American House-Carpenter, 1845

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Google This is a digital copy of a book that was preserved for generations on library shelves before it was carefully scanned by Google as part of a project to make the world’s books discoverable online. It has survived long enough for the copyright to expire and the book to enter the public domain. A public domain book is one that was never subject to copyright or whose legal copyright term has expired. Whether a book is in the public domain may vary country to country. Public domain books are our gateways to the past, representing a wealth of history, culture and knowledge that’s often difficult to discover. Marks, notations and other marginalia present in the original volume will appear in this file - a reminder of this book’s long journey from the publisher to a library and finally to you. Usage guidelines Google is proud to partner with libraries to digitize public domain materials and make them widely accessible. Public domain books belong to the public and we are merely their custodians. Nevertheless, this work is expensive, so in order to keep providing this resource, we have taken steps to prevent abuse by commercial parties, including placing technical restrictions on automated querying. We also ask that you: + Make non-commercial use of the files We designed Google Book Search for use by individuals, and we request that you use these files for personal, non-commercial purposes. + Refrain from automated querying Do not send automated queries of any sort to Google’s system: If you are conducting research on machine translation, optical character recognition or other areas where access to a large amount of text is helpful, please contact us. We encourage the use of public domain materials for these purposes and may be able to help. + Maintain attribution The Google “watermark” you see on each file is essential for informing people about this project and helping them find additional materials through Google Book Search. Please do not remove it. + Keep it legal Whatever your use, remember that you are responsible for ensuring that what you are doing is legal. Do not assume that just because we believe a book is in the public domain for users in the United States, that the work is also in the public domain for users in other countries. Whether a book is still in copyright varies from country to country, and we can’t offer guidance on whether any specific use of any specific book is allowed. Please do not assume that a book’s appearance in Google Book Search means it can be used in any manner anywhere in the world. Copyright infringement liability can be quite severe. About Google Book Search Google’s mission is to organize the world’s information and to make it universally accessible and useful. Google Book Search helps readers discover the world’s books while helping authors and publishers reach new audiences. You can search through the full text of this book on the web atthtto: //books.gqoogle.com/

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S/gu_4 yi AE a Oliver. R Erant. 7 A e 2235 SOUTHERN BOUL: y i hN eo Neve YOR At 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. rt ew aa BY R..G. HATFIELD, ARCHITECT. SECOND EDITION. T ä. 7 Rilustrated by more than three hundred Engravings, NEW-YORK & LONDON: WILEY AND PUTNAM. 1845. Ole R. Era 7. : + 2239 SVU TH eho op i o * aa NEW X L : YY

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PREFACE. Tuts book is intended for carpenters—for masters, journeymen and apprentices. It has long been the complaint of this class that architectural books, in- tended for their instruction, are of a price so high as to be placed beyond their reach. This is owing, in ù great measure, to the costliness.of the plates with which they are illustrated: an unnecessary expense, as illustrations upon wood, printed on good paper, answer every useful purpose. 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 sub- ject matter has been gleaned from works of the first authority, and subjected to the most careful examina- tion. The explanations have all been written out from the figures themselves, and not taken from any he figures have all been drawn ex- doing tis the utmost care VE as plain as the

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iv PREFACE. 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 points; a rule to determine the projection of eave cornices ; a new RRR et proportioning a cor- nice to a larger given one; a way to determine fhe lengths and bevils of rafters for hip-roofs ; a way to proportion the rise to the tread in 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 introduced. Much labour has been bestowed upon the section on stairs, in which the subject of hand-railing is presented, in many re- spects, in a new, and, it is hoped, more practical form than in previous treatises on that subject. | l ~ The author has endeavoured to present a fund of useful 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.

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PREFACE TO THE SECOND EDITION. . In preparing a second edition, the author regrets his want of leisure to give the work that thorough revision, which is demanded by the importance and intricacy of the subjects treated of. He has corrected all the typo- graphical errors in the references, &c., that he could discover, and added a Section on the subject of Shadows. He cannot refrain from giving expression to the satisfaction which he feels at the unexpected suc- cess of his undertaking. But it is evident that in a wide-spread, new country like ours, works of a practical character, adapted to the wants of the people, and cal- culated to instruct our operative citizens in the every- day employment of their heads and hands, cannot but meet with a favourable reception. In another edition, perhaps, opportunity will be given for additions and rovements of a still more important nature.

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TABLE OF CONTENTS. INTRODUCTION. 2| To use the avin = . ae Articles necessary for ati, 2 5 | Directions for drawing, - 13 To prepare the paper, - SECT. I.—PRACTICAL GEOMETRY. DEFINITIONS Tofinda fourth point incircle, 86 Lines, - - - - 17|To describe a segment oa ~ Angles, Bb julie A 23| circle by a set- le, > 87 Angular point, - - - 27| Do. by intersection f s Polygons, - = >~ 28 | To curve an angle, 3 The circle, - - - 47) To inscribe a circle within eee The cone, - =- -> 56| given triangle, - 90 Conic sections, - é - 58| Tomake triangle about circle, 91 The ellipsis, - - - 61| To find the length of a cir- The cylinder, - - - 68] cumference, - 92 PROBLEMS. To describe a triangle, hea, . To bisect a line, - 71| gon, &e., - - 93 To erect a mernebdioalay, - 72|Todrawan octagon, - 94 To let fall a perpendicular, 73| To eight-square a rail, &c., 94 To erect ditto on a end of line, 74|'To describe any polygon ‘in Six, eight and ten rule, - 74| acircle, 95 To square end of board, - 74|To “ta equilateral triangle; 96 To square foundations, ‘be. «a | 74|To draw a squire by com- To let fall a perpendicular passes, 97 -~ _meartheendofaline; - 75|To draw any polygon on a To make equal angles, - 76| given line, - 98 i go bisect an angl TEI a ate! a a fotangle of required |! i SOL i 99 To. rae any right ned eure, 100 To make a parallelogram | aqual to a triangle, - 101 To fi d the area of a a 101

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Art. To find the length of a brace, 103 To ascertain the pitch of a roof, - - ° - 103 To ascertain the rake of a step-ladder, - - 103 To describe one circle equal to two others, - To make one polygon equal to two or more, - To make a square equal to a rectangle, - - To make a square equa to a triangle, - To find a third proportional, To find a fourth proportional, To proportion one i ai to another, - To divide a line as another, To find a mean proportional, Definitions of conic sections, To find the axes of an ellipti- cal section, - 2 To find the axes and base of the parabola, - To find the height, base ‘and axes of the hyperbola, - To find foci of ellipsis, - To describe an ellipsis with a string, - - - To describe an ellipsis with a trammel, - - To construct a trammel, - To describe an eg by or- dinates, - To trace a curve through given points, - - - To describe an ellipsis by in- tersection of lines, - 104 115 116 116 117 117 118 CONTENTS. Do. from conjugate diameters, Do. by intersecting arcs, - To describe an oval by com- passes, - Do. in the proportion, 7 x 9, 5 x 7, &c, - - To draw a tangent to an el- lipsis, - To find the point of contact, To find a conjugate to the given diameter, - To find the axes from given diameters, - : To find axes S onal to given ones, - To describe a parabola byi in- tersection of lines, - - To describe hyperbola by do., DEMONSTRATIONS. Definitions, axioms, &c., 130. Addition of angles, - Equal triangles, - Angles at base of isoceles tri- angle equal, - - Parallelograms divided equal- ly by diagonal, .- - Equal parallelograms, - Parallelogram equal triangles, To make mangle Pque, poly- gon, - Opposite angles i - Angles of triangle equal two right angles, - - - Corollaries from do., 150. Angle in semi-circle a right angle, - š P Hecatomb problem, - - SECT. II.—ARCHITECTURE. HISTORY. Antiquity of its origin, - Its cultivation among the an- cients, - 3 Among the Greeks, - Among the Romans, - .159 | Ruin caused by Goths and Vandals, - : š 160 | Of the Gothic, : ° 161 | Of the Lombard, - 162 163 164 165

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CONTENTS. | ix Art. Of the Byzantine andOriental, 166 Moorish, Arabian ane Modern Gothic, - 167 Of the English, - 168 Revival of the art in the sixth century, 169 The art improved i in n the 14th and 15th centuries, - 170 Roman styles cultivated, 171 STYLES. Origin of different styles, Stylobate and pedestal, - Definitions of an order, - Of the several oe of an order, - 175. 185 To proportion an order, - 186 The Grecian orders, $ 187 Origin of the Doric, - - 188 Intercolumniation, - - 189 Adaptation, - - - 190 Origin of the Ionic, - I91 Characteristics, - - 192 Intercolumniation, - - 193 Adaptation, - - 194 To describe the volute, - Origin of the Corinthian, - Adaptation, - : Persians, - - å F Caryatides, - - - The Roman orders, - 7 Art. Extent of Roman structures, 202 Roman styles copied from Grecian, - - E 208 Origin of the Tuscan, - 204 Adaptation, - - - 205 Characteristics of the Egypt- ian, Extentof Egyptian stiüctüres; 206 Adaptation, - 207 Appropriatenessof design, 208. 211 Durable structures, - - 212 Plans of dwellings, &c., 213 SECT. II.—MOULDINGS, CORNICES, &c. MOULDINGS, &c. Elementary forms, - - 229 Characteristics, - - 230 Grecian and Boman; - - 231 Profile, - 232 To describe the torus and scotia, - = - 233 To describe the aohiniss: 234 To describe the cavetto, 235 To describe tne cyma-recta, 236 To describe the cyma-reversa, 237 Directions for designing, 218, 214 PRINCIPLES. Origin ofthe art, - - 215 Arrangement and design, - 216 Ventilation and cleanliness, 217 Stability, - - - 218 Ornaments, - = 219 Scientific knowledge neces- sary, - - 220 The foundation, - - 221 The column, - - ° 222 The wall, ° - - 223 The lintel, - - 2 224 The arch, = - - 225 The vault, - - - 226 The dome, - - - 227 The roof, - - - 228 Roman mouldings, - 238 Modern mouldings, - - 239 Ante caps, - - - 240 CORNICES. Designs, - - - -~ 241 To proportion an eave cornice, 242 Do. from a smaller given one, - : 243 Do. from a larger given ° one, - 244 To find shape of angle-bracket, 245 To find form of raking cornice, 246

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x CONTENTS. SECT. IV.—FRAMING. Jaws of pressure, - - Parallelogram of forces, - To measure the pressure on rafters, `- P : Do. on tie-beams, - The effect of position, - The composition of forces, Best position for a strut, - Nature of ties and struts, - To distinguish ties from struts, Lattice-work framing, - Direction of pressure in raft- ers, - Oblique thrust of lean- to wh, Pressure on floor-beams, - Kinds of pressure, - - Resistance to compression, Area. of post, . - Resistance to tension, . Area of suspending piece, Resistance to cross-strains, Area of bearing timbers, Area of stiffest beam, = - Bearers narrow and deep, Principles of framing, : FLOORS. Single-joisted, - To find area of floor- üinbeis, Dimensions of trimmers, &c., Strutting between beams, Cross-furring and denfening, Double floors, - Dimensions of binding-joists; Do. of bridging-joists, Do. of ċeiling-joists; - Framed floors, - .- 5 Dimensions of girders, - Girders sawn and bolted, - Trussed girders, - - Floors in general, - - PARTITIONS. Nature of their construction, Designs for partitions, - Superfluous timber, - - Improved method, - Weight of partitioning, - ROOFS. Lateral strains, - š Pressure on roofs, ‘ j Weight of covering, - Definitions, . > Relative size of timbers, To find the area of a king- ei Of a queen-post, - Of a tie-beam, - - - Of a rafter, - - “ Of a straining-beam, š Of braces, - - - Of purlins, - a : Of common rafters, - To avoid shrinkage, - -~ Roof with a built-rib, - Badly-constructed roofs, - To find the length and ome in hip-roofs, - To find the packing a hip- rafter, - DOMES. With horizontal ties, - Ribbed dome, - j Area of the ribs, - - Curve of equilibrium, - To describe a cubic parabola, Small domes for stairways, To find the curves of the ribs, To find the shape of the cover- ing for spherical domes, Do. when laid horizontally, To find an angle-rib, - - BRIDGES. Wooden bridge with tie-beam, Do. without a tie-beam, Do. with a built-rib, Table of least rise in bridges, Rule for built-ribs, - Pressure on arches, - To form bent-ribs, - - Elasticity of timber, - To construct a framed rib, Width of roadway, &c.,_ - Stone abutments and piers, Piers constructed of piles,

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CONTENTS. Art. . Piles in ancient bridges, 321 ARE Art. Centring for stone bridges, 322 | Scarfing, or splicing, 332. 334 Pressure of arch-stones, - 322| To proportion the parts, - 835 Centre without a tie at the Joints in beams and posts, - 336 base, - - 323 | Joints in floor-timbers, - 337 Construction of centres, - 324| Timber weakened by framing, 338 General directions, Š 325| Joints for rafters and braces, 339 Lowering centres, - - 326| Evil of shrinking avoided, - 340 Relative size of timbers, - 327 | Proper joint for collar-beam, 341 Short rule for do. - - 328] Pins, nails, bolts and straps, 342 . Joints between arch-stones, 329 | Dimensions of straps, - 342 Do. in elliptical arch, - 330} To prevent ie rusting of Do. in parabolic arch, - 331] straps, - - += 342 SECT. V.—DOORS, WINDOWS, &c. DOORS., WINDOWS. Dimensions of doors, - - 343 | To determine the size, - 349 To. proportion height to width, 344 Width of stiles, rails and panels, . - - Example of trimming, - Elevation of a door and trim- mings, - General directions for hang- ing doors, - - - 345 346 347 348 To find dimensions of frame, 350 To proportion box to Hap shutter, - - 351 To proportion and arrange windows, - - - 352 Circular-headed windows, 353 To find the form of the soffit, 354 SECT. VI.—STAIRS. 356 357 Their position, - Principles of the pitch- bua To proportion the rise to the tread, - -358 The angle of ascent, - ~- 859 Length of steps, - 360 To construct a pitch-board, 361 To lay-out the string, - 362 Section of step, - - 363 PLATFORM STAIRS. To construct the cylinder, - 364 365 366 To cut the lower edge of do., To place the balusters, - Do. in a circular wall, - 355 To find the moulds for the rail, - - - 3670 Elucidation of this method, 368 Two other examples, 369, 370 To apply the mould to the plan - - 371 te bore for the balusters, - 872 Face-mould for moulded rail, 373 To apply this mould to plank, 374 To ascertain thickness of stuff, 875 WINDING STAIRS. Flyers and winders, - To construct winding stairs, 876 877

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xii CONTENTS. Art. Timbers to support winding To find bevils for splayed- stairs, - - - - 378 work, š - - To find falling-mould of rail, 379| Another method for face- To find face-mould of do., 380| moulds, - - - 392 391 . Position of butt-joint, - 380 | To apply face-mould to plank, 394 To ascertain thickness of To apply falling-moul@, - 395 stuf, - - - .- 881 To apply the mould to plank, “383 SCROLLS. Elucidation of the butt-joint, 384 | General rule, - -= 396 Quarter-circle stairs, - 385 | To describe scroll for rail, 398 Falling-mould for do., - 386 For curtail-step, - + 899 Face-mould for do., - 387 | Balusters under soroll, - 400 Elucidation of this method, 388, Falling-mould for scroll, - 401 To bevil edge of plank, - 389 | Face-mould for do., -- 402 To apply moulds without be- Round rails over winders, - 403 villing plank, - - 390] To find form of newel-cap, 404 SECT. VII.—SHADOWS. Inclination of line of shadow, 407 | Shadow in a recess, - 419 Shadows on mouldings, 408 | Do, with wall inclined, - 420 Shadow ofa shelf, - - 409 | Shadow ina fireplace, - 421 Of a shelf of varying width, 410 | Shadow of window lintel, - 422 Of do. with oblique end, 411 | Shadow of step-nosing, - 423 Of an inclined shelf, - 412 | Of pedestal upon steps, - 424 Of do. inclined in section, 413 | Of square abacus on column, 425 Of do. having a curved edge, 414 | Of circular abacus on do. 426 Of do. curved in elevation, 415 | On the capital of a column, 427 Shadow on cylindrical wall, 416 | Of column and entablature, 428 Do. on inclined wall, - 417 | Shadows on Tuscan cornice, 429 Shadow of a beam, - - 418° Reflected light, - - 430 APPENDIX. Glossary of Architectural Terms, - - - - reeg Table of Squares, Cubes and Roots, z - - - 14 Rules for extending the use of the foregoing table, - - 21 Rule for finding the roots of whole numbers with decimals, - 23 Rules for the reduction of Decimals, - - å 23 Table of Areas and Circumferences of Circles, - “ - 25 Rules for extending the use of the foregoing table, - - 28 Table showing the Capacity of Wells, Cisterns, &c., —- - 29 Rules for finding the Areas, &c., of Polygons, 3 - 30 Table of Weights of Materials, - - - - - 31

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Po INTRODUCTION. Arr. 1.—A knowledge of the. properties and principles of lines can bést be acquired by practice. Although the various problems throughout this work may be understood by inspection, yet they will be impressed upon the mind with much greater force, if they are actually performed with pencil and paper by the student. Science is acquired by study—art by practice: he, therefore, who | would have any thing more than a theoretical, (which must of - necessity be a superficial,) knowledge of Carpentry, will attend to the following directions, provide himself with the articles here specified, and perform all the operations described in the follow- ing pages. Many of the problems may appear, at the first read- ing, somewhat confused and intricate; but by making one line at a time, according to the explanations, the student will not only succeed in copying the figures correctly, but by ordinary attention will learn the principles upon which they are based, and thus be able to make them available in any uħexpected case to which they may apply. 2.—The following articles are necessary for drawing, viz: a drawing-board, paper, drawing-pins or mouth-glue, a sponge, a T-square, a set-square, two straight-edges, or flat rulers, a lead _ pencil, a piece of india-rubber, a cake of india-ink, a set of draw- ing-instruments, and a scale of equal parts. 3.—The size of the drawing-board must be regulated accord- ing to the size of the drawings which are to be made upon it. Yet for ordinary practice, in learning to draw, a board about 15 1

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2 AMERICAN HOUSE CARPENTER. by 20 inches, and one inch thick, will be found large enough, and more convenient than a larger one. This board should be well-seasoned, perfecfly square at the corners, and without clamps on the ends. A board is better without clamps, because the little service they are supposed to render by pre@enting the board from warping, is overbalanced by the consideration that the shrinking of the panel leaves the ends of the clamps Project- ing beyond the edge of the board, and thus interfering with the proper working of the stock of the T-square. When the. stuff is well-seasoned, the warping of the board will be but trifling ; and by exposing the rounding side to the fire, or to the sun, it may be brought back to its proper shape. | 4.—For mere line drawings, the paper need not ¢ommoaly be what is called drawing-paper; a8 ‘this is rather costly, and will, where much is used, make quite an item of expense. Cartridge-paper, as it is called, of akout 20 by 26 inches, and of . as good a quality nearly as drawing-paper, can be bought for . about 50 cts. a quire, of 2 pence a sheet ; and each sheet may be cut in halves, or even quarters, for practising. If the drawing is to be much used, as working drawings generally are, cartridge- paper is much better than the other kind. 6.—A drawing-pin is a small brass button, having a steel pin projecting from the under side. By having one of these at each corner, the paper can be fixed to the board ; but this can be done in a much better manner with mouth-glue. The pins will pre- vent the paper from changing its position on the board ; but, more than this, the glue keeps the paper perfectly tight and smooth, thus making it so much the more pleasant to work on. To attach the paper with mouth-glue, lay it with the bottom side up, on the board; and with a straight-edge and penknife, cut off the rough and uneven edge. With a sponge moderately wet, rub all the surface of the paper, except a strip around the edge about half an inch wide. As soon as the glistening of the water disappears, .turn the sheet over, and place it upon the

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INTRODUCTION. 3 board just where you wish it glued. Commence upon one of the longest sides, and proceed thus: lay a flat ruler upon the paper, parallel to the edge, and within a quarter of an inch of it. With a knife, or any thing similar, turn up the edge of the paper against the edge of the ruler, and put one end of the cake of mouth-glue between your lips to dampen it. Then holding it upright, rub it against and along the entire edge of the paper ‘that ig turned up against the ruler, bearing moderately against the edge of the ruler, which must be held firmly with the left hand. Moisten the glue as often as it becomes dry, until a sufficiency of it is rubbed on the edge of the paper. Take away the ruler, restore the turned-up edge to the level of the board, and lay upon jt a strip of pretty stiff paper. By rubbing upon this, not very hard'bħt pretty rapidly, with the thumb nail of the right hand, so as to gause a gentle friction, and heat to be imparted to.the glue that, is on the edge of the faper, you will make it adhere to the board. The other edges in succession must be treated in the same manner. ə Some short distances along one or more of the edges, may afterwards be found loose: if so, the glue must again be applied, and the paper rubbed until it adheres. The board must then be — laid away in a warm or dry place; and in a short time, the sur- face of the paper will be drawn out, perfectly tight and smooth, and ready for use. The paper dries best when the board is laid level. When the drawing is finished, lay a straight-edge upon the paper, and cut it from the board, leaving the glued strip still attached. This may afterwards be taken off by wetting it freely with the sponge; which will soak the glue, and loosen the paper. Do this as soon as the drawing is taken off, in order that the board may be dry when it is wanted for use again. Care must be taken that, in applying the glue, the edge of the paper does not become damper than the rest: if it should, the paper must be laid aside to dry, (to use at another time,) and another sheet be used in its place.

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4 AMERICAN HOUSE CARPENTER. Sometimes, especially when the drawing board is new, the paper will not stick very readily; but by persevering, this diffi- culty may be overcome. In the place of the mouth-glue, a strong solution of gum-arabic may be used, and on some accounts is to be preferred; for the edges of the paper need not be kept dry, and it adheres more readily. Dissolve the gum in a sufficiency of warm water to make it of the consistency of linseed oil. It must be applied to the paper with a brush, when the edge is turned up against the ruler, as was described for the mouth-glue. - If two drawing-boards are used, one may be in use while the other is laid away to dry ; and as they may be cheaply made, it is advisable to have two. The drawing-board having a frame around it, commonly called a panel-board, may afford rather more facility in attaching the paper when this is of the size to suit; yet it has objections which overbalance that con- sideration. 6.—A T-square of mahogany, at once simple in its construc- tion, and affording all necessary service, may be thus made. Let the stock or handle be seven inches long, two and a quarter inches wide, and three-eighths of an inch thick: the blade, twenty inches long, (exclusive of the stock,) two inches wide, and one-eighth of an inch thick. In joining the blade to the stock, a very firm and simple joint may be made by dovetailing it—as shown at Fig. 1.

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INTRODUCTION. 5 7.—The set-square is in the form of a right-angled triangle ; and is commonly made of mahogany, one-eighth of an inch in thickness. The size that is most convenient for general use, is six inches and three inches respectively for the sides which con- tain the right angle; although a particular length for the sides is by no means necessary. Care should be taken to have the square corner exactly true. This, as also the T-square and rulers, should have a holg bored through them, by which to hang them upon a nail when not in use. 8.—One of the rulers may be about twenty inches long, and the other six inches. The pencil ought to be hard enough to. retain a fine point, and yet not so hard as to leave ineffaceable marks.. It should be used lightly, so that the extra marks that are not needed when the drawing is inked, may be easily rubbed off with the rubber. The best kind of india-ink is that which will easily rub off upon the plate; and, when the cake is rub- bed against the teeth, will be free from grit. 9.—The drawing-instruments may be purchased of mathe- matical instrument makers at various prices: from one to one hundred dollars a set. In choosing a set, remember that the lowest price articles are not always the cheapest. A set, com- prising a sufficient number of instruments for ordinary use, well made and fitted in a mahogany box, may be purchased at Pike and Son’s, (Broadway, near Maiden-lane, N. Y.,) for three or four dollars. The compasses in this set have a needle point, which is much preferable to a common point. 10.—The best scale of equal parts for carpenters’ use, is one that has one-eighth, three-sixteenths, one-fourth, three-eighths, one-half, five-eighths, three-fourths, and seven-eighths of an inch, and one inch, severally divided into ¢welfths, instead of being divided, as they usually are, into tenths. By this, if it be required to proportion a drawing so that every foot of the object represented will upon the paper measure one-fourth of an inch, use that part of the scale which is divided into one-fourths ofan

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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 parallel 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 diréction 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, of the set-square even with said line; then place the ruler, g 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 toa b. 12.—To draw a line, as k į, (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 firmly, and slide the set- square along until the side, e d, touches the point, ķ ; then the line, Z k, drawn by it, will be perpendicular to a b. In like

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INTRODUCTION. 7 manner, the drawing of other problems may be facilitated, as will 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 water, 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 the nibs of the drawing-pen, and screw the nfbs together unti n makes a fine line. Beginning with the curved lines, procee all the lines of the figure ; being careful now to make everylineRgpits 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 @eatness and accuracy which is indispensable to a good drawing. When the ink is dry, efface the pencil-marks with the india-rubber. If

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8 AMERICAN HOUSE-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 sought, 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 separated 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.

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THE AMERICAN HOUSE-CARPENTER. SECTION I.—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.) P a a Fig. 4. Fig. 5. Fig. 6

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10 AMERICAN HOUSE-CARPENTER. 24.—A right angle has one line perpendicular to the other. (Fig. 5.) 25.—An oblique angle is either greater or less than a right angle. (Fig. 4 and ô.) 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 b 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.) AA Fig. 1. Fig. 8. 29.—An equi-lateral triangle has its three sides equal. (Fig. 7.) i i 30.—An isoceles triangle has only two sides equal. (Fig. 8.) 31.—A scalene triangle has all its sides unequal. - (Fig. 9) LZ Fig. 9. . Fig. 10. . 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 obiuse-angled triangle has one obtuse angle. (Fig. 9.) 35.—A quadrangle has four sides and four angles. (Fig. 11 to 16.)

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PRACTICAL GEOMETRY. ll | | Fig. 11. Fig. 12. 36.—A parallelogram is a quadrangle having its opposite sides parallel. (Fig. 11 to 14.) , ' 37.—A fectangle 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. 40.—A rhomboid is a parallelogram having oblique angles. (Fig. 14.) 41.—A trapezoid is a quadrangle having only two of its sides parallel. (Fig. 15.) Fig. 15. Fig. 16. x - 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. 44.—A regular polygon has its sides and angles 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

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12 AMERICAN HOUSE-CARPENTER. five, (Fig. 17;) a hexagon 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 0000 Fig. 17. Fig. 18 Fig. 19. Fig. 20. 47.—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. l 48.—The radius of a circle 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. 49.—The dtameter is a right line passing through the centre, and terminating at two opposite points in the circumference.. Hence it 1s twice the length of the radius. (cd, Fig. 21.) 50.—An arc of a circle is a part of the circumference. (cb or be d, Fig. 21.) 51.—A chord is a right line joining the extremities of an arc. (b d, Fig. 21.) |

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PRACTICAL GEOMETRY. 13 52.—A segment is any part of a cirele 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 arc. (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.) 5 d h Fig. 22. Fig. 23. 57.—The avis 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 if a cone be cut by a plane, parallel to a plane touching the curved surface. (c d, Fig. 23— c d being parallel to f g.) 60.—An hyperbola is described if a cone be cut by a plane, parallel to any plane within the cone that passes through its vertex. (eh, Fig. 23.) i 61.—Foci are the points at which the pins are placed in de- scribing an ellipse. (See Art. 115, and f, f, Fig. 24.)

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14 AMERICAN HOUSE-CARPENTER. 62.—The transverse aris 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. (cd, 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. (gh and g t, Fig. 24.) 65.—A diameter of an ellipsis is any right line passing through the centre, and terminated by the curve. (kl, or m n, Fig. 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, k J. 67.—A double ordinate is any right line, crossing a diameter of an ellipsis, and drawn paralle} to a tangent at the extremity of that diameter. (èt, Fig. 24.) | 68.—A cylinder is a solid generated by the revolution of a right-angled parallelogram, or rectang.e, about one of its sides ; and consequently the ends of the :ylinder are equal circles. (Fig. 25.)

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PRACTICAL GEOMETRY. 15 A in Fig. 25. Fig. 26. 69.—The azis 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.)

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PROBLEMS. RIGUT LINES AND ANGLES. 71. -Uh biseet a line. Upon the ends of the line, a b, (Fig. ¥7,) va eontres, with any distance for radius greater than halt Re. n wb, dexeribe area cutting each other in ¢ and d; draw the line, e d, and the point, e, where it cuts a b, will be the middle of the line, a b. Tn practice, a line is generally divided with the compasses, or dividers ; but thia prob tiom is useful where it is desired to draw, rien middle of another line, one at right angles to it. (See ri. B d 0 a b Fig. 28 72,— To erect a perpendicular. From the point, a, (Fig. 28,)

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PRACTICAL GEOMETRY. 17 set off any distance, as a b, and the same distance from a to c Á upon ç, 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 as bit when the operation is so large that a set-square cannot used. 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 aline. Let a, (Fig. 30,) at the end of the line, c a, be the point at which the perpendicular is to be erected. ‘Take any point, as b, above the 3

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18 AMERICAN HOUSE-CARPENTER. line, c a, and with the radius, bò a, describe the arc, d ae; through d and b, draw the line, d e ; join e and a, then e e will be the perpendicular required. The principle here made use of, is a very important one ; and is applied in many other cases—(see Art. 81, b, and Art. 84. For proof of its correctness, see Art. 156.) T Fig. 31. TA, @.-- A second method. Let b. (Fig. 31,) at the end of ths line, a A, be the point at which it is required to erect a perpen- dheular, Upon à with any radius less than b a, describe the are, eed; npon e, with the same radius, describe the small arc at e, and upon e, another at d ; upon e and d, with the same or any ather radins greater than half e d, describe arcs intersecting at f ; join fand b, and the line, f b, will be the perpendicular required. Fig. 32. 74, b.—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, describe an arc at e, and upon e, another at c ;

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PRACTICAL GEOMETRY. 19 through d and e, draw d c, cutting the arc inc; join c and b, then c b will be the perpendicular required. This problem can be solved by the siz, 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 note to 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 ca 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 toc 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 anda; then e a will be the perpendicular required. c S u v d g e Fig. 33. 76.— To make an angle, (as e d f, Fig. 33,) equal to a given angle, (as b a c.} From the angular point, a, with any radius, describe the arc, b c ; and with the same radius, on the line, d e,

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96 AMERICAN HOUSE-CARPENTER. and from the point, d, describe the arc, f g ; take the distance, b c, and upon g, describe the small arc 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, ac; 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. 18.— To trisect a right angie. ‘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-

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PRACTICAL GEOMETRY. 91 fore, is equal to three, and a quadrant to one and a half of those parts. The radius, therefore, is equal to $ of a quadrant; and this is equal to a right angle. °] a b d E c 79.— Through a given point, to draw a line parallel to a 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. o J Fig. 37 d 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 5; through the points, 1, 2, 3 and 4, draw 1 e,2 J, 3 g and 4 h, parallel to c b; which will divide the line, a 6, as was required. The lines, a b and ae, 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,

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33 AMERICAN HOUSE-CARPENTER. (Fig. 38,) and bisect it with the perpendicular, c d ; bisect c d with the line, e f, as at g ; then g is the centre as was required. Fig. 39. 81, a.—A second method. Upon any two points in the cir- cumference nearly opposite, as a and b, (Fig. 39,) describe arcs cutting each other at cand d ; take any other gwo points, as e and f, and describe arcs intersecting as at g dnd h ; join g and h, and c and d ; the intersection, o, is the centre. This is upon the same principle as Art. 85. Fig. 4u. 81, b.—A third method. Draw any chord, as a b, (Fig. 40,)

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PRACTICAL GEOMETRY. 23 and from the point, a, draw a c, at right angles to a b ; 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 6 and a c, represent,) draw lines cutting the circle, as at b and c ; join b and c ; then if b c is bisected, as at d, the point, d, will be the centre. (See Art. 156.) C r Be 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 cd; cdis the tangent required. Fig. 42. 83.— 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 b e; through a and d, draw a line; this will be the tangent required. 84.—A circle and a tangent given, to find the point of con- tact. From any point, as a, (Fig. 43,) in the tangent, b c, draw

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bp | SESC N 307-2 owes a ne v Že cence i NST e EK? pm + wih fe ndam, e e. deserfte be uv. ¿ 7i “7s be wmm of rmt mied. E f and 2 wem ‘cmest ie ine vaud ien zet mees with the tangent j o Wa £~. LW. §5.— Through any three goiniy nat in a xrzizàkt line, te draw a circle. Let a,b ani z. Fig. the three given points. Upon a and 6, with any ralius greater than half e b, describe arcs intersecting at d and e; upon 3 and e. with any radius greater than half b c, describe ares intersecting at f and ¢ ; through d and e, draw a right line, also another through fand g ; upon the intersection, A, with the radius, Å a, describe the circle, a b c, and it will be the one required.

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PRACTICAL GEOMETRY. 25 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, a c b ; bisect the angle, cb a, (Art. 77,) with the line, b d ; also bisect c a in e, and erect e d, perpendicular to a c, cutting b d in d ; then d is the fourth point required. A fifth point may be found, as at f, by assuming a, d and b; 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. 311.) c d .e Fig. 46. f 87.— To descri a Segment of a circle by a set-triangle. Let a b, (Fig. 46,) be the chord, and c d the height of the seg- ment. Secure two straight-edges, or in the position, ¢ e and c f, by nailing them together at c, te a brace from eto f; put in pis at a and b; move the angular point, c, m the direction, a © b; keeping the edges of the triangle hard against the pins, a and b; a pencil held at c will describe the arc, ac b. Tf the angle formed by the rulers at ¢ be a right angle, the segment described will be a semi-circle. This problem is useful in describing centres for brick arches, when they are required 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. 3 Ge

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26 AMERICAN HOUSE-CARPENTER. 88.— To describe the segment of a circle by intersection of lines. Let a b, (Fig. 47,) be the chord, and c d the height of the segment. Through c, draw e f, parallel to a b ; draw b fat right angles to c b; make c e equal toc f; draw a g and b h, at right angles to a b ; divide c e, c f, d a, d b, a g and b h, each into a like number of equal parts, as four; draw the lines, 1 1, 2 2, &c., and from the points, o, o and o, draw lines to c; at the intersection of these lines, trace the curve, a c b, which will be the segment required. In very large work, or in laying out ornamented gardens, this will be found useful ; and where the centre of the le L arc of a circle is inaccessible, it will be invaluable. (To trace ' the curve, see note at Art. 117.) b ‘ ig. a 89.—In a given angle, to describe a tanged curve. Leta b c, (Fig. 48,) be the given angle, and 1 in the line, a b, and 5 in the line, b c, the hion of the curve. Divide 1 b and b 5 into a like number of equal parts, as at 1, 2, 3, 4 and Š ; join 1 and 1, 2 and 2, 3 and 3, &c. ; and a regular cutve will be formed that will be tangical to the line, a b, at the point, 1, and to b c at 5. This is of much use in stair-building, in easing the angles formed between the wall-string and base of the hall, also 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. 93;) yet in large angles the difference is not perceptible. This problem can be applied to describing segments of circles for door-

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PRACTICAL GEOMETRY. 4 heads, window-heads, &c., to rather better advantage than Art. 87. For instance, let a b, (Fig. 49,) be the width of the open- ing, and c d the height of the arc. Extend c d, and make d e equal to c d ; join a and e, also e and b; and proceed as direct- ed at Art. 89. 7 LEDS 90.—To describe a circle within any given triangle, so that the sides of the triangle shall be tangical. Let abc, (Fig. 50,). be the given triangle. Bisect the angles, a and b, according to Art. 77; upon d, the point of intersection of the bisecting lines, with the radius, d e, describe the required circle. go s h S c g Fig. 51. 91.—About a given circle, to describe an equi-lateral tri- angle. Let a d b c, (Fig. 5],) be the given circle. Draw the - diameter, c d ; upon d, with the radius of the given circle, de- scribe the arc, a e b; join a and b ; draw f g, at right angles to dc; make f c and c g, each equal to a b; from f, through a, draw f h, also from g, through b, draw g h; then f g h will be the triangle required. #

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28 . AMERICAN HOUSE-CARPENTER. 92.— To find a right line nearly equal to the circumference of a circle. Let a b c d, (Fig. 52,) be the given circle. Draw the diameter, a c ; on this erect an equi-lateral triangle, a ec, according to Art. 96 ; draw g f, parallel to a c ; extend e ¢ tof, also e a to g ; then g f will be nearly the length of the semi- circle, a d c ; and twice œ f will nearly equal the circumference of the circle, a b c d, as was required. Lines drawn from e, through any points in the circle, as o, 0 and o, to p, p and p, will divide g fin the same way as the semi-: circle, a d c, is divided. So, any portion of a circle may be transferred to a straight line. This is a very useful problem, and should be well studied; as it is frequently used to solve problems on stairs, domes, &c. Fig. 53 92, a.—Another method. Let a b f c, (Fig. 53,) be the given circle. Draw the diameter, ac; from d, the centre, and at right angles to a c, draw d b; join b andc; bisect bc at e; from d, through e, draw d f ; then e f added to three times the diameter,

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PRACTICAL GEOMETRY. 29 will equal the circumference of the circle within the ,',z part of its length. | POLYGONS, &Cc. 93.— Within a given circle, to inscribe an equt-lateral tri- angle, hexagon or dodecagon. Let a b cd, (Fig. 54,) be the Fig. 54. ` given circle. Draw the diameter, b d ; upon b, with the radius of the given circle, describe the arc, a ec; join a and c, also a and d, and c and d—and the triangle is completed. For the hexagon : from a, also from c, through e, draw the lines, a f and c g ; join a and b, b and c, c and f, &c., and the hexagon is completed. The dodecagon may be formed by bisecting the sides of the hexagon. Each side of a regular hexagon is exactly equal to the radius of the circle that circumscribes the figure. For the radius is equal to a chord of an arc of 60 degrees ; and, as every circle is supposed to be divided into 360 degrees, there is just 6 times 60, or 6 arcs of 60 degrees, in the whole circumference. A line drawn from each angle of the hexagon to the centre, (as in the figure,) divides it into six equal, equi-lateral triangles. |

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30 AMERICAN HOUSE-CARPENTER. 94.— Within a square to inscribe an octagon. Leta bcd, (Fig. 55,) be the given square. Draw the diagonals, a d and bc; upon a, b, c and d, with a e for radius, describe arcs 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 c, upon the end of it, after it has been squared-up. Set a gauge to the dis- tance, a 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. 95.— Within a given circle to inscribe any regular polygon. _ Let a b c 2, (Fig. 56, 57 and 58,) be given circles. Draw the diameter, a c; upon this, erect an equi-lateral triangle, a e c, according to Art. 96; divide a c into as many equal parts as the polygon is to have sides, as at 1, 2, 3, 4, &c.; from e, through each even number, as 2, 4, 6, &c., draw lines cutting the circle in the points, 2, 4, &c.; from these points and at right angles to a c, draw lines to the opposite part of the circle; this will give the remaining points for the polygon, as b, f, &c. In forming a hexagon, the sides of the triangle erected upon ac, (asat Fig. 57,) mark the points, b and f. 96.— Upon a given line to construct an equi-lateral triangle. Let a b, (Fig. 59,) be the given line. Upon a and b, with a b

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PRACTICAL GEOMETRY. 31 c a Fig. 59. > a for radius, describe arcs intersecting at c; join a and c, also c and b ; then a cb will be the triangle required. c d a b Fig. 60 97.— To describe an equi-lateral rectangle, or square. Let a b, (Fig. 60,) be the length of a side of the proposed square. Upon a and b, with a b for radius, describe the arcs, a d and 6c ; bisect the arc, a e, in f ; upon e, with e f for radius, describe the arc, c f d; join a and c, c and d, d and b; then a c d b will be the square required. 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 b, with a b for radius, describe the arcs, a c d and

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32 AMERICAN HOUSE-CARPENTER. feb; divide ac into as many equal parts as the polygon is to have sides, and extend those divisions from c towards d ; from the second point of division counting from c towards a, as 3, (Fig. 61,) 4, (Fig. 62,) and 5, (Fig. 63,) draw a line to b; take the distance from said point of division to a, and set it from b to e; join e and a; upon the intersection, o, with the radius, o a, describe the circle, a f d 6; then radiating lines, drawn from b through the even numbers on the arc, a d, will cut the circle at the several angles of the required figure. In the hexagon, (Fig. 62,) the divisions on the arc, a d, are not necessary ; for the point, 0, is at the intersection of the arcs, a d and f b, the points, f and d, are determined by the intersec- tion of those arcs with the circle, and the points above, g and h, can 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 intersection, o, comes above the arcs ; in such case, therefore, the lines, a e and b 5, (Fig. 63,) have to be extended before they will intersect. a a ann Fig. 64. 99.— To construct a triangle whose sides shall be severally equal to three given lines. Let a, 6 and c, (Fig. 64,) be the given lines. Draw the line, d e, and make it equal to c ; upon e, with b for radius, describe an arc atf; upon d, with a for radius, describe an arc intersecting the other at f; join d and f, also f and e ; then d f e will be the triangle required. Fig. 65.

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Res PRACTICAL GEOMETRY. 33 100.— To construct a figure equal to a given, right-lined figure. Let a b c-d, (Fig. 65,) be the given figure. Make ef, (Fig. 66,) equal to c d ; upon f, with d a for radius, describe an arc at g; upon e, with ¢ a for radius, describe an arc intersécting the other at g ; join g and e ; upon f and g, with d b and a 6 for radius, describe arcs intersecting at h; join g and h, also k and f ; then Fig. 66 will every way equal Fig. 65. So, right-lined figures of any number of sides may be copied, oy first dividing them into triangles, and then proteins 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. 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 bc; from b and c, draw b f and 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 æ, 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.

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34 AMERICAN HOUSE-CARPENTER. 102.—A parallelogram being given, to construct another equal to it, and having a side equal to a given line. 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.) 103.— To make a square equal to two or more given squares. Let A and B, (Fig. 69,) be two givén squares. Place them so 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 — b Fig. 70. the side of a given square, c a, placed at right angles to a b, be the side of another given square, and c d, placed at right angles to

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PRACTICAL GEOMETRY. 35 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 the hypothenuse. Rule.—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 EE is the other of the sides that contain the right angle. Procee as directed by the rule: 17 9 17 9 119 81 = square of 9. 17 289 = square of 17. 289 = square of 17. 370 Product. 1 ) 370 ( 19:235 + = square-root of 370 ; equal 19 feet, 2% in. 1 1 nearly: which would be the required — — length of the rafter. 382) - -900 3 764 3843 ) 13600 3 11529 38465): 207100 192325 (By reference to the table of square-roots in the appendix, the root ot almost any number may. be found ready calculated.)

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36 AMERICAN HOUSE-CARPENTER. Again: suppose it be required, in a frame building, to find 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— : 3 3 E = 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 4ths. full. | (2.)—The hypothenuse and one side being known, to find the other side. Kule.—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. 69 Product: the square-root of which is 8 ‘3066 + feet, or 8 feet, 3 inches and *ths. 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, 11 inches and #,ths.—the answer, as required.

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R -© PRACTICAL GEOMETRY. 37 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 length, 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, 13% nearly, will be the length of the rafter in feet; viz, 13 feet and 7ths of a foot. If the dimensions are large, as 30 feet and 20, take the half 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 ‘acctrate 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 rhaps several inches difference in the length of the rafter. For shorter dimensions, however, the result will be true enough. n 4 m — Fig. 71 104.— To make a circle equal to two given circies. Let A and B, (Fig. 71,) be the given citeles. In the right-angled tri- angle, a b c, make a b equal to the diameter of the circle, B, and c b equal to the diameter ofthe circle, A ; then the hypothenuse, LN Fig. 72.

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38 AMERICAN HOUSE-CARPENTER. 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 hype- ar ° thenuse of a right-angled triangle, will be equal to two simi figures,” as B and C, formed on the two legs of the triangle. 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 b c equal tob e; bisect a c in f, and upon f, with the radius, f a, describe the semi-circle, a gc; 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. 105, a.—Another method. Let A, (Fig. 74,) be the given rectangle. Extend the side, a b, and make a d equal to a c; * Similar figures are such as have their several angles respectively equal, and their sides respectively proportionate.

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PRACTICAL GEOMETRY. 39 bisect a d in e; upon e, with the radius, e a, describe the semi- circle, a f d ; extend g b till it cuts 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 proceed as directed at Art. 105. A B a a 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 c and b ; from d, draw d e, parallel to c 6 ; then a e will be the third proportional re- quired. That is, a e bears the same proportion to B, as B does to A. | e 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 6; from e, draw e f, parallel to c b; then a f will be the fourth proportional required. That is, a f bears the same proportion to C, as B does to A.

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40 AMERICAN HOUSE-CARPENTER. To 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- gis, can be found ; so that it will bear the same ae to the longer axis, as the shorter of the given ellipsis does to its longer. (See also, Art. 126.) A . B a 1 2 73 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 a c 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. 243 and 244.) So of a pilaster, architrave, &c. A > d ] a o (29 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 equal to A, and b c equal to B ; bisect a cine; upon e, with e a for radius, describe the semi-cirele, a d

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PRACTICAL GEOMETRY. 41 c ; at b, erect © d, at right angles to a c; then b d will be the mean proportional between A and B. For an application of this problem, see Art. 105. CONIC 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 isoceles triangle ; (as a b c, Fig.79;) and the base will be a semi-circle. If a cone be cut by a plane in the direction, e f, the section will be an ellipsis ; if in the direction, m l, the section will be a para- bola ; 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, a b c, then— 112.— To find the ares of the ellipsis, bisect e f, (Fig. 79,) in g ; through g, draw h i, parallel to a b ; bisect ¿in j ; upon j, with j h for radius, describe the semi-circle, A k i; from g, draw g k, at right angles to hi; then twice g k will be the conjugate axis, and e f the transverse. 6

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42 AMERICAN HOUSE-CARPENTER. 113.— To find the axis and base of the parabola. Let m l, (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 Z m will be the axis and height, and m d.an ordinate and half the base; asat Fig. 92, 93. 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 Fig. 94. Fig. 80. 115.— The axes being given, to find the foci, and to describe an ellipsis with a string. Let a b, (Fig. 80,) and cd, be the given axes. Upon c, with ae orb e for radius, describe the arc, ff ; then fand f, the points at which the arc cuts the transverse . axis, will be the foci. At f and fplacetwo pins, and another at c ; tie a 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, cg a; it will then describe the required ellipsis. The lines, fg 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 use 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 notroper for the purpose: a linen, or flaxen cord is much better.

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PRACTICAL GEOMETRY. l 43 116.— The axes being given, to describe an ellipsis with a trammel. Let a b and 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, c bd; thenthe pencil at e will describe the required ellipse. A trammel may be constructed thus: take two straight strips of 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 Fug. 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 =e i 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 one another. Lay them so as to coincide with the axes of the proposed ellipse, having the angular point at the centre. Then, in a rod having a hole for the pencil at one end, place two brad-awls at the distances described at Art. 116. While the po is moved in the direction of the curve, keep the brad-awls ard 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 saa a may be made use of, in place of the straight- ges. An improved method of constructing the trammel, is as fol- lows: make the sides of the grooves bevilling grom the face of the stuff, or dove-tailing instead of square. Prepare two slips of wood, each about two inches long, which shall be of a shape to just fill the groove when slipped in at the end. ‘These, instead of

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44 AMERICAN HOUSE-CARPENTER. 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. 62. 117.— To describe an ellipsis by ordinates. Leta b and cd, (Fig. 82,) be given axes. With a e ore b for radius, de- scribe the quadrant, f g h; divide f 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 g ; take the distance, 1i, and place it at 1 Z, transfer 2 j to 2 m, and3 k to3 n; through the points, a, n, m, l and c, 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 Sette is mostly used in tracing face-moulds for stair hand- railing. kia 118.— To describe an ellipsis by intersection of lines. Let

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PRACTICAL GEOMETRY. 45 a b and c d, (Fig. 83,) be given axes. Through c, draw fg, parallel to a b; from a and b, draw a f and b g, at right angles to ab; divide fa, g b, aeand eb, each into a like number of equal parts, as at 1, 2, 3 and o, 0, 0; from 1, 2 and 3, draw lines toc; through o, o and o, draw lines from d, intersecting those drawn toc ; then a curve, traced through the points, i, i, ¢, will be that of an ellipsis. Where neither trammel nor string is at hand, this, perhaps, is the most 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 aad its conjugate be given. Thus, a 6 and cd, (Fig. 84,) are conjugate diameters: f g is drawn parallel to a b, instead of being at right angles to c d ; also, f a and g b are drawn parallel to c d, instead of being at right angles to a b. ) 119.7 describe an ellipsis by intersecting arcs. Let ab

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46 AMERICAN HOUSE-CARPENTER. and cd, (Fig. 85,) be given axes. Between one of the foci, f and f, and the centre, e, mark any number of points, at random, as 1,2 and 3; upon fand f, with b 1 for radius, describe arcs at g,g,g and g ; upon f and f, with a 1 for radius, describe arcs inter- secting the others at g, æ, g and g ; then these points of intersection will be in the curve of the ellipsis. T'he other points, h and i, are found in like manner, viz: h is found by taking b 2 for one radius, and a 2 for the other; ¿is found by taking b 3 for one radius, and a 3 for the other, always using the foci for centres. Then 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. . 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 int; bisect i cin o ; from o, draw og, at right angleggr to į c, meeting c d extended to g ; join i and g, cutting the trans- verse axis in r; make A j equal toh g,and h k equal to Ar; from j, through r and k, draw j m and j n ; also, from g, through k, draw g l; upon g and j, with g c for radius, describe the

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PRACTICAL GEOMETRY. 47 arcs, l and mn; upon rand k, with r a for radius, describe the arcs, m iand / n ; this will complete 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 mm; and the intersection of e d with the transverse axis, will be the centre for describing the arc, m i, &c. As the elliptic curve is continually changing its course from that of a circle, a true ellipsis cannot be descrited with a pair of compasses. The above, therefore, is only an «approximation. 121.— To draw an oval in the proportion, seven by nine. Let cd, (Fig. 87,) be the given conjugate axis. Bisect cd ino, and through o, draw a b, at right angles to c d ; bisectc o ine ; upon 0, 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 / and f, draw g kand gl; upon g, with g c for radius, describe the are, kl; upon e, with e d for radius, describe the arc, j ¿i ; upon A and f, with A k for radius, describe the arcs, j Æ and /7; this will complete the figure. This is a very near approximation to an ellipsis; and perhaps no nfthod can be found, by which a well-shaped oval can be drawn with greater facility. By a little variation in the process, ovals of different proportions may be obtained. If quarter of 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.

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48 AMERICAN HOUSE-CARPENTER. 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, s r ; then s r will be the tangent required. 123.—An ellipsis with a tangent given, to detect the point of contact. Letag bf, (Fig. 89,) be the given ellipsifand tan- gent. Through the centre, e, draw a b, parallel to the tangent; any where between e and J, draw c d, parallel to ab ; bisected in 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 the diameter required.

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PRACTICAL GEOMETRY. _ 49 125.—Any diameter and its conjugate being given, to as- certain the two azes, and thence to describe the ellipsis. Let a band c d, (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 to e f ; make c g equal to a h or h b ; join g and Å; upon g, with g c for radius, describe the arc, è k cj; upon A, with the same radius, describe the arc, J n ; through the intersections, 2 and n, draw n o, cutting the tangent, e f, in o ; upon o, with o g for radius, describe the semi-circle, e ig f; join e and g, also g and f, cutting the arc, ¿ c j, ink and ¢; from e, through A, draw e m, also from f, through h, draw f p ; from k and ¢, draw kr and ¢ s, parallel to œ h, cutting e m in r, and f p in s; mak h m equal to hr, and A p equal to hs; then r m and s p will be the axes required, by which me ellipsis may be drawn in the usual way. 126.— To describe an ellipsis, whose axes shall be pror: tionate to the azes of a larger or smaller given ome. 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 ¢ e, parallel to a c; make o f equal to oe; then e f will be 7

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50 AMERICAN HOUSE-CARPENTER. the conjugate axis required, and will bear the same proportion to ij, as c d does to ab. (See Art. 108.) d 1l 33m3? l d Fig. 92. 127.— To describe a parabola by intersection of lines. Let 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 l, draw aa, parallel: to dd ; 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, 11, 22, &c., parallel to m l; from each point of division in d a, draw lines to 7; then a curve traced through the points of intersection, 0, 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 J @ equal to m l ; join a and d, and a and d; divide a d and ad, each into a like number of equal parts, as at 1, 2, 3, &c.; join 1 and 1, 2 and 2, &c., and the parabola will be completed.

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PRACTICAL GEOMETRY, 51 VP VA 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; from p, draw a p, parallel tor 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, 0, 0, &c., will be that of an hyperbola. The parabola and hyperbola afford handsome curves for various mouldings.

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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. c ò Fig. 95. - 130.—Definition. If a straight line, as a b, (Fig. 95,) stand Ron another straight line, as c d, so that the two angles made at a Tt

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PRACTICAL GEOMETRY. 53 the point, b, are equal—a b c toa 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, é&c. Fig. 96. 132.— Definition. The measure of an angie 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, e b a; and ad, of theangle, a b d. 133.—Cerollary. 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. @ e + tT [a i Fig. 97. T 135.—Aziom. If two straight lines, parallel to one another,

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54 AMERICAN HOUSE-CARPENTER. as a b andc d, (Fig. 97,) stand upon another straight line, as e f, the angles, a b f and c d f, are equal; and the angle, a b e, is 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 cd, then one of the an- gles, as a b c, is called an obtuse angle, and the other, as ab d, an acute angle. 137.—Aziom. '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 cb e, are together equal to two right angles. l 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 point in that line, or all the angles that can be made about a point, are together equal to four right angles. t Fig 98. 140.—Proposition. If to each of two equal angles a third angle be edded, their sums will be equal. Let abecandd ef, _ (Fig. 98,) be equal angles, and the angle, 4 j k, the one to be added. Make the angles, g b a and hed, each equal to the given angle, i 7 k ; then the angle, g b c, will be equal to the angle, h e f; for, if ab cand d efbe angles of 90 degrees, and i jk, 30, then the angles, g bc and h e f, will be each equal to 90 and 30 added, viz: 120 degrees.

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PRACTICAL GEOMETRY. 55 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, æ 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 equal to the triangle, e d f. b a 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 a c, are equal. Bisect the angle, (Art.

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56 AMERICAN HOUSE-CARPENTER. 77,) b a c, by the line, ad. Then the line, b a, being equal to 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 6 must be equal to the angle at c. a h b @ c w w e Fig. 101. 143.—Proposition. A diagonal crossing a parallelogram di- vides it into two equal triangles. Let a bcd, (Fig. 101,) bea given parallelogram, and 6 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. : c 6 a Fig. 102 144.—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 5 ; through the point, g, the intersection of the lines, b c and e f, draw À i, parallel to b d. In every paral- lelogram thus divided, the parallelogram, A, is equal to the paral- lelogram, B. According to Art. 143, the triangle, a be, 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

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PRACTICAL GEOMETRY. 57 in one must be equal to what remains in the other; therefore, the ‘parallelogram, A, is equal to the parallelogram, B. Fig. 103. 145.—Proposition, Parallelograms standing upon the same base and between the same parallels, are equal. Leta b cd and e f cd, (Fig. 103,) be given parallelograms, standing upon the same base, c d, and between the same parallels, a f and cd. 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 bf, and the angle, c 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 g 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 b e, parallel toa d; extend c dto e ; join a and e ; then the triangle, a e c, will be equal in area to the quadrangle, a bc d. Since the triangles, a d b and a d e, stand upon the sarne base, a d, and between the same paral- -8

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58 AMERICAN HOUSE-CARPENTER. Z d e Fig. 104. c lels, a d and be, 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 ec, is equal to the quadrangle, a b c d. . b Fig. 105. 148.—Proposition. If two straight lines cut each other, as' 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 right angles.: Leta b c, (Fig. 106,) be a given tri- angle, with its sides extended to f, e, and d, and the line, cg,

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PRACTICAL GEOMETRY. 59 e Fig. 106. drawn parallel tobe. As g cis parallel to e b, the angle, g c d, is, equal, (Art. 135,) to the angle, e bd; as the lines, fc and b e, cut one another at a, the opposite angles, f a e and b a c, are equal, (Art. 148 ;) as the angle, f a e, is equal, (Art. 135,) to the angle, a cg, the angle, a c 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 atc are equal, (Art. 137,) to two right angles, the three angles of the triangle, a 6 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. e 151.—Corollary. If one angle of a triangle be a right angle, and the two remaining angles are 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 angle is also 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.

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60 AMERICAN HOUSE-CARPERYER. e Fig. 107. 156.—Proposttion. If from the extremities of the diameter of a semi-circle, two straight lines be drawn to any point in the cir- cumference, the angle formed by them at that point will be a right angle. Leta bc, (Fig. 107,) be a given semicircle ; and ab and b c, lines drawn from the extremities of the diameter, a c, to the given point, 6 ; 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 dc, being radii of the same circle, are equal ;. the angle at a is therefore equal, (Art. 142,) to the angle, a b d, also, the angle at c is, for the same reason, equal to the angle, d b c; the angle, a bc, being equal to the angles at a and c taken together, must therefore, (Art. 153,) be a right angle. . Fig. 108. 157.—Proposition. 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,) bea given right-angled triangle, having a square formed on each of its sides: then, the square, b e, is equal to the squares, hc and g b, taken together. Thiscan be

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PRACTICAL GEOMETRY. 61 proved by showing that the parallelogram, b /, is equal to the square, g b ; and that the parallelogram, c l, is equal to the square, h e. The angle, c 6 d, is a right angle, and the angle, a 6 f, is aright angle ; add to each of these the angle, a b c ; then the angle, f b c, 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 same parallels, f b and gc, 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, 6 d and al, the parallelogram, b J, 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, g b, is equal to the parallelo- gram, b}, 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 by 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- btem. It is of great value in the exact sciences, more especially in Mensuration and Astronomy, in which many otherwise intri- cate calculations are by it made easy of solution. These demonstrations, which relate mostly to the problems pre- viously given, are introduced to ‘satisfy the learner in regard to their mathematical accuracy. By studying and thoroughly un- derstanding 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 sufficiently intelligible to be understood by the young mechanic.

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62 AMERICAN HOUSE-CARPENTER. The house-carpenter, especially, needs information of this kind, and were he thoroughly acquainted with the principles of geome- try, he would be much less liable to commit mistakes, and be better qualified to excel in the execution of his often difficult un- dertakings.

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SECTION I.—ARCHITECTURE. HISTORY OF ARCHITECTURE. 158.—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. 159.—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 style 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- a ? a é

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64 AMERICAN HOUSE-CARPENTER. morable event, which 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 temple of Belus was square at its base, each side measuring one furlong, 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. 160.—Historical accounts of ancient cities, of which there are now but few remains—such as Babylon, Palmyra and Ninevah of the Assyrians; Sidon, Tyre, Aradus and Serepta of the Phe- 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 eharacteristics of their works, are gigantic dimensions, immoveable solidity, and, in some instances, harmonious splendour. The extraordinary size of same 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 11} acres, and its height is nearly 500 feet. The stones of which it is built are immense—the smallest being full-thisetix feet long. ve *161.— Among the Greeks, architagiffe was cultivated as a fine ay

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ARCHITECTURE. 65 art, and rapidly advanced towards perfection. Dignity and 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 Leautiful 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. 162.—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- 9

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' 66 “AMERICAN HOUSE-CABPENTER. tained in Italy. Umder 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 building—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 adorned. “After Augus- tus, his successors continued to 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 architect st much of its elegance and dignity. The Romans, a. Of celebrated for their scientific knowledge as displayed in the construction of their various edifices, were Met capable of appr@iating 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 i" simplicity of the Grecian style, the artists of his time aimed a’

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ARCHITECTURE. 67 inventing new ones, and added to the already redundant 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 improve- 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. . 163.—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 inverted, columns-.etanding on their wrong ends, and other ridi- culous arrangements characterized their clumsy work. The vast - number of columns which the ruins around them afforded, they 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 are not of the classical styles, and which were ae

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68 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. 164.—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 Gorxic style; by some called the old Gothic to distinguish it from the pointed style, which is generally called modern Gothic. 165.—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 classic architecture which they found standing, and erected in their stead huge build- ings of stone which were greatly destituteof proportion, elegance or utility—their characteristics being scarcely any thing more thay stability and immensity combined with ornaments of a puerile cha- racter. Their churches were disfigured with rows of small 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. | : 166.—The Byzantine architects, from Byzantium, Constantino- ple, erected many spacious edifices; among which are included the cathedrals of Bamberg, Worms and Mentz, and the most an cient part of the minster at Strasburg ; in all of these they com-- bined the Reman-lonic order with the Gothic of the Lombards» `

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ARCHITECTURE. 69 This style is called the Lomparp-ByzanTINE. To the last style there were afterwards added cupolas similar to those used in the east, together with numerous slender pillars with tasteless capi- tals, and the many minarets which are the characteristics of the proper Byzantine, or Oriental style. 167.—In the eighth century, when the Arabs and Moors de- stroyed the ra ie 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 Moorrs# 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 seen in all its splendour in the ancient palace of the Mohammedan monarchs at Grenada, called the Alhambra, or red- house. The Mopern-Goruic 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 lancet 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. | '168.— 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

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70 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 13th 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 M@hcentury. 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. 169.—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. Thecathedral 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 toinspire 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 of Europe, the Gothic, or pointed style, was prevalent. The cathedral at Strasburg, designed by Irwin Steinbeck, was erected

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ARCHITECTURE. 71 in the 13th and 14th centuries.. In France and England during the 14th century, many very superior edifices were erected in this style. 170.—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 splen- did buildings wéte erected, which displayed a purer taste in the science. Among others, St. Peters of Rome, which was built about this time, is a lasting monument of the architectural skill of theage. 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 cou- -pled columns, the rounded pediment, the many curved-and-twisted enrichments, and the convex frieze, were unknown to pure Gre- cian architecture. Yet their efforts were serviceable in correcting, joa good degree, the very impure taste that had prevailed since the overthrow of the Roman empire. 171.—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 at the present time, the pure Grecian style is more or less studied ; and perhaps the sim- plicity of its principles is better adapted to a republican country, than the intricacy and extent of those of the Gothic. STYLES OF ARCHITECTURE. 172.—It is generally acknowledged that the various styles in architecture, were originated in accordance with the different pur-

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y2 AMERICAN HOUSE-CARPENTER. suits 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 kut was the offspring of their wants; while the shepherd,“who followed his. flocks and was compelled to traverse large tracts of country for pasture, found the ¢ent 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 Chi- nese 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-Gothiec, or pointed style, which was most generally confined to ecclesiastical structures, is said by some to have originated in an attempt $ imitate the bower, or grove of trees, in which the ancients per- formed their idol-worship. 173.—There are numerous styles, or orders, in architecture } and a knowledge of the peculiarities of each, is important to the student in the art. The Sryiosare is the substructure, or base- ment, upon which the columns of an order are arranged. In Roman architecture—especially in the interior of an edifice—it frequently occurs that each column has a separate substructure ; this is called a pedestal. If possible, the pedestal should be avoided in all cases; because it gives to the column the appear- ance of having been originally designed for a small building, and afterwards pieced-out to make it long enough for a larger one. 174..-An OrDeER, in architect is composed of two primei: pal parts, viz: the column and the entablature.

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ARCHITECTURE. 72 175.—The Cotumn is composed of the base, shaft and capital. 176.—The ENTABLATURE, above and supported by the columns, is horizontal ; and is composed of the architrave, frieze and cornice. ‘These principal parts are again divided into various members and mouldings. (See Sect. II.) 177.—The Base of a column is so called from basis, a founda- tion, or footing. 178.—The SsaFr, 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. 179.—The Capitat, from kephale or caput, the head, is the uppermost and crowning part of the column. 180.—The ArcHiITRAVE, from archi, chief or principal, and trahs, a beam, is that part of the entablature which lies in imme- diate connection with the column. 181.—The Frieze, from fibron, a fringe or border, is that part of the entablature which is immediately above the architrave and Wpeath the cornice. It was called by some of the ancients, zophorus, -because it was usually enriched with sculptured animals. 182.—The Cornice, from corona, to crown, is the upper and projecting part of the entablature—being also the uppermost and crowning part of the whole order. 183.—The Pepimenrt, above the entablature, is the triangu- lar 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. 184.—The Tympanvo is the perpendicular triangular surface which is enclosed by the cornice of the pediment. 185.—The Attic is a small order, consisting of pilasters and entablature, raised above a larger order, instead of a pedi- ment. An attic story is the upper story, its windows being usually square, 10

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74 AMERICAN HOUSE-CARPENTER. 186.—An order, in architecture, has its several parts and mem- bers proportioned to one another by a scale of 60 equal parts, which are called minutes. If the height of buildings were al- ways the same, the scale of equal parts would be a fixed quan- tity—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 num- ber of diameters which is directed for the order under considera- tion. 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, including column and entablature, is 8 diameters. Suppose now it were desirable to construct an exam- ple of this order, forty feet high. Then 40 feet divided by 8 gives 5 feet for the length of the scale; and this being divided b 60, the scale is completed. The upright columns of figures, marked Hand P, by the side of the drawings illustrating the ord 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 entablature. The figures represent minutes, or 60ths, of the major diameter of the shaft of the column. 187.—Grecian SryLes. The original method of building among the Greeks, was in what iscalled the Doric order: to this were afterwards added the Jonic and the Corinthian. These three were the only styles known among them. Each is distinguished from the other two, by not only a peculiarity of 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

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> ARCHITECTURE. 75 their general character, they are alike famous for grace and dig- nity, elegance and grandeur, to a high degree of perfection. 188.—The Doric OrpeR is so ancient that its origin is un- xnown—although some have pretended to have discovered it. But the most general opinion is, that it is an improvement upon the original log huts of the Grecians. 'These no doubt were very man and RR 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 perpendicu- lar 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 dithe roof in front, to sereen the entrance from the weather, gave origin to the portico. The peculiarities of the Doric order are the triglyphs—those parts of the frieze which have perpendicular channels cut in their surface; the absence of a base to the column—as also 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. 189.—The intercolumniation, or space between the columns, is regulated by placing the centres of the columns under the cen- tres of the triglyphs—except at the angle of the building ; where, as may be seen in Fig. 110, one edge of the triglyph must be over the centre of the column. Where the columns are so dis- posed that one of them stands beneath every other triglyph, the arrangement is called, mono-triglyph, and is most common.

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DORIC ORDER. 76 Ss : Ty ts ti ig ELLI] ` jee} be ra je -u o ae al = A ee aj 23| 37 g BR3SRA| O io a Land | PIRI | ® 171°] a “saqqjoureyp fS Pig. 110.

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ARCHITECTURE. - 77 $ When a column is placed beneath every third triglyph, the ar- rangement is called diastyle; and when beneath every fourth, aræostyle. This last style is the worst, and is seldom practised. 190.—The Doric order is suitable for buildings that are des- tined for national purposes, for banking-houses, &c. Its appear- ance, though massive and grand, is nevertheless rich and grace- ful. ‘The Custom-House and the Union Bank, in New-York city, are good specimens of this order. 191.—The Ionic OrnvER. 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 at- tained. ‘heir 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 proportions, to represent the character of their male deities rather than the female, there seems to have been a necessity for another style which should be 2mblematical of feminine graces, and with which they might Jecorate such temples as were dedicated to the goddesses. Hence re 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 possession of the Ionians, the order was called, the Ionic. i 192.—The distinguishing features of this order are the volutes, or spirals of the capital; and the dentils among the bed-mould- ings of the cornice: although in some instances, dentils are want- ing. -The volutes are said to have been designed as a represen- tation of curls of hair on the head of a matron, of whom the whole column is taken as a semblance. 193.—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 4 diameters, it is called, pycnostyle, or columns thick-set; when. `’ two diameters, systyle ; when two and a quarter diameters, eustyle ; when three diameters, diastyle ; and when more than

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78 IONIC. TAPT =<. some re 50 57 29 E a s| SelSee

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ARCHITECTURE. 79 three diameters, ar@ostyle, orcolumns thin-set. In all the orders, when there are four columns in one row, the arrangement is called, ¢etrastyle ; when there are six in a row, herastyle ; and when eight, octastyle. | 194.—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 frontof the Merchants’ Exchange, New-York city, is a good specimen of this order. c Fig. 1n.

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. $0 AMERICAN HOUSE-CARPENTER. NC | LLI m ALAALA] Fig. 113. 195.— To describe the Ionic volute. Draw a perpendicular from a to s, (Fig. 112,) and make a s equal to 20 min. or to + of the whole height, a c ; draw s o, at right angles to s a, and equal to 1} min. ; upon o, with 2, min. for radius, describe the eye of the volute; about o, the centre of the eye, draw the square, ré1 2, with sides equal to half the diameter of the eye, viz., 2, min., and divide it into 144 equal parts, as shown at Fig. 113. The several centres in rotation are at the angles formed by the heavy lines, as figured, 1, 2, 3, 4, 5, 6, &c. The position of these an- gles 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. 112;) 2 is thecentre for the arc, b c; and so onto the last. The inside spiral line is to be described from the centres, v, 7, z, &c., (Fig. 113,) being the centre of the first small square towards the middle of the eye frcm the centre for the outside arc. The breadth of the fillet at a j, is to be made equal to 2;3; min. This is for a spiral of three tevolutions; but one of any number of revolutions, as 4 or 6,

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ARCHITECTURE. 81 may be drawn, by dividing o f, (Fig. 113,) into a corresponding number of equal parts. Then divide the part nearest the centre, o, into two parts, as ath; join o and 1, also oand 2; draw A 3, pa- rallel to o 1, and / 4, parallel to o 2; then the lines, o 1,0 2, h 3, h 4, will determine the length of the heavy lines, and the place of the centres. (See Art. 396.) 196.—The CorinTH1IAN ORDER 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 generally supposed to have had its origin in the capitals of the columns of Egyptian temples ; which, though not approaching it in elegance, have yet a similari- ty of form with the Corinthian. The oft-repeated story of its origin which is told by Vitruvius—an architect who flourished in Rome, in the days of Augustus Casar—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 foliage ; some of which, reaching the tile, turned gracefully over in the form of a volute. A celebrated sculptor, Calima chus, saw the basket thus decorated, and from the hint which it sug- gested, conceived and constructed a capital for a column. ‘This was called Corinthian from the fact that it "was invented and first made use of Pe at Corinth. 197.—The Corinthian being the gayest, the richest and most lovely of all the @rders, it is appropriate for edifices which are 11

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PETF] | 6 gssagls gis x lsi pera H E

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ARCHITECTURE. 83 dedicated to amusement, banqueting and festivity—for all places where delicacy, gayety and splendour are desirable. 198.—In addition to the three regular orders of architecture, it was sometimes customary among the Greeks—and afterwards — among other nations—to employ representations of the human form, instead of columns, to support entablatures ; these were . called Persians and Caryatides. 199.—PERsIANs are statues of men, and are so called in com- memoration of a victory gained over the Persians by Pausanias. 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. 200.—CaRYATIDEs 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 van- quished, 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 Corinthian 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 pro- portioned to a statue in like manner as toa column of the same height. . 201.—These semblances of slavery have been in frequent use among moderns as well as ancients; and as a telief 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. 202.—Roman Stvrues. 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

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84 AMERICAN HOUSE-CARPENTER. and merit, which were erected by architects from Etruria ; but Rome was principally indebted to Greece for what she acquired of the art. Although there is no such thing as an architecture of Roman invention, yet no nation, perhaps, ever was so devoted to the cultivation of the art as the Roman. 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. Aqueducts 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 suc- cession 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 splendid; indeed, the exuberance of the ornaments alone was offensive to good taste. So overloaded with enrichments were the baths of Diocletian, that on an occasion of public festivity, great quantities of sculp- ture fell from the ceilings and entablatures, killing many of the people. 203.—The three orders of Greece were introduced into Rome in all the richness and elegance of their perfection. But the luxu- rious Romans, not satisfied with the simple elegance of their re- fined proportions, sought to improve upon them by lavish displays of ornament. They transformed in many instances, the true ele- gance of the Grecian art into a gaudy splendour, better suited to their less refined taste. 'The Romans remodelled each of the orders: the Doric was modified by increasing the height of the column to 8 diameters; by changing the echinus of the capital for an ovolo, or quarter-round, and adding an astragal and neck ~

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ARCHITECTURE. 85 below it; by placing the centre of the first triglyph, instead of one edge, over the centre of the column; and intro@ucing hori- zontal instead of inclined mutules in the cornice. The Ionic was modified by diminishing the size of the volutes, and, in some specimens, introducing a new capital in which the volutes were diagonally arranged. This new capital has been termed modern Ionic. The favorite order at Rome and her colonies was the Co- rinthian. The Roman artists, in their search for novelty, sub- jected it to many alterations—especially in the foliage of its capi- tal. Into the upper part of this, they introduced the modified Ionic capital; thus combining the two inone. This change was dignified with the importance of an order, and received the ap- pellation CompostTE, or Roman: the best specimen of which is found in the Arch of Titus. This style was not much used among the Romans themselves, and is but slightly appreciated now. Its decorations are too profuse—a standing monument of the luxury of the age in which it was invented. 204.—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 Ita'y 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 imitation of that style: this is very probable, since his- tory informs us that the Etruscans held intercourse with the Greeks at a remote period. The rudeness of this order prevented its extensive use in Italy. All that is known concerning it is from Vitruvius—no remains of buildings in this style being found among ancient ruins. 205. For mills, factories, markets, barns, stables, &c., where utility and strength are of more importance than beauty, the im- prove! modification of this order, called the modern Tuscan, (Fig. 116,) will be useful; and its simplicity recommends it where economy is desirable. 206.—Ecyrtian STYLE. The architecture of the ancient

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TUSCAN. IG 9 Fig. 116.

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ARCHITECTURE. rf Egyptians—to which that of the ancient Hindoos bears some re- semblance—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 long 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 short 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 14 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 of 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 blocks are 40

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EGYPTIAN.

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ARCHITECTURE. 89 feet long, and polished with emery to a surprising degree. It is conjectured that the stone for these pyramids was brought, by rafts and canals, from a distance of 6 or 7 hundred miles. 207.—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. 117 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. 208.—That style of architecture is to he preferred in which 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. 209.—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 12

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90 AMERICAN HOUSE-CARPENTER. English cottage-style is rather more appropriate, and is becom- ing extensively practised for small buildings in the country., 210.—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 for achurch, 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. 211.—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 site, 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 ‘heatly trimmed, and the grounds well cul- tivated. 212.—Huropeans express surprise that so many houses in thigay 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 build with a more durable material. Wooden cornices and gut- ters, attached to brick houses, are objectionable—not only on ac- count of their frail nature, but also because they render the build- ing liable to destruction by fire.

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92 AMERICAN HOUSE-CARPENTER. 213.—Dwelling houses are built of various dimensions and styles, according to their destination; and to give designs and di- rections for their erection, it is necessary to know their situation 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, &c.; nor would a house designed for the city, be appropriate for the country. For city houses, arrangements that would be convenient for one fa- mily, might be very inconvenient for two or more. Fig.118, 119, 120 and 121, represent the ichnographical projection, or ground- plan, of the floors of an ordinary city house, designed to be occupied by one family only. Fig. 122 is an elevation, or front-view, of the same house: all these plans are drawn at the same scale— which is that at the bottom of Fig. 122. Fig. 118 is a plan of the basement. a is the dining-room. b—kitchen. c—wash-room. d, d, d,—-wash-troughs. e, e,—pantries with shelving. f—passage having shelves, drawers, &c., on one side, and clothes-hooks on the other. g—kitchen-dresser. h, i,—front and rear areas. Fig. 119—plan of the first-story. J, jy —parlours, : k—iibrary. l—portico. | Fig. 120—plan of the second-story. a—toilet and sitting room. b—principal bed-chamber. c—bath-room. d, d,—bed-chambers. e—passage with wardrobe and clothes-hooks,

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QUITS Is [ f | A ao Ro MANIE g Fig.

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94 AMERICAN HOUSKE-CARPENTER. Fig. 121—plan of the attic-story. f—nursery, 2, 2, 2, &,—bed-chambers, h, h, h, h, h,—wardrobes, i—pantry with shelves, j—step-ladder leading to roof. Fig. 122—front elevation. a—section, b—front, These are introduced to give some general ideas of the princi- ples to be followed in designing city houses. 'The width of city lots is ordinarily 25 feet, but asit has become a common practice to reduce this size, on account of the enhanced value of land, the plans here given are designed for a lot only 20 feet wide—the, or- dinary width of many buildings of this class. In placing the chimneys, make the parlours of equal size, and set the chimney- breast in the middle of the space between the sliding-door parti- tion and the front (and rear) walls. The basement chimney- breasts may be placed 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 base- ment 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 pos- sible, it may be placed a few inches towards that point from over the breast below. So in arranging those of the stories above, always make provision for the flues from below. 214.—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, let the winders of

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pa HN eoa AO — 17 ppe E O SS HES — — : — 7 i BE | - 95 °

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96 AMERICAN HOUSE-CARPENTER. the other stories be placed perpendicularly over and under them; and be careful to provide for head-room. 'T'o 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. T'o dispose the windows properly, the 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 room ; because 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 objects cannot be attained, an approximation to each must be attempted. The piers should in no case be less in width than the window open- ings, else the blinds or shutters when thrown open will interfere with one another; in general practice, it is well to make the out- side piers $ of the width of one of the middle piers. When this is desirable, deduct the amount of the three dpenings from the width of the front, and the remainder will be the amount of the width of all the piers ; divide this by 10, and the product will be 3 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. 215.—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 ideato 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 disproportioned work of his own hands, he was led to make gradual changes ; till his abode was rendered af 4

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ARCHITECTURE. 97 not only commodious and durable, but pleasant in its appearance; and building became a fine-art, having utility for its basis. 216.—In all designs for buildings of importance, utility, dura- bility and beauty, the first great principles of architecture, should be pre-eminent. In order that the edifice be useful, commodious and comfortable, the arrangement of the apartments should be such as to fit them for their several destinations; for public as- semblies, oratory, state, visitors, retiring, 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 import- ant, and each should have due attention; yet when both cannot be obtained, the latter should in most caseg give place to the for- mer. A building that is neither convenient nor regular, whatever other good qualities it may possess, will be sure of disappro- bation. 217.—The utmost importance should be attached to such ar- rangements as are calculated to promote health : among these, ven- tilation 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—especially for those rooms which are used for sleeping apartments. All the apartments should be so arranged as to sequre their being easily kept dry and clean. In dwellings, suitable apartments should be fitted up for bathing, with all the necessary apparatus for conveying the water. @ 218.—To insure stability in an edifice, it should be designed upon well-known geometrical principles: such as science has de- monstrated to be necessary and sufficient for firmness and dura- 13 ;

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98 AMERICAN HOUSE-CARPENTER. bility. Itis 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. 219.—The 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. ) 220.—The builder should be intimately acquainted with the principles upon which the essential, elementary parts of & build- ing are founded. A scientific knowledge of these will insure certaify and security, and enable the mechanic to erect 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

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ARCHITECTURE. 99 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. 221.—"In laying the FounpatTion 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. 222.—The simplest member in any building, though by no means an essential one to all, is the CoLumn, 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, ANNE

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100 AMERICAN HOUSE-CARPENTER. rather than of a cone. This figure is the joint result of twovcal- 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 en- 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. 223.—The Wa t, 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 of cheapness, have only one face of the stones squared, the inner half of the wall being completed with brick; so that they can,

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ARCHITECTURE. 101 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. Rubble 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.

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102 AMERICAN HOUSE-CARPENTER. 224.—The LINTEL, 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 vertical 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. l 225.—The Arcu 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 wedgeg—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 it is 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 ¢wo of the bricks, however, by the disposition of their mortar, cannot collective- 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

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ARCHITECTURE. 103 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 lateral 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 it is 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

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104 AMERICAN HOUSE-CARPENTER. described usually from two centres outside the arch. It isa strong arch for supporting vertical pressure. The rampant arch is one in which the two ends spring from unequal heights. The horse-shoe or Moorish arch 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 voussoirs, 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 to this 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 of 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 be strong 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-

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4 ARCHITECTURE. 105 rable that each of the piers should possess sufficient horizontal strength to resist the lateral pressure of the adjoining arches. 226.—The Vavtt 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 tothe 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 Gothic architecture. 227.—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 capable 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 14 |

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106 AMERICAN HOUSE-CARPENTER. rests upon acircular wall, which is supported, in its turn, by archeg 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 Art. 303.) : 228.—The Roor is the most common and cheap method of covering buildings, to protect them from rain and other effects of the weather. Itis 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.

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ARCHITECTURE. 107 The roof is supported by rafters, which abut on the walls on each side, like the extremities of an arch. 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 tie-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 other by the process called scarfing. Mortices, in-

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108 AMERICAN HOUSE-CARPENTER. tended to truss 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 orearthern 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.”—Ency. Am. (See Art. 285.)

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SECTION IIJ].—MOULDINGS, CORNICES, &c. MOULDINGS. 229.—A moulding is so called, because of its being of the same determinate shape along 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 aficient architecture, are eight in number ; and are known by the following names : C] Annulet, band, cincture, fillet, listel or square. Fig. 123. D Astragal or bead. Fig. JU. 0) Torus or tore. Fig. 125. (C Scotia, trochilus or mouth. Fig. 126. J Ovolo, quarter-round or echinus. Fig. 197

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110 AMERICAN HOUSE-CARPENTER. | | Cavetto, cove or hollow. _ Fig. 128. Cymatium, or cyma-recta. > Ogee. Inverted cymatium, or —_—,, Fig. 130. ~ 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 resemblea, when on the base of a column. 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. 230.—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, to

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MOULDINGS, CORNICES, &C. 111 separate the upper and lower torus, and to produce a pleasing variety and relief. The form of the bead, and that of the torus, is the same; 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 never 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 cyma- recta and cyma-reversa are combinations of the ovolo and cavetto. 231.—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 of 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. 232.—A profile is an assemblage of essential parts and mould- ings. That 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. 233.—To describe the Grecian torus and scotia. Join the extremities, a and b, (Fig. 131;) and from f, the given projection of the moulding, draw f o, at right angles to the fillets; from b, draw 6 h, at right angles to a b; bisect a b inc; join f and c, and upon c, with the radius, c f, describe the arc, f h, cutting b h inh; through c, draw d e, parallel with the fillets ; make dc and c 6, each equal to bh; then d e and a b will be conjugate diame-

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112 AMERICAN HOUSE-CARPENTER. ters of the required ellipse. 'To describe the curve by intersec- tion of lines, proceed as directed at Art. 118 and note ; by a trammel, see Art. 125; and to find the foci, in order to describe it with a string, see Art. 115. 234.— Fig. 132 to 139 exhibit various modifications of the Grecian ovolo, sometimes called echinus. Fig. 132 to 136 are

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MOULDINGS, CORNICES, &C. 113 Fig. 135. Fig. 137. Fig. 139. elliptical, a b and b c being given tangents to the curve; parallel to which, the semi-conjugate diameters, a d and d c, are drawn. In Fig. 132 and 133, 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 Art. 118. In Fig. 137, the curve is para- bolical, and is drawn according to Art. 127. In Fig. 138 and 139, 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 a c. 15

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114 AMERICAN HOUSE-CARPENTER. TAN Fig. 14L Fig. 140. 235.— To describe the Grecian cavetto, (Fig. 140 and 141,) having the height and projection given, see Art. 118. 236.— To describe the Grecian cyma-recta. When the pro- jection is more than the height, as at Fig. 142, make a b equal to the height, and divide a 6 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. 143,) at right angles to ab; complete the rectangle, a b c d ; divide this into 4 equal rectangles, and proceed according to Art. 118. = ` d Fig. 145. 237.— To describe the Grecian cyma-reversa. e When the

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MOULDINGS, CORNICES, &C. 115 projection is more than the height, as at Fig. 144, 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. 145,) at right angles to the fillet; make a d equal to the projection of the moulding: then proceed as directed for Fig. 142. 238.—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. 146 and 147, which re- semble the elliptical curve. The ovolo and cavetto are generally a quadrant, but often less. When they are less, as at Fig. 150, the centre is found thus: join the extremities, a and b, and bisect abinc; from c, andat right angles to a b, draw c d, cutting a level line drawn from a ind; then d will be the centre. This moulding projects less than its height. When the projection is more than the height, as at Fig. 152, extend the line from c until inn Fig. 146. Fig. 147. k NI . Fig. 148. Fig. 149.

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116 AMERICAN HOUSE-CARPENTER. CA Fig. 151. j Fig. 153. . Fig. 138. } j Fig. 156. Fig. 157.

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Š \ MOULDINGS, CORNICES, &C. 117 Fig. 158. Fig. 159. 7 A d Fig. 160. Fig. 161. 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. 147, 151, 153, 156, 157, 158 and 159. The centres for the curves at Fig. 160 and 161, are found thus : bisect the line, a b, at c ; upon a, cand b, successively, with a c or c b for radius, describe arcs intersecting at d and d ; then those intersections will be the centres. . 239.—Fig. 162 to 169 represent mouldings of modern inven- tion. They have been quite extensively and successfully used in inside finishing. Fig. 162 is appropriate for a bed-moulding under a low, projecting shelf, and is frequently used under man- tle-shelves. The tangent, i h, is found thus: bisect the line, a b, ` at c; antl:b cat d; from d, draw de, at right angles to e b; from: b, diaw b f, parallel toe d ; upon b, with b d for radius, describe the arc,-d-f ; divide this arc into 7 equal parts, and set oneof the parts from s, the limit of the projection, to o ; make o h equal to oe; from h, through c, draw the tangent, h i; divide b h, h c, ci and i a, each intoa like number of equal parts, and draw the in-

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118 AMERICAN HOUSE-CARPENTER. Fig. 162.

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MOULDINGS, CORNICES, &C. 119 Fig. 167. Fig. 168, Fig. 169 tersecting lines as directed at Art. 89. If a bolder form is desired, draw the tangent, è h, nearer horizontal, and describe an elliptic curve as shown in Fig. 131, 164, 175 and 176. Fig. 163 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. 162. In this case, however, where the moulding has so little projection

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120 AMERICAN HOUSE-CARPENTER. in comparison with its height, the point, e, being found as in the last figure, A s may be made equal to s e, instead of o e as in the last figure. Fig. 164 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 of the diagonal, e f,is taken at pleasure; and dc is parallel toa e. To find the length of dc, draw b h, 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 to bh.* , To draw the curve, see note to 47#.118. Fig. 165 to 169 are peculiarly distinct from ancient mouldings, being composed principally of straight lines; the few curves they possess are quite short and quick. Fig. 170. Fig. 171: 240.—Fig. 170 and 171 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. 131, 175 and 176, is new, and is important in this application. It is founded upon well-known mathematical principles, viz: All the parallelograms that may be circumscribed about an ellipsis are equal to one another, and consequently 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.

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MOULDINGS, CORNICES, &C. 121 diameter of the ante is divided into 20 equal parts, and the height and projection of the members, are regulated in accordance with those parts, as denoted under Hand P, height and projection. The projection is measured from the middle of the ante. These will be found appropriate for porticos, door-ways, mantle-pieces, door and window trimmings, &c. The height of the ante for mantle-pieces, should be from 5 to 6 diameters, having an entab- lature of from 2 to 24 diameters. This is a good proportion, it being similar to the Doric order. But for a portico these propor- tions are much too heavy ; an ante, 15 diameters high, and an en- tablature of 3 diameters, will have a better appearance. CORNICES. 241.— Fig. 172, 173 and 174, are designs for eave cornices, and Fig. 175 and 176, for stucco cornices for the inside finish of rooms. The projection of the uppermost member from the facia, is divided into 20 equal parts, and the various members are pro- portioned according to those parts, as figured under H and P. H. P. 14] 20 Fig. 172. 16

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AMERICAN HOUSE-CARPENTER. Fig. 17% Fig. 174

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124 AMERICAN HOUSE-CARPENTER. or os | é \ 50 60 e m rr i et! o 70 a b 1 &@ S84 e Fig. 177. 242.— To proportion an eave cornice in accordance with the height of the building. Draw the line, a c, (Fig. 177,) and make 6 c and b a, each equal to 18 inches; from b, draw b d, at right angles to a c, and equal in length to # of ac; bisect bd in e, and from a, through e, draw a f; upon a, with a c for radius, describe the arc, c f, and upon e, with e f for radius, describe the arc, f d ; divide the curve, d f c, into 7 equal parts, as at 10, 20, 30, &c., and from these points of division, draw lines to b c, pa- rallel tod b ; then the distance, 6 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 are between 30 and 40 into 10 equal parts, and from the fourth point from 30, draw a line to the base, 6 c, parallel with bd ; 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 18 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. 172,173 and 174.

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MOULDINGS, CORNICES, &C. 125 243.-—To proportion a cornice according to a smaller given one. Let the cornice at Fig. 178 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 be 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 6c; parallel with cd, draw lines from each projection of the given cornice to the line, a d; 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. 244.— To proportion a cornice according to a larger given one. Let A, (Fig. 179,) be the given cornice. Extenda o to b, and draw c d, at right angles to ab; 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, a d ; from d, draw d b, parallel to f e; 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

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196 AMERICAN HOUSE-CARPENTER. Fig. 19. d projection of the different members of the given cornice to a o, 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, f e; 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 oa for radius, describe the quadrant, a d ; join d and b ; from f, draw f e, parallel to d b ; theno 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. 245.— To find the angle-bracket for a cornice. Let A, (Fig. 180,) 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-

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MOULDINGS, CORNICES, &C. 127 with the wall, fd, draw the line, ab; make ec equal a f, 1 through c, draw c b, parallel with e d ; join d and b, and from : several angular points in B, draw ordinates to cut d b in 1, 2 13; at those points erect lines perpendicular to d b; from h, w h g, parallel tof a; take the ordinates, 1 0, 2 0, &c., at B, i transfer them to C, and the angle-bracket, C, will be defined. the same manner, the angle-bracket for an internal cornice, or angle-rib of a coved ceiling, or of groins, as at Fig. 181, can ound. '46.—A level crown moulding being given, to find the raking ulding and a level return at the top. Let A, (Fig. 182,) be given moulding, and A b the rake of the roof. Divide the ve of the given moulding into any number of parts, equal or qual, as at 1, 2, and 3 ; from these points, draw horizontal 38 to a perpendicular erected from c; at any convenient place the rake, as at B, draw a c, at right angles to A b ; also, from lraw the horizontal line, b a; place the thickness, d a, of the ulding at A, from 6 to a, and from a, draw the perpendicular „a e; from the points, 1, 2, 3, at A, draw lines to C, parallel 4b; make a1, a 2 and a 3, at B and at C, equal to a 1, &c., { ; through the points, 1, 2 and 3, at B, trace the curve—this | be the proper form for the raking moulding. From1, 2 and

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128 AMERICAN HOUSE-CARPENTER. Fig 182. 3, at C, drop perpendiculars to the corresponding ordinates fr 1, 2 and 3, at A; through the points of intersection, trace ` curve—this will be the proper form for the return at the top.

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t SECTION IV.—FRAMING. 247.—This subject is, to the carpenter, of the highest impor- Atxe; and deserves more attention and a larger place in a volume >F this kind, than is generally allotted to it. Something, indeed, Ras been said upon the geometrical principles, by which the seve- “al lines for the joints and the lengths of timber, may be ascer- ‘ined ; yet, besides this, there is much to be learned. For how- Sver precise or workmanlike the joints may be made, what will it avail, should the system of framing, from an erroneous position of its timbers, é&c., change its form, or become incapable of sus- ‘gaining even its own weight? Hence the necessity for a know- edge of the laws of pressure and the strength of timber. These yeing once understood, we can with confidence determine the best yosition and dimensions for the several timbers which compose a loor or a roof, a partition or a bridge. As systems of framing wre more or less exposed ‘to heavy weights and strains, and, in sase of failure, cause nof only a loss of labour and material, but frequently that of @ i. it is very important that the materials employed be of the proper quantity and quality to serve their des- tination. And, on the other hand, 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 17 _@ e

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130 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 consider 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. 248.—Laws oF Pressure. (1.) A heavy body always exerts a pressure equal to its own weight in a vertical direction. Example: Suppose an iron ball, weighing 160 lbs., be supported upon the top of a perpendicular post, (Fig. 196;) then the pressure exerted upon that post will be equal to the weight of the ball; viz., 100 lbs. (2.) But if two inclined posts, (Fig. 183) 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, and 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, W, (Fig. 183,) 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

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FRAMING. 131 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 supports, (as in the case of a roof supported by its rafters,) it becomes important, therefore, to know the exact amount of pressure anv certain weight is capable of exerting upon oblique supports. This can be ascertained by the following process. Let a b and b c, (Fig. 183,) represent the axes of two sticks of timber supporting the weight, W; and let the weight, W, be equal to 6 tons. Make the vertical line, b d, equal to 6 inches; from d, draw d f, parallel to a b, and de, parallel toc b; then the line, b e, will be found to be 34 inches long, which is equal to the number of tons that the weight, W, exerts upon the post, a b. The pressure upon the other post is represented by b f, which in this case is of the same length as b e. The posts being inclined at equal angles to the vertical line, b d, the pressure upon them is equal. Thus it will be found that the weight, which weighs only 6 tons, exerts a pressure of 7 tons; the amount being in- creased because of the oblique position of the supports. The lines, e b, b f, f d and d e, compose what is called the parallelo- gram of forces. The oblique strains exerted by any one force, therefore, may always be ascertained, by making b d equal, (upon any scale of equal parts,) to the number of lbs., cwts., or tons contained in the weight, W, and b e will then represent the num- ber of lbs., cwts., or tons with which the timber, a b, is pressed, and b f that exerted upon b c. Fig. 184

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132 AMERICAN HOUSE-CARPENTER. Correct ideas of the comparative pressure exerted upon timbers according to their position, will be readily formed by drawing various designs of framing, and estimating the several strains in accordance with these principles. In Fig. 184, the struts are framed into a third piece, and the weight suspended from that. The struts are placed at a different angle to show the diverse pressures. Thè length of the timber used as struts, does not alter the amount of the pressure. But it may be observed that long timbers are not so capable of resistance as short ones. Fig. 185. 249.—In Fig. 185, the weight, W, exerts a pressure on the ‘struts in the direction of their length ; their feet, n, n, have, there- fore, a tendency to move in the direction, n 0, and would so move, were they not opposed by a sufficient resistance from the blocks, Aand A. If apiece of each block be cut off at the horizontal line, a 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 number of tons (or pounds) with which the strut is pressed ; construct the parallelogram of forces

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FRAMING. 133: by drawing o e parallel to a n, and o f parallel to bn, then n f, (by the. same scale,) shows the number of tons (or pounds) pres- sure that is exerted by the strut in the direction, 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, however much one strut may be in- clined more than the other; and also, that the united vertical pressure of the two strutsis exactly equal to the weight, W.: (In this calculation, the weight of the timbers is not taken into con- sideration.) 250.—Suppose that the two struts, B and B, (Fig. 185,) 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 desira- ble to remove the horizontal pressure. This may be.done by uni- ting the feet of the rafters with a rope, iron rod, or piece of tim- ber, as in Fig. 186. This figure is similar to the truss of a roof. Fig. 186. 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 & $

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134 AMERICAN HOUSE-CARPENTER. rafter, as was shown at Fig. 185; but it can be more readily and as accurately measured, by drawing from f and e horizontal lines tothe vertical line, b d, meeting itin o and o ; then f o will be the horizontal thrust at B, and eoat 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 pres- sure is beneficial rather than otherwise, as a roof thus formed tends to steady the walls. i wW Fig. 187.- Fig. 188. 251.—-Fig. 187 and 188 exhibit methods of framing for sup- porting the equal weights, W and W. Suppose it be required to _ Measure and compare the strains produced on the pieces, A B and .4 C. Construct the parallelogram of forces, e b f d, ac- cording to Art. 248. Then b f will show the strain on A B, and b e the strain on A C. By comparing the figures, b d being equal in each, it will be seen that the strains in Fig. 187 are about three

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FRAMING. 135 times as great as those in Fig. 188: the position of the pieces, A Band A C, in Fig. 188, is therefore far preferable. This and the preceding examples exemplify, in a measure, the resolution of fogags ; viz., the finding of two or more forces, which, acting in differ Biadirections, shall exactly balance the pressure of any given single force. Thus, in Fig. 185, supposing the weight, W, to be the greatest force that the two timbers, in their present position, are capable of sustaining, then the weight, W, is the given force, and the timbers are the ¿wo forces just equal to the given force. * 252.—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 m the same manner. For example; let A and B, (Fig. 189,) be two pieces of timber, pressed in the direction of their length to- wards b—A by a force equal to 6 tons weight, and B*equal to 9. To find the direction and amount of pressure they would unitedly exert, draw the lines, b e and b f, in a line with the axes of the . timbers, and make b e equal to the pressure exerted by B, viz., 9; also make 6 f equal to the pressure on A, viz.,6, and complete the parallelogram of forces, eb fd; then b d, the diagonal of the

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136 AMERICAN HOUSE-CARPENTER. parallelogram, will be the direction, and its length will be the amount, of the united pressures of A and of B. ‘The line, b d, is termed the resultant of the twoforces, b fandbe. If A and Bareto be supported by one post, C, the best position in the direction of the diagonal, b d; and sufficiently. strong to support the united presswig# of A and of B. Fig. 290. 253.—Another example: let Fig. 190 represent a piece of framing commonly called a crane, which is used for hoisting heavy weights by means of the rope, Bb f, which passes over a pulley atb. This is similar to Fig. 187 and 188, yet it is mate- rially different. In those figures, the strain is in ene direction only, viz., from b to d; but in this there are two strains, from A to Band from Ato W.° The strain in the direction, A B, is evi- dently equal to that in the direction, A W. To ascertain the best position for the strut, A C, make b 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 required. Should the foot, C, of the strat be placed either higher or lower, the strain on A C 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 strut a trifle lower than where the diagonal, 6 d, would in- dicate, but never higher

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FRAMING. l 137 254.— Ties and Struts. Timbers in a state of tension are called ¢tes, 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. 191,) represent beams of timber supporting 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 ato b. 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 bearn be very little injured by it. But if, on the other hand, the cuts be made in the bottom edge of the timbers, from c tob, B will be seriously in- jured, while A will scarcely beaffected. 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 ina state of tension ; butif 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 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 com- pressed in the direction of their length. By this, it appears highly important that.pieces in a state of tension should be distinguished’ < m

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ad 138 AMERICAN HOUSE-CARPENTER. from such as are compressed, in order that the former may be pre- served continuous. A strut may be constructed of two or mor pieces; yet, where there are many joints, it will not resist com |. pression so firmly. a. 255.— To distinguish ties from struts. ,'Fhis may be done by the following rule. In Fig. 183, the timbers, a.b and b c, are the sustaining forces, and the weight, W, is the straining force ; and, if the support be removed, the straining force would move from the point of support, b, towards d. Let it be required to ascer- tain whether the sustaining forces are stretched or pressed by the straining force. Rule: upon the direction of the straining force, b d, as a diagonal, construct a parallelogram, e b f d, whose sides shall be parallel with the direction of the sustaining forces, a b and cb ; through the point, b, draw a line, parallel to the diag- onal, e f; 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 occupy if unre- sisted, are compressed, while those on the other side of the divi- ding line are stretched. In Fig. 183, the supports are both compressed, being on that side of the dividing line which the straining force would occupy if unresisted. In Fig. 187 and 188, in which A B and AC are the sustaining forces, A C is compressed, whereas A B is in a state of tension; A C being on that side of the line, hi, 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. 186, the foot of the rafter at A is sus- tained by two forces, the wall and the tie-beam, one perpendicular and the other horizontal: the direction of the straining force is indicated by the line,’ a. The dividing line, h i, ascertained by the rule, shows that the wall is pressed and the tie-beam stretched. 256.—Another example: let E A B F, (Fig. 192,) represent a gate, supported by hinges at A and Æ. In this case, the strain-

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FRAMING. 139 ing force is the weight of the materials, and the direction of course vertical. Ascertain the dividing line at the several points, G, B, I, J, Hand F. It will then appear that the force at G is sustained by A G and G E, and the dividing line shows that the former is stretched and the latter compressed. The force at H is supported by A H and H #—the former stretched and the latter compressed. The force at B is opposed by H B and A B, one pressed—the other stretched. The force at F'is sustained by G Fand F' E, G F being stretched and F' E pressed. By this it appears that A B isin a state of tension, and Æ F, of compres- sion; also, that A Hand G F are stretched, while B H and G E are compressed: which shows the necessity of having A H and G F, each in one whole length, while B Hand G E may be, as they are shown, each in two pieces. The force at Jis sus- tained by G' Jand J H, the former stretched and the latter com- pressed. ‘The piece, C D, is neither stretched nor pressed, and could be dispensed with if the joinings at Jand J could be made as effectually without it. In case A B should fail, then C D would be in a state of tension. 257.— The pressure of inclined beams. The centre of gravi- ty of a uniform prism or cylinder, is in its axis, at the middle of its length. In irregular bodies with plain sides, the centre of

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140 AMERICAN HOUSE-CARPENTER. gravity may be found by balancing them upon the edge of a prism 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. Fig. 193. An inclined post or strut, supporting some heavy pressure ap- plied at its upper.end, as at Fig. 186, 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. 193,) be a beam lean- ing against the wall, Bc, and supported at its foot by the abut- ment, A, in the beam, A c, and let o be the centre of gravity of the beam. Through o, draw the vertical line, b d, and from B, draw the horizontal line, B b, cutting b d in b; join band 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 tob A. The amount of pressure will be found thus: let b d, (by any scale of equal parts,) equal the number of tons, cwts., or pounds weight upon the beam, A B; draw d e, parallel to B b ; then b e, (by the same scale,) equals the pressure in the direc- tion, b A; and ed, the pressure against the wall at B—and also the horizontal thrust at A, as these are always equal in a construc- tion of this kind. Fig. 194 represents two equal beams, sup- ported 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 pre- cisely the position of the beam in Fig. 193. The horizontal

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FRAMING. 14] Fig. 194. pressures at B, being equal and opposite, balance one another; and their horizontal thrusts at the tie-beam are also equal. (See Art. 250—Fig. 186.) When the inclination of a roof, (Fig. 194,) is one-fourth of the span, or of a shed, (Fig. 193,) 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 dis- tributed uniformly over its surface. The inclination, in a rafter uniformly loaded, which will produce the least oblique pressure, (6 e, Fig. 193,) is 35 degrees and 16 minutes. Fig. 195. 258.—In shed, or lean-to roofs, as Fig. 193, the horizontal pressure will be entirely removed, if the bearings of the rafters, as A, B, (Fig. 195,) are made horizontal—provided, however, 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. . -259.—A beam laid horizontally, supported at each end and uniformly loaded, is subject to the greatest strain at the middle

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142 AMERICAN HOUSE-CARPENTER. of its length. The amount of pressure at that point is equal to half of the whole load sustained. The greatest strain coming upon the middle of such a beam, mortices, large knots and other defects, 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 that it will bear at or near the ends. (See Art. 282.) 260.—The resistance of timber. When the stress that a given load exerts in any particular direction, has been ascertain- ed, before the proper size of the timber can be determined for the resistance of that pressure, the strength of the kind of timber to be used must be known. The following rules for calculating the resistance of timber, are based upon the supposition that the tim- ber used be of what is called “merchantable” .quality—that is, strait-grained, seasoned, and free from large knots, splits, decay, Se. The strength of a piece of timber, is to be considered in ac- cordance with the direction in which the strain is applied upon

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FRAMING. 143 it. When itis compressed in the direction of its length, as in Fig. 196, its strength is termed the resistance to compression. When the force tends to pull it asunder in the direction of its length, (A, Fig. 197,) it is termed the resistance to tension. And when strained by a force tending to break it crosswise, as at Fig. 198, its strength is called the resistance to cross strains. 261.— Resistance to compression. When the height of a piece of timber exceeds about 10 times its diameter if round, or 10 times its thickness if rectangular, it will bend before crushing. The first of the following cases, therefore, refers to such posts as would be crushed if overloaded, and the other two to such as would bend before crushing. In estimating the strength of tim- ber for this kind of resistance, it is provided in the following rules that the pressure be exactly in aline with the axis of the Case 1.—To find the area of a post that will safely bear a given weight—when the height of the post is less than 10 times its least thickness. Rule.—Divide the given weight in pounds by 1000 for pine and 1400 for oak, and the quotient will be the least area of the post in inches. This rule requires that the area of the abutting surface be equal to the result: should there be, there- fore, a tenon on the end of the post, this quotient will be too small. Example.—What should be the least area of a pine post that will safely sustain 48,000 pounds? 48,000, divided by 1000, gives 48—the required area in inches. Such a post may be 6x8 inches, and will bear to be of any length within 10 times 6 inches, its least thickness. ; Case 2.—To find the area of a rectangular post that will safely bear a given weight—when its height is 10 times its least - thickness or more. ule.—Multiply the given weight or pres- sure in pounds by the square of the length in feet; and multi- ply this product by the decimal, 0015, for oak, ‘0021, for pitch pine and ‘0016 for white pine; then divide this product by the . breadth in inches, and the cube-root of the quotient will be the

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144 AMERICAN HOUSE-CARPENTER. thickness in inches. Lzxample.—What should be the thickness of a pine post, 8 feet high and 8 inches wide, in order to support a weight of 12 tons, or 26,880 pounds? ‘The square of the length is 64 feet; this, multiplied by the weight in pounds, gives 1,730,320; this product, multiplied by the decimal, -0016, gives 2768:512 ; and this again, divided by the breadth in inches, gives 346-064; by reference to the table of cube-roots in the appendix, the cube-root of this number will be found to be 7 inches large— which is the thickness required. 'The stiffest ss ee post is that in which the sides are as 10 to 6. Case 3.—To find the area of a round, or cylindrical, post, that will safely bear a given weight—when its height is 10 times its least diameter or more. Aule.—Multiply the given weight or pressure in pounds by 1:7, and the produet by ‘0015 for oak, -0021 for pitch pine and ‘0016 for white pine ; then multiply the square- root of this product by the height in feet, and the square-root of the last product will be the diameter required, in inches. Hzam- ple.—What should be the diameter of a cylindrical oak post, 8 feet high, in order to support a weight of 12 tons, or 26,880 pounds? ‘This weight in pounds, multiplied by 1-7, gives 45,696; and this, by ‘0015, gives 68:544 ; the square-root of this product is (by the table in the appendix) 8°28, nearly—which, multiplied by 8, gives 66°24; the square-root of this number is 8°14, nearly ; therefore, 8:14 inches is the diameter required. Experiments have shown that the pressure should never be more than 1000 pounds per square inch on a joint in yellow pine —when the end of the grain of one piece is pressed against the side of the grain of the other. 262.— Resistance to tension. A bar of oak of an inch square, pulled in the direction of its length, has been torn asunder by a weight of , - - - - 11,500 lbs. Of white pine - - - 11,000 Of pitch pine - s - 10,000

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FRAMING. 145 ‘Therefore, when the strain is applied in a line with the axis of the piece, the following rule must be observed. To find the area of a piece of timber to resist a given strain in the direction of its length. Rule.—Divide the given weight to be sustained, by the weight that will tear asunder a bar an inch square of the same kind of wood, (as above,) and the product will be the area in inches of a piece that will just sustain the given weight; but the area should be at least 4 times this, to safely sustain a constant load of the given weight. Hzample—What should be the area of a stick of pitch pine timber, which is re- quired to sustain safely a constant load of 60,000 pounds? 60,000, divided by 10,000, (as above,) gives 6, and this, multiplied by 4, give 24 inches——the answer. 263.—Resistance to cross strains. To find the scantling of a piece of timber to sustain a given weight, when such piece is supported at the ends in a horizontal position. _ Case 1.—When the breadth is given. Rule.—Multiply the square of the length in feet by the weight in pounds, and this product by the decimal, :009, for oak, ‘(011 for white pine and 016 for pitch pine ; divide the product bythe breadth in inches, and the cube-root of the quotient will be the depth required in inches. Ezxample.—What should be the depth of a beam of white pine, having a bearing of 24 feet and a breadth of 6 inches, in order to support 900 pounds? The square of 24is 576, and this, multiplied by 900, gives 518-400; and this again, by ‘011, gives 5702-400 ; this, divided by 6, gives 950°400; the cube-root of which is 9-83 inches—the depth required. Case 2.—When the depth is given. KRule.—Multiply the square of the length in feet by the weight in pounds, and multi- ply this product by the decimal, ‘009, for oak, -011 for white pine and ‘016 for pitch pine; divide the last product by the cube of the depth in inches, and the quotient will be the breadth in inches required. Example.—What should be the breadth of a beam of oak, having a bearing of 16 feet and a depth of 12 inches, in 19

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Ss, 146. ° AMERICAN HOUSE-CARPENTER. order to support a weight of 4000 pounds? The square of [6 is 256, which, multiplied by 4000, gives 1,024,000 ; this, multiplied by ‘009, gives 9216; and this again, divided by 1728, the cube of 12, gives 5} inches—which is the breadth required. Case 3.—When the breadth bears a certain proportion to the depth. When neither the breadth nor depth is given, it will be best to fix on some proportion which the breadth should have to the depth; for instance, suppose it be convenient to make the breadth to the depth as 0°6 is to 1, then the rule would become as follows: Rule.—Multiply the weight in pounds by the decimal, 009, for oak, ‘011 for white pine and ‘016 for pitch pine; divide the product by 0°6, and extract the square-root ; multiply this root by the length in feet, and extract the square-root a second time, which will be the depth in inches required. The breadth is equal to the depth multiplied by the decimal, 0°6. It is obvious that any other proportion of the breadth and depth may be ob- tained by merely changing the decimal, 0°6,in the rule. Eram- ple.—What should be the depth and breadth of a beam of pitch pine, having a proportion to one another as 0°6 to 1, and a bearing of 22 feet, in order to sustain a ton weight, or 2240 pounds? This, multiplied by ‘016, gives 35°84, which, divided by 0°6, gives 59°73; the square-root of this is 7:7, which, multiplied by 22, the length, gives 169:4; the square-root of this is 13—-which is the depth required. Then 13, multiplied by 0-6, gives 7'8 inches—the required breadth. -~ Case 4.—When the. beam is inclined, as A B, Fig. 193. Rule.—Multiply together the weight in pounds, the length of the beam in feet, the horizontal distance, A c, between the supports, in feet, and the decimal, ‘009, for oak, -011 for white pine, and ‘016 for pitch pine; divide this product by 0°6, and the fourth root of the quotient will give the depth in inches. The breadth is equal to the depth multiplied by the decimal, 0'6. Ezample.— What should be the size of an oak beam, the sides to bear a pro- portion to one another as 0°6 to 1, in order tosupport a ton weight

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FRAMING. . 4147 or 2240 pounds, the beam being inclined so that, its length being 20 feet, its horizontal distance between the points of support will be 16 feet? 2240, multiplied by 20, gives 44,800, which, multi- plied by 16, gives 716,800; and this again, by the decimal, -009, gives 6451°2; this last, divided by 0°6, gives 10,752, the fourth root of which is 10°18, nearly ; and this, multiplied by 0°6, gives 6'1; therefore, the size of the beam should be 10°18 inches by 6°1 inches, 264.— To ascertain the scantling of the stiffest beam that -can be cut from a cylinder. Letd ac b, (Fig. 199,) be the sec- ‘tion, and e the centre, of a given cylinder. Draw the diameter, a b; upon a and b, with the radius of the section, describe the arcs, 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. 265.—The greater the depth of a beam in proportion to the thickness, the greater the strength. But when the difference be- tween the depth and the breadth is great, the beam must be stayed, (as at Fig. 202,) to prevent its falling over and breaking sideways. Their shrinking is another objection to deep beams ; but where these evils can be remedied, the advantage of increas- ing the depth is considerable. The following rule is, to find the strongest form for a beam out of a given quantity of timber. Rule.—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

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148 a AMERICAN HOUSE-CARPENTER. 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 be3 inches. A beam 16 inches by 3 would bear twice as much as a square beam of the same area of section; which shows how im- portant it is to make beams deep and thin. In many old build- ings, and even in new ones, in country places, the very reverse of this has been practised ; the principal beams being oftener laid on the broad side than on the narrower one. 266.—Systems of Framing. Inthe various parts of framing known as floors, partitions, roofs, bridges, &c., each has a specific object; and, in all designs for such constructions, this object should be kept clearly in view; the various parts being so dis- posed 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 de- rangement by multiplied compressions, shrinkage, and, in conse- quence, highly increased oblique strains; by which its stability and durability are greatly lessened. FLOORS. 267.—Floors have been constructed in various ways, and are known as single-joisted, double, and framed. In a single- joisted floor, the timbers, or floor-joists, are disposed as is shown in Fig. 200, Where strength is the principal object, this manner ` of disposing the floor-joists is far preferable ; as experiments have proved that, with the same quantity of material, single-joisted floors are much stronger than either double or framed floors. To obtain the greatest strength, the joists should be thin and deep. 268.—To find the depth of a joist, the length of bearing and thickness being given, when the distance from centres is

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FRAMING. 149 12 inches. Rule.—Divide the square of the length in feet, by the breadth in inches; and the cube-root of the quotient, multi- plied by 2:2 for pine, or 2:3 for oak, will give the depth in inches. Example.—What should be the depth of floor-joists, having a bearing of 12 feet and a thickness of 3 inches, when said joists are of pine and placed 12 inches from centres? The square of 12 is 144, which, divided by 3, gives 48; the cube-root of this number is 3°63, which, multiplied by 2°2, gives 7-986 inches, the depth required ; or 8 inches will be found near enough for practice. . 269.—Where chimneys, flues, stairs, &e., occur to interrupt the bearing, the joists are framed into a piece, (b, Fig. 201,) called a trimmer. The beams, a, a, into which the trimmer is framed, are called ¢rimming-beams, trimming-joists, or car- rtage-beams. ‘They need ta be stronger than the common joists, in proportion to the number of beams, c, c, which they support. The trimmers have to be made strong enough to support half the weight which the joists, c, c, support, (the wall, or another trim- mer, at the other end supporting the other half,) and the carriage- + a É. Fig. 208. d

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150 AMERICAN HOUSE-CARPENTER. beams must each be strong enough to support half the weight which the trimmer supports. In calculating for the dimensions of floor-timbers, regard must be had to the fact that the weight which they generally support—such as persons of 150 pounds moving over the floor—exerts a much greater influence than equal weights at rest. When the trimmer, b, is not more dis- tant from the bearing, d, than is necessary for ordinary hearths, é&c., it will be sufficient to add } of an inch to the thickness of the carriage-beam for every joist, c, that is supported. Thus, if the thickness of c is 3 inches, and the number of joists supported be 6, add 6 eighths, or of an inch, making the carriage-beams 32 inches thick. It is generally the practice in dwellings to make the carriage-beam, in all situations, one inch thicker than the common joists. But it is well to have a rule for determining the size More accurately in extreme cases. . 270.—When the bearing exceeds 8 feet, there should be struts, as a and a, (Fig. 202,) well nailed between the joists. ‘These will prevent the turning or twisting of the floor-joists, and will greatly stiffen the floor. For, in the event of a heavy weight resting upon one of the joists, these struts will prevent that joist from settling below the others, to the injury of the plastering

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FRAMING. 15° Fig. 202. upon the underside. When the length of bearing is great, struts should be inserted at about every 4 feet. 271.—Single-joisted floors may be constructed for as great a length of bearing as timber of sufficient depth can be obtained ; but, in such cases, where perfect ceilings are desirable, either double or framed floors are considered necessary. Yet the ceil- ings under a single-joisted floor may be rendered more durable by cross-furring, as it is termed—which consists of nailing a series of narrow strips of board on the under edge of the beams and at right angles to them. To these, instead of the beams, the laths are nailed. The strips should be not over 2 inches wide—enough to join the laths upon is all that is wanted in width—and not more than 12 inches apart. It is necessary that all furring for plastering be narrow, in order that the mortar may have a sufii- cient clinch. When it is desirable to prevent the passage of sound, the open- ings 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 upon which mortar is laid to the depth of about 2 inches, leaving but about half an inch from its upper surface to the under side of the floor-plank. 272.—Double floors. A double floor consists, as at Fig. 203, of three tiers of joists or timbers; viz., bridging-joists, a, a, binding-joists, b, b, and ceiling-joists, cp c. The binding-joists. © ) @ `

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1 AMERICAN HOUSE-CARPENTER. are the principal support, and of course reach from wall to wall. The bridging-joists, which support the floor-plank, are laid upon the binding-joists, to which they are nailed; sometimes they are notched into the binding-joists, but they are sufficiently firm when well nailed. The ceiling-joists are notched into the under side of the binders, and nailed; they are the support of the lath and plastering. 273.—Binders are faid 6 feet apart. At this distance the fol- lowing rules will give the scantling. Case 1.—To find the depth of a binding-joist, the length and breadth being given. Rule.—Divide the square of the length in feet, by the breadth in inches ; and the cube-root of the quotient, multiplied by 3:42 for pine, or by 3:53 for oak, will give the depth in inches. Exrample.—What should be the depth of a binding- joist, having a length of 12 feet and a breadth of 6 inches, when the kind of timber is pine? The square of 12 is 144, which, di- vided by 6, gives 24; the cube-root of this is 2°88, which, multi- ‘plied by 3°42, gives 9°85, the depth in inches, Case 2.—To find the breadth, when the depth and length are given. Rule.—Divide the square of the length in feet, by the

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FRAMING. | 159 cube of the depth in inches ; and multiply the quotient by 40 for pine, or by 44 for oak, which will give the breadth in inches. Example. —What should be the breadth of a binding-joist, hav- ing a length of 12 feet and a depth of 10 inches, when the kind of wood is pine? The cube of 10 is 1000; the square of 12 is 144; this, divided by 1000, gives a quotient of ‘144; and this quotient; multiplied by 40, gives 5°76, the breadth in inches. 274.—Bridging-joists are laid from 12 to 20 inches apart. The scantling may be found by the rule at Art. 268. 275.—Ceiling-joists are generally placed 12 inches apart from centres. They are arranged to suit the length of the lath; this being, in most cases, 4 feet long. What is said at Art. 271, in regard to the width of furring for plastering, will apply to the thickness of ceiling-joists. ° To find the depth of a ceiling-joist, when the length of bearing and thickness are given. KRule.—Divide the length in feet by the cube-root of the breadth in inches; and multiply the quotient by 0°64 for pine, or by 0°67 for oak, which will give the depth in inches. zample.—What should be the depth of a ceiling-joist of pine, when the length of bearing is 6 feet and the thickness 2 inches? 'The length in feet, 6, divided by the cube-root of the breadth in inches, 1:26, gives a quotient of 4°76, which, being multiplied by the decimal, 0°64, gives 3 inches, the depth re- quired. When the thickness of a ceiling-joist is 2 inches, the depth in inches will be equal to half the length of bearing in feet. Thus, if the bearing is 6 feet, the depth will be 3 inches; bearing 8 feet, depth 4 inches, &c. 276.—Framed floors. When a good ceiling is required, and the distance of bearing is great, the binding-joists, instead of reaching from wall to wall, are framed into girders. These are heavy timbers, as d, (Fig. 204,) which reach from wall to wall, being the chief support of the floor. Such an arrangement is termed a framed floot. The binding, the bridging and the ceil- 20 `

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. 184 AMERICAN HOUSE-CARPENTER. Fig. 201. -. ing-joists in these, are the same as those in double floors just described. The distinctive feature of this kind of floor is the girder. 277.—Girders should be made as deep as the timber will allow: if their being increased in size should reduce the height of a story ` afew inches, it would be better than to have a house suffer from defective ceilings and insecure floors. In the following rules for the scantling of girders, they are supposed to be placed at 10 feet apart. Case 1.—To find the depth, when the breadth of the girder and the length of bearing are given. Rule.—Divide the square of the length in feet, by the breadth in inches; and the cube-root of the quotient, multiplied by 4:2 for pine, or by 4:3 for oak, will give the depth required in inches. Ezample.—What should be - the depth of a pine girder, having a length of 20 feet and a breadth of 13 inches? The square of 20 is 400, which, divided by 13, gives 30°77 ; the cube-root of this is 3:12, which, multiplied by 4:2, gives 13 inches, the depth required.

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FRAMING. 155 Case 2.—To find the breadth, when the length of bearing and depth are given. Rule.—Divide the square of the length in feet, by the cube of the depth in inches; and the quotient, multiplied by 74 for pine, or by 82 for oak, will give the breadth in inches. Example.—What should be the breadth of a pine girder, having a length of 18 feet and a depth of 14inches? The square of the length in feet, 324, divided by the cube of the depth in inches, 2744, gives '118; and this, multiplied by 74, gives 8°73 inches, the breadth required. 278.—When the breadth of agirder is more than about 12 inches, it is recommended to divide it by sawing from end to end, verti€ally through the middle, and then to bolt it together with the sawn sides outwards. ‘This is not to strengthen the girder, as some have supposed, but to reduce the size of the timber, in order that it may dry sooner. The operation affords also an op- portunity to examine the heart of the stick—a necessary precau- tion; 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 several 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 asare not exactly tight, nor yet far enough apart to permit the escape of moisture. i 279.— 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 operation has positively weak- ened the girder. . A girder may be strengthened by mechanical contrivance, when

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186 AMERICAN HOUSE-CARPENTER. Fig. 205, its depth is required to be greater than any one piece of timber will allow. Fig. 205 shows a very simple yet scientific method of doing this. The two pieces of which the girder 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 equal in depth, in order that the joint between them may be in the neutral, line. (See Arf, 254.) The thickness of the keys should be about half their breadth, and the amount of their united thick- nesses should be equal to a trifle over the depth and one-third of the depth of the girder. Instead of bolts or pins, jron hoops are sometimes used; and when they can be procured, they are far preferable. In this case, the girder is diminished at the ends, and the hoops driven from each end towards the middle. 280.—Beams may be spliced, if none of a sufficient length can be obtained, though not at or near the middle, if it can be avojded. (See Art. 259 and 332.) 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 evaporate. F'loor-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 une- qual settlements will derange the level of the floor. As all floors, however well-constructed, settle in some degree, it is advisable to

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FRAMING. 137 frame the joists a little higher at the middle of the room than at its sides,—as also the ceiling-joists and cross-furring, when either are used. In single-joisted floors, for the same reason, the rounded edge of the stick, if it have one, should be placed up- permost. 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 shrink again ; as itis the nature of wood to shrink after every planing however dry it may have been before. PARTITIONS. 281.—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 parti- tioning weighs about half a ton, a greater weight, when added to its customary load, such as furniture, storage, &c., than any ordinary floor is calculated to sustain. Hence the timbers bend, the ceilings and cornices crack, and the whole interior part of the house 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 settle, 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 ceiling is an important object in the con- struction of a partition. 282.—The joists in a partition should 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 object, are much better omitted. Fig. 206 represents a partition having a door in the middle. Its

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I 158 AMERICAN HOUSE-CARPENTER. MNU l ill ë i P = 5 hye = È Lat Í Fig. 207. construction is simple but effective. Fig. 207 shows the manner of constructing a partitidh having doors near theends. T'he truss is formed above the door-heads, and the lower parts are suspended from it. The posts, a and b, are halved, and nailed to the tie, c d, and the sill, ef. ‘The braces 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, the principal timbers may be 3x4 inches for a 20 feet span, 34x5 for 30 feet, and 4x6 for 40. The thickness of the filling-in stuff may be regulated according to what is said at Art. 271, in regard to the width of furring for plastering. The we

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` Ea FRAMING. 159 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 tg injure the stability of the plaster- ing ; for, asthe strength of th plastering depends, in a great mea- sure, upon its clinch, formed by pressing the mortar through the space between the laths, the narrower the surface, therefore, upon which the laths are nailed, the less will be the quantity of plas- tering unclinched, and hence its greater security from fractures. For this reason, the principal timbers of the partition should have their edges reduced, by chamfering off the corners. Fig. %8. 283.—When the principal timbers of a partition require to be lagge 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 hori- zontal pieces; in order, upon these and the principal timbers, to nail upright battens at the proper distances for lathing, as in Fig. 208. A partition thus constructed 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 con- versation 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 |

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` 4 180 AMERICAN HOUSE-QQRPENTER. the timbers may be ascertained, by applying the principles which regulate the laws of pressure and those of the resistance of tim- ber, as explained at the first part of this section. The following data, however, may assist in calculating the amount of pressure upon partitions: 284.—The weight of a square, (that is, a hundred square feet,) of partitioning may be estimated at from 1500 to 2000 Ibs.; a square of single-joisted flooring, at from 1200 to 2000 Ibs.: a square of framed flooring, at from 2700 to 4500 lbs.; and the weight of a square of deafening, (as described at the latter part of Art. 271,) at about 1500 lbs. When a floor is supported at two opposite extremities, and bya partition introduced midway, one-half of the weight of the whole floor will then be supported by the partition. As the settling of partitions 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 executed. ROOFS.” 285.—In ancient buildings, the Norman and the Gothic, the 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 rafters. But in modern buildings, the walls are so slightly built as to be incapable of resisting scarcely any oblique pressure ; anf 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 contri- bute to bind and steady them. 286.—In estimating the pressures upon any certain roof, for the purpose of ascertaining the proper sizes for the timbers, calcula- tion must be made for the pressure exerted by the wind, and, if * See also Art. 228.

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SPRAMING. - a ; : 16} in a cold climate, for the weight of snow, in addition to the weight of the materials of which the roof is composed. The force of wind may be calculated at 40 lbs. on a square foot. The weight of snow will be of course according to the depth it acquires. (See weight of materials, in Appendix.) Ina severe climate, roofs ought 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 theroof. The inclina- _ tion 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 inclination, and in estimating the weight of the various kinds of covering: MATERIAL. INCLINATION. WEIGHT UPON A SQUARE FOOT. Tin, - - Rise 1 inch to a foot. & to 14 lbs. Copper, - 8 as A ltol4 “ Lead, -_--- “ 2inches “ 4to7 «“ Zinc, = á “. g3 « 66 1} to 2. “ Short pine shingles, ae EE af 14, to 2} “ Long cypress shingles, “ 6 “ “ 4to5 « Slate, - - Le | a s 5to9 «“ The weight of the covering, as above estimated, is that of the material only, added to the weight of whatever is used to fix it to. the roof, such as nails, &c. ; what the material is laid on, such as plank, boards or lath, is not included. 287.—Fig. 209 to 212 give a general idea of the usual manner of constructing trusses for roofs: c, (Fig. 209,) is a comman

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a è FRAMING. 163 rafter; R is a principal rafter; Æ is a king-post; sis a strut; S, (Fig. 211,)is a straining-beam ; Q is a queen-post; T is a tie- beam; and P, P, (Fig. 212,) are purlins. [In constructing a roof of importance, the trusses should be plaċed not over 10 feet apart, the principal rafter supported by a strut at every purlin, the purlin notched on instead of being framed into the principal rafters, and the tie-beam supported at proper distances, according to the weight of the ceiling or whatever else it is required to support. 288.—The dimensions of the timbers may be found in accord- ance with the principles explained at the first part of this section ; but for general purposes, the following rules, deduced from the experience of practical builders and from scientific principles, may be found useful: these rules give the dimensions of the piece at its smallest part. 289.— To find the dimensions of a king-post. Rule.—Mul- tiply the length of the post in feet by the span in feet. Then multiply this product by the decimal, 0:12, for pine, or by 0°13 for oak, which will give the area of the king-post in inches; and divide this area by the breadth, and it will give the thickness ; or by the thickness for the breadth. Hzrample.—What should be the dimensions of a pine king-post, 8 feet long, fora roof having a span of 25 feet? 8 times 25 is 200; this, multiplied by the decimal, 0°12, gives 24 inches for the area; 4x6, therefore, would be a good size at the smallest part. 290.— To find the dimensions of a queen-post. Rule.—Mul- tiply the length in feet, of the queen-post or suspending-piece, by that part of the length of the tie-beam it supports, also in feet. This product, multiplied by the decimal, 0°27, for pine, or by 0°32 for oak, will give the area of the post in inches; and dividing this aa by the thickness will give the breadth. Hrample.— The queen-posts in Fig. 210 support each } of the tie-beam, which is 124 feet. To make them of pine, 6 feet long, what should be their dimensions? 12%, multiplied by 6, gives 76,

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164 AMERICAN HOG@SE-CARPENTER. which, multiplied by 0°27, gives 20:52; which indicates a size of about 4x54. 291.— To find the dimensions of a tie-beam, that is required to support a ceiling only. Rule.—Divide the length of the longest unsupported part by the cube-root of the breadth ; and the quotient, multiplied by 1:47 for pine, or by 1°52 for oak, will give the depth in inches. EH’rample.—The length of the longest un- supported part of the tie-beam in Fig. 210 is 123 feet. What should be the depth of the tie-beam, the breadth being 6 inches, and the kind of wood, pine? The cube-root of 6 is 1-82, and 124, divided by 1°82, gives a quotient of 6°956; this, multiplied by 1-47, gives 10°225. The size of the tie-beam, therefore, may be 6x10}. When there are rooms in the roof, the dimensions for the tie-beam can be found by the rule for girders, (Art. 277.) 292.—To find the dimensions of a principal rafter when there is a king-post in the middle. Rule.—Maultiply the square of the length of the rafter in feet, by the span in feet ; and divide the product by the cube of the thickness in inches. For pine, multiply the quotient by °096, which will give the depth in inches. Ezample.—What should be the depth of a rafter of pine, 22°36 feet long, and 6 inches thick, the roof having a span of 40 feet? The square of 22°36 is 500 nearly, this, multiplied by 40, gives 20000; and this, divided by 216, the cube of the thick- ness, gives 92:59; which, multiplied by -096, equals 8:888. The sizeof the rafter should, therefore, be 6x83. 293.— To find the dimensions of a principal rafter when two queen-posts are used instead of a king-post. Rule.—The same as the last, except that the decimal, 0°155, must be used instead of °096. Hzample.—What should be the dimensions of a principal rafter, having a length of 14 feet, (as in Fig. 210,) and a thickness of 6 inches, when the span of the roof is 38 feet and the wood is pine? The square of 14 is 196, which, multi- plied by 38, gives 7448; this, divided by 216, the cube of 6, gives

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FRAMING. 165 31:48, which, multiplied by 0:155, gives 5°34. The size of the rafter should, therefore, be 6x5}. 294.— To find the dimensions of a straining-beam. In or- der that this beam may be the strongest possible, its depth should be to its thickness as 10 is to7. Rule.—Multiply the square-root of the span in feet, by the length of the straining-beam in feet, and extract the square-root of the product. Multiply this root by 0:9 for pine, which will give the depth in inches To find the thickness, multiply the depth by the decimal, 0:7. Hrample.— What should be the dimensions of a pine straining-beam, 12 feet rg, fora span of 38 feet? The square-root of the span is 6'164, which, multiplied by 12, gives 73:968 ; the square-root of this is nearly 8°60, which, multiplied by 0:9, gives 7°74—the depth. This, multiplied by 0°7, gives 5-418—the thickness. 'Therefore, the beam should be 53x73, or 51x8. 295.— To find the dimensions of struts and braces. Rule.— Multiply the square-root of the length supported in feet, by the length of the brace or strut in feet; and the square-root of the product, multiplied by 0'8 for pine, will give the depth in inches; and the depth, multiplied by the decimal, 0-6, will give the thick- ness Ín inches. Ezample.—in Fig. 210, the part supported by the brace or strut, a, is equal to half the length of the principal rafter, or 7 feet; and the length of the brace is 6 feet: what should be the size of a pine brace? The square-root of 7 is 2°65, which, multiplied by 6, gives 15-9 ; the square-root of this is 3°99, which, multiplied by 0°8, gives 3:192—the depth. This, multi- plied by 0°6, gives 1:9152, the thickness. Therefore, the brace should be 2x3 inches. It is customary to make the principal rafters, tie-beam, posts and braces, all of the same thickness, that the whole truss may be of the same thickness throughout. 296.— To find the dimensions of purlins. Rule.—Multiply the cube of the length of the purlin in feet, by the distance the purlins are apart in feet; and the fourth root of the product for pine will give the depth in inches ; or multiply by 1°04 to obtain

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166 AMERICAN HOUSE-CARPENTER. the depth for oak; and the depth, multiplied by the decimal, .0°6, will give the thickness. Lzxample.—What should be the dimen- sions of pine purlins, 9 feet long and 6 feet apart? The cube of 9 is 729, which, multiplied by 6, gives 4374; the fourth root of this is 8°-13—the required depth. This, multiplied by 0:6, gives 4‘878—the thickness. A proper size for them would be about 5x8 inches. Purlins should be long enough to extend over two, three or more trusses, | 297.—To find the dimensions of common rafters. 'The fol- lowing rule is for slate roofs, having the rafters placed 12 inches apart. Shingle roofs may have rafters placed 2 feet apart. TI dimensions of rafters for other kinds of covering may be found by reference to the table at Art. 286, and the lawsof pressure at the first part of this section. Rule.—Divide the length of bearing in feet, by the cube-root of the breadth in inches; and the quotient, multiplied by 0°72 for pine, or 0-74 for oak, will give the depth in inches. Lizample.—What should be the depth of a pine rafter, 7 feet long and 2 inches thick? 7 feet, divided by 1:26, the cube- foot of 2, gives 5°55, which, multiplied by 0°72, gives nearly 4 inches—the depth required. 298.—If, instead of framing the principal rafters and straining- beam into the king und the queen posts, they be permitted to abut ‘against each other, and the king and the queen posts be made in halves, notched on and bolted, or strapped to each other and tq the tie-beam, much of the ill effects of shrinking in the heads of the king and the queen posts will be avoided. (See Art. 339 and 340.) Fig. 213.

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FRAMING. 167 299.— Fie. 213 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, 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, a b, is just half of a c, the curve to pass through the middle of the rib. The rib is composed of two series of abutting pieces, bolted together. These pieces should be as long as the dimen- sions of the timber will admit, in order that there may be but few joints. The suspending 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 construction needs, for ordinary roofs, no diagonal braces between the suspending pieces, but if extra strength is required the braces may be added. The best place for the suspending pieces 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 strain- ing-beam fora roof of the same size. The proportion of the depth to the thickness should be about as 10 is to 7. Fig. 214. 300.—Some writers have given designs for roofs similar to Fig. 214, having the tie-beam omitted for the accommodation of an arch in the ceiling. This and all similar designs are seriously objectionable, and should always be avoided ; as the small height gained by the omission of the tie-beam can never compensate for the powerful lateral strains, which are exerted by the oblique posi- tion of the supports, tending to separate the walls. Where an arch

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t68 AMERICAN HOUSE*CARPENTER. 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 entirely removed. Many a pub- lic building in this place and vicinity, has been. all but ruined by the settling of the roof, consequent upon a defective plan in the formation of the truss in this respect. It is very necessary, there- fore, that the horizontal tie-beam be used, except where the walls are made so strong and firm by abutments, or other support, as to prevent a possibility of their separating. ns Fig. 215. 301.— Fig. 215 is a method of obtaining the proper lengths and bevils for rafters in a hip-roof, a band b care walls at the angle of the building ; be is the seat of the hip-rafter and œ f of a jack or cripple rafter. Draw e h, at right angles to b e, and miake it equal to the rise of the roof; join b and h, and A b will be the length of the hip-rafter. Through e, draw d i, at right angles tob c; upon b, with the radius, b h, describe the arc, h 4, cutting d i ini; join b andi, and extend g f to meet b iin j ; then g j will

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FRAMING. : 169 be the length of thejack-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 fl, at right ‘angles to be ; make f k equal to fl by the arc, lk, or make g k equal to g j by the arc, j k; then the angle at 7 will be the top- devil of the jack-rafters, and the one at & will be the down-bevil.* 302.— To find the backing of the hip-rafter. At any con- venient place in b e, (Fig. 215,) as o, draw m n, at right -angles to be; from o, tangical to b h, describe a semi-circle, cutting b e in $; join m and e and n and s ; then these lines will form at s the proper angle for beviling the top of the hip-rafter. DOMES.t Fig. 217. _ * The lengths and bevils of rafters for roof-valleys can also be found by the above process t See also Art. 227 22

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170 AMERICAN HOUSE-CARPENTER. 303.—The most usual form for domes is that of the sphere, the base being circular. When the interior dome does not rise too high, a horizontal tie may be thrown across, by which any de gree of strength required may be obtained. Fig. 216 shows a section, and Fig. 217 the plan, of a dome of this kind, a b being the tie-beam in both. Two trusses of this kind, (Fig. 216,) pe rallel to each other, are to be placed one on each side of the qpen- ` 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. 286.) 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. 219. 304.— When the interior dome rises tou high to admit of a level

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FRAMING. 171 »-beam, the framing may be composed of a succession of ribs ganding upon a continuous circular curb of timber, as seen at “Zo. 218 and 219,—the latter being a plan and the former a sec- on. This curb must be well secured, as it serves in the place f a tie-beam to resist the lateral thrust of the ribs. In small Omes, these ribs may be easily cut from wide plank ; but, where V extensive structure is required, they must be built in two hicknesses so as to break joints, in the same manner as is descri- bed for a roof at Art. 299. They should be placed at about two eet apart at the base, and strutted as at a in Fig. 218. 305.—The scantling of each thickness of the rib may be as ‘ollows : For domes of 24 feet diameter, 1x8 inches. t 6 Æ a & 14x10 “ te ¢ 60 (23 2x13 (1s rT (13 90 6c 24 x13 te (14 ee 108 <3 3x13 66 306.—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. 220 ;) and for one having a tower, &c., above it, a curve approaching 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, Fug. 218,) the pressure will have a tendency to burst the dome outwards at about one-third of its height. Therefore, when this form is used-

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172 AMERICAN HOUSE-CARPENTER. in the construction of an extensive dome, an iron band should 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 less 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. ajihg J e a Fig. 220. 307.— To describe a cubic parabola. Let a b, (Fig. 220,) 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 144, g h12}- h i 10%, i j 94, and the balance, 83, to j a; divide b ¢ into 8 equal parts, and, from the points of division, draw lines parallel to a bs to meet perpendiculars from the several points of division in a by at the points, 0, 0, o, &c. Then a curve traced through these points will be the one required. 308.—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. 309.— To find the curves for the ribs of an elliptical dome. Let a bc d, (Fig. 221,) be the plan of a dome, and e f the seat

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FRAMING. 173 Fig. 221 of one of the ribs. Then take e f for the transverse aris and twice the rise, o g, of the dome for the conjugate, and describe, (according to Art. 115, 116, &c.,) the semi-ellipse, e g f, which will be the curve required for the rib, eg f. The other ribs are found in the same manner. Fig. 222. 310.— To find the shape of the covering for a spherical dome. Let A, (Fig. 222,) 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,

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174 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 c d, describe the arcs, od 0, 1e 1, 2 f 2, &c.; make d o equal to half the width of one of the boards, and draw o s, parallel to ac; 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 ¢ 4, 7 3, &c., parallel to a c; make d o, e 1, &c., on the lower side of a c, equal tod o, e 1, &c., on the upper side; trace a curve through the points, o, 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. : D Fig. 223. 311.— To find the curve of the boards when laid in horizon- tal courses. Let A B C, (Fig. 223,) 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, &c.; through 1 and 2, draw a line to meet the axis extended at a; then a will be the centre for describing the edges of the board, E. Through 3 and 2, draw 36 ; then 6 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, L o. If this method is taken to find

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FRAMING. 175 the centres for the boards at the base of the dome, they would occur so distant as to make it impracticable : the following method 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. 224,) a f H 8 4 c 4 Fig. 224. 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 fo large to be described easily, reduce it by finding intermediate points in the curve, as at Art. 86. H Fig. 225. 312.— To find the shape of the angle-rib for a polygonal dome. Let A G H, (Fig. 225,) be the plan of a given dome, and

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176 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 to the vertex, their shape may be found in a similar manner to that shown at Fig. 222. | BRIDGES. 313.—Various plans have been adopted for the construction of bridges, of which perhaps the following are the most useful. Fig. 226 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. 226. 314.— Fig. 227 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, it

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FRAMING. 177 Fig. 227. should not be attempted, as it is evident that such a system of framing is capable of a tremendous lateral thrust. 315.— Fig. 228 represents a wooden bridge in which a built-rib, (see Art. 299,) is 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 approaches 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 fect. |Spanin feet.{ Least rise in feet. oot, vee | eee eee | -erane ee CD g 120 7 280 24 140 8 300 28 160 10 320 32 180 11 390 39 200 12 380 47 220 14 400 33 240 17 260 20 The rise should never be made less than this, but in all cases 23

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178 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, proba- ~ bly, that of a dense crowd of people. This may be estimated at 120 pounds per square foot, and the framing and gravelled road- way at 180 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. Aule.—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, multiplied by the number of ribs; the quotient, multiplied by the decimal, 0:0011, will give the area of each rib in feet. When the road- way is only planked, use the decimal, 0°0007, instead of 0:0011. ÆErample.—What should be the area of the ribs for a bridge of 200 feet 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. 229. 316.—The above rule gives the area of a rib, that would be re- quisite to support the greatest possible uniform load. But in 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

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FRAMING. 179 bridge will be likely to encounter, may be estimated at from about 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 b, (Fig. 229.) 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 bv; then, by constructing the paral- lelogram of forces, e b f d, according to Art. 248, b f will show the pressure in the direction, b c. Then the scantling for the rib may be found by the following rule. Rule.—Multiply the pressure in pounds in the direction, b c, by the decimal, 0°0016, for white pine, 0:0021 for pitch pine, and 0-0015 for oak, and the product by the decimal representing the sine of the angle, g b h, to a radius of unity. Divide this pro- duct by the united breadth in inches of the several ribs, and the cube-root of the quotient, multiplied by the distance, b c, in feet, will give the depth of the rib. Erample.—Iin a bridge of 200 feet span, 15 feet rise, having 3 ribs each 24 inches thick, or 72 inches whole thickness, the pressure in the direction, 6 c, is found to be 166,000 lbs., and the sine of the angle, g b h, is 0°-1—what should be the depth of the rib for white pine? 166,000, mul- tiplied by 0:0016, gives 265:6, which, multiplied by 0'1, gives 26°56 ; this, divided by 72, gives 0:3689. Thecube-root of the last sum is 0°717 nearly, and the distance, b c, is 135 feet: then, 0:717, multiplied by 135, gives 962 inches, the depth required. By this, each rib will require to be 24x97 inches, in order to en- counter without injury the greatest possible load. 317.—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, thev may be constructed by bending the pieces to the proper curve,

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180 AMERICAN HOUSE-CARPENTER. and bolting them together. In this case, where timber of suffi. cient 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. 238.) To ascertain the greatest depth for the pieces which compose the rib, so that the process of bend- ing may not injure their elasticity, multiply the radius of curvature in feet by the decimal, 0-05, and the product will be the depth in inches. Ezxample.—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 ifiches. White pine or oak timber, 5 inches thick, would freely bend to the above curve ; and, if the required depth of such a rib be 20 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. 230. 318.—When the span is over 250 feet, a framed rib, formed as in Fig. 230, 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 notched 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 about 8 feet. The roadway

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FRAMING. 181 ‘should be supported by vertical standards bolted to the ribs at 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- nal 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. 319.—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. 315.) The rise of the roadway should 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 itis termed, should not be below high-water mark : and the bridge should be placed at right angles with the course of the current. 320._-The best material for the abutments and piers of a bridge, is stone ; and, if possible, stone should be procured for the purpose. The following rule is to determine the extent of the abutments, they being rectangular, and built with stone weighing {20 Ibs. to a cubic-foot. Rule.——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 multiply the root, so diminished, by

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182 AMERICAN HOUSE-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. Example.—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, A400, 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 whieh is 36 ; 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 iii the ep- tire thrust of the other. : 321.—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 sooner 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 to

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FRAMING. 183 alternate dryness and moisture, than when it is either continually wet or continually dry. It has been ascertained that the piles under London bridge, after having been driven about 600 years, were not materially decayed. These piles are chiefly of elm, and vholly immersed. 322.—Centres for stone bridges. Fig. 231 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. T'o 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 upon 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 point, the pressure is inconsiderable,

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184 AMERICAN HOUSE-CARPENTER. but gradually increases towards the crown. At an angle of 45 degrees, the pressure equals about one-quarter the weight of the stone; at 57 degrees, half the weight; and when a vertical line, as a b, (Fig. 232,) passing through the centre of gravity of Ks Fig. 232. the arch-stone, does not fall within its bed, c d, the pressure may be considered equal to the whole weight of the stone. This will be the case at about 60 degrees, when the depth of the stone is double its breadth. 'The direction of these pressures is consid- ered in a line with the radius of the curve. The weight upon a centre being known, the pressure may be estimated and the tim- ber calculated accordingly. But it must be remembered that the whole weight is never placed upon the framing at once—as seems to have been the idea had in view by the designers of some cen- tres. In building the arch, it should be commenced at each but- tress 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. 323.—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 intermediate supports, and without a continued horizontal tie at the base ; such a centre is shown at Fig. 233. In laying the stones from the base up to a and c, the pieces, b d and b d, act as ties to prevent any rising at b. ~ After this, while the stones are being laid from a and from c to b, they act as struts: the piece, f g, is added for

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FRAMING. 185 Fig. 233. additional-security. Upon this plan, with some variation to suit circumstances, centres may be constructed for any span usual in stone-bridge building. 324.—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 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. Ties should be introduced across, especially where many timbers meet; and as the centre is to serve but a temporary pur- pose, the whole should be designed with a view to employ the timber afterwards for other uses. For this reason, all unneces- sary cutting should be avoided. 325.—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 cen- tre should be lowered a trifle immediately after the key-stone- is laid, in order that the stones may take their bearing 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 24 gel

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186 AMERICAN HOUSE-CARPENTER. arch. Between every two trusses, diagonal braces should be in- troduced to prevent lateral motion. 326.—In order that thecentre may be easily lowered, the frames, or trusses, should be placed upon wedge-formed sills ; as is shown at d, (Fig. 233.) These are contrived so as to admit of the settling of the frame by driving the wedge, d, with a maul, or, in large centres, a piece of timber mounted as a battering-ram. The operation of lowering a centre should be very slowly performed, in order that the parts of the arch may take their bearing uni- formly. 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 Fig. 233. - 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 lower- ing a centre, by which nine men were killed. 327.—To give some idea of the manner of estimating the pressures, in order to select timber of the proper scantling, calcu- late the pressure of the arch-stones from ito b, (Fig. 233,) and suppose half this pressure concentrated at a, and acting in the direction, a f. Then, by reference to the laws of pressure and the resistance of timber at Art. 248, 260, &c., the scantlings of the several pieces composing the frame, b d a, may be computed. Again, calculate the pressure of that portion of the arch included between a and c, and consider half of it collected at b, and acting in a vertical direction; then the amount of pressure on the beams, b d and b d, may be found by reference to the first part of this section, as above. Add the pressure of that portion of the arch which is included between i and b to half the weight of the cen- tre, and consider this amount concentrated at d, and acting in a vertical direction; then, by constructing the parallelogram ot forces, the pressure upon dj may be ascertained. 328.—As a short rule for calculating the scantlings of the tim- bers, let every strut be sufficiently braced, so that it. will yield to ag eS -t = BET b <1. —_— 2 -E =

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FRAMING. 187 crushing before it will bend under the pressure—( Art. 261.) Then divide the pressure in pounds by 1000, and the quotient will be the area of the strut in inches. For example, let the pressure upon a strut, in the direction of its axis, be 60,000 Ibs. This, divided by 1000, gives 60, the area of the strut in inches; the size of the strut, therefore, might be 6x10. This rule is based upon experiments by which it has been ascertained, that 1000 pounds is the greatest load that can be trusted upon a square inch of timber, without more indentation than would be compatible with the stability of the framing. The area ascertained by the rule, therefore, must have reference to the actual amount of sur- face upon which the load bears ; and should the strut have a tenon on the end, the area of the shoulders, instead of a section of the whole piece, must be equal to the amount given by the rule. 329.—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. In a circular arch, the ‘oints tend toward the centre of the circle: in an elliptical arch, the joints may be found by the following process : RoN f Fig. 234. f 330.— To find the direction of the joints for an elliptical arch. A joint being wanted at a, (Fig. 234,) draw lines from that point to the foci, fand f; bisect the angle, f a f, with the line, a b ; then a b will be the direction of the joint. 331.— To find the direction of the joints for a parabolic arch. A joint being wanted at a, (Fig. 235,) draw a e, at right angles to the axis, e g ; make c g equal toce,and join aand g ; draw ah, at right angles toa g ; then ah will be the direction of the joint.

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188 AMERICAN HOUSE-CARPENTER. The direction of the joint from 6 is found in the same manner. The lines, a g.and b f, are tangents to the curve at those points respectively ; and any number of joints in the curve may be ob- tained, by first ascertaining the tangents, and then drawing lines at right angles to them. JOINTS. —— Fig. 236. 332.— Fig. 236 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 Fig. 237; or by placing ae ee Fig. 237. keys in the joints, as at the lower edge in the same figure. This process, however, weakens the beam in proportion to the depth of the indents. 333.— Fig. 238 shows a method of scarfing, or splicing, a tie- beam without bolts. The keys are to be of well-seasoned, hard

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FRAMING. 189 sen cee E, Fig. 238. 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. Fig. 239. 334.—Fig. 239 shows about as strong a splice, perhaps, as . can well be made. It isto be recommended for its simplicity ; as, on account of their 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 recommend them. 335.—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 com- bined, twice the depth of the beam. The 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. 240. 336.—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

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190 AMERICAN HOUSE-CARPENTER. wood in the upper part are compressed ; and therefore a simple butt joint at that place, (as in Fig. 240,) is far preferable to any other. In such case, an oblique joint is the 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 under the heads of the bolts, gives a great addition of strength. Scarfing 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. Fora temporary purpose, Fig. 236 would do very well; it would be improved, however, by having a piece bolted on all four sides. F%g. 237, and indeed each of the others, since they have no oblique joints, would resist compression well. 337.—In framing one beam into another for bearing purposes, such as a floor-beam into a trimmer, the best place to make the mortice in the trimmer, is in the neutral line, (see Art. 254,) which is in the middle of its depth. Some have thought that, as the fibres of the upper edge are compressed, a mortice migh® be made there, and the tenon be driven in tight enough to make the parts as capable of resisting the compression, 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 bottomedge. For the nearer this is placed to the upper edge, the greater is the liability for it to splinter off; if the joint is formed, therefore, as at Fig. 241, it will combine all the advantages that can be obtained. Double tenons are objec- tionable, because the piece framed into is needlessly weakened,

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FRAMING. 191 Fig. 241. and the tenons are seldom so accurately made as to bear equally. For this reason, unless the tusk at a 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. 338.—T hus it will be seen that framing weakens both pieces, more or less. It should, therefore, be avoided as much as possi- ble ; and where it is practicable one piece should rest upon the other, rather than be framed into it. This remark applies to the bridging-joists in a framed floor, to the purlins and jack-rafters of a roof, &c. Fig. 242. Fig. 243. Fig. 244. 339.—In a framed truss for a roof, bridge, partition, &c., the joints should be so constructed as to direct the pressures through the axes of the several pieces, and also to avoid every tendency of the parts to slide. To attain this object, the abutting 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. 242 for the foot of a rafter, (see Art. 257,) in Fig. 243 for the head of a rafter, and in Fig. 244 for the foot of a strut or brace. The joint at Fig. 242 is not cut completely across the tie-beam, but a narrow lip is left

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192 AMERICAN HOUSE-CARPENTER. standing in the middle, and a corresponding indent is made i the rafter, to prevent the parts from separating sideways. The abutting surface should be made as large as the attainment other necessary objects will admit. The iron strap is added to prevent the rafter from sliding out, should the end of the tie-beam, by decay or otherwise, splinter off In making the joint shown at Fig. 243, it should be left a little open at a, so as to bring the parts to a fair bearing at the settling of the truss, which mus necessarily take place from the shrinking of the king-post and fy 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. v go X i 340.—If the rafters and struts were made to abut end to end, as in Fig. 245, 246 and 247, and the king or queen post notched on in halves and bolted, the ill effects of shrinking would be avoided. ‘This method has been practised with success, in some of the most celebrated bridges and roofs in Europe; 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.

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FRAMING. 193 Fig. 248. Fig. 249. 341.—Fig. 248 isa 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 employed. The shrinking of the timber, if only to a small degree, permits the tie to withdraw—as is shown at Fig. 249. ‘The dotted line shows the position of the tie after it has shrunk. 342.—-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 used in preference wherever the strength of iron is 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 withoutthem. 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 ordi- nary 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 ṣẹ thick. | 15 ce 3 © l} c } «“ 20 te (13 2 4 4 iT In fastening astrap, 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, 25

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194 AMERICAN HOUSE-CARPENTER. 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 wMl give it a coating which dampness will not penetrate.

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SECTION V.—DOORS, WINDOWS, &c. DOORS. 343.—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. 'Thedimensions 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

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196 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@mo case be less than about 6 feet 3 inches; and generally not less than 6 feet 8 inches. 344.—The proportion between the width and height of single doors, for a dwelling, should be as 2 is to 5; 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 by 2; but, if the height is 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 ¢ ôf 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. . 345.— 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

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DOORS, WINDOWS, &c. 197 others, of the same width as the stiles. The moulding on the panel should be equal in width to 3 of the width of the stile. Fig. 250. 346.— Fig. 250 shows an approved method of trimming doors : a is the door stud ; b, the lath and plaster; c, the ground ; d, the jamb; e, the stop ; f and g, architrave casings ; and h, 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 as at 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. 347.—Fig. 251 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- rience, however, has proved that the omission of the lock rail is at the expense of the strength and durability of the door; a four-panel door, therefore, is the best that can be made. 348.—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.

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198 AMERICAN HOUSE-CARPENTER. Fig. 91. It may be observed, however, that a bed-room door should be hung so that, when half open, it will screen the bed; ahd a door leading from a hall, or passage, to a principal room, should screen the fire. WINDOWS. 349.—A 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

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DOORS, WINDOWS, &C. 199 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 product by the height in feet; then the square-root of this product will show the required number of square feet of glass. 350.—To ascertain the dimensions of window frames, add 4} inches to the width of the glass for their width, and 62 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—4} for the stiles of the sash, and 34 for hanging stiles— and the height between the stone sill and lintel is about 103 inches more than the height of the glass, it being varied according to the thickness of the sill of the frame. 351.—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 4 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, 3times 11 is 33, and 4} added for the wood of the sash, gives 37} 372 and‘1} is 39,

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2 & 200 AMERICAN HOUSE-CARPENTER. and 39, divided by 4, gives 93; to which add half an inch, and the result will be 10} inches, the width required for the box shutter. 352.—In disposing and proportioning windows for the walls of a building, the rules of architectural 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. 214.) 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. ce 66 principal -story t 2% & 6 6 second-story “ow 7; = ái third-story “ & Eog fourth-story “ 1} s i s 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 isto 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. 353.—The present style of finish requires the heads of win- dows in general to be horizontal, or sguare-headed ; yet, it is well to be possessed of information for trimming circular-headed win- dows, as repairs of these are occasionally needed. 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 stretch-out of the

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DOORS, WINDOWS, &C. | , 201 circle, (Art. 92 ;) but, when the jambs are placed obliquely to the face of ‘the wall, occasioned by the demand for light in an oblique direction, the forggof the soffit will be obtained as in the following article: and, when the face of the wall is circular, as in the succeeding one. Fig. 252. 354.—To find the form of the soffit for circular window- heads, when the light is received in an oblique direction. Let a bcd, (Fig. 252,) 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 a g, at right angles to a b ; obtain the stretch-out of e f a, and make e g equal to it; divide e g 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, c h, is found in the same manner. 355.— To find the form of the soffit for circular window- heads, when the face of the wall is curved. Letab cd, (Fig. 253,) 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. 26

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202 AMERICAN HOUSE-CARPENTER. Proceed as in the foregoing article to obtain the line, d g ; then 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, c b, (Fig. 252;) and, having drawn ordinates at right angles to c b, transfer them toe fa; in this way, a section at right angles to the jambs can be obtained.

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SECTION VI.—STAIRS. 356.—The strarrs is that mechanical arrangement in a build- ing by which access is obtained from one story to another. Their 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. 357.—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 of 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 ;

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204 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 tg the angle at which the flight rises. If it is 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 problem is constructed. 358.— To proportion the rise and tread to one another: Make the line, a b, (Fig. 254,) equal to 24 inches; from b, erect b c, at right angles to a b, and make 6 c equal to 12 inches; join & and c, and the triangle, a b c, will form a scale upon which t® graduate the sides of the pitch-board. For example, suppose æ 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 y 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 into 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

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STAIRS. 205 the whole height of a story into a certain number of rises, sup- pose the length of each is found to be 6 inches. Place this length from b to h, and draw h i, parallel to a b; then A 4, 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, Z k, from lto k; then! kb will be the proper proportion for the sides of a pitch-board for that run. i 359.—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 g 124 “ In second-class “ s i “ In third-class “ and cottages “ 9 « Step-ladders to ascend to scuttles, &c., should have from 10 to 11 inches run on the rake of the string. (See notes at Art. 103.) 360.—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 being very small, they are frequently executed at 24 feet. 361.— 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 of the floor in the lower story to the top of the floor in 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

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206 _ 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 160 inches. The proper rise for a stairs in a house of this class is about 6 inches. Then, 160 divided by 6, gives 263 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, } and y, of an inch. Then, if the space for the extension of the stairs is not limited, the tread can be found asat Art. 358. 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.’ Waving obtained this rise and tread, the pitch-board may be made in the follow- ing manner. Upon a piece of well-seasoned board about ý of an inch thick, having one edge jointed straight and square, lay the corner of a carpenters’-square, as shown at Fig. 255. Make ab FA Fig. 255. 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. ° 362.— To lay out the string. The space required for timber

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STAIRS. 207 c d e f i ESN j Fig. 256. 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 6, (Fig. 256.) Commencing at one end, lay the longest side of the pitch-board against the gauge-mark, a b, as 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. 957. 363.— Fig. 257 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. 364.—A platform stairs ascends from one story to another in two or more flights, having platforms between for resting and to change their direction. This kindof stairs is the most easily constructed, and is therefore the most common. The cylin-

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208 AMERICAN HOUSE-CARPEXTER. a Fig. TR. e der is generally of small diameter, in most cases about 6 inches. It may be worked out of one solid piece, buba better way is to glue together three pieces, as in Fig. 258; 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. 365.— To obtain the form of the lower edge of the cylinder. Find the stretch-out, d e, (Fig. 259,) 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 h g, parallel to d e, and make h f and g i, each equal to one rise; from i and f, draw 7 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, & A andi l; parallel to these, draw m n and o p, at the requisite distance for the dimen- sions of the string; from s, the centre of the plan, draws q, parallel tod f; divide A 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; join n and o, cutting q sinr; then the angles, u n r and r o t, 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 u z, at right angles tomn,; fromr, drawr zv, at right angles to n o; then z will be the centre for the curve, u r. The centre for the curve, r £, is found in the same manner.

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STAIRS. l | 209 366.—To find the position for the balusters. Place the centre of the first baluster, (b. Fig. 260,) 4 its diameter from the face of the riser, c d, and $ 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. 260.

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240 AMERICAN HOUSE-CARPENTER. - is, 7 Ap ait ; A] [J 367.—-To find the face-mould for a round hand-rail to plat- form stairs. Case 1.— When the cylinder is small. In. Fig. 261, 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 ¢ is the plan of a line passing through the centre of the rail around the cylinder. Through 7 and d, drawik, and through j and e, draw j k ; from k, draw kl, parallel to fe; from b,draw b m, parallel to g d; from l, draw l r, parallel to Æ j ; from n, draw n t, at right angles to 7 k ; on the line, o b, make o ¢ equal to n ¢ ; join c and ¢; on the line, j c, (Fig. 262,) make e c equal to e n at Fig. 261; from c, draw c é, at right angles to j c, and make cĉ Fig, 261. c

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Š BTAIRS: 213 equal to ¢ t at Fig. 261; through ¢, draw p J, parallel to j c, and make ¢ J equal toż / at Fig. 261 ; join l and c, and complete the parallelogram, e c l s ; find the points, o, 0, o, according to Art. 118; upon e, 0, e, 0, and /, 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 c l. 368.—Elucidation of the foregoing method. This excellent plan for obtaining tlie 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. 263; 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 c 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, which 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, ec, and level in the direction, c 0; suppose this plane, e ao ec,

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212 AMERICAN HOUSE-CARPENTER. Fig. 263. be revolved on e c as an axis, in the manner indicated by the arcs, o nand a zv, until it coincides with the plane, e r ¢ c; the line, a o, will then be represented by the line, z n ; then add the paral- lelogram, z r ¿ n, and the triangle, c £l, deducting the triangle, er s; and the edgesof the plane, e s l c, inclined inthedirection, ec, and also in the direction, c Z, will lie perpendicularly over the plane, e abf. From this we gather that the line, co, being at right angles to

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| STAIRS. : 213 ust, in order to reach the point, /, be lengthened the distance, ad the right angle, e c t, be made obtuse by the addition to he angle, ¿c ł. By reference to Fig. 261, it will be seen his lengthening is performed by forming the right-angled le, c o t, corresponding to the triangle, co ¢, in Fig. 263. ine, c ¢, is then transferred to Fig. 262, and placed at right 3to ec; this angle, e c t, being increased by adding the an- c l, corresponding to t c l, Fig. 263, the point, Z, is reached, 1e proper position and length of the lines, e c and c J ob- lL To obtain the face-mould for a rail over a cylindrical iole, the same process is necessary to be followed until the ngth and position of these lines are found ; then, by forming wallelogram, e c ls, and describing a quarter of an ellipse n, the proper form will be given. 2 Fig 2. —Case 2.— When the cylinder is large. Fig. 264 re-

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214 AMERICAN HOUSE-CARPENTER. presents a plan and a vertical section of a line passing through the centre of the rail as before. From 6, draw 6 k, parallel to cd ; ex- tend the lines, 7 d and j e, until they meet ķ b in k and f ; from n, draw n l, parallel too b ; through J, draw } ¢, parallel to 7 k ; from k, draw k t, at right angles to 7 k ; on the line, o b, make o ¢ equal tokt. Makee c, (Fig. 265.) equal to ¢ kat Fig, 264; from c, Si es i Fig. 265. draw ct, at right angles to e c, and equal toc ¿at Fig. 264; from t, draw ¢ p, parallel toc e, and make ¢/ equal to ¢/ at Fig. 264 ; complete the parallelogram, e c l s, and find the points, o, o, o, as before; then describe the circles and complete the mould as in Fig. 262. The difference between this and Case.1 is, that the line, c ¢, instead of being raised and throwy out, is lowered and drawn in. Fig. 266. c 370.—Case 3.— Where the rake meets the level. In Fig.

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STAIRS. 215 266, a bc isthe plan of aline passing through the centre of the 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 hin d, and through d, draw d f, paraliel to A g ; bisect f n in J, and from J, draw / ¢, parallel to nj; from n, draw n t, at right angles to j n ; on the line, o b, make o ¢ equal to n ¢ Then, to obtain a mould for the twist going up the flight, procedd as at Fig. 262; making e cin that figure equal to e n in Fig. 266, 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 b o, (Fig. 266,) toi; make o į equal to f l, and join iandc; make c i, (Fig. 267,) equal to ciat Fig. 266; through draw c d, at o ¥ d c Fig. 267. right angles toci; make d c equal to d f at Fig. 266, and com- plete the parallelogram, o d c i; then proceed as in the previous cases to find the mould. 371.—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. 372.— To bore for the balusters of a round rail before round- tng it. Make the angle, o c t, (Fig. 268,) equal to the angle, o c t, at Fig. 261; 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 tangical to the circle; from c, draw c a, at right angles to oc; then, bd being the bottom of the rail, set a gauge from b to a, and run it the whole length of the stuff; in boring, place the centre of the

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216 AMERICAN HOUSE-CARPENTER. = Fig. 268. bit in the gauge-mark at a, and bore in the direction, a c. Tdo 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. | S Ñ j Aw PUS = & Fer e N

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STAIRS. 217 873.—To obtain, by the foregoing principles, the face-mould for the twists of a moulded rail upon platform stairs. In Fig. 269, 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. 261. Set half the width of the rail from b to f and from b to r, and from f and r, draw f e aug r d, parallel toca. At Fig. 270, the centre lines of the S S rail, k c and c n, are obtained as in the previous examples. Make c i and c j, each equal to c i at Fig. 269, and draw the lines, 2 m - and j g, parallel to c k; make n e and n d equal to n e and n d at Fig. 269, and draw d o 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 vertically. If the rail crosses the platform rather steep, a butt joint will be preferable, to obtain which see Art. 405.

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218 AMERICAN HOUSE-CARPENTER. 374.— 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. 271. Before applying the mould, the edge, e f, must be bevilled according to the angle, c ¢ x, at Fig. 269 ; if the rail is to be canted wp, the edge must be bevilled at an obtuse angle with the upper face; but if it is to be canted down, the angle that the edge makes with the upper face mnst be acute. From the spring of the curve, a, and the end, craw 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. 375.—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 must be thicker than that for the straight. In Fig. 269, draw a section of the rail be- -~ tween the lines, d r and e 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; ilies the distance that these lines are apart, will be the saie anoa 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. 376.— Winding stairs have steps tapering narrower at one end than at the other. In some stairs, there are steps of parallel width incorporated with tapering steps ; the former are then called flyers and the latter winders. : 377.— To describe a regular geometrical winding stairs. In Fig. 272, a b 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 g h the cylinder, or face of the front string. The line,

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STAIRS. 219 d Fig. T12. a e, is given as the face of the first riser, and the point, j, for the limit of the last. Make e i equal to 18 inches, and upon o, with ọ å 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. 361, and divide the arc, ij, 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 of each riser, and determine the form and width of the steps. Allow the necessary projection for the nosing beyond a e, which should be equal to the thickness of the step, and then a e l 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. 256, gauging from the upper edge of the string for the line at which to set the pitch-board. Upon the back of the string, with a 1} inch dado plane, make

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220 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 cut 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 gjued 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 a centre-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 l 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. 273. 378. To find the shape and position of the timbers neces- sary to support a winding stairs. The dotted lines in Fig. 272 shew the proper position of the timbers as regards the plan : the shape of each is obtained as follows. In Fig. 273, the line, 1 a, is equal to ariser, less the thickness of the floor, and the lines, 2 m, 3 n, 4 0,5-p and 6 q, are each equal to one riser. The,

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STAIRS. 221 line, a 2, is equal to a m in Fig. 272, the line, m 3 to m n in that figure, &c. In drawing this figure, commence at a, and make the lines, a1 and a 2, of the length above specified, and draw them at right angles to each other; draw 2m, 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 theend. Then, through the points, 1, 2, 3, 4, 5 and 6, trace the line, 1 6; upon the Pints, 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. 272 ; 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. 379.— To find the falling-mould for the rail of a winding stairs. In Fig. 274, 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 ac 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 ¢ and f? ; from e t f° and J, 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 g°, drawg i; make g x equal to about three-fourths of a rise, (the top of the newel, z, should be 34 feet from the floor ;) draw z u, at right angles to e z, and ease off the angle at u ; at a distance equal to the thickness of * In the above, the references, a2, b2, &e:, 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 may then be mistaken for a 2, b 2, &c. This can be avoided in reading by giving ts sound corresponding to their meaning, which Am anannt L Bean awa ennmi h o a

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222 AMERICAN HOUSE-CARPENTER. the rail, draw v w y, parallel to z u i; from the centre of the plan, o, draw o l, at right angles to e d ; bisect hn in p, and through p, at right angles to g i, draw a line for the joint; in the same manner, draw the joint at k ; then z y will be the falling-mould for that part of the rail which extends from s to b on the plan. 380.— To find the face-mould for the railof awinding-stairs. From the extremities of the joints in the falling-mould, as 4, z and y, (Fig. 274,) draw k a’, z 6’ land y d, at right angles toe d ; make b e equal to f d. Then, tp obtain the direction of the joint, a’ c’, or b° d’, proceed as at Nig. 275, at which the parts are

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STAIRS, 228 Fig. 275. wn at half their full size. Ais the plan of the rail, and B is falling-mould ; in which > z is the direction of the butt-joint. m k, draw k b, parallel tol o, and k e, at right angles tok b; n b, draw b f, tending to the centre of the plan, and from f, draw parallel to b Æ; from 1, through e, draw ¿ i, and from 7, draw t arallel toe f ; join d and b, and d b will be the proper direction

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$ j Zei AMERICAN HOUSE-CARPENTER. for the joint on the plan. The direction of the joint on the other side, a c, can be found by transferring the distances, ac Dee: tozaandoc. (See Art. 384.) Fig. 276. Having obtained the direction of the joint, make s r d b, (Fig. 276,) equal to s r œ b'in Fig. 274; through r and d, draw ta; through s and from d, draw ¢ u and d e, at right angles to? a; make ¢ u and d e equal to ¢ u and b° m, respectively, in Fig. 274; from u, through e, draw u o ; through b, from r, and from as many other points in the line, ¢ a, as is thought necessary, as f, h and j, draw the ordinates, rc, f g, h i,j k and ao; from u, c, g, i, k, e and o, draw the ordinates, u 1, c 2, g 3, i 4, #6, e 6 ando 7, at right angles to wo; make u 1 equal to £ s, c 2 equal to r 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’ b, on the plan at Fig. 274. | To ascertain the mould for the next quarter, make a c je, (Fig.

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STAIRS. i ‘205 277,) equal to a* cj e at Fig. 274; at any convenient height on the line, d i, in that figure, draw gq 7’, parallel to e d; through c and j, (Fig. 277,) draw b d ; through a, and from j, draw b k and j o, at right angles to bd; make b k andj o equal to # k and q i, respectively, in Fig. 274; from k, through o, draw kf; and proceed as in the last figure to obtain the face-mould, A. 381.— To ascertain the requisite thickness of stuff. Case 1.— When the falling-mould is straight. Make o h and k m, (Fig. 277,) equal to i y at Fig. 274; draw h i and m n, parallel tob d; through the corner farthest from k f, as n or i, draw n i, parallel to Æ f ; then the distance between & f and ni will give the thickness required. 382.—Case 2.— When the falling-mould is curved. In Fig. 278, sr d bis equal to s r œ b° in Fig. 274. Make ac equal to the stretch-out of the arc, s b, according to Art. 92, and divide a c and s b each into a like number of equal parts ; from a and c, and from each point of division in the line, ac, draw a k, e l, &c., at right an- gles to ac ; make a kequal tot u in Fig. 274, and c jequal tob* m 29

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4 226 AMERICAN HOUS&CARPENTER. 7 SS. eee e e e e e e a de At 4 / ee ae A j | & ae en == taa m F se fg ‘ies a r Fig. 278, in that figure, and complete thetalling-mould, k 7, every way equal to u min Fig.274; from the points of division in the arc, sb, draw lines radiating towards the centre of the circle, dividing the are, r d, in the same proportion as $ b is divided ; from d and b, draw d t and b u, at right angles to a d, and from j and v, draw j u and v w, at right angles to j c; then z ¢ u w will bea vertical projection of the joint, d b. Supposing every radiating line across s r d b— corresponding to the vertical lines across Æ% 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 be a helinet, 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. 383.— To apply the face-mould to the plank. In Fig. 279, 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

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. STAIRS. | . ART | Fig. 279. 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 shapé of the twist; from a and o, draw the lines, a b and o c, across the edge of the plank, the angles, e a b and e o c, corresponding with Æ fd at Fig.277; turning the plank up as 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 6 ; 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 onter, 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. 384.— Elucidation of the process by which the direction of the butt-joint is obtained in Art. 380. 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. 280, 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.

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= 228 AMERICAN HOUSE-CARPENTER. 382. Bisect r ¢, part of an ordinate from the centre of the plang A and through the middle, draw c b, at right angles to g v; fro 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 c 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 0, e m or i j, are level; for instance, the level line, w v, on the top of the a % \

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STAIRS. 299 n the helinet, is a true representation of the radiating line, j i, ne plan. The line, b h, therefore, on the top of the rail in elinet, is a true representation of e m on the plan, and & c on ottom of the rail truly represents do. From k, draw k l, lel to c b, and from h, draw Af, parallel to b c; join l and iocand f ; then c k lb will be a true representation of the of the lower piece, B, and c fh b of the end of the upper ,A;and fk orh lwill show how much the joint is open on ner, or concave side of the rail.

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230 AMERICAN HOUSE-CARPENTER. To show that the process followed in Art. 880 is correct, let do and e m, (Fig. 281,) be the direction of the butt-joint found as at Fig. 275. 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 k to w and h, and join wand kh; then it will be evident that w À isa true representation jn the helinet of j k on the plan, it being in the same plane as j &, and also in the same winding surface as w v. The line, 2 n, also, is a true repre- sentation on the bottom of the helinet of the line, j &, in the plan. The line of the joint, e m, therefore, is projected in the same way and truly by 7 b on the top of the helinet; and the line, d o, by c & on the bottom. Join a andi, and then it will be seen that the lines, c a, ai and ¿ b, exactly coincide with c b, the line of the joint on the convex side of the rail; thus proving the lower end 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. 275, it will be seen that-the line, 7 i, corresponds to zi in Fig. 281; anà that e k in that figure is a representation of f b, andi k of db. Fig. 262. In getting out the twists, the joints, before the falling-mould is

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STAIRS. n 231 applied, are cut perpendicularly, the face-mould being long enough to include the overplus necessary for a butt-joint. 'The face-mould for A, therefore, would have to extend to the line, ¿ b ; and that for B, to the line, y z. 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. 282,) at right angles to i f, and make i s equal to me at Fig. 281; through s, draw s g, pa- rallel. to i f, and make s b equal to s b at Fig. 281; join b and i; make if equal toi fat Fig. 281, and from f,draw f g, parallel to ¢ b ; then 7 b g f will be a perpendicular section of the rail over the line, e m, on the plan at Fig. 281, corresponding toi b 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, 7 b, and the bottom to the line, f g. | | | 385.— 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. 283, 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 hiand ij, parallel to a b, bec andc d; make g kand gl 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 to the 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, k p m, according to Art. 92; extend a b to ¢, and k m to s; make c s equal to the length of the steps, and 7 u equal to 18 inches, and de- scribe the arcs, $ ¢ and u 6, parallel to m p ; from #, draw t w, tend-. ing to thecentre of the cylinder ; from 6, and on the line, 6 u z, run off the regular tread, as at 5, 4, 3,2, 1 and v; make u z equal to half the arc, u 6, and make the point of division nearest to z, 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 0, and equal to one rise;

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233: AMERICAN HOUSE-CARPENTER. Fig. 283. from a°, draw a’ s, parallel to » o, and equal to one tread ; from s, through o, draw s b°. m Then from w, draw w c*, at right angles to n o, and set up, on the line, w c°, 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 e’, draw e f°, at right angles to n o, and through 5, (on the line,

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STAIRS. 233 wc’,) draw g° f°, parallel tomo; through 6, (on the line, w c’*,) and f°, draw the line, h* 6’; make 6 c* equal to half a rise, and from c* and 6, draw c° 7’ and 6 7’, parallel to n o ; make h* č equal to k f°; from 7, draw 7 K, at right angles to 7 h°, and from f°, draw f?’, at right angles to f° h’; upon Æ, with Æ f° for radius, describe the arc, f°? ©; make b’ P equal to b* f°, and ease off the angle at b by the curve, f? P. 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 n o, meeting the curve in m°, n’, o’ and p°; from these points, draw lines at right angles to n o, and meeting it in 2°, 7°, $° and #; from z* and 7°, draw lines tending to «°, and meeting the convex side of. the rail in 4° and z*; make m 1° equal to r s*, and m w° equal tor # ; from y’, 2°, v*, and w°, through 4, 3, 2 and 1, draw lines meeting the line of the wall-string in a’, b°, 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 n o, and from f°, through 6, draw f° o°; then the heavy lines, f’? 2°, ed, ya, 2} b, vc, w d, and z y, will be the lines for the risers, | which, being extended to the line of the front string, b ec d, will give the dimensions of the winders, and the grading of the front string, as was required. 386.— To obtain the falling-mould for the twists of the last- mentioned stairs. Make č g* and © h°’, (Fig. 283,) each equal to half the thickness of the rail; through A’ and g*, draw A # and g* 7, parallel to 7? s ; assuming k k’ and m m’ on the plan as the amount of straight to be got out with the twists, make n q equal to k k*, and r P equal to m m°; from n and F, draw lines at, right angles to n o, meeting the top of the falling-mould in n’ 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 wu’ 8, at right angles to n o ; through 8, draw 7 9, tending to i? ; then n? 7 will be the falling-mould for the upper twist, and 7 œ the falling-mould for the ou awl

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234 AMERICAN HOUSE-CARPENTER. 387.— To obtain the face-moulds. The moulds for the twists į" of this stairs may be obtained as at Art. 380; 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 above a line joining the other two. Having found at Fig. 283 the direction of the joint, w s° and p e, according to Art. 380, make k p e a, (Fig. 284,) equal to Æ p'e p in Fig. 283; join b and c, and from o, draw o h, at right angles to b c ; obtain the stretch-out of d g, as d f, and at Fig. 283, place it from the axis of the cylinder, p, to q ; from ¢ in that figure, draw q° r°, at right angles to n o ; also, at a convenient height on the line, n n’, in that figure, and at right angles to that line, draw w’ v’; from b and c, in Fig. 284,

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STAIRS. 225. draw b j and c l, at right angles to b c ; make b j equal to u’ n’ in Fig. 283, i h equal to w° r° in that figure, and c 7 equal to v’ 9; from l, through j, draw Im; from h, draw À n, parallel to c b; from n, draw n r, at right angles tọ b c, and join r and s ; through hahe lowest corner of the plan, as p, draw v e, parallel to bc; 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 r s; through k, draw w z, 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 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. 283 is of this nature. The face-mould for this is found at Fig. 285 in a manner similar to that at Fig. 284. The heights are all taken from the top of the falling-mould at Fig. 283; b j being equal to w 6 in Fig. 283, i h equal to 2°¥ in that figure, and c/tol’o*. Draw a line through j and J, and from h, draw h n, parallel to bc; from n, draw n r, at right angles to 6 c, and join r ands; then r s will be the bevil for the lower ordinates. From A, draw %4 v, at right an- gles to j l; upon n, with r s for radius, describe an intersecting arc at 7, and join nand z ; then n z will be the bevil for the upper ordinates, upon which the faee-mould is found as in Case 1. 388.— 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. 286 and 287 repre- sent solid blocks, or prisms, standing upright on a level base, b d ; the upper surface, j a forming oblique angles with the face, 6 —

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236 AMERICAN HOUSE-CARPENTER. in Fig. 286 obtuse, and in Fug. 287 acute. Upon the base, de- scribe the semi-circle, b s c ; from the centre, 7, draw 7s, at right angles to b c; from s, draw s z, at right angles to e d, and from i, draw i h, at right angles to b c ; make ¿i Å equal to s z, and join hand z ; then, h and v being of the same height, the line, A z, joining them, is a level line. From A, draw A n, parallel to b c, and from n, draw n r, at right angles to b c; join r and s, also n

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STAIRS. 237 | | l L] J | | I 1 i | al E Fig. 286. and z; then, n and z being of the same height, n zis a level 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 n x, and perpendicularly over corresponding lines drawn parallel to r s on 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 x l, may be 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. 284 and 285 is founded. 'The plan 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 v and r s, in those figures, eens ton zr and r s in Fig. 286 and 287. In Fig. 288, 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 h v in the last two figures, and that will lie level ` and in the upper surface; so that all ordinates at right angles to dr on the base, will correspond to those that are at right angles

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238 AMERICAN HOUSKE-*CARPENTER. LY s : Sea wee ew ew ee Fig. 288 r toeconthetop. This elucidates Fig. 276; at which the nes, h 9 and i 8, correspond pees i 8 in this figure. p Fig. 289. 389.— 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. 289,) at right angles to ¿ s, and equal to w hat Fig. 284; make i s equal toz s in that figure, and join w and s ; then sw p will be the bevil required in order to apply the face-mould at Fig. 284. In Fig. 285, the middle height being below the line joining the other two, the bevil is therefore acute. 'To determine this, draw i s, (Fig. 290,) at

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STAIRS. 239 Fig. 290. right angles to ¿ p, and equal toi s in Fig. 285; make s w equal toh win Fig. 285, and join w andi; then w ¿i p will be the bevil required in order to apply the face-mould at Fig. 285. Al- though the falling-mould in these cases is curved, yet, as the plank is sprung, or bevilled on its edge, the thickness necessary to get out the twist may be ascertained according to Art. 381— taking the vertical distance across the falling-mould at the joints, and placing it down from the two outside heights in Fug. 284 or 285. After bevilling the plank, the moulds are applied as at Art. 383—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 J, in Fig. 284 or 285. Fig. 201. 390.—To apply the moulds without bevilling the plank. Make w p, (Fig. 291,) equal to w p at Fig. 289, and the angle, b c d, equal to b j lin Fig. 284; make pa equal to the thick- ness of the plank, as w a in Fig. 289, and from @ draw a o, pa- rallel to w d ; from c, draw c e, at right angles to w d, and join e

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= 240 AMERICAN HOUSE-CARPENTER. » “and b ; then the angle, b e o, ona square edge of the plank, hav- ing a line on the upper face at the distance, p a, in Fig. 289, 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, to be above the plane of the paper as much as a, in Fig. 289, 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, ¢ 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. 391.— 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 introduced in this connection. In Fig. 292, a bc 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 j, equal to a bc, and from f, draw f h, parallel to e 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 theangle, a e d, will be the proper bevil for the ends from the inside, or Æ d e from the outside., If a mitre-joint is re-

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STAIRS. 241 quired, set f g, the thickness of the stuff on the level, from e to m, and join m and d ; then k d m will be the proper bevil for a mitre-joint. : If 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, f gj, being the same as a b c, will be the proper bevil for that purpose. Suppose, therefore, that a piece indicated by the lines, k g, gf and fh, 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, o,—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. 392.—Another method for face-moulds. It will be seen by reference to Art. 388, 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. It can be best introduced, perhaps, by the following explanation : In Fig. 293, j d represents a prism standing upon a level base, b d, its upper surface forming an acute angle with the face, b J, as at Fig. 287. Extend the base line, bc, and the raking line, j J, to meet at f; also, extend e d and g a, to meet at k; from f, through k, draw fm. If wesuppose the prism to stand upon a level floor, o f m, and the plane, j g al, 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, fk; 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 k f, parallel to f k, must be a level line. By finding the position of a perpendicular plane, at right angles to the raking plane, 7 f k g, we shall greatly shorten the process for obtaining ordinates. 31

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bo | 242 AMERICAN HOUSE-CARPENTER. This may be done thus: from f, draw f o, at right angles to fm; extend e b too, and g j, to¢; from o, draw o £, at right angles to o f, and join ¢ and f ; then ¢o f will be a perpendicular plane, at night angles to the inclined plane, ¢ g 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 j, be equal to p b. Proceed, in the same way, from the points, s and c, to find v and J. Now, to apply the principle here explained, let the curve, b s ¢, (Fig. 294,) 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 2 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 /, draw a line to meet the chord line extended to f ; from s, draw s k, parallel to b f, and from z, draw z k, parallel toj f; from f, through k, draw fm ; then f m will be the intersecting line of the plane of the section with the

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STAIRS. 243 Fig. 294. > of the base. - This line can be proved to be the intersection iese planes in another way ; from b, through s, and from j, igh v, draw lines meeting at m ; then the point, m, will be æ intersecting line, as is shown in the figure, and also at , 293. | om f, draw f p, at right angles to f m ; from b and c, and .as many other points as is thought necessary, draw ordinates, llel tof m; make p q equal to b j, and join q and f; from points at which the ordinates meet the line, q f, draw others zht angles to q f; make each ordinate at A equal to its cor- mnding ordinate at C, and trace the curve, g n i, through the ts thus found. , ow it may be observed that A is the plane of the section, B plane of the segment, corresponding to the plane, q p f, of . 293, and C is the plane of the base. To give these planes : proper position, let A be turned ong f as an axis until it

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id 244 AMERICAN HOUSE-CARPENTER. stands perpendicularly over the line, g f, and at right angles to the plane, B ; then, while A and B are fixed at right angles, let B be turned on the line, p f, 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, i, resting at J, the point, n, at xz, and g at j ; and the curve, g n i, lying perpendicularly over b s c—as was required. If we suppose the cylinder tobe | 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 f and p f would meet in /; and the plane of the section, 4, the plane of the segment, B, and the piap of the base, C, would all meet in that point. 393.— To find the face-mould PA a hand-rail according to the principles explained in the previous article. In Fig. 295, a e cf is the plan of a hand-rail over a quarter of a cylinder ; and in Fig. 296, a b c d is the falling-mould; f e being equal to the stretch-out of a df in Fig. 295. From c, draw c h, parallel 1 OC e f; bisect h c in 2, and find a point, as b, in the arc, d f, (Fig - 295,) corresponding to į in the line; hc; from i, (Fig. 296,) te? the top of the falling-mowld, draw ¢ j, at right anglesto fc ; at Fig- — 295, from c, through b, draw c g, and from b and ¢, draw b j anc ck, at right angles to g c; make c k equal tohg at Fig. 2965 and b j equal tot 7 at that figure; from k, thrqugh j, draw k goa and from g, through a, draw g p ; then g p will be the intersecting” line, corresponding to f m in Fig. 293 and 294; through e, draw” p 6, at right angles to gp, and from c, draw.c q, parallel to g p ; make r q equal toh g at Fig. 296; joinp and gq, and proceed as in the previous examples to find the face-mould, A. The joint of the face-mould, uù v, will be more accurately determined by finding the projection of the centre of the plan, o, as at w ; joining s and w, an drawing u v, parallel to s w. It may be noticed that c Æ and b j are not of a length corres- ponding to the above directions : they are but $ the length given.

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246 AMERICAN HOUSE-CARPENTER. Fig. 296. 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 in and the other in m; then a line drawn from m, through i, will give the point, g, as was required. The point, g, may also be

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STAIRS. i 247 ained thus: at Fig. 296, make h 7 equal toc b in Fig. 295 ; n J, draw J k, at right angles toh c; from j, draw j k, parallel h c; from g, through k, draw g n; at Fig. 295, make b g tal to ln in Fig. 296 ; then g will be the point required. | [he reason why the points, a, b and c, inthe plan of the rail at v, 295, are taken for resting points instead of e, ¿i and f, is this: top of the rail being level, it is evident that the points, a and e, he section a e, are of the same height ; also that the point, i, is of same height as b, and cas f. Now, if a is taken for a point the inclined plane rising from the line æ p, e must be below t plane ; if bis taken for a point in that plane, 7 must be below and if cis in the plane, f must be below it. The rule, then, taking these points, is to take in each section the one that is west to the line, g p. Sometimes the line of intersection, g p, ypens to come almost in the direction of the line,er: in such e, after finding the line, see if the points from which the ghts were taken agree with the above rule; if the heights re taken at the wrong points, take them according to the rule yve, and then find the true line of intersection, which will not -y much from the one already found. Fig. 297, 394.— To apply the face-mould thus found to the plank. he face-mould, when obtained by this method, is to be applied a square-edged plank, as directed at Art. 383, with this differ- ce: instead of applying both tips of the mould to the edge of

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248 AMERICAN HOUSE-CARPENTER. the plank, one of them is to be set as far from the edge of the plank, as z, in Fig. 295, is from the chord of the section p g—as is shown at Fig. 297. 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 tog x in Fig. 295, and the angle, e a c, on the edge, equal to the angle, p q r, at Fig. 295. 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 b, 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 d h, at right angles to i d ; then revolve the mould, C, upon J, until the point, A, arrives at j, causing the line, f h, to coincide with f j, and the line, ¿ d, to coincide with k 1; then the tips of the mould will be at k and Z. The rule for doing this, then, will be as foHlows : sake the an- gle, i f k, equal to the angle q v z, at Fig. 295; make f k equal to fz, and through k, draw k l, parallel to ij; then apply the corner of the mould, 4, at k, and the other corner d, at the line, k l. The thickness of stuff is found as at Art. 381. 395.— To regulate the application of the falling-mould. Obtain, on the line, h c, (Fig. 296,) the several points, r, q, p, l and m, corresponding to the points, b’, a’, z, y, &c., at Fig. 295; from r q p, &c., draw the lines, r £, q u, p v, &c., at right angles tohc; makeh s,r t, qu, &c., respectively equal to 6c’, r q, 5 d’, &c., at Fig. 295; through the points thus found, trace the curve, sw c. Then get out the piece, & s c, attached to the fall- ing-mould at several places along its length, as at z, z, z, &c. In applying the falling-mould with this strip thus attached, the edge, s w c, will coincide with the upper surface of the rail piece

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STAIRS. 249 ` before it is squared ; and thus show the proper position of the fall- ing-mould along its whole length. (See Art. 403.) SCROLLS FOR HAND-RAILS. 396.—General rule for finding the size and pasition of the regulating square. 'The breadth which the scroll is 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 equal 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. 3 Fig. 298. 397.—To find the proper centres in the regulating square. Let a 2 1 b, (Fig. 298,) 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- 32

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250 AMERICAN ‘HOUSE-CARPENTER. mine these centres being each one part less in length than its pre 1 c Fig. 299. 398.— To describe the scroll for a hand-rail over a curtail step. Leta b, (Fig. 299,) be the given breadth, 1} the given | number of revolutions, and let the relative size of the regulating square to the eye be $ 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; bisect a c in e; 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 toa b ; set b c from 1 to 2, | and upon 1 2; complete the regulating square; divide this square as at Fig. 298; 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 € h, &c.; make d lequal to the

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STAIRS. | 251 width of the rail, and upon 1, describe / m ; upon 2, aescribe m n, &c.; describe the eye upon 8, and the scroll is completed. 399.— To describe the scroll for a curtail step. Bisect d l, (Fig. 299,) in o, and make o v equal to 4 of the diameter of a baluster; make v w equal to the projection of the nosing, and e x equal to w l; upon 1, describe w y, and upon 2, describe y z ; also upon 2, describe z i; upon 3, describe ij, and so around to z ; and the scroll for the step will be completed. 400.— To determine the position of the balusters under the scroll. Bisect dl, (Fig. 299,) 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 o, g, r, $, t, u, &c., as far as g*; 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 ae as shown in the figure. 401.— To obtain the falling-mould for the raking part of the scroll. Tangicalto the rail at h, (Fig. 299,) 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

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252 AMERICAN HOUSE-CARPENTER. e°, find the position of the point, k, asat Æ ; at Fig. 300, make ed equal toe’ d in Fig. 299, and d c equal tod c’ 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 c in J, and through /, draw m q, parallel to e h ; m q is the height of the level part of a scroll, 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 2 j, and at a distance equal to the thickness of the rail; at a convenient place for the joint, as i, draw in, at right angles to b j; through n, draw jh, at right angles to eh; make d k equal to d &° in Fig. 299, und from k, draw k o, at right angles to e h; at Fig. 299, make d h* equal to d h in Fig. 300, and draw h’ b°, at right angles tod h* ; then k a* and k’ @ will be the position of the joints on the plan, and at Fig. 300, o p and i n, their position on the falling- mould; and p o in, (Fig. 300,) will be the falling-mould re- quired. Fig. 301. 402.— To describe the face-mould. At Fig. 299, from k, draw kr’, at right angles tor’ d; at Fig. 300, make h r equal to $ r in Fig. 299, and from r, draw r s, at right angles to r h ; from the intersection of r s with the level line, m q, through i, draw s t; at Fig. 299, make h’ 5° equal to q t in Fig. 300, and join & and 7°; 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. 301,) equal in its length and in its divisions to the line, 7° b, in Fig. 299; fromr, n, o, p,q

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STAIRS. 253 and /, draw the lines, r k, n d, oa, pe, gfand lc, at right an- gles to r 6, and equal to r° k, Ë s, fa’, &., in Fig. 299; through the points thus found, trace the curves, k 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, ¿Z 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 t h, in Fig. 300. ‘The thitkness of stuff required for this mould is shown at Fig. 300, between the lines s ¢ and u v—u v being drawn pe- rallel to s ¢. | 403.—AĦ 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. 262, 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. |

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254 AMERICAN HOUSE-CARPENTER. 404.—To ascertain the form of the newel-cap from a section of the rail. Draw a b, (Fig. 302,) through the widest part of the given section, and parallel to c d; bisect ab in e, and through a, e and b, draw h 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, ¿ 7 g; make r l equal toeborea; join land j, also / andi; from the curve, f b, to the line, Z j, draw as many ordinates as is thought necessary, parallel to f g; from the points at which these ordinates meet the line, 7 j, and upon the centre, o, describe arcs in continuation to meet o p ; from n, t, x, &c., draw n s, t u, &c., parallel to f g; make n s, t u, &c., equal to e f, w v, &c.; make x y, &c., equal to z d, &c.; make 02, 03, &c., equal to o n, o t, &c. ; make 24 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. nl ee Fig. 303. 405.—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. 373, and make the line a b, Fig. 303, equal to ac, Fig. 269; from b, draw b c, at right angles to a b, and equal in length to n m, Fig. 269; join a 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 ok each equal to half e h at Fig. 269; through e and f, draw hi and g j, parallel to a c. At Fig. 270, make n a equal to e d, Fig. 303, 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- ann ADA

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STAIRS. 255 cally as described at Art. 374, but the joint is to be sawn square through the plank. The moulds obtained for round rails, (Art. 371,) give the line for the joint, when applied to either side of the ` plank; but here, for moulded rails, the line for the joint can be obtained from only one side. When 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 down, the vertical joint, obtained as at Art. 373, will be a butt joint, and therefore, in such a case, the process described in this article will be unnecessary.

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SECTION VII—SHADOWS. 406.—The art of drawing consists in representing solids upon a plane surface: so that a curious and nice adjustment of lines is made 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 elementagy 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. 407.— 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. 304, 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 the plane, B, the »

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SHADOWS. . 257 Fig. 304. , a b, is drawn at an angle of 45 degrees with the horizon, and line, c b, at the same angle with the vertical plane, B. The ie, B, being a rectangle, this makes the true direction of the 's rays to be in a course parallel to d b; which direction has 1 proved to be at an angle of 35 degrees and 16 minutes with horizon. It is convenient, in shading, to have a set-square 1 the two sides that contain the right angle of equal length ; will make the two acute angles each 45 degrees; and will . the requisite bevil when worked upon the edge of the T- ire. One reason why this angle is chosen in preference to her, is, that when shadows are properly made upon the draw- by it, the depth of every recess is more readily known, since breadth of shadow and the depth of the recess will be equal. ‘o distinguish between the terms shade and shadow, it will be erstood that all such parts of a body as are not exposed to the ĉt action of the sun’s rays, are in shade; while those parts ch are deprived of light by the interposition of other bodies, in shadow. 33

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258 AMERICAN HOUSE-CARPENTER. Fig. 307. Fig. 208. 408.—To find the line of shadow on mouldings and other ho- rizontally straight projections. Fig. 805, 306, 307 and 308, 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, a 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. 409.—To find the line of shadow cast by a shelf. In Fig. 309, 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 perpéndicular 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 me the shelf to f, and make c f equal to the projection of the shelf

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SHADOWS. l 264 the wall; from f, draw f g, at the customary angle, and from 3p the vertical line, c g, intersecting f g at g; from g, draw parallel to the shelf, and from c, draw c d, at the usual angle ; the lines, c d and d e, will determine the extent of the shadow fore. & Fig. 310. 0.—To find the shadow cast by a shelf, which is wider at end than at the other. In Fig. 310, A is the plan, and B levation. Find the point, d, as in the previous example, and any other point in the front of the shelf, as a, erect the perpen- ar, a e; from a and e, draw a b and e c, at the proper angle, from b, erect the perpendicular, b c, intersecting ec in c;

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260 AMERICAN HOUSE-CARPENTER. from d, through c, draw d o ; then the lines, z d and d o, will give the limit of the shadow cast by the shelf. Fig. 311. 411.—To find the shadow of a shelf having one end acute or obtuse angled. Fig. 311 shows the plan and elevation of a acute-angled shelf. Find the line, e g, as before; from a, erdt the perpendicular, a b; join b and e; then b e and e g will define the boundary of shadow. __ E Fig. 312. 412.—To find the shadow cast by an inclined shelf. In Fig. $12, 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. 310, and find the points, d and c; then ad and d c will be the shadow: required. If the shelf had been

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SHADOWS. 261. yarallel in width on the plan, then the line, d c, would have been sarallel with the shelf, a b. Fig. 313. . Fig. 314. 413.—To find the shadow cast by a shelf inclined in its ver- "cal section either upward or downward. From a, (Fig. 313 and 314,) draw a b, at the usual angle, and from b, draw b parallel with the shelf; obtain the point, e, by drawing a line from d, at the usual angle. In Fig. 313, join eand ï; then ie e c will define the shadow. In Fig. 314, from o, draw o i, parallel with the shelf; join: and e; then ze and e c will be the shadow required. -The projections in these several examples are bounded by atraight lines; but the shadows of curved lines may be found -in the same manner, by projecting shadows from several points in the curved line, and tracing the curve of shadow through these points. Thus—

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. WR AMERICAN HOUSE-CARPENTER. - 414.—To find the shadow of a shelf having its front edge, or end, curved on the plan. In Fig. 315 and 316, A.and A show an example of each kind. From several points, as a, a, in the plan, and from the corresponding points, o, o, in the elevation, draw ‘mays and perpendiculars intersecting at e, e, &c.; through these points of intersection trace the curve, and it will define the shadow. 9 415.—To find the shadow of a shelf curved in the elevation. In Fig. 317, find the points of intersection, e, e and ¢, as in the last examples, and a curve traced through them will define the shadow. - The preceding examples show how to find shadows when catt upon a vertical plane; shadows thrown upon curved surfaces at ascertained in a similar manner. Thus—

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‘SHADOWS. | mB . 416.—To find the shadow cast upon a cylindrical wall by a projection of any kind. By an inspection of Fig. 318, 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. 319. - 417.—To find the shadow cast by à shelf upon an inclined wall. Cast the ray, a b, (Fig. 319,) from the end of the shelf to Be 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 ¢ 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 ipstances are found. Let us attend now to the application of this principle to such familiar objects as are likely to occur in practice.

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264 AMERICAN HOUSE-CARPENTER. 418.—To find the shadow of a projecting horizontal beam. From the points, a, a, &c., (Fig. 320,) 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 beam be in- clined, either on the plan or elevation, at any angle other thana sight angle, the difference in the manner of proceeding can be seen by reference to the preceding examples of inclined shelves, &c. ; Fig. 321. 419.—To find the shadow in a recess. From the poit, a, (Fig. 321,) in the plan, and b in the elevation, draw the rays, a c and be; from c, erect the perpendicular, c e, and from 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 parallel with the face of the wall. Fig. 322. 420.—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. 322, A shows the section and B the elevation ¢& a recess of this

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SHADOWS. 265 kind. From b, and from any other point in the line, b a, as a, draw the rays, bc andae; from c, a, and e, draw the horizontal lines, cg, a f, andeh; from d and f, cast the rays, d i and fh; from 1, through h, draw ts; then sz and tg will define the shadow. d | il "i T Wala AW | | | | Iii ii 1i LAINN ili | I 421.—To find the shadow in a fireplace. From a and b, (Fig. 323,) cast the rays, a c and be, and from c, erect the per- pendicular, c e; from e, draw the horizontal line, e o, and join o and d ; then c e, e o, and o d, will give the extent of the shadow. 422.—To find the shadow of a moulded window-lintel. Cast rays from the projections, a, 0, &c., in the plan, (Fig. 324,) and d, e, &c., in the elevation, and draw the usual perpendiculars in- tersecting the rays at 7,7, and č; these intersections connected 34

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266 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 from 7 to s, and drawing the horizontal line, $ n. 423.—To find the shadow cast by the nosing of a step. From a, (Fig. 325,) 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 vo, meeting the mitre-line, a g, in o ; cast a ray from o to 2, and from i, erect the perpendicular, + 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 h, draw the horizontal lines, kn and hs; then the limit of the shadow will be completed. | 424.—To find the shadow thrown by a pedestal upon steps. From a, (Fig. 326,) 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 hz that on the second tread, as at D; which completes the shadow of the left-hand pedestal, both on the plan and elevation. A mere inspection of the figure will be suf-

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Fig. 396. : ficient to show how the shadow of the right-hand pedestal is Fig. 327. 425.—To find the shadow thrown ona column by a square abacus. From a and b, (Fig. 327,) draw the rays, ac 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, h o; take any point between a and f, as i, and from this, as also from a corresponding point, n, draw the rays, è r and ns; from r, and from d, erect the perpendiculars, r s and do; through the points, e, $, and o, trace the curve as shown in the figure ; then the extent of the shadow will be defined. 426.—To find the shadow thrown on a column by a circular abacus. This is so near like the last example, that no explanation will be necessary farther than a reference to the preceding article.

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2068 AMERICAN HOUSE-CARPENTEk. —— << ——_ lid 427.—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

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SHADOWS. 260 nature—cutting it in the lines, a b, c d, &c., (Fig. 329,) 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 6, the plane cuts the ovolo at e, and therefore f will be the corresponding point upon the elevation; k corresponds with g, è with j, 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 h, 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, p, r, t, u, v, w, x, as also 1, 2, 3, &c., are successively found, and the lines of shadow ‘traced through them. | Fig. 330 is an example of the same capital with all the shadows finished in accordance with the lines obtained on Fig. 329. 428.—To find the shadow thrown on a vertical wall by a column and entablature standing in advance of said wall. Cast

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270 AMERICAN HOUSE-CARPENTER. Fig. 331. rays from a and b, (Fig. 331,) 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 z and j, erect the perpendiculars, + / and 7 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 t; the shadow as required will then be completed.

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Fig. 332. Fig. 332 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. 333. 429.—F ig. 333 and 334 are examples of the Tuscan cornice. The manner of obtaining the shadows is evident.

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272 AMERICAN HOUSE-CARPENTER. Fig. 334. 430.—Reflected light. In shading, the finish and life of an object depend much on reflected light This is seen to advantage in Fig. 330 and on the column in Fig. 332. 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. 330 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 be seen in the other examples. |

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APPENDIX.

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Bs 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 of 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 ge rdens. 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. Amphitheatre.—A double theatre, employed. by the ancients for the exhibition of gladiatorial fights and other shows. ‘Ancones.—Trusses employed as an apparent support to a cornice upon the flanks of the architrave. 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. Ante.—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. Arch.—An arrangement of stones or other material in a curvilin 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.

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4 APPENDIX. Archivolt.—The ceiling of a vault: the under surface of an arch. Area.—Superficial measurement. An open space, below the level of the ground, in front of basement windows. __ Arsenal.—A public establishment for the deposition of arms and rlike 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-relievo. 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.—aA circular moulding. Bed-mouldings.—Those mouldings which are between the corona and the frieze. . Belfry.—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. Brick-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. 4", Campanile.—A tower for the reception of bells, usually, in Italy, fseparated 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.

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APPENDIX. 5 Caravansera.—In the East, a large public building for the 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. e 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. Chantry.—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. Chapel.—A building for religious worship, erected separately from a church, and served by a chaplain. Chaplet.—A moulding carved into beads, olives, é&c. Cincture.—The ring, listel, or fillet, at the top and bottom of a co- 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. 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 or large church, having a peristyle or ambulatory around it, covered with a range of buildings. 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 framed between two principal rafters above the tie-beam. Collonade.—A range of columns. Columbarium.—A_ pigeon-house.

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6 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 lants. 3 Consoles.—The same as ancones, which see. . Contour.—The external Tines which bound and terminate a figure. Convent.—A building for the reception of a society of religious per- sons. i 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 @ 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, with 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, &c. Dead-shoar.—A piece of timber or stone stood vertically in briok- 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—(From 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.

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APPENDIX. Te Die.—That part of a pedestal included between the base and the cornice ; it is also called a dado. 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 trame being placed 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. Fagade.—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- tion of one moulding from another, and to give breadth and firmness to the edges of mouldings. e Flutes.—Upright channels on the shafts of columns. - Flyers.—Steps in a flight of stairs that are parallel to each other. Forum.—In 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 moulds or shapes. 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 roof, from the level of the eaves to the summit. Gain.—A recess made to receive a tenon or tusk. Gallery.—A common passage to several rooms in an upper story. A long room for the reception of pictures. A platform raised on co- lumns, pilasters, or piers. 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.

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8 APPENDIX. Granary.—A building for storing grain, especially that intended to be kept for a considerable time. Groin.—The line formed by the intersection of two arches, which cross each other at any angle. Gutte.—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.—Originally, a space measured out and covered with sand for the exercise of 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 or village. Helix.—The small volute, or twist, under the abacus in the Corin- thian capital. Hem.—The projecting spiral fillet of the Ionic capital. Herastyle.—A building having six columns in front. Hip-rafter.—A piece of timber placed at the angle made by two ad- jacent inclined roofs. Homestali.—A mansion-house, or seat in the country. Hotel, or Hostel.—A large inn or place of public entertainment. A large house or palace. Weehouse 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. Impost.—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. l Jail.—A place of legal confinement. Jambs.—The vertical sides of an aperture. Joggle-piece.—A post to receive struts. Joists.—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. Kiin.—A building for the accumulation and retention of heat, in or- 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.

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APPENDIX, 9 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 each 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 rrincipal 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 Jever-boards, 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. 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. Mosaic.—A mode of representing objects by the inlaying of small cubes of glass, stone, marble, shells, &c. 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, é&c., called muniments. 1

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40 APPENDIX. Museum.—A repository of natural, scientific and literary, oųriosities, or of works of art. Mutule.—A projecting ornament of the Doric cornice supposed to- represent the ends of rafters. Nave.—The main body of a Gothic church. Negwel.— A post at the starting or landing of a flight of stairs. Niche.—A cavity or hollow place in a wall for the reception of a statue, vase, &o. _ Nogs.—Wooden bricks. Nosing.—The rounded and projecting edge of a step in stairs. Nunnery.—A building or buildiigs appropriated for the reception of nuns. © _ Obelisk.—A lofty pillar of a rectangular form. . Octastyle,—A building with eight columns in front. : - Odeam.—Among the Greeks, a species of theatre wherein the poe and musicians rehearsed their compositions previous to the public pro- duction of them. Ogee.—See Cyma. ` 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 ether apartment. Ovolo.—A convex projecting moulding whose profile is the quad- fant of a circle. | ` Pagoda.—A temple ‘or place of worship in India. Palisade.—A fence of pales or stakes driven into the ground. . .Parayet.—A small well of any material for protection on the sides of bridges, quays, or high buildings. . Puvilion.—A turret or small building generally insulated and com- prised under a single roof. . Pedestal.—A square foundation used to elévate and sustain.a co- lumn, statue, &c. Peditrent.—The triangular crowning part of à portico or aperture which terminates vertically the sloping parts of the roof: this, in Gothic architecture, is called à gadle. Penitentiary.— A prison for the confinement of criminals whose Grimes are not of a very heinous nature. Piaza.—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 places, or in the bed of a river. . Pillar —A column of irregular form, always disengaged, and al-

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APPENDLX. 41 ways deviating from the proportions of the orders ; whence the distinc- tion between a pillar and a column. Pinnacle.—A small spire used to ornament Gothic buildings. Planceer.—The same as soffit, which see. Plinth.—The lower square member of the base of a column, pedes- tal, or wall. Porch.—An exterior appendage to a building, forming a covered approack 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 one 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. 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 intercoluniniation of one and a half diameters. Pyramid.—A solid body standing on a square, tangolar 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 v2rtical board forming the front of a step. Rostrum.—An elevated platform from which a speaker addresses an audienes. - 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 ville Rustie-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.

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12 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. Scantling.—The measure 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. Scroll.—A carved curvilinear ornament, somewhat resembling in profile the turnings of a ram’s horn. hre.—A grave, tomb, or place of interment. Sewer.—A drain or conduit for carrying off soil or water from any lace. j 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 orga ed wall. Sill.—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. Talus.—T he 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, &c. Theatre.—A building appropriated to the representation of drama..c spectacles. Tile.—A thin piece or plate of baked clay or other material used for the external covering of a roof. Tomb.—A grave, or place for the interment of a human body, ìn- cluding also any commemorative monument raised over such a place. Torus.—A moulding of semi-circular profile used in the bases of columns. | Tower. —A lofty building of several stories, round or polygonal. Transept.—The transverse portion of 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.

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APPENDIX. 13 Tread.—That part of a step which is included between the face of its 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 the 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 arrangement 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 level 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 Ionic and the Composite capitals. Voussoirs.—Arch-stones Wainscoting.— Wooden lining of walls, generally in panels. Water-table——The stone covering to the projecting foundation or other walls of a building. Weill.—The space occupied by a flight of stairs. ‘The space left beyond the ends of the steps is called the wedl-hole. l Wicket.—A small door made in a gate. Winders.—In stairs, steps not parallel to each other. Zophorus.—The same as frieze, which see. Zystos.—Among the ancients, a portico of unusual length, common- ly appropriated to gymnastic exercises.

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ES<RLSSSSSSSSG LORS S SSS SS LSE LB EE ERE EUES ESI SEE ee Secsaceew-| E] TABLE OF SQUARES, CUBES, AND ROOTS. te (From Hutton's Mathematics.) : r Sq. Root. Cube. | Bq- Root. 314432) 32462113, 4081654 14142136 328509 68-30662733 4101566 17320508 H3000 A 3666003) 4-121285 2:0000000 357911) 84261498) 414061: 2 B-4852A 14 4-160168 24494897 889017} A-5440037| 4-179333 2645751 405224) §-6029253 4-1980R 282542771 421875) B-6602540, 4-217163 435976) B71779799 4235824 31622777 456533) R- 7749644 425422 33166249 474552) 8-8317609, 427265 34641016 493039) 86-8881944) 4-290840 36055513 512000| 89442719) 4-303865 J7416574 531441) 9-0000000) 4326749 38729833. 951368) 90553851) 4344451 4 571787) 9-1104336/ 4-362071 41231056 592704 9-1651514) +7795 426407 614125) 92195445 +3968% 435839989 636055) 9-2736185, 4414005 44721 } WIAA 9 aT 1 4S Ths 45825757 681472) 9 4447960 46904158 704969) 9 11| 4-464745 47959315 39000 9 4868990) 4481405 4 753571) 9-5393920) 44975 5 8688) 9-5916630| 451435 50990195 B0357) 9-6496508) 4-530655) 51961524 B3054) 96959507) 4546536 § 2915026 857375) 9°7467943, 4-562904 53851648) 884736) 9-7979590| 4-578957 54772256 912673) 9-B488578) 459470 55677644) 941192) 9-s994949) 4-6 [ME 6568542) 7 99498744! 462606: §°7445696 1000000) 10-0000000) 4-64.15 58309519) | 1030301 756) 46570 59160798 1061208 1 HERI +4 26/725 60000000 1092727| 101488916, 4-687: 60827625 1124864) 101950390 470265 61644140 1157625) 10-2469506) 471763 62449980 1191016) 102956301) 4732623 6°3245553 1225043) 103440804) 4-74745% 64031 1259712] 10-9999048) 4-762203 64807407 1295029) 10-4403065) 4-776856 65574385 1331000) 10-4880885) 4°79142 66392496 1367631) 10-5356538) 4:2058. 6-7082039 1404928] 10:5830052) 482025 67823300 1442897] 10-6301458) 4-831585 68556546 1481544! 106770783) 4-848806 69232033 20875| 10°7238053) 4- 8629H 7-0000000 1560896} 10°7703296| 4°87699% 7-07 10678 1601613) 10°8166533) 489097: 71414254 549039) LO-A627805) 4-904865 72111026 1685159 | 4-918685 72301099 1728000] 10°9544512| 493242 73484692 1771561) 11-0000000) 4-94608 74161985 1815644] 1104153610) 495967 74893 148 1860867) 11-0905365, 4-97319 75499344 1906624! 11°1355287) 4-989664 76157731 1953125) 11°1803399) 5000000 7°63.11457 2000376) 122497292) 501329 TTS IAT 2048383) 112694277| 5-026526) 73102497 2097152| 11-3137085, 5-03968 78740079 2146689) 11-3578167| 5052774 79372539 2197000) 114017543) 506575 Bt 2249091) 114455231) 5-078754 80622577 2299968| 114891253 5-0916 8124034 2352637| 11:5325626| 5-104465 a1 2406104) 11:5758369| 5:117230

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APPENDIX. Sq. Root. |CubeRoot, No. Square, 11-6189500 11-6619035 | 12-0000000 4625) 120415946) i 120830460 a 12124355 12-1655251 12-2065556 12-2474487| 12-7279221 12-96 14814 13-0000000 13 IRERIER 13-0766968) 148| 13-1148770 | 13°1529464 13-1909060 5 141067360 | 141421356 14°1774469) * ]20998 o°142 5155137 516764! 5-180101 | SA rete BEM ER EEE REC EC ERNIE R EE HEHE WEL EEE y Die TI 40804 41206 41616 ey 14172488 aoe} 14212670 | 143478068 14 rTP a] 16°1245155 161554944 16-1864141 Ji p 7 6000000 bsi j et T e

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r p - a i7 ni 2 “at es a = Tæ ~*~ e 0 On O Oe GG GO Le p Pr i a 7 i F- - m CS LO Oo =k To Dor roe an = ry 310 311 314 Cube. | Sq. Root. CubeRoot. | 16-4012195] | APPENDIX FAJUL 39651621 40001688 4035360 40707554 41063625 41421736 — 41781923 42144192 42508549 42875000 43243551 43614208) - 43985977 44738875 45118016 Aa j| 18 JH 19 err L E 18-439 1889 ah) me a ual | 19-0000000 191049782 | 191311265 191572441 19-1833261 19-209378 19-9499373 19-97 498 UU > a C ama ad ~ Ta t Te z i E -E-i -L-1 -i-i - 1-1- 1-1-1- -CTT TETT: =. TG G : = re mr E = Pe Rak = 5 PAP AY AT eaa 47-37-10 =l VIII Oe eee rere ere re rere rere rere rere VIYIIIIIIIII za —=—= y E. " F " -x I = FA J FF... i | {HSS mt = = =g 7 f- eo a — — ea Pe "a Ce "eee =o A os “a D ho BS BS = i oe oe ee = a Ti iz DE a F = “4 - + LS oe BS Ce - a — pF a į , = a d E = T i = i or ed Ga hoe be E eo gE PEER Pe DS pai ; a s E. Ls Gi A p , Š i AL wll BRET RERESEE BREE a : ; : P. a tW EN i os = Ch 2 65 A5 GA Be DD fs OO be m= | ea

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APPENDIX. 17 TOEO a 96424 | 20-1990099| 7-416859|| 475) 225625] 107171875) 21-7944947| 7-802454 20-2237484 7°422914|| 476 226576) 107850176) 21-6174242) 7-807925 20-2484567| 7428959 477) 227529) 108531333) 21-8403297| 7-813385 202731349) 7-434994|| 478 228484) 109215352) 21-863211)| 7-8186 20-297783 g ! 0902239 ARGOGE 0 4 7691372 “697002 7702625) 524) 274576) 143577824 7°708239| 525) 275625) 144703125 7713345) 526| 276676) 145531576 | 29| 146363183 | 147197952 9437| 230867925 8-107913 B| 7-758402] 534| 285156 152273304| 231084400, B-11298 102503232) 21-6333077| 7: i S 236225) 153130375) 231300670, 8-11404]] 103161709 216561073) 7760402) $36 287296 159990656] 23:1516738, 8-1230% |

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361201 362404 ` 186169411 _ 187149218: ' 183132517, 189119224 25, 190109375 776 191102976 ' 192100033 334034! 193100552! 194104539! 336400; 195112000 | 1961229141 24i 197137368 198155287 199176704 206125971 207474688 208527857 20958458 4 210614875 211708736 212776173 213847192 214921799 216000000 217081801 218167208 219256227, 23° 1732605 23 1733902 * 23°1945502 | 273-515952) 5 5 23-7065392 23-8356063 23-9165215 239374184 23-9592971 23-9791576 24-0000000" 24-0205243, 21-0416306: 24-0624188 24-0631891' 24-1033416, 8 24- 1216762 24-1453929, 24-1660919 24-1867732! 24-2974369! 24-2230829, 24214937113, 24-2693222: 24-2899156 24-3104916; 24-3310501! 24-3515913; 243721152: 24-3326218! 24-4131112' 24-4335834 24-4540385 24.4744765 24-4918974| 245153013 APPENDIX. : Sy Roce CubeRoot ee eee eee eee DEE © 3133145 "133197 sv 13233 ‘344341 8-349126. 8°353905 650 8-358678, 8-363417, 8368209; 8372967! 8-377719], 8-332165 8-337206 8391942 8-396673 8-401398- 8-406118 8-410933 8-415542 8-420246! 8°424945) 8-429633), 8-139010] | | 8439010 629 ; 651! 423301 652 425104! 43:336 431619 432964' 656, 657| 653 659) 434281 286191179! 25 660| 435600 | 661! 436921! 662) 438244! 663, 439569 664: 440896 665, 442225 666! 443556 C68: 446224 298077632! 258456960 422500| 27 Cube. Sq. Root. Cubelo 24-5764115 & 221445125 24-5967473 9-1570 21-6170673 8-162315 24-637 370) §- Dau x t 8: Wilde 249193716 8-5316 21-9399278 8-5361 244140625 25 27589445 277167808 282300416, 283593393) 234890312, | 25°377 1551 25°396850: ` 246576560 §-4TICC | UOT RIO 9 215797106 8-5284% 25 25-1197134 8577153 25-1396102 8-591681 25-1594913 8-536205 25-1793566 8-593072 25-1992063 8-595238 25.2982213 8617735 25°3377 189. 8-626706 25°3574447' 8-631183 8°63565: 25°6515107| 8-69 6709953 8-702188 2874960001 25-6904652: 8-706588 288804781| 25-7099203 8-710983 290117528 25°7293607' 8-715373 291434247 25-7487864; 8-719760 292754944| 25-7681975! 8-72414 294079625" 25-7875939: 8-728518 295108296| 25-8069758: 8-732892 667! 444889| 296740963| 258263431 21:5356883| 8:443688!! 669! 447561] 299418309! 25-8650343 24:5560583| 8-448360|! 670! 448900| 300763000| 25-8843582 87 6/200 8°74162 874596 8°750340

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333 | F 333 22022228222333 ere ' --J g NO fant 522729 524 176, 525625; 529984 531441 532900 534361 541696 543169 302111711 458329; 310288733 527076] < 528529} < 535824) < APPENDIX. *Cube. 25-9036677 25-9229628 25-9422435 8754691 8759033)! 303464448 304821217 306182024 307546875 308915776 8°767719]| 26°0192237 26°0384331 26.0576284 260763096 26-0959767 26:1151297 261342687 26°1533937 26°1725047 322828856] 26-1916017 26-2106848 26-2297541 262483095 ‘2678511 ‘2868789 26°30538929 311665752 313046839 314432000 315821241 317214568 26°4575131 26°4764046 264952826 26°5141472 265329983 26°55 18361 26-5706605 265894716 26-6082694 26-6270539 266458252 266645833 26°6833281 26-7020598 267207784 267394839 267581763 267768557 267955220 26°8141754 26°8328157 26°85 14432 26°8700577 26°8886593| 8975241 26°9072481| 8-979377 26°9258240| 8°983509}! 269443872! 8-987637. 270000000 27-0185122 27°0370117 27-0594985 270739727 270924344 27.1108834 271293199 271477439 9-008223 9012329). 398688256 400315553 9028715 9-004113]| 797 | 798 Sq. Root. CubeRoot.|| No. | Square. 738 739 8:763331|| 740 741 742 743 744 745 746 747 743 749 750 751 753 799 803| 644809 9-032802\| 804 Cube. 401947272! 403583419 405224000 406369021 4085 18488 410172407 411830784! 413493625' j 415160936| 416832723 418508992] 420189749 421875000 423564751 425259008 426957777 423661064 430368875 432081216 433798093 435519512 437245479 438976000 440711081 442450728 444194947 445943744 447697125 96} 449455096 451217663 452984832 454756609 456533000 458314011 460099648 461889917 463684824 465484375 467288576 469097433 470910992 472729139 474552000 476379541 475211768 480048687 481890304 483736625 | 485937656 487443403 489303372 491169069 493039000 494913671 34: 496793088 498677257 - 500566184 502459879 j| 504358336 506261573 508169592 510082399 517781627 519718464 8q. Root. 27°1661554 19 CubeRoot. 9-036886 27-1845544! 9-04096 272029410 272213152 272396769 272580263 272763634 272946881 7°3 130006 27-3313007! 27:3195887 27°3678644 27:3861279 27°4043792 27:4226 184 27°4408455 274590604 27°4772633 276767050 27°6947648 27°7128129 27°730849z 27°7488739 27°7668865 27°7345880 278028775 278208555 2T'538B8218 278567766 278747197 278926514 279105715 27-9284801 279463772 279642629 279821372 280000000 28 0178515 28°0356915 28°0535203 28°0713377 28°089 1438 23-1069336 W 1247222 23142446 231602557 281780056 22W: 1957444 282134720 28:231 1354 28248893S 28-2665881 28°2842712 283019434 28°3196045 28°3372546 28°3548938 9-045042 9-049114 9053183 : 9-057249 9-06131 9065360 9-063422 9073173 907752 9-Us 156: 9- 85603) 9-084963 9-093672 9097701 9-101726 9-2599 ka 9-263797 9-267680 927155 9275435 9-279300 y-2%w83175 9-237044 9-290907 9.294767 9-298624

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B52) 725004 870 871) 758641 Cube. Bq. Root. |CubeRoot | | 521660125) 28 3725219 523600616 28-3901391| 9-306323 929957943) 28-4077454 | 527514112) 28 4253408 | 529475129) 259-442925; 531441000) 28-4604985 533411731) 28-4730617 387328) 24-4956137 59| 537367797, 29-5131549 539353144 23-5306852 625026375) 29-2403830 627222016, 29-2574777 629429793, 20-2745623 9 631624712, 292916370 636056000 29-3257566, 638277381) 2934423015 640509928 642795047, 293768616) 644972544) 29-393376y| 9 647214625| 29-4103823 649461896 29-4278779 651714363) 29-4448637 653972032| 29-4618397 656234909! 2394738059 658503000) 2941957624 660776311) 29-5127091] APPENDIX. 9302477 Saio 9531750 9535417 9-539067 FHH 9'546403/ 9550059 29.6954 m al 10094 20-7153] 73213 337 29-74894 idi 29-7657521 HiS “7R95459) iyi 29-7993239) 9-611791 29-816 IE ) 9-61 i 298329678) 961! #8496231) 9 299666481 of 29-9833287 9651317 70-O000000 9-65459" 300166620) 9-655468 30-0333 | 1446) 3662040 300499584) 9-66561¢ diq 9-669176 30-51292926| 9-76 8) 30-5286750 9-76 30-5450 9-771

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APPENDIX. 21 Now Square. | Cube, | Sq. Root. CubeR oot. | No, | Square. Cube. 939| 881721 827936019) 30-6431069) 9-792386 | 970| 940900, 912673000 940| 883600, 830584000) 30-6594194| 9:795861) 971| 942841) 91549861 } 941 942) | | B33237621) 30-6757233) 9°799334) 972) 944784) 918330048 835896888) 30-6920185) 9802804 | 973) 946729) 921167317 31- 838561807) 30-7083051) 9806271 974) M8676) 924010424 BA1L232384) 307245830 9809736 | 975| 950825) 926859375 945| 843908625 30-7408523/ 9813199 | 976| 952576| 929714176 J46 846590536| 30°7571130| 9:816659|| 977| 954529| 932574833 947 849278123' 30-7733651| 9820117 | 978) 956484) 935441352 948 851971392) 307896086) 9823572 | 979) 958441| 938313739 S49 854670349) 30-8058436| 9827025) 980) 960400) 94119200 | 950 j- 0| 9830476 | 981| 962361| 944076141 951 860085351| 30:8382879| 9833924 | 982| 964324| 94696616 952 862801408 30:8544972/ 9°837369) 983 9498620 953 865523177) 30°8706991| 9840813 | 984| 968256) 95276390 954 30- Pa| 985) 970225) 955671625 955 870983875| 30-9030743 9°847692| 986| 972196) 958585256 956 873722816 30-9192497) 9851128 | 987| 974169) 96150480 957 876467493 30:9354166] 9°854562| 288) 976144 958 879217912) 30-9515751) 9857993 | 989| 978121) 967361669 959 | 881974079| 30-9677251| 9-861422 | 990/ 980100) 970299000 50 0| 884736000) 30°9838668| 9°864848 | 991| 982081) 9732427271 961 887503681) 31-0000000| 9-868272'| 992) 984064! 976191485 962 890277128) 31-0161248| 9-871694| 993) 986049) 979146657 963 893056347) 31-0322413) 9-875113| 994| 988036) 982107784 964) 895841944 31-0493494| 9°878530| 995 985074875 (| 965) 9 898632125) 31-0644491 9881945 | 996) 992016] 988047936 | 966) 901428696) 31-0805405 sesan 997) 994009) 991026973 967| í 904231063) 31-0966236| 9888767 998) 996004| 994011992 968 907039232) 31:1126984| 9892175 | 999) 998001) 99700299% | 969) 909853209 31-1287648) 9895580 | 1000) 1000000) 1000000000) The allowing 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.—W hat is the square of 2,000? 2,000, divided by 1,000, a number found 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 di- 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 divisible without a remainder. Rule.—Divide the given number by such a number, from the forego-

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22 APPENDIX. ing table, as will divide it without a remainder ; then, the cube of the 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, therefore : 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? In 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 number 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 2 18522 18400 9200 9261 As 27,722 is to 27,661, so is 21 to 20:953 + the Ans. 21 27661 55322 27722)580881(20:953 + - 05444 264410 249498 149120 138610 105100 83166 21934

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APPENDIX. 93 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. Example.—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, therefore : 3°4641 3°3166 °1475 "14 5900 1475 -020650 3:3166 3:33725: the Ans. RULES FOR THE REDUCTION OF DECIMALS. To reduce a fraction to its equivalent decimal. Rule.—Divide the numerator by the denominator, annexing cyphers as required. Example.—What is the decimal of a foot equivalent to 3 inches ? 3 inches is ,3, of a foot, therefore : rs: 12) 3-00 -25 Ans. Another example.— W hat is the equivalent decimal of 7 of an inch? Z... . 8) 7:000 °875 Ans. To reduce a compound fraction to its equivalent decimal. Rule.—TIn accordance with the preceding rule, reduce each fraction, commen- cing at the lowest, to the decimal of 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 proceed to the last ; and the final product will be the answer. Example.—What is the decimal of a foot equivalent to 5 inches, 3 and %4 of an inch ? The fractions in this case are, 4 of an eighth, § of an inch, and -& of a foot, therefore :

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24 APPENDIX. Boece ee 2910 5 3: eighths. 5: inches. fy eee 12) 5°-487500 ‘453125 Ans. 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 j of re thus : a...... 8| 35000 r... ee 12| 5437500 °453125 Ans. To reduce a decimal to its equivalent in terms of lower denominatwns. Rule.—Multiply the given decimal by the number of parts in the next less denomination, and point off from the product as many figures at 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 at the left will be the answer. Example.—W hat is the expression in inches of 0°390625 feet ? 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 4 and q}. Another example.—What is the expression, in fractions of an inch, of 0:6875 inches ? Inches 0:6875 8 eighths in an inch. Eighths 5:5000 2 sixteenths in an eighth. Sixteenth 1:0 Ans., $ and py.

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_ TABLE OF CIRCLES. (From Gregory’s Mathematics.) ‘From this table may be found by inspection the area or circumfe- rence of a circle 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. é Bide of | Bide of | Diam. Area. Circum. | equgl sq. || Diam. Area, Circum. equal aq. 25 “4908 “78539 “20150 “715 19645 157079 44311)| 11- io “44178 290619 66467 25 1 "78539 314159 5 1-223715 392699 110778 “75 176714 471238 132934|| 12- | 75 9-40528 549778} 155089 -25 2 314159 -28315 PT5 D | 25 397607 706858 199401 "T9 -5 490873 785398 221556 13 ‘To 593057 863937 243715 25 2 706855) 9-42477 265568 4 “25 829576 10-21017 2889023 "5 9632113 10-99557 3°10179)| 14: “75 1104466 1178097. 332335 25 RAE AEB : 5 1963495, 15°70706) 443113 21°64753| 1649336, 465269 2375829) 1727875; 487424 25°96722) 18°06415) 509580 6 2827433) 1884955) 531736 “29 3067961; 1963495) 553891 5 aJ 18307) 2042035) 5°76047 "To 35-78470|/ = 21-20575) 598203 7 3848456; 21°99114) 620358 25 41°28249) 2277654) 642514 “5 4417864) 9 23-56194) 664670 tn ae 4717297, 2434734) 686925 8 50-2658 25139274) 7 1 25 5345616) 25°91813) 731137 5 5674501) 2670353 T5 75 6019204) 27-48893) 7°75445 g- 6361725) 2927433 7-97604 “29 6720063) 2905973 819759 5 7O-88218) 29°84513) 841915 75 7466191) 3063053 86407! 10 73-53981) 3141592) 8 86226 25 8251539; 32°29132) 9-08382 5 B6-59014) 3208672! FIAI) I- | 4*

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=J cee el NT sau go gog Eo.. b gag ee JA 5 ne ¥. 8 ; ee. ell oa SA ha SER ASR Circum. 66-75884| 67-4444) 63°32064 | 69'11503 69-9043) 70-68583 71:47123 83-25220) 8403760! APPENDIX. Side o 18-83232 190537 19-27543 1949699) 1971854 1994010 20-16166 20-38321' 20-6047 | 2082633 21 04788 2126944 21-49100 2171255 219A 2215567 22397722 22-59878 2282034 2304190) 20-2645, 23-4850], 2370657 2392812 24 14968, 24-37 124) 24-59279| 24-81135) 2503591 2525746 2547902] 2570058 25°92213| 26° 14369, 2036525 26:59680/ 26'R0L3G 27-02992 2725147 2747303 2769459] 2791614 2813770 JJA r) k È er a Zz iJ i E a e r e ————————— equal sq. | Diam. — — | 35° = = 3 abe. i = — th te [=z] i È = È E + . * F *# 8 oe = F z a & oT ian! We SOR BSR SSE SOR ASK ASR SSR BOR FSS iR BOR Sas S fn TOR ROR ai ts A "PGs be G EH: _.e ah Area, 113411494 114908660 1164°15642 1179-32442 1194°59060 1209-95495 122541748 1210-97818 125663704 12644910 127-2450 12801990 128:80529 129-523069 130-397609 13116149 171°21679) SEP aah ews ae: a he a - me T *) A h 38 TRSASRAGGHGRE SSS A FE ; FERLLIIEER 41°43110 41-652

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n ey e e e . . ~ dr TAR a oO ro tn an en © e e e bas e e e Q e e e re) ry s Ww e e. e ~t SERS ae © GI ~ren ed ar hyd ao” oO Ot wn” OH ta Or Or 25 5 - 75 63: e e . R e e s = e e es Sp al fad Area. ~ 2354-28008 2375°82244 2397-47698 2419-22269 2141-06657 2463-00864 2185-04887 2507-18728 2520-42387 2551-75863 2574°19156 2596-72267 2619-35196 2642-07942 2664 90505 2687-82886 2710-85084 2733-97100 2757-18933 2780-50584 2803-92053 2827-43338 2851-04442 2874-75362 2898-56100 2922-46656 2946-47029 2370-57220 2994-77228 3019-0754 3043-4€697 3067-96157 3092-55435 3117-24531 3142-03444 3166-92174 3191-90722 3216-99087 3242-17270 3267-45270 3292-83088 3318-30724 3313-88176 3369-55447 3395-32534 3421-19439 3417-16162 3473-22702 3199-39060 3525-65235 3552-01228 3578-47033 3605-02665 3631-68110 3658 ‘43373 3766-42597 3793-66947 3321-01115 3848°45 100 3375-98902 3903-62522 3931-35959 3959-19214 3987-12286 Circum, 175°14379 175°92918 176-71458 177:49998 178°28538 179-07078 179-856 17 180-64157 181:42687 182 21237 182:99777 183-78317 184:56856 18535396 186: 13936 186-92476 187° 71016 216:76989 217: 99529 equal sy. APPENDIX. Side of | 48-52092'| 71 48-74248 48-96403 49- 18559. 49-40715, 49-62870, 49-85026, 50-07182: 50-29337, 50-51493. 50-73649- 50-95804 | 51-17960, 51-40116' 51-6227], 51 84427) 52-06583, 5228738 5250894 3 ~? ~ E ee ee OP os SO es ee OE a a e eo 9% t o OR POR SOR SOR" BOR FOR" BOB ~J © g op ag 3 -l Ja z | Diam. | 5 75 63:14366)] 88- Area. ~ 4015: 19176 4043-27883 4071-50407 409982750 4128-24909 4156°76886 4185 38681 4214-10293 4242-91722 4271-82969 4300-84034 4329-94916 4359: 15615 4388-46132 4417-86466 4147-36618 4476-96588 4506-66374 4536-45979 4566-35400 4536 34640 4626-43696 4656-62571 4870-79579 4901-66992 4932-74225, 4963-91274 4995-18140 5026-54824 5058-01325 5089-57644 5121-23781 5152-99735 5184-85506 5216-81095 5248-86501 5281-01725 5313-26766: 5345-61624 5378-06301 5410-60794 5443-25105 5475-99234 5508-83180 5541-76944 5574-80525 5607-93923 5641-17139| 5674-50173! 5707-93023! 5741-45692: 5775-08178 5944 67869 5978-89260 6013-20468 6017-61494 6082-12337 Circum. 274: 19335 274:°88935 275°67475 276°46015 ide o. equal sq.

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Area. 611672998 61514476 Circum. 97724555 27803004 27881634 279°60174) 250-387 14) APPENDIX. Area, 6976-74097 7013-80194 7050-96 109 7088-21842 7125-57992 Circum. | equal eq. | 7163-02759 7200-57944 7238-22947| W 7275°97767| 30237829 7313-82404 735176859 7389-81131 7427-95221 746619129 750452853 754296396] 7581:49755| 7620-12933 7658:85927| 31023227 7697-68739 7736°61369| 311-80307 7775°63816| 312-58346 781476081) 313-37336 7853-98163) 2314-15926) The following rules are for extending the use of the above table. To find the area, circumference, or side of equal square, of a circle having a diameter of more than 100 inches, feet, gc. 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 divided by the square of the aforesaid mul- tiplier, will give the answer.

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APPENDIX, i 29 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)38-484 19:24? 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 : . l 5 X 5 = 25)201:0619(8:0424 + feet. Ans. 200 106 100 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 is required to contain 8 pounds of pure water; and since a cubic foot of pure water weighs 62:5 pounds, the gallon contains 221-184 cubic inches. Upon these data the following table is computed. One foot in depth of a cistern of 3 feet diameter will contain = - - - 55223 gallons. 34 do. do. - - - 75'164 do. 4 do. do. - - - 98:174 do. 44 do. do. - - - 124-252 do. 5 do. do. - - - 153-39. do. 54 do. do. - - - 185°611 do. 6 do. do. - - - 220°893 do. 6} do. do. - - : 259242 do. 7 do. do. - - - 30066 do. 8 do. do ° - ° - 392-699 do. 9 do. do. - - - 497:009 do. 10 do. do. - - - 613-592 do. 12 do. do. - - e- 883:573 do. Note.—To reduce cubic feet to gallons, divide by the decimal, "128.

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TABLE OF POLYGONS. (From Gregory’s Mathematics.) Multipliers for | Radius of cir- areas. cum. circle. Names. No. of rides eerste | ese re ee EES gh SEED | eE ED | Ec Factors for sides. 3 Trigon - - 0:4330127) 0:5773503| 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 8 Octagon 4:8284271| 1:3065628| 0-765367 9 Nonagon 6:1818242| 1:4619022| 0:684040 Decagon 7:6942088| 1°6180340) 0:618034 Undecagon 9:-3656399| 1:7747324| 0:563465 Dodecagon 11:1961524| 1:9318517| 0-517638 Fi 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. Example.-—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. Ex- ample.—What is the side of a regular octagon that may be inscribed within a circle, whose radius is 5 feet ? "765367 5 3-826835: Ans.

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WEIGHT OF MATERIALS. Woods. ee Metals. e ha, Appleg - 2 . - 49 Wire-drawn brass, - 534 Ash, à - - 45| Cast brass, - - 506 Beach, - - - - 40| Sheet-copper, - - 549 Birch, - ʻ - 45| Pure cast gold, - - 1210 Box, - - s - 60| Bar-iron, -. 475 to 487 Cedar, š - - 28| Castiron, - - 450 to 475 Virginian red cedar, - 40| Milled lead, - - - 718 Cherry, á = á 38| Cast lead, : P 709 Sweet chestnut, i - 36 Pewter, : - - 453 Horse-chestnut, - - 84| Pure platina, - - 1345 Cork, - - . - 15| Pure cast silver, - - 654 Cypress, - - - 28| Steel, - - 486 to 490 Ebony, - - - - 83| Tin - - - - 456° Elder, - - - 43| Zinc, - - z 439 Elm, - - - - 34| Stone, Earths, &c. Fir, (white spruce,) - 29| Brick, Phila. stretchers, 105 Hickory, - - - 52| North river common hard Lance-wood, - - 59 brick, - - - 107 Larch, - - ‘ - 31 Do. salmon brick, 100 Larch, (whitewood,) - 22| Brickwork, about - 95 Lignum-vite, - - - 83| Cast Roman cement, - 100 Logwood, - z 3 57| Do. and sand inequal parts, 113 St. Domingo mahogany, - 45| Chalk, - 144 to 166 Honduras, or bay mahogany, 35| Clay, - s - - 119 Maple, - - - 47| Potter’sclay, - 112 to 130 e White oak, - - 43to53{ Common earth, 95 to 124 Canadian oak, - - 54) Flit, - - - - 163 Red oak, - - - 47| Plate-glass, ae 172 Live oak, + « - 76| Crown-glass, - - - 157 White pine, - - 23to 30} Granite, - ` - 158 to 187 Yellow pine, - 34 to 44, Quincy granite, - - 166 Pitch pine, - - 461058; Gravel, - - - 109 Poplar, - - ‘ 25| Grindstone, - - - 134 Sycamore, - - - 36| Gypsum, (Plaster-stone,) 142 Walnut, œ - 40| Unslaked lime, - - 52

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&£P PENDIX. lbs. ina cubic foot. Limestone, - - 118 to 198| Common blue stone, Marble, - - 161to177| Silver-gray flagging, New mortar, - - - 107| Stonework, about, Dry mortar, - 90| Common plain tiles, Mortar with hair, (Plaster- Sundries. ing,) - - - - 105| Atmospheric air, Do. dry, - > 86| Yellow beeswax, - Do. do. including lath and nails, from 7 to 11 lbs. per superficial foot. Birch-charcoal, Oak-charcoal, Pine-charcoal, Crystallized quartz, - 165| Solid gunpowder, Pure quartz-sand, - 171| Shaken gunpowder, Clean and coarse sand, 100| Honey, : Welsh slate, - - - 180| Milk, š < Paving stone, - - 151| Pitch, - - Pumice stone, š - 56| Sea-water, - Nyack brown stone, - 148| Rain-water, - Connecticut brown stone, 170| Snow, - : Tarrytown blue stone, - 171| Wood-ashes, - THE END. KS Ht

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