'■ af'swwK'^iB-fgagawBMtwwHwiijapWHiawHf 
 
^6 
 
 THE 
 
 Imerican house-carpenter : 
 
 A TREATISE UPON 
 
 ARCHITECTURE, • 
 
 CORNICES AND MOULDINGS 
 
 FRAMING, 
 
 DOORS, WINDOWS, AND STAIRS 
 
 TOGETHER WITH 
 
 THE MOST IMVORTAKT PRINCIPLES 
 
 OF 
 
 PRACTICAL GEOMETRY. 
 
 BY R. G. HATFIELD, 
 
 ARCHITECT. 
 FIFTH EDITION, WITH ADDITIONS, 
 
 JUustrateD bp more t{)dn V^x^t i)un^reti 2inarc-;biua3 
 
 NEW YORK: 
 JOHN WILEY, 16V BROADWAY 
 
 1 8 52. 
 
X 
 
 .A 
 
 Entered, according to Act of Congress, in tiie year 1844, by 
 
 R. G. HATFIELD, 
 
 in tlift Clerk's Office of the District Court of the Southern District of New York. 
 
 R, CRAIGHEAD, PRINTER, 
 
 53 Vesey Street, X Y 
 
 f\ 
 
 ^ 
 
 J 
 
PREFACE. 
 
 This 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 a 
 great measure, to the costliness of the plates with 
 which they are illustrated : an imnecessary 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 
 other work ; and the figures have all been drawn ex- 
 pressly for this book. In doing this, the utmost care 
 has been taken to make every thing as plain as the 
 nature of the case would admit. 
 
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 method of proportioning a cor- 
 nice to a larger given one ; a way to determine the 
 lengths and bevils of rafters for hip-roofs ; a way to 
 proportion the rise to the tread 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-rjiiling is presented, in many re- 
 spects, in a new, and, it is hoped, more practical form 
 than in previous treatises on that subject. 
 
 The author has endeavoured to present a fund of 
 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. 
 
PREFACE TO THE FIFTH EDITION. 
 
 Since the first edition of this work was pubUshed, I 
 have received numerous testimonials of its excellent 
 practical value, from the very best sources, viz. from 
 the workmen themselves who have used it, and who 
 have profited by it. As a convenient manual for 
 reference in respect to every question relating either 
 to the simpler operations of Carpentry or the more 
 intricate and abstruse problems of Geometry, those who 
 have tried it assure me that they have been greatly 
 assisted in using it. And, indeed, to the true workman, 
 there is, in the study of the subjects of which this 
 volume treats, a continual source of profitable and 
 pleasurable interest. Gentlemen, in numerous instances 
 have placed it in the hands of their sons, who have 
 manifested a taste for practical studies ; and have also 
 procured it for the use of the workmen upon their 
 estates, as a guide in their mechanical operations. I 
 was not, then, mistaken in my impressions, that a work 
 of this kind was wanted ; and this evidence of its 
 usefulness rewards me in a measure for the pains 
 taken in its preparation. 
 
 R. G. H. 
 
 New York, Oct. 1, 1852. 
 
TABLE OF CONTENTS. 
 
 INTRODUCTION. 
 
 Articles necessary for drawing, 
 To prepare the paper, 
 
 Art. 
 •2 
 5 
 
 To use the set-square, 
 Directions for drawing, 
 
 Art. 
 11 
 13 
 
 SECT. I.— PRACTICAL GEOMETRY. 
 
 DEFINITIONS. 
 
 Lines, - - . . 17 
 
 Angles, - - - 23 
 
 Angular point, . - - 27 
 
 Polygons, - - - 28 
 
 The circle, - - - 47 
 
 The cone, ... 56 
 
 Conic sections, - - - 58 
 
 The ellipsis, - - - 61 
 
 The cylinder, . - . 68 
 
 PROBLEMS. 
 
 To bisect a line, - - 71 
 
 To erect a perpendicular, - 72 
 
 To let fall a perpendicular, 73 
 
 To erect ditto on end of line, 74 
 
 Six, eight and ten rule, - 74 
 
 To square end of board, - 74 
 
 To square foundations, &c., 74 
 To let fall a perpendicular 
 
 near the end of a line, - 75 
 
 To make equal angles, - 76 
 
 To bisect an angle, - . 77 
 
 To .risect a right angle, 78 
 
 To draw parallel lines, - 79 
 To divide a line into equal 
 
 parts, ... - 80 
 
 To find the centre of a circle, 81 
 
 To draw tangent to circle, 82 
 
 Do. without using centre, 83 
 
 To find the point of contact, 84 
 To draw a circle through three 
 
 given points, - - 85 
 
 To find a fourth point in circle, 86 
 To describe a segment of a 
 circle by a set-triangle, - 87 
 Do. by intersection of lines, 88 
 To curve an angle, - 89 
 
 To inscribe a circle within a 
 
 given triangle, - - 90 
 To make triangle about circle, 91 
 To find the length of a cir- 
 cumference, - - 92 
 To describe a triangle, hexa- 
 gon, &c., ... 93 
 To draw an octagon, - 94 
 To eight-square a rail, &c., 94 
 To describe any polygon in 
 
 a circle, ... 95 
 
 To draw equilateral triangle, 96 
 To draw a square by com- 
 passes, - - - 97 
 To draw any polygon on a 
 
 given line, ... 98 
 To form a triangle of required 
 
 size, . . - . 99 
 
 To copy any right-lined figure, 100 
 To make a parallelogram 
 
 equal to a triangle, - 101 
 
 To find the area of a triangle, 101 
 To make one parallelogram 
 
 equal another, - - 102 
 To make one square equal to 
 
 two others, - - - 103 
 To find the length of a rafter, 103 
 
Vlll 
 
 CONTENTS. 
 
 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, - - 104 
 
 To make one polygon equal 
 
 to two or more, - - 104 
 To make a square equal to 
 
 a rectangle, - - 105 
 
 To make a square equal to 
 
 a triangle, - - - 106 
 To find a third proportional, 107 
 To find a fourth proportional, 108 
 To proportion one ellipsis to 
 
 another, - - - 108 
 To divide a line as another, 109 
 To find a mean proportional, 110 
 Definitions of confc sections, 111 
 To find the axes of an ellipti- 
 cal section, - - - 112 
 To find the axes and base of 
 
 the parabola, - - 113 
 
 To find the height, base and 
 
 axes of the hyperbola, - 114 
 To find foci of ellipsis, - 115 
 To describe an ellipsis with 
 
 a string, - - - 115 
 
 To describe an ellipsis with 
 
 a trammel, - - 116 
 
 To construct a trammel, - 116 
 To describe an ellipsis by or- 
 
 dinates, - - - 117 
 To trace a curve through 
 
 given points, - - - 117 
 To describe an ellipsis by in- 
 tersection of lines, - 118 
 
 120 
 
 121 
 
 122 
 128 
 
 124 
 
 125 
 
 126 
 
 127 
 128 
 
 Art. 
 
 Do. from conjugate diameters, 118 
 
 Do. by intersecting arcs, - 119 
 
 To describe an oval by com- 
 passes, - - - 
 
 Do. in the proportion, 7x9, 
 5x7, &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 proportionate to 
 given ones, 
 
 To describe a parabola by in- 
 tersection of lines, - 
 
 To describe hyperbola by do., 
 
 DEMONSTRATIONS. 
 
 Definitions, axioms, &;c.. ISO. 139 
 Addition of angles, - 140 
 Equal triangles, - - 141 
 Angles at base of isoceles tri- 
 angle equal, - - 142 
 Parallelograms divided equal- 
 ly by diagonal, - - 143 
 Equal parallelograms, - 144 
 Parallelogram equal triangles, 146 
 To make triangle equal poly- 
 gon, - - . . 147 
 Opposite angles equal, • - 148 
 Angles of triangle equal two 
 
 riglit angles, - - - 149 
 
 Corollaries from do., 150. 155 
 Ani^le in semi-circle a rijrht 
 
 angle, - . - 156 
 
 Hecatomb problem, - - 157 
 
 SECT. II.— ARCHITECTURE. 
 
 HISTORY. 
 
 Antiquity of its origin, - 159 
 Its cultivation among the an- 
 cients, - - - 160 
 Among the Greeks, - - 161 
 
 Among the Romans, - 162 
 Ruin caused by Gollis and 
 
 Vandals, - - - I'^S 
 
 Of the Gothic, . - 164 
 
 Of the Lombard, - - 165 
 
CONTENTS. 
 
 IX 
 
 Art. 
 
 Of the Byzantine and Oriental, 166 
 Moorish, Arabian and Modern 
 
 Gothic, - - - 167 
 
 Of the English, - - 168 
 Revival of the art in the sixth 
 
 century, - - - 169 
 The art improved in the 14th 
 
 and 15th centuries, - 170 
 
 Roman styles cultivated, 171 
 
 STYLES. 
 
 Origin of different styles, 172 
 
 Stylobate and pedestal, - 173 
 
 Definitions of an order, - 174 
 Of the several parts 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, - 191 
 
 Characteristics, - - 192 
 
 Intercolumniation, - - 193 
 
 Adaptation, - - - 194 
 
 To describe the volute, - 195 
 
 Origin of the Corinthian, - 196 
 
 Adaptation, - - - 197 
 
 Persians, - - - - 199 
 
 Caryatides, - - - 200 
 
 The Roman orders, - - 202 
 
 Ah. 
 Extent of Roman structures, 203 
 Roman styles copied from 
 
 Grecian, - - . 208 
 Origin of the Tuscan, - 204 
 
 Adaptation, - - . 205 
 Characteristics of the Egypt- 
 ian, .... 206 
 Extent of Egyptian structures, 206 
 Adaptation, - - . 207 
 Appropriateness of design, 208. 211 
 Durable structures, - - 212 
 Plans of dwellings, &c., 213 
 
 Directions for designing, 213, 214 
 
 PRINCIPLES. 
 
 Origin of the 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, - - - 224 
 The arch, - - . 225 
 The vault, - - - 226 
 The dome, - - - 227 
 The roof, - - - 228 
 
 SECT. HI.— MOULDINGS, CORNICES, &c. 
 
 MOULDINGS, &C. 
 
 Elementary forms, - - 229 
 
 Characteristics, - - 230 
 
 Grecian and Roman, - - 231 
 
 Profile, - - - 332 
 
 To describe the torus and 
 
 scotia, - - - - 233 
 
 To describe the echinus, 234 
 
 To describe the cavetto, 235 
 
 To describe the cyma-recta, 236 
 
 To describe the cyma-reversa, 237 
 
 Roman mouldings, - 238 
 
 Modern mouldings, - - 239 
 Antse 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 
 
CONTENTS. 
 
 SECT. IV.— FRAMING. 
 
 Art. 
 
 Laws of pressure, - - 248 
 
 Parallelogram of forces, - 248 
 To measure the pressure on 
 
 rafters, - - - 249 
 Do. on tie-beams, - 250 
 The effect of position, - 251 
 The composition of forces, 252 
 Best position for a strut, - 253 
 Nature of ties and struts, - 254 
 To distinguish ties from struts, 255 
 Lattice-work framing, - 256 
 Direction of pressure in raft- 
 ers, - - - - 257 
 Oblique thrust of lean-to roofs, 258 
 Pressure on floor-beams, - 259 
 Kinds of pressure, - - 260 
 Resistance t» compression, 261 
 Area of post, - - 261 
 Resistance to tension, - 262 
 Area of suspending piece, 262 
 Resistance to cross-strains, 263 
 Area of bearing timbers, 263 
 Area of stiffest beam, - 264 
 Bearers narrow and deep, 265 
 Principles of framing, - 266 
 
 FLOORS. 
 
 Single-joisted, - - 267 
 
 To find area of floor-timbers, 268 
 
 Dimensions of trimmers, &c., 269 
 
 Strutting between beams, 270 
 
 Cros8-furring and deafening, 271 
 
 Double floors, - - - 272 
 
 Dimensions of binding-joists, 273 
 
 Do. of bridging.joists, 274 
 
 Do. of ceiling-joists, - 275 
 
 Framed floors, - - - 276 
 
 Dimensions of girders, - 277 
 
 Girders sawn and bolted, - 278 
 
 Trussed girders, - * 279 
 
 Floors in general, - - 280 
 
 PARTITIONS. 
 
 Nature of their construction, 281 
 
 Designs for partitions, - 282 
 
 Superfluous timber, - - 282 
 
 Improved method, - - 283 
 
 Weight of partitioning, - 284 
 
 ROOFS. 
 
 Lateral strains, 
 Pressure on roofs, 
 Weight of covering, 
 •Definitions, 
 Relative size of timbers, 
 
 Art 
 285 
 
 286 
 286 
 
 287 
 288 
 
 To find the area of a king- post, 289 
 Of a queen-post, - - 290 
 Of a tie-beam, - - - 291 
 Of a rafter, - - - 292 
 Of a straining-beam, - 294 
 
 Of braces, - - - 295 
 Of purlins, - - - 296 
 
 Of common rafters, - 297 
 
 To avoid shrinkage, - - 298 
 Roof with a built-rib, - 299 
 Badly-constructed roofs, - 300 
 To find the length and bevils 
 
 in hip-roofs, - - 301 
 
 To find the backing of a hip- 
 rafter, ... - 302 
 
 DOMES. 
 
 With horizontal ties, - 303 
 Ribbed dome, - - - 304 
 Area of the ribs, - - 305 
 Curve of equilibrium, - 30^^ 
 To describe a cubic parabola, 307 
 Small domes for stairways, 308 
 To find the curves of the ribs, 309 
 To find the shape of the cover- 
 ing for spherical domes, 310 
 Do. when laid horizontally, 311 
 To find an angle-rib, - - 312 
 
 BRIDGES. 
 
 Wooden bridge with tie-beam, 313 
 
 Do. without a tie-beam, 314 
 
 Do. with a built-rib, 315 
 
 Table of least rise in bridges, 315 
 
 Rule for built- ribs, - - 315 
 
 Pressure on arches, - 316 
 
 To form bent-ribs, - - 317 
 
 Elasticity of timber, - 317 
 
 To construct a framed rib, 318 
 
 Width of roadway, &c., - 319 
 
 Stone abutments and piers, 320 
 
 Piers constructed of piles, 321 
 
CONTENTS. 
 
 XI 
 
 Art. 
 
 Piles in ancient bridges, 321 
 
 Centring for stone bridges, 322 
 
 Pressure of arch-stones, - 322 
 Centre without a tie at the 
 
 base, - - - 323 
 
 Construction of centres, - 324 
 
 General directions, - 325 
 
 Lowering centres, - - 326 
 
 Relative size of timbers, - 327 
 
 Short rule for do. - - 328 
 
 Joints between arch-stones, 329 
 
 Do. in elliptical arch, - 330 
 
 Do. in parabolic arch, - 331 
 
 JOINTS. 
 
 Art. 
 Scarfing, or splicing, 332. 334 
 To proportion the parts, - 335 
 Joints in beams and posts, - 336 
 Joints in floor-timbers, - 337 
 Timber weakened by framing, 338 
 Joints for rafters and braces, 339 
 Evil of shrinking avoided, - 340 
 Proper joint for collar-beam, 341 
 Pins, nails, bolts and straps, 342 
 Dimensions of straps, - 342 
 To prevent the rusting of 
 straps, . - . . 342 
 
 SECT, v.— DOORS, WINDOWS, (fee. 
 
 DOORS 
 
 Dimensions of doors, - - 343 
 To proportion height to width, 344 
 Width of stiles, rails and 
 
 panels, . . . 345 
 Example of trimming, - 346 
 
 Elevation of a door and trim- 
 mings, - - - 347 
 General directions for hang- 
 ing doors, - - - 348 
 
 WINDOWS. 
 
 To determine the size, - 349 
 To find dimensions of frame, 350 
 To proportion box to flap 
 
 shutter, - - - 351 
 To proportion and arrange 
 
 windows, - - - 352 
 
 Circular-headed windows, 353 
 
 To find the form of the soffit, 354 
 
 Do. in a circular wall, - 355 
 
 SECT. VI.— STAIRS. 
 
 Their position, - - - 356 
 
 Principles of the pitch-board, 357 
 To proportion the rise to the 
 
 tread, - - - 358 
 
 The angle of ascent, - - 359 
 
 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 
 
 To cut the lower edge of do., 365 
 
 To place the balusters, - 366 
 
 To find the moulds for the 
 
 rail, - - - - 367 
 
 Elucidation of this method, 368 
 
 Two other examples, 369, 370 
 To apply the mould to the 
 
 plank, - - - 371 
 
 To bore for the balusters, - 372 
 
 Face-mould for moulded rail, 373 
 
 To apply this mould to plank, 374 
 
 To ascertain thickness of stuff, 375 
 
 WINDING STAIRS- 
 
 Flyers and winders, - 376 
 
 To construct winding stairs, 377 
 
Xll 
 
 CONTENTS. 
 
 Art. 
 
 Timbers to support winding 
 
 stairs, ... - 378 
 To find falling-mould of rail, 379 
 To find face-mould of do., 380 
 Position of butt-joint, - 380 
 To ascertain thickness of 
 
 stufl:; - - . - 381 
 To apply the mould to plank, 383 
 Elucidation of the butt-joint, 384 
 Quarter-circle stairs, - 385 
 Falling-mould for do., , - 386 
 Face-mould for do., - 387 
 Elucidation of this method, 388 
 To bevil edge of plank, - 389 
 To apply moulds without be- 
 villing plank, - -. 390 
 
 AH. 
 
 To find bevils for splayed- 
 
 work, - - - 391 
 
 Another method for face- 
 moulds, - - - 892 
 
 To apply face-mould to plank, 394 
 
 To apply falling-mould, . 395 
 
 SCROLLS. 
 
 General rule, - - 396 
 
 To describe scroll for rail, 398 
 
 For curtail-step, - - 399 
 
 Balusters under scroll, - 400 
 
 Falling-mould for scroll, - 401 
 
 Face-mould for do., - 402 
 
 Round rails over winders, - 403 
 
 To find form of newel-cap, 404 
 
 SECT. VII.— SHADOWS. 
 
 Inclination of line of shadow, 497 
 Shadows on mouldings, 408 
 
 Shadow of a shelf, - - 409 
 Of a shelf of varying width, 410 
 Of do. with oblique end, 411 
 
 Of an inclined shelf, - 412 
 
 Of do. inclined in section, 413 
 Of do. having a curved edge, 414 
 Of do. curved in elevation, 415 
 Shadow on cylindrical wall, 416 
 Do. on inclined wall, - 417 
 Shadow of a beam, - - 418 
 
 Shadow in a recess, - 419 
 
 Do. with wall inclined, - 420 
 
 Shadow in a fireplace, . 421 
 
 Shadow of window lintel, - 422 
 
 Shadow of step-nosing, - 423 
 
 Of a pedestal upon steps, - 424 
 Of square abacus on column, 425 
 
 Of circular abacus on do. 426 
 
 On the capital of a column, 427 
 
 Of column and entablature, 428 
 
 Shadows on Tuscan cornice, 429 
 
 Reflected light, - - 430 
 
 APPENDIX. 
 
 Page. 
 
 Glossary of Architectural Terms, - - ' - - 3 
 
 Table of Squares, Cubes and Roots, - - • - 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, - - - 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, - - 30 
 
 Table of Weights of Materials, - - - - - 31 
 
INTRODUCTION. 
 
 Art. 1. — A knowledge of the properties and principles of lines 
 can best 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, 
 £ind thus be able to make them available in any unexpected 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 16 
 
 I 
 
4 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, perfectly 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 preventing 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 stufi 
 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 commonly 
 be what is called drawing-paper ; as this is rather costly, and 
 will, where much is used, make quite an item of expense. 
 Cartridge-paper, as it is called, of about 20 by 26 inches, and of 
 as good a quality nearly as drawing-paper, can be bought for 
 about 50 cts. a quire, or 2 pence a sheet ; and each sjieet 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. 
 
 5. — 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 oflf 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 
 
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 papei 
 against the edge of the ruler, and put one end of the cake o1 
 mouth-glue between your lips to dampen it. Then holding it 
 upright, rub it against and along the entire edge of the paper 
 that is 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 it a strip of pretty stiff paper. By rubbing 
 upon this, not very hard but pretty rapidly, with the thumb nail 
 of the right hand, so as to cause a gentle friction, and heat to be 
 imparted to the glue that is on the edge of the paper, you will 
 make it adhere to the board. The other edges in succession 
 must be treated in the same manner. 
 
 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. 
 
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. 
 
 Fig. L 
 
INTRODUCTION. 
 
 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 hole 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, w^hen 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 twelfths, instead ot 
 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 mch, 
 use that part of the scale which is divided into one-fourths of an 
 
" 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. 
 
 ll.-In drawing p^allel 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 
 direction oblique to either side of the board, the set-square must 
 be used. Let a 6, {Fig. 2,) be a line, parallel to which it is 
 
 Fiff. 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/ ; and a line drawn by the edge, if, will be parallel to a b. 
 12.— To draw a line, as k I, {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, k; then the 
 line, I k, drawn by it, will be perpendicular to a b. In like 
 
INTRODUCTION. 
 
 manner, the drawing of other problems may be facilitated, as will 
 be d' tcovered in using the instruments. 
 
 Fig. 3. 
 
 13. — In drawing a problem, proceed, with the pencil sharpened 
 » 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, 
 imtil a sujSiciency 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 camePs-hair pencil, add a little water to the 
 ink that was rubbed on the plate, and mix it well. It should be 
 diluted sufiiciently to flow freely from the pen, and yet be thick 
 enough to make a black line. With the hair pencil, place a 
 little of the ink between the nibs of the drawing-pen, and screw 
 the nibs together until the pen makes a fine line. Beginning 
 with the curved lines, proceed to ink all the lines of the figure ; 
 being careful now to make every line of its requisite length. If 
 they are a trifle too short or too long, the drawing will have a 
 ragged appearance ; and this is opposed to that neatness and 
 accuracy which is indispensable to a good drawing. When the 
 ink is dry, efiace the pencil-marks with the india-rubber. If 
 
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. 
 
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. i^Fig. 4.) 
 
 Fig. 4. Fig. 5. Fig. & 
 
 2 
 
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 6.) 
 
 26. — An acute angle is less than a right angle. [Fig. 4.) 
 
 27. — An obtuse angle is greater than a right angle. [Fig. 6.) 
 
 When an angle is denoted by three letters, the middle one, in 
 the order they stand, denotes the angular point, and the other 
 two the sides containing the angle ; thus, let a 6 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.) 
 
 Fig. 7. 
 
 Fig. & 
 
 29. — An equi-lateral triangle has its three sides equal. 
 {Pig. 7.) 
 
 30. — An isoceles triangle has only two sides equal. [Fig. 8.) 
 31. — A scalene triangle has all its sides unequal. [Fig. 9) 
 
 Fix. 9. 
 
 Fig. 10. 
 
 32. — A right-angled triangle has one right angle. [Fig. 10.) 
 
 33. — An acute-angled triangle has all its cingles acute. 
 [Fig. 7 and 8.) 
 
 34. — An obtuse-angled triangle has one obtuse angle. 
 [Fig. 9.) 
 
 35. — \ quadrangle has four sides and four angles. [Fig, 11 
 to 16.) 
 
PRACTICAL GEOMETRY. 
 
 11 
 
 Fig. 11. 
 
 Fig. 12. 
 
 36. — A parallelogram is a quadrangle having its opposite 
 sides parallel. {Fig. 11 to 14.) 
 
 37. — A rectangle is a parallelogram, its angles being right 
 angles. {Fig. 11 and 12.) 
 
 38. — A square is a rectangle having equal sides. {Fig. 11.) 
 
 39. — A rhombus is an equi-lateral parallelogram having ob- 
 lique angles. {Fig. 13.) 
 
 Fig. 13. 
 
 Fig. 14. 
 
 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. 
 
 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 
 
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 
 sides. 
 
 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. 
 
 48. — The radius of a circle is a right line drawn from the 
 
 centre to any point in the circumference, {a 6, Fig. 21.) 
 
 All the radii of a circle are equal. 
 
 Fi-. 21. 
 
 49. — The diameter is a right line passing through the centre, 
 and terminating at two opposite points in the circumference. 
 Hence it is twice the length of the radius, (c d, Fig. 21.) 
 
 50. — An arc of a circle is a part of the circumference, (c b or 
 bed, Fig. 21.) 
 
 51. — A chord is a right line joining the extremities of an arc 
 (6 d, Fig. 21.) 
 
PRACTICAL GEOMETRY. 
 
 13 
 
 52. — ^A segment is any part of a circle bounded by an arc and 
 its chord. {A, Fig. 21.) 
 
 53. — A sector is any part of a circle bounded by an arc and 
 two radii, drawn to its extremities, (i?, 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. [f g^ 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.) 
 
 Fig. 22. 
 
 57.— The axis of a cone is a right line passmg 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 6, Fig. 23.) 
 
 59. — A parabola is described if a cone be cut by a plane, 
 parallel to a plane touching the curved surface, (c rf, Fig. 23 — 
 c d being parallel to / 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, (e A, Fig. 23.) 
 
 61. — Foci are the points at which the pins are placed in de- 
 scribing an ellipse. (See Art. 115, and/,/, Fig, 24.) 
 
14 
 
 AMERICAN HOUSE-CARPENTER. 
 
 62. — The transverse axis is the longest diameter of the 
 ellipsis, [a b, Fig. 24.) 
 
 63. — The conjugate axis is the shortest diameter of the 
 ellipsis ; and is, therefore, at right angles to the transverse axis. 
 (c d, Fig. 24.) 
 
 64. — The parameter is a right line passing through the focus 
 of an ellipsis, at right angles to the transverse axis, and termina- 
 ted by the curve, {g h and g t, Fig. 24.) 
 
 65. — A diameter of an ellipsis is any right line passing 
 through the centre, and terminated by the curve, {k I, 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 I. 
 
 67. — A double ordinate is any right line, crossing a diameter 
 of an ellipsis, and drawn parallel to a tangent at the extremity of 
 that diameter, {i t, Fig. 24.) 
 
 68. — A cylinder is a solid generated by the revolution of a 
 right-angled parallelogram, or rectangle, about one of its sides ; 
 and consequently the ends of ^Lie :ylinder are equal circles. 
 (Fig. 25.) 
 
PRACTICAL GEOMETRY. 
 
 15 
 
 Fig. 25. 
 
 Fig. 26. 
 
 69. — The axis 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.) 
 
PROBLEMS. 
 
 RIGHT LINES AND ANGLES. 
 
 Tl. — To bisect a line. Upon the ends of the hne, a 6, {Fig. 
 ^5) as centres, with any distance for radius greater than hall 
 
 a b, describe arcs cutting each other in c and d ; draw the line, 
 
 c d, and the point, e, where it cuts a b, will be the middle of the 
 
 line, a b. 
 
 In practice, a line is generally divided with the compasses, or 
 dividers ; but this problem is useful where it is desired to draw, 
 at the middle of another line, one at right angles to it. (See 
 Art. 85.) 
 
 d 
 
 a 
 Fig. 28. 
 
 72, — To erect a perpendicular* From the point, a, {Fig* 28,) 
 
PRACTICAL GEOMETRY. 
 
 ir 
 
 set oflf any distance, as a 6, and the same distance from a to c ; 
 upon c, as a centre, with any distance for radius greater than c a, 
 describe an arc at d ; upon 6, with the same radius, describe 
 another at d ; join d and a, and the Une, d a, will be the per- 
 pendicular required. 
 
 This, and the three following problems, are more easily per- 
 formed by the use of the set-square — (see Art. 12.) Yet they 
 are useful when the operation is so large that a set-square cannot 
 be used. 
 
 Fig. 29. 
 
 73. — To let fall a 'perpendicular. Let a, [Fig. 29,) be the 
 point, above the line, h c, from which the perpendicular is re- 
 quired to fall. Upon a, with any radius greater than a d, de- 
 scribe an arc, cutting 6 c at e and/; upon the points, e and/, 
 with any radius greater than e d, describe arcs, cutting each 
 other at g ; join a and g, and the line, a d, will be the perpen- 
 dicular required. 
 
 Fig. 30. 
 
 74. — To erect a perpendicular at the end of a line. Let a, 
 (Fig. 30,) at the end of the line, c a, be the point at which the 
 perpendicular is to be erected. Take any point, as 6, above the 
 
 3 
 
18 
 
 AMERICAN HOUSE-CARPENTER. 
 
 line, c a, and with the radius, b a, describe the arc, d a e, 
 
 through d and Z>, draw the hue, d e ; join e and a, then e a 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, 6, and Art. 84. 
 For proof of its correctness, see Art. 156.) 
 
 74, a. — A second method. Let &, {Fig. 31,) at the end of th* 
 line, a b, be the point at which it is required to erect a perpen- 
 dicular. Upon 6, with any radius less than b a, describe the arc, 
 c e d ; upon c, 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 
 other radius greater than half e d, describe arcs intersecting at / ; 
 join/ and 6, and the line,/ 6, will be the perpendicular required. 
 
 d 
 
 Fig. 32. 
 
 74^ 5. — 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 ef; upon d, with 
 the same radius, describe an arc at e, and upon e, another at c ; 
 
PRACTICAL GEOMETRY. 19 
 
 through d and e, draw d c, cutting the arc in c ; join c and 6, 
 then c h will be the perpendicular required. 
 
 This problem can be solved by the six^ 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 c?, [Fig. 30,) equal eight, and a e, six ; then, \i d e equals ten, 
 the angle, e a c?, is a right angle. Because the square of six 
 and that of eight, added together, equal the square of ten. thus : 
 6 X 6 = 36, ai*d 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 c a be the edge of the board, 
 and a the point at which it is required to be squared. Take the 
 point, 6, as near as possible at an angle of forty-five degrees, or on 
 ami^re-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 h to a, and then bring the leg at a around to d ; 
 draw a line from c?, through 6, out indefinitely ; take the dis- 
 tance, d b, and place it from b to e ; join e and a ; then e a will 
 be at right angles to c 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 c?, 
 
 obliquely to the line, c a ; bisect e d at b ; upon 6, with the 
 
 radius, b e, describe the arc, e a d ; join e and a ; then e a will 
 
 be the perpendicular required. 
 
 76. — To make an angle, (as e df Fig. 33,) eqnal to a given 
 angle, (as b a c.) From the angular point, a, with any radius 
 describe the arc, be; and with the same radius, on the line, d e, 
 
20 
 
 AMERICAN HOUSE-CARPENTER. 
 
 and from the point, c?, describe the arc,/^; take the distance, 
 
 b c, and upon g, describe the small arc at/; join /and d ; and 
 
 the angle, e df, will be equal to the angle, b a c. 
 
 If the given line upon which the angle is to be made, is situa- 
 ted parallel to the similar line of the given angle, this may be 
 performed more readily with the set-square. (See Art. 11.) 
 
 Fig. 34. 
 
 77. — To bisect an angle. Let a b c, {Fig. 34,) be the angle 
 
 to be bisected. Upon b, with any radius, describe the arc, a c ; 
 
 upon a and c, with a radius greater than half a c, describe arcs 
 
 cutting each other at d ; join b and d ; and b d will bisect the 
 
 angle, a 6 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. 
 
 Fig. 35. 
 
 78. — To trisect a right angle. Upon a, {Fig. 35,) with any 
 
 tadius, describe the arc, b c ; upon b and c, with the same radius, 
 
 describe arcs cutting the arc, 6 c, at c? 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- 
 
PRACTICAL GEOMETRY. ^tl 
 
 fore, is equal to three, and a quadrant to one and a half of those 
 parts. The radius, therefore, is equal to f of a quadrant ; and 
 this is equal to a right angle. 
 
 
 Fig. 3G. 
 
 79. — Through a given pointy 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 </, in the line, h 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 a c ; through e and o, 
 draw the line, e a ; which will be the line required. 
 
 This is upon the same principle as Art. 76. 
 
 80. — To divide a given line into any number of equal parts. 
 
 Let a bj {Fig. 37,) be the given line, and 5 the number of parts. 
 
 Draw a c, at any angle to a b ; on a c, from a, set off 5 equal 
 
 parts of any length, as at 1, 2, 3, 4 and c ; join c and b ; through 
 
 the points, 1, 2, 3 and 4, draw 1 e, 2/, 3 ^ and 4 h, parallel to 
 
 c b ; which will divide the line, a -6, as was required. 
 
 The lines, a b and a c, 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, 
 
22 
 
 4MERICAN HOUSE-CARPENTER. 
 
 {Fig. 38,) and bisect it Avith the perpendicular, c d ; bisect c d 
 with the line, e f, as at g ; then g is the centre as was required. 
 
 81, a. — A second method. Upon any two points in the cir- 
 cumference nearly opposite, as a and h. {F\s:. 39,) describe arcs 
 cutting each other at c and d : take any other two points, as e 
 and/, and describe arcs intersecting as at g and h ; join g and A, 
 and c and d ; the intersection, o, is the centre. 
 
 This is upon the eame princi|)!e as Art. 85. 
 
 81, b. — A third method. Draw any chord, as a b, (Pig, 40,) 
 
PRACTICAL GEOMETRY. 23 
 
 and from the point, a, draw a c, at right angles to a b ; join 
 
 c and b ; bisect c 6 at a — 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 at a ; by the edges of the square, (which the lines, a b and 
 a c, represent,) draw lines cutting ihe circle, as at b and c ; join 
 b and c ; then if 6 c is bisected, as at c?, the point, d, will be the 
 centre. (See Art. 156.) 
 
 82. — At a given jjoiiil 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 bj draw c d ; c dis the tangent required. 
 
 Fig. 42. 
 
 S3. — The same, without making use of the centre of the 
 circle. Let a, {Fig. 42,) be the given point. From a, set oiF 
 any distance to 6, and the same from b to c ; join a and c ; 
 upon a, with a b for radius, describe the arc, d b e ; make d b 
 equal to be; through a and d, draw a line ; this will be the 
 tangent required. 
 
 84. — A circle and a tangent given, to find the jpoint of con- 
 tact. From any point, as a, {Fig, 43,) in the tangent, b c, draw 
 
u 
 
 AAIERJCA X 1 :(irSl>CARPB:NTER. 
 
 a line to the centre d ; bisect a d at e ; ^ipon e, with the radius, 
 
 e a, describe the arc, afd;fis the point of contact required. 
 
 If/ and d were joined, the line would form right angles with 
 the tangent, b c. (See A/^t. 156.) 
 
 b 
 
 Fig. 44. 
 
 85. — Through any three points not in a straight line, to 
 draw a circle. Let a, h and c, [Fig. 44,) be the three given 
 points. Upon a and b, with any radius greater than half a b, 
 describe arcs intersecting at d and e ; upon b and c, with any 
 radius greater than half b c, describe arcs intersecting at /and g ; 
 through d and e, draw a right line, also another through /and g ; 
 upon the intersection, A, with the radius, h a, describe the cn*cie, 
 ah c, and it will be the one required. 
 
 Fig. 45. 
 
PRACTICAL GEOMETRY. 
 
 26 
 
 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 ain e, 
 
 and erect e d, perpendicular to a c, cutting b d in d ; then d is 
 
 i\iQ fourth point required. 
 
 A fifth point may be found, as at/, 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.) 
 
 87. — To describe a segment of a circle by a sei-triaiigle. 
 Let a b, {Fig. 46,) be the chord, and c d the height of the seg- 
 ment. Secure two straight-edges, or rulers, in the position, c e 
 and cf by nailing them together at c, and afiixing a brace from 
 e to/; put in pins at a and b ; move the angular point, c, in 
 the direction, a c b ; keeping the edges of the triangle hard 
 against the pins, a and b ; a pencil held at c will describe the 
 arc, a c b. 
 
 If the angle formed by the rulers at c 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. 
 
 g 1 
 
 1 h s 
 
 :^^rr 1^^:^; 
 
 3 (2 3 
 
 Fig. 47. 
 
 4 
 
26 
 
 AMERICAN HOUSE-CARPENTER. 
 
 88. — To describe the segment of a circle hy intersection of 
 
 lines. Let a b, {Fig. AT,) be the chord, and c d the height of 
 
 the segment. Through c, draw ef parallel to a b ; draw 6 /at 
 
 right angles to c b ; make c e equal to c /; 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 cb, which will be 
 
 the segment required. 
 
 In very large work, or in laying out ornamented gardens, &c., 
 this will be found useful ; and where the centre of the proposed 
 arc of a circle is inaccessible, it will be invaluable. (To trace 
 the curve, see note at Art. 117.) 
 
 Fig. 48. 
 
 89. — In a given angle, to describe a tanged curve. Let a 
 
 b c, {Fig. 48,) be the given angle, and 1 in the line, a b, and 5 
 
 in the line, b c, the termination of the curve. Divide 1 b and b 5 
 
 into a like number of equal parts, as at 1, 2, 3, 4 and 5 ; join 1 
 
 and 1, 2 and 2, 3 and 3, &c. ; and a regular curve will be formed 
 
 that will be tangical to the line, a b, at the point, 1, and to 6 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- 
 
PRACTICAL GEOMETRY. 
 
 2T 
 
 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 t, d ; join a and e, also e and b ; and proceed as direct- 
 ed at Arf 89. 
 
 Vv:. 
 
 90. — fo describe a circle within any given triangle^ so that 
 the sides of the triangle shall be tangical. Let a b c, [i'^ig- 
 50,) be the given tiiangle. Bisect the angles, a and 6, according 
 to Art. 77 ; upon d, the point of intersection of the bisecting 
 lines, with the radius d e, describe ihe required circle. 
 
 91. — About a given circle, to describe an equi-lateral tri- 
 angle. Let a d b c, {Fig. 51 ,) 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 ; drsiW f g, at right angles to 
 d c ; make/c and c g, each equal to a b ; from/, through a, 
 draw / h, also from gj through 6, draw g h; thenfg h will be 
 the triangle required. 
 
28 
 
 AMERICAN HOUSE-CARPENTER. 
 
 g P 
 
 92. — To find a right line nearly equal to the circumference 
 of a circle. Let ah c d^ [Fig. 52,) be the given circle. Draw 
 the diameter, a c ; on this erect a.n equi-lateral triangle, a e c, 
 according to Art. 96 ; draw gf parallel to a c ; extend e c to/, 
 also e ato g ; then ^/ will be nearly the length of the semi- 
 circle, a d c ; and twice g f will nearly equal the circumference 
 of the circle, ab c d^as was required. 
 
 Lines drawn Irom e, through any points in the circle, as o, o 
 and 0, top, p and/>, will divide gf in the same way as the semi- 
 circle, a d c, is div^ioed. So, any portion of a circle may be 
 transferred to a straight line Tni> i : \ very useful problem, 
 and should be well studied ; as it is frequently used to solve 
 problems on stairs, domes, <5lc. 
 
 Fig. 53. 
 
 92, a. — Another method. Let a bf c, {Pig. 53,) be the given 
 circle. Draw the diameter, a c ; from d, the centre, and at right 
 angles to a c, draw d b ; join b and c ; bisect be at e; from rf, 
 through e, draw df; then e/ added to three times the diameter, 
 
PRACTICAL GEOMETRY. 
 
 29 
 
 will equal the circumference of the circle within the 4^1)0 part of 
 its length. 
 
 POLYGONS, &c. 
 
 93. — Within a given circle^ to inscribe an equi-lateral tri- 
 angUy hexagon or dodecagon. Let abed, {Fig. 54,) be the 
 
 Ti., 
 
 given circle. Draw the diaaieter, b d ; upon 6, with the radius 
 
 v»f the given circle, dcFcribe the arc, a e c ; join a and c, also a 
 
 eind d, and c and d — and the triangle is completed. For the 
 
 I;e*xagon : from a, also from c, through e, draw the lines, a f 
 
 and eg; join a and b, b and c, c and/, (fcc, 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, 
 vxr 6 arcs of 60 degrees, in the whole circumference. A line 
 c'rawn from each angle of the hexagon to the centre, (as in the 
 figure,) divides it into six equal, equi-lateral triangles. 
 
30 
 
 AMERICAN HOUSE-CARPENTER. 
 
 94. — Within a square to inscribe an octagon. Let a b c d, 
 {Fig. 55,) be the given square. Draw the diagonals, a d and 
 6c; upon a, 6, c and c?, 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, (fcc, 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 oif, in order to make the piece octagonal. 
 
 Fif?. 56. 
 
 Fig. 57. 
 
 95. — Within a given circle to inscribe any regular polygon. 
 
 Let a b c 2, [Fig. 56, 57 and 58,) be given circley. 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 equ.^'. parts as the 
 
 polygon is to have sides, as at 1, 2, 3, 4, (fee. ; ficii e, through 
 
 each even number, as 2, 4, 6, &c., draw lines cuttmg the circle 
 
 in the points, 2, 4, (fee. ; 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, /, (fee. 
 
 In forming a hexagon, the sides of the triangle erected upon 
 a c, (as at Fig. 57,) mark the points, b and/. 
 
 96. — Upon a given line to construct an equi-lateral triangle. 
 
 Let a bj {Fig. 59,) be the given line. Upon a and b, with a b 
 
PRACTICAL GEOMETRY. 
 
 31 
 
 for radius, describe arcs intersecting at c ; join a and c, also c 
 and b ; then a cb will be the triangle required. 
 
 Fig. 60 
 
 97. — To describe an equi-latei^al j^ectangle, 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 be; 
 bisect the arc, a e, in / ; 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 db will 
 be the square required. 
 
 Fig- 61. 
 
 Fig. 62. 
 
 Fig. 63. 
 
 98. — Upon a given line to describe any regular polygon. 
 Let a 6, {Fig. 61, 62 and 63,) be given lines, equal to a side of 
 the required figure. From 6, draw b c, at right angles to a 6 ; 
 upon a and b, with a b for radius, describe the arcs, a c d and 
 
32 AMERICAN HOUSE-CARPENrER. 
 
 f eh ] divide a c 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, 
 
 a. describe the circle, a f d b ; then radiating lines, drawn 
 
 from b through the even numbers on the arc, a c?, will cut the 
 
 circle at the several angles of the required figure. 
 
 [n the hexagon, {Fig. 62,) the divisions on the arc, a d, are 
 not necessary ; for the point, o, is at the intersection of the arcs, 
 a d and/ 6, the points, /and d, are determined by the intersec- 
 tion of those arcs with the circle, and the points above, g and A, 
 can be found by drawing lines from a and b, through the centre, 
 0. 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 h 5, {Fig. 63,) have to be extended before they 
 will intersect. 
 
 Fiff. 64. 
 
 99. — To construct a triangle whose sides shall be severally 
 equal to three given lines. Let a, h and c, {Fig. 64,) be the 
 given lines. Draw the line, d e, and make it equal to c ; upon 
 e, wit}i b for radius, describe an arc at/; upon c?, with a for 
 radius, describe an arc intersecting the other at/; join d and/, 
 also/ and e ; then dfe will be the triangle required. 
 
 Fig. 65. Fig. 66. 
 
PRACTICAL GEOMETRY. 
 
 S3 
 
 100. — To construct a figure equal to a given^ right-lined 
 
 figure. Let abed, {Fig. 65,) be the given figure. Make ef, 
 
 {Fig. 66,) equal to c d ; upon /, with d a for radius, describe an 
 
 arc at g ; upon e, with c a for radius, describe an arc intersecting 
 
 the other at g ; join g and e ; upon / and g, with d b and a b 
 
 for radius, describe arcs intersecting at h ; join g and A, also h 
 
 and/; 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 proceeding as 
 above. The shape of the floor of any room, or of any piece of 
 land, (fee, may be accurately laid out by this problem, at a scale 
 upon paper ; and the contents in square feet be ascertained by 
 the next. 
 
 101. — To make a parallelogram equal to a given triangle. 
 
 Let ab c, {Fig. 67,) be the given triangle. From a, draw a rf, 
 
 at right angles to be; bisect a d in e ; through e, draw fg^ 
 
 parallel to b c ; from b and c, draw b f and c g, parallel to d e ; 
 
 then bfgc will be a parallelogram containing a surface exactly 
 
 equal to that of the triangle, a b c. 
 
 Unless the parallelogram is required to be a rectangle, the lines, 
 b f and c g, need not be drawn parallel to d e. If a rhomboid is 
 desired, they may be drawn at an oblique angle, provided they 
 be parallel to one another. To ascertain the area of a triangle, 
 multiply the base, b c, by half the perpendicular height, da. In 
 doing this, it matters not which side is taken for base. 
 
 A ^/ 
 
 ^^ e 
 ^^ C 
 
 i 
 Fig. 68. 
 
 5 
 
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, 6, c and d ; make 
 e d equal to B ; through c?, draw c /, parallel to g b ; through 
 e, draw the diagonal, c a ; from a, draw a /, parallel to e d 
 then C will be equal to A. (See Art. 144.) 
 
 A 
 
 a ^"V,^, 
 
 
 B 
 
 Fig 69. 
 
 103. — To make a square equal to two or fjtore given squares. 
 Let A and B, {Fig. 69,) be two given 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 
 
 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 
 
PRACTICAL GEOMETRY. 35 
 
 c 6, be the side of a third given square ; then the square, A, 
 formed upon the line, d 6, 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 height, is the other of the sides that contain the right angle. 
 Proceed as directed by the rule : 
 
 17 
 17 
 
 
 9 
 9 
 
 119 
 17 
 
 - 
 
 81 = square of 9. 
 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. 
 
 29 ) 270 
 9 261 
 
 382)- -900 
 2 764 
 
 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.) 
 
36 AMERICAN HOUSE-CARPENTER. 
 
 Ag^ir : 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 
 poixit oi" the right angle. The length of the sides containing the 
 right angle will be each 3 feet : then, as before — 
 
 3 
 3 
 
 9 = square of one side. 
 3 times 3=9 = square of the other side. 
 
 ] 8 Product : the square-root of which is 4*2426 -f ft., 
 or 4 feet, 2 inches and :ths. full. 
 
 (2.) — The hypothenuse and one side being known, to find the 
 other £ide. Rnle. — Subtract the square of the given side from 
 the square of tlie 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- 
 quijed, the strings of which are to be of plank, 12 feet lone ; and 
 it is desirable to know the greatest run such a length of string 
 will aflford. 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 -f 
 feet, or 8 feet, 11 inches and i^gths. — the answer, as re<iuire4« 
 
PRACTICAL GEOMETRY. 
 
 3f 
 
 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, (fcc, 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 |ths 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 accurate as the preceding ; but when 
 the length of a very long rafter is sought, it requires great care 
 and precision to ascertain the fractions. For the least variation 
 on the square, or in the length taken on the rule, would make 
 perhaps several inches difference in the length of the rafter. 
 For shorter dimensions, however, the result will be true enough. 
 
 104. — To make a circle equal to tioo given circles. Let .4 
 and i?, [Fig. 71,) be the given circles. In the right-angled tri- 
 angle, ah c, make a b equal to the diameter of the circle, B, and 
 c b equal to the diameter of the circle, A ; then the hypothenuse. 
 
 rig. 72. 
 
3a 
 
 AMERICAN HOUSE-CARPENTER. 
 
 a Cy will be the diameter of a circle, C, which will be equal in 
 
 area to the two circles, A and B, added together. 
 
 Any polygonal figure, as A, {Fig. 72,) formed on the hypo- 
 thenuse of a right-angled triangle, will be equal to two similar 
 figures,* as B and C, formed on the two legs of the triangle. 
 
 ^ ~^ 
 
 T h 
 
 f '/^ 
 
 \ 
 
 xV 
 
 J 
 
 Fig. 73 
 
 105. — To construct a square equal to a given rectangle. 
 Let A, [Fig. 73,) be the given rectangle. Extend the side, a 6, 
 and make b c equal to 6 e; bisect a c in/, and upon/, with the 
 radius, / «, describe the semi-circle, age; extend e b, till it 
 cuts the curve in g ; then a square, b g h d, formed on the line, 
 b gy will be equal in area to the rectangle, A. 
 
 B 
 
 (Z 
 
 \ 
 
 A 
 
 b 
 
 Fig. 74. 
 
 g 
 
 105, a. — Another method. Let A, {Fig. 74,) be the given 
 rectangle. Extend the side, a 6, and make a d equal to a c ; 
 
 • Similar figures are such as have their several angles respectively equal, and their 
 sides respectively proportionate. 
 
PRACTICAL GEOMETRY. 39 
 
 bisect a d in e ; upon e, with the radius, e a, describe the semi- 
 circle, a f d ; extend^ h till it cuts the curve in/; join a and 
 /; then the square, B^ formed on the line, a/, 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 a 6, 
 {Fig. 73,) equal the base of the given triangle, and h e equal 
 half its perpendicular height, (see Fig. 67 ;) then proceed as 
 directed at Art. 105. 
 
 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 h equal to A ; from a, draw a c, at any angle with ab ; 
 make a c and a d each equal to B ; join c and b ; from d, draw 
 d e, parallel to c b ; 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. 
 
 108. — Three right lines being given, to find a fourth pro 
 portional thereto. Let A, B and C, {Fig. 7^,) be the given 
 lines. Make a b equal to A ; from a, draw a c, at any angle 
 with a b; make a c equal to B, and a e equal to C ; join c and 
 b ; from e, draw e f parallel to c 6 ; then a f will be the fourth 
 proportional required. That is, a f bears the same proportion 
 to C, as B does to A. 
 
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- 
 sis, can be foimd ; so that it will bear the same proportion to the 
 longer axis, as the shorter of the given ellipsis does to its longer. 
 (See also, Art. 126.) 
 
 109. — A li7ie 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 Xo 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, (fee. 
 
 110. — Between two given right lines, to find a mean pro- 
 portional. Let A and /?, [Fig. 78,) be the given lines. On 
 the line, a c, make a b eqwul to A, and b c equal to B ; bisect a 
 c me ; upon e, with e a for radius, describe the semi-circle, a d 
 
PRACTICAL GEOMETRY. 
 
 41 
 
 c ; at 6, erect b 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/, the section will be 
 an ellipsis ; if in the direction, m I, the section will be a para- 
 bola ; and if in the direction, r o, an hyperbola. (See Arl. 56 
 to 60.) If the cutting planes be at right angles with the plane, 
 a 6 c, then — 
 
 112.— To find the axes of the ellipsis, bisect e /, [Fig. 79,) 
 in g ; through g, draw h i, parallel to a b ; bisect hiinj ; upon 
 j, with j h for radius, describe the semi-circle, h k i ; from g, 
 draw g kj at right angles to hi ; then twice g k will be the 
 conjugate axis, and e f the transverse. 
 
 6 
 
42 
 
 AMERICAN HOUSE-CARPENTER. 
 
 113. — To find the axis and base of the parabola. Let m /, 
 {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 I m will be the 
 axis and height, and m 6^ an ordinate and half the base ; as at 
 Fig. 92, 93. 
 
 114. — To find the height ^ base and transverse axis of an 
 hyperbola. Let o r, {Fig. 79,) be the direction of the cutting 
 plane. Extend o r and a c till they meet at n ; from o, draw 
 p, at right angles to a b; then r o Avill be the height, nr the 
 transverse axis, and o p half the base ; as at Fig. 94. 
 
 Fig. 80. 
 
 115. — The axes being given, to find the foci, and to describe 
 
 an ellipsis ivith a string. Let a b, {Fig. 80,) and c d, be the 
 
 given axes. Upon c, with a e ovb e for radius, describe the arc, 
 
 ff; then /and/, the points at which the arc cuts the transverse 
 
 axis, will be the foci. At/ and /place two pins, and another at c ; 
 
 tie a string about the three pins, so as to form the triangle, //c ; 
 
 remove the pin from c, and place a pencil in its stead ; keeping the 
 
 string taut, move the pencil in the direction, eg 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 not proper for the purpose : a linen, or flaxen cord is 
 much better. 
 
PRACTICAL GEOMKTRV, 43 
 
 116. — The axes being given, to describe an ellipsis mith a 
 trammel. Let a b and c d, {Fig. 81,) be the given axes. Place 
 the trammel so that a line passing tlirongh the centre of the 
 grooves, would coincide with the axes ; make the distance from 
 the pencil, e, to the nut,/, equal to half c d ; 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 b d ; then the pencil at e will describe the required ellipse. 
 
 A trammel maybe 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 seen at Fig. 81. A rod is then to be 
 prepared, having two moveable nuts made of v/ood, with a mor- 
 tice through them of the size of the rod, and pins under them 
 large enough to fill the grooves. Make a hole at one end of the 
 rod, in which to place a pencil, in the absence of a regular tram- 
 mel, a temporary one may be made, Vv'hich, 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 
 pencil is moved in the direction of the curve, keep the brad-awls 
 hard against the straight-edges, as directed for using the tram- 
 mel-rod, and one-quarter of the ellipse will be drawn. Then, 
 by shifting the straight-edges, the other three quarters in succes- 
 sion may be drawn. If the required ellipse be not too large, a 
 carpenters'-square may be made use of, in place of the straight- 
 edges. 
 
 An improved method of constructing the trammel, is as fol- 
 lows : make the sides of the grooves bevilling from the face 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 
 
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. 
 
 Fiff.82. 
 
 117. — To describe an ellipsis by ordinates. Let a b and c d, 
 
 [Fig. 82,) be given axes. With c e on e d for radius, de- 
 scribe the quadrant,/^ h ; divide f h^ a e and e 6, each into a 
 like number of equal parts, as at ] , 2 and 3 ; through these 
 points, draw ordinates, parallel io c d andfg- ; take the distance, 
 1 i, and place it at 1 /, transfer 2 ; to 2 m, and 3 k to3 n; through 
 the points, a, n^ 7n, I 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, 7n, Z, (fcc, and a thin slip of wood bent 
 around by them, the curve can be made quite correct. This 
 method is mostly used in tracing face-moulds for stair hand- 
 railini 
 
 'ir>' 
 
 118. — To describe an ellipsis by intersection of lines. Let 
 
PRACTICAL GEOMETRY. 
 
 45 
 
 a b and c d, {Fig. 83,) be given axes. Through c, draw f g, 
 parallel to a b ; from a and b, draw a / and b g, at right angles 
 to ab ; divide f a, g b, a e and e 6, each into a like number of 
 equal parts, as at 1, 2, 3 and o, o, o ; from 1, 2 and 3, draw lines 
 tocy through o, o and o, draw lines from c?, intersecting those 
 drawn to c ; then a curve, traced through the points, i, 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 and its conjugate be given. Thus, a b and c d, {Fig- 84,) are 
 conjugate diameters : f g is drawn parallel to a 6, instead of 
 being at right angles to c d ; also, / a and g b are drawn paraUel 
 to c dj instead of being at right angles to a b. 
 
 119. — To describe an ellipsis by intersecting arcs. Let a h 
 
46 
 
 AMERICAN HOUSE-CARPENTER. 
 
 and c d, {Fig. 85,) be given axes. Between one of the foci, / 
 and/, and the centre, e, mark any number of points, at random, 
 as 1, 2 and 3 ; upon /and/, with b 1 for radius, describe arcs at 
 ^) ^5 §" ^^^ S ; upon/ and/ with a 1 for radius, describe arcs inter- 
 secting the others at^,^,^ and g ; then these points of intersection 
 will be in the curve of the ellipsis. The other points, h and i, are 
 found in like manner, viz : h is found by taking h 2 for one radius, 
 and a 2 for the other ; i 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, 6, &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 Avhich c/ equals a e, the half of the 
 transverse axis. 
 
 /I 
 
 '\ ^\ 
 
 
 ^^ 
 
 
 i 
 
 ^ 
 
 ^ 
 
 i/ "/ 
 
 
 g 
 
 Fiff. 8G. 
 
 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. From c, draw c e, parallel to ab ; from a, draw a e, 
 parallel to c d; join e and c?; bisect e a in/; join / and c, inter- 
 secting e dini; bisect i cin o ; from o, draw og, at right angles 
 to i c, meeting c d extended to g ; join i and g, cutting the trans- 
 verse axis in r ; make h j equal to h g, and h k equal to h r ; 
 from J, through r and A:, draw j m and j n; also, from g, through 
 k, araw g I; upon g and j, with g c for radius, describe the 
 
PRACTICAL GEOMETRY. 
 
 47 
 
 arcs, i I and m n ; upon r and k, with r a for radius, describe 
 the arcs, m i and I 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, ^ Zand m ?i; 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 described with a pair of 
 compasses. The above, therefore, is only an ^ipproximation. 
 
 121. — To draw an oval in the jjroportion^ seven by nine. 
 Let c c?, {Fig. 87,) be the given conjugate axis. Bisect c d in o, 
 and through o, draw a b, at right angles to c d ; bisect c o in e ; 
 upon 0, with o e for radius, describe the circle, e f g- h ; from e, 
 through h and/, draw ej and e i; also, from g, through h and/, 
 draw g k and g I ; upon g, with g c for radius, describe the arc, 
 kl ; upon e, with e d for radius, describe the arc, j i ; upon h and 
 / with h k for radius, describe the arcs, j k and I i; this will 
 complete the figure. 
 
 This is a very near approximation to an ellipsis ; and perhaps no 
 method 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, efg A, one will be 
 drawn in the proportion, five by seven. 
 
46 
 
 AMERICAN HOUSE-CARPENTER. 
 
 122. — To draw a tangent to an ellipsis. Let abed, {Fig. 
 88j) be the given ellipsis, and d the point of contact. Find the 
 foci, [Art. 115,)/ and/, and from them, through d, draw/e and 
 f d ; bisect the angle, [Art. 77,) e d o, with the line, 5 r ; then 
 s r will be the tangent required. 
 
 c Fig. 89. 
 
 123. — An ellipsis with a tangent given, to detect the point 
 of contact, hetagbf, {Fig. 89,) be the given ellipsis and tan- 
 gent. Through the centre, e, draw a h, parallel to the tangent j 
 any where between e and/, draw c d, parallel to ab ; bisect c d in 
 ; through o and e, dmwfg; then g will be the point of con- 
 tact required. 
 
 124. — A diameter of an ellipsis given, to find its conjugate. 
 Let a b, {Fig. 89,) be the given diameter. Find the line,/^, by 
 the last problem; then/^ will be the diameter required. 
 
PRACTICAL GEOMETRY. 
 
 49 
 
 125. — An]/ diameter and its conjugate being given, to as- 
 certain the two axes, a7id thence to describe the ellipsis. Let 
 a b and c d, {Fig. 90,) be the given diameters, conjugate to one 
 another. Through c, draw e /, parallel to a b ; from c, draw c 
 g, at right angles to ef; make c g equal to a h or h b ; join g 
 and h ; upon g, with g c for radius, describe the arc, i k c j ; 
 upon A, with the same radius, describe the arc, I n ; through the 
 int*"rsections, I and n, draw n o, cutting the tangent, ef, in o ; 
 upon 0, with o gfov radius, describe the semi-circle, eig f ; join 
 e and^, also g and/, cutting the arc, i c j, in k and t ; from e, 
 through h, draw e 7?i, also from/, through A, draw//? ; from A; 
 and ^, draw k r and ^ 5, parallel to^ A, cutting e ni in r, and/p 
 in s ; make A ni equal to A r, and h p equal to h s ; then r m 
 and 5 /> will be the axes required, by which the ellipsis may be 
 drawn in the usual way. 
 
 126. — To describe an ellipsis, whose axes shall be propor- 
 tionate to the axes of a larger or smaller given one. Let a 
 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 ; trom i, 
 Iraw i e, parallel to a c ; make o f equal to oe ; then e/ will be 
 
5C 
 
 AMERICAN HOUSE-CARPENTER. 
 
 the conjugate axis required, and will bear the same proportion to 
 ij, asc d does to a b. (See Art. 108.) 
 
 2 3 m 3 2 
 Fi-. 92. 
 
 127. — To describe a parabola by intersection of lines. Lee 
 m Ij {Fig. 92,) be the axis and height, (see Fig. 79,) and d dy & 
 double ordinate and base of the proposed parabola. Through / 
 draw a a, parallel to d d ; through d and d, draw d a and d a, 
 parallel to ni I ; divide a d and d m^ each into a like number of 
 equal parts ; from each point of division in d m, draw the lines, 
 1 1, 2 2, (fee, parallel Xoml; from each point of division in d 
 a, draw lines to I ; then a curve traced through the points of 
 intersection, o, o and o, will be that of a parabola. 
 
 127, a. — Another method. Let rn, I, {Fig. 93,) be the axis and 
 height, and d d the base. Extend m I, and make I a equal to m 
 I /join a and d, and a and d ; divide a d and a d. 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. 
 
PRACTICAL GEOMETRY. 
 
 51 
 
 iff 
 
 T 
 
 3/ 
 
 V 
 
 a/ 
 
 V 
 
 1/ 
 
 V 
 
 m 
 Fig. 93. 
 
 j>l2.<o3 'Z L p 
 
 Fig. 94. 
 
 128. — To describe an hyperbola by intersection of lines. 
 Let r 0, (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 to r ; divide a p and p 0, 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, (fcc, will be that of an hyperbola. 
 
 The parabola and hyperbola afford handsome curves for various 
 mouldings. 
 
^DEMONSTRATIONS. 
 
 129. — To impress more deeply upon the mind of the learner 
 some of the more important of the preceding problems, and to 
 indulge a veiy 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 ac^.iom 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. 
 
 b 
 Fie. 95. 
 
 130. — Definition. If a straight line, as a 6, [Fig. 95,) stt.nd 
 upon another straight line, as c d^ so that the two angles made at 
 
PRACTICAL GEOMETRY. 53 
 
 the point, 6, are equal — a b do a b d, (see note to Art. 27,) then 
 each of the two angles is called a right angle. 
 
 131. — Defifiition. 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, <fec. 
 
 132. — Definition. The measure of an angle is the number of 
 degrees contained between its two sides, using the angular point 
 as a centre upon which to describe the arc. Thus the arc, c e, 
 {Fig. 96,) is the measure of the angle, c b e ; e a, of the angle, 
 e b a ; and a d, of the angle, ab d. 
 
 133. — Corollary. As the two angles at 5, {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 r. right angle, 45 ; of 
 one-third of a right angle, 30, &c. 
 
 134. — Defio'Ation. In measuring an angie, {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. 
 
 135.^— Axiom. If two straight lines, parallel to one another, 
 
m 
 
 AMERICAN HOUSE-CARPENTER. 
 
 9ts a b and c d, {Fig, 97,) stand upon another straight line, as e/, 
 the angles, a bf and c d fj are equal ; and the angle, a 6 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 c d, then one of the an- 
 gles, as a b c, is called an obtuse angle, and the other, as ab dj 
 an acute angle. 
 
 137. — Axiom. The two angles, a b d and a b c, {Fig. 96,) are 
 together equal to two right angles, {Ai^t. 130, 133;) also, the 
 three angles, a b d, e b a and c 6 e, are together equal to two right 
 angles. 
 
 138. — Corollary. Hence all the angles that can be made upon 
 one side of a line, meeting in a point in that line, are together 
 equal to two right angles. 
 
 139. — Corollary. Hence all the angles that can be made on 
 both sides of a line, at a point in that line, or all the angles that 
 can be made about a point, are together equal to four right angles. 
 
 6 d 
 
 Fia 98. 
 
 140. — Proposition. If to each of two equal angles a third 
 angle be added, their sums will be equal. Let ab c and d e f, 
 {Fig. 98,) be equal angles, and the angle, i j k, the one to be 
 added. Make the angles, gb a and hed, each equal to the given 
 angle, ij k ; then the angle, g b r, will be equal to the angle, h e 
 f; for, if a b c and d e/be angles of 90 degrees, and i j k, 30, 
 then the angles, g b c and h ef, will be each equal to 90 and 
 30 added, viz : 120 degrees. 
 
PRACTICAL GEOMETRY. 
 
 66 
 
 Ki-' !)i>. 
 
 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 efhe two given triangles, having the angle 
 at a equal to the angle at d, the side, a b, equal to the side, d e, 
 and the side, a c, equal to the side, df; then the third side of 
 one, b c, is equal to the third side of the other, ef; the angle at b 
 is equal to the angle at e, and the angle at c is equal to the angle 
 at/. For, if one triangle be applied to the other, the three points, 
 b, a, c, coinciding with the three points, e, d, /, the line, b c, must 
 coincide with the line, e f; the angle at b with the angle at e ; 
 the angle at c with the angle at/; and the triangle, 6 a c, be every 
 way equal to the triangle, e df. 
 
 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, 
 
56 
 
 AMERICAN HOUSE-CARPENTER. 
 
 77,) b a Cjhy the line, a d. 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 conmion to each ; the angle, b 
 a dj being equal to the angle, d a c ; the line, b d, must, accord- 
 ing to ArL 141, be equal to the line, dc ; and the angle at b must 
 be equal to the angle at c. 
 
 B 
 
 Fi?. Mi. 
 
 143. — Proposition. A diagonal crossing a parallelogram di- 
 vides it into two equal triangles. Let abed, [Fig. 101,) be a 
 given parallelogram, and 6 c, a line crossing it diagonally. Then, 
 as a c is equal to b cZ, and a b to c d, the angle at a to the angle 
 at d, the triangle. A, must, according to Art. 141, be equal to the 
 triangle, B. 
 
 a 
 
 
 /, 
 
 A 
 
 ^^^ 
 ^^f 
 
 C /^ 
 ^^ D 
 
 B 
 
 I'l-. 10- 
 
 144. — Proposition. Let a h c d^ (^Vo-- 102,) be a given pa- 
 rallelogram, and 6 c a dingoiial. At any distance between a b and 
 c c?, draw e/, parallel to a b ; through the point, g, the intersection 
 of the lines, b c and ef, draw h «, parallel to b d. In every paral- 
 lelogram thus divided, the parallelogram, J, is equal to the paral- 
 lelogram, B. According to xirt. 143, the triangle, a 6 c, is 
 equal to the triangle, bed; the triangle, C, to the triangle, D ; 
 and E to F ; this being the case, take D and F from the triangle, 
 bed. and C and E from the triangle, a b c, and what remains 
 
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. Let ah c d and 
 ef cd^ {Fig. 1()3,) be given parallelograms, standing upon the 
 same base, c d, and between the same parallels, a f and c d. 
 Then, ah and e/ being equal to c c/, are equal to one another: 
 h e being added to both a h and e f^ a e equals h f ; the line, a c. 
 being equal to h d, and a e to hf, and the angle, c a e, being 
 equal, {Art. 135,) to the angle, d h f, the triangle, a e c, must be 
 equal, {Art. 141,) to the triangle, hfd; these two triangles being 
 equal, take the same amount, the triangle, beg, from each, and 
 what remains in one, a h g c, must be equal to what remains in 
 the other, efdg; 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, ah c d, is equal to the paral- 
 lelogram, efcd. 
 
 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 dj {Fig. 103,) is double, {Art. 143,) the triangle, 
 c e d. 
 
 147. — Proposition. Let a h c d, {Fig. 104,) be a given quad- 
 rangle with the diagonal, a d. From h, draw b e, parallel to a d; 
 extend cdto e ; join a and e ; then the triangle, <z e c, will be equal 
 in area to the quadrangle, a he d. Since the triangles, adh and 
 a d e^ stand upon the same base, a d, and between the same paral- 
 
 8 
 
58 
 
 AMERICAN HOUSE-CARPENTER. 
 
 Fig. 104. 
 
 lels, a d and h e, they are therefore equal, {^Art. 145, 146 ;) and 
 since the triangle, C, is common to both, the remaining triangles, A 
 and 5, are therefore equal ; then B being equal to yl, the triangle, 
 a e c, is equal to the quadrangle, a h c d. 
 
 Fig. 105. 
 
 148. — Proposition. If two straight lines cut each other, as 
 a h and c c?, {Fig. 10.5,) the vertical, or opposite angles, A and 
 C, are equal. Thus, a e, standing upon c d., forms the angles, 
 jBand C, which together amount, {Art. 137,) to two right angles ; 
 in the same manner, the angles, A and 5, form two right angles ; 
 since the angles, A and B^ are equal to B and C, take the same 
 amount, the angle, J5, 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. Let a b c, {Fig'. 106,) be a given tri- 
 angle, with its sides extended to/, e, and d, and the line, eg. 
 
PRACTICAL GEOMETRY. 69 
 
 drawn parallel to h e. As ^ c is parallel to e b, the angle, g c d^ 
 is, equal, [Art. 135,) to the angle, e b d ; as the lines, /c and b e, 
 cut one another at a, the opposite angles, / a c and b a c, are 
 equal, {Art. 14S ;) as the angle, /« c, is equal, [Art. 135,) to the 
 angle, a eg, the angle, a c g, is equal to the angle, b a c ; 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 ate are equal, 
 {Art. 137,) to two right angles, the three angles of the triangle, a 
 b c, must likewise be equal to two right angles. Any triangle 
 can be subjected to the same proof. 
 
 150. — Corollary. Hence, if one angle of a triangle be a right 
 angle, the other two angles amount to just one right angle. 
 
 151. — Corollary. If one angle of a triangle be a right angle. 
 and the two remaining angles 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. 
 
60 
 
 AMERICAN HOUSE-CARPENTER. 
 6 
 
 \m.~-Proposition. If from the extremities of the diameter of 
 a semi-circle, two straight hues be drawn to any point in the cir- 
 cumference, the angle formed by them at that point will be a 
 right angle. Let a h c, [Fig. 107,) be a given semi-circle ; and 
 a 6 and 6 c, lines drawn from the extremities of the diameter, a 
 c, to the given point, h ; the angle formed at that point by these 
 lines, is a right angle. Join the point, h, and the centre, d; the 
 lines, d a,dh and d c, being radii of the same circle, are equal ; 
 the angle at a is therefore equal, {Art. 142,) to the angle, a h d, 
 also, the angle at c is, for the same reason, equal to the angle, d I 
 c; the angle, a h c, being equal to the angles at a and c taken 
 together, must therefore, {Art. 153,) be a right angle. 
 
 ^^7 .—Proposition. The square of the hypothenuse of a 
 right-angled triangle, is equal to the squares of the two remaining 
 sides. Let a h c, {Fig. 108,) be a given right-angled triangle, 
 having a square formed on each of its sides : then, the square, h e, is 
 equal to the squares, A c and ^ 6, taken together. This can be 
 
PRACTICAL GEOMETRY. 61 
 
 proved by showing that the parallelogram, h l, is equal to the square, 
 ^b ; and that the parallelogram, c /, is equal to the square, h c. The 
 angle, c 6 (/, is a right angle, and the angle, abf, is aright angle ; 
 add to each of these the angle, ab c ; then the angle,/ b c, will evi- 
 dently be equal, {Art. 140,) to the angle, abd ; the triangle,/ 6 c, 
 and the square, g- 6, being both upon the same base,/6, and between 
 the same parallels, / b and^ c, the square, g- b. is equal, {Art. 146,) 
 to twice the triangle,/ 6 c ; the triangle, a b d, and the parallelo- 
 gram, b I, being both upon the sa.me base, b d, and between the 
 same parallels, b d and a I, the parallelogram, b Z, is equal to twice 
 the triangle, abd; the triangles,/ 6 c and abd, being equal to 
 one another, {Art. 141,) the square, g- b, is equal to the parallelo- 
 gram, b I, either being equal to twice the triangle, /6 c or a b d. 
 The method of proving h c equal to c Z is exactly similar — thus 
 proving the square, b e, equal to the squares, h c and ^ 6, 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-oifer 
 ing for its discovery he sacrificed a hundred oxen to the gods. 
 From this circumstance, it is sometimes called the hecatomb pro- 
 blem. It is of great value in the exact sciences, more especially 
 in Mensuration and Astionomy, in which many otherwise intri- 
 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. 
 
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. 
 
SKCTlOiN JL— AROHITECTTTRE. 
 
 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, (fee, 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 
 
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 Babnl. The tower of the temnle of Belus was square 
 at its base, eacn side measuring one iurlong, and consequently 
 half a mile in circumference. Its form was that of a pyramid 
 and its height was 660 feet. It had a winding passage on the 
 outside from the base to the summit, wliich was wide enough for 
 two carriages. 
 
 1^0.- -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 Phoe- 
 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 j 
 the subterraneous temples of the Hindoos upon the islands, Ele- 
 phanta and Salsetta ; the ruins of Persepolis in Persia ; pyramids, 
 obelisks, temples, palaces and sepulchres in Egypt — all prove that 
 the architects of those early times were possessed of skill and 
 judgment highly cultivated. The principal characteristics of 
 their works, are gigantic dimensions, immoveable solidity, and, in 
 some instances, harmonious splendour. The extraordinary size 
 of some is illustrated in the pyramids of Egypt. The largest of 
 these stands not far from the city of Cairo : its base, which is 
 square, covers about 111 acres, and its height is nearly 500 feet. 
 The stones of which it is built are immense — the smallest being 
 full thirty feet long. 
 
 161. — Among the Greeks, architecture was cultivated as a fine 
 
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 beautiful temple of Minerva, 
 erected upon the acropolis of Athens, the Propyleum, the Odeum 
 and others, were lasting monuments of their success. The Ionic 
 and Corinthian orders Avere 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 st«:ict 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 Caesar, many large edifices were erected in various cities of 
 Italy. But imder Caesar Augustus, at about the beginning of the 
 christian era, the art arose to the greatest perfection it ever at- 
 
 9 
 
66 AMERICAN HOUSE-CARPENTER. 
 
 tained in Italy. Under his patronage, Grecian artists were en- 
 couraged, and many emigrated to Rome. It was at about this 
 time that Solomon's temple at Jerusalem was rebuilt by Herod — 
 a Roman. This was 46 years in the erection, and was most pro- 
 bably of the Grecian style of 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 to a great height ; 
 the whole being built of white marble exquisitely polished. This 
 IS the building concerning which it was remarked — " Master, see 
 vvhat 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 architecture lost much of its 
 elegance and dignity. The Romans, though justly celebrated 
 for their scientific knowledge as displayed in the construction of 
 their various edifices, were not capable of appreciating the simple 
 grandeur, the refined elegance of the Grecian style ; but sought 
 to improve upon it by the addition of luxurious enrichment, and 
 thus deprived it of true elegance. In the days of Nero, whose 
 palace oif gold is so celebrated, buildings were lavishly adorned. 
 Adrian did much to encourage the art ; but not satisfied with the 
 simplicity of the Grecian style, the artists of his time aimed at 
 
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. Tlie 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 Yandals, 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 standing on their wrong ends, and other ridi- 
 culous arrangements characterized their clumsy work. The vast 
 number of columns which the ruins around them afforded, 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 ar3 not of the classical styles, and which were 
 
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 Gothic 
 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 wjiich were greatly destitute of proportion, elegance 
 or utility — their characteristics being scarcely anything more than 
 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, and a lavish display of incongruous orna- 
 ments. This khid of architecture is called, the Lombard style, 
 and was employed in the 7th century in Pa via, 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 Roman-Ionic order with the Gothic of the Lombards, 
 
ARCHITECTURE. 69 
 
 This style is called the Lombard-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 kingdom of the Goths, the arts and sciences were 
 mostly in possession of the Musselmen-conquerors ; at which 
 time there were three kinds of architecture practised ; viz : the 
 Arabian, the Moorish and the modern-Gothic. The Arabian 
 style was formed from Greek models, having circular arches 
 added, and towers which terminated with globes and mir.arets. 
 The Moorish 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 Modern-Gothic 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 zigzags being the 
 most common. Buildings in this style were erected in the 12th 
 
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, a-nd the peculiarly graceful character of its foliage. This 
 style is of the 18th century. The Decorated style, as its name 
 implies, is characterized by a great profusion of enrichment, 
 which consists principally of the crocket, or feathered-ornament, 
 and ball-flower. It was mostly in use in the 14th century. The 
 Perpendicular style, which dates from the 15th century, is distin- 
 guished by its high towers, and parapets surmounted with spires 
 similar in number and grouping to oriental minarets. 
 
 169. — Thus these several styles, which have been erroneously 
 terme Jl Got kic, were distinguished bypeculiar characteristics as well 
 as by different names. The first symptoms of a desire to return to a 
 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 Yenice, which 
 arose in the 10th or 11th century, was the work of Grecian archi- 
 tects, and resembles in magnificence the forms of ancient archi- 
 tecture. The cathedral at Pisa, a v.^onderful structure for the age, 
 was erected by a Grecian architect in 1016. The marble with 
 which the walls of this building were faced, and of which the four 
 rows of columns that support the roof are composed, is said to be 
 of an excellent character. The Campanile, or leaning-tower as it 
 is usually called, was erected near the cathedral in the 12th cen- 
 tury. Its inclination is generally supposed to have arisen from 
 a poor foundation ; although by some it is said to have been thus 
 constructed originally, in order to inspire in the minds of the 
 beholder sensations ot b;ublimity 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 
 
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 were erected, which displayed a purer taste in the 
 science. Among others, St. Peter's of Rome, which was built 
 about this time, is a lasting monument of the architectural skill of 
 the age. Giocondo, Michael Angelo, Palladio, Yignola, and other 
 celebrated architects, each in their tm-n, 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 efibrts were serviceable in correcting, 
 to a 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 m 
 architecture, were originated in accordance with the different pur- 
 
72 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 hut 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 teiii 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 caver 71 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-Gothic, or 
 pointed style, which was most generally confined to ecclesiastical 
 structures, is said by some to have originated in an attempt to 
 imitate the bower, or grove of trees, in which the ancients per- 
 formed their idol-worship. 
 
 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 Stylobate 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 tlie 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 Order, in architecture, is composed of tw i princi- 
 pal parts, viz : the column and the entablature. 
 
ARCHITECTURE. 73 
 
 175. — The Column 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. III.) 
 
 177. — The Base of a column is so called from basis, a founda- 
 tion, or footing. 
 
 178. — The Shaft, the upright part of a column standing upon 
 the base and crowned with the capital, is from shaflo, to dig — 
 in the manner of a well, whose inside is not unlike the form of a 
 column. 
 
 179. — The Capital, from kephale or caput, the head, is the 
 uppermost and crowning part of the column. 
 
 180. — The Architrave, from archi, chief or principal, and 
 trahs, a beam, is that part of the entablature which lies in imme- 
 diate connection with the cokimn. 
 
 181. — The Frieze, from^^iron, a fringe or border, is that part 
 of the entablature which is immediately above the architrave and 
 beneath 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 Pediment, 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 Tympanum 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 
 
M AMERICAN HOUSE-CARPENTER. 
 
 186. — All 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 by 
 60, the scale is completed. The upright columns of figures, 
 marked ^and P, by the side of the drawings illustrating the orders, 
 designate the height and the projection of the members. The 
 projection of each member is reckoned from a line passing through 
 the axis of the column, and extending above it to the top of the 
 entablature. The figures represent minutes, or 60ths, of the 
 major diameter of the shaft of the column. 
 
 187. — Grecian Styles. The original method of building 
 among the Greeks, was in what is called the Doric order : to 
 this were afterwards added the Ionic 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 
 
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 Order is so ancient that its origin is un- 
 known — 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 
 rude, and perhaps not unlike the following figure. 
 
 The trunks of trees, set 
 
 perpendicularly to support 
 the roof, may be taken for 
 columns ; the tree laid upon 
 the tops of the perpendicu- 
 lar ones, the architrave ; the 
 ends of the cross-beams 
 
 Fig. 109. 
 
 which rest upon the architrave, the triglyphs ; the tree laid on 
 the cross-beams as a support for the ends of the rafters, the bed- 
 moulding of the cornice ; the ends of the rafters which project 
 beyond the bed-moulding, the mutules; and perhaps the projection 
 of the roof in front, to screen the entrance from the weather, gave 
 origin to the portico. 
 
 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 i7itercohiinniatio7i^ 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. 
 
 * See note, page 108. 
 
DORIC ORDER. 
 
 Fig. 110. 
 
ARCHITECTURE. 77 
 
 When a column is placed beneath every third triglyph, the ar- 
 rangement is called diastyle ; and when beneath every fourth, 
 arcp.ostyle. 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, (fcc. 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 Order. 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. Their temples were the principal objects upon which 
 their skill in the art was displayed ; and as the Doric order seems 
 to have been well fitted, by its massive 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 
 emblematical of feminine graces, and with which they might 
 decorate such temples as were dedicated to the goddesses. Hence 
 the origin of the Ionic order. This was invented, according to 
 historians, by Hermogenes of Alabanda ; and he being a native 
 of Caria, then in the possession of the lonians. the order was 
 called, the Ionic. 
 
 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 
 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 
 
78 
 
 IONIC. 
 
 Fig. 111. 
 
ARCHITECTURE. 
 
 79 
 
 three diameters, arceostyle, or columns thin-set. In all the orders, 
 when there are four columns in one row, the arrangement is 
 called, tetrastyle ; when there are six in a rov/, hexastyle ; 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 front of the 
 Merchants' Exchange, New- York city, is a good specimen of this 
 order. 
 
 Fip. 112. 
 
SQ 
 
 AMERICAN HOUSE-CARPENTER. 
 
 Fiff. 113. 
 
 195. — To describe the Ionic volute. Draw a perpendicular 
 from a to 5, [Fig. 112,) and make a s equal to 20 min. or to f of 
 the whole height, a c ; draw s 0, at right angles to s a, and equal 
 to 11 min. ; upon 0, with 2, min. for radius, describe the eye of 
 the volute ; a!)'wi;t 0, the centre of the eye, draw the square, r 1 1 
 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 the centre for 
 the arc, be; and so on to the last. The inside spiral line is to be 
 described from the centres, x, x^ x^ (fee, (Fig. 113,) being the 
 Cv ! !(' of the first small square towards the middle of the eye 
 f r< m the centre for the outside arc. The breadth of the fillet at 
 aj^ is to be made equal to 2-j\ min. This is for a spiral of three 
 revolutions ; but one of any number of revolutions, as 4 or 6, 
 
ARCHITECTURE. 
 
 81 
 
 may be drawn, by dividing o/, {Fig- 113,) into h corresponding 
 number of equal parts. Then divide the part nearest the centre, 
 0, into two parts, as at h ; join o and 1, also o and 2 ; draw h 3, pa- 
 rallel to 1, and h 4, parallel to o 2 ; then the lines, o 1, o 2, A 3, A 
 4, will determine the length of the heavy lines, and the place of 
 the centies. (See Art. 396.) 
 
 196. — The Corinthian 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 Yitruvius — an architect who flourished in 
 Rome, in the days of Augustus Caesar — 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 
 at Corinth. 
 
 197. — The Corinthian being the gayest, the richest and most 
 lovely of all the orders, it is appropriate for edifices which ai 
 
 11 
 
 Fig. 114. 
 
82 
 
 CORINTHIAN. 
 
 13 
 
 ~4 
 1 
 5 
 
 ~4 
 6 
 5 
 5 
 
 24^ 
 
 45 
 41 
 39 
 
 15 
 ~5 
 15 
 
 3i 
 
 J5 
 
 27 
 
 28* 
 
 25 
 
 33 
 35 
 35 
 33 
 38 
 
 43 
 
 mmm 
 
 L 
 
 1 
 
 )_ 
 
 Fig. 115 
 
ARCHITECTURE. S3 
 
 dedicated to amusement, banqueting and festivity — for ail 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 female& 
 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 to a column of the same 
 height. 
 
 201. — These semblances of slavery have been in frequent use 
 among moderns as well as ancients : and as a relief from the 
 stateliness and formality of the regular orders, are capable ot 
 forming a thousand varieties ; yet in a land of liberty such marks 
 of human degradation ought not to be perpetuated. 
 
 202. — Roman Styles. 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 
 
g4 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, <fec. — 
 were, in extent and grandeur, beyond any thing attempted in 
 modern times. Aqueducts Avere 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 
 
ARCHITECTURE. 85 
 
 belvow it ; by placing the centre of the first triglyph, instead of 
 one edge, over the centre of the column ; and introducing 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 in one. This change was 
 dignified with the importance of an order^ and received the ap- 
 pellation Composite, 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 
 ;he luxury of the age in which it was invented. 
 
 204. — The Tuscan Order 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 I modification of this order, called the moc^er/i Tuscan, 
 [Fig. 116,) will be useful ; and its simplicity recommends it 
 where economy is desirable. 
 
 206. — Egyptian Style. The architecture of the ancient 
 
86 
 
 TUSCAN. 
 
 Fig. 116. 
 
ARCHITECTURE. 87 
 
 Egyptians— to which tliat of the ancient Hindoos bears some re- 
 semblance — is characterized by boldness of outline, solidity and 
 grandeur. The imazing 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 11 diameters and seldom exceeding 2|. 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 wert 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 
 
88 
 
 EGYPTIAN. 
 
 HP. 
 
 I-'ii:. 117. 
 
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, (fee. ; 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 be preferred in whicn 
 utility, stability and regularity, are gracefully blended with gran- 
 deur and elegance. But as an arrangement designed for a warm 
 country would be inappropriate for a colder climate, it would seem 
 that the style of building ought to be modified to suit the wants 
 of the people for whom it is designed. High roofs to resist the 
 pressure of heavy snows, and arrangements for artificial heat, are 
 indispensable in northern climes ; while they would be regarded 
 as entirely out of place in buildings at the equator. 
 
 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 buildingSjporticos, porches, <fec., 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 
 
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 a church, 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 adjacent country. A splendid country-seat should 
 have the out-houses and fences in the same style with itself, the 
 trees and shrubbery neatly trimmed, and the grounds well cul- 
 tivated. 
 
 212. — Europeans express surprise that so many houses in this 
 country are built of wood. And yet, in a new country, where 
 wood is plenty, that this should be so is no cause for wonder. 
 Still, the practice should not be encouraged. Buildings erected 
 with brick or stone are far preferable to those of wood ; they are 
 more durable ; not so liable to injury by fire, nor to need repairs ; 
 and will be found in the end quite as economical. A wooden 
 house is suitable for a temporary residence only ; and those who 
 would bequeath a dwelling to their children, will endeavour to 
 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. 
 
91 
 
 Fig. 118 
 
 Fig. 119. 
 
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 diiferent 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 icJuio graphical 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. 
 
 c, e, — pantries with shelving. 
 
 / — passage having shelves, drawers, &c.j on one side, and 
 clothes-hooks on the other. 
 g — kitchen-dresser. 
 A, ^, — front and rear areas. 
 
 Fig. 119 — plan of the first-story, 
 j^i— parlours. 
 k — library. 
 I — 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. 
 
93 
 
 ~^ 
 
 - ». ' 
 
 J=^. 
 
 r 
 
 =n c 
 
 rh c 
 
 /M 
 
 ^^=^ ?=^ i 
 
 I f I 
 
 ■i= i^ v-^ — "- ^ * 
 
 / 
 
 '» r 
 
 1 
 
 ji 
 
 s 
 
 7. 
 
 l-T 
 
 a 
 
 h ±h i;. 1 
 
 I \ ' 
 
 7i 
 
 g 
 
 r '1 
 
 .1 1 . 
 
 Ti-. IJU. 
 
 Fijj. 121. 
 
94 AMERICAN HOUSE-CARPENTER. 
 
 Fig. 121 — plan of the attic-story. 
 
 /—nursery, 
 
 g^ g, gy ^ — bed-chamber«= 
 
 hj A, h, h, /?, — wardrobes 
 
 I — pantry with shelves, 
 
 ; — 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 as it 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) v/alls. 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 Avinders of 
 
96 
 
 Ee 
 
 . _— r i ^ ) i 
 
 L 
 
 n 
 
 L i 
 
 
 i c 
 
 
 ~n 
 
 
 1 
 
 [L_ 
 
 - 
 
 
 
 — 
 
 J 
 
 T 
 
 
 mil III! 
 3 43210 
 
 10 
 
 15 
 
 2fj/Y 
 
 Fig. 122. 
 
96 AMERICAN HOUSP>CARPENTER. 
 
 tlie Other stories be placed perpendicularly over and under them ; 
 and be careful to provide for head-room. To ascertain this, when 
 it is doubtful, it is well to draw a vertical section of the whole 
 stairs ; but in ordinary cases, this is not necessary. To dispose 
 the windows properly, the middle Avindow 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, 
 ill 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 f of the width of one of the middle piers. When this 
 is desirable, deduct the amount of the three openings from the 
 width of the front, and the remainder will be the amount of the 
 width of all tlie piers ; divide this by 10, and the product will be 
 -^- 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 idea to durability. 
 And when in his walks abroad, the symmetry, proportion and 
 beauty of nature met his admiring gaze, contrasting so strangely 
 with the misshapen and disproportioned work of his own hands, 
 he was led to make gradual changes ; till his abode was rendered 
 
ARCHITECTURE. 97 
 
 not only commodious and durable, but pleasant in its appearance ; 
 and building became a fine-art, having utility for its basis. 
 
 2] 6. — 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 pubUc as- 
 semblies, oratory, state, visitors, retiring, eating, reading, sleeping, 
 bathing, dressing, (fee. — 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 cases 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 puUies, 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 secure their being easily kept dry and clean. In dwellings, 
 suitable apartments should be fitted up for bathings 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 
 
P8 AMERICAN HOUSE-CARPENTER. 
 
 bility. It is well, also, that it have the appearance of stability as 
 well as the reality ; for should it seem tottering and unsafe, the 
 sensation of fear, rather than those of admiration and pleasure, 
 will be excited in the beholder. To secure certainty and accu- 
 racy in the application of those principles, a knowledge of the 
 strength and other properties of the materials used, is indispensa- 
 ble ; and in order that the whole design be so made as to be 
 capable of execution, a practical knowledge of the requisite 
 mechanical operations is quite important. 
 
 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 em-ichment : 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 a build- 
 ing are founded. A scientific knowledge of these will insure 
 certainty 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 
 
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 Foundation 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 betv/een 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 bottoiu 
 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, ox pillar. This is 
 a perpendicular part, commonly of equal breadth and thickness, 
 not intended for the purpose of enclosure, but simply for tlie 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 Avhich presses equally on 
 the whole column, the thickness should gradually decrease from 
 bottom to top. The outline of columns should be a little curved, 
 so as to represent a portion of a very long spheroid, or paraboloid, 
 
100 AMERICAN HOUSE-CARPENTER. 
 
 rather than of a cone. This figure is the joint result of two cal- 
 culations, independent of beauty of appearance, " One of these 
 is, that the form best adapted for stability of base is that of a 
 cone; the other is, that the figure, which would be of equal 
 strength throughout for supporting a superincumbent weight, 
 would be generated by the revolution of two parabolas round the 
 axis of the column, the vertices of the curves being at its ex- 
 tremities. The swell of the shafts of columns was called the 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 Wall, 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 structures as are intended to possess great 
 strength and durability, and, in some 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 
 ^alf of the wall being completed with brick ; so that they can, 
 
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 wall is made of earth, rendered compact by ramming it in 
 moulds or cases. This method is called building in pise^ 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. 
 
102 AMERICAN HOUSE-CARPENTER. 
 
 224. — The Lintel, or heam^ 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. 
 
 225. — The Arch is a transverse member of a building, an- 
 swering the same purpose as the lintel, but vastly exceeding it in 
 strength. The arch, unlike the lintel, may consist of any num- 
 ber of constituent pieces, without impairing its strength. It is, 
 however, necessary that all the pieces should possess a uniform 
 shape, — the shape of a portion of a wedge, — and that the joints, 
 formed by the contact of their surfaces, should point towards a 
 common centre. In this case, no one portion of the arch can be 
 displaced or forced inward ; and the arch cannot be broken by 
 any force which is not sufficient to crush the materials of which 
 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 two 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 it is 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 tlie 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 
 
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 Unk, 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 Aveight 
 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 
 
104 AMERICAN HOUSE-CARPENTER. 
 
 described usually from two centres outside the arch. It is a 
 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 llie 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 voiissoirs, 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. It is 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 
 side way 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 w^ant 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- 
 
ARCHITECTURE. 105 
 
 rable that each of the piers should possess sufficient horizontal 
 strength to resist the lateral pressure of the adjoining arches. 
 
 226. — The Vault is the lateral continuation of an arch, serving 
 to cover an area or passage, and bearing the same relation to the 
 arch that the wall does to the column. A simple vault is con- 
 structed on the principles of the arch, and distributes its pressure 
 equally along the walls or abutments. A complex or groined 
 vault is made by two vaults intersecting each other, in whicji 
 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 somewliat iilve ihe 
 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 alwa^^s 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 
 
106 AMERICAN HOUSE-CARPENTER. 
 
 rests upon a circular wall, which is supported, in its turn, by 
 arches upon massive pillars or piers. This construction is called 
 •building upon pendentives^ and gives open space and room for 
 passage beneath the dome. The remarks which have been made 
 ill 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 
 ^leat 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 
 vrails, 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 Roof is the most common and cheap method of 
 covering buildings, to protect them from rain and other effects of 
 the weather. It is sometimes fiat, 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 ^nansard roof. In modern times, roofs 
 are made almost exclusively of Avood, though frequently covered 
 with incombustible materials. The internal structure or carpen- 
 try of roofs is a subject of considerable mechanical contrivance. 
 
ARCHITECTURE. 107 
 
 The roof is supported by rafters^ which abut on the walls on 
 each side, Uke 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-heam^ as it is called, extends 
 across, receiving the ends of tlic rafters, and protecting the wall 
 from their horizontal thrust. To prevent the tie-beam from 
 sagging^ or bending dovv^nward 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 tritssing. Qiueeii-posts aie : omctimcs 
 added, parallel to the king-post, in iarge roofs ; also various other 
 connecting timbers. In Gothic buildings, where the vauUs 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 niateria! 
 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 to approach 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- 
 
108 AMERICAN HOUSE-CARPENTER. 
 
 tended to truss or suspend one piece by another, should be formed 
 "ipon similar principles. 
 
 Roofs in the United States, after being boarded, receive a se- 
 condary covering of shingles. When intended to be incombustible, 
 they are covered with slates or earth ern 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.) 
 
 NOTE TO ARTICLE 189. 
 
 Gi'ECiAN DoiJic Orper. When the xcidth to be occupied by the whole front is limited ; to deter- 
 mine tiie dianutLr of ihe colmnn. 
 The relation between the parts may be expressed thus : 
 
 _ 00 o 
 
 '^ ~ «:(i + c) + (60 — c) 
 
 Where a eq'i;il« tii • width in feet occupied by the cohTmns, and their intercolumuiations taken 
 collectively. ni.'.isi'jvJ at tlie base ; b eci'.ials the width of the metope, in miniUes; c equ;tls the width 
 of the trigiypij-- in iuiuuteij; d equals the number of metopes, iX'-A z equals the di;iinet(=r in feet. 
 
 ExampU. — A ir.mi. of six columns— hexiistyle— 61 feet wide; thi- frieze having one triyjlyph over 
 each intercolumtiirnion, or mono-triij!y;)h. In this case, there hexw.; five intercoluminatious aud two 
 metopes over e:.c!). tiierefore there ai-o 5 X 2=^ 10 metope:?. Let ihe metispe equal 4'2 minutes and 
 (he triglyph op;;.; 2-^. T:ien = 01; Z' = 42; r = W; auJ d=z 10; and the formula above becoiaos, 
 
 ' - iTi-^-jr^— ^ (GiT^ITo^ - i0^;) + 32= 732"= ^ *"^' ^ ^'"^ ^'^'"'^'' x^^'^\r^(^. 
 
 Example. — Aii uct-istyit' fror^t, 8 coiunins, 1S4 feel wide, threi- metopes over each inter col urania- 
 lion, 21 in all, nud the metope and triglyph 42 and 28, as bel'orr. Then, 
 
 ' ^ 21(42 + 2dH-~(6(ri28y = 1^ = '-^^TS o 2 fcct = the d.ameter required. 
 
SKCTION III.— 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 ancient architecture, are eight 
 in number ; and are known by the following names : 
 
 Annulet, band, cincture, fillet, listel or square. 
 
 Fip. ]23. 
 
 Fig. 124. 
 
 _) Astragal or bead. 
 
 Fig. 125. 
 
 ^ Torus or tore. 
 
 L 
 
 Fig: 126. 
 
 Scotia, trochilus or mouth. 
 
 Ovolo, quarter-round or echinas. 
 
 Fig. !27. 
 
110 
 
 AMERICAN HOUSE-CARPENTER. 
 
 Cavetto, cove or hollow. 
 
 Fij-. 128. 
 
 ^ 
 
 Cymatium, or cyma-recta. 
 
 Fig. 129. 
 
 / 
 
 Inverted cymatiumj or cyma-reversa 
 
 Fit^. 130. 
 
 ) 
 
 Some of the terms are derived thus : fillet, from the French 
 word Jil, thread. Astragal, from astragalos^ a bone of the heel 
 — or the curvature of the heel. Bead, because this moulding, 
 when properly carved, resembles a string of beads. Torus, or 
 tore, the Greek for rope^ which it resembles, when on the base of 
 a column. Scotia, from shotia, darkness, because of the strong 
 shadow which its depth produces, and Avhich 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 ^"^/ma^o«, 
 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 
 
MOULDINGS, CORNICES, &C. 1 I I 
 
 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 ; tlie reasons for giving distinct names to tliem 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- 
 rectaand 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, bat 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 6, {Fig. 131 ;) and from/, the given projection 
 of the moulding, draw/ o, at right angles to the fillets ; from 6, 
 draw h A, at right angles to a b ; bisect a b in c ; join / and c, 
 and upon c, with the radius, c/ describe the arc, / h, cutting b h 
 in h ; through c, draw d e, parallel with the fillets ; make d c and 
 c e, each equal to b h ; then d e and a b will be conjugate diame- 
 
112 
 
 AMERICAN HOUSE-CARPENTER. 
 
 Fig. 131. 
 
 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. 
 
 d 
 
 \ 
 
 ^ 
 
 
 a 
 
 Fig. 133 
 
 234. — Fig. 132 to 139 exhibit various modifications of the 
 Grecian ovolo, sometimes called echinus. Fig. 132 to 136 eire 
 
MOULDINGS, CORNICES, &C. 
 
 113 
 
 Fii:. 134. 
 
 Vl'r. 135. 
 
 Fiar.lSe. 
 
 Fig. 13* 
 
 ^ b 
 
 Fig. 138. 
 
 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 6, being taken at pleasure 
 in order to flatten the curve, a b should be made short in propor- 
 tion to a c. 
 
 15 
 
114 
 
 AMERICAN HOUSE-CARPENTER. 
 
 Fig. 141. 
 
 Fig. 140. 
 
 235. — To describe the Grecian cavetto, {Fig. 140 and 141,) 
 having the height and projection given, see Art. 118. 
 
 a 
 
 I 
 
 \iw 
 
 1^ 
 
 Wi 
 
 V 
 
 c 
 
 Fiff. 142. 
 
 Fig. 143. 
 
 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 ah c d into 4 equal parallelograms ; 
 then proceed cis directed in note to Art. 118. When the projec- 
 tion is less than the height, draw d a, {Fig. 143,) at right angles 
 to a b ; complete the rectangle, abed; divide this into 4 equal 
 rectangles, and proceed according to Art. 118. 
 
 Fi-. 114 
 
 237. — To describe the Grecian cyma-reversa. When the 
 
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 quaxi- 
 rants, having different radii, as at Fig. 1 46 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 6, and bisect 
 ahin c ; from c, and at right angles to a b, draw c d^ cutting a 
 level line drawn from a in d ; then d will be the centre. This 
 moulding projects less than its height. When the projection is 
 more than the height, as at Fig. 152, extend the line from c until 
 
 Fig. 146. 
 
 b\s. 14' 
 
 n 
 
 Fif. 148. 
 
 Rf. 14». 
 
116 
 
 AMERICAN HOUSE-CARPENTER. 
 
 Fig. 151. 
 
 d 
 
 K 
 
 
 :>^ 
 
 
 a 
 
 Fig. 152. 
 
 Fig. 153. 
 
 
 
 r 
 
 J 
 
 
 
 
 
 : 
 
 Fig. 154. 
 
 Fig. 155. 
 
 rig. 190. 
 
 Fif. urr 
 
MOULDINGS, CORNICES, &C. 
 
 117 
 
 Fig. 158. 
 
 Fig. 159. 
 
 fig. 160. 
 
 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 ait 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 6, at c ; upon a, c and b, successively, with a c 
 01 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 6, 
 at c, and b c aX d; from d, draw d e, at right angles to e b ; from 
 b, draw b /, parallel to e d ; upon b, with b d for radius, describe 
 the arc, df; divide this arc into 7 equal parts, and set one of the 
 parts from 5, the limit of the projection, to o ; make o h equal to 
 e; from A, through c, draw the tangent, hi; divide b h, h c,ci 
 and i a, each into a like number of equal parts, and draw the in- 
 
118 
 
 AMERICAN HOUSE-CARPENTER. 
 
 Fig. 162. 
 
 Far 163, 
 
 Fig. 164. 
 
MOULDINGS, CORNICES, &C. 
 
 119 
 
 Fiff. 165. 
 
 Fig. 166. 
 
 Fig 167. 
 
 Fig. 168, 
 
 Fig. 169 
 
 tersecting lines as directed at Art. 89. If a bolder form is desired, 
 draw the tangent, i A, 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 
 
120 
 
 AMERICAN HOUSE-CARPENTER. 
 
 in comparison with its height, the point, c, being found as in the 
 last figure, h 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 /, is taken at pleasure ; and d cis parallel to a 
 e. To find the length of d c, draw b h, at right angles to a b ; 
 upon 0, with o f for radius, describe the arc,/ h, cutting b h in 
 h ; then make o c and o d, each equal to b h* To draw the curve, 
 see note to A?^t. 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. 
 
 H. P. 
 
 5 15 
 
 125^1 
 
 2i 11 
 
 9l0i' 
 
 10 
 
 J 
 
 Fig. 170. 
 
 H. P. 
 
 U 15 
 
 Hi 
 
 14i 
 
 13 
 
 iU 
 
 lU 
 
 lO.j 
 
 10 
 
 I 
 
 Fig. 171. 
 
 240.— K^. 170 and 171 are designs for antse caps. The 
 
 * The manner of ascertaining the length of the conjugate diameter, dc/m this figure, 
 and also in Fig. 131, 175 and 176, is new, and is important in tliis application. Ic is 
 founded upon well-known mathematical principles, viz : All the parallelograms that may- 
 be circumscribed about an ellipsis are equal to one another, and consequeudy any one 
 is equal to the rectangle of the two axes. And again : the sum of the squares of every 
 psdr of conjugate diameters is equal to the sum of the squares of the two axes. 
 
MOULDINGS, CORNICES, &C. 
 
 121 
 
 diameter of the antse is divided into 20 equal parts, and the height 
 and projection of the members, are regulated in accordance with 
 those parts, as denoted under ^Tand P, height and projection. 
 The projection is measured from the middle of the antas. These 
 will be found appropriate for porticos, door- ways, mantle-pieces, 
 door and window trimmings, <fec. The height of the antse for 
 mantle-pieces, should be from 5 to 6 diameters, having an entab- 
 lature of from 2 to 2i 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 antae, 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. 
 
 
 
 
 
 
 
 n 
 
 20 
 
 ..... , 
 
 1 
 
 5 
 
 
 
 
 
 J 
 
 f 
 
 
 i7i 
 
 
 
 
 _J 
 
 
 
 J 
 
 
 3 
 
 8 
 
 16; 
 
 
 
 
 
 
 ..-.'' ^ 
 
 c^ *'**•. 
 
 
 1 
 25 
 
 1 
 
 ) 
 
 
 
 
 
 
 
 
 
 Fig. 172. 
 
 16 
 
122 
 
 AMERICAN HOUSE-CARPENTER. 
 
 H. P. 
 
 [U;20 
 
 1 
 1 
 
 16 
 
 3i 
 
 U 
 
 3i 
 
 u 
 
 ^4^ 
 
 U 
 
 2} 
 
 2^ 
 
 Fig. 173. 
 
 :;:^ 
 
 H.P. 
 
 3J20 
 
 3i,16 
 1 
 
 . 2 ■ 
 
 Fig. Hi. 
 
MOULDI.NC.S, CORNICES, &C. 
 
 123 
 
 Fh. 175. 
 
 Fijr. 176. 
 
124 
 
 AMERICAN HOUSE-CARPENTER. 
 
 b 12 3 4c 
 
 Fig. 177. 
 
 242.-' — To proportioti an eave cornice in accordance with the 
 height of the building. Draw the line, a c, {Fig. 177,) and 
 make b 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 a c ; bisect b din 
 e, and from a, through e, draw a f; upon a, with a c for radius, 
 describe the arc, c/, and upon e, with e/for radius, describe the 
 'dvcfd; divide the curve, df c, into 7 equal parts, as at 10, 20, 
 30, (fee, and from these points of division, draw lines to b c, pa- 
 rallel to d b ; then the distance, b 1, is the projection of a cornice 
 for a building 10 feet high ; b 2, the projection at 20 feet high ; 
 b 3, the projection at 30 feet, ttc. If the projection of a cornice for 
 a building 34 feet high, is required, divide the arc between 30 and 
 40 into 10 equal parts, and from the fourth point from 30, draw a 
 line to the base, b c, parallel with b d ; then the distance of the 
 point, at which that line cuts the base, from b, will be the projec- 
 tion required. So proceed for a cornice of any height within 70 
 feet. The above is based on the supposition that 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, 
 maybe 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. 
 
MOULDINGS, CORNICES, &C. 
 b 
 
 125 
 
 Fig. 178. 
 
 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 h ; join a and h ; then h 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 be; parallel with c d, 
 draw lines from each projection of the given cornice to the line, 
 ad; then e d will be the required projection for the proposed 
 cornice, and the perpendicular Imes falling upon e d will indicate 
 the proper projection for the members. 
 
 244. — To proportioii a cornice according to a larger given 
 one. Let J., {Fig. 179,) be the given cornice. Extend a 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 / and e ; upon o, with the 
 projection of the given cornice, o a, for radius, describe the quad- 
 rant, ad; from d, draw d b, parallel iofe; upon o, with o b for 
 radius, describe the quadrant, be; then o c will be the proper pro- 
 jection for the proposed cornice. Join a and c ; draw lines from the 
 
126 
 
 AMERICAN HOUSE-CARPENTER. 
 
 Fig. 179. 
 
 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, of, 
 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 by a different process. Thus : upon o, 
 with c for radius, describe the quadrant, c b ; upon o, with o a 
 for radius, describe the quadrant, a d ; join d and b ; from/, draw 
 / e, parallel to d b ; then o 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 a larger ; 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- 
 
MOULDINGS, CORNICES, &C. 
 
 A \e 
 
 l^iT 
 
 g Fig. 180 
 
 Fi'j. 181, 
 
 lei with the wal],/c?, drav/ the line, ab ; make e c equal a /, 
 and through c, draw c b, parallel with e d ; join d and 6, and from 
 the several angular points in B, draw ordinates to cut ^ 6 in 1, 2 
 and 3 ; at those points erect lines perpendicular to d b ; from h, 
 draw h g, parallel to/ a ; take the ordinates, 1 o, 2 o, &c., at B, 
 and transfer them to C, and the angle-bracket, C, will be defined. 
 In the same manner, the angle-bracket for an internal cornice, or 
 the angle-rib of a coved ceiling, or of groins, as at Fig. 181, can 
 be found. 
 
 246. — A level crown moulding being given , to find therakins;' 
 m,ouldi7ig and a level return at the top. Let A, {Fig. 182.) be 
 the given moulding, and A b the rake of the roof. Divide the 
 curve of the given moulding into any number of parts, equal or 
 unequal, as at 1, 2, and 3 ; from these points, draw horizontal 
 lines to a perpendicular erected from c; at any convenient place 
 on the rake, as at 5, draw a c, at right angles to A b ; also, from 
 6, draw the horizontal line, b a ; place the thickness, d a, of the 
 moulding at A, from b to a, and from a, draw the perpendicular 
 line, a e ; from the points, 1, 2, 3, at J., draw lines to C, parallel 
 to Ab ; make al, a 2 and a 3, at J5 and at C, equal to a 1, &c., 
 at A ; through the points, 1, 2 and 3, at B, trace the curve — this 
 will be the proper form for the raking moulding. From 1, 2 and 
 
128 
 
 AMERICAN HOUSE-CARPENTER. 
 
 Fig 182. 
 
 3. at C. drop perpendiculars to the corresponding ordinates from 
 1. 2 and 3, at J. /through the points of intersection, trace the 
 curve — this will be the proper tbrm for the return at the top. 
 
SECTION IV.— FRAMING. 
 
 247. — This subject is, to the carpenter, of the highest impor- 
 tance ; and deserves more attention and a larger place in a volume 
 of this kind, than is generally allotted to it. Something, indeed, 
 has been said upon the geometrical principles, by which the seve- 
 ral lines for the joints and the lengths of timber, may be ascer- 
 tained ; yet, besides this, there is much to be learned. For how- 
 ever precise or workmanlike the joints may be made, what will 
 it avail, should the system of framing, from an erroneous position 
 of its timbers, <fec., change its form, or become incapable of sus- 
 taining even its own weight ? Hence the necessity for a know- 
 leds:e of the laws of pressure and the strength of timber. These 
 bemg once understood, we can with confidence determine the best 
 position and dimensions for the several timbers which compose a 
 floor or a roof, a partition or a bridge. As systems of framing 
 are more or less exposed to heavy weights and strains, and, in 
 case of failure, cause not only a loss of labour and material, but 
 frequently that of life itself, 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 
 
130 
 
 AMERICAN HOUSE-CARPENTER. 
 
 the least quantity of timber that will suffice for strength. The 
 greatest fciult 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 bo constructed, let us attend to a few of 
 the elementary laws iji Mechanics ^ w^hich 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 weiglit in a vertical direction. 
 Example: Suppose an iron ball, weighing lUU 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 Vv^eight, and will 
 be in proportion to their position. The farther apai t their feet are 
 spread the greater will be the pressure, nnd vice versa. Hence 
 tremendous strains m.ay 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, h d ; and the weight, W^ would fall in that line, it 
 the two posts were removed, hence the best position for a support 
 
 w 
 
 Fig. 183. 
 
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 any certain 
 weight is capable of exerting upon oblique supports. This can 
 be ascertained by the following process. 
 
 Let a h and h 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, h d, equal to 6 inches ; 
 from d, draw df, parallel to a b, and d e, parallel Xo c b ; then 
 the line, b e, will be found to be Sg inches long, which is equal tc 
 the number of tons that the weight, W, exerts upon the post, a b. 
 The pressure upon the other post is represented by bf, 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, bf,fd 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 coequal, (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 6, is pressed, 
 and bf that exerted upon b c. 
 
 Fig. 184 
 
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. The 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. 165. 
 
 249. — In Fig. 185, the weight, W^ exerts a pressure on the 
 struts in the direction of their length ; their feet, w, w, have, there- 
 fore, a tendency to move in the direction, n o, and would so move, 
 were they not opposed by a sufficient resistance from the blocks, 
 A and A. If a piece of each block be cut off at the horizontal 
 line, a ri, 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 6, then the struts would 
 descend vertically. To estimate the horizontal and the vertical 
 pressures exerted by the struts, let w o be made equal (upon any 
 scale of equal parts) to the number of tons (or pounds) with 
 .vhich the strut is pressed ; construct the parallelogram of forces 
 
FRAMING. 
 
 133 
 
 by drawing© e parallel to a n, and 0/ parallel to 6 ii ; then 71 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, ?i h. 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 struts is 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 jB, {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 asunde' 
 in the direction of its length, may be measured at the foot of th^ 
 
134 
 
 AMERICAN HOUSE-CARPENTER. 
 
 rafter, as was shown at Fig. 185 ; but it can be more readily 
 and as accurately measured, by drawing from /and e horizontal 
 lines to the vertical line, h d, meeting it in o and o; then/ o will be 
 the horizontal thrust at B, and e ooX 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. 
 
 Fig. 188. 
 
 251 — Fig. 187 and 188 exhibit methods of framing for sup- 
 porting the equal weights, Tl^^and W. Suppose it be required to 
 measure and compare the strains produced on the pieces, A B 
 and A C. Construct the parallelogram of forces, e h f d^ ac- 
 cording to Art. 248. Then h f will sliow the strain on A B, and b 
 5 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 
 
FRAMING. 135 
 
 times as great as those in Fig. 188 : the position of the pieces, 
 A B and A C, in Fig. 188, is therefore far preferable. 
 
 This and the preceding examples exemplify, in a measure, the 
 resolution of forces ; viz., the finding of two or more forces, which, 
 acting in different directions, shall exactly balance the pressure 
 of any given sijigle force. Thus, in Fig. 185, supposing the 
 weight, Wj 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 two forces just equal to 
 the given force. 
 
 C Fig. 189 
 
 252. — The composition of forces consists in ascertaining the 
 direction and amount of one force, which shall be just capable of 
 balancing two or inore given forces, acting in different directions. 
 This is only the reverse of the resolution of forces, and the two 
 are founded on one and the same principle, and may be solved in 
 the same manner. For example ; let A and B, {Fig. 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 b f equal to the pressure on A, viz., 6, and complete 
 the parallelogram of forces, ebfd; then b d, the diagonal of th^ 
 
136 
 
 AMERICAN HOUSE-CARPENTER. 
 
 parallelogram, will be the direction, and its length will be the 
 amoujitj of the united pressures of A and of B. The line, b d, is 
 termed the resultant of the two forces, h f and he. HA and B are to 
 be supported by one post, C, the best position for that post will be 
 in the direction of the diagonal, h d ; and it will require to be 
 sufficiently strong to support the united pressures of A and of B 
 
 't*^9 
 
 Fig. 190. 
 
 253. — Another example : let Fig. 190 represent a piece oi 
 framing commonly called a crane, which is used for hoisting 
 heavy weights by means of the rope, Bh f, which passes over a 
 pulley at h. This is similar to Fig. 187 and 188, yet it is mate- 
 rially different. In those figures, the strain is in one direction 
 only, viz., from b to d ; but in this there are two strains, from A 
 to B and from A to W. The strain in the direction, A B, is 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 bfd; then draw the diagonal, b d, 
 and it will be the position required. Should the foot, C, of the 
 strut 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 
 
FRAMING. 
 
 137 
 
 r\ 
 
 •w/lltv^ 
 
 
 B 
 
 L 
 
 
 
 
 flc 
 
 
 
 
 ^^ . 
 
 
 Sa«», 
 
 
 v_y 
 
 '^V/V 
 
 w 
 
 254. — Ties and Struts. Timbers in a state of tension are 
 called ties, while such as are in a state of compression are termed 
 struts. This subject can be illustrated in the following manner. 
 
 Let A and B, [Fig. 191,) represent beams of timber supporting 
 the weights, TF, W and W; A having but one support, which is 
 in the middle of its length, and B two, one at each end. To 
 show the nature of the strains, let each beam be sawed in the 
 middle from a to 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 beam 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 be affected. By this it appears evident 
 that, in a piece of timber subject to a pressure across the direction 
 of its length, the fibres are exposed to contrary strains. If the tim- 
 ber is supported at both ends, as at B, those from the top edge down 
 to the middle are compressed in the direction of their length, while 
 those from the middle to the bottom edge are in a state of tension ; 
 but if the beam is supported as at J., 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 
 
 18 
 
13S AMERICAN HOUSE-CARPENTER. 
 
 from siich as are compressed, in order that the former may be pre- 
 served continuous. A strut may be constructed of two or more 
 pieces; yet, where there are many joints, it will not resist com- 
 ression so firmly. 
 
 255. — To distinguish ties from struts. This may be done 
 by the following rule. In Fig. 183, the timbers, a h and h 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, 6, towards d. Let it be required to ascer- 
 tain whether the sustaining forces are stretched or j^ressed by the 
 straining force. Rule : upon the direction of the straining force, 
 b c?, 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 c 6 ; through the point, 5, draw a line, parallel to the diag- 
 onal, cf; 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 A C 
 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, h i, 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, b 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 P, {Fig. 192,) represent 
 a gate, supported by hinges at A and E. In this case, the strain- 
 
Fi?. 19-2. 
 
 i?ig force is the weight of the materials, and the direction ol 
 course verticaL Ascertain the dividing line at the several points, 
 G, B, I, J, Hnud 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 Hand HE — 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 jPis sustained by G 
 Fwiid F E, G Fhe'mg stretched and FE pressed. By this it 
 appeals that A B is in a state of tension, and E F, of compres- 
 sion; also, that AHnud G Fare stretched, wiiile B H and G 
 E are compressed : which sliows tlie necessity of having A H 
 and G F, each in oue whole length, while B Hand G E may 
 be, as they are shown, each in two pieces. The force at /is sus- 
 tained by G J and 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 J and i 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 cent'e ol 
 
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, B c, 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 Ime, B b, cattirig b d in b ; join b and A^ and b A 
 will be the direction of the thrust. To prevent the beam from 
 loosing its footing, the joint at A should be made at right angles 
 to b 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 e d, 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 
 
FRAMING. 
 
 141 
 
 Fig. 194. 
 
 pressures at 5, 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 ashed, [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 surf^ice. The inclination, in a rafter 
 uniformly loaded, which will produce the least oblique pressure, 
 {6 e. Fig. 193,) is 35 degrees and 16 minutes. 
 
 ^■^rA 
 
 Fig. 195. 
 
 %8. — In shed, or lean-to roofs, as Fig. 193, the horizontal 
 j>:it*<5sure will be entirely removed, if the bearings of the rafters, as 
 A, Bj {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 horizoptaPy supported at each end and 
 uniformly loaded, is subjec* to tJi«' greatest strain at the middle 
 
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 othei 
 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 tvtnher. 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, 
 &c. 
 
 Fig. 196. 
 
 Fig. 197 
 
 Fig. 198. 
 
 The strength of a })iece of timber, is to be considered in ac* 
 cordance witti the direction in which the strain is applied upon 
 
FRAMING. 143 
 
 it. When it is compressed in the direction of its length, as in 
 Fig. 196j its strength is termed the resistance to compression. 
 When the force tends to pull it asunder in the direction of its 
 length, (^, 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 hend 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 a line v^^ith the axis of the 
 post. 
 
 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. Ride. — 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. Rule. — 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 
 
144 AMERICAN HOUSE-CARPENTER. 
 
 thickness in inches. Example. — 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 rectangular post is 
 that in which the sides are as 10 to 6. 
 
 Case 3.— To find the area of a row?ic?, or cylindrical, post, that 
 will safely hear a given weight — when its height is 10 times its 
 least diameter or more. Rule. — Multiply the given weight or 
 pressure in pounds by 1*7, and the product by -0015 for oak, -0021 
 for j)itch 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. Exam- 
 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 - - - 10,000 
 
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. Example. — 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, 'Oil for white pine and '016 
 for pitch pine ; divide the product by the breadth in inches, and 
 the cube-root of the quotient will be the depth required in inches. 
 Example. — 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 24 is 576, and this, multiplied 
 by 900, gives 518-400; and this again, by -Oil, 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. Rule. — Multiply the 
 square of the length in feet by the weight in pounds, and multi- 
 ply this product by the decimal, -009, for oak, -Oil 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. Exarnple. — What should be the breadth of a beam of 
 oak, having a bearing of 1 6 feet and a depth of 12 inches, in 
 
 19 
 
146 AMERICAN HOUSE-CARPENTER. 
 
 order to support a weight of 4000 pounds? The square of 16 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-g 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, 'Oil 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, 
 v/hich 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 tiie breadth and depth may be ob- 
 tained by merely changing the decimal, 0'6, in the rule. Exam- 
 ple. — What should be the depth and breadth of a beam of pitch 
 pine, having a proportion to one another as 06 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-G, 
 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, '0095 for oak, -Oil 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. Example. — 
 What should be the size of an oak beam, the sides to bear a pro- 
 portion to ci.e another as 0*6 to 1, in order to support a ton weight 
 
FRAMING. 
 
 147 
 
 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 1018, 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. 
 
 Fig. 199. 
 
 264. — To ascertain the scantling of the stiffest beam that 
 can he cut from a cj/linder. Let d a c b, [Fig- 199,) be the sec- 
 tion, and e the centre, of a given cylinder. Draw the diameter, 
 a b ; upon a and 6, with the radius of the section, describe the 
 arcs, d e and e c ; join d and «, a and c, c and h. and h and d ; 
 then the rectangle, d a cb^ 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 
 
148 AMERICAN HOUSE-CARPENTER. 
 
 ii'icheSj and length of bearing 20 feet ? The length m feet, 20. 
 multiplied by the decimal, 0*6, gives 12; the given area in inches, 
 48, divided by 12, gives a quotient of 4, the square of which is 
 16 — this is the depth in inches ; and the breadth must be c 
 inches. A beam 1(3 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- 
 in[;s, 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. In the 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 smgle-joisted, double., and framed. In a single- 
 joisted floor, the timbers, or fioor-joists, are disposed as is shown in 
 Fig. 200. Where strength is the principal object, this manner 
 of disposing the fioor-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 hearing 
 and thickness being given, when the distance from centres is 
 
FRAMING. 
 
 149 
 
 Fig. 20a 
 
 12 inches. Rule. — Divide the square of the length in feet, by 
 the breadth in inches ; and the cube-root of the quotient, multi- 
 pUed 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, <fec., occur to interrupt 
 the bearing, the joists are framed into a piece, (6, Fig. 201,) 
 called a trimmer. The beams, a, a, into which the trimmer is 
 framed, are called tri?n7?iing-b earns, trimming-joists, or car- 
 riage-beams. They need to 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- 
 
160 
 
 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, 
 &c., it will be sufficient to add I 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 f of an inch, making the carriage-beams 
 3| 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 acciuately 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 stifien 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 
 
FRAMING. 
 
 16' 
 
 Fig. 202. 
 
 upon the undex^'ide. 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 sufli- 
 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, 6, 6, and ceiling-joists, c, c. The binding-joists 
 
152 
 
 AMERICAN HOUSE-CARPENTEB. 
 
 Fig. 203. 
 
 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 sufliciently 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 laid 6 feet apart. At this distauce the fol- 
 lowing rules will give the scantling. 
 
 Case 1.— To find the depth of a binding-joist, the length and 
 breadth being given. jR/^/e.— 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. Example.— 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 
 §iven. Rule.—'DiYide the square of the length in feet, by the 
 
FRAMING. 153 
 
 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. Rule. — 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. Examiile. — What should be the depth of a ceiling-joibt 
 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, (fcc. 
 
 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 c?, {Fig. 204,) which reach from wall to wall, 
 being the chief support of the floor. Such an arrangement is 
 termed di framed floor. The binding, the bridging and the ceil 
 
 20 
 
154 
 
 AMERICAN HOUSE-CARPENTER. 
 
 Fig. 204. 
 
 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 
 a few 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. i?z/ie.— 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. Example.— 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 312, which, multiplied by 
 4-2, gives 13 inches, the depth required. 
 
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 14 inches ? 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 a girder is more than about 12 
 inches, it is recommended to divide it by sawing from end to end, 
 vertically through the middle, and then to bolt it 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 
 as are not exactly tight, nor yet far enough apart to permit the 
 escape of moisture. 
 
 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 
 
156 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 Art. 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, iron 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 avoided. 
 (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. Floor-timbers are 
 supported at their ends by walls of considerable height. They 
 should not be permitted to rest upon intervening partitions, which 
 are not likely to settle as much as the walls ; otherwise the une- 
 qual settlements will derange the level of the floor. As all floors, 
 however well-constructed, settle in some degree, it is advisable to 
 
FRAMING. 157 
 
 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- 
 per most. 
 
 If the floor-plank are laid down tempoiarily 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 ot 
 the plank are planed after the first laying, they will shrink again ; 
 as it is 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, 
 (fcc, 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 
 
168 
 
 AMERICAN H0USE-CARPE:NTER. 
 
 1 
 
 
 
 
 
 
 
 
 
 
 
 
 
 •WW 
 
 
 
 . J_ 
 
 
 /. 
 
 _ 
 
 1 
 
 ^ 
 
 1 
 
 i 
 
 1 
 
 
 
 
 
 i 
 
 1 
 
 m 
 
 f 
 
 \ 
 
 ^ 
 
 1 
 
 % 
 
 d-, 
 
 M-N^ 
 
 
 
 
 
 
 
 
 
 
 
 
 
 tJ 
 
 Fi£r. 206. 
 
 \}.\\'i\ J 
 
 Fijr. 207. 
 
 construction is simple but effective. Fig- 207 shows the manner 
 of constructing a partition having doors near the ends. The truss 
 is formed above the door-heads, and the lower parts arc suspended 
 from it. The posts, a and Z>, 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, 3ix5 for 30 feet, and 4x6 for 40. The thickness of 
 the fiUing-in stuff may be regulated according to what is said at 
 Art. 271, in regard to the width of furring for plastering. The 
 
FRAMING. 
 
 159 
 
 filling-in pieces should be stiffened at about every three feec by- 
 short struts between. 
 
 All superfluous timber, besides being an unnecessary load upon 
 the points of support, tends to injure the stability of the plaster- 
 ing ; for, as the strength of the 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. 
 
 cr 
 
 (^ 
 
 '/ 
 
 ■/ 
 
 K 
 
 ^ 
 
 di 
 
 El 
 
 ^ 
 
 !^ 
 
 feJ 
 
 Fiff. 2U8. 
 
 283. — When the principal timbers of a partition require to be 
 large for the purpose of greater strength, it is a good plan to omit 
 the upright filling-in pieces, and in their stead, to place a few hori- 
 zontal pieces ; in order, upon these and the principal timbers, to 
 nail upright battens at the pi-oper 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 dimensionK. oi 
 
160 AMERICAN HOUSE-CARPENTER. 
 
 the timbers ma^ be ascertained, by applying the principles which 
 regulate the laws of pressure and those of the resistance of tim- 
 ber, as explained at the lirst part of this section. The following 
 data, however, may assist in calculating the amount of pressure 
 upon partitions: 
 
 284.— The weight of a square, (tliat is, a hundred square feet,) 
 of partitioning maybe estimated at from 1500 to 2000 lbs.; a 
 square of single-joisted flooring, at from 1200 to 2000 lbs. ; a 
 square of framed flooring, at from 2700 to 4500 lbs.: and the 
 weight of a square of deafenings (as described at the latter part 
 of Art. 271,) at about 1500 lbs. 
 
 When a floor is supported at two opposite extremiti(?s, and by a 
 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 ; 
 and hence the necessity of constructing the roof so that all 
 oblique and lateral strains may be removed; as, also, that instead 
 of having a tendency to separate the walls, the roof may 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. 
 
FRAMING. 
 
 161 
 
 in a cold climatej 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 weio-Jit of materials, in Appendix.) In a 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 the roof. 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. 
 
 f to l\ lbs. 
 
 Copper, 
 
 ;; 1 " 
 
 1 to 1^ " 
 
 Lead, 
 
 " 2 inches ** 
 
 4 to 7 " 
 
 Zinc, 
 
 '• 3 " " 
 
 li to 2 " 
 
 Short pine shingles, 
 
 '• 5 " " 
 
 lito2^ " 
 
 Loner cypress shingles. 
 
 a Q u 
 
 4 to 5 " 
 
 Slate, 
 
 u (3 u a 
 
 5 to 9 " 
 
 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, <fcc. ; 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 common 
 
162 
 
 AMERICAN HOUSE-CARPENTEK. 
 
 !< '^^ feet. 
 
 Fig. 211. 
 
FRAMING. 163 
 
 'after ; J? is a principal rafter ; A: is a king-post ; 5 is 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 placed 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. Exainple. — What should be 
 the dimensions of a pine king-post, 8 feet long, for a 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 di?nensions 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 area by the thickness will give the breadth. Fxa??iple. — 
 The queen-posts in Fig. 210 support each |- of the tie-beam, 
 which is 12f feet. To make them of pine, 6 feet long, what 
 should be their dimensions I 12f j multiplied by 6, gives 76, 
 
164 AMERICAN HOUSE-CARPENTER. 
 
 which, multiplied by 0'2T, gives 20*52 ; which indicates a size of 
 about 4x5^. 
 
 291. — To fold 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. Example. — The length of the longest un- 
 supported part of the tie-beam in Fig. 210 is 12f feet. What 
 should be the depth of the tii^-luMm, the breadth being 6 inches, 
 and the kind of wood, pine? The cube-root of 6 is 1*82, and 12f, 
 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, maybe 
 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 dimeiisions of a principal rafter when 
 there is a king-post in the middle. Rule. — Multiply 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. Example. — 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 
 size of the rafter should, therefore, be 6x8|. 
 
 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. Example. — What should be the dimensions ot 
 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 74 18 ; this, divided by 216, the cube of 6, gives 
 
FRAMING. 165 
 
 34 IS, 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 to 7. Rule. — Multiply the square-root 
 of the span in feet^ by the length of the straining-beam in feet, 
 and extract the sqi^ere-root of the product. Multiply this root by 
 ()'9 for pine, which will give the depth in inches To find the 
 thickness, multiply the depth by the decimal, 0"7. Example. — 
 What should be the dimensions of a pine straining-beam, 12 feet 
 long, for a 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 5|x7|, or 5|x8. 
 
 295. — To find the dim^ensions of stmts 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 in inches. Example. — 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 dim,ensions 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 
 
166 
 
 AMERICAN HOUSE-CARPENTER. 
 
 the depth for oak ; and the depth, multiplied by the decimal, 0*6, 
 will give the thickness. Example. — 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*87S — 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 dbnensions of co9nmon 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. The 
 dimensions of rafters for other kinds of covering may be found by 
 reference to the table at Art. 286. and the laws of 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. Exaiiiple. — 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 ;md 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 to 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.) 
 
FRAMING. 
 
 167 
 
 299. — Fig. 213 shows a method of constructing a truss having 
 a huilt-rih 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 6, 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 sufliciently strong, if the area of its section contain 
 about one-fourth more timber, than is required for that of a strain- 
 ing-beam for a 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 
 
168 
 
 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. 
 
 FliT. 2iJ. 
 
 301. — Fig' 215 is a meihod of obtaining the proper lengths and 
 bevils for rafters in a hip-roof, a b and b c are walls at the angle 
 of the building : & e is the seat of the hip-rafter and g f of a. 
 jack or cripple rafter. Draw e h, at right angles to b e, and make 
 it equal to the rise of the roof; join b and h, and h b will be the 
 length of the hip-rafter. Through e, draw d i, at right angles 
 to 6 c; upon 6, with the radius, b A, describe the arc, h i, cutting 
 d i'mi ; join b and i, and extend gf to meet biin j ; then gj will 
 
FRAMING. 
 
 169 
 
 be the length of the jack-rafter. The length of each jack-rafter is 
 found in the same manner — by extending its seat to cut the line, 
 b i. From/, draw fk, at right angles tofg, also//, at right 
 angles to be; make/ A: equal to /Z by the arc, I k, or make g- k 
 equal to gj by the arc, J k ; then the angle atj will be the top- 
 bevil of the jack-rafters, and the one at k 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 A, describe a semi-circle, cutting b e in 
 5 ; join ni and s and n and s ; then these lines will form at 5 the 
 proper angle for beviling the top of the hip-rafter. 
 
 DOMES.t 
 
 Fig. 216. 
 
 Fig. 217. 
 
 ♦ The lengths and bevils of rafters for roof-valleys can also be found by the abov? 
 process t See also Art. 227 
 
 22 
 
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 h being 
 the tie-beam in both. Two trusses of this kind, [Fig. 216,) pa- 
 rallel to each other, are to be placed one on each side of the open- 
 ing in the top of the dome. Upon these the whole framework is to 
 depend for support, and their strength must be calculated accord- 
 ingly. (See the first part of this section, and Art. 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. 
 
 Fiff. 918. 
 
 Fig. 219. 
 
 304. — When the interior dome rises too high to admit of a level 
 
FRAMING. 171 
 
 tie-beam, the framing may be composed of a succession of ribs 
 standing upon a continuous circular curb of timber, as seen at 
 Fig. 218 and 219,— the latter being a plan and the former a sec- 
 tion. This curb must be well secured, as it serves in the place 
 of a tie-beam to resist the lateral thrust of the ribs. In small 
 domes, these ribs may be easily cut from wide plank ; but, where 
 an extensive structure is required, they must be built in two 
 thicknesses so as to break joints, in the same manner as is descri- 
 bed for a roof at Art. 299. They should be placed at about two 
 feet apart at the base, and strutted as at a in Fig. 218. 
 
 305. — The scantling of each thickness of the rib may be as 
 follows : 
 
 For domes of 24 feet diameter, 1x8 inches. 
 " '• 36 " lixlO " 
 
 " ' 60 '' 2x13 " 
 
 " " 90 " 2|xl3 " 
 
 " " 108 " 3x13 " 
 
 306. — Although the outer and the inner surfaces of a dome 
 may be finished to any curve tliat may be desired, yet the framing 
 should be constructed of such a form, as to insure that the curve 
 of eqiiilibriiun 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 whicli the v/hole fabric may be exposed. Tiie 
 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 Icudern^ tower or cupola above 
 it, a aihic 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. Fig. 218,) 
 ihe pressure will have a tendency to burst the dome outwards at 
 -bout one-third of its height. Therefore, when this form is used 
 
172 
 
 AMERICAN HOUSE-CARPENTER. 
 
 in the construction of an extensive dome, an iron band should be 
 placed around the framework at that height ; and whatever may- 
 be the form of the curve, a band or tie of some kind is necessary 
 around or across the base. 
 
 If the framing be of a form 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 continuous horizontal circles. 
 
 
 
 
 ^ 
 
 
 
 
 
 
 / 
 
 
 
 / 
 
 
 
 
 
 
 J 
 
 / 
 
 
 
 
 
 Oj 
 
 / 
 
 
 
 
 
 
 / 
 
 
 
 
 
 
 
 A 
 
 
 
 
 
 
 
 a j I h 
 
 uo. 
 
 307. — To describe a cubic parabola. Let a b, {Fig. 220,) be 
 the base and b c the height. Bisect a b at d, and divide a d into 
 100 equal parts; of these give d e 26, ef lSi,f g 14|, g h 12^, 
 h i lOf, ij 9i, and the balance, 8|, to ; a; divide b c into 8 equcil 
 parts, and, from the points of division, draw lines parallel to a 6, 
 to meet perpendiculars from the several points of division in a b, 
 at the points, o, o, o, ifcc. 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 6 c d, [Fig. 221,) be the plan ot a dome, and e f the seat 
 
FRAMING. 
 
 173 
 
 Fig. 2-1 
 
 of one of the ribs. Then take e f for the transverse axis 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, e g f. The other ribs are 
 found in the same manner. 
 
 b 4 
 
 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 a c, 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, 6, and the line, d c, each into a like number of equal parts, 
 
174 
 
 AMERICAN HOUSE-CARPENTER. 
 
 as 5, (a large number will insure greater accuracy than a small 
 one ;) uponc, through the several points of division in c d^ describe 
 the arcs, o c? o, 1 e 1, 2/ 2, &c. ; make d o equal to half the width 
 of one of the boards, and draw o s, parallel to a c ; join s and a, 
 and from the points of division in the arc, o b, drop perpendicu- 
 lars, meeting a 5 in ij k I ; from these points, draw i 4, j 3, (fee, 
 parallel to a c; make d o, e l,&c., on the lower side of a c, equal 
 to d 0, e 1, &c., on the upper side ; trace a curve through the 
 points, 0, 1, 2, 3, 4, c, on each side o{ d c ; then o c o 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,o{ 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. 
 
 311. — Tojiiid the ciirve of the hoards when laid in horizon- 
 tal cottrses. Let ABC, {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, Sec. ; 
 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, F. 
 Through 3 and 2, draw 36; then b will be the centre for describing 
 F. Through 4 and 3, draw Ad; then d will be the centre for G 
 B is the centre for the arc, 1 o. If this method is taken to find 
 
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 /, meeting 
 the arc, A B, in/; join /and 4, /and 5 and/ and 6, cutting the 
 axis, D B, in s, n and ?n ; from 4, 5 and 6, draw lines parallel to 
 A Cand cutting the axis in c, p and r; make c 4, {Fig, 224,) 
 
 equal to c 4 in the preyious figure, and c s equal to c 5 also in the 
 previous figure ; then describe the inner edge of the board, H^ 
 according to Art. 87 : the outer edge can be obtained by gauging 
 from the inner edge. In like manner proceed to obtain the next 
 board — taking p .5 for half the chord and p n for the height of the 
 segment. Should the segment be too large to be described 
 easily, reduce it by finding intermediate points in the curve, as at 
 Art. 86. 
 
 312. — To find the shape of the angle-rib for a polygonal 
 dome. Let AG H^ {Fig. 225,) be the plan of a given dome, and 
 
176 
 
 AMERICAN HOUSE-CARPENTER. 
 
 O Z> a vertical section taken at the line, e f. From 1, 2, 3, (fee, 
 in the arc, C JD, draw ordinates, parallel to A Z>, to meet/ G ; 
 from the points of intersection on / G^ draw ordinates at right- 
 angles to/ G ; make s 1 equal to o 1, 5 2 equal to 2, &c. ; ^hen 
 GfB^ 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. — Yarious plans have been adopted for the construction of 
 bridges, of which perhaps the following are the most useful. 
 Fig. 22G shows a method of constructing wooden bridges, where 
 the banks of the river are higli enough to permit the use of the 
 tie-beam, a b. The upright pieces, c (/, are notched and bolted 
 on ni pairs, for the support of the tie-beam. A bridge of this 
 construction exerts no lateral pressure upon the abutments. This 
 method maybe 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 
 
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. 
 
 Fig. 22a 
 
 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 feet. 
 
 7 1 
 
 Span in feet. 
 
 Least rise in feet. 
 
 30 
 
 0-5 
 
 120 
 
 280 
 
 24 
 
 40 
 
 0-8 
 
 140 
 
 8 
 
 300 
 
 28 
 
 50 
 
 1-4 
 
 160 
 
 10 
 
 320 
 
 32 
 
 60 
 
 2 
 
 180 
 
 11 
 
 350 
 
 39 
 
 70 
 
 2i 
 
 200 
 
 12 
 
 380 
 
 47 
 
 80 
 
 3 
 
 220 
 
 14 
 
 400 
 
 53 
 
 90 
 
 4 
 
 240 
 
 17 
 
 
 
 100 
 
 5 
 
 260 
 
 20 
 
 
 
 The rise should never be made less than this, but in all cases 
 
 23 
 
L78 
 
 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. Rule. — 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, 
 O'OOll, will give the area of each rib in feet. When the road- 
 way is only planked, use the decimal, 0*0007, instead ot 
 O'OOll. Example. — 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 6666f , 
 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 
 
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 weigl^t, may be considered to be in the direction 
 of the chord lines, b a and be; 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. Example. — In 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. The cube-root of the 
 last sum is 0-717 nearly, and the distance, b c, is 135 feet : then, 
 0-'^ 17, multiplied by 135, gives 96f 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 spems, where the rib will be such as to 
 render it difficult to procure timber of sufficient breadth, they 
 may be constructed by bending the pieces to the proper curv€^ 
 
)80 
 
 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 m 
 inches. Example. — Suppose the curve of the rib to be described 
 with a radius of 100 feet, then what should be the depth ? The 
 radius in feet, 100, multiplied by 0*05, gives a product of 5 inches. 
 White pine or oak timber, 5 inches thick, would freely bend to 
 the above curve ; and, if the required depth of such a rib be 2( 
 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, ?l 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 
 
FRAMING. 18i 
 
 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- 
 na' braces should be framed between the standards, over the space 
 between the ribs, to prevent lateral motion. The timbers for the 
 roadway may be as light as their situation will admit, as all use- 
 less timber is only an unnecessary load upon the arch. 
 
 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. The commencement of the arch at the abutments — the 
 springs as it is termed, should not be below high-water mark : 
 and the bridge should be placed at right angles with the course of 
 the current. 
 
 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 
 120 lbs. 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 
 
AMERICAN HOUSE-CARPENTER. 
 
 half the span of the arch, and by the weight of a square-foot ot 
 the arch. Divide the last product by 120 times the height of the 
 abutment, and the quotient will be the thickness of the abutment. 
 Exain'ple. — Let the height of the abutment from the base to the 
 springing of the arch be 20 feet, half the span 100 feet, the weight 
 of a square foot of the arch, including the greatest possible load 
 upon it, 300 pounds, and the rise of the arch 18 feet — what should 
 be its thickness ? The square of the height of the abutment, 
 400, multiplied by 100, gives 64,000, and 300 by 18, gives 5400 ; 
 64,000, divided by .5400, gives a quotient of 11*852, one added to 
 this makes 12*852, the square-root of which is 3*6 ; this, less one, 
 is 2*6 ; this, multiplied by 100, gives 260, and this again by 300, 
 gives 78,000 ; this, divided by 120 times the height of the abut- 
 ment, 2400, gives 32 feet 6 inches, the thickness required. 
 
 The dimensions of a pier will be found by the same rule. 
 For, although the thrust of an arch may be balanced by an ad- 
 joining arch, when the bridge is finished, and while it remains 
 uninjured ; yet, during the erection, and in the event of one arch 
 being destroyed, the pier should be capable of sustaining the en- 
 tire thrust of the other. 
 
 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 oif 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 is a 
 well-established fact, that timber is less durable when subject to 
 
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, 
 vere 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 swept away. The most usual error in badly-constructed 
 centres is, that the timbers are disposed so as to cause the framing 
 to rise at the crown, during the laying of the arch-stones up the 
 sides. To remedy this evil, some have loaded the crown with 
 heavy stones ; but a centre properly constructed will need no 
 such precaution. 
 
 Experiments have shown that an arch-stone does not press 
 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 pumt, the pressure is inconsiderable, 
 
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 67 degrees, half the weight ; and when a vertical line, 
 as a 6, {Fig. 232,) passing through the centre of gravity of 
 
 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 riv^cr 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 h 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, /^, is added foi 
 
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 togetlier 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 settijig. 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 
 
186 AMERICAN HOUSE-CARPENTER. 
 
 arch. Between every two trusses, diagonal braces should be h* 
 troduced to prevent lateral motion. 
 
 326. — In order that the centre maybe easily lowered, the frames, 
 or trusses, should be placed upon wedge-formed sills ; as is shown 
 at dj {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 i to 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, <fec., 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 in^l;ded 
 between a and c, and consider half of it collected at 6, 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 tiiat it will yield Ic 
 
FRAMING. 187 
 
 crushing before it will bend under the pressure — {A rt. 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 lbs. 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 tlie 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 voussolrs, 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 : 
 
 330. — To find the direct iou of the joints for an elliptical 
 arch. A joint being wanted at «, [Fig. 234,) draw lines from 
 that point to the foci, /and/; bisect the angle, /a/ with the 
 line, ah ; 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, [Pig. 235,) draw a e, at right angles to 
 the axis, eg; make c g equal to c e, and join a and g ; draw a A, at 
 right angles to a ^ ; then a h will be the direction of the joint 
 
188 
 
 AMERICAN HOUSE-CARPENTER. 
 
 
 g 
 
 f 
 
 ^/L/^ 
 
 e \ 
 
 / ^ 
 
 \ 
 
 Fig. 235. 
 
 The direction of the joint from h is found in the same mannei . 
 The hnes, a g and 6/, are tangents to the curve at those pomts 
 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. 
 
 Fis. 236. 
 
 332. — Fig' 236 shows a simple and quite strong method ot 
 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 
 
 ^ 
 
 -[z^ 
 
 ■ G^ 
 
 FiL'. 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 
 
FRAMING. 189 
 
 Fii 
 
 wood, andj 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. 
 
 Fiff. 239. 
 
 334. — Fig. 239 shows about as strong a splice, perhaps, as 
 can well be made. It is to 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 ihere 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 
 pme and similar soft woods, depending wholly on indents, should 
 be about 1-2 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. 
 
 1 — — — - — — -- — -x 
 
 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 
 
190 AMERICAN IIOUSE-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. For a temporary purpose, Fig. 236 would do very 
 well ; it would be improved, however, by having a piece bolted 
 on all four sides. Fig. 237, and indeed each of the others, since 
 they have no oblique joints, would resist compression v^ell. 
 
 337.— In framing one beam into another for bearing pui poses, 
 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 might 
 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 bottom edge. For the nearer this is placed to 
 the upper edge, the greater is the liability for it to splinter off; if 
 the joint is 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, 
 
FRAMING. 
 
 191 
 
 1 
 
 t'lg. '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. — Thus 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 njw?i 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, tfec. 
 
 Fig. 242 
 
 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 
 \s not cut completely across the tie-beam, but a narrow lip is left 
 
192 
 
 AMERICAN HOUSE-CARPENTER. 
 
 Standing in the middle, and a corresponding indent is made in 
 the rafter, to prevent the parts from separating sideways. The 
 abutting surface should be made as large as the attainment of 
 other necessary objects will admit. The iron strap is added to 
 prevent the rafter from sliding out, should the end of the tie-beamj 
 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 must 
 necessarily take place from the shrinking of the king-post and 
 other parts. If the joint is made fair at first, when the truss 
 settles it will cause it to open at the under side of the rafter, thus 
 throwing the whole pressure upon the sharp edge at a. This will 
 cause an indentation in the king-post, by which the truss will be 
 made to settle further ; and this pressure not being in the axis of 
 the rafter, it will be greatly increased, thereby rendering the rafter 
 liable to split and break. 
 
 h 
 
 Fiff. 245. 
 
 Fig. 246. 
 
 c:^^' 
 
 Fig. 247. 
 
 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 equdize the pressure. 
 
FRAMING. 193 
 
 Fig. 248. Fig. 249. 
 
 341. — Fig. 248 is a proper joint for a collar-beam in a small 
 roof: the principle shown here should characterize all tie-joints. 
 The dovetail joint, although extensively practised in the above 
 and similar cases, is the very worst that can be 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 without them. They can 
 be used to advantage, however, at the foot of suspending-pieces, 
 and for the rafter at the end of the tie-beam. In roofs for 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 j\ thick. 
 15 " " U " i " 
 
 20 " " 2 " i " 
 
 In fastening a strap, its hold on the suspending-piece will be much 
 increased, by turning its ends into the wood. Iron straps should 
 be protected from rust ; for thin plates of iron decay very soon 
 
 26 
 
IQ'* 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 will give it a coating whici' 
 dampness will not penetiat>3. 
 
SECTION v.— DOORS, WINDOWS, &^. 
 
 DOORS. 
 
 343. A.mong 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. The dimensions 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 suflii- 
 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 
 
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 to 8 feet, folding or sliding doors are adopted. 
 As to the height of a door, it should in no case be less than about 
 6 feet 3 inches ; and generally not less than 6 feet 8 inches. 
 
 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 ttie 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 | of the height ; yet this 
 rule needs some qualification : for, if the width of the opening 
 be greater than one-half the width of the room, there will not be 
 a sufficient space left for opening the doors ; also, the height 
 should be about one-tenth greater than that of the adjacent single 
 doors, 
 
 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 : in a 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 4^ inches wide, each panel being 8 inches. 
 The bottom rail and the lock rail ought to be each equal in 
 width to tV of the height of the door ; and the top rail, and all 
 
DOORS, WINDOWS, &C. 
 
 197 
 
 others, of the same width as the stiles. The moulding on the 
 panel should be equal in width to i of the width of the stile. 
 
 Fig. 250. 
 
 346. — Fig. 250 shows an approved method of trimming doors : 
 a is the door stud ; 6, the lath and plaster ; c, the ground ; d, tfie 
 jamb ; e, the stop ; /and g^ architrave casings ; and A, the door 
 stile. It is customary in ordinary work to form the stop for the 
 door by rebating the jamb. But, when the door is thick and 
 heavy, a better plan is to nail on a piece 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 5 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 ; 9 
 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 aflix the hinges, no general rule can be assigned 
 
198 
 
 AMERICAN HOUSE-CARPENTER. 
 
 aK 
 
 liliaiLIfLMl^tl^mmtJi^JiiAiiiiiiyfnffmiiiJi^j^uMiti mM^um i Lf.mmMJL i LJtnii 
 
 C 
 
 _ 
 
 Fig. 251. 
 
 It may be observed, however, that a bed-room door should be 
 hung so that, when half open, it will screen the bed ; and 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 
 
DOORS, WINDOWS, &C. l99 
 
 can well be given, that will answer in all cases ; yet, as an ap- 
 proximation, the following has been used for general purposes. 
 Multiply together the length and the breadth in feet of the apart- 
 ment to be lighted, and the product by the height in feet ; then 
 the square-root of this product will show the required number of 
 square feet of glass. 
 
 350. — To ascertain the dimensions of window frames, add 4J 
 inches to the width of the glass for their width, and 6^ inches to 
 the height of the glass for their height. These give the dimen- 
 sions, in the clear, of ordinaiy frames for 12-light windows ; the 
 height being taken at the inside edge of the sill. In a 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 3^ for hanging stiles — 
 and the height between the stone sill and lintel is about 10 g 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 boxes, 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 i of 
 an inch at each hinging ; which being added to the margin, the 
 entire width of the shutters will be 1^ inches more than the width 
 of the frame in the clear. Then, to ascertain the width of the 
 box shutter, add 1^ inches to the width of the frame in the clear, 
 between the pulley stiles ; divide this product by 4, and add 
 half an inch to the quotient ; and the last product will be the re- 
 quired width. For example, suppose the window to have 3 
 lights in width, 11 inches each. Then, 3 times 11 is 33, and 4^ 
 added for the wood of the sash, gives 37^ 2>7i and 1^ is 39 
 
200 AMERICAN HOUSE CARPENTER. 
 
 and 39; divided by 4, gives 9| ; 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 taste require that they be of 
 different heights in different stories, but of the same width. The 
 windows of the upper stories should all range perpendicularly 
 over those of the first, or principal, story; and they should be 
 disposed so as to exhibit a balance of parts throughout the front 
 of the building. To aid in this, it is always proper to place the 
 front door in the middle of the front of the building ; and, where 
 the size of the house will admit of it, this plan should be adopted. 
 (See the latter part of Art. 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. 
 " " principal-story " 2| " 
 
 " " second-story " If " 
 
 « " third-story " 1| « 
 
 " " fourth-story " If " 
 
 " " attic-story " the same as the width. 
 
 But, in determining the height of the windows for the several 
 stories, it is necessary to take into consideration the height of the 
 story in which the window is to be placed. For, in addition to 
 the height from the floor, which is generally required to be from 
 28 to 30 inches, room is wanted above the head of the window 
 for the window-trimming and the cornice of the room, besides 
 some respectable space which there ought to be between these. 
 
 353. — The present style of finish requires the heads of win- 
 dows in general to be horizontal, or square-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 sojfif, or surface of the head, would be 
 straight, and its length be found by getting the stretch-out of the 
 
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 form of the soffit will be obtained as in the 
 ollowing article : and, when the face of the wall is circular, as in 
 the succeeding one. 
 
 / 
 
 Fig. 252. 
 
 354. — To find the for ?n of the soffit for circular window- 
 heads^ when the light is received in an oblique direction. Let 
 a b cdj {Fig. 252,) be the ground-plan of a given window, ande/ 
 a, a vertical section taken at right angles to the face of the jambs. 
 From a, through e, draw ag, at right angles to a b ; obtain the 
 stretch-out of efa^ 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 eg; 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 A, 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. Let abed, {Fig. 
 253,) be the ground-plan of a given window, and efa.B. vertical 
 section of the head taken at right angles to the face of the jambs. 
 
 26 
 
202 
 
 AMERICAN HOUSE-CARPENTER. 
 
 Fi',^ x:53. 
 
 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 6, [Fig. 252 ;) and, having drawn ordinates at 
 right angles to c 6, transfer them to ef a ; in this way, a section 
 at right angles to the jambs can be obtained. 
 
SECTION VL— STAIRS. 
 
 356. — The stairs is that m»3chtinlciil 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 ; 
 
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 to 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 followiug problem 
 is constructed. 
 
 3.58. — 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 6, and make b c equal to 12 inches ; join a 
 and c, and the triangle, a b c, will form a scale upon which to 
 graduate the sides of the pitch-board. For example, suppose a 
 very easy stairs is required, and the tread is fixed at 14 inches. 
 Place it from b to/, and from/; draw f g^ at right angles to a b ; 
 then the length oi 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 
 
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 h to A, and draw h i, parallel to a b ; then h i, or b j will be 
 the proper tread for that rise, and J b i will show the angle of as- 
 cent. On the other hand, if the angle of ascent be given, as a 
 b l^ {b I being lOJ inches, the proper length of run for a step- 
 ladder,) drop the perpendicular, I /;, from I Xo k ; then I kb will 
 be the proper proportion for the sides of a pitch-board for that 
 run. 
 
 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 " 12^ " 
 
 In second-class " "11 " 
 
 In third-class " and cottages " 9 " 
 
 Step-ladders to ascend to scuttles, (fee, should have from 10 to 
 11 inches run on the rake of the string. (See notes at Ai^t. 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 2 J feet. 
 
 361. — To find the dimensions of the pitch-board. The first 
 thing in commencing to build a stairs, is to make the /^zVcA-board ; 
 this is done in the following manner. Obtain very accurately, in 
 feet and inches, the perpendicular height of the story in which 
 the stairs are to be placed. This must be taken from the top ot 
 the floor in the lower stoiy 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 
 
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 26f 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 ~ of an inch. 
 
 Then, if the space for the extension of the stairs is not limited, 
 the tread can be found as at 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. Having obtained this 
 rise and tread, the pitch-board may be made in the follow- 
 ing manner. Upon a piece of well-seasoned board about | of an 
 inch thick, having one edge jointed straight and square, lay the 
 corner of a carpenters'-square, as shown at Fig. 255. Make a h 
 
 Fig. 255. 
 
 equal to the rise, and h 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 
 
2ivr 
 
 Fitr. 25(j. 
 
 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.) Connnencing at one end, 
 lay the longest side of the pitch-board against the gauge-mark, a 
 6, as at c, and draw by the edges the lines for the first rise and 
 tread; then place it successively as at d, e and/, until the re- 
 quired number of risers shall be laid down. 
 
 F7 
 
 UJ. 
 
 Fig. 257. 
 
 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 cnange their direction. This kind of stairs is the most easily 
 constructed, and is therefore the most common. The cyliu 
 
20n 
 
 AMERICAN HOUSE-CAIIPENTBE. 
 
 Fig. 258. 
 
 der is generally of small diameter, in most cases about 6 inches. 
 It may be worked out of one solid piece, but a better way is to 
 glue together three pieces, as in Fig. 258; in which the piece.^j 
 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 at o and r? is the most proper 
 for that kind of joint. 
 
 36.5. — 7^0 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 6 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 hf and 
 g i, each equal to one rise; from i and/, draw ij and/ A:, paral- 
 lel to h g ; place the tread of the pitch-board at these last lines, 
 and draw by the lower edge the lines, k h and i I ; parallel to 
 tnese, draw m n and o p, at the requisite distance for the dimen- 
 sions of the string : from 5, the centre of the plan, draw s q. 
 parallel to df; divide h qand q g, each into 2 equal parts, as at 
 V and w; from v and w, draw v n and w o, parallel to/ c?; join n 
 and 0, cutting q s m r ; then the angles, u n r and rot, 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 w, draw u :r, at right angles 
 to mn; from r, draw r a:, at right angles to no ; then x will be 
 the centre for the curve, u r. The centre for the curve, r ^, is 
 found in the same manner. 
 
STAIRS. 
 
 209 
 
 Fig. 259. 
 
 366. — To find the position for the balusters. Place the 
 centre of the first baluster, (6. Fig. 260,) 5 its diameter from the 
 face of the riser, c c?, and i 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. 
 
 a- 
 
 ^ 
 
 Piir sm. 
 
 27 
 
210 
 
 AMERICAN HOUSE-CARPENTER. 
 
 Fig. 261. 
 
 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 /, being the platform ; and a 6 c is 
 the plan of a line passing through the centre of the rail around 
 the cylinder. Through i and d, draw i k, and through J and e, 
 draw 7 k ; from A;, draw k Z, parallel to /e ; from 6, draw h m, 
 parallel tog d; from /, draw I r, parallel to kj ; from n, draw n 
 /, at right angles to jf k : on the line, o b. maki. o t equal to w ^ ; 
 join c and ^ : on ie line, j c, {Fig. 262,) make e c equal to e w at 
 Fig. 261 ; from c, draw c t^ ai nt'ht angles Xoj c, and make c t 
 
STAIRS. 21J 
 
 i I 
 
 Fig. 262. 
 
 equal to c ^ at Fig. 261 ; through t, drawp /, parallel ioj c, and 
 make 1 1 equal to ^ / at Fig. 261 : join I and c, and complete the 
 parallelogram, eels; find the points, o, o, o, according to ^r<f. 
 118 ; upon e, o, o, o, 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 
 I is drawn at right angles to c I. 
 
 368. — Elucidation of the foregoing method. This excellent 
 plan for obtaining the face-moulds for the hand-rail of a platform 
 stairs, has never before been pnblished. 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, i? 
 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 wi, 
 m c and cj, 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 Z, is all that will need ex- 
 planation. 
 
 Suppose, then, that the parallelogram, e a o c, represent a plane 
 lying perpendicularly over e ah f being inclined in the direction, 
 e c, and level in the direction, c o ; suppose this plane, e a o c, 
 
212 
 
 AMERICAN HOUSE-CARPENTER. 
 
 Fi?. 263. 
 
 be revolved on e c as an axis, in the manner indicated by the arcs, 
 o n and a x^ until it coincides with the plane, e r t c ; the line, a 
 0, will then be represented by the line, x n ; then add the paral- 
 lelogram, xrt n^ and the triangle, ctl^ deducting the triangle, ers; 
 and the edges of the plane, e s I c. inclined in the direction, ec, and 
 also in the direction, c I, will lie perpendicularly over the plane, e 
 a bf. From this we gather that the line, c o, being at right angles to 
 
STAIRS. 
 
 213 
 
 e c, must, in order to reach the point, Z, be lengthened the distance, 
 n t^ and the right angle, e c ^, be made obtuse by the addition to 
 it of the angle, t c I. By reference to Fig. 261, it will be seen 
 that this lengthening is performed by forming the right-angled 
 triangle, c o t^ corresponding to the triangle, c o ^, in Fig, 263. 
 The line, c Z, is then transferred to Fig. 262, and placed at right 
 angles to e c ; this angle, e c t^ being increased by adding the an- 
 gle, t c Ij corresponding to ^ c /, Fig. 263, the point, Z, is reached, 
 and the proper position and length of the lines, e c and c I ob- 
 tained. To obtain the face-mould for a rail over a cylindrical 
 well-hole, the same process is necessary to be followed until the 
 the length and position of these lines are found ; then, by forming 
 the parallelogram, eels, and describing a quarter of an ellipse 
 therein, the proper form will be given. 
 
 Fig 264. 
 
 369. — Case 2. — When the cylinder is large. Fisr. 264 re- 
 
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 h k, parallel toed; ex- 
 tend the lines, i d and J e, until they meet khiuk and/; from n, 
 draw n I, parallel to o 6 ; through I, draw 1 1, parallel to j k ; from 
 k, draw k t, at right angles to j k ; on the line, o 6, make o t equal 
 to A; t. Make e c, (Fi^. 265,) equal to e A: at Fig. 264 ; from c, 
 
 Fisr. 263. 
 
 draw c ^, at right angles to e c, and equal to c ^ at Fig, 264 ; from 
 t, draw ^ ^, parallel to c e, and make 1 1 equal to ^ Z at Fig. 264 ; 
 complete the parallelogram, eels, and find the points, o, o, o, as 
 before : then describe the circles and complete the mould as in 
 Fio\ 262. The difference between this and Case 1 is, that the 
 line, c t, instead of being raised and thrown out, is lov/ered and 
 drawn in. (See note at y^^^. 2.55.^ 
 
 Fig. -^eu c 
 
 370._Oase 3. — Where the rake meets the level. In Fig. 
 
STAIRS. 215 
 
 266, a b cis the plan of a line 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, h g. Bisect e hin d, and 
 through d, draw df, parallel to h g ; bisect/ n in Z, and from /, 
 draw / t, parallel to nj; from w, draw n t, at right angles to / n ; 
 on the line, o 6, make o t equal to 71 1. Then, to obtain a mould 
 for the twist going up the flight, proceed as at Fig. 262 ; making 
 c cin that figure equal to e n in Fig, 266, and the other lines ot 
 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 6 
 0, {Fig. 266,) to i ; make i equal to/ Z, and join i and c ; make 
 c i, {Fig. 267,) equal to c i at Fig. 266 ; through c, draw c c?, at 
 
 d 
 
 Fig. 267. 
 
 right angles to c i ; make d c equal to tZ/ at Fig. 266, and com- 
 plete the parallelogram, odd; 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- 
 ing it. Make the angle, c t^ [Fig. 268,) equal to the angle, 
 c Z, at Fig. 261 ; upon c, describe a circle with a radius equal to 
 half the thickness of the rail ; draw the tangent, b rf, parallel to 
 t c, and complete the rectangle, e b df having sides tangical to 
 the circle ; from c, draw c a, at right angles to oc ; then, b d 
 being the bottom of the rail, set a gauge from b to a, and run it 
 the whole length of the stufl*; in boring, place the centre of the 
 
216 
 
 AM ER iCA \ [ T O U P I:>CA RPENTER. 
 
 h 
 
 d 
 
 i- lu. -I'S. 
 
 bit in the gauge-mark at a. and bore in the direction, a c. To do 
 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 
 ef, 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. 
 
 Fig. 269 
 
STAIRS. 
 
 217 
 
 373. — To obtain, by the foregoing principles, the face-mould 
 for the twists of a moulded rail upon platform stairs. In Fig. 
 269, a b c 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, v\^ith 
 the hnes 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 / and r, draw/ 
 e and r d, parallel to c a At Fig. 270, the centre lines of the 
 
 s 
 
 d 
 
 n e 
 
 ml ^ 
 
 /t~~ 
 
 iHv 
 
 \ Ic/ 
 
 i 
 c 
 
 // 
 
 { 1 ^^ 
 
 , 
 
 g 
 
 J 
 
 
 Fig. 270. 
 
 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, i 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 I, parallel to n c ; also, through k, 
 draw s g, parallel to n c ; then, in the parallelograms, m s d o and 
 g s e I, 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. 
 
218 AMERICAN HOUSE-CARPENTEJl. 
 
 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/, must be bevilled according to the angle, c t x^dX Fig. 
 269 ; if the rail is to be canted up, 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, c, draw 
 vertical lines across the edge of the plank by applying the pitch- 
 board, a b c ; then, in applying the mould to the other side, place 
 the points, a and c, at b and/; 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 A, parallel to d e; 
 then the distance that these lines are apart, will be the thickness 
 required for the twists of a moulded rail. . 
 
 The foregoing method of finding moulds for rails is applicable 
 to all stairs which have continued rails around cylinders, and are 
 without winders. 
 
 WINDING STAIRS. 
 
 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 ^yer* 
 and the latter tiinders. 
 
 377. — To describe a regular geometrical winding stairs. 
 In Fig. 272, abed represents the inner surface of the wall en- 
 closing the space allotted to the stairs, a e the length of the steps, 
 and efgh the cylinder, or face of the front string. The line, 
 
STAIRS. 
 
 219 
 
 Fig. 272. 
 
 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 
 i for radius, describe the arc, ij; obtain the number of risers 
 and of treads required to ascend to the floor at j, according to Art. 
 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 el 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 
 
220 AMERICAN HOUSE-CARPENTER. 
 
 a succession of grooves IJ 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, abed. 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 o a ; 
 upon this the string must be bent, and the grooves filled with strips 
 of wood, called ke^s, which must be very nicely fitted and glued 
 in. After it has dried, a board thin enough to bend around on the 
 outside of the string, must be glued on from one end to the other, 
 and nailed with clout nails. In doing this, be careful not to nail 
 into any place where a riser or step is to enter on the face. 
 
 After the string has been on the drum a sufficient time for the 
 glue to set, take it off, and cut the mortices for the steps and 
 risers on the face at the lines previously made ; which may be 
 done by boring with 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 I 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 show 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 a riser, less the thickness of the floor, and the 
 lines, 2 w, 3 w, 4 0, 5 /? and 6 q, are each equal to one riser. The 
 
STAIRS. 2li\ 
 
 line, a 2, is equal to am in Fig. 272, the line, m 3 to m ?i in that 
 figure, &c. In drawing this figure, commence at a, and make 
 the lines, a 1 and a 2, of the length above specified, and draw 
 them at right angles to each other ; draw 2 m, at right angles to 
 a 2, and m 3, at right angles to m 2, and make 2 m and m 3 of 
 the lengths as above specified : and so proceed to the end. Then, 
 through the points, 1, 2, 3, 4, 5 and 6, trace the line, 1 b ; upon 
 the points, 1, 2, 3, 4, &c., with the size of the timber for radius, 
 describe arcs as shown in the figure, and by these the lower line 
 may be traced parallel to the upper. This will give the proper 
 shape for the timber, a b, in Fig. 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 sufiicient, 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 loinding 
 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 a c b, according to Art. 92; from o, through the point of 
 the mitre at the newel-cap, draw o s ; obtain on the tangent, e d, 
 the position of the points, s and h'* as. at t and/^ ; from e tf^ and 
 /, draw e a;, t u,/^ g'^ and /A, all at right angles Xo e d ; make e 
 g equal to one rise and/- ^^ equal to 12, as this line is drawn 
 from the 12th riser ; from g^ through g^.^ draw^ i; make g x equal 
 to about three-fourths of a rise, (the top of the newel, x, should 
 be 3^ feet from the floor ;) draw x w, at right angles to e x^ and 
 ease off the angle at w ; at a distance equal to the thickness of 
 
 • In the above, the references, a^, U^, &c., are introduced for the first time. During the 
 time taken to refer to the figure, the memory of the /otto of these may pass from the mind, 
 while that of the sound alone remains ; they may then be mistaken for a 2, 6 2, &c. This 
 can be avoided in reading by giving them a sound corresponding to their meaning, which 
 IS second a second h, &c. or a second, b second. 
 
222 
 
 AMERICAN HOUSE-CARPENTER. 
 
 r[ie rail, draw v w y, parallel to x ii i; from the centre of the plan, 
 0, draw o I, at right angles to e d ; bisect h n in p, and through 
 />, at right angles to,^' i, draw a line for the joint; in tlie same 
 manner, draw the joint at k ; then x y will be the falling-mould 
 for that part of the rail which extends from 5 to 6 on the plan. 
 
 380. — To find the face-mould for the railof axoinding-stairs. 
 From the extremities of the joints in the falling-mould, as A', z 
 and y, (Fig. 274,) draw k a^, z 6^ and y c?, at right angles Xo e d : 
 make h ^ equal to / d. Then, to obtain the direction of \\v?. 
 joint, c^ c^, or 6^ d\ proceed as at Fig. 275, at which the parts are 
 
STAIRS. 
 
 223 
 
 Fig. 275. 
 
 shown at half their fill! size. A is the plan of the rail, and B is 
 the falling-monid : in which k z is the direction of the butt-joint. 
 From kj draw k b, parallel to I o, and k e, at right angles to k b ; 
 from 6, draw b /, tending to the centre of the plan, and from/, draw 
 / e, parallel to b k ; from Z, through e, draw I ^, and from i, draw i 
 d, parallel toef; join d and 6, and d b will be the proper direction 
 
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, x h and o rf, 
 Xo X a and o c. (See Art. 384.) 
 
 Fi-. 27G. 
 
 Having obtained the direction of the joint, make s r d b, (Fig. 
 276,) equal to 5 r db'^ in Fig. 27 A ; through r and d, draw t a ; 
 through 6' and from d, draw t u and d e, at right angles to t a ; 
 make t u and d e equal to t u and 6^ m, respectively, in Fig. 274 ; 
 from w, through e, draw u ; through 6, from r, and from as many 
 other points in the line, t a, as is thought necessary, as/, h and ;, 
 draw the ordinates, r c,fg, h i,j k and a ; from w, c, ^, ^J k^ e 
 and 0, draw the ordinates, ii 1, c 2, g 3, i A, k 5, e 6 and 7, at 
 right angles to it, ; make ?i 1 equal to ^ 5, c 2 equal to r 2, ^ 3 
 equal to/ 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, t a, and the arc, r d ; join 1 
 and c, also e and 7 ; then, I 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. 27 L 
 
 To ascertain the mould for the next quarter, make acje, (i'V^. 
 
STAIRS. 
 
 22k 
 
 Fig. 277. 
 
 277,) equal to a' c^j e^ at Fig. 274 ; at any convenient height on 
 the h'ne, d i, in that figure, draw q i\ parallel to e d ; through c 
 and J, {Fig. 277,) draw b d ; through a, and from 7, draw b k and 
 ; 0, at right angles to b d ; make b k an&j equal to ^ k and q 
 i, respectively, in Fig. 27 A ; from k, through 0, 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 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 ^, draw n ^, 
 parallel to kf ; then the distance between k f and n i will give 
 the thickness required. 
 
 382. — Case 2. — When the falling-mould is curved. In FHg. 
 278, sr db'is, equal tosr d^Wm. Fig. 27 L Make a c equal to the 
 stretch-out of the arc, s 6, 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 ak^el^ (fee, at right an- 
 gles to a c ; make a A; equal to ^ w in Fig. 274, and c; equal to 6' wi 
 
 29 
 
22t 
 
 AMERICAN HOUSE-CARPENTER. 
 
 in that figure, and complete the ral ling-mould, kj, every way equal 
 to M m in Fig. 274 ; from the points of division in the arc, sb, draw 
 lines radiating towards the centre of the circle, dividing the arc. 
 r d, in the same proportion ass bis divided ; from d and b, draw 
 dtanAb u, at right angles to a d, and from j and v, dmwj u and v 
 w, at right angles toj c ; then xtuwwiWhea. vertical projection 
 of the joint, d b. Supposing every radiating line across s r d b~ 
 corresponding to the vertical lines across A: ^-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, oisrdb. 
 To find the thickness of plank necessary to get out this part of 
 the rail, draw the line, z t, touching the upper side of the helinet 
 m two places : through the corner farthest projecting from that 
 line, as w, draw y w, parallel to ^ ^- then the distance between 
 those Imes 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. 
 i»i— To apply the face-mould to the plank. In Pig 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 
 
STAIRS. 
 
 227 
 
 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, (^,) and 
 mark out the shape of the twist ; from a and o, draw the lines, a 
 h and o c, across the edge of the plank, the angles, e a h and e o 
 Cj corresponding with kfd at Fig. 277 ; turning the plank up as 
 at Bj apply the tips of the mould at b and c, and mark it out as 
 shown in the figure. In sawing out the twist, the saw must be 
 be moved in the direction, a b ; which direction will be perpen- 
 dicular when the twist is held up in its proper position. 
 
 In sawing by the face-mould, the sides of the rail are obtained ; 
 the top and bottom, or the upper and the lower surfaces, are ob- 
 tained by squaring from the sides, after having bent the falling- 
 mould around the outer, or convex side, and marked by its edges. 
 Marking across by the ends of the falling-mould will give the 
 position of the butt-joint. 
 
 384. — Elucidation of the process by which the direction of 
 the butt-joint is obtained in Art. 380. Mr. Nicholson, in his 
 Carpenter^s 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 i'ls, the plan of a part of the rail about the 
 joint, 5 w 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 
 
328 
 
 AMERICAN HOUSE-CARPENTER. 
 
 gR 
 
 Fig. 'iSO. 
 
 382. Bisect r t, part of an ordinate from the centre of the plan, 
 and through the middle, draw c b, at right angles to g v ; from 
 b and c, draw c d and b e, at right angles to s u ; from d and e, 
 draw lines radiating towards the centre of the plan : then d o 
 and e m will be the direction of the joint on the plan, according to 
 Nicholson, and 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 c? o, e m or 
 ij^ are level j for instance, the level line, w t?, on the top of the 
 
STAIRS. 
 
 229 
 
 rail in the helinet, is a true representation of the radiating line, j i 
 on the plan. The line, b A, therefore, on the top of the rail in 
 the helinet, is a true representation of e mon the plan, and A: c on 
 the bottom of the rail truly represents d o. From k, draw k /, 
 parallel to c b, and from h, draw hf, parallel to 6 c ; join I and 
 b, also c and/; then c k I b will be a true representation of the 
 end of the lower piece, B^ and c fh b of the end of the upper 
 piece, A ; and/ k oi h I will show how much the joint is open on 
 the inner, or concave side of the rail. 
 
 Fig. '981. 
 
230 
 
 AMERICAN HOUSE-CARPENTER. 
 
 To show that the process followed in Art. 380 is correct, let d o 
 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 
 kj draw vertical lines from J and k to w and A, and join w and h ; 
 then it will be evident that whissi true representation in the helinet 
 of j k on the plan, it being in the same plane as ; k, and also in the 
 same winding surface as w v. The Hue, I w, also, is a true repre- 
 sentation on the bottom of the helinet of the line, j' A:, in the plan. 
 The line of the joint, e m^ therefore, is projected in the same way 
 and truly by i 6 on the top of the helinet ; and the line, d o, by 
 c a on the bottom. Join a and i, and then it will be seen that 
 the lines, c a, a i and i b, exactly coincide with c 6, 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, I i, corresponds to a; i in Fig. 281 ; and 
 that e k in that figure is a representation of/ b, and i k of d b. 
 
 Fig.ses. 
 
 In getting out the twists, the joints, before the falling-mould is 
 
STAIRS. 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, i b ; and that for 
 B, to the line, yz. Being sawed vertically at first, a section of the 
 joint at the end of the face-mould for A, would be represented in 
 the helinet by h if g. To obtain the position of the line, h i, on 
 the end of the twist, draw i .«?, {Fig-. 282,) at right angles to if, 
 and make i s equal to m e at Fig. 281 ; through 5, draw 5 g, pa- 
 rallel to if, and make s h equal to 5 6 at Fig. 281 ; join h and i ; 
 make i/equal to i /at Fig. 281, and from f, draw/^, parallel to i 
 b ; then i b gf will be a perpendicular section of the rail over the 
 line, e m, on the plan at Fig. 281, corresponding toi b gf in the 
 helinet at that figure ; and when the rail is squared, the top, or 
 back, must be trimmed off to the line, i b, and the bottom to the 
 line, fg. 
 
 385. — To grade the front string of a stairs, havi?ig 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, bee 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, y*^, 
 g h i and ij, parallel to a b, b e c and c d; make g k and g I 
 each equal to half the width of the rail, and through k and /, 
 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 no; obtain the stretch-out, g r, of the semi-circle, k 
 J) m, according to Art. 92 ; extend ab to t, and k mio s ; make c s 
 equal to the length of the steps, and i u equal to 18 inches, and de- 
 scribe the arcs, s t and u 6, parallel to mp ; from t, draw t w, tend- 
 ing to the centre of the cylinder ; from 6, and on the line, 6ux, run 
 ofl" the regular tread, as at 5, 4, 3, 2, 1 and v ; make it x equal to 
 half the arc, u 6, and make the point of division nearest to ar, as 
 v, the limit of the parallel steps, or flyers ; make r o equal iomz ; 
 from 0, draw o a^. at right angles to n o, and equal to one rise ; 
 
232 
 
 AMERICAN HOUSE-CARPENTER. 
 
 713 A3 
 
 r8 h^ 
 
 Fig. 283. 
 
 from a^5 draw a^ s, parallel to it o, and equal to one tread ; from s, 
 through 0, draw 5 6\ 
 
 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, J., is above 
 the first winder, B, as at 1, 2, 3, 4, 5 and 6; through 5, (on the 
 arc, 6 u,) draw dr e^, tending to the centre of the cylinder ; from 
 c^, draw s^/^, at right angles to n o, and through 5, (on the line. 
 
STAIRS. 233 
 
 w c',) draw ^'/^ parallel Xono ; through 6, (on the line, w c^,) 
 and/^, draw the line, K^ ¥ ; make 6 coequal to half a rise, and 
 from c' and 6, draw c^ r and 6/, parallel ion o ; make h^ i^ equal 
 to h^p ; from i^, draw iH-*^, at right angles to i^ A^, and from/-, 
 draw/*^^, at right angles to/Vi%' upon F, with Ic^ p for radius, 
 describe the arc,/^ i%- make 6^ Z^ equal to h"^ f^-, and ease off the 
 angle at h^ by the curve, f^ P. In the figure, the curve is de- 
 scribed from a centre, but in a full-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 7i o, meeting the curve in 
 m^, 7i^, 0^ and p"^ ; from these points, draw lines at right angles to 
 n 0, and meeting it in x"^, r^, s"^ and f; from x^ and r\ draw lines 
 tending to w^, and meeting the convex side of the rail in y^ and 
 z"^ ; make ni v^ equal to r s"^, and tu w'^ equal to r f ; from y-, z''. 
 ^;^ and iv"^, through 4, 3, 2 and 1, draw lines meeting the line of 
 the wall-string in a^, 6^, c^ and cP ; from e^, where the centre-line of 
 the rail crosses the line of the floor, draw e^f^, at right angles to n 
 0, and from /^, through 6, draw/^ g"^ ; then the heavy lines,/^^', 
 e^ (P, if' d^ z^ 6^, v^ &^ \c^ (P^ and zy^ will be the lines for the risers, 
 which, being extended to the line of the front string, b e c d^ will 
 give the dimensions of the winders, and the grading of the front 
 string, as was required. 
 
 386. — To obtain the falling-mould for the twists of the last- 
 mentioned stairs. Make P g^ and P h^, {Fig. 283,) each equal 
 to half the thickness of the rail ; through h^ and g^^ draw h^ r 
 dind g^f, parallel to P s ; assuming k k^ and m w^ on the plan as 
 the amount of straight to be got out with the twists, make 7i q 
 equal to k k^, and r P equal to ?n m^ ; from n and Z^, draw lines at 
 right angles to n o, meeting the top of the falling-mould in rv' and 
 0^ ; from o^, draw a line crossing the falling-mould at right angles 
 to a chord of the curve, /^ V ; through the centre of the cylinder, 
 draw u^ 8, at right angles to 7i o : through 8, draw 7 9, tending to 
 k^ ; then n^ 7 will be the falling-mould for the upper twist, and 7 
 
 0^ the falling-mould for the lower twist. 
 
 30 
 
234 
 
 AMERICAN HOUSE-CARPENTER. 
 
 387. — To obtain the face-moulds. The moulds for the twists 
 of this stairs may be obtained as at Art. 380 ; but, as the faUing- 
 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 stuiF. 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. 
 
 Fig, 284. o 
 
 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 W p^ e p in Fig. 283 ; join h and c, and from o, 
 draw A, at right angles to b c ; obtain the stretch-out of d g, as 
 dfj and at Fig. 283, place it from the axis of the cylinder, p, to 
 q^ ; from q^ in that figure, draw q^ r\ at right angles ion o ; also, 
 at a convenient height on the line, n n% in that figure, and at 
 right angles to that line, draw u^ v^ ; from b and c, in Fig. 284, 
 
STAIRS. 235 
 
 draw h j and c Z, at right angles to b c ; make b j equal to u^ n' in 
 Fig. 283, i h equal to lu^ r^ in th?it figure, and c Z equal \o v^ ^ \ 
 from Zj through j*, draw / m ; from A, draw /i n, parallel to c b ; 
 from 71. draw n r, at right angles to b c, and join ?' and 5 ; through 
 the lowest corner of the plan, as />, draw v e, parallel to b c ; from 
 a, e, II, p, k, t, and from as many other points as is tliought ne- 
 cessary, draw ordinates to the base-line, v e, parallel to ?' s : 
 through A, draw w x, at right angles to in I ; upon n, with r s for 
 radius, describe an intersecting arc at x, and join n and x ; from 
 the points at wliich the ordinates from the plan meet the base- 
 line, V e, draw ordinates to meet the line, m /, at right angles to v 
 e ; and from the points of intersection on in Z, draw correspond- 
 ing ordinates, parallel to nx ; make the ordinates which are pa- 
 rallel to n :r of a length corresponding to those Vv^hich 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 ot 
 the falling-mould at Fig. 283 ; b j being equal to lo 6 in Fig. 283. 
 i h equal to x^ if" in that figure, and cl\ol^ ol Draw a line 
 through J and Z, and from A, draw h tz, parallel to 6 c ; from n. 
 draw n i\ at right angles to b c, and join r and s ; then r s will be 
 the bevil for the lower ordinates. From A, draw h x, at right an- 
 gles toj I ; upon ?i, with 7' s for radius, describe an intersecting 
 arc at x, and join n and x : then n x will be the bevil for the upper 
 ordinates, upon which the face-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 : 
 'he upper surface, j a forming oblique angles with the face, b I — 
 
236 
 
 AMERICAN HOUSE-CARPENTER. 
 
 Fig. 285. 
 
 in Fig. 286 obtuse, and in Fig. 287 acute. Upon the base, de- 
 scribe the semi-circle, h s c ; from the centre, i, draw i s, at right 
 angles to 6 c ; from 5, draw 5 x, at right angles to e d, and from i, 
 draw i h, at right angles to b c ; make i h equal to 5 x, and join 
 h and x ; then, h and x being of the same height, the line, h x, 
 joining them, is a level line. From A, draw h w, parallel to h c, 
 and from?^, draw n r, at right angles tob c; join r and 5, also n 
 
STAIBS. 
 
 237 
 
 Fiff. 286. 
 
 !■ ijr. vJr^7. 
 
 and ^; then, n and x being of the same height, n x is a level hne ; 
 and this line lying perpendicularly over r 5, n x and r s must be 
 of the same length. So, all lines on the top, drawn parallel to n 
 or, and perpendicularly over corresponding Unes drawn parallel to 
 r 5 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 Z, 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 x and r 5, in those figures, correspond to n x 
 and r 5 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 x in the last two figures, and that will lie level 
 and in the upper surface ; so that all ordinates at right angles to 
 dron the base, will correspond to those that are at right angles 
 
238 
 
 AMERICAN HOUSE-CARPENTER. 
 
 F\-^ 2S8. r 
 
 to e c on the top. This ekicidates Fig: 276 ; at which the lines, 
 h 9 and i 8, correspond to h 9 and i 8 in this figure. 
 
 Fig. 289. 
 
 389. — To find the hevil 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 i s, and equal to w hat Fig. 284 ; make i s equal to t 5 in 
 that figure, and join iv and s ; then sw p will be the bevil required 
 m 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 
 
STAIRS. 
 
 239 
 
 Fig. 290. 
 
 right angles to i p, and equal to i 5 in Fig". 285 ; make 5 w equal 
 \o h w in Fig. 285, and join w and i ; then tv 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 Fig. 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, j /, 
 in Fig. 284 or 285. 
 
 Fig. 291. 
 
 390. — To apply the moulds ivithout bevilling the plank. 
 Make w p, {Fig. 291,) equal to w p at Fig, 289, and the angle, 
 bed, equal to 6 j I in Fig. 284 ; make p a equal to the thick- 
 ness of the plank, as w? a in Fig. 289, and from a draw a o, pa- 
 rallel tow d ; from c, draw c e, at right angles to w d, and join e 
 
240 
 
 AMERICAN HOUSE'CARPENTER. 
 
 and h ; then the angle, b e o,ona square edge of the plank, hav- 
 ing a Ime 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 6 as 
 an axis until the line, p w, falls below the plane of the paper, and 
 the line, p a, arrives in it ; then, it is evident that the point, c, 
 will fall, in the line, c e, until it lies directly behind the point, e, 
 and the line, b c, will lie directly behind b e. 
 
 k 
 
 Fig. 292. 
 
 391. — To find the b evils 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 6 c is the angle at which the 
 work is splayed, and b c?, on the upper edge of the board, is at 
 right angles to a b ; make the angle, /^^', equal to a b c, and 
 from/, draw /A, parallel to e a; from 6, draw b o, at right an- 
 gles Xoab ; through o, draw i e, parallel to c 6, and join e and 
 d ; then the angle, a e d, will be the proper bevil for the ends from 
 the inside, ox k d e from the outside. If a mitre-joint is re- 
 
STAIRS. 211 
 
 quired, setfg, the thickness of the stuff on the level, from e to 
 m, and join th 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, 
 fgj, being the same as a 6 c, will be the proper bevil for that 
 purpose. Suppose, therefore, that a piece indicated by the lines, 
 k g, g-f and /A, 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 k d, it would pass through the point, n. 
 
 392. — Another method for face-'inoulds. 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 
 diflferent 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,^' d represents a prism standing upon a level base, 
 h </, its upper surface forming an acute angle with the face. 
 h /, as at Fig. 287. Extend the base line, h c, and the raking 
 line,^' Z, to meet at/; also, extend e d and ^ a, to meet at k ; 
 from /, through A*, draw / m. If we suppose the prism to stand 
 upon a level floor, o f m^ and the plane, ; g a /, to be extended 
 to meet that floor, then it will be obvious that the intersection 
 between that plane and the plane of the floor would be in the line, 
 f k ; and the line, /A:, being in the plane of the floor, and also in 
 the inclined plane, J ^ A;/, any line made in the plane, ^'^ A*/, 
 parallel to/ A:, must be a level line. By finding the position of a 
 perpendicular plane, at right angles to the raking plane, jf k g, 
 we shall greatly shorten the process for obtaining ordinates. 
 
 31 
 
242 
 
 AMERICAN HOUSE-CARPENTER. 
 
 Fig. 293, 
 
 This may be done thus : from/, draw/ o, at right angles iofm; 
 extend e 6 to o, and^ /, to t ; from o, draw o t, at right angles to 
 ofj and join t and/; then t of will be a perpendicular plane, at 
 right angles to the inclined plane, t g kf; because the base of 
 the former, o/ is at right angles to the base of the latter,/ ^*, both 
 these lines being in the same plane. From 6, draw h p, at right 
 angles to o/ or parallel iofm ; from/;, draw p q, at right angles 
 to o/ and from q, draw a line on the upper plane, parallel to fin, 
 or at right angles to tf; then this line will obviously be drawn 
 to the point, ^*, and the line, g j, be equal top b. Proceed, in the 
 same way, from the points, 5 and c, to find x and I. 
 
 Now, to apply the principle here explained, let the curve, 65 c, 
 {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 a 
 plane passing through three given points in its curved surface : 
 one perpendicularly over 6, at the h'eight, bj; one perpendicu- 
 larly over 5, at the height, s x ; and the other over c, at the height, 
 c I — these lines being drawn at right angles to the chord of the 
 base, b c. Fromj, through Z, draw a line to meet the chord line 
 extended to/; from 5, draw 5 A:, parallel to b /, and from x. 
 draw X k, parallel to jf ; from/ through k, draw/m; then/w 
 will be the intersecting line of ^he plane of the section with the 
 
STAIRS. 
 
 213 
 
 Fig. 294. 
 
 plane of the base. This line can be proved to be the intersection 
 of these planes in another way ; from 6, through 6\ and from^*, 
 through .-r, draw lines meeting at rn ; then the point, tu, will be 
 in the intersecting line, as is shown in the figure, and also at 
 Fig. 293. 
 
 From/, draw/p, at right angles to f m : from b and c, and 
 from as many other points as is thought necessary, draw ordinates, 
 parallel to fm; make p q equal to h j^ and join q and/; from 
 the points at which the ordinates meet the line, qf. draw others 
 at right angles to q f; make each ordinate at A equal to its cor- 
 responding ordinate at C, and trace the curve, g n i, through the 
 points thus found. 
 
 Now it may be observed that A is the plane of the section, B 
 the plane of the segment, corresponding to the plane, q p/ of 
 Fig. 293, and C is the plane of the base. To give these planes 
 their proper position, let A be turned on qf as an axis until it 
 
244 AMERICAN HOUSE-CARPENTER. 
 
 Stands perpendicularly over the line, qf^ 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, ]> /, as an axis until it stands perpendicu- 
 larly over p/, and at right angles to the plane, C ; then the plane, 
 J, will lie over the plane, C, with the several lines on one corres- 
 ponding to those on the other ; the point, i, resting at /, the point, 
 ^^, at x^ and g at j ; and the curve, g n i, lying perpendicularly 
 over b s c — as was required. If we suppose the cylinder to be 
 cut by a level plane passing through the point, Z, (as is done in 
 finding a face-mould,) it will be obvious that lines corresponding 
 to Q'/ and jo/ would meet in / ; and the plane of the section. A, 
 the plane of the segment, B, and the plane of the base, C, would 
 all meet in that point. 
 
 393. — To find the face-mould for a limid-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 ; / e being equal to the 
 stretch-out of a df in Fig. 295. From c, draw c A, parallel to 
 ef; bisect h c in i, and find a point, as 6, in the arc, df {Fig. 
 295.) corresponding to i in the line, h c; from i, {Fig. 296,) to 
 the top of the falling-mould, draw i j, at right angles to Ac; at Fig. 
 295, from c, through 6, draw c g, and from b and c, draw bj and 
 c k, at right angles to ^ c ; make c k equal io h g at Fig. 296, 
 and bj equal to ij at that figure ; from k, through j, draw k g, 
 and from^, through a, drsiw g p ; then ^p will be the intersecting 
 line, corresponding to/m in Fig. 293 and 294 ; through e, draw 
 p 6, at right angles to g j^, and from c, draw c q. parallel to g p ; 
 make r q equal to h g at Fig. 296 ; join p and q^ and proceed as 
 in the previous examples to find the face-mould, A. The joint 
 of the face-mould, u v, will be more accurately determined by 
 finding the projection of the centre of the plan, o, as at w ; 
 joining 5 and w^ and drawing ii v, parallel to s w. 
 
 It may be noticed that c k and b j are not of a length corres- 
 ponding to the above directions : they are but^ the length given. 
 
iiiAlRfe. 
 
 24S 
 
 Fig. 295. 
 
246 
 
 AMERICAN H0r5'Ji'''?ARPENTEIl. 
 
 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 c 
 k and h j to be the full length of the given lines, bisect one in i 
 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 
 
STAIRS. 
 
 247 
 
 obtained thus : at Fig. 296, make h I equal to c 6 in Fig. 295 ; 
 from Z, draw I k, at right angles to h c ; from 7, draw^* k, parallel 
 to A c ; from g, through k, draw g n; at Fig. 295, make b g 
 equal io In in Fig. 296 ; then g will be tlie point required. 
 
 The reason why the points, a, b and c, in the plan of the rail a^ 
 Fig. 295, are taken for resting points instead of e, i and/, is this : 
 the top of the rail being level, it is evident that the points, a and e, 
 in the section a e, are of the same height ; also that the point, i, is of 
 the same height as 6, and c as/. Now, if a is taken for a point 
 in the inclined plane rising from the line g ;:>, e must be below 
 that plane ; if b is taken for a point in that plane, i must be below 
 it ; and if c is in the plane,/ must be below it. The rule, then, 
 for taking these points, is to take in each section the one that is 
 nearest to the line, g p. Sometimes the line of intersection, g p, 
 happens to come almost in the direction of the line, e r : in such 
 case, after finding the line, see if the points from which the 
 heights were taken agree with the above rule ; if the heights 
 were taken at the wrong points, take them according to the rule 
 above, and then find the true line of intersection, which will not 
 vary nmch from the one already found. 
 
 Fig. -297. 
 
 394.— To appl]/ the face-mould thus found to the plank. 
 The face-mould, when obtained by this method, is to be applied 
 to a square-edged plank, as directed at Art. 383, with this differ- 
 ence : instead of applying both tips of the mould to the edge of 
 
248 AMERICAN HOUSE-CARPENTER. 
 
 the plank, one of them is to be set as far from the edge of the 
 plank, as x^ in Fig. 295, is from the chord of the section p q — as 
 is shown at Fig. 297. A. in this figure, is the mould applied on 
 the upper side of the plank, i5, the edge of the plank, and C, the 
 mould applied on the under side ; a b and c d being made equal 
 to q 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 lips of the mould, e and 6, 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, 6, arrives at g^ causing the line, e b, 
 to coincide with eg: the mould upon the under side of the 
 plank must now be revolved upon a point that is perpendicularly 
 beneath e, as /; from/, draw / A, parallel to i d. and from (/, 
 draw d A, at right angles to i d ; then revolve the mould, C, upon 
 /, until the point. A, arrives at J, causing the line,/ A, to coincide 
 with/^', and the line, i d, to coincide with k I j then the tips of 
 the mould will be at k and I. 
 
 The rule for doing this, then, will be as follows : make the an- 
 gle, ^/A', equal to the angle q v x, at Fig. 295 ; make /A' equal 
 to/i, and through A-, draw k /, parallel to ij ;. then apply the 
 corner of the mould, i, at /r, and the other corner c?, at the line, k I. 
 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,2h I 
 and m, corresponding to the points, J^, a^^ z^ y, &c., at Fig. 295 ; 
 from r q Pj (fee, draw the lines, r t, q ii, p v, (fee, at right angles 
 to he; make A .«?, r t, q u, &c., respectively equal to 6 c"; r q, 5 
 d', &c., at Fig. 295 ; through the points thus found, trace the 
 curve, s w c. Then get out the piece, g 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, sw c, will coincide with the upper surface of the rail piece 
 
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 a7id jjosition of the 
 regulating square. The breadth which the scroll is to occupy, 
 the number of its revolutions, and the relative size of the reguJa 
 ting square to the eye of the scroll, being given, multiply tlie 
 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 th^ 
 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. 
 
 
 
 
 
 
 
 
 
 6 
 
 
 
 
 5 
 
 
 
 
 
 
 
 
 
 
 
 
 8 
 
 
 
 
 
 I- 
 
 
 
 
 
 
 
 
 
 
 
 4 
 
 
 1 
 
 
 
 
 
 
 
 Fiff. 298. 
 
 397. — To find the proper centres in the regulating square. 
 Let a 2 1 6, {Fig. 29S,) be the size of a regulating square, found 
 according to the previous rule, the required number of revolu- 
 tions being If. Divide two adjacent sides, as a 2 and 2 1, 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 
 
250 
 
 AMERICAN HOUSE-CARPENTER. 
 
 mine th«se centres being each one part less in length than its pre- 
 ceding line. 
 
 ^ M 
 
 Fig. 299. 
 
 398. — To describe the scroll for a hand-rail over a curtail 
 step. Let a b, (Fig. 299,) be the given breadth. If 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, If 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 6, therefore, into 10 equal parts, and set one from 
 b to c ; bisect a c in e ; then a e will be the length of the longest 
 ordinate, (1 c/ or 1 e.) From a, draw a d, from e, draw e 1, and 
 from b, draw 6/, all at right angles to a b ; make e 1 equal to e 
 a, and through 1, draw 1 d, parallel to a b ; set b c from 1 to 2, 
 and upon 1 2, complete the regulating square ; divide this square 
 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/^ ; upon 4, describe g A, &c. ; make d I equal to the 
 
STAIRS. 
 
 251 
 
 width of the rail, and upon 1, describe Im ; upon 2, aescribe m 
 w, <fec. ; describe the eye upon 8, and the scroll is completed. 
 
 399. — To describe the scroll for a curtail step. Bisect d I, 
 {Fig". 299,) in o, and make o v equal to ^ of the diameter of a 
 baluster ; make v w equal to the projection of the nosing, and e 
 X equal to w I; upon 1, describe w y, and upon 2, describe y z , 
 also upon 2, describe x i ; upon 3, describe i j, and so around to 
 z ; and the scroll for the step will be completed. 
 
 400. — 7b determine the 'position of the balusters under the 
 scroll. Bisect c?Z, [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, q^ r, 5, ^, u^ &c., as far as q^ ; upon 2, 
 3, 4 and 5, describe the centre-line of the rail around to the eye 
 of the scroll ; from the points of division in the circle, o r ?/, draw 
 lines to the centre-line of the rail, tending to the centre of the 
 eye, 8 ; then, the intersection of these radiating lines with the 
 centre-line of the rail, will determine the position of the balusters, 
 as shown in the figure. 
 
 Fijr. 300. 
 
 401. — To obtain the falling-mould for the raking part of the 
 scroll. Tangical to the rail at h, {Fig. 299,) draw h k, parallel to d 
 a; then k d^ will be the joint between the twist and the other part 
 of the scroll. Make d e^ equal to the stretch-out of d e, and upon d 
 
252 
 
 AMERICAN HOUSE-CARPENTER. 
 
 e'^, find the position of the point, k^ as at ^ ; at Fig. 300, make e d 
 equal to e^ d in Fig. 299, and d c equal to d & in that figure ; 
 from c, draw c a, at right angles to e c, and equal to one rise ; 
 make c h equal to one tread, and from 6, through a, draw 6 j , 
 bisect a c in Z, and through Z, draw ?7i 9-, parallel to e h ; m q is 
 the height of the level part of a scroll, which should always be 
 about 3A feet from the floor ; ease off the angle, m/J, according 
 to Art. 89, and draw g w n, parallel to m x j^ and at a distance 
 equal to the thickness of the rail ; at a convenient place for the 
 joint, as i, draw i n, at right angles to b j ; through n, draw / A, 
 at right angles to c h ; make d k equal to d k"^ in Fig. 299, and 
 from A*, draw k 0, at right angles to e h ; at Fig. 299, make d 
 h^ equal to c? ^ in Fig. 300, and draw h^ 6^, at right angles to d 
 h^ ; then k a^ and h^ i^ will be the position of the joints on the 
 plan, and at Fig. 300, p and i n, their position on the falling- 
 mould ; and p i ?i, {Fig. 300,) will be the falling-mould re- 
 quired. 
 
 Fig. SOL 
 
 402. — To describe the face-mould. At Fig. 299, from/:, draw 
 k r'^, at right angles to r'- d ; at Fig. 300, make h r equal to h"^ r^ 
 in Fig. 299, and from r, draw r s, at right angles to r h ; from 
 the intersection of r 5 with the level line, m q^ through ^, draw s 
 t ; at Fig. 299, make K^ b'^ equal to ^ ^ in Fig. 300, and join b"^ 
 and r'^ ; from a^, and from as many other points in the arcs, a^ I 
 and k d, as is thought necessary, draw ordinates to r'^ d, at right 
 angles to the latter ; make r b, {Fig. 301,) equal in its length and 
 in its divisions to the line, r"^ b% in Fig. 299 ; from r, n, 0, /?, q 
 
STAIRS. 
 
 253 
 
 and Z, draw the lines, r k, n d, o a, p e, ^/and I c, at right an- 
 gles to r b, and equal to r^ k, d' s\ f a\ &c., in Fig. 299 ; 
 through the points thus found, trace the curves, A; Z and 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, I 6, 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 thickness of stuff required for this mould is shown at 
 Fig. 300, between the lines s t and u v—u v being drawn pa- 
 rallel to s t. 
 
 403.— All the previous examples given for finding face-moulds 
 over winders, are intended for ?noulded rails. For roimd 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. 
 
 g 
 
 Fig. 302. 
 
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 io c d ; bisect a 6 in e, and through 
 a, e and b, draw h i,fg, and kj, at right angles to a 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, i j g ; make r I equal 
 to e b or e a ; join I and j, also I and i ; from the curve, / b, to 
 the line, I j, draw as many ordinates as is thought necessary, 
 parallel to f g ; from the points at which these ordinates meet 
 the line, I j, and upon the centre, o, describe arcs in continuation to 
 meet op; from n, t, x, &c., draw n s, t u, (fee, parallel to f g ; 
 make n s, t u, &c., equal to e f lo v^ &c. ; make x y, &c., equal 
 to z dy &c. ; make o 2, o 3, &:c., equal to on, o t, &,c. ; make 2 4 
 equal to n s, and in this way find the length of the lines crossing 
 m ; through the points thus found, describe the section of the 
 newel-cap, as shown in the figure. 
 
 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 a c, Fig. 269 ; from b, draw b c, at right angles to a b, 
 and equal in length to n ??i, Fig. 269 ; join a and c, and bisect a c 
 in ; through o draw e f at right angles to a c, and d k, parallel 
 to cb ; make o d and o k each equal to half e h at Fig. 269 ; 
 through e and /, draw h i 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- 
 
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, th'; line for the joint can be 
 obtained from only one side. Whe i 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 dowr?. the vertical joint, 
 obtained as at A7't. 373, will be a butt joint, and therefore, in such 
 a case, the process described in this article will be unnecessary 
 
 NOTE TO ARTICLE 369. 
 
 Platform stairs vrith a large cylinder. Instead of 
 placing ihe platform-risers at the spring of the cyl- 
 inder, a more easy and graceful appearance may be 
 given to the rail, and the necessity of canting either 
 of the twists entirely obviated, by fixing the place of 
 the above risers at a certain distance within the cyl- 
 inder, as shown in the annexed cut — the lines indi- 
 cating the face of the risers cutting the cylinder at k 
 and I, instead of at p and q, the spring of the cylin- 
 der. To ascertain the position of the risers, let a 6 c 
 be the pitch-board of the lower flight, and c d e that 
 of the upper flight, these being placed so that b c 
 and c d shall form a right line. Extend a c to cut 
 de in f; draw / g parallel to d b, and of indefinite 
 length : draw g o at right angles to / g, and equal 
 in length to the radius of the circle formed by the 
 centre of the rail in passing around the cylinder; 
 on o as centre describe the semicircle j g i ; make 
 h equal to the radius of the cylinder, and describe 
 on o the face of the cylinder p h q ; then extend d b 
 across the cylinder, cutting it in I and k — giving the 
 position of the face of the risers, as required. To 
 find the face-mould for the twists is simple and ob- 
 vious : it being merely a quarter of an ellipse, hav- 
 ing j for semi-minor axis, and the distance on the 
 
 rake corresponding to o g, on the plan, for the semi-major axis, found thus,— extend i j to 
 meet a /, then from this point of meeting to / is the semi-major axis. 
 
SECTION VIL— 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 
 elementary principles upon which shading, in architectural sub- 
 jects, is based. The necessary knowledge of drawing, prelim- 
 inary to this subject, is treated of in the Introduction, from Art. 
 1 to 14. 
 
 407. — TTie 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, By the 
 
SHADOWS. 
 
 257 
 
 Fig. 304. 
 
 line, a b, is drawn at an angle of 45 degrees with the horizon, and 
 the hne, c b, at the same angle with the vertical plane, B. The 
 plane, B, being a rectangle, this makes the true direction of the 
 sun's rays to be in a course parallel to d b ; which direction has 
 been proved to be at an angle of 35 degrees and 16 minutes with 
 the horizon. It is convenient, in shading, to have a set-square 
 with the two sides that contain the right angle of equal length ; 
 this will make the two acute angles each 45 degrees ; and will 
 £^ive the requisite bevil when worked upon the edge of the T- 
 square. One reason why this angle is chosen in preference to 
 another, is, that when shadows are properly made upon the draw- 
 ing by it, the depth of every recess is more readily known, since 
 the breadth of shadow and the depth of the recess will be equal. 
 
 To distinguish between the terms shade and shadow, it will be 
 understood that all such parts of a body as are not exposed to the 
 direct action of the sun's rays, are in shade ; while those parts 
 which are deprived of light by the interposition of other bodies, 
 are in shadow. . 
 
 33 
 
258 
 
 AMERICAN HOUSE-CARPENTER. 
 
 Fig. 305. 
 
 Fig. 306. 
 
 Fig. 307. 
 
 Fig. 308. 
 
 408. — To find the line of shadow on mouldings and other ho- 
 rizontaUy straight projections. Fig. 305, 306, 307 and 308, 
 lepresent 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 h, is drawn from the 
 projections at an angle of 45 degrees. Wliere 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 shadoio 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 6, erect a perpendicular intersecting c ddX 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 
 of me shelf to/, and make c/ equal to the projection of the shelf 
 
SHADOWS. 
 
 25^ 
 
 Fig. 309. 
 
 from the wall ; from/, draw/^, at the customary angle, and from 
 c, drop the vertical line, c g, intersecting f g ^X g ; from g^ draw 
 g e, parallel to the shelf, and from c, draw c d, at the usual angle ; 
 then the lines, c d and d e, will determine the extent of the shadow 
 as before. 
 
 
 
 
 1 
 
 e 
 
 
 
 iir '" 1 
 
 
 1 
 
 ''i'iiiiiiiililllli!iii||i{;ii 
 
 C 
 
 , 
 
 lliijiiiiiiiini 
 
 iiiiiiiiiiiiiiiiiii 
 
 liilllitllllllllll 
 
 
 
 / 
 
 y^b 
 
 
 
 
 c 
 
 Fig. 310 
 
 
 - 
 
 B 
 
 410. — To find the shadow cast hy a shelf , which is wider at 
 one end than at the other. In Fig. 310, A is the plan, and B 
 the elevation. Find the point, d, as in the previous example, and 
 from any other point in the front of the shelf, as a, erect the perpen- 
 dicular, a e ; from a and e, draw a h and e c, at the proper angle, 
 and from 6, erect the perpendicular, h c, intersecting e c m c ; 
 
^260 
 
 AMERICAN nOUSE-CARPENTEll. 
 
 from d, through c, draw d o ; then the hnes, i 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 an 
 acute-angled shelf. Find the line, e g, as before ; from a, erect 
 the perpendicular, ab ; join 6 and e ; then b e and e g will define 
 the boundary of shadow. 
 
 Fig. 312. 
 
 412. — To find the shadow cast by an inclined shelf. In Fig. 
 312, 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 a d 
 and d c will be the shadow required. If the shelf had been 
 
SHADOWS. 
 
 261 
 
 parallel in width on the plan, then the line, d c, would have been 
 parallel with the shelf, a b. 
 
 
 
 
 
 
 / 
 
 
 
 
 
 1 
 d 
 
 
 
 
 \l 1 1 illi ililll! 
 
 D 
 
 
 
 e 
 
 c 
 
 Fig. 313. 
 
 Fig. 314. 
 
 413. — To find the shadow cast hy a shelf inclined in its ver- 
 tical section either upward or downward. From a, {Fig, 313 
 and 314,) draw a b, at the usual angle, and from b, draw b c, 
 parallel with the shelf; obtain the point, e, by drawing a line 
 from c?, at the usual angle. In Fig. 313, join e and i ; then i e 
 and e c will define the shadow. In Fig. 314, from o, draw o i, 
 parallel with the shelf; join i and e ; then i e and e c will be the 
 shadow required. 
 
 The projections in these several examples are bounded by- 
 straight 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 — 
 
 Fig. 316. 
 
 Fig. 315. 
 
262 
 
 AMERICAN HOTjSE-CARPENTEU. 
 
 414. — To find the shadow of a shelf having its front edge^ or 
 endy 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 
 rays and perpendiculars intersecting at e, e, &c. ; through these 
 points of intersection trace the curve, and it will define the shadow. 
 
 e 
 
 ^.v 
 
 e 
 
 11 
 
 e 
 
 lliiiiiiiiiiiiiiiii 
 
 11^ 
 
 lllllllllllllll;! 
 
 iiiiiiiiiiiiiiii 
 
 
 / 
 
 
 
 Fig. 31- 
 
 415. — To find the shadow of a shelf curved in the elevation. 
 In Fig. 317, find the points of intersection, e, e and e, as in the 
 last examples, and a curve traced through them will define the 
 shadow. 
 
 The preceding examples show how to find shadows when cast 
 upon a vertical plarie ; shadows thrown upon curved surfaces are 
 ascertained in a similar manner. Thus — 
 
 Fig. 318. 
 
SHADOWS. 
 
 263 
 
 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 a shelf upon an inclined 
 loaU. Cast the ray, a b, {Fig. 319,) from the end of the shelf to 
 the face of the wall, and from b, draw b c, parallel to the shelf; 
 cast the ray, d e, from the end of the shelf ; then the lines, d e 
 and e c, will define the shadow. 
 
 These examples might be multiplied, but enough has been 
 given to illustrate the general principle, by which shadows in all 
 instances are found. Let us attend now to the application of this 
 principle to such familiar objects as are likely to occur in practice. 
 
264 
 
 AMERICAN IiaUSE-CARPENTER. 
 
 418. — To find the shadow of a projecting horizontal becmi. 
 From the points, a, a, 6cc., (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, eitlier on the plan or elevation, at any angle other than a 
 light 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 point, a, 
 [Fig. 321,) in the plan, and h in the elevation, draw the ra3'S, a c 
 and h e ; 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 tlie 
 extent of the shadow. This applies only where the back of the 
 recess is parallel with the face of the wall. 
 
 FiL'. 32^. 
 
 420. — To find the shadow in a recess, when the face of the 
 wall is inclined, and the hack of the recess is vertical. In Fig 
 322, A shows the section and B the elevation of a recess of this 
 
SHADOWS. 
 
 265 
 
 kind. From h, and from any other point in the Hne, h a, as a, 
 draw the rays, h c and a e ; from c, a, and e, draw the horizontal 
 lines, eg", a f, and eh; from J and /, cast the rays, d i and/ A ; 
 from i, through h^ draw i s ; then s i and i g will define the 
 shadow. 
 
 d 
 
 mmi, 
 
 Fig. 323. 
 
 421. — To find the shadow in a fireplace. From a and h, 
 {Fig. 323,) cast the rays, a c and b e, 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. 
 
 ^miiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiii|{ 
 "11 s 
 
 iiiiiinimiiii !iiMiiiiiiiii!iii;i! 
 
 Fig. 324. 
 
 422. — To find the shadow of a moulded window-lintel. Cast 
 rays from the projections, a, o, &c., in the plan, {Fig. 324,) and 
 d, e, &c., in the elevation, and draw the usual perpendiculars in- 
 tersecting the rays at ^, i, and i ; these intersections connected 
 
 34 
 
206 
 
 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 i to s, and drawing the horizontal line, s n. 
 
 Fig. 325. 
 
 423. — To find the shadow cast hy the nosing of a step. From 
 «, {Fig. 325,) and its corresponding point, c, cast the rays, a h 
 and c d, and from h, erect the perpendicular, h d ; tangical to the 
 curve at e, cast the ray, e /, and from e, drop the perpendicular, 
 e o, meeting the mitre-line, a g, in o ; cast a ray from o to z, and 
 from i, erect the perpendicular, i f ; from h, draw the ray, h k; 
 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, k n and h s ; 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 a o will show the extent of the shadow 
 on the first riser, as at ii ; f g will determine the shadow on the 
 second riser, as at ^ ; c d gives the amount of shadow on the 
 first tread, as at C, and h i 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- 
 
SHADOWS. 
 
 267 
 
 [ll!l!l||ll 
 
 iiiiiiiiiiiiijiiiiiiiiiiiiiiiiiiimiiiiiii!iii!iiiiiiiiiiiiiii 
 
 D 
 
 T 
 
 Miiiiiiiiinmi::'''' '!;''^!':'!TJiiiii|iiii 
 
 Fig. 326. 
 
 ficient to show how the shadow of the right-hand pedestal is 
 obtained. 
 
 Fig. 327. Fig. 3-i8, 
 
 425. — To find the shadow thrown on a column by a square 
 abacus. From a and b, {Fig. 327,) draw the rays, a c and b e, 
 and from c, erect the perpendicular, c e ; tangical to the curve at 
 c/, draw the ray, d f, and from It, corresponding to / in the plan, 
 draw the ray, h o ; take any point between a and/, as i, and from 
 this, as also from a corresponding point, ?2, draw the rays, i r and 
 n s ; from r, and from d, erect the perpendiculars, r s and d o : 
 through the points, e, s, and o, trace the curve as shown in the 
 figure : then the extent of the shadow v/ill 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 
 
268 
 
 AMERICAN HOUSE-CARPENTEK. 
 
 Fisr. 329. 
 
 427. — To find the shadows on the capitcd 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 may sup- 
 pose the column to be sliced, so to speak, with planes of this 
 
5?I1AD0\YS. 
 
 269 
 
 IH.UI 
 
 I 1 1 I 1 1 1 1 I ' I 
 
 II II I I I M I I I I I I 1 
 M I M I I M I 1 I I I I I I I I ; 1 I ' 
 
 FiK. 330. 
 
 nature — cutting it in the lines, a h, c d, &c., {Fig. 329,) and, m 
 tlie 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 Z), the plane 
 cuts the ovolo at e, and therefore /will be the corresponding point 
 upon the elevation ; h corresponds with g, i with^, o with s, and 
 I with h. Now, to find the. shadows upon this line of section, cast 
 from m, the ray, r/i 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 eacli line 
 of section, and in that way the points, p, r, t, u, v, tv, 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 
 
270 
 
 AMERICAN HOUSE-CARPENTER. 
 
 Fig. 331. 
 
 rays from a and h, {Fig. 331,) and find the point, c, as in tlie 
 previous examples ; from d^ draw the ray, d e, and from e, the 
 horizontal line, e f ; tangical to the curve at g and A, draw the 
 rays, g j and h z, and from i and j^ erect the perpendiculars, i I 
 and^' k ; from m and n^ draw the rays, m/and n k^ and trace the 
 curve between k and /; cast a ray from o to p, a. vertical line 
 from p to 5, and through s, draw the horizontal line, s t ; the 
 shadow as required will then be completed. 
 
SHADOWS. 
 
 271 
 
 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. — Fig. 333 and 334 are examples of the Tuscan cornice. 
 The manner of obtaining the shadows is evident. 
 
272 
 
 A.MERICAN HOUSE-CARPENTER. 
 
 Fig. 334. 
 
 430. — Reflected light. In shading, the finish and Ufe 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. 
 
APPENDIX 
 
GLOSSARY. 
 
 Tenns not found here can be found in the lists of definitions m other parts of this booV.i 
 
 or m fommon dictionaries. 
 
 Abacus. — The uppermost member of a capital. 
 
 Ahhatoir. — 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, beards-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 frorp 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 esirade, or partition of 
 columns. Recess with seats, &c., in gerdens. 
 
 Altar. — A pedestal whereon sacrifice was offered. In modern 
 churches, the area within the railing in front of the pulpit. 
 
 Alto-relievo. — High i-elief ; 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 o'' the architrave. 
 
 A.nnulet. — A small square moulding used to separate others ; the 
 fill^'ts in the Doric capital under the ovolo, and those which separate 
 tiiG flutings of columns, are known by this term. 
 
 Antce. — 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 curvilinear 
 form, so as to perform the office of a lintel and carry superincumbent 
 weights. 
 
 Architrave. — That part of the entablature which rests upon the 
 capital of a column, and is beneath the frieze. The casing and 
 mouldings about a door or window. 
 
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 
 warlike stores. 
 
 Astragal. — A small moulding consisting of a half-round with a fillet 
 on each side. 
 
 Attic. — A low story erected over an order of architecture. A low 
 additional story immediately under the roof of a building. 
 
 Aviary. — A place for keeping and breeding birds. 
 
 Balcony. — An open gallery projecting from the front of a building. 
 
 Baluster. — A small pillar or pilaster supporting a rail. 
 
 Balustrade. — A series of balusters connected by a rail. 
 
 Barge-course. — That part of the covering which projects over the 
 gable of a building. 
 
 Base. — The lowest part of a wall, column, &c. 
 
 Basement-story. — That which is immediately under the principal 
 story, and included within the foundation of the building. 
 
 Basso-relievo. — Low relief; sculptured figures projecting from a 
 surface one-half their thickness or less. See Alto-relievo. 
 
 Battering. — See Talvs. 
 
 Battlement. — Indentations on the top of a wall or parapet. 
 
 Bay-window. — A window projecting in two or more planes, and not 
 forming; the segment of a circle. 
 
 Bazaar. — A species of mart or exchange for the sale of various ar- 
 ticles of merchandise. 
 
 Bead. — A circular moulding. 
 
 Bed-mouldings. — Those mouldings which are between the corona 
 and the frieze. 
 
 Belfry. — That part of a steeple in which the bells are hung : an- 
 ciently called camjMnile. 
 
 Belvedere. — An ornamental turret or observatory commanding a 
 pleasant prospect. 
 
 Boiv-ivindow. — A window projecting in curved lines. 
 
 Bressummer. — Abeam or iron tie supporting a wall over a gateway 
 or other opening. 
 
 Brick -Hogging. — The brickwork between studs of partitions. 
 
 Buttress. — A projection from a wall to give additional strength. 
 
 CaMe. — A cylindrical moulding placed in flutes at the lower part of 
 the column. 
 
 Camber. — To give a convexity to the upper surface of a beam. 
 
 Campanile. — A tower for the reception of bells, usually, in Italy, 
 separated from the church. 
 
 Canopy. — An ornamental covering over a seat of state. 
 
 Cantalivers. — The ends of rafters under a projecting roof. Pieces 
 of wood or stone supporting the eaves. 
 
 Capital. — The uppermost part of a column included between the 
 shaft and the architrave. 
 
APPENDIX. 5 
 
 Carmmnsera. — 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. Th^ 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 arc interred. 
 
 Cenotaph. — A monument erected to the memory of a person buried 
 in another place. 
 
 Centring. — The temporary woodwork, or framing, whereon any 
 vaulted work is constructed. 
 
 Cesspool. — A well under a drain or pavement to receive the waste- 
 water and sediment. 
 
 Chamfer. — The bevilled edge of any thing originally right-angled. 
 
 Chancel. — That part of a Gothic church in which the altar is placed. 
 
 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, &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. 
 
 Clerestory. — 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. 
 
G APPENDIX. 
 
 Column. — A vertical, cylindrical support under the entablature of 
 an order. 
 
 Common-rafters. — The same as jack-rafters, which see 
 
 Conduit. — A long, narrow, walled passage underground, for secret 
 communication between different apartments. A canal or pipe for the 
 conveyance of water. 
 
 Conservatory. — A building for preserving curious and rare exotic 
 plants. 
 
 Consoles. — The same as ancones, which see. 
 
 Contour. — The external lines which bound and terminate a figure. 
 
 Convent. — A building for the reception of a society of religious per- 
 sons. 
 
 Coping. — Stones laid on the top of a wall to defend it from the 
 weather. 
 
 Corbels. — Stones or timbers fixed in a wall to sustain the timbers of 
 a floor or roof. 
 
 Cornice. — Any moulded projection which crowns or finishes the 
 part to which it is affixed. 
 
 Corona. — That part of a cornice which is between the crown- 
 moulding and the bed-mouldings. 
 
 Cornucopia. — The horn of plenty. 
 
 Corridor. — An open gallery or communication to the difforent aparr- 
 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. 
 \ny 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-revrsa. 
 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 brick- 
 work, to support a superincumbent weight until the brickwork which 
 is to carry it has set or become hard. 
 
 Decastyle. — A building having ten columns in front. 
 
 Dentils. — (From the Latin, denies, teeth.) Small rectangular blocks 
 used in the bed-mouldings of some of the orders. 
 
 Diaslyle. — An intercolumniation of three, or, as some say, four 
 diameters. 
 
APPENDIX. ' 
 
 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 o^' necessity. 
 
 Dooks. — A Scotch term given to wooden bricks. 
 
 Dormer. — A window placed on the roof of a house, the frame 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. 
 
 Entallature. — That part of an order which is supported by the co- 
 lumns ; consisting of the architrave, frieze, and cornice. 
 
 Eustyle. — An intercolumniation of two and a quarter diameters. 
 
 Exchange. — A building in which merchants and brokers meet to 
 transact business. 
 
 Extrados. — The exterior curve of an arch. 
 
 Facade. — The principal front of any building. 
 
 Face-mould — The pattern for marking the plank, out of which hand- 
 railing is to be cut for stairs, &c. 
 
 Facia, or Fascia. — A flat member like a band or broad fillet. 
 
 Falling-mould. — The mould applied to the convex, vertical surface 
 of the rail-piece, in order to form the back and under surface of the 
 rail, and finish the squaring. 
 
 Festoon. — An ornament representing a wreath of flowers and leaves. 
 
 Fillet. — A narrow flat band, listel, or annulet, used for the separa- 
 tion of one moulding from another, and to give breadth and firmness 
 :.o the edges of mouldings. 
 
 Flutes. — Upright channels on the shafts of columns. 
 
 Flyers. — Steps in a flight of stairs that are parallel to each other. 
 
 Forum. — -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 moulde 
 or shapes. 
 
 Frieze. — That part of an entablature included between the archi- 
 trave and the cornice. 
 
 Gahle. — The vertical, triangular piece of wall at the end of a rooi. 
 from the level of the eaves to the summit. 
 
 Gain. — A recess made to receive a t^non or tusk. 
 
 Gallery. — A common passage to several rooms in an upper storv. 
 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. 
 
B 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. 
 
 Gv.it(E. — 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 
 Notting/ia77i, Bucking^a7?i, &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. 
 
 Hexastyle. — A building having six columns in front. 
 
 Hip-rafter. — A piece of timber placed at the angle made by two ad- 
 jacent inclined roofs. 
 
 Homestall. — A mansion-house, or seat in the country. 
 
 Hotel, or Hostel. — x\ large inn or place of public entertainment. A 
 large house or palace. 
 
 Hot-house. — A glass building used in gardening. 
 
 Hovel. — An open shed. 
 
 Hut. — A small cottage or hovel generally constructed of earthy 
 materials, as strong loamy clay, &c. 
 
 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. 
 
 Jail. — A place of legal confinement. 
 
 Jamhs. — 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. 
 Kiln. — 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. 
 
APPENDIX. V 
 
 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 -hoards. — 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 
 principal room or rooms of a house. 
 
 Lodge. — A small house near and subordinate to the mansion. A 
 cottage placed at the gate of the road leading to a mansion. 
 
 Loop. — A small narrow window. Loophole is a term applied to the 
 vertical series of doors in a warehouse, through which goods are de- 
 livered by means of a crane. 
 
 Luffer -boar ding. — The same as lever-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. 
 
 ModilUon. — 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 u 
 Gothic window. 
 
 Muniment-house. — A strong, fire-proof apartment for the keeping 
 and preservation of evidences, charters, seals, &c., called muniments. 
 
 1* 
 
10 APPENDIX. 
 
 Museum. — A repository of natural, scientific and literary, curiosities, 
 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. 
 
 Newel. — A post at the starting o:" landing of a flight of stairs. 
 
 Niche. — A cavity or hollow place in a wall for the reception of a 
 statue, vase, &;c. 
 
 Nogs. — Wooden bricks. 
 
 Nosing. — The rounded and projecting edge of a step in stairs. 
 
 Nunnery. — A building or buildings appropriated for the reception of 
 nuns. 
 
 Obelisk. — A lofty pillar of a rectangular form. 
 
 Octastyle. — A building with eight columns in front. 
 
 Odemn. — Among the Greeks, a species of theatre wherein the poets 
 and musicians rehearsed their compositions previous to the public pro- 
 duction of them. 
 
 Ogee. — See Cyma. 
 
 Orangery. — A gallery or building in a garden or parterre fronting 
 the south. 
 
 Oriel-window. — A large bay or recessed window in a hall, chapel, or 
 other apartment. 
 
 Ovolo. — A convex projecting moulding whose profile is the quad- 
 rant of a circle. 
 
 Pagoda. — A temple or place of worship in India. 
 
 Palisade. — K fence of pales or stakfes driven into the ground. 
 
 Parapet. — A small wall of any material for protection on the sides 
 of bridges, quays, or high buildings. 
 
 Pavilion. — A turret or small building generally insulated and com- 
 prised under a single roof. 
 
 Pedestal. — A square foundation used to elevate and sustain a co- 
 lumn, statue, &c. 
 
 Pediment. — The triangular crowning part of a portico or aperture 
 which terminates vertically the sloping parts of the roof : this, in 
 Gothic architecture, is called a gable. 
 
 Penitentiary. — A prison for the confinement of criminals whose 
 crimes are not of a very heinous nature. 
 
 Piazza. — A square, open space surrounded by buildings. This 
 term is often improperly used to denote a 'portico. 
 
 Pier. — 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. 
 
 PUes. — 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- 
 
APPENDIX. -1 
 
 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. 
 
 Phnth. — The lower square member of the base of a column, pedes- 
 tal, or wall. 
 
 Porch. — An exterior appendage to a building, forming a covered 
 appioach 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 intercolumniation of one and a half diameters. 
 
 Pyramid. — A solid body standing on a square, triangular or poly- 
 gonal basis, and terminating in a point at the top. 
 
 Quarry. — A place whence stones and slates are procured. 
 
 Quay. — (Pronounced, key.) A bank formed towards the sea or on 
 the side of a river for free passage, or for the purpose of unloading 
 merchandise. 
 
 Quoin. — An external angle. See Rustic quoins. 
 
 Rabbet, or Rebate. — A groove or channel in the edge of a board. 
 
 Ramp. — A concave bend in the back of a hand-rail. 
 
 Rampant arch. — One having abutments of different heights. 
 
 Regula. — The band below the tsenia in the Doric order. 
 
 Riser. -^\n stairs, the vertical board forming the front of a step. 
 
 Rostrum. — An elevated platform from which a speaker addresses an 
 audience. 
 
 Rotunda. — A circular building. 
 
 Rubble-wall. — A wall built of unhewn stone. 
 
 Rudenture. — The same as cable, which see. 
 
 Rustic quoins. — The stones placed on the external angle of a build- 
 ing, projecting beyond the face of the wall, and having their edges 
 bevilled. 
 
 Rustic-work. — A mode of building masonry wherein the faces of the 
 stones are left rough, the sides only being wrought smooth where the 
 union of the stones takes place. 
 
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 measu/e to which a piece of timber is to be or has 
 been cut. 
 
 . Scarfing. — The joining of two pieces of timber by bolting or nailing 
 transversely together, so that the two appear but one. 
 
 Scotia. — The hollow moulding in the base of a column, between the 
 fillets of the tori. 
 
 Scroll. — A carved curvilinear ornament, somewhat resembling in 
 profile the turnings of a ram's horn. 
 
 Sepulchre. — A grave, tomb, or place of interment. 
 
 Sewer. — A drain or conduit for carrying off soil or water from any 
 place. 
 
 Shaft. — The cylindrical part between the base and the capital of a 
 column. 
 
 Shoar. — A piece of timber placed in an oblique direction to support 
 a building or wall. 
 
 Sill. — The horizontal piece of timber at the bottom of framing ; the 
 timber or stone at the bottom of doors and windows. 
 
 Soffit — 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. 
 
 TcBnia. — The fillet which separates the Doric frieze from the archi- 
 trave. 
 
 Talus. — The slope or inclination of a wall, among workmen called 
 battering. 
 
 Terrace. — An area raised before a building, above the level of the 
 ground, to serve as a walk. 
 
 Tesselated pavement. — A curious pavement of Mosaic work, com- 
 posed of small square stones. 
 
 Tetrastyle. — A building having four columns in front. 
 
 Thatch. — A covering of straw or reeds used on the roofs of cottages, 
 barns, &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. 
 
 Tomh. — A grave, or place for the interment of a human body, in- 
 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 clerestory window. 
 
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 
 .he two vertical edges, and having two channels in the middle. 
 
 Tripod. — A table or seat with three legs. 
 
 Trochilus. — The same as scotia, which see. 
 
 Truss. — An 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-tahle. — The stone covering to the projecting foundation or 
 other walls of a building. 
 
 Well. — The space occupied by a flight of stairs. The space left 
 beyond the ends of the steps is called the well-hole. 
 
 Wicket. — A small door made in a gate. 
 
 Winders. — In stairs, steps not parallel to each other. 
 
 Zophorus. — The same as frieze, which see. 
 
 Zyitos. — Among the ancients, a portico of unusual length, common 
 Iv appropriated to gymnastic exercises. 
 
TABLE OF SaUARES, CUBES, AND ROOTS. 
 
 (From Hutton's Mathematics.) 
 
 No. 
 
 1 Square. 
 
 ! Cube. 
 
 ' Sq. Root. CubeRoot. 
 
 No. 
 68 
 
 Square. 
 
 Cube. 
 
 Sq. Root. 
 
 CubeRoot. 
 
 1 
 
 1 
 
 1 
 
 1-0000000 1000000' 
 
 4624 
 
 314432 
 
 8-2462113 
 
 4-081655 
 
 2 
 
 4 
 
 8 
 
 l-4142136j 1-259921! 
 
 69 
 
 4761 
 
 323509 
 
 1 8-3066239 4 101566 
 
 3 
 
 9 
 
 27 
 
 1-73205081 1-442250 
 
 70 
 
 4900 
 
 343000 
 
 8-3666003 4-121285 
 
 4 
 
 16 
 
 64 
 
 1 20000000; 1-587401 
 
 71 
 
 5041 
 
 357911 
 
 8-4261498 4-140818 
 
 5 
 
 25 
 
 125 
 
 1 2-23606801 1-709976 
 
 72 
 
 5184 
 
 373-248 
 
 8-4852814 4-160168 
 
 6 
 
 35 
 
 216 
 
 2 -44948971 1-817121! 
 
 73 
 
 5329 
 
 389017 
 
 8-5440037 4-179339 
 
 7 
 
 49 
 
 343 
 
 : 2-6457513^ 1-912931 i 
 
 74 
 
 5476 
 
 405224 
 
 8-6023253 4-198336 
 
 8 
 
 64 
 
 512 
 
 ; 2-82842711 2-000000: 
 
 75 
 
 5625 
 
 421875 
 
 8-6602540 4 217163 
 
 9 
 
 81 
 
 729 
 
 i 30000000J 2-080084' 
 
 76 
 
 5776 
 
 438976 
 
 1 8-7177979 4^35824 
 
 10 
 
 100 
 
 1000 
 
 { 3-162-2777i 2-154435 
 
 77 
 
 5929 
 
 456533 
 
 8-7749644 4-254321 
 
 11 
 
 121 
 
 1331 
 
 3-3 166243; 2-223980! 
 
 78 
 
 6084 
 
 474552 
 
 8-8317609 4-272659 
 
 12 
 
 144 
 
 1728 
 
 3-4641016 2-289429 
 
 79 
 
 6241 
 
 493039 
 
 8-8381944 4-29t3840 
 
 13 
 
 169 
 
 2197 
 
 3-6055513 2351335 
 
 80 
 
 6400 
 
 512000 
 
 8-944-2719! 4-30H869 
 
 14 
 
 196 
 
 2744 
 
 : 3-7416574 2-410142 
 
 81 
 
 6561 
 
 531441 
 
 9-0000000 
 
 i-'s^S'lVJ 
 
 15 
 
 225 
 
 3375 
 
 ; 3-8:29833 2-466212 
 
 82 
 
 6724 
 
 551368 
 
 9-0553851 
 
 4-344481 
 
 16 
 
 256 
 
 4096 
 
 4-00;)0000 
 
 2-519342 
 
 83 
 
 6839 
 
 571787 
 
 9-1104336 
 
 4-36-2071 
 
 17 
 
 289 
 
 4913 
 
 41231056 
 
 2-571232 
 
 84 
 
 7056 
 
 592704 
 
 9-1651514 
 
 4-379519 
 
 18 
 
 324 
 
 5832 
 
 4-2426407 
 
 2-620741 
 
 85 
 
 7225 
 
 6141-25 
 
 9-2195445 
 
 4-396830 
 
 19 
 
 361 
 
 6859 
 
 4-358-i989 
 
 2-668402 
 
 86 
 
 7396 
 
 636056 
 
 9-2736185 
 
 4-414005 
 
 20 
 
 400 
 
 8000 
 
 4-4721360 
 
 2-714418 
 
 87 
 
 7569 
 
 658503 
 
 9-3273791 
 
 4-131048 
 
 21 
 
 441 
 
 9261 
 
 4-58-25757 
 
 2-758924 
 
 88 
 
 7744 
 
 631472 
 
 9-3808315 
 
 4-447960 
 
 22 
 
 484 
 
 10648 
 
 4-6901158 2-802039 
 
 89 
 
 7921 
 
 704969 
 
 9-4339811 
 
 4-464745 
 
 23 
 
 529 
 
 12167 
 
 4-7958315 
 
 2-843S67 
 
 90 
 
 8100 
 
 729000 
 
 9-4888330 
 
 4-481405 
 
 24 
 
 576 
 
 13824 
 
 4-8939795 
 
 2-881499!, 
 
 91 
 
 8-281 
 
 753571 
 
 9-5393920 
 
 4-497941 
 
 25 
 
 625 
 
 15625 
 
 5-0000000 
 
 2-924018 
 
 92 
 
 8464 
 
 778638 
 
 9-5916630 
 
 4 514357 
 
 26 
 
 676 
 
 17576 
 
 5-0990195 
 
 2-962496 
 
 93 
 
 8649 
 
 8-04357 
 
 9-6436508 
 
 4-6^0655 
 
 27 
 
 729 
 
 19683 
 
 5 196 1524 
 
 3 000000 
 
 94 
 
 8836 
 
 830534 
 
 9-6953597 
 
 4 546836 
 
 28 
 
 784 
 
 21952 
 
 5 29151)26 
 
 3036589 
 
 95 
 
 9025 
 
 857375 
 
 9-7467943 
 
 4-562903 
 
 29 
 
 841 
 
 24389 
 
 5-3351648 
 
 3072317:' 
 
 96 
 
 9216 
 
 884736 
 
 9-7979590 
 
 4-578357 
 
 30 
 
 900 
 
 27000 
 
 5-4772256 
 
 3-107-232! 
 
 97 
 
 9409 
 
 912673 
 
 9-8483578 
 
 4-59i701 
 
 3i 
 
 961 
 
 29791 
 
 5 5677644 
 
 31413ji; 
 
 98 
 
 9604 
 
 941192 
 
 9-8994949 
 
 4-G.10436 
 
 32 
 
 1024 
 
 3-2768 
 
 5-6568542 
 
 3-174802 : 
 
 99 
 
 9301 
 
 970299 
 
 9-9493744 
 
 4-625065 
 
 33 
 
 1089 
 
 35937 
 
 5-7445626 
 
 3-207534 
 
 100 
 
 10000 
 
 1000000 
 
 10-0000000! 4 641589 
 
 ^ 
 
 1156 
 
 39304 
 
 5-8309519 
 
 3 239612 
 
 101 
 
 10201 
 
 1030301 
 
 10-04937561 4-657009 
 
 35 
 
 1225 
 
 42875 
 
 5-9160798 
 
 3-271066 
 
 102 
 
 10404 
 
 1061-208 
 
 10-09950491 4-672329 
 
 36 
 
 1296 
 
 - ,. - ^ 
 
 46656 
 
 6 0000000 
 
 3-301927, 
 
 103 
 
 10609 
 
 1092727 
 
 10- 14889 1G[ 4-687548 
 
 37 
 
 1369 
 
 50653 
 
 0-OS27625 
 
 3-332222; 
 
 104 
 
 10816 
 
 1124864 
 
 10-1980390' 4-702669 
 
 38 
 
 1444 
 
 54S72 
 
 6-1644140 
 
 3-361975 
 
 105 
 
 110-25 
 
 1157625 
 
 10-2469508^ 4-717694 
 
 '^•j 
 
 *; 
 
 1521 
 
 59319 
 
 6-24 19980 3-391211 
 
 106 
 
 11236 
 
 1191016 
 
 10-2956301: i -732623 
 
 40 
 
 1600 
 
 61000 
 
 6-3245553 3 419952 
 
 107 
 
 11449 
 
 1225043 
 
 10-314080'i; 4-747459 
 10-3923048 4-76-2203 
 
 i^ 
 
 1681 
 
 68921 
 
 6-4U31212 3 U8217 
 
 108 
 
 11664 
 
 1259712 
 
 42 
 
 1764 
 
 74088 
 
 6-4807407 3-476027i, 
 
 109 
 
 11381 
 
 1295029 
 
 10-4403065' -,-776856 
 
 43 
 
 1319 
 
 79507 
 
 6-5574385 3-503398 
 
 110 
 
 12100 
 
 1331000 
 
 10-4880835 4-7914-20 
 
 44 
 
 1936 
 
 85184 
 
 6-6332496 3-53;J3l8 
 
 111 
 
 12321 
 
 1367631 
 
 10-53565381 4-805895 
 
 45 
 
 2025 
 
 91125 
 
 6-7032039 3-556893 
 
 112 
 
 12544 
 
 1404928 
 
 10-583O052' 4 •820-284 
 
 46 
 
 2116 
 
 97335 
 
 6-7823300 3-583J48^ 
 
 113 
 
 12769 
 
 144-2897 
 
 10-63014581 4-834538 
 
 4; 
 
 2209 
 
 103823 
 
 6-8556548 3-(k)8S2"j 
 
 114 
 
 12996 
 
 1431514 
 
 10-67707.^^3 4-8iS308 
 
 48 
 
 2304 
 
 1 10592 
 
 6-9282032 3-634241 
 
 115 
 
 13-225 
 
 1520875 
 
 10 7-238053: 4-862944 
 
 49 
 
 2401 
 
 117649 
 
 7-0000000 3-659306 
 
 116 
 
 13455 
 
 1560896 
 
 10-77032v^6; 4-876999 
 
 5U 
 
 2500 
 
 125000 
 
 7-0710678 3-6 S 1031 
 
 117 
 
 13689 
 
 1601613 
 
 10-816!;53.Si 4-890973 
 
 51 
 
 26U1 
 
 132651 
 
 7-1414284 3-70843 j' 
 
 118 
 
 13924 
 
 1643032 
 
 10-8627805' 4-904868 
 
 52 
 
 2704 
 
 140608 
 
 7-2111026 3-732511 
 
 119 
 
 14161 
 
 1685159 
 
 10-9087121J 4-918635 
 10-9544512 4-9324-24 
 
 53 
 
 2309 
 
 14S877 
 
 7-2S01099 3-756285 
 
 120 
 
 14400 
 
 1728000 
 
 54 
 
 2916 
 
 157464 
 
 7-3481692 3779763 i 
 
 121 
 
 14641 
 
 1771561 
 
 11-0000001) 4-946037 
 
 55 
 
 3025 
 
 166375 
 
 7-4161985 3-502952 
 
 122 
 
 14884 
 
 1815348 
 
 ll-0453i>lO 4'959676 
 
 56 
 
 3136 
 
 175616 
 
 7-483314S 3-825862i 
 
 123 
 
 15129 
 
 1860867 
 
 11-0905365 4-973190 
 
 57 
 
 3249 1 
 
 185193 
 
 7-5498344 3-848501! 
 
 124 
 
 15376 
 
 19066-24 
 
 11-1355-287 4-986631 
 
 58 
 
 3364 
 
 195112 
 
 7-6157731 
 
 3-8708771 
 
 125 
 
 156-25 
 
 1953125 
 
 11-1803399 5-000000 
 
 59 
 
 3481 
 
 205379 
 
 7-68114571 
 
 3-89-2996 1 
 
 1-26 
 
 15376 
 
 2000376 
 
 11-2249723 5-013298 
 
 60 
 
 3600 
 
 216000 
 
 7-7459667, 
 
 3-914868! 
 
 127 
 
 16129 
 
 2048383 
 
 11-2694277 
 
 5-026526 
 
 61 
 
 3721 
 
 226981 
 
 7-8102497i 
 
 3-936497ii 
 
 128 
 
 16334 
 
 2097152 
 
 11-3137085 
 
 5 039681 
 
 62 
 
 3^44 
 
 238328 
 
 7-87400791 
 
 3-957891!' 
 
 129 
 
 16641 
 
 2146689 
 
 11-3578167 
 
 5-052774 
 
 63 
 
 3969 
 
 250047 
 
 7-9372539' 
 
 3-979057!' 
 
 130 
 
 16900 
 
 2197000 
 
 11-4017543 
 
 5 065797 
 
 64 
 
 40% 
 
 262144 
 
 8-0000000 
 
 4-000000! i 
 
 131 
 
 17161 
 
 224Sim\ 
 
 1 J -4455231 
 
 5-07y753 
 
 65 
 
 4^25 
 
 274r)25 
 
 8-06225771 
 
 4-020726! 
 
 132 
 
 17424 
 
 2299968; 
 
 n-4891253 
 
 5-0916i3 
 
 66 
 
 4356 
 
 2874'j6 
 
 8-1240384 
 
 4-041-240 1 
 
 133 
 
 17689! 
 
 2352637 
 
 11.53-25626 
 
 5-104469 
 
 67 
 
 4489 
 
 300763 
 
 8-1853528 
 
 4-061548 1 
 
 134 
 
 17956' 
 
 2406104 
 
 11-57583691 
 
 511T280 
 
APPENDIX. 
 
 15 
 
 No. 
 
 Square. 
 
 Cube. 
 
 Sq. Root. CubeRooU 
 
 No. 
 
 Square. 
 
 Cube. 
 
 Sq. Root. 
 
 CubeRoot. 
 
 135 
 
 18225 
 
 2160375 
 
 11-6189500 5-129928 
 
 202 
 
 4U804 
 
 8242408 
 
 14-2126704 
 
 5-867464 
 
 136 
 
 18496 
 
 2515456 
 
 11-661903S 5-142563 
 
 203 
 
 41209 
 
 8365427 
 
 14-2478068 
 
 5-877131 
 
 137 
 
 18769 
 
 2571353 
 
 11-7046999 
 
 5-155137 
 
 204 
 
 41616! 84896641 14-2o23569l 5-836765| 
 
 133 
 
 39044 
 
 2628072 
 
 11-7473401 
 
 5-167649 
 
 205 
 
 42025 
 
 8615125 
 
 1 14-31782111 5-3963681 
 
 139 
 
 19321 
 
 2635619 
 
 11-7893-261 
 
 5-180101 
 
 206 
 
 4243t 
 
 8741816 
 
 1 14-3527001 
 
 5-905941 
 
 140 
 
 19600 
 
 2744000 
 
 11-8321596 
 
 5-19-2494 
 
 207 
 
 42849 
 
 8869743 
 
 1 14-3874946 
 
 5-915482 
 
 141 
 
 li)881 
 
 2803221 
 
 11-8743422 
 
 5 •2043-28 
 
 208 
 
 43264 
 
 8998912! 14-4222051! 5 9249921 
 
 142 
 
 20164 
 
 2S63288 
 
 11-9163753 
 
 5-217103 
 
 209 
 
 43681 
 
 9129329 
 
 1 14-4568323! 5-934473 
 
 143 
 
 20449 
 
 2924207 
 
 11-9532607 
 
 5-229321 
 
 210 
 
 44100 
 
 9-261000 
 
 14-4913767 5943922 
 
 144 
 
 2,)736 
 
 2985934 
 
 12-0000000 
 
 5-241483 
 
 211 
 
 44521 
 
 9393931 
 
 1 14-52583901 5-953342 
 
 145 
 
 al025 
 
 3043625 
 
 12 0415946 
 
 5-253533 
 
 212 
 
 44944 
 
 9528128 
 
 I 14-5602198 5-96-2732 
 
 146 
 
 21316 
 
 3112136 
 
 12-0830460 
 
 5-265637 
 
 213 
 
 45369 
 
 9663597 
 
 1 14-5945195 5-972093 
 
 147 
 
 21609 
 
 31715523 
 
 12-1243557 
 
 5-277632! 214 
 
 45796 
 
 9300344 
 
 i 14-6-287338! 5-931424 
 
 148 
 
 21904 
 
 3211792 
 
 12-1655251 
 
 5-239572 215 
 
 46225 
 
 9933375 
 
 14-6623783: 5990726 
 
 149 
 
 22201 
 
 3307949 
 
 12-2065556 
 
 5 301459 1 216 
 
 46656 
 
 10077696 
 
 14-6969385; 6-000000 
 
 150 
 
 22500 
 
 3375000 
 
 12-2174437 
 
 5-313293i 217 
 
 47089 
 
 10218313 
 
 14-7309199 
 
 6-009245 
 
 151 
 
 22S01 
 
 3442951 
 
 12-2832057 
 
 5-325074 218 
 
 47524 
 
 10^360232 
 
 14-7613231 
 
 6-013462 
 
 152 
 
 23104 
 
 3511803 
 
 12-3238280 
 
 5-336803 219 
 
 47961 
 
 10503459 
 
 14-7986486 
 
 6-027650 
 
 153 
 
 23409 
 
 3581577 
 
 12-3693169 
 
 5-318481' 220 
 
 48400 
 
 10643000 
 
 14-8323970 
 
 6-036811 
 
 154 
 
 23716 
 
 3652264 
 
 12-4096736 
 
 5-36010S:j 221 
 
 43S41 
 
 10793861 
 
 14-8660687i 6-045943 
 
 155, 
 
 24025 
 
 3723375 
 
 12-449399.'- 
 
 5-37I6S5i 222 
 
 49284 
 
 10941048 
 
 14-8996644! 6-055049 
 
 156 
 
 2433?i 
 
 3796416 
 
 12-4399J60 
 
 5-333213 
 
 ; 223 
 
 49729 
 
 11089567 
 
 14-93318451 6064127 
 
 157 
 
 24649 
 
 3869393 
 
 12-5299641 
 
 5-394691; 
 
 224 
 
 50176 
 
 1 1239424 
 
 14-9666295 6073178 
 
 158; 
 
 24964 
 
 3944312 
 
 12-5698051 
 
 5-406120i 
 
 225 
 
 50625 
 
 11390625 
 
 15-0000000 6-082202 
 
 159! 
 
 25281 
 
 4019679 
 
 12-6095202 
 
 5-417501 
 
 226 
 
 51076 
 
 11513176 
 
 150332964! 6-091199 
 
 160; 
 
 25600 
 
 4096000 
 
 12-6491106 
 
 5 ■428335; 1 227 
 
 51529 
 
 11697083 
 
 15-0665192 6-100170 
 
 16i 
 
 25921 
 
 4173281 
 
 12-6335775 
 
 5 -440 1-22: 223 
 
 51984 
 
 11852352 
 
 15-09966391 6-109115 
 
 1621 
 
 25244 
 
 4251523 
 
 12-7279-221 
 
 5-451362: 229 
 
 5-2441 
 
 12008939 
 
 151327460 6-118033 
 
 163' 
 
 26569 
 
 4330747 
 
 12-7671153 
 
 5-46-2556 230 
 
 52900 
 
 12167000 
 
 I51657509i 6-126926 
 
 164j 
 
 26896 
 
 4410944 
 
 12-8062485 
 
 5-473704 i 231 
 
 53361 
 
 12326391 
 
 15- 193634-2! 6-135792 
 
 165 
 
 27225 
 
 4492125 
 
 12-8452326 
 
 5-434807:; 232 
 
 53324 
 
 12487163 
 
 15-2315462! 6-144634 
 
 166 
 
 S7556 
 
 4574296 
 
 12-8840987 
 
 5-495365 ; 233 
 
 51-289 
 
 12649337 
 
 15-2643375 6-153449 
 
 1671 
 
 27SS9 
 
 4657463 
 
 12 9228480 
 
 5-506878 i 234 
 
 54756 
 
 12812904 
 
 15-2J70585' 6-16-2240 
 
 1631 
 
 28224 
 
 4741632 
 
 12-9614814 
 
 5 -5 17848 235 
 
 55225 
 
 12977875 
 
 15-3297097! 6-171006 
 
 169! 
 
 2S561 
 
 4826809 
 
 13 0000000 
 
 5-523775;! 236 
 
 55696 
 
 13144256 
 
 15 3622915' 6-179747 
 
 170' 
 
 %3d00 
 
 4913000 
 
 13-0384048 
 
 5-539653;: 237 
 
 56169 
 
 13312053 
 
 15-3948013 
 
 6-183463 
 
 17l! 
 
 2nn 
 
 5000211 
 
 13-0766968 
 
 5 •550499:; 233 
 
 56644 
 
 13181272 
 
 15 4272136 
 
 6-197154 
 
 172: 
 
 2'J^84 
 
 5033448 
 
 13-1143770 
 
 5-561293:1 239 
 
 57121 
 
 13651 J I'j 
 
 15-4596248 
 
 6-205822 
 
 173 
 
 29929 
 
 5177717J 13-1529464 
 
 5-572055 I 240 
 
 57600 
 
 1332401)0 
 
 15-4919334 
 
 6-214465 
 
 174: 
 
 30276 
 
 5268024' 13-1909060 
 
 5 53-2770:. 241 
 
 53081 
 
 139.)7521 
 
 15-5211747 
 
 6-223034 
 
 175 
 
 30625 
 
 5359375 13-2237568 
 
 5-593445! 242 
 
 53564 
 
 14172433 
 
 15-5563192 
 
 6-231630 
 
 176; 
 
 30976 
 
 5451776 13-2664992 
 
 5-604079;; 243 
 
 59049 
 
 14318907 
 
 15-5384573 
 
 6-240-251 
 
 1771 
 
 31329 
 
 5545233 133041347 
 
 5-614672!! 244 
 
 59536 
 
 145-26734 
 
 15-6-204994 
 
 6-243800 
 
 178; 
 
 3168i 
 
 5639752 13-3416641 
 
 5-625226i 245 
 
 60025 
 
 147061-^5 
 
 15-6524753 
 
 6-257325 
 
 179i 
 
 32041 
 
 5735339 13-3790832 
 
 5-635741! 246 
 
 00516 
 
 14836936 
 
 15-6843371 
 
 6-265327 
 
 1801 
 
 32100 
 
 5832000 
 
 13-4164079 
 
 5-646216;: 247 
 
 6100;* 
 
 15069223 
 
 15-7162336 
 
 6-274305 
 
 181! 
 
 32761 
 
 59-29741 
 
 13-4536-240 
 
 5-656653 1 248 
 
 61504 
 
 15252992 
 
 15-7480157 
 
 6-232761 
 
 182i 
 
 33124 
 
 6023568 
 
 134907376 
 
 5-6670sIi' 249 
 
 6-2001 
 
 15433249 
 
 15-7797338 
 
 6-291195 
 
 1831 
 
 334S9 
 
 6128487 
 
 13-5277493 
 
 5-677411; 250 
 
 6-2500 
 
 15625000 
 
 15-8113333 
 
 6-299605 
 
 184! 
 
 33336 
 
 62295U4 
 
 13-5646600 
 
 5-637734; 251 
 
 63001 
 
 15313^51 
 
 15-842)795 
 
 6-3J7994 
 
 185 
 
 34225 
 
 6331625 
 
 136014705 
 
 5-6930l9|; 252 
 
 63504 
 
 16003008 
 
 15-8745079 
 
 6-316360 
 
 18r. 
 
 34596 
 
 6434356 
 
 13-6331317 
 
 5-708267 253 
 
 64009 
 
 16194277 
 
 15-9059737 
 
 6-324704 
 
 187 
 
 34969 
 
 6539203 
 
 13-6747943 
 
 5-718479|i 254' 
 
 64516 
 
 16337064 
 
 15-9373775 
 
 6-333026 
 
 183 
 
 35344 
 
 6644672 
 
 13-7113092 
 
 5-723654!. 255' 
 
 650-25 
 
 16581375 
 
 15-9687194 
 
 6-341326 
 
 189; 
 
 35721 
 
 6751269 
 
 13-7477271 
 
 5-733794!! 256! 
 
 65536 
 
 16777216 
 
 16-0000090 
 
 6-349604 
 
 190 
 
 36100 
 
 6859000 
 
 13-7840433 
 
 5-748397; 
 
 257 
 
 66049 
 
 16J74593 
 
 16-0312195 
 
 6 357861 
 
 191! 
 
 364-31 
 
 6967871 
 
 13-8202750 
 
 5-753965 
 
 253 
 
 66564 
 
 17173312 
 
 16-0623734 
 
 6-366097 
 
 19-i 
 
 36864 
 
 7077833' 
 
 13-8564065 
 
 5-763993 
 
 259 
 
 67031 
 
 17373979 
 
 16-0934769 
 
 6-374311 
 
 193i 
 
 372i9 
 
 7189057; 
 
 13-8924 i40' 
 
 5-778996 
 
 260! 
 
 67G09 
 
 17576000 
 
 16-1245155 
 
 6-38-2504 
 
 1941 
 
 37,i36 
 
 7301334I 
 
 13-9283883 
 
 5-783960 
 
 261 
 
 63121 
 
 17779581 
 
 16-1554944 
 
 6-390676 
 
 195; 
 
 38025 
 
 7414875 
 
 13-964-2400 
 
 5-793390 
 
 262 
 
 6-644 
 
 17984723 
 
 16-1861141 
 
 6-398823 
 
 1%: 
 
 3S416 
 
 75-29536 
 
 14-0000000 
 
 5-308786 
 
 263; 
 
 69169 
 
 18191447 
 
 16-2172747 
 
 6-406958 
 
 197: 
 
 38809 
 
 7645373 
 
 14-0356683 
 
 5-818648 
 
 264| 
 
 69C)96 
 
 18399744 
 
 16-2480768 
 
 6-415069 
 
 19Si 
 
 39204 
 
 7762392 
 
 14-0712473 
 
 5-323477 
 
 265 
 
 70225 
 
 18609625 
 
 16-2738206 
 
 6-4231D8 
 
 1991 
 
 39601 
 
 7880599 
 
 14-1067360 
 
 5-833272 
 
 266 
 
 70756 
 
 18821096 
 
 16-3095061 
 
 6-431223 
 
 201.' 
 
 40000 
 
 8000000 
 
 14-1421356 
 
 5 -818035 
 
 267 
 
 71289 
 
 1U034163 
 
 16-3401346 
 
 6-439277 
 
 201 
 
 40401 
 
 8120601 
 
 14-1774469 
 
 5-857766 
 
 268 
 
 71824 
 
 19248832 
 
 16-3707055 
 
 6-447306 
 
16 
 
 APPENDIX 
 
 No. 
 
 j Square. 
 
 Cube. 
 
 Sq. Root. 
 
 CubeRoot. 
 
 1 
 
 No. 
 
 Square. 
 
 Cube. 
 
 Sq Root. 
 
 CubeRoo*. 
 
 269 
 
 72361 
 
 19165109 
 
 16-4012195 
 
 6-455315' 
 
 336 
 
 1 12896 
 
 37933055 
 
 18-3303028 
 
 6-952053 
 
 270 
 
 72900 
 
 19633000 
 
 16-4316767 
 
 6-463304' 
 
 337 
 
 113559 
 
 38-272753 
 
 18-3575598 
 
 6-958943 
 
 271 
 
 73441 
 
 19902511 
 
 16-46-20776 
 
 6-471274 
 
 33:5 
 
 114^44 
 
 33<)14472 
 
 18-3847753 
 
 6-965320 
 
 272 
 
 73984 
 
 20123618 
 
 16-49-24225 
 
 6-4792-24 
 
 339 
 
 114921 
 
 38953219 
 
 18-4119526 
 
 6-97-2683 
 
 273 
 
 74529 
 
 20346417 
 
 16-5227116 
 
 6-437151' 
 
 310 
 
 11550;; 
 
 39304000 
 
 18-4390839 
 
 6979532 
 
 274 
 
 75076 
 
 20570824 
 
 16-55-29454 
 
 6-495065 
 
 341 
 
 110231 
 
 39551821 
 
 184661853 
 
 6-936368 
 
 275 
 
 75625 
 
 20796375 
 
 16-5331240 
 
 6-502957 
 
 342 
 
 116964 
 
 40001633 
 
 18-4932420 
 
 6-993191 
 
 276 
 
 76176 
 
 21024576 
 
 16-6132177 
 
 6-510330| 
 
 343 
 
 117649 
 
 40353607 
 
 18-5202592 
 
 7 000000 
 
 277 
 
 75729 
 
 21253933 
 
 16-6433170 
 
 6-5 135 ill 
 
 314 
 
 1 13336 
 
 40707531 
 
 18-5472370 
 
 7-006796 
 
 278 
 
 77234 
 
 21484952 
 
 16-67333-20 
 
 6-5-25519, 
 
 345 
 
 1 19025 
 
 4105.3625 
 
 18-5741756 
 
 7-013579 
 
 279 
 
 77311 
 
 2171763.) 
 
 16-7032931 
 
 6534335! 
 
 346 
 
 119716 
 
 41421736 
 
 18-6!)1()752 
 
 7-0-20349 
 
 280 
 
 784(X) 
 
 21952000 
 
 16-7332005 
 
 6-512133 
 
 317 
 
 12)409 
 
 41781923 
 
 18-6279360 
 
 7027106 
 
 281 
 
 78961 
 
 22183041 
 
 167630546 
 
 6-5499 12' 
 
 34^ 
 
 121104 
 
 42144192 
 
 18 6517531 
 
 7-033350 
 
 282 
 
 79524 
 
 2242576^ 
 
 16-7923556 
 
 6-557672 
 
 349 
 
 121801 
 
 425 )3549 
 
 13-6815417 
 
 7-040531 
 
 283 
 
 80089 
 
 22665187 
 
 16 822603-^ 
 
 6-555414' 
 
 350 
 
 1-22500 
 
 42375 )00 
 
 18-7032859 
 
 7-047299 
 
 284 
 
 80656 
 
 229063>1 
 
 16 8522995 
 
 6-573139 
 
 351 
 
 123-29! 
 
 43243551 
 
 13-7319940 
 
 7-051(M)4 
 
 235 
 
 81225 
 
 23149125 
 
 16-33 !9 130 
 
 6-5<0>j4' 
 
 352 
 
 123901 
 
 43614208 
 
 18-761663) 
 
 7-060697 
 
 286 
 
 81796 
 
 23393656 
 
 16 9115315 
 
 6 5SS532I 
 
 353 
 
 1215:)9 
 
 43-J3-.977 
 
 18-733-2942 
 
 7-067377 
 
 287 
 
 82369 
 
 ^23639903 
 
 16-9410743 
 
 6 5'J5202' 
 
 354 
 
 125316 
 
 44361864 
 
 18-8148877 
 
 7074044 
 
 238 
 
 82944 
 
 23337872 
 
 16-9705527 
 
 6-t;;.3;54; 
 
 355 
 
 126025 
 
 44733875 
 
 18-8414437 
 
 7-030699 
 
 289 
 
 83V21 
 
 241375G9 
 
 170000000 
 
 6-511439 
 
 356 
 
 126736 
 
 45118016 
 
 18-8679623 
 
 7-037341 
 
 290 
 
 84100 
 
 24389000 
 
 17-0293364 
 
 6-519106 
 
 357 
 
 127449 
 
 45199293 
 
 18-8944436 
 
 7-093)7] 
 
 291 
 
 84681 
 
 24642171 
 
 17-0537221 
 
 6-626705 
 
 353 
 
 128164 
 
 45382712 
 
 18-92')8379 
 
 7-K)0533 
 
 292 
 
 85264 
 
 24897083 
 
 17-0830075 
 
 6-53 12 S7. 
 
 359 
 
 123881 
 
 46263279 
 
 18-9472953 
 
 7-107194 
 
 293 
 
 85349 
 
 251*^3757 
 
 17-1172423 
 
 6 641352 
 
 350 
 
 129600 
 
 46656000 
 
 18-9736660 
 
 7-113737 
 
 294 
 
 86136 
 
 25112184 
 
 17-14542^2 
 
 u-549400 
 
 361 
 
 130321 
 
 470 15 -331 
 
 19-0000000 
 
 7-120367 
 
 295 
 
 87025 
 
 25672375 
 
 171755640 
 
 6C)56930 
 
 352 
 
 131041 
 
 47437928 
 
 19 0262976 
 
 7-126935 
 
 296 
 
 87616 
 
 25934336 
 
 l7-20465>)5 
 
 6-664444; 
 
 363 
 
 131769 
 
 47832147 
 
 19-0525589 
 
 7-133492 
 
 297 
 
 83209 
 
 26198073 
 
 17-2336379 
 
 6-571940 
 
 354 
 
 132495 
 
 43223544 
 
 19-0737840 
 
 7-1400.37 
 
 293 
 
 83304 
 
 261G3592 
 
 17-2626765 
 
 6-67912) 
 
 355 
 
 133225 
 
 43627125 
 
 19-1049732 
 
 7-146569 
 
 299 
 
 89401 
 
 26730899 
 
 17-2916165 
 
 6-685333 
 
 355 
 
 133.%6 
 
 49027395 
 
 19-1311265 
 
 7 15309^^ 
 
 300 
 
 90000 
 
 270000CO 
 
 17-3205031 
 
 6-69432.) 
 
 357 
 
 134639 
 
 49430863 
 
 19-1572441 
 
 7-15959^^ 
 
 301 
 
 90601 
 
 27270901 
 
 17-3493516 
 
 6-701759 
 
 363 
 
 135424 
 
 49835032 
 
 lv^-1833261 
 
 7-166095 
 
 302 
 
 91204 
 
 27543603 
 
 17-3731472 
 
 6-70.) 173 
 
 369 
 
 136161 
 
 5)243409 
 
 19 2093727 
 
 7-172531 
 
 303 
 
 91809 
 
 27318127 
 
 17-4063952 
 
 6-71557^), 
 
 370 
 
 135900 
 
 5065390',) 
 
 19-2353341 
 
 7-179054 
 
 304 
 
 92416 
 
 28094464 
 
 17-4355953 
 
 6-723.}5!| 
 
 371 
 
 137641 
 
 51054311 
 
 19-2613803 
 
 71855iC) 
 
 305 
 
 93025 
 
 28372625 
 
 17-4642492 
 
 6-7313151 
 
 372 
 
 133334 
 
 51473343 
 
 19-2373)15 
 
 7-191965 
 
 3)6 
 
 93636 
 
 28652616 
 
 17-4923557 
 
 6-73>!654' 
 
 373 
 
 139129 
 
 51895117 
 
 19-31.32079 
 
 7-193405 
 
 307 
 
 94249 
 
 28934443 
 
 17-5214155 
 
 6-745997, 
 
 374 
 
 139376 
 
 523136-24 
 
 19-3390796 
 
 7-204332 
 
 308 
 
 94864 
 
 29218112 
 
 17-5499-233 
 
 6-753313, 
 
 375 
 
 1405-25 
 
 52734375 
 
 19-3649167 
 
 7-2 1 1248 
 
 309 
 
 95481 
 
 29533529 
 
 17 573335 -i 
 
 6-76:!6l4l 
 
 376 
 
 in 376 
 
 53157376 
 
 19-39' >7194 
 
 7-217652 
 
 310 
 
 96100 
 
 2979101)0 
 
 17-6063169 
 
 6-7'6739.7 
 
 37: 
 
 142129 
 
 535S-2f,33 
 
 19-4164878 
 
 7-224045 
 
 311 
 
 96721 
 
 3(V)30231 
 
 17-635i;>21 
 
 6-77515:)' 
 
 378 
 
 1423-^4 
 
 54010152 
 
 19-44-22221 
 
 7-23)427 
 
 312 
 
 97344 
 
 30371323 
 
 17-6635217 
 
 6-732123 
 
 379 
 
 143541 
 
 54439939 
 
 19-4679223 
 
 7-236797 
 
 313 
 
 97969 
 
 30664297 
 
 17-6913060 
 
 6-739661 
 
 330 
 
 144490 
 
 5437-21)00 
 
 19-4935337 
 
 7-213156 
 
 3U 
 
 98596 
 
 30959144 
 
 17-7200451 
 
 6-795831 
 
 3S1 
 
 145161 
 
 5530C)3il 
 
 19-519-2213 
 
 7-249504 
 
 315 
 
 99225 
 
 31255375 
 
 17-74323J3 
 
 6-304;);!2 
 
 332 
 
 145.124 
 
 55742963 
 
 19-54432i)3 
 
 7-255311 
 
 31B 
 
 99856 
 
 315.54496 
 
 17-7763S-33 
 
 6-31 12 35 
 
 333 
 
 1466 39 
 
 56131.387 
 
 19-5703353 
 
 7-262167 
 
 317 
 
 100489 
 
 31855013 
 
 17-804493S 
 
 6-313462 
 
 384 
 
 147456 
 
 55623104 
 
 19-5959179 
 
 7-263432 
 
 318 
 
 101124 
 
 32157432 
 
 17-8325545 
 
 6-8-25524 
 
 3^5 
 
 143225 
 
 57066625 
 
 19-6214169 
 
 7-274786 
 
 319 
 
 101761 
 
 32461759 
 
 17-3605711 
 
 6-832771 
 
 336 
 
 143996 
 
 57512456 
 
 19-6463327 
 
 7-231079 
 
 320 
 
 102400 
 
 32763000 
 
 17-83^5t33 
 
 6339.)04 
 
 337 
 
 149769 
 
 57950603 
 
 19 67-23156 
 
 7-287362 
 
 321 
 
 103041 
 
 33076161 
 
 17-9164729 
 
 6-347-)21 
 
 338 
 
 1.50514 
 
 58 11 1072 
 
 19-6977156 
 
 7 293633 
 
 322 
 
 103684 
 
 333362 J3 
 
 17-9443534 
 
 6 854124 
 
 339 
 
 151321 
 
 53^^63859 
 
 19-72308-29 
 
 7-299394 
 
 323 
 
 104329 
 
 33598267 
 
 17-9722008 
 
 6-88 !212 
 
 390 
 
 152100 
 
 59319000 
 
 19 7434177 
 
 7-306341 
 
 3i4 
 
 104976 
 
 34012224 
 
 18-OOOOOOi) 
 
 6-353235 
 
 391 
 
 15-23 SI 
 
 59776471 
 
 19-7737199 
 
 7 312333 
 
 325 
 
 105625 
 
 34323125 
 
 13-0277564 
 
 6-875344 
 
 392 
 
 153564 
 
 69236233 
 
 19-7989899 
 
 7-318611 
 
 326 
 
 106276 
 
 34645976 
 
 13-0551701 
 
 6-332339 
 
 393 
 
 154449 
 
 60593457 
 
 19-3-24-2-276 
 
 7-32 48-29 
 
 327 
 
 106929 
 
 34965733 
 
 180331413 
 
 6-339419 
 
 394 
 
 155236 
 
 61162934 
 
 19-8494332 
 
 7-331037 
 
 328 
 
 107581 
 
 35237552 
 
 13- 1107703 
 
 6-8.)6135 
 
 395 
 
 156925 
 
 6162.)375 
 
 19-8746069 
 
 7-337234 
 
 329 
 
 108241 
 
 35611239 
 
 18-1333571 
 
 6 903136 
 
 396 
 
 15->816 
 
 62099135 
 
 19-8997437 
 
 7-3434-20 
 
 330 
 
 108900 
 
 35937000 
 
 18-1659021 
 
 6-910423 
 
 397 
 
 157699 
 
 62570773 
 
 19-9243538 
 
 7-349597 
 
 331 
 
 109561 
 
 36264691 
 
 18-1931054 
 
 6-917395 
 
 398 
 
 158404 
 
 63044792 
 
 19 9499373 
 
 7-355752 
 
 332 
 
 110224 
 
 36594368 
 
 18-2208572 
 
 6-924355 
 
 399 
 
 159201 
 
 63521199 
 
 19-9749814 
 
 7361918 
 
 333 
 
 110839 
 
 35926037 
 
 13-2482 S76 
 
 6-9313)1 
 
 400 
 
 160000 
 
 64UO00UO 
 
 20-0000000 
 
 7 3630(;3 
 
 334 
 
 111556 
 
 37259704 
 
 18-2756569 
 
 6-933232 
 
 1 401 
 
 160301 
 
 64481201 
 
 1 20 0249844 
 
 7-374193 
 
 335 
 
 112225 
 
 37595375 
 
 18-3030052 
 
 6-945150 
 
 4'.)2 
 
 161604 
 
 64954308 
 
 ! 20 049-J377 
 
 7-330323 
 
APPENDIX. 
 
 17 
 
 No. 
 
 Square. 
 
 Cube. 
 
 Sq. Root. 
 
 CubeRooU 
 
 No. 
 
 Square. 
 
 Cube. 
 
 Sq. Root. 
 
 CubeRoot. 
 
 403 
 
 162409 
 
 65450827 
 
 20-0748599 
 
 7-386437 
 
 470 
 
 220900 
 
 103323000 
 
 21-6794834 
 
 7-774980 
 
 404 
 
 163216 
 
 65939264 
 
 200997512 
 
 7-392542 
 
 471 221841 
 
 104487111 
 
 21-7025344 
 
 7-780490 
 
 405 
 
 164025 
 
 66430125 
 
 20-1246118 
 
 7-398636 
 
 472 222784 
 
 105154048 
 
 21-7255610 
 
 7-785993 
 
 406 
 
 164836 
 
 66923416 
 
 20-1494417 
 
 7-404721 
 
 473 223729 
 
 105823817 
 
 21-7485632 
 
 7-791487 
 
 407 
 
 165n49 
 
 67419143 
 
 20-1742410 
 
 7-410795 
 
 474 
 
 224676 
 
 106496424 
 
 21-7715411 
 
 7-796974 
 
 408 
 
 166464 
 
 67917312 
 
 20-1990099 
 
 7-416859 
 
 475 
 
 225625 
 
 107171875 
 
 21-7944947 
 
 7-802454 
 
 409 
 
 167281 
 
 68417929 
 
 20-2237484 
 
 7-422914 
 
 476 
 
 226576 
 
 107850176 
 
 21-8174242 
 
 7-807925 
 
 410 
 
 168100 
 
 68921000 
 
 20-2484567 
 
 7-428959 
 
 477 
 
 227529 
 
 108531333 
 
 21-8403297 
 
 7-813389 
 
 411 
 
 168921 
 
 69426531 
 
 20-2731349 
 
 7-434994 
 
 478 
 
 228484 
 
 109215352 
 
 21-8632111 
 
 7-818846 
 
 412 
 
 169744 
 
 69934528 
 
 20-2977831 
 
 7-441019 
 
 479 
 
 229441 
 
 109902239 
 
 21-8860686 
 
 7-824294 
 
 413 
 
 170569 
 
 70444997 
 
 20-3224014 
 
 7-447034 
 
 480 
 
 230400 
 
 110592000 
 
 21-9089023 
 
 7-829735 
 
 414 
 
 171396 
 
 70957944 
 
 20-3469899 
 
 7-453040 
 
 431 
 
 231361 
 
 111284641 
 
 21-9317122 
 
 7-835169 
 
 415 
 
 173225 
 
 71473375 
 
 20-3715488 
 
 7-459036 
 
 482 
 
 232324 
 
 111980168 
 
 21-95449S4 
 
 7-840595 
 
 41f. 
 
 173056 
 
 71991296 
 
 20-3960781 
 
 7-465022 
 
 483 
 
 233289 
 
 112678537 
 
 21-9772610 
 
 7-846013 
 
 417 
 
 173889 
 
 72511713 
 
 ■20-4205779 
 
 7-470999 
 
 484 
 
 234256 
 
 113379904 
 
 22-0000000 
 
 7-851424 
 
 418 
 
 174724 
 
 73034632 
 
 20-4450483 
 
 7-476966 
 
 485 
 
 235225 
 
 114084125 
 
 22-02-27155 
 
 7-856828 
 
 419 
 
 175561 
 
 73560059 
 
 20-4694895 
 
 7-482924 
 
 486 
 
 236196 
 
 114791256 
 
 22 0454077 
 
 7-862224 
 
 420 
 
 176400 
 
 74088000 
 
 20-4939015 
 
 7-488872 
 
 487 
 
 237169 
 
 115501303 
 
 22-0680765 
 
 7-867613 
 
 421 
 
 177241 
 
 74618461 
 
 20-5182845 
 
 7-494811 
 
 488 
 
 238144 
 
 116214272 
 
 22-0907220 
 
 7-872994 
 
 422 
 
 178084 
 
 75151448 
 
 20-5426386 
 
 7-500741 
 
 489 
 
 239121 
 
 116930169 
 
 22-1133444 
 
 7-878368 
 
 423 
 
 178929 
 
 75686967 
 
 20-5669638 
 
 7-506661 
 
 490 
 
 240100 
 
 117649000 
 
 22 1359436 
 
 7-883735 
 
 424 
 
 179776 
 
 76225024 
 
 20-5912603 
 
 7-512571 
 
 491 
 
 241031 
 
 118370771 
 
 22-1585198 
 
 7-889095 
 
 425 
 
 180625 
 
 76765625 20-6155281 
 
 7-518473 
 
 492 
 
 212064 
 
 119095488 
 
 22-1810730 
 
 7-894447 
 
 426 
 
 181476 
 
 77308776 20-6397674 
 
 7-524365 
 
 493 
 
 243049 
 
 119823157 
 
 22-203d033 
 
 7-899792 
 
 427 
 
 182329 
 
 77854483 20-6639783 
 
 7-530248 
 
 494 
 
 244036 
 
 120553784 
 
 22-225 llO:i 
 
 7-905129 
 
 428 
 
 183184 
 
 78402752 20-6881609 
 
 7-536122 
 
 495 
 
 245025 
 
 121287375 
 
 22-2185955 
 
 7-910460 
 
 429 
 
 184041 
 
 78953589 20-7123152 
 
 7-541987 
 
 496 
 
 246016 
 
 122023936 
 
 22-2710575 
 
 7-915783 
 
 430 
 
 184900 
 
 79507000 20-7364414 
 
 7-547842 
 
 497 
 
 247009 
 
 122763473 
 
 22-2934968 
 
 7-921099 
 
 431 
 
 185761 
 
 80062991 20-7605395 
 
 7-553689 
 
 498 
 
 248004 
 
 123505992 
 
 2-2-315913r) 
 
 7-926403 
 
 432 
 
 1866*24 
 
 80621568 20-7846097 
 
 7-559526 
 
 499 
 
 249001 
 
 124-251499 
 
 22-3383079 
 
 7-931710 
 
 433 
 
 187489 
 
 81182737 20-8086520 
 
 7-565355 
 
 500 
 
 250000 
 
 125000000 
 
 22-3606798 
 
 7-937005 
 
 434 
 
 188356 
 
 81746504 20-8326667 
 
 7-571174 
 
 501 
 
 251001 
 
 125751501 
 
 22-383U293 
 
 7-942293 
 
 435 
 
 189225 
 
 82312875 20-8566536 
 
 7-576985 
 
 502 
 
 252001 
 
 126506008 
 
 22-4053565 
 
 7-947574 
 
 436 
 
 190096 
 
 82881856! 20-8806130 
 
 7-582786 
 
 503 
 
 253009 
 
 127263527 
 
 22-4276615 
 
 7-95-2848 
 
 437 
 
 190969 
 
 83453453j 20-9045450 
 
 7-588579 
 
 504 
 
 254016 
 
 128024064 
 
 22-4499443 
 
 7-953114 
 
 438 
 
 191844 
 
 84027672 20-9284495 
 
 7-5J4363 
 
 505 
 
 255025 
 
 128787625 
 
 22-4722051 
 
 7-963374 
 
 439 
 
 192721 
 
 84604519 20-9523268 
 
 7-600138 
 
 506 
 
 256036 
 
 129554216 
 
 22-4944438 
 
 7-9686-27 
 
 440 
 
 193600 
 
 85184000 20-9761770 
 
 7-605905 
 
 507 
 
 257049 
 
 13032.3843 
 
 22-51666U5 
 
 7-973873 
 
 441 
 
 194481 
 
 857661211 21-0000000 
 
 7-611663 
 
 508 
 
 253064 
 
 13109b512 
 
 22-5388553 
 
 7-97911-4 
 
 4421 195364 
 
 86350888: 21-0237960 
 
 7-617412 
 
 509 
 
 259081 
 
 131872229 
 
 22-5610283 
 
 7-984344 
 
 443 196249 
 
 86938307 21-0475652 
 
 7-623152 
 
 510 
 
 260100 
 
 132651000 
 
 22-5331796 
 
 7-989570 
 
 444 
 
 197136 
 
 87528384i 21-0713075 
 
 7-628884! 511 
 
 261121 
 
 133432831 
 
 22-6053091 
 
 7-994788 
 
 445 
 
 198025 
 
 88121125' 21-0950231 
 
 7-634607 
 
 512 
 
 262144 
 
 134217728 
 
 22-6274170 
 
 8000000 
 
 446 
 
 198916 
 
 887165361 21-1187121 
 
 7 640321 
 
 513 
 
 263169 
 
 135005697 
 
 22-6495033 
 
 8-005205 
 
 447 
 
 199809 
 
 89314623 21-1423745 
 
 7-646027 
 
 514 
 
 264196 
 
 135796744 
 
 22-67151381 
 
 8-010403 
 
 448 
 
 200704 
 
 89915392: 21-1660105 
 
 7-651725 
 
 515 
 
 265225 
 
 136590875 
 
 22-6936114 
 
 8-015595 
 
 449 
 
 201601 
 
 90518849: 21-1896201 
 
 7-657414 
 
 516 
 
 266256 
 
 13738S096 
 
 22-7156334 
 
 8-020779 
 
 450i 202500 
 
 91125000! 21-2132034 
 
 7-663094 
 
 517 
 
 267289 
 
 13818,^413 
 
 22-7376340 
 
 8-025957 
 
 451 203401 
 
 9173385li 21-2367606 
 
 7-6687661 
 
 518 
 
 268324 
 
 138991832 
 
 22-7596134 
 
 8-031129 
 
 452 204304 
 
 92345408 21-2802916 
 
 7-674430 
 
 519 
 
 269361 
 
 139793359 
 
 22-7815715 
 
 8 036293 
 
 453i 205209 
 
 92959677 21-2837967 
 
 7-680086 520 
 
 270400 
 
 140608000 
 
 22-8035085 
 
 8-041451 
 
 454! 206116 
 
 93576664; 21-3072758 
 
 7-685733,! 521 
 
 271441 
 
 141420761 
 
 22-8254-244 
 
 8-046603 
 
 45 3 i 207025 
 
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 7-691372 522 
 
 272484 
 
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 22-84731i^3 
 
 8-051748 
 
 4561 207936 
 
 94818816 21-3541565 
 
 7-697002! 523 
 
 273529 
 
 143055667 
 
 22-8691933 
 
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 95443993 21-3775583 7-702625;! 524 
 
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 22-9128785 
 
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 22-9346 S99 
 
 8-072262 
 
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 146363183 
 
 22-9564806 
 
 8-077374 
 
 461 212521 
 
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 7-725032,; 528 
 
 278784 
 
 147197952 
 
 22-9782506 
 
 8-082480 
 
 4621 213444 
 
 986111-28 21-4941853 
 
 7-730614.! 529 
 
 279841 
 
 148035889 
 
 23-0000000 
 
 8-087579 
 
 463: 214369 
 
 99252847, 21-5174348 
 
 7-736188 1 530 
 
 280900 
 
 148877000 
 
 23-0^1728j 
 
 8092672 
 
 464' 215296 
 
 99897344 215406592 
 
 7-741753:| 531 
 
 281961 
 
 149721291 
 
 23-0134372 
 
 8-097759 
 
 465; 216225 
 
 100544625; 21-5638587 
 
 7-74731 Ifj 532 
 
 283024 
 
 150568768 
 
 23-065125^ 
 
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 466! 217156 
 
 101194696: 21-5870331 
 
 7-752861 li 533 
 
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 23-0867928 
 
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 467 218089 
 
 101847563 21-6101828 
 
 7-7584021 534 
 
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 23-1084400 
 
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 23-1300670 
 
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 103161709 21-6564078 
 
 7-769462! 536 
 
 287296 
 
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 23-1516738 
 
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18 
 
 APPENDIX. 
 
 Xc, £:^irr. 
 
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 : -1532?4 
 
 551 
 
 552 
 
 553 
 
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 556 
 
 557 
 
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 55^ 
 
 560 
 
 561 
 
 562 
 
 563 
 
 564 
 
 565- 
 
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 569 
 570 
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 594 
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 3cKj763l>'w>j 
 
 2547547>4 5-'~. 
 
 25-495 :'976 5-6-= ■-:?r:! 
 
 25-5147016 5-666i31 
 
 25-534-291.17 5-671266 
 
 25-553>647 5 675697 
 
 25-57ci4-::r- 3^*1104 
 
 : ■ ; ■ : h-il 
 
 . - 76 
 
 ^ - "34 
 
 c.v--"-, 5t:-"'^s 
 
 Tv'.-^v-'.S 3 710.--33 
 
 25-7-293507 ; 715373 
 
 2n-4-rs64 5719760 
 
 ""- .:5 3 724141 
 
 - -.39 5-7-25513 
 
 : -^ -.'55 5 732^2 
 
 : ?--':;-43i 8-737260 
 
 : -.: ■.6<.) 3-7416-25 
 
 :>: 10343 6-745955 
 
 2o^^43552 5 75:'34i» 
 
APPENDIX. 
 
 19 
 
 No. 
 
 Square. 
 
 Cube. 
 
 Sq. Root. 
 
 CubeRoot. 
 
 No. 
 
 738 
 
 Square.! Cube. i Sq. Root. CubeRooU 
 
 671 
 
 450241 
 
 302111711 
 
 25-9036677 
 
 8-754691 
 
 544644 
 
 401947272' 27-1661554 9-036886 
 
 672 
 
 451584 
 
 303464448 
 
 25-9229628 
 
 8-759038 
 
 739 
 
 546121 
 
 403583419 '27-1845544 9040965 
 
 673 
 
 452929 
 
 304821217 
 
 25-9422435 
 
 8-763381 
 
 740 
 
 547600 
 
 405224000 27-20-29410 9-0450i-i 
 
 674 
 
 454276 
 
 306182024 
 
 25-9615100 
 
 8-767719 
 
 741i 5490811 4U6369021 27-2213152 9-0iJiU| 
 
 675 
 
 455625 
 
 307546875 
 
 25-9807621 
 
 8-77205311 742i 550564] 408518488 27-23967691 y-053l83| 
 
 676 
 
 456976 
 
 308915776 
 
 26 0000000 
 
 8-776383 
 
 743 
 
 552049 410172407; 272580263' 9-0J7-.48 
 
 677 
 
 458329 
 
 310288733 
 
 26-0192237 
 
 8-780708 
 
 744 
 
 553536; 41L>30784' 272763634; 9-06i^Ki 
 
 678 
 
 459684 
 
 311665752 
 
 26-0381331 
 
 8-785030 
 
 745 
 
 5550-25i 4134^3625 -47-2946881^ 9-Oi;-53oo 
 
 679 
 
 461041 
 
 313046839 
 
 26.0576284 
 
 8-7893471 746| 
 
 556516j 415160936. -i7-31300o6! ^06^422 
 
 680 
 
 462400 
 
 314432U00 
 
 26-0768096 
 
 8-793659' 1 7471 
 
 558009; 4168327-23, 27-3313007! ^ 07 3473 
 
 681 
 
 463761 
 
 315821241 
 
 26-0959767 
 
 8-797968! 
 
 748 
 
 559504 418508992^ -27-34958871 9 077520 
 
 682 460124 
 
 317214568 
 
 26-1151297 8-8022721 
 
 749 
 
 56100r 420189749 27-3678644 sOol563 
 
 683 
 
 466489 
 
 318611987 
 
 26-1342687 8-8065721 
 
 750 562500: 42l87.!)000: 273861279 9'085l.03 
 
 684 
 
 467856 
 
 320013504 
 
 26-1533937 
 
 8-8108681 
 
 751 564uOi: 4-23564751! 27-404371-2! 9-0o;.63i. 
 
 685 
 
 469225 
 
 321419125 
 
 26-1725047 
 
 8-815160 
 
 752| 565504 4:i5-259UU8 •27-4226184 9-Ut'o67-^ 
 
 686 
 
 470596 
 
 322828856 
 
 26-1916017 
 
 8-819447 
 
 753 567009J 426957777 -xi/ -4408455 i?-Ui,7V0l 
 
 687 
 
 471969 
 
 324242703 
 
 26-2106818 
 
 8-823731 
 
 754 568516 4-3661064' -^7-45906^4: 9-lOl7-.it) 
 
 688 
 
 473344 
 
 325660672 
 
 26-2297541 
 
 8-8260101: 755 
 
 5700-251 430ii68875; 27-4772633; a-iwj.-io 
 
 689 
 
 474721 
 
 327082769 
 
 26-248S095 
 
 8-832285'| 756 
 
 571536! 43:i08l21b 27-4'j54542 i7-io9767 
 
 690 
 
 476100 
 
 328509U00 
 
 26-2678511 
 
 8-836556 
 
 757 
 
 573U49 4337980^3 •47-5i3b33-j '.;-j-i3/c2 
 
 691 
 
 477481 
 
 329939371 
 
 26-2868789 
 
 8-840823 
 
 758 
 
 574564 4355195U 27-5317^^8 y-il77b3 
 
 692 
 
 478864 
 
 331373888 
 
 26-3058929 
 
 8-845085 
 
 759 
 
 576081: 437-245479! •27-549'j54b ■i>-rzLi<Jl 
 
 693 
 
 480249 
 
 332812557 
 
 26-3248932 
 
 8-84y344 
 
 760 
 
 57760o! 438ii7600O 27-568-J975 i>-i25d05 
 
 694 
 
 481636 
 
 334255384 
 
 26-3438797 
 
 a -853598 
 
 761 
 
 57^121 440711081 -27-5802284 y-i-2^'b'.6 
 
 695 
 
 483025 
 
 335702375 
 
 26-3628527 
 
 8-857849 
 
 762 
 
 580644 44-2450728 27-6043475| 
 
 y-l33;50b 
 
 696 
 
 4»4416 
 
 337153536 
 
 26-3S18119 
 
 8-862095 
 
 763 
 
 552169! 444194947 
 
 Z7 6224546 
 
 y-lb77..'V 
 
 697 
 
 485809 
 
 338608873 
 
 26-4007576 
 
 8.866337 
 
 1 764 
 
 583696; 445943744 
 
 2r64u549-j 
 
 •./-i41To< 
 
 698 
 
 487204 
 
 340068392 
 
 26-41968^6 
 
 8-870576 
 
 765 
 
 585-225! 447697125 
 
 •27-658u334 
 
 i;-i4 j77-i 
 
 699 
 
 488601 
 
 341532099 
 
 26-4386081 
 
 8-074810 
 
 766 
 
 58b756j 44ij455096 
 
 27 67o705o 
 
 y-i4-j7.Jc 
 
 700 
 
 49U000 
 
 343000000 
 
 26-4575131 
 
 8-879040 
 
 767 
 
 588;;o9j 45l2i'i663 
 
 '47'6;^4764o 
 
 y- 153737 
 
 701 
 
 491401 
 
 344472101 
 
 26-4764046 
 
 8-883266 
 
 768 
 
 589824 45-29«483--i 
 
 27-712812;/ 
 
 ty-iD(7i4 
 
 702 
 
 492804 
 
 345948408 
 
 26-495-2826 
 
 8-887488 
 
 769 
 
 591361] 454756609 
 
 27-730849Z 
 
 9-iuloo'i 
 
 703 
 
 494209 
 
 347428927 
 
 26-5141472 
 
 8-891706 
 
 1 770 
 
 592900, 456533000 
 
 27-7488739 
 
 y-l6o65b 
 
 704 
 
 495616 
 
 348913664 
 
 26 5329983 
 
 8-895920 771 
 
 594441 458314011 
 
 '27 -7668860 
 
 y-i6Jo24 
 
 705 
 
 497025 
 
 35040^625 
 
 26-5518361 
 
 8-9U0130 772 
 
 5959841 4b009'.<648 
 
 27-704^860 
 
 y-17358.j 
 
 706 
 
 498436 
 
 351895816 
 
 26-5706605 
 
 8-904337 773 
 
 5975-29 461889917 
 
 27-8028775 
 
 y-i77544 
 
 707 
 
 499849 
 
 353393243 
 
 26-5894716 
 
 8-908539^ 774 
 
 59^076 
 
 ■^63684^24 
 
 27-8408555 
 
 y-i5ioOi^ 
 
 708 
 
 501264 
 
 354894912 
 
 26-6082694 
 
 8-912737 
 
 i 775 
 
 600625 
 
 465484375 
 
 47-o38621o 
 
 y-lo545c 
 
 709 
 
 502681 
 
 356400829 
 
 26-6270539 
 
 8-916931 
 
 776 
 
 602176 
 
 4o728b576 
 
 ;^7 b.5677bo 
 
 y-l8»d0.4 
 
 710 
 
 504100 
 
 357911000 
 
 26-6458-^52 
 
 8-921121 
 
 777 
 
 6037-29 
 
 46yoy7433 
 
 •47-8747197 
 
 9 1^3347 
 
 711 
 
 505521 
 
 359425431 
 
 26-6645833 
 
 8-925308 
 
 778 
 
 605-284 
 
 ^709ioy;)2 
 
 -47-oy265l4 
 
 y-iy7-~yo 
 
 712 
 
 506944 
 
 360944128 
 
 26-6833281 
 
 8-929490 
 
 779 
 
 606841 
 
 4727-.i9i3y 
 
 •27 y 1057 15 
 
 y -40 122b 
 
 713 
 
 508369 
 
 362467097 
 
 26-7020598 
 
 8-933669 
 
 780 
 
 60840 
 
 47455-2000 
 
 -27-y2840ui 
 
 9 -'205 164 
 
 714 
 
 509796 
 
 363994344 
 
 26-7207784 
 
 8-937843 
 
 781 
 
 60jy61 
 
 47637y541 
 
 -47-y4b3772 
 
 9-2oy09b 
 
 715 
 
 511225 
 
 365525875 
 
 S' -7394839 
 
 b-9 12014 
 
 782 
 
 6115^4 
 
 47c2ll768 
 
 '27yo4-26-4y 
 
 y-^i3025 
 
 716 
 
 512656 
 
 367061696 
 
 26-7581763 
 
 8-946181 
 
 783 
 
 613089 
 
 480048687 
 
 •47-y821372 9-416950 
 
 717 
 
 514089 
 
 368601813 
 
 26-7768557 
 
 8-950344 
 
 784 
 
 614656 
 
 48189u304 
 
 28-0000000 9 -2208 7 b 
 
 718 
 
 515524 
 
 370146232 
 
 26-7955220 
 
 8-954503 
 
 785 
 
 6162-25 
 
 483736625 
 
 28 0l7o5l5 xi-^Z-il'^i 
 
 719 
 
 516961 
 
 371694959 
 
 26-8141754 
 
 8-958658 
 
 786 
 
 6l77i:<6 
 
 4855^7656 
 
 28-0356yl5 y-42o7o', 
 
 720 
 
 518400 
 
 373248000 
 
 26-8328157 
 
 8-962809 
 
 1 78) 
 
 61936^.: 
 
 l«74434o3 
 
 -^8-0535-403 9-2.3'2619 
 
 721 
 
 519841 
 
 374805361 
 
 26-8514432 
 
 8-9669571^ 788 
 
 620944 
 
 48^303872 
 
 2807 1337 ( y-2ot 52b 
 
 722 
 
 521284 
 
 37636704^ 
 
 26-8700577 
 
 1 8-971101 
 
 1 789 
 
 62-2521 
 
 491l6ii069 
 
 '28-0a9l43Oj 9-24^435 
 
 723 
 
 522729 
 
 j 377933067 
 
 26-8886593 
 
 ; 8-975241 
 
 : 790 
 
 6:i41oO 
 
 4y3o3yooo 
 
 26 -106^386 y-44i33b 
 
 724 
 
 524176 
 
 ' 379503424 
 
 26-907-2481 
 
 : 8-979377 
 
 1 791 
 
 6-25681 
 
 494913671 
 
 28-12472-^2 y -44843-1 
 
 725 
 
 525625 
 
 ' 381078125 
 
 26-9-25S240 
 
 ' 8-983509 
 
 , 792 
 
 627264 
 
 496793080 
 
 •28-l4-24a4o y-452i3i. 
 
 726 
 
 527076 
 
 382657176 
 
 26-9443872 
 
 ! 8-987637 
 
 1 793 
 
 62884y 
 
 4y8677257 
 
 28-i602557 y-456022 
 
 727 
 
 528529 
 
 1 384240533 
 
 26-9629375 
 
 ' 8-991762 
 
 ; 791 
 
 630436 
 
 50056618-1 
 
 -4o-i7800D6i y-2J9ylI 
 
 728 
 
 529984 
 
 ; 385828352 
 
 26-9814751 
 
 8-995883 
 
 795 
 
 63-2025 
 
 502459875 
 
 -48-1 957444 i y-26.i797 
 
 729 
 
 531441 
 
 ! 387420489 
 
 27-0000000 
 
 9-000000 
 
 1 796 
 
 633616 
 
 50435O336 
 
 2«-213472o| 9-267680 
 
 730 
 
 532900 
 
 i 389017000 
 
 27-0185122 
 
 1 9-004113 
 
 j 79/ 
 
 63520b 
 
 506--i61573 
 
 28-231 lo84i 9-471559 
 
 731 
 
 534361 
 
 390617891 
 
 27-0370117 
 
 i 9-008223 
 
 j 798 
 
 6368«-M 
 
 508169592 
 
 48-2488y3oi 9-275435 
 
 732 
 
 535824: 392223168 
 
 27-0554985 
 
 9-012329 
 
 ; 799 
 
 638401 
 
 51008-4399 
 
 28-4665081! 9-479308 
 
 733 
 
 1 527289| 393832837 
 
 27-0739727 
 
 9-016431 
 
 1 800 
 
 640000 
 
 512000000 
 
 28-28427 r^l 9-203178 
 
 734 
 
 538756' 395446904 
 
 27-0924344 
 
 9-020529 
 
 , 801 
 
 641601 
 
 5139-2-2401 
 
 28-b0ly434 
 
 9-207044 
 
 735 
 
 540225' 397065375 
 
 27.1108834 
 
 : 9-01^624 
 
 802 
 
 643204 
 
 51584y608 
 
 28-3l9c040 
 
 y-290907 
 
 736 
 
 541G96! 39S688256 
 
 27-1293199 
 
 9-028715 
 
 1 803 
 
 644809 
 
 517781627 
 
 28-3372540 
 
 9-29476i 
 
 737 
 
 543169J 400315553 
 
 27-1477439 
 
 1 9-032802 
 
 1 804 
 
 646416 
 
 519718464 
 
 28-3548938 
 
 9-298624 
 
20 
 
 APPENDIX. 
 
 No. 
 
 b05 
 
 806 
 
 ti07 
 
 808 
 
 80'j| 
 
 rjUH 
 
 bill 
 
 6U\ 
 
 6U\ 
 
 814' 
 
 81:')! 
 
 bi; 
 
 818 
 81'J: 
 8-i0| 
 821; 
 8221 
 823, 
 8241 
 825' 
 826 
 o27 
 828! 
 82<J 
 8301 
 
 83 1: 
 
 832' 
 
 833 1 
 83 4 1 
 835 
 836! 
 
 8371 
 838i 
 
 83yi 
 
 840' 
 8411 
 8421 
 843 
 
 844 
 845' 
 846 
 
 847 i 
 8481 
 8491 
 850, 
 851 
 852' 
 853i 
 8541 
 855 
 856 
 857 
 8581 
 859 1 
 
 860; 
 
 861 
 852 
 8631 
 864 
 865 
 866 
 867 
 868 
 869 
 870 
 871 
 
 Square. 
 
 Cube. 
 
 Sq. Root. 
 
 648025 521660125; 
 649636 52360C616! 
 651249 525557943: 
 652864 527514112, 
 654481 529475129i 
 656100 5314410001 
 657721 533411731! 
 65<;344 535387328 
 660969 537367797' 
 662596 539353144! 
 664225 541343375: 
 6.i5856 543338496 
 6r,74H9 54533S513 
 669124 547313432' 
 670761 549353259i 
 672400 5513680001 
 674U41 55338766 Ij 
 675684 5554122481 
 67732y 557441 767 i 
 678976 559476224| 
 680625 5615156251 
 682276 563559976: 
 6:)3929 565609283! 
 685584 567663552 
 687241 569722789 
 6889(0 571787000 
 G90561 573856191 
 69-224 575930368 
 693889 578009537 
 6'.*5556 580093704 
 697225 582182875 
 698896 584277056 
 700569 586376253 
 702244 5884804721 
 703921 590589719! 
 705600 592704000| 
 707281 594823321! 
 708i^64 596947688! 
 710649 599077107] 
 7123361 601211534! 
 714025! 6U335 11251 
 7157161 605495736 
 717409! 607645423 
 
 CubeRoot. 
 
 719104 
 
 7j''1801 
 
 n.'^K) 614125000 
 
 609800192 
 611960049 
 
 724201 
 725904 
 727609 
 729316 
 7310^5 
 732736 
 734449 
 736164 
 737881 
 739600 
 741321 
 743044 
 744769 
 746496 
 748225 
 749956 
 751689 
 753424 
 755161 
 756900 
 758641 
 
 616295051 
 
 618470208; 
 
 620650477 
 
 622835864 
 
 625026375 
 
 6272220161 
 
 629422793' 
 
 631628712! 
 
 633839779! 
 
 636056000! 
 
 638277381! 
 
 640503928! 
 
 642735647! 
 
 614972544 
 
 647214625! 
 
 649461896^ 
 
 6517143631 
 
 653972032| 
 
 656234909! 
 
 658503000' 
 
 660776311! 
 
 28 3725219 
 
 28 3901391 
 
 28-4077454 
 
 28 4253408 
 
 28-4429253 
 
 28-4604989 
 
 28-4780617 
 
 28-4956137 
 
 285131549 
 
 28-5306852 
 
 28-5482048 
 
 28-5657137 
 
 28-5832119 
 
 28-6006993 
 
 28-6181760 
 
 28-6356421 
 
 28-6530976 
 
 28-6705124 
 
 28-6879766 
 
 28-7054002 
 
 28-72-28132 
 
 28-7402157 
 
 28-7576077 
 
 28-7749891 
 
 28-79236011 
 
 28-8097206 
 
 28-8270706' 
 
 28-8144102 
 
 28-8617394 
 
 28-8790582 
 
 2S-8963666 
 
 28-9136646 
 
 28-9309523 
 
 28-9482297 
 
 28-9654967 
 
 28-9827535 
 
 29O0000O0! 
 
 29-0172363! 
 
 29-0344623 
 
 29-0511)781' 
 
 29-0688837 
 
 290860791 
 
 29-1032644 
 
 29-1204396 
 
 29-1376046 
 
 29- 1547595 
 
 29-1719043 
 
 29 1890390 
 
 29-2061637! 
 
 29-2232784 
 
 29-2403830! 
 
 29-2574777 
 
 29-2745623 
 
 29 2916370 
 
 29-3087018 
 
 29 -3257566 
 
 29-3428015 
 
 29-3598365 
 
 29-3768616 
 
 29-3938769 
 
 29-4108823 
 
 2^-4278779 
 
 29-4448637 
 
 29-4618397 
 
 29-4788059 
 
 29-4957624 
 
 29-5127091 
 
 No. 
 
 Square. 
 
 9-302477! 
 9-306328' 
 9-310175 
 9-314019 
 
 9-317860' 
 9-321697; 
 9-325532 
 9-329363 
 9-333192 
 9-337017 
 9-340839 
 9-344657 
 9-348473 
 9-3522^8:! 
 9-356095 1 
 9-359902 I 
 9-363705' 
 9-367505! 
 9-371302:1 
 9 3750961 
 9-378887i| 
 9-38-2675!! 
 9-3S6460 
 9-390242 
 9-394021 
 9-3.-7796! 
 9-401569! 
 9-4053391 
 9-4091051 
 9-412869: 
 9 -4166301 
 9-420.387 
 9-4241421 
 9 -427894; 
 9-431642' 
 9-4353881 
 9-439131; 
 9-442870 
 9 •446607! 
 9-450341 
 9-454072; 
 9-457800! 
 9-461525! 
 9-465247; 
 9-468966 
 9-472682! 
 9-476396! 
 9-480106' 
 9-483814 
 9-487518. 
 9-491220 
 9-494919 
 9-498615 
 9-5023U8: 
 9-505998 
 9-509685 
 9-513370 
 9-517051 
 9-5-20730 
 9-524406 
 9-528079 
 9-531750 
 9-535417 
 9-539082 
 9-542744 
 , 9-546403 
 ' 9-550059 
 
 872 
 
 873; 
 
 874 
 
 875! 
 
 876 
 
 877 
 
 878 
 
 879 
 
 880 
 
 881 
 
 882! 
 
 883: 
 
 S-M 
 
 8b5 
 
 886 
 
 887 
 
 888 
 
 889 
 
 890 
 
 891 
 
 892 
 
 893 
 
 894 
 
 895 
 
 896 
 
 897 
 
 898 
 
 899 
 
 900 
 
 901 
 
 902 
 
 903 
 
 904 
 
 905 
 
 906 
 
 907 
 
 908 
 
 909 
 
 910 
 
 911 
 
 912 
 
 913 
 
 914 
 
 915 
 
 916 
 
 91 
 
 918 
 
 919 
 
 92u 
 
 921 
 
 922 
 
 923 
 
 924 
 
 925 
 
 926 
 
 92 
 
 928 
 
 929 
 
 930 
 
 931 
 
 932 
 
 933 
 
 934 
 
 935 
 
 936 
 
 937 
 
 938 
 
 Cube. 
 
 760384 
 
 762129 
 
 763376 
 
 765625 
 
 767376 
 
 769129 
 
 770884 
 
 772641 
 
 774400 
 
 776161 
 
 777924 
 
 779689 
 
 78145(: 
 
 783225 
 
 78499^: 
 
 786769 
 
 788544 
 
 790321 
 
 7'j2100 
 
 793881 
 
 795664 
 
 797449 
 
 799236 
 
 801025 
 
 802816 
 
 804609 
 
 806404 
 
 808201 
 
 810000 
 
 811801 
 
 813604 
 
 815409 
 
 817216 
 
 819025 
 
 820836 
 
 822649 
 
 824464 
 
 826281 
 
 828100 
 
 8-29921 
 
 831744 
 
 833569 
 
 835396 
 
 837225 
 
 839056 
 
 840889 
 
 842724 
 
 844561 
 
 846400 
 
 818-241 
 
 850084 
 
 851929 
 
 853776 
 
 855625 
 
 857476 
 
 859329 
 
 8611841 
 
 863041 
 
 864900' 
 
 8667611 
 
 868624. 
 
 8704891 
 
 872356! 
 
 874225 
 
 876096 
 
 8779691 
 
 8798441 
 
 Sq. Root. CubeRoot, 
 
 663054848 
 
 665338617 
 
 667627624 
 
 669921875 
 
 tw2-221376 
 
 674526133 
 
 676836152 
 
 679151439 
 
 681472000 
 
 683797841 
 
 686128968 
 
 688465387 
 
 690807104 
 
 693154125 
 
 695506456 
 
 697864103 
 
 700227072 
 
 702595369 
 
 704969000 
 
 707347971 
 
 709732288 
 
 712121957 
 
 714516984 
 
 716917375 
 
 719323136 
 
 721734-273 
 
 724150792 
 
 726572699 
 
 729000000 
 
 73143-2701 
 
 7338708081 
 
 736314327 
 
 738763264 
 
 741217625 
 
 743677416 
 
 746142643 
 
 748613312 
 
 751089429 
 
 753571000 
 
 756058031 
 
 758550528 
 
 761048497 
 
 763551944 
 
 766060875 
 
 768575296 
 
 771095213 
 
 773620632 
 
 776151559 
 
 778688000 
 
 781229961 
 
 783777448 
 
 786330467 
 
 788889024 
 
 791453125 
 
 794022776 
 
 796597983 
 
 799178752 
 
 801765089 
 
 804357000 
 
 806954491 
 
 809557568 
 
 812166237 
 
 814780504 
 
 817400375 
 
 820025356 
 
 822656953 
 
 825293672 
 
 29-5296461 
 
 29-5465734 
 
 29-5634910 
 
 29-5803989 
 
 29-5972972 
 
 29-6141858 
 
 29-6310648 
 
 29-6479342 
 
 29-6647939 
 
 29-6816442 
 
 29.6984848, 
 
 29-7153159! 
 
 29-73213751 
 
 29-7489496 
 
 29-7657521 
 
 29-78254521 
 
 29-7993289 
 
 29-8161030 
 
 29-83-28678 
 
 29-8496231 
 
 29-8663690 
 
 29-8831056 
 
 29-8998328 
 
 29-9165506 
 
 29-933-2591 
 
 29-9499583 
 
 29-9666481 
 
 29-9833^7 
 
 30-0000000 
 
 30-0166620 
 
 30-0333148 
 
 30-0499584 
 
 30-0665928 
 
 30-0832179 
 
 30-0998369 
 
 30-1164407 
 
 30-1330383 
 
 30-1496269 
 
 30-1662063 
 
 30-18-27765 
 
 30-1993377 
 
 30-2158899 
 
 30-2324329 
 
 30-2489669 
 
 30-2654919 
 
 30-2820079 
 
 30-2985148 
 
 30-3150128 
 
 30-3315018 
 
 30-3479818 
 
 30-3644529 
 
 30-3809151 
 
 30-3973683 
 
 30-4138127 
 
 30-4302481 
 
 30-4466747 
 
 30-4630924 
 
 30-4795013 
 
 30-4959014 
 
 30-512-2926 
 
 30-5286750 
 
 30-5450487 
 
 30-5614136 
 
 30-5777697 
 
 30-5941171 
 
 30-6104557 
 
 30-6267857 
 
 9-553712 
 9-557363 
 9-561011 
 9-564656 
 
 9-568298 
 
 9-571938 
 
 9-575574 
 
 9-579208 
 
 9;582840 
 
 9-:86468 
 
 9-590094 
 
 9-593717 
 
 9-597337 
 
 9-600955 
 
 9-604570 
 
 9-608182 
 
 9-611791 
 
 9-615398 
 
 9-619002 
 
 9-622603 
 
 9-626-202 
 
 9-62979 
 
 9-633391 
 
 9-636981 
 
 9-64056^ 
 
 9-644154 
 
 9-647737 
 
 9-651317 
 
 9-654894 
 
 9-658468 
 
 9-66-2040 
 
 9-665610 
 
 9-669176 
 
 9-672740 
 
 9-676302 
 
 9-679860 
 
 9-683417 
 
 9-686970 
 
 9-690521 
 
 9-694069 
 
 9-697615 
 
 9-701158 
 
 9-701699 
 
 9-708237 
 
 9-711772 
 
 9-715305 
 
 9-718835 
 
 9-7-22363 
 
 9-725888 
 
 9-7-29411 
 
 9-732931 
 
 9-736448 
 
 9-739963 
 
 9-743476 
 
 9-746986 
 
 9-750493 
 
 9-753998 
 
 9-757500 
 
 9-761000 
 
 9-764497 
 
 9-767992 
 
 9-771484 
 
 9-774974 
 
 9-778462 
 
 9-781947 
 
 9'7854 
 
 9-788909 
 
APPENDIX. 
 
 21 
 
 No. 
 
 Square.; Cube. 
 
 Sq, Root. 
 
 CubeRoot. 
 
 ! 
 
 No. 
 
 970 
 
 Square. 
 
 Cube. 
 
 Sq. Root. 
 
 CubeRoot. 
 
 939 
 
 881721; 827936019 
 
 30-6431069 
 
 9-792386 
 
 940900 
 
 912673000 
 
 31-1448230 
 
 9-898983 
 
 910 
 
 883600 830584000 
 
 30-6594194 
 
 9-795861; 
 
 971 
 
 942841 
 
 915498611 
 
 31-1608729 
 
 9-902333 
 
 941 
 
 885481 
 
 833237621 
 
 30-6757233 
 
 9-799334 
 
 972 
 
 944784 
 
 918330048 
 
 31-1769145 
 
 9-905782 
 
 942 
 
 887364 
 
 835896888 
 
 30-6920185 
 
 9-802804 
 
 973 
 
 946729 
 
 921167317 
 
 31-1929479 
 
 9-90917"^ 
 
 943 
 
 889249 
 
 838561807 
 
 30-7083051 
 
 9-806271 
 
 974 
 
 948676 
 
 924010424 
 
 31-2089731 
 
 9-912571 
 
 944 
 
 891136 
 
 841232384 
 
 30-7245830 
 
 9-809736 
 
 975 
 
 950625 
 
 926859375 
 
 31-2249900 
 
 9 -915962 
 
 945 
 
 893025 
 
 843908625 
 
 30-7408523 
 
 9-813199 
 
 976 
 
 952576 
 
 929714176 
 
 31-2409987 
 
 9-919351 
 
 946 
 
 894916 
 
 846590536 
 
 3U-;571130 
 
 9-816659 
 
 977 
 
 954529 
 
 93-2574833 
 
 31-2569992 
 
 9-9-2273 -. 
 
 947 
 
 896809 
 
 849278123 
 
 30-7733651 
 
 9-820117 
 
 978 
 
 956484 
 
 935441352 
 
 31-2729915 
 
 9-926122 
 
 94b 
 
 898704 
 
 851971392 
 
 30-7896086 
 
 9-823572 
 
 979 
 
 958441 
 
 938313739 
 
 31-2889757 
 
 9-929504 
 
 949 
 
 900601 
 
 854670349 
 
 30-8058436 
 
 9-827025 
 
 980 
 
 960400 
 
 941192000 
 
 31-3049517 
 
 9-932834 
 
 yso 
 
 902500 
 
 857375000 
 
 30-8220700 
 
 9-830476 
 
 981 
 
 962361 
 
 944076141 
 
 313-209195 
 
 9-936-261 
 
 951 
 
 904401 
 
 860085351 
 
 30-8382879 
 
 9-833924 
 
 982 
 
 964324 
 
 946966168 
 
 31-3368792 
 
 9-939636 
 
 952 
 
 906304 
 
 862801408 
 
 30-8544972 
 
 9-837369 
 
 983 
 
 966289 
 
 949862087 
 
 31-35-28308 
 
 9-9430'J9 
 
 953 
 
 908209 
 
 865523177 
 
 30-8706981 
 
 9-840813 
 
 984 
 
 968256 
 
 952763904 
 
 31-3687743 
 
 9-916380 
 
 954 
 
 910116 
 
 868250664 
 
 30-8868904 
 
 9-844254 
 
 985 
 
 970225 
 
 955671625 
 
 31-3847097 
 
 9-949748 
 
 955 
 
 912025 
 
 870983875 
 
 30-9030743 
 
 9-847692 
 
 986 
 
 972196 
 
 958585256 
 
 31-4006369 
 
 9-953114 
 
 956 
 
 913936 
 
 873722816 
 
 30-9192497 
 
 9-851128 
 
 987 
 
 974169 
 
 961504803 
 
 31-4165561 
 
 9-956477 
 
 957 
 
 915849 
 
 876467493 
 
 30-9354166 
 
 9-854562 
 
 288 
 
 976144 
 
 964430272 
 
 31-4324673 
 
 9-95983U 
 
 958 
 
 917764 
 
 879217912 
 
 30-9515751 
 
 9-857993 
 
 989 
 
 978121 
 
 967361669 
 
 31-4483704 
 
 9-963 198 
 
 959 
 
 919681 
 
 881974079 
 
 30-9677251 
 
 9-861422 
 
 990 
 
 980100 
 
 970299000 
 
 31-464-2654 
 
 9-966555 
 
 -A)0 921S00 
 
 884736000 
 
 30-9838668 
 
 9-864848 
 
 991 
 
 982081 
 
 973242271 
 
 31-4801525 
 
 9-9699.!.' 
 
 961 
 
 923521 
 
 887503681 
 
 31-0000000 
 
 9 •868-272 
 
 992 
 
 984064 
 
 976191488 
 
 31-4960315 
 
 9-973i6.: 
 
 962 
 
 925444 
 
 S90277128 
 
 31-0161248 
 
 9-871694 
 
 993 
 
 986049 
 
 979146657 
 
 31-5119025 
 
 9-976612 
 
 963 
 
 927369 
 
 893056347 
 
 31-0322413 
 
 9-875113 
 
 994 
 
 988036 
 
 982107784 
 
 31-5277655 
 
 9-9799G0 
 
 964 
 
 929296 
 
 8958413441 31-0483494 
 
 9-878530 
 
 995 
 
 990025 
 
 985074875 
 
 31-5436206 
 
 9-933305 
 
 965 
 
 931225 
 
 898632125 
 
 31-0644491 
 
 9-881945 
 
 996 
 
 992016 
 
 988047936 
 
 31-5594677 
 
 9-98664y 
 
 966 
 
 933156 
 
 901428696 
 
 310805405 
 
 9-885357; 
 
 997 
 
 994009 
 
 991026973 
 
 31-5753068 
 
 9-98999(. 
 
 967 
 
 935089 
 
 904231063 
 
 31-0966236 
 
 9-8887671 
 
 998 
 
 996004 
 
 994011992 
 
 31-5911380 
 
 9-993;.-2.. 
 
 968 
 
 937024 
 
 907039232 
 
 31-1126984 
 
 9-892175 
 
 999 
 
 998001 
 
 99700-2999 
 
 31-6069613 
 
 9-996666 
 
 969 
 
 938961 
 
 909853209 
 
 31-1287648 
 
 9-895580 
 
 1000 
 
 1000000 
 
 1000000000 
 
 31-6227766 
 
 10-OOUOOU 
 
 The following rules are for finding the squares, cubes and roots, of 
 numbers exceeding 1,000. 
 
 To find the square of any number divisible without a remainder. 
 Rule. — Divide the given number by such a number, from the forego- 
 ing table, as will divide it without a remainder ; then the square of the 
 quotient, multiplied by the square of the number found in the table, 
 will give the answer. 
 
 Example- -What is the square of 2,000 ? 2,000, divided by 1,000, 
 a number lound 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 bo 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, v*'ill 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— i = 2,211,169: the Ans. 
 
 To find the cube of any number aiv/sible without a remainder. 
 Rule. — Divide the given number by sucn a number, from the forego- 
 
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. 
 
 Exam.pl e,--Whsit 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. Mule. — Select, in the column of squares or cubes, as the case 
 may require, that number which is nearest the given number ; then 
 t!ie 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, eearly. 
 
 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 tJie 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 numbei' selected 
 to the root of the number given. 
 
 Example. — What is the cube-root of 9,200 ? The nearest number 
 in th-e 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 4- the Ans. 
 21 
 
 27661 
 55322 
 
 27722)580881(20-953 + 
 55444 
 
 264410 
 249498 
 
 149120 
 138610 
 
 105100 
 3316rt 
 
 21934 
 
APPENDIX. 23 
 
 To find the square or cube root of a whole numher 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 
 
 the Ans. 
 
 (See page 32. 
 
 
 5900 
 1475 
 
 
 •020650 
 3-3166 
 
 
 3-33725 : 
 
 App, 
 
 ctULES 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 -^^ of a foot, therefore : 
 /j ... 12) 3-00 
 
 •25 Ans. 
 Another example. — What is the equivalent decimal of f of an inch ? 
 ^ .... 8) 7^000 
 
 •875 Ans. 
 
 To Tpd"ce a compound fraction to its equivalent decimal. Rule. — In 
 dccoraance with the preceding rule, reduce each fraction, commen- 
 cmg 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, | 
 
 and j\ of an inch ? 
 
 The fractions in this case are, ^ of an eighth, f of an inch, and -^ 
 of a foot, therefore : 
 
el 
 
 APPEND 
 
 IX. 
 
 2) 1-0 
 
 •5 
 3- 
 
 8) 3-5000 
 
 eighths. 
 
 •4375 
 5- 
 
 inches. 
 
 -i- 12) 5-437500 
 
 •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 left of its numerator, thus : 
 
 i 2 
 
 f 8 
 
 -a- 12 
 
 12 
 
 1-0 
 
 3-5000 
 
 5-437500 
 
 •453125 Ans. 
 
 To reduce a decimal to its equivalent in terms of lower denominations. 
 Rule. — Multiply the given decimal by the number of parts in the next 
 less denomination, and point off from the product as many figures 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 ; th<»n ^h-' -eve- 
 ral figures pointed off at the left will be the answer. 
 
 Example. — What is the expression in inches of 0-390625 feet '? 
 Feet 0-3906-25 
 
 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 f and -V- 
 Another example. — What is the expression, in fractions of ar inch, 
 ©f 0-6875 inches ? 
 
 Inches 0-6875 
 
 8 eighths in an inch. 
 
 Eighths 5-5000 
 
 2 sixteenths in an eighth. 
 
 Sixteenth 1-0 
 
 Ans., I and j\. 
 
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, 6zc. 
 
 
 
 
 Side of 
 
 
 
 
 Side oi" 
 
 Diam. 
 
 Area. 
 
 Circum. 
 
 equal sq. 
 
 Diam. 
 
 Area. 
 
 Circum. 
 
 equal sq. 
 
 ^ 
 
 •04908 
 
 •78539 
 
 •22155 
 
 ^ 
 
 90-76257 
 
 33-77212 
 
 9-526J3 
 
 •5 
 
 •19635 
 
 1-57079 
 
 -44311 
 
 11- 
 
 95-03317 
 
 34-55751 
 
 9-74SI.; 
 
 •75 
 
 •44178 
 
 2-25619 
 
 •66467 
 
 •25 
 
 99-40195 
 
 35-34291 
 
 9-'.i700o 
 
 1- 
 
 •78539 
 
 314159 
 
 •88622 
 
 •5 
 
 103-86890 
 
 3;r 1-2331 
 
 10-1916 ; 
 
 •25 
 
 122718 
 
 3-92699 
 
 1-10778 
 
 •75 
 
 108-43403 
 
 36-91371 
 
 10-41316 
 
 •5 
 
 1-76714 
 
 4-71-238 
 
 1-32934 
 
 12- 
 
 1130'.)733 
 
 37-69911 
 
 i;;-63472 
 
 •75 
 
 2-40528 
 
 5-49778 
 
 1-5508.) 
 
 •25 
 
 117-85SS1 
 
 33-48451 
 
 10 85827 
 
 2- 
 
 314159 
 
 6-28318 
 
 1-77-245 
 
 -5 
 
 122-71816 
 
 39-26990 
 
 11-077-^-^ 
 
 •25 
 
 3-97607 
 
 7-06858 
 
 1-99401 
 
 •75 
 
 1-27-67628 
 
 40 05530 
 
 ll-2..'-..3.; 
 
 •5 
 
 4-90873 
 
 7-8539S 
 
 2-21556 
 
 13- 
 
 1.32-7322S 
 
 40-84070 
 
 li -5 j( ;'.).'): 
 
 •75 
 
 5-93957 
 
 8-63937 
 
 2-43712 
 
 •25 
 
 137-83646 
 
 41 -6-2610 
 
 ll-7i:;5- 
 
 3^ 
 
 7-06853 
 
 9-42477 
 
 2-65SGS 
 
 -5 
 
 1431338] 
 
 42-4 11 50 
 
 ii-'jr.4 its 
 
 •25 
 
 8-29576 
 
 10-21017 
 
 2-88023 
 
 •75 
 
 148-43934 
 
 43-19689 
 
 12-Ir.5;i- 
 
 •5 
 
 9-62112 
 
 10-99557 
 
 3-10179 
 
 14- 
 
 153-93804 
 
 43-93-2-29 
 
 12-407 17! 
 
 •75 
 
 11-04466 
 
 11-78097 
 
 3-32335 
 
 •25 
 
 159-43491 
 
 44-76769 
 
 12-r>2sT3! 
 
 4^ 
 
 12-56637 
 
 12-56637 
 
 3-54490 
 
 •5 
 
 165- 1-2996 
 
 45-55309 
 
 12-350-i'jJ 
 
 •25 
 
 14-18625 
 
 13-35176 
 
 3-766461 
 
 •75 
 
 170-87318 
 
 46-33-S49 
 
 13-07 l^;i 
 
 •5 
 
 15-90431 
 
 14-13716 
 
 3-98802| 
 
 15- 
 
 176-71458 
 
 47-12333 
 
 13-2V'34j 
 
 •75 
 
 17-7-2054 
 
 14-92256 
 
 4-209571 
 
 •25 
 
 182-65416 
 
 47-U092S 
 
 13-5I^-.J;i 
 
 5- 
 
 19-63195 
 
 15-70796 
 
 4-43113; 
 
 ■f) 
 
 188-69190 
 
 43-69463 
 
 13-73o5i 
 
 •25 
 
 21-64753 
 
 16-49336 
 
 4-65269; 
 
 -75 
 
 191-8-2783 
 
 49-48003 
 
 13-'J5S..'7 
 
 •5 
 
 23-75829 
 
 17-27875 
 
 4-87424 
 
 16- 
 
 201-06192 
 
 50-26548 
 
 14-17y63 
 
 •75 
 
 25-96722 
 
 18-06415 
 
 5-09580J 
 
 •25 
 
 207-394-20 
 
 51-05088 
 
 14-401 Is 
 
 6- 
 
 28-27433 
 
 18-84955 
 
 5-31736 
 
 •5 
 
 213-82464 
 
 51-83G-27 
 
 14-62-27i 
 
 •25 
 
 30-67961 
 
 19-63495 
 
 5-53891 
 
 •75 
 
 220-35327 
 
 5262167 
 
 14-S443,) 
 
 •5 
 
 38-18307 
 
 20-42035 
 
 5-76047 
 
 17^ 
 
 226-98006 
 
 53-40707 
 
 15-0t'5-i5 
 
 •75 
 
 35-78470 
 
 21-20575 
 
 5-982031 
 
 •25 
 
 233-70504 
 
 54-19-247 
 
 15--287 4i 
 
 7- 
 
 38-48456 
 
 21-99114 
 
 6-20358 
 
 •5 
 
 240-5-28181 
 
 54-97737 
 
 15-5'J897 
 
 •25 
 
 41-28249 
 
 2!?'7765i 
 
 6-42514 
 
 •75 
 
 •247-44950 
 
 5576326 
 
 15-73-.;-52 
 
 •5 
 
 44-17864 
 
 23-56194 
 
 6-64670 
 
 13^ 
 
 264-46900 
 
 56-54366 
 
 15-95-20S 
 
 •75 
 
 47-17297 
 
 24-31731 
 
 6-86S25 
 
 -25 
 
 266 -536671 
 
 57-33406 
 
 16-17364 
 
 8^ 
 
 50-26548 
 
 25-13274 
 
 7-08981 
 
 •5 
 
 268-80252 
 
 53-11946 
 
 1639519 
 
 •25 
 
 53-45616 
 
 25-91813 
 
 7-31137 
 
 •75 
 
 276-116541 
 
 53-90436 
 
 1()-61()75 
 
 •5 
 
 56-74501 
 
 26-70353 
 
 7-53292 
 
 19^ 
 
 283-528731 
 
 59-69026 
 
 16-83331 
 
 •75 
 
 60-13204 
 
 27-48893 
 
 7-7544S 
 
 •25 
 
 291-03910! 
 
 60-47565 
 
 17-05936 
 
 9^ 
 
 63-61725 
 
 28 27433 
 
 7-97604 
 
 -5 
 
 298-64765! 
 
 61-26105 
 
 17-28142 
 
 •25 
 
 67-20063 
 
 29-05973 
 
 8-19759 
 
 •75 
 
 306-35437 
 
 62-04645 
 
 17-50298 
 
 •5 
 
 70-88218 
 
 29-84513 
 
 8-41915 
 
 20^ 
 
 314 159-26 
 
 62-83185 
 
 17-72453 
 
 •75 
 
 74-66191 
 
 30-63052 
 
 8-64071 
 
 •25 
 
 322-06233 
 
 63-61725 
 
 17-94609 
 
 10- 
 
 78-53981 
 
 31-41592 
 
 8-86226 
 
 •5 1 
 
 330 06357 
 
 64-40264 
 
 18-16765 
 
 •25 
 
 82-51589 
 
 32-20132 
 
 9-08382 
 
 •75 1 
 
 338-16-299 
 
 65-18804 
 
 18-38920 
 
 5 
 
 86-59014 
 
 32-98672 
 
 9-30538 
 
 21- i 
 
 346-36059 
 
 65-97344| 
 
 18-61076 
 
26 
 
 APPENDIX, 
 
 
 
 
 Side of ' 
 
 
 
 
 Side of 1 
 
 Diam. 
 
 Area. 
 
 Circum. 
 
 equal sq. 
 
 Diam. 
 
 Area. 
 
 Circum. 
 
 equal sq. | 
 
 Tl^ 
 
 354-65f)35 
 
 66^75884 , 
 
 18-83232 
 
 38- 
 
 1134-11494 
 
 119-38052 
 
 33-67662 
 
 •5 
 
 363-05030 
 
 67^54424 i 
 
 19-05387 1 
 
 •25 
 
 1149-08660 
 
 120-16591 
 
 33-89817 
 
 •75 
 
 371-54241 
 
 68-32964 
 
 19^27543 
 
 •5 
 
 1164-15642 
 
 120-95131 
 
 34-11973 
 
 22- 
 
 380-13271 
 
 69-115031 
 
 19^49699 
 
 •75 
 
 1179.32442 
 
 121-73671 
 
 34 34129 
 
 •25 
 
 388-82117 
 
 69-900431 
 
 19-71854 
 
 39- 
 
 1194-59060 
 
 122-52211 
 
 34-56*285 
 
 •5 
 
 3 J7 -60782 
 
 70-68583' 
 
 19-94010 
 
 •25 
 
 1209-95495 
 
 123-30751 
 
 34-78440 
 
 •75 
 
 406-49263 
 
 71-47r23 
 
 20-16166 
 
 •5 
 
 h225 41748 
 
 124-09290 
 
 35(10596 
 
 23- 
 
 415-475fi2 
 
 72-25663; 
 
 20-38321- 
 
 •75 
 
 1240-97818 
 
 124-87830 
 
 35-22752 
 
 •25 
 
 424-55679 
 
 7304202 
 
 20-60477 
 
 40- 
 
 r25f;-63704 
 
 125-66370 
 
 35-44907 
 
 •5 
 
 433-73613 
 
 73-82742 
 
 20-82633 
 
 •25 
 
 1272-39411 
 
 126-44910 
 
 35-67063 
 
 •75 
 
 443-01365 
 
 74-61282^ 
 
 21 04788 
 
 -5 
 
 1-288-21933 
 
 127-23450 
 
 35-89219 
 
 21- 
 
 452-38934 
 
 75-398"22 
 
 21-26944 
 
 •75 
 
 1304-20273 
 
 128-01990 
 
 36-11374 
 
 -25 
 
 461-8!")3-20 
 
 76-18362 
 
 21-49100 
 
 41- 
 
 13-20-25431 
 
 128-80529 
 
 36-33530 
 
 •5 
 
 471-43524 
 
 76-96902 
 
 2171255 
 
 -25 
 
 1336-40406 
 
 129-59069 
 
 36-55686 
 
 •75 
 
 481-10546 
 
 77-75441 
 
 21-93411 
 
 •5 
 
 1352-65198 
 
 130-37609 
 
 36-77841 
 
 25- 
 
 490-87385 
 
 78-53J81i 
 
 22-15567 
 
 •75 
 
 1368-99808 
 
 131-16149 
 
 .36-999<)7 
 
 •25 
 
 500-74041 
 
 79-32521! 
 
 22-377-22 
 
 42^ 
 
 1385-44*2.36 
 
 131-94689 
 
 37-22153 
 
 •5 
 
 510^705 15 
 
 80-11061 
 
 22-59878 
 
 •25 
 
 1401-98480 
 
 132-73-2-28 
 
 37-44308 
 
 •75 
 
 520-76^06 
 
 80-89601: 
 
 22-82034 
 
 •5 
 
 1418-62543 
 
 133-51768 
 
 37-66464 
 
 •.6^ 
 
 530-9-2915 
 
 81-68140 
 
 •23-04190 
 
 •7fS 
 
 1435-3&423 
 
 134-30308 
 
 37-88620 
 
 ••25 
 
 541-58842 
 
 82-46680 
 
 23-26345 
 
 43- 
 
 1452-20120 
 
 135-08348 
 
 3810775 
 
 •5 
 
 551-51586 
 
 ^ 83--25220 
 
 23-48501 
 
 •25 
 
 146.)-13635 
 
 135-87388 
 
 38-3-2931 
 
 •75 
 
 562-00147 
 
 84-03760 
 
 23-70657 
 
 -5 
 
 1486-16967 
 
 136-65928 
 
 38-5508"*/ 
 
 27- 
 
 572-55526 
 
 84-82300. 
 
 23-92812 
 
 •75 
 
 1503-30117 
 
 137-44467 
 
 33-77-242 
 
 •25 
 
 583-20722 
 
 85-60839' 
 
 24- 14968 
 
 44- 
 
 1520-53084 
 
 138-23007 
 
 38-99398 
 
 •5 
 
 5'j3-95736 
 
 86-39379 1 
 
 •24-371'24 
 
 -25 
 
 1537-85869 
 
 139-01547 
 
 39-21554 
 
 •75 
 
 604-80567 
 
 87-17919 
 
 24-59279 
 
 •5 
 
 1556-28471 
 
 139-80087 
 
 39-43709 
 
 28- 
 
 615-75216 
 
 87-96459. 
 
 24-81435 
 
 •75 
 
 1572-80890 
 
 140-58627 
 
 39 65865 
 
 •25 
 
 626-79682 
 
 88-74999; 
 
 25-03591 
 
 45- 
 
 1590-43128 
 
 141-37166 
 
 39-88021 
 
 •5 
 
 637-93965 
 
 89-53539 
 
 25-25746 
 
 •25 
 
 1608- 15182 
 
 142-15705 
 
 4010176 
 
 •75 
 
 649^ 18066 
 
 90-32078; 
 
 25-47902 
 
 •5 
 
 1625-97054 
 
 142-94246 
 
 40-32332 
 
 29- 
 
 660 51985 
 
 91-10613 
 
 25-70058 
 
 •75 
 
 1643 88744 
 
 143-72786 
 
 40-54488 
 
 •25 
 
 671-95721 
 
 91-89158' 
 
 25-92213 
 
 46- 
 
 1661-90251 
 
 144-513-26 
 
 40-76643 
 
 •5 
 
 683-49275 
 
 92-67698 
 
 •26-14369 
 
 •25 
 
 1680-01575 
 
 145-29866 
 
 40-98799 
 
 •75 
 
 6 J5- 12646 
 
 93-4623-^ 
 
 26-365-25 
 
 •5 
 
 1698-22717 
 
 146-08405 
 
 41-20955 
 
 3J- 
 
 706-85834 
 
 94-24777. 
 
 26 58680 
 
 •75 
 
 1716-53677 
 
 146-86945 
 
 41-43110 
 
 •25 
 
 718 68840 
 
 95-03317 
 
 26-80836 
 
 47- 
 
 1734-9445-1 
 
 147-65485 
 
 41-65266 
 
 •5 
 
 730-61664 
 
 95-81857 
 
 27-02992 
 
 -25 
 
 1753-45048 
 
 148-44025 
 
 41-87422 
 
 •75 
 
 742 64305 
 
 96-60397 
 
 27-25147 
 
 -5 
 
 1772-05460 
 
 149-22565 
 
 42-09577 
 
 31- 
 
 75r76763 
 
 97-38937 
 
 27-47303 
 
 -75 
 
 1790-75689 
 
 150-01104 
 
 42-31733 
 
 •25 
 
 766-99039 
 
 98-17477 
 
 27-69459 
 
 48- 
 
 1809 55736 
 
 150-79644 
 
 42-53889 
 
 •5 
 
 779-31132 
 
 98 96016 
 
 27-91614 
 
 -25 
 
 1828 -4 560 J 
 
 151-58184 
 
 42-76044 
 
 •75 
 
 791-73043 
 
 99-74556 
 
 28-13770 
 
 •5 
 
 1847-45282 
 
 152-36724 
 
 42-98200 
 
 32- 
 
 804-24771 
 
 100-53096 
 
 28 •:! 59-26 
 
 •75 
 
 1866-54782 
 
 153-15-264 
 
 43-20356 
 
 •'25 
 
 816^.><6317 
 
 101-31636 
 
 2^. ■58081 
 
 49^ 
 
 1885-74099 
 
 153-93804 
 
 43-42511 
 
 •5 
 
 8-29-57681 
 
 102^ 10176 
 
 2Sh0237 
 
 •25 
 
 1J05-83233 
 
 154-72343 
 
 43-64667 
 
 •75 
 
 812-38861 
 
 102-88715 
 
 29 02393 
 
 -5 
 
 19-24-42184 
 
 155-50883 
 
 43-86823 
 
 33- 
 
 855-29859 
 
 103-67255 
 
 29-21548' 
 
 •75 
 
 1943-90954 
 
 156-29423 
 
 44-08978 
 
 •25 
 
 868-30675 
 
 104-45795 
 
 29-46704 
 
 50- 
 
 1963-49540 
 
 157-07963 
 
 44-31134 
 
 •5 
 
 881-41308 
 
 105-21335 
 
 29-68860; 
 
 •25 
 
 1983-17944 
 
 157-96503 
 
 44-53290 
 
 •75 
 
 894-61759 
 
 106-02875 
 
 29-910l5j 
 
 •5 
 
 2002-96166 
 
 158-65042 
 
 44-75445 
 
 31- 
 
 907-92027 
 
 106-81415 
 
 30-13171! 
 
 •75 
 
 2022-84205 
 
 159-43582 
 
 44-97601 
 
 25 
 
 921-32113 
 
 107-59954 
 
 30-353-27; 
 
 51- 
 
 -2042-82062 
 
 160-22122 
 
 45-19757 
 
 •5 
 
 93i-8^2016 
 
 108-3S494 
 
 30-574821 
 
 -25 
 
 2062-89736 
 
 161-0(662 
 
 45-41912 
 
 •75 
 
 948 41736 
 
 109-17034 
 
 30-79638' 
 
 1 '5 
 
 2083-07227 
 
 161-79-202 
 
 45-64068 
 
 35^ 
 
 962- J 1275 
 
 109-95574 
 
 31-01794' 
 
 ' -75 
 
 2103-3 1536 
 
 16-2-57741 
 
 45-86-2-24 
 
 •25 
 
 975-90630 
 
 110-74114 
 
 31-23949; 
 
 52- 
 
 21-23-71663 
 
 163-36281 
 
 46-08380 
 
 •5 
 
 989-79803 
 
 111-52653 
 
 31-46105! 
 
 •25 
 
 2144-18607 
 
 164-14821 
 
 46-30535 
 
 •75 
 
 1003-78794 
 
 112-31193 
 
 31-68261; 
 
 -5 
 
 2164-75368 
 
 184-93361 
 
 46 5-2691 
 
 35^ 
 
 1017-87601 
 
 11309733 
 
 31-90416 
 
 •75 
 
 2185-41947 
 
 165^71901 
 
 46-74847 
 
 •25 
 
 1032-06'2-27 
 
 113-88-273 
 
 32- 1-2572 j 
 
 53- 
 
 2-206-18344 
 
 166^50441 
 
 46-97002 
 
 •5 
 
 1046-34670 
 
 114-66813 
 
 32-347281 
 
 •25 
 
 2-2-27-04557 
 
 167-28980 
 
 47-19158 
 
 •75 
 
 1060-72930 
 
 115-4^353 
 
 32-56883; 
 
 •5 
 
 -2248-00589 
 
 168-07520 
 
 47-41314 
 
 37^ 
 
 1075-2100S 
 
 116-23892 
 
 32-79039 
 
 •75 
 
 2269-06438 
 
 168-86060 
 
 47-63469 
 
 •25 
 
 1089-78903 
 
 117^02432 
 
 33-01195 
 
 54- 
 
 j -^290-22104 
 
 169-64600 
 
 47-85()-25 
 
 •5 
 
 1104-46616 
 
 117-80972 
 
 33-23350 
 
 •25 
 
 1 2311-475-^8 
 
 170-43140 
 
 48-07781 
 
 •75 
 
 11 19^24147 
 
 1 18-59572 
 
 33-45506 
 
 i -5 
 
 2332-8-2889 
 
 171-21679 
 
 48-29936 
 
APPENDIX. 
 
 27 
 
 1 — 
 
 
 
 Side of 
 
 1 
 
 
 
 Side of 
 
 iDiam. 
 
 Area. 
 
 Circura. 
 
 equal sq. 
 
 1 Diam. 
 
 Area. 
 
 Circum. 
 
 equal sq. 
 
 "5475 
 
 2354-28CH)3 
 
 17200219 
 
 48-52092 
 
 71-5 
 
 4015-15176 
 
 2-24-62337 
 
 63-365-22 
 
 55- 
 
 2375-82944 
 
 172-78759 
 
 48-74248 
 
 •75 
 
 4043-27833 
 
 2-25-40927 
 
 63-53678 
 
 •25 
 
 2397-47698 
 
 173-57-299 
 
 48-96403 
 
 72- 
 
 4071-50407 
 
 226-19467 
 
 63-80833 
 
 •5 
 
 2419-22269 
 
 1 174-35839 
 
 49-18559 
 
 •25 
 
 4099-82750 
 
 226-98006 
 
 64-0-2989 
 
 •75 
 
 2441-06657 
 
 175-14379 
 
 49-40715 
 
 •5 
 
 4123-24909 
 
 227-76546 
 
 64-25145 
 
 55- 
 
 246300864 
 
 175-9-2918 
 
 49-62870 
 
 •75 
 
 4156-76386 
 
 2-28-55086 
 
 64-47300 
 
 •25 
 
 2185-04887 
 
 176-71458 
 
 49-85026 
 
 73- 
 
 4185-33681 
 
 229-336-26 
 
 64-69456 
 
 •5 
 
 2507- 187-28 
 
 177-49998 
 
 50-07182 
 
 •25 
 
 4214-10-293 
 
 230-12166 
 
 64-91612 
 
 •75 
 
 25-20-42387 
 
 178-28538 
 
 50-29337 
 
 ■5 
 
 4242-917-22 
 
 230-907C6 
 
 65- 13767 
 
 57- 
 
 2551 -75863 
 
 i 179-07078 
 
 50 51493 
 
 •75 
 
 4271-82969 
 
 231-69245 
 
 65-35923 
 
 •25 
 
 2574-19156 
 
 179-85617 
 
 50-73649 
 
 74- 
 
 4300-84034 
 
 232-47785 
 
 65-58079 
 
 •5 
 
 2596-7-2267 
 
 1 180-64157 
 
 50-95304 
 
 •25 
 
 4329-94916 
 
 233-26325 
 
 65-80234 
 
 •75 
 
 2619-35196 
 
 181-4-2697 
 
 51-17960 
 
 •5 
 
 4359-15615 
 
 234-04865 
 
 66-02390 
 
 53- 
 
 264207942 
 
 182 21237 
 
 51-40116 
 
 •75 
 
 433s -46 132 
 
 234-83105 
 
 66-24546 
 
 •25 
 
 2664 90505 
 
 182-99777 
 
 51-62-271 
 
 75^ 
 
 4417 86466 
 
 235-61944 
 
 66-46701 
 
 •5 
 
 2687-82886 
 
 183-78317 
 
 51 844-27 
 
 -25 
 
 4447-36618 
 
 236-40484 
 
 66-68357 
 
 •75 
 
 2710-85084 
 
 184 -56855 
 
 52-06533 
 
 •5 
 
 4476-'..C5-<8 
 
 237-19024 
 
 66-91043 
 
 59^ 
 
 2733-97100 
 
 185-35396 
 
 52-28733 
 
 •75 
 
 4506-66374 
 
 237-97564 
 
 67-13168 
 
 •25 
 
 2757-18933 
 
 186-13936 
 
 52-50394 
 
 76^ 
 
 4536-45979 
 
 233-76104 
 
 67-35324 
 
 •5 
 
 2780-50584 
 
 186-9-2476 
 
 52-73050 
 
 -25 
 
 4566-35100 
 
 239-54643 
 
 67-57480 
 
 •75 
 
 28039-2053 
 
 187-71016 
 
 52 •95-205 
 
 •5 
 
 45.t6 34640 
 
 240-33133 
 
 67-79635 
 
 60- 
 
 23-27-43338 
 
 183-49555 
 
 53-17364 
 
 •75 
 
 46-26-43696 
 
 241-117-23 
 
 63-01791 
 
 •25 
 
 2851-04442 
 
 189-28095 
 
 53 39517 
 
 77- 
 
 4656 6-2571 
 
 241-90-263 
 
 68-23947 
 
 •5 
 
 2874-75362 
 
 19J-06635 
 
 53-61672 
 
 •25 
 
 4636-91-262 
 
 242-63803 
 
 68-46102 
 
 •75 
 
 2898-56100 
 
 190-85175 
 
 53-833-28 
 
 -5 
 
 4717-29771 
 
 243-47343 
 
 68-68253 
 
 61 
 
 2922-46656 
 
 191-63715 
 
 54 05'.''84 
 
 •75 
 
 4747-78093 
 
 244-25332 
 
 68-90414 
 
 •25 
 
 2946-47029 
 
 192-4-2255 
 
 54-28139 
 
 78- 
 
 4773-36-242 
 
 245 04422 
 
 69-12570 
 
 •5 
 
 2970-572-20 
 
 193-20794 
 
 54-50295 
 
 -25 
 
 4809-04204 
 
 245-82962 
 
 69 34725 
 
 •75 
 
 2994-772-23 
 
 193-99334 
 
 54-7-2451, 
 
 •5 
 
 4839-81983 
 
 246-61502 
 
 69-56381 
 
 62- 
 
 3019-07054 
 
 194-77374 
 
 54-94606 
 
 •75 
 
 4370-79579 
 
 247-40042 
 
 69-79037 
 
 •25 
 
 3043-46697 
 
 195^56414 
 
 55-16762 
 
 79^ 
 
 4901-66993 
 
 243-13581 
 
 70-01192 
 
 •5 
 
 3067-96157 
 
 196-34S54 
 
 55-33918 
 
 •25 
 
 4932-74225 
 
 248-97121 
 
 70-23343 
 
 •75 
 
 3092-55435 
 
 197-13493 
 
 5561073 
 
 •5 
 
 4963-91274 
 
 249-75661 
 
 70-45504 
 
 63- 
 
 3117-24531 
 
 197-92033 
 
 55-832-29 
 
 •75 
 
 4995-18140 
 
 250-34201 
 
 70-67659 
 
 •25 
 
 3142-03444 
 
 198-70573 
 
 56-05335, 
 
 80- 
 
 5026-51824 
 
 251-32741 
 
 70-39315 
 
 •5 
 
 3166-92174 
 
 199-49113 
 
 56-27540 
 
 •25 
 
 5053-013-25 
 
 252-11-231 
 
 71-11971 
 
 •75 
 
 3191-90722 
 
 200-27653 
 
 56-49696 
 
 -5 
 
 5039-57644 
 
 252-898-20 
 
 71-341-26 
 
 64- 
 
 3216-9-9U87 
 
 201-06192 
 
 56-71852 
 
 -75 
 
 5121-23781 
 
 253-63350 
 
 71-56282 
 
 •25 
 
 3242-17270 
 
 201-84732 
 
 56-94007 
 
 81- 
 
 5152-99735 
 
 254-46900 
 
 71-78433 
 
 •5 
 
 3-267-45270 
 
 20263272 
 
 57-16163 
 
 -25 
 
 5134-85506 
 
 255-25440 
 
 72-00593 
 
 •~5 
 
 3-292 83088 
 
 203-41812 
 
 57-33319 
 
 •5 
 
 5216-81095 
 
 256-03.^30 
 
 72-2-2749 
 
 65- 
 
 3318-30724 
 
 204-20352 
 
 5760475 , 
 
 •75 
 
 5-243-86501 
 
 256 82579 
 
 72-44905 
 
 •25 
 
 3313-88176 
 
 204-98392 
 
 57-82630 1 
 
 82^ 
 
 5-231 -017-25 
 
 257-61059 
 
 ■2-67060 
 
 •5 
 
 3369-55447 
 
 205-77431 
 
 53 04786 
 
 •25 
 
 531326766 
 
 258-39599 
 
 72-89216 
 
 •75 
 
 3395-3-2534 
 
 206-551^71 
 
 53-26942 
 
 •5 
 
 5345-61621 
 
 259-18139 
 
 7311372 
 
 66- 
 
 3421-19439 
 
 207-34511 
 
 53-49097 
 
 •75 
 
 5373-06301 
 
 259-96679 
 
 73-33527 
 
 25 
 
 344716162 
 
 203-13051 
 
 53-71-253 
 
 83^ 
 
 5410-60794 
 
 260-75219 
 
 73-55683 
 
 •5 
 
 3473-227u2 
 
 203-9 1591 
 
 58-93409 
 
 •25 
 
 5443-25105 
 
 261-53753 
 
 73-77839 
 
 •75 
 
 3499 39060 
 
 209-70130 
 
 59-15564 
 
 •5 
 
 5475-99234 
 
 262-32298 
 
 73-99994 
 
 67- 
 
 3525-65235 
 
 210 48670 
 
 59-37720, 
 
 •75 
 
 5508-83180 
 
 263-10338 
 
 74-22150 
 
 25 
 
 3552-01228 
 
 211-27210 
 
 59-59876 
 
 84- 
 
 5541-76944 
 
 263-89378 
 
 74-44306 
 
 •5 
 
 3573-4703S 
 
 212-05750 
 
 59-8-2031! 
 
 •25 
 
 5574-80525 
 
 264-67918 
 
 74-66461 
 
 •75 
 
 360502665 
 
 212-84-290 
 
 60-041871 
 
 •5 
 
 5607-93923 
 
 265-46457 
 
 74-88617 
 
 68- 
 
 3631-68110 
 
 213-62530 
 
 60-26343 
 
 •75 
 
 5641-17139 
 
 266-24997 
 
 75-10773 
 
 •25 
 
 3658-43373 
 
 214-41369 
 
 60-4S498 
 
 85^ 
 
 5674-50173 
 
 267-03537 
 
 75-32923 
 
 •5 
 
 3685-28453 
 
 215-19909 
 
 60-70654 
 
 •25 
 
 5707-93023 
 
 267-82077 
 
 75-55034 
 
 •75 
 
 3712-23350 
 
 215-984491 
 
 60-92810 
 
 •5 
 
 5741-45692 
 
 268-60617 
 
 75-77240 
 
 69- 
 
 3739-28065 
 
 216 76939 
 
 61-14965 
 
 •75 
 
 5775-08178 
 
 269-39157 
 
 75-99395 
 
 •25 
 
 37G6-42597 
 
 217-555-29i 
 
 61-37121 
 
 86^ 
 
 5808-80481: 
 
 270-17696 
 
 76-21551 
 
 •5 
 
 3793-66947 
 
 213340681 
 
 61-59277 
 
 •25 
 
 5842-6-2602' 
 
 270-96236 
 
 76-43707 
 
 •75 
 
 3821-01115 
 
 219- 12608! 
 
 61-81432 
 
 •5 
 
 5876-54540 
 
 271-74776 
 
 76-65362 
 
 70^ 
 
 3348-45100 
 
 219-91148 
 
 62-03538 
 
 •75 
 
 5910-56296 
 
 272-53316 
 
 76-33013 
 
 •25 
 
 3875-98902 
 
 2-20-69683I 
 
 62-25744 
 
 87^ 
 
 5944 67369 
 
 273-31856 
 
 77-10174 
 
 •5 
 
 3903-62522 
 
 22 1 -48-2281 
 
 62-47899 
 
 •25 
 
 5978-89260 
 
 274-10395 
 
 77-32329 
 
 •75 
 
 3931-35959 
 
 2-22-207681 
 
 62-70055 
 
 •5 
 
 6013-20468 
 
 274-88935 
 
 77-54485 
 
 7r 
 
 3959-19214 
 
 223-053071 
 
 62-9-2211 
 
 •75 
 
 6047-61494 
 
 275-67475 
 
 77-76641 
 
 •25 
 
 3987-12286 
 
 223 83347 1 
 
 63-14366 
 
 88^ 
 
 6082-12337 
 
 276-46015 
 
 77-98796 
 
28- 
 
 APPENDIX. 
 
 
 
 
 Side of 
 
 
 
 
 Side of 
 
 Diam. 
 
 Area. 
 
 Circum. 
 
 equal sq. 
 
 Diam. 
 
 Area. 
 
 Circum. 
 
 equal sq. 
 
 88-25 
 
 6116-72998 
 
 277-24555 
 
 78-20952 
 
 "94"^ 
 
 6976-74097 
 
 296-09510 
 
 83-52688 
 
 •5 
 
 6 151 •43476 
 
 278-03094 
 
 78-43103 
 
 •5 
 
 7013-80194 
 
 296-88050 
 
 83-74<944 
 
 •75 
 
 6186^23772 
 
 278-81634 
 
 78 •65-263 
 
 •75 
 
 7050-96109 
 
 297-66590 
 
 83-97000 
 
 89- 
 
 6221-13885 
 
 279-60174 
 
 78^87419 
 
 95- 
 
 7';)88-2]842 
 
 298-45130 
 
 84-19155 
 
 •25 
 
 6-256-13815 
 
 280-33714 
 
 79-09575 
 
 -25 
 
 7125-57992 
 
 299-23670 
 
 84-41311 
 
 •5 
 
 6291-23563 
 
 281-17254 
 
 79-31730 
 
 •5 
 
 7163-02759 
 
 300-02209 
 
 84^63467 
 
 •75 
 
 6326-43129 
 
 281-95794 
 
 79-53836 
 
 •75 
 
 7200-57944 
 
 300-80749 
 
 84-85622 
 
 90- 
 
 6361-72512 
 
 282-74333 
 
 79-76042 
 
 96- 
 
 7-238-22947 
 
 301-59289 
 
 85-07778 
 
 •25 
 
 6397-11712 
 
 283-52873 
 
 79-98198 
 
 .4/0 
 
 7-275-97767 
 
 302-37829 
 
 85-29934 
 
 •5 
 
 6432-60730 
 
 284-31413 
 
 80-20353 
 
 •5 
 
 7313-82404 
 
 303-16369 
 
 85-52089 
 
 •75 
 
 6468-19566 
 
 285-09953 
 
 80-1-250'J 
 
 •75 
 
 7351-76859 
 
 303-9 1908 
 
 85- 74-245 
 
 91- 
 
 6503-88219 
 
 285-8^493 
 
 80-646(39 
 
 97- 
 
 7389-81131 
 
 304-73448 
 
 85-96401i 
 
 .25 
 
 6539-66689 
 
 286-67032 
 
 80-86820 
 
 •25 
 
 7427-95221 
 
 305-51988 
 
 86-18556 
 
 •5 
 
 6575-51977 
 
 2S7-45572 
 
 81-08976 
 
 -5 
 
 7466-19129 
 
 30(5-30528 
 
 86-40712 
 
 •75 
 
 6611-53082 
 
 288-24112 
 
 81-31132 
 
 •75 
 
 7504-52853 
 
 .307-09068 
 
 86-6-2868 
 
 92^ 
 
 6547-61005 
 
 289-02652 
 
 81-53^87 
 
 93^ 
 
 7542-96396 
 
 307-87(')08 
 
 86-85023 
 
 •25 
 
 6683-78745 
 
 28.)-8il92 
 
 81-75443 
 
 •25 
 
 7581-49755 
 
 308-66147 
 
 87-07179 
 
 •5 
 
 67-20-06303 
 
 290-597:;-2 
 
 8I-975J9 
 
 •5 
 
 7620-12933 
 
 309-44637 
 
 87-29335 
 
 •75 
 
 6756-43678 
 
 291-3^271 
 
 82] 9754 
 
 •75 
 
 765S-85'.i27 
 
 310-23227 
 
 87-51490 
 
 93- 
 
 6792-90871 
 
 292-16811 
 
 82-41910 
 
 99^ 
 
 7(;vi7-68739 
 
 311-01767 
 
 87-73646 
 
 •25 
 
 68-29-478S1 
 
 292-9535] 
 
 8201066 
 
 ■25 
 
 7736-61369 
 
 3 11 -80307 
 
 87-95802 
 
 •5 
 
 6866-14709 
 
 293-7389] 
 
 82-86221 
 
 •5 
 
 7775-63816 
 
 312-58846 
 
 88^ 17957 
 
 •75 
 
 6902-91354 
 
 2'.)4-5243i 
 
 83-08377 
 
 •75 
 
 7814-76081 
 
 313-37336 
 
 88-40113 
 
 9^ 
 
 6939-77817 
 
 295 -30970 
 
 83 3i.''5;jo 
 
 !100- 
 
 7-^53-98iG3 
 
 314-159-26 
 
 88-6-2-269 
 
 The followinsf rules are for extfTidiiia the use of the above table. 
 
 To jind the area, circumference^ or side of equal square, of a circle 
 having a diameter of more than 100 inches, feet, SfC. 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 tliat divisor, or, the 
 area opposite that diameter, multiplied by the square of the aforesaid 
 divisor, will give the answer. 
 
 Example. — What 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 dianneters in 
 the table, the area of which is 176-71458, therefore : 
 
 176-71458 
 
 100=- 10 X 10 
 
 17,671-45800 inches. Ans. 
 To fnd the area, circumference, or side of equal square, of a circle 
 having an intermediate diameter to those in the tahle. 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. 
 
APPENDIX. 
 
 29 
 
 Example. — What 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 3S-484, therefore : 
 
 2)38-484 
 
 19-242 inches. Ans. 
 Another example. — What is the area of a circle, the diameter of 
 which is 3-2 feet"? 3-2, multiplied by 5, gives 16, and the area of 16 
 is 201-0619, therefore : 
 
 5 X 5 — 25)201-0619(8-0424 + feet. Ans. 
 200 
 
 106 
 100 
 
 61 
 50 
 
 119 
 100 
 
 19 
 Note. — The diameter of a circle, multiplied by 3-14159, vv^ill 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 2211S4 cubic 
 inches. Upon these data the following table is computed. 
 
 One foot in depth of a cistern of 
 3 feet diameter will contain 
 
 H 
 
 do. 
 
 do. 
 
 4 
 
 do. 
 
 do. 
 
 H 
 
 do. 
 
 do. 
 
 5 
 
 do. 
 
 do. 
 
 5i 
 
 do. 
 
 do. 
 
 6 
 
 do. 
 
 do. 
 
 6i 
 
 do. 
 
 do. 
 
 7 
 
 do. 
 
 do. 
 
 8 
 
 do. 
 
 do. 
 
 9 
 
 do. 
 
 do. 
 
 10 
 
 do. 
 
 do. 
 
 12 
 
 do. 
 
 do. 
 
 
 - 55-223 
 
 gallons 
 
 - 
 
 75-164 
 
 do. 
 
 
 - 98-174 
 
 do. 
 
 - 
 
 124-252 
 
 do. 
 
 
 . 153-39 
 
 do. 
 
 . 
 
 185-611 
 
 do. 
 
 
 - 220-893 
 
 do. 
 
 . 
 
 259-242 
 
 do. 
 
 
 - 300-66 
 
 do. 
 
 . 
 
 392-699 
 
 do. 
 
 
 . 497-009 
 
 do. 
 
 . 
 
 613-592 
 
 do. 
 
 - 
 
 - 883-573 
 
 do. 
 
 Note. — To reduce cubic feet to gallons, divide by the decimal, 
 128. 
 
TABLE OF POLYGONS. 
 
 (From Gregory's Mathematics.) 
 
 
 Names. 
 
 Multipliers for 
 areas. 
 
 Radius of cir- 
 cum. circle. 
 
 Factors for 
 sides. 
 
 3 
 
 Trigon 
 
 0-4330127 
 
 0-5773503 
 
 1-732051 
 
 4 
 
 Tetragon, or Square 
 
 l-OOOOOOO 
 
 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 
 
 10 
 
 Decagon 
 
 7-6942088 
 
 1-6180340 
 
 0-618034 
 
 11 
 
 Undecagon 
 
 9-3656399 
 
 1-7747324 
 
 0-503465 
 
 12 
 
 Dodecagon 
 
 11-1961524 
 
 1-9318517 
 
 0-517638 
 
 To Jind the area of any regular polygon, ivhose sides do not exceed 
 twelve. Rule. — 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 = 2X2 
 
 14-5356496: Ans. 
 To jind the radius of a circle which will circumscrihe any regular 
 polygon given, whose sides do not exceed twelve. Rule. — Multiply a 
 side of the given polygon by the number in the column termed Radius 
 of circumscrihing 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 he 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 
 
 ' 3-826835: Ans. 
 
WEIGHT OF MATERIAI^. 
 
 U)s. in a 
 Woods. cubic foot. 
 
 Apple, . - - - 49 
 
 Ash, . - - 45 
 
 Beach, - - - - 40 
 
 Birch, ... 45 
 
 Box, .... 60 
 
 Cedar, . - - 28 
 
 Virginian red cedar, - 40 
 
 Cherry, ... 38 
 
 Sweet chestnut, - - 36 
 
 Horse-chestnut, - - 34 
 
 Cork, .... 15 
 
 Cypress, - - - 28 
 Ebony, - - - -83 
 
 Elder, - - - 43 
 
 Elm, - . - - 34 
 
 Fir, (white spruce,) - 29 
 
 Hickory, - - - 52 
 
 Lance-wood, - - 59 
 
 Larch, - - - - 31 
 
 Larch, (whitewood,) - 22 
 
 Lignum-vitae, - - - 83 
 
 Logwood, ... 57 
 
 St. Domingo mahogany, - 45 
 
 Honduras, or bay mahogany, 35 
 
 Maple, - - - . 47 
 
 White oak, - - 43 to 53 
 
 Canadian oak, - - 54 
 
 Red oak, . - - 47 
 
 Live oak, - - - 76 
 
 White pine, - - 23 to 30 
 
 Yellow pine, - 34 to 44 
 
 Pitch pine, - - 46 to 58 
 
 Poplar, - - - 25 
 
 Sycamore, - - - 36 
 
 Walnut, - - 40 
 
 Metals. cubic foot. 
 
 Wire-drawn brass, - 534 
 
 Cast brass, - , - 506 
 
 Sheet-copper, - - 549 
 
 Pure cast gold, - - 1210 
 Bar-iron, - 475 to 487 
 
 Cast iron, - - 450 to 475 
 Milled lead, - - .713 
 Cast lead, - - 709 
 
 Pewter, - - - 453 
 
 Pure platina, - - 1345 
 Pure cast silver, - - 654 
 Steel, - - 486 to 490 
 
 Tin, - - - - 456 
 Zinc, - - - 439 
 
 Stone, Earths, cf*c. 
 
 Brick, Phila. stretchers, 105 
 North river common hard 
 brick, - - - 107 
 
 Do. salmon brick, 100 
 
 Brickwork, about - 95 
 
 Cast Roman cement, - 100 
 Do. and sand in equal parts, 113 
 Chalk, - 144 to 166 
 
 Clay, - - - - 119 
 Potter's clay, - 112 to 130 
 Common earth, 95 to 124 
 
 Flint, . - - - 163 
 Plate-glass, - - 172 
 
 Crown-glass, ... 157 
 Granite, - - 158 to 187 
 Quincy granite, - - 166 
 Gravel, - - - 109 
 Grindstone, - - - 134 
 Gvpsum, (Plaster-stone,) 142 
 Unslaked lime, - - 52 
 
32 
 
 ! APPENDIX. 
 
 
 ths. 
 
 in a 
 
 
 lbs. in a 
 
 cubic foot. 
 
 
 cvinc foot. 
 
 Limestone, - - 11-8 to 198 
 
 Common blue stone. 
 
 160 
 
 Marble, - - 161 to 177 
 
 Silver-gray flagging. 
 
 - 185 
 
 New mortar, - 
 
 107 
 
 Stonework, about. 
 
 120 
 
 Dry mortar, 
 
 90 
 
 Common plain tiles. 
 
 - 115 
 
 Mortar with hair, (Plaster- 
 
 
 Sundries. 
 
 
 ing,) ... - 
 
 105 
 
 Atmospheric air. 
 
 - 0-075 
 
 Do. dry, 
 
 86 
 
 Yellow beesxvax, - 
 
 - 60 
 
 Do. do. including Jath 
 
 
 Birch-charcoal, - 
 
 34 
 
 and nails, from 7 to 11 
 
 
 Oak-charcoal, 
 
 - 21 
 
 lbs. per superficial foot. 
 
 
 Pine-charcoal, 
 
 17 
 
 Crystallized quartz. 
 
 165 
 
 Solid gunpowder, - 
 
 - 109 
 
 Pure quartz-sand. 
 
 171 
 
 Shaken gunpowder. 
 
 58 
 
 Clean and coarse sand. 
 
 100 
 
 Honey, 
 
 . 90 
 
 Welsh slate, - - - 
 
 180 
 
 Milk, 
 
 64 
 
 Paving stone, 
 
 151 
 
 Pitch, - 
 
 . 71 
 
 Pumice stone. 
 
 56 
 
 Sea-water, 
 
 64 
 
 Nyack brown stone, - 
 
 148 
 
 Rain-water, - 
 
 - 62-5 
 
 Connecticut brown stone, 
 
 170 
 
 Snow, 
 
 8 
 
 Tarrytown blue stone, - 
 
 171 
 
 Wood.ashes, 
 
 - 58 
 
 ROOTS OF DECIMALS. 
 
 Rule. — Seek for the given decimal in the column of numbers, and opposite in the 
 columns of roots will be found the answer, correct as to the figures, but requiring 
 the decimal point to be shifted. The transposition of the decimal point is to be per- 
 formed thus : For every place the decimal point is removed in the root, remove it in 
 the number two places for the sq^iare root and three places for the cube root. 
 
 Examples. — By the table, the square root of 86-0 is 9'2736, consequently, by the 
 rule the square root of 0*86 is 0-92736. The square root of 9- is 3-, hence the square 
 root of 0"09 is 0-3. For the square root of 0*0657 we have 0-25fi? " found opposite 
 No. 657. So, also, the square root of 0-000927 is 0-030446, foui. -^site No. 927. 
 
 And the square root of 873 (whole number with decimals) is 2-95-- 1 opposite 
 
 No. 873. The cube root of 0-8 is 0-928, found at No. 800 ; the cube -f 0-08 is 
 
 0-4308, found opposite No. 80, and the cube root of 0-008 is 0*2, as 2-0 is the Cal)e 
 root of 8-0. So also the cube root of 0-047 is 0-36088, found opposite No. 47. 
 
 THE END. 
 
 \ 
 
 iN 589 
 
 Q-)