRACTICAL TREATISE ON METALLURGY. Adap- ted from the last German edition of Prof. Kerl’s Metallurgy. By William Crookes and Ernst Rohrig. In three vols. thick Die Pricen eer Toot e Omen a $30 00 Separately. Vol. 1. Lead, Silver, Zinc, Cadmium, Tin, Mer- cury, Bismuth, Antimony, Nickel, Arsenic, Gold, Platinum, and Sulphur... . smissu eeens a a SI DOC $10 00 Vol. 2. Copper and Iron............-- Rave siete sasan 10 00 Vol. 3. Steel, Fuel, and Supplement. .....eseseeeso - 10 00 CAST AND WROUGHT IRON FOR BUILDING. By Wm. Fairbairn. 8vo, cloth.......esssesesosseeseo . $2 00 HISTORY OF IRON TRADE, FROM 1621 TO 1857. By B. F. French. 8vo, cloth.......... eB ate e e N $2 00 COLLECTION OF REPORTS (CONDENSED) AND OPINIONS OF CHEMISTS IN REGARD TO THE USE OF LEAD PIPE FOR SERVICE PIPE, in the Distribution of Water for the Supply of Cities. By I. P. Kirkwood, C.E. 8vo, cloth........scseecceecee .... Hl 50 THE: IRON MANUFACTURER'S GUIDE TO THE FURNACES, FORGES, AND ROLLING-MIULS OF THE! UNITED STATES. By J. P. Lesley. With maps and plates. 1 vol. 8vo, cloth........... aa E $3 00 MACHINISTS—MECHANICS. THE BOSTON MACHINIST. A complete School for the Apprentice and Advanced Machinist. By W. Fitzgerald, 1 vol. 18mo, cloth.......... A. E T T $0 75 SAW FILING. The Art of Saw Filing Scientifically Treated and Explained. With Directions for putting in order all kinds of Saws, from a Jeweller’s Saw to a Steam Saw-mill. Illus- trated by forty-four engravings. Tbird edition. By i w. Holly. 1 vol 18mo, cloth... .. -E T snes $0 75 THH MECHANISM AND ENGINEER INSTRUCTOR. Comprising Forging, Planing, Lining. Slotting, Shaping, Turning, Screw-Cutting, etc., etc. By Cameron wit” a 5]

Page 451

100 TURNING, &c. TURNING, &c. WILLIS, BOOTH. CELDARD. BULL. FRANCKE. GREEN. JOHN WILEY & SON’S LIST OF PUBLICATIONS. LATHE, THE, AND ITS USES, ETC.; or, Instruction in the Art of Turning Wood and Metal. Including a descrip- tion of the most modern appliances for the ornamentation of plane and curved surfaces, with a description also of an entirely novel form of Lathe for Eccentric and Rose Engine Turning, a Lathe and Turning Machine combined, and other valuable matter relating to the Art. 1 vol. 8vo, copiously illustrated. Including Supplement. 8vo, cloth...... $7 00 “The most complete work on the subject ever pnblished.”—Amertcan Artisan. “ lIere isan invalnable book to the practical workman and amatcur.”—London Weekly Times. SUPPLEMENT AND INDEX TO LATHE AND ITS USES. Large type. Paper. Svo.............<+++:- $0 90 PRINCIPLES OF MECHANISM. Designed for the use of Students in the Universities and for Engineering Students generally. By R-bert Willis, M.D., F.R.S., President of the British Association for the Advancement of Science, &c., &c. Second edition, enlarged. 1 vol. 8vo, cloth........... $7 50 *,* It onght to be in every large Machine Workshop Office, in every School of Mechanical Engineering at least, and in the hands of every Professor of Mechanics, &«.—Prof. S. EDWARD WARREN, MANUFACTURES. NEW AND OOMPLETE CLOCK AND WATCH MAKERS’ MANUAL. Comprising descriptions of the various gearings, escapements, and ‘Compensations now in * use in French, Swiss, and English clocks and watches, Patents, Tools, etc., with directions for cleaning and repairing. With numerous engravings. Compiled from the French, with an . Appendix containing a History of Clock and Watch Making in America. By Mary L. Booth. With numerous plates. 1 Vol, plod; ION gina he a eee ase esa hone eae $2 00 HANDBOOK ON COTTON MANUFACTURE; or, A Guide to Miachine-Building, Spinning, and Weaving. With practical examples, all needful calculations, and many useful and important tables. The-whole intended to be a complete vet compact authority for the manufacture of cotton. By James Geldard. “With steel engravings. 1 vol. MNO, WOlOLNY aeaee e osu E ie Ch ee eae ere $2 50 MEDICAL, &c. HINTS TO MOTHERS FOR THE MANAGEMENT OF HEALTH DURING THE PERIOD OF PREG- NANCY, AND IN THE LYING-IN ROOM. With an exposure of popular errors in connection with those subjects. By Thomas Bull, M.D. 1 vol. 12mo, cloth........... $1 00 OUTLINES OF A NEW THEORY OF DISEASE, applied to Hydropathy, showing that water is the only true remedy. With observations on the errors committed in the practice of Hydropathy, notes on the cure of cholera by cold water, and a critiqne on Priessnitz’s mode of treatment. Intended for popular use. By the late H. Francke. Translated from the German by Robert Blakie, M.D. 1 vol 12mo, cloth...$1 50 A TREATISE ON DISEASES OF THE AIR PASSAGES. Comprising an inquiry into the History, Pathology, Causes, and Treatment of those Affections of the Throat called Bron- chitis, Chronic Laryngitis, Clergyman’s Sore Throat, ete., etc, By Horace Green, M.D. Fourth edition, revised and enlarged. I vol Svolti. stars << steal sete ers2s. ore soso. /c-0 0 6 /0ie care $3 00 A PRACTICAL TREATISE ON PULMONARY TUBER- CULOSIS, embracing its History, Pathology, and Treat- ment. By Horace Green, M.D. Colored plates 1 vol. 8vo, Old ere rs e a a a a e ee E E $5 00

Page 452

CREEN. (23 TILT. VON DUBEN. BRUSH. DANA. DANA & BRUSH. DANA. (9 BAILEY. CARLYLE. CATLIN. CHEEVER. 6 (23 CONYBEARE. CHIiD’S BOOK JOHN WILEY & SON’S LIST OF PUBLICATIONS. 1u1 OBSERVATIONS ON THE PATHOLOGY OF CROUP. With Remarks on its Treatment by Topical Medications, By Horace Green, M.D. 1 vol. 8vo, cloth............. $1 25 ON THE SURGICAL TREATMENT OF POLYPI OF THE LARYNX, AND CEDEMA OF THE GLOTTIS. By Horace Green, M.D, 1 vol. 8vo.................. $1 25 FAVORITE PRESCRIPTIONS OF LIVING PRACTI- TIONERS. With a Toxicological Table, exhibiting the Symptoms of Poisoning, the Antidotes for each Poison, and the Test proper for their detection. By Horace Green, 1 VOL. 8yo, CHOON... 3... cas. Re te... oe $2 50 ON THE PRESERVATION OF THE HEALTH OF WOMEN AT THE CRITICAL PERIODS OF LIFE. By E. G. Tilt, M.D. 1 vol. 18mo, cloth.............. $0 50 GUSTAF VON DUBEN’S TREATISE ON MICRO- SCOPICAL DIAGNOSIS. With 71 engravings, Trans- lated, with additions, by Prof. Louis Bauer, M.D. 1 vol. 8vo, cloth. ee CO a ge $1 00 MINERALOGY. ON BLOW-PIPE ANALYSIS. By Prof. Geo. J. Brush. (In preparation. ) DESCRIPTIVE MINERALOGY. Comprising the most re- cent Discoveries. Fifth edition. Almost entirely re-written and greatly enlarged. Containing nearly 900 pages 8vo, and upwards of 600 wood engravings, By Prof. J. Dana. 0) rr. WR ne: $10 00 “ We have used a good many works on Mineralogy, but have met with none that begin to compare with this in fulness of plan, detail, and execntion.”— American Journal of Mining. APPENDIX TO DANA’S MINERALOGY, bringing the work down to 1872. By Prof. G. J. Brush. 8vo..... $0 50 DETERMINATIVE MINERALOGY. 1 vol. (In prepa- ration.) A TEXT-BOOK OF MINERALOGY. 1 vol. (In prepa- ration.) MISCELLANEOUS. THE NEW TALE OF A TUB. An adventure in verse. By F. W. N. Bailey. With illustrations. 1 vol. 8vo...... $0 75 ON HEROES, HERO-WORSHIP, AND THE HEROIC IN HISTORY. Six Lectures. Reported, with emendations and additions. By Thomas Carlyle. 1 vol. 12mo, cloth...$0 75 THE BREATH OF LIFE; or, Mal-Respiration and its Effects upon the Enjoyments and Life of Man. By Geo. Catlin. With numerous wood engravings. 1 vol. 8vo, $0 75 CAPITAL PUNISHMENT. A Defence of. By Rev. “fo 20 B” Chee vei Ie D” we ClO tgs, rage nig: sger sine eg os ei s HILL DIFFICULTY, and other Miscellanies. By Rev. George B. Cheever, D.D. 1vol 12mo, cloth......... $1 00 JOURNAL OF THE PILGRIMS AT PLYMOUTH ROCK. By Geo. B. Cheever, D.D. 1 vol. 12mo, cloth........ $1 09 j e B WANDERINGS OF A PILGRIM IN THE ALPS y George B. Cheever, D.D. 1 vol. 12mo, cloth......... $1 00 p R OF THE WATER OF ki A TE B. Cheever. 1 vol. 12mo, cloth AEE ese ses etae ngisi = ejar tetea e S ON INFIDELITY: 12mo, cloth......... poo saccances OF FAVORITE STORIES. Large colored plates. si clo

Page 453

102 EDWARDS. GREEN, GOURAUD. (73 HEIGHWAY. t KELLY. MAVER’S NOTLEY. PARKER. ib STORY OF TUPPER. WALTON & COTTON. WARREN, WILLIAMS, JOHN WILEY & SON’S LIST OF PUBLICATIONS. FREE TOWN LIBRARIES. The Formation, Manageiet.t, and History in Britain, France, Germany, and America.” Together with brief notices of book-collectors, and vf the respective places of deposit of their surviving collections. By Edward Edwards. 1 vol. thick Svo............... $8 00 THE PENTATHEUCH VINDICATED FROM THE AS: PERSIONS OF BISHOP COLENSO. By Wm. Henry Green, Prof. Theological Seminary, Princeton, N. J. 1 vol. 12mo, cloth...... Agee eee e E a a Graver tien: $1 25 PHRENO-MNEMOTECHNY; or, The Art of Memory. The series of Lectures explanatory of the principles of the system. By Francis Fauvel-Gouraud. 1 vol. 8vo, cloth, $2 00 PHRENO-MNEMOTECHNIC DICTIONARY. Being a Philosophical Classification of all the Homophonic Words of the English Language. To be used in the application of the Phreno-Mnemotechnic Principles. By Francis Fauvel-Gou- reud: 1 vol Sve, Cloth a a ee a e a $2 00 LETRA ADA. demo, cloth. ....2s<ccsssasevsees ona 1 00 LEILA ADA’S RELATIVES. 12mo, cloth..... saec 1 00 CATALOGUE OF AMERICAN BOOKS. The American Catalogue of Books, from January, 1861, to January, 1866. Cu mpiled by James Kelly. 1 vol. 8vo, net cash....... $5 00 CATALOGUE OF AMERICAN BOOKS. The American Catalogue of Books from January, 1866, to January, 187i. Compiled by James Kelly. 1 vol. 8vo, fete ..4s. 1455 $7 50 COLLECTION OF GENUINE SCOTTISH MELODIES. For the Piano-Forte or Harmonium, in keys suitable for the voice. Harmonized by C. H. Morine. Edited by Geo. Alex- ander. 1 vol! 4to, half calf nense seanaire enes secs 3 $10 00 A COMPARATIVE GRAMMAR OF THE FRENCH, ITALIAN, SPANISH. AND PORTUGUESE LAN- GUAGES. By Edwin A. Notley. 1 vol. s cloth rests $5 00 POLAR MAGNETISM. First and Second ioe By John Am ark cram Del rege Seren cies aa averse eaters $0 25 NON-EXISTENCE OF PROJECTILE FORCES IN NATURE. By John@® Pier... ...cacee seceanaet $0 25 A POCKET BIBLE. Illustrated. 12mo, cloth........ $1 00 PROVERBIAL PHILOSOPHY. 12mo............... 1 00 THE COMPLETE ANGLER; or, The Contemplative Man’s Recreation, by Isaac Walton, "and Instructions how to Angle for a Trout or Grayling in a Clear Stream, by Charles Cotton, with copious notes, for the most part original. A bibliographical preface, giving an account of fishing and Fishing Books, from the earliest antiquity to the time of Walton, and a notice of Cotton and his writings, by Rev. Dr. Bethune. To which is added an appendix, including ` the most complete catalogue of books in angling ever printed, &c. Also a genera] index to the whole work. 1 vol. 12mo, cloth.. $3 00 NOTES ON POLYTECHNIC OR SCIENTIFIC SCHOOLS IN THE UNITED STATES. Their Nature, Position Aims, and Wants. By S. Edward Warren. Paper....$0 4 TEE MIDDLE KINGDOM. A Survey of the Geography, Government, Education, Social Life, Arts, Religion, &c., of the Chinese Empire and its Inhabitants. With a new map of the Empire. By S. Wells Williams. Fourth edition, in 2 vols eeecvresreeeeene Pee eReesoereeeeteeeeereatteetestse 4 00

Page 454

RUSKIN. 6b 66 ét “ RUSKIN. ét =% JOHN WILEY & SON’S LIST OF PUBLICATIONS. 103 RUSKIN’S WORKS. Uniform tn size and style. MODERN PAINTERS. 5 vols. tinted paper, bevelled coe, Plates; $n WOK, cse esiee ca seth es. oss. | $18 00 MODERN PAINTERS. 5 vols. half calf............ 27 00 s i ‘* without plates. ...... 12 00 ef &é 66 és half calf, 20 00 Vol. 1.—Part 1. General Principles. Part2. Truth. Vol. 2.—Part 3. Of Ideas of Beauty. Vol. 3.—Part 4. Of Many Things. Vol. 4.—Part 5. Of Mountain Beauty. Vol. 5.—Part 6. Leaf Beauty. Part 7. Of Cloud Dealing! Part 8. Ideas of Relation of Invention, Formal. Part 9. Ideas of Relation of Invention, Spiritual. STONES OF VENICE. 3 vols., on tinted paper, bevelled boards; iw bok Morc. te. a ROOM Otte. Means ot $7 PEE OF VENICE. 3 vols., on tinted paper, vt Cb eserse. s ss A ONO e oe inc co a a ge $12 STONES OF VENICE. 3 vols., cloth..... eoguggocj: - 6 00 Vol. 1.—The Foundations. Vol. 2.—The Sea Stories, Vol. 3.—The Fall. SEVEN LAMPS OF ARCHITECTURE. With illustrations, drawn and etched by the authors. 1 vol. 12mo, cloth, $1 3 LECTURES ON ARCHITECTURE AND PAINTING. With illustrations drawn by the author. 1 vol. 12mo, (6110): | o ea aisan E a 5 oo Gin GEES on. c OO a a $1 50 THE TWO PATHS. Being Lectures on Art, and its Appli- cation to Decoration and Manufacture. With plates and cuts; i1 Wol. Mino, clot? Eeee - aE. oe see ee se $1 25 THE ELEMENTS OF DRAWING. In Three Lettars to Beginners. With illustrations drawn by the author. 1 vol. 12mo, cloths... «Agee E a 2... ee $1 00 THE ELEMENTS OF PERSPECTIVE. Arranged for the use of Schools. 1 vol. 12mo, cloth..... a o AEE o afte $1 00 THE POLITICAL ECONOMY OF ART. 1 vol. 12mo, Cloths. o osease s nie emer eee ooa o TREE o cee $1 00 PRE-RAPHABELITISM. NOTES ON THE CONSTRUCTION OF |; vol, ones SHEEPFOLDS. cloth, $1 00 KING OF THE GOLDEN RIVER; or, The Black Brothers. A Legend of Stiria. SESAME AND LILIES. Three Lectures on ¥ooks, Women, &c. 1. Of Kings’ Treasuries. 2. Of Queens’ Gardens. 3. Of the Mystery of Life, 1 vol. 12mo, cloth.......... $1 50 INTO SOME OF THE CONDITIONS AT AN ORSSENT AFFECTING “THE STUDY OF AR- CHITECTURE” a. OUR SCHOOLS. 1 vol. 12mo, PAPE.. 0. cece cece ct cce cc cnccccccsocscnwecccscsses $0 15 THE ETHICS OF THE DUST. Ten Lectures to Little Housewives, on the Elements of Crystallization. 1 vol, 12mo, cloth. OMFS sor >cagaa@ec 00 a Gono CO RUONG “UNTO THIS LAST.” Four Essays on the First Principles of Political Economy. 1 vol. 12mo, cloth.......++s+++s -$1 00

Page 455

104 JOHN WILEY & SON’S LIST OF PUBLICATIONS. RUSKIN. THE CROWN OF WILD OLIVE. Three Lectures on ea Traffic, and War. 1 vol. 12mo. cloth............ ssa eel a TIME AND TIDE BY WEARE AND TYNE. a al five Letters to a Workingman on the Laws of Work. 1 vol. PPO ClO D e caiecirste secre herore sa a a he citer $1 00 at THE QUEEN OF THE AIR. Being a Study of the Greek Myths of Cloud and Storm. 1 vol. 12mo, cloth ...... $1 00 $ LECTURES ON ART. 1 vol. 12mo, doth. T areae 1 66 os FORS CLAVIGERA. Letters to fhe Workmen and Labourers of Great Britain. Part 1. 1 vol. 12mo, cloth, plates, $1 00 es FORS CLAVIGERA. Letters to the Workmen and Labourers of Great Britain. Part2. 1 vol. 12mo, cloth, plates, $1 60 st MUNERA PULVERIS. Six Essays on the Elements of Political Economy. 1 vol. 12mo, cloth............... $1 00 =s ARATRA PENTELICI. Six Lectures on the Elements of Sculpture, given before the University of Oxford. By John Ruskin. 12mo, cloth, $1 50, or with plates......... $3 00 i THE EAGLES NEST. Ten Lectures on the relation of Natural Science to Arb. i vol. 12mo............... $1 50 Ce THE POETRY OF ARCHITECTURE: Villa and Cottage. With numerous alates. 1 vol. 12mo, cloth (shortly). ...$2 00 Ge FORS CLAVIGERA. Letters to the Workmen and Laborers of Great Britain. Part 3. 1 vol. 12mo, cloth (shortly).$1 50 BEAUTIFUL PRESENTATION VOLUMES, Printed on tinted paper, and eleganliy bound in crape cloth extra, bevelled boards, gilt head. RUSKIN. THE TRUE AND THE BEAUTIFUL IN NATURE, ART, MORALS, AND RELIGION. Selected from the Works of John Ruskin, A.M. With a notice of the author by Mrs. L. C. Tuthill. Portrait. 1 vol. 12mo, cloth extra, prultineatt em, co eee e a a EE a e EE E ce a $2 50 D THE TRUE AND THE BEAUTIFUL IN NATURE, ART, MORALS, AND RELIGION. Selected from the Works of John Ruskin, A.M. With a notice of the author by Mrs. L. C. Tuthill. Portrait. 1 vol. 12mo, plain cloth, $2 00 ce PRECIOUS THOUGHTS: Moral and Religions. Gathered from the Works of Jobn Ruskin, A.M. By Mrs. C. L. Tuthill. 1 vol. 12mo, cloth extra, gilt SF ip ai EN: $2 06 s i PRECIOUS THOUGHTS: Moral and Religious. Gathered from the Works of John Ruskin, A.M. By Mrs. L. C. Tuthill. I vol 12m0, plain Cloths -sanee 155) < cain as waa wale © ome $1 50 SELECTIONS FROM THE WRITINGS OF JOHN RUSKIN. 1 vol. 12mo, cloth extra, gilt head........ $2 50 is SELECTIONS FROM THE WRITINGS OF JOHN RUSKIN. 1 vol. 12mo, plain cloth............... ~.$2 00 Se SESAME AND LILIES. 1 vol.“i2mo................ $1 75 ce ETHICS OF THE DUST. 12mo...... eee orerace cesar vane 175 “ CROWN OF WILD OLIVE. i2mo.................. 1 50 RUSKIN’S BEAUTIES. “ THE TRUE AND BEAUTIFUL A vols. , in box, cloth sii e a ilt head..........26 00 CHOICE SELECTIONS. do., half calf. ..10 00 RUSKIN’S POPULAR VOLUMES. CROWN OF WILD OLIVE. SESAME AND LILIES. QUEEN OF THE AIR. ETHICS OF THE DUST. . 4 vols. in box, cloth extra, gilt head.............0.05 $6 00

Page 456

JOHN WILEY & SON’S LIST OF PUBLICATIONS. 105 RUSKIN'’S WORKS. : Revised edition. RUSKIN. Vol. 1.—_SHSAME AND LILIES. Three Lectures. By John Ruskin, LL.D. 1. Of Kings’ Treasuries. 2. Of Queens’ Gardens. 38. Of the Mystery of Life. 1 vol. 8vo, cloth, $2 00. Large paper......... 00.0.0 1 ee ee $2 50 bd Vol. 2.—MUNERA PULVERIS. Six Essays on the Elements of Political Economy. By John Ruskin. 1 volume &vo, cloth: sepa... Bis Ror gs kk a $2 00 EE EGEE o 000 00 Dh 50 OR EEE eres 2 50 “ Vol. 3—ARATRA PENTELICI. Six Lectures on the Fle. Š ments of Sculpture, given before the University of Oxforu By John Ruskin. 1 vol. 8vo..... | > EE, E $4 00 Large paper........... CE E a a E 4 50 RUSKIN—COMPLETE WORKS. THE COMPLETE WORKS OF JOHN RUSKIN. 27 vols., extra cloth, in a box..$40 00 Ditto 27 vols., extra cloth. Plates... 48 00 Ditto Bound in 17 vols., half calf. do.... 70 00 SEAMANSHIP. l ALSTON. SEAMANSHIP. New edition, revised and enlarged, by Com- mander R. H. Harris, R. N. With a Treatise on Nantical hundred illustrations. 1 vol. 12mo, cloth, price....... $5 00 SHIP-BUILDING, &c. BOURNE. A TREATISES ON THE SCREW PROPELLER, SCREW VESSELS, AND SCREW ENGINES, as adapted for Purposes of Peace and War. Illustrated by numerous wood- cuts and engravings. By John Bourne. New edition. 1867. 1 vol. 4to, cloth, $18.00; half russia........... e... Prd 00 WATTS. RANKINE (W. J. M.) AND OTHERS. Ship-Building, Theo- retical and Practical, consisting of tug Hydraulics of Ship- Building, or Buoyancy, Stability, Speed and Design—The Geometry of Ship-Building, or Modelling, Drawing, and Laying Off—Strength of Materials as applied to Ship-Building —Practical Ship-Building—Masts, Sails, and Rigging—Marine Steam Engineering—Ship-Building for Purposes of War. By Isaac Watts, C.B., W. J. M. Rankine, C.B., Frederick K. - Barnes, James Robert Napier, etc. Ilustrated with numerous fine engravings and woodcuts. Complete in 30 numbers, boards, $35.00; 1 vol. folio, cloth, $37.50; half russia, $40 00 WILSON (T. D.) SHIP-BUILDING, THEORETICAL AND PRACTICAL. In Five Divisions. — Division I. Naval Architecture. II. Lay- ing Down and Taking off Ships. III. Ship-Building IV. Masts and Spar Making. V. Vocabulary of Terms uscd— intended as a Text-Book and for Practical Use in Public and Private Ship-Yards. By Theo. D. Wilson, Assistant Naval Constructor, U. S. Navy; Instructor of Naval Construction, U. S. Naval Academy; Member of the Institution of Naval ‘ Architects, England. With numerous plates, lithographic and wood, 1 vol. 8vo. Shortly........-..s002 seeeeeeee SOAP. MORFIT. A PRACTICAL TREATISE ON THE MANUFACTURE OF SOAPS. With numerous wood-cuts and elaborate work- ing drawings. By Campbell Morfit, M.D., F.C.S. 1 voi. BVO... 06 0 ae BGS Gor Dace wee... sd OS cas coset $20 00 STEAM ENGINE. TROWBRIDCE. TABLES, WITH EXPLANATIONS, OF THE NON- CONDENSING STATIONERY STEAM ENGINE, and of High-Pressure Steam Boilers. By Prof. W. P. Trow- hridge, of Yale College Scientific School. 1 vol. 4to, pates Ge, e coc eee cee eee ee aa cw sone s $3 50

Page 457

106 JOHN WILEY & SON’S LISI OF PUBLICATIONS. TURNING, &c. THE LATHE, AND ITS USES, ETC. On Instructions in the Art of Turning Wood and Metal. Including a description of the most modern appliances for the ornamentation of plane and curved surfaces. With a description, also, of an entirely novel form of Lathe for Eccentric and Rose Engine Turning, a Lathe and Turning Machine combined, and other valuable matter relating to the Art. 1 vol. 8vo, copiously illustrated, cloth.......... $7 00 “ SUPPLEMENT AND INDEX TO SAME. Paper...$0 99 VENTILATION. LEEDS (L. W.). A TREATISE ON VENTILATION. nee Seven Lec tures delivered before the Franklin Institute, showing the great want of improved methods of Ventilation in our build- ings, giving the chemical and physiological process of res- piration, comparing the effects of the various methods of heating and lighting upon the ventilation, &c. Ilustrated by many plans of all classes of public and private buildings, showing their present defects, and the best means of im- proving them. By Lewis W. Leeds, 1 vol. 8vo, with nu- merous wood-cuts and colored plates. Cloth......... $2 50 * It ought to be in the hands of every family in the country.”— Technologist. & Nothing could be clearer than the author's exposition of the principles of the principles and practice of both Bo aud bad ventilation,” — Van Nostrand’s Engineering Magazine, f “ The work is every way worthy of ie widest circulation."—Scientijic American. REID. VENTILATION IN AMERICAN DWELLINGS. Witha series of diagrams presenting examples in different classes of habitations. By David Boswell Reid, M.D. To which is added an introductory outline of the progress of improvement in ventilation, By Elisha Harris, M.D. 1 vol. 12mo, $1 5u WEIGHTS, MEASURES, AND COINS. TABLES OF WEIGHTS, MEASURES, COINS, &c., OF -THE UNITED STATES AND ENGLAND, with their Equivalents in the French Decimal System. Arranged by T. T. Egleston, Professor of Mineralogy, School of Mines, Columbia College. 1 vol. 18mo0............. 0 cece eee $0 75 “It is a most nscful work for all chemists and others who have occasion to make the conversions from one system te another.”"—American Chemist. “Every mechanic should have these tables at hand.”"—American Horologica} Journal. J. W. & SON are Agents for and keep in stock SAMUEL BAGSTER & SONS’ PUBLICATIONS, LONDON TRACT SOCIETY PUBLICATIONS, COLLINS’ SONS & CO.’S BIBLES, MURRAY’S TRAVELLER’S GUIDES, WEALE’S SCIENTIFIC SERIES. Full Cataogues gratis on application. J. W. & SON import to order, for the TRADE AND PUBLIC, BOOKS.) i OD LOA LS. ac... FROM ENGLAND, FRANCE, AND GERMANY. * .* JOHN WILEY & SON'S Complete Classified Catalogue of the most valu- abie and latest scientific publications supplied gratis to order.

Page 458

Page 459

Page 460

Page 461

res Me TE