\ ^\ THE AMERICAN HOUSE-CARPENTER: A TREATISE UPON ARCHITECTURE, 3Z CORNICES AND MOULDINGS, FRAMINO, DOORS, WINDOWS, AND STAIRS. TOGETHER WITH THE MOST IMPORTANT PRINCIPLES PRACTICAL GEOMETRY. ^■. BY K G. HATFIELD, ARCHITECT. Sllustvafea lis more Qan tf)rcc fjuntrrttt 2Snsrab(ns», NEW-YORK & LONDON : WILEY AND PUTNAM. 1844. ^^^-^ t^Z ^ ^6 j^^^^r ^fu^c^^.f^^'^-o e-^^-^^^ ,< Y ^tiffei Entered according to the Act of Congress, in the year 1844, BY K. G. HATFIELD, In the Clerk's office of the District Court of the Southern District of New- York. NEW-YORK E WILLIAM OSBORN, PRINTER, 88 WiLLIAM-STBRBT, 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 unnecessary expense, as illustrations upon wood, printed on good paper, answer every useful purpose. Wood engravings, too, can be distributed among the letter-press ; an advantage which plates but partially possess, and one of great importance to the reader^ Considerations of this kind induced the author to undertake the preparation of this volume. The sub- ject matter has been gleaned from works of the first €iuthority, 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 laalure of the case would admits 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-railing 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. TABLE OF CONTENTS. INTRODUCTION. Art. Articles necessary for drawing, 2 To prepare the paper, - 5 To use the set-square, Directions for drawing, AH. 11 13 SECT. I.— PRACTICAL GEOMETRY. DEFINITIONS. Lines, - . . . Angles, - - - Angular point, - Polygons, - - - The circle, The cone. Conic sections, - - - The ellipsis, ... The cylinder, PROBLEMS. To bisect a line. To erect a perpendicular, - To let fall a perpendicular, To erect ditto on end of line, Six, eight and ten rule, - To square end of board. To square foundations, dsc. To let fall a perpendicular near the end of a line, To make equal angles, - To bisect an angle, - To trisect a right angle, To draw parallel lines, - To divide a line into equal parts, . . . - To find the centre of a circle, To draw tangent to circle. Do. without using centre. To find the point of contact, To draw a circle through three given points, 17 23 27 28 47 50 58 61 71 72 73 74 74 74 74 75 76 77 78 79 80 81 82 83 84 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 VI 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 conic 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- dinatQs, - - - 117 To trace a curve through given points, - - - 117 To describe an ellipsis by in- tersection of lines, - 118 Arl. Do. from conjugate diameters, 118 Do. by intersecting arcs, - 119 To describe an oval by com- passes, - - - 120 Do. in the proportion, 7x9, 5x7, &c., - - - 121 To draw a tangent to an el- lipsis, - - - 122 To find the point of contact, 123 To find a conjugate to the given diameter, - 124 To find the axes from given diameters, - - - 125 To find axes proportionate to given ones, - - 126 To describe a parabola by in- tersection of lines, - - 127 To describe hyperbola by do., 128 DEMONSTRATIONS . Definitions, axioms, &c., 130. 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. right angles, - - - 149 Corollaries from do., 150. 155 Angle in semi-circle a right angle, - - - 156 Hecatomb problem, - - 157 SECT. II.— ARCHITECTURE. HISTOKY, Antiquity of its origin. Its cultivation among the an- cients, ... Among the Greeks, - 1.59 160 Among the Romans, Ruin caused by Goths Vandals, Of the Gothic, and 161 Of the Lombard, 162 163 164 165 CONTENTS. Vll 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 Art. Extent of Roman structures, 202 Roman styles copied from Grecian, - - - 203 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. To proportion an order. 186 Origin of the art, 215 The Grecian orders. 187 Arrangement and design, - 21ff Origin of the Doric, - 188 Ventilation and cleanliness. 2ir Intercolumniation, - 189 Stability, 218 Adaptation, 190 Ornaments, - - - 219 Origin of the Ionic, 191 Scientific knowledge neces- Characteristics, 192 sary. 220 Intercolumniation, - 193 The foundation. 221 Adaptation, 194 The column, - - - 222 To describe the volute, - 195 The wall, 22a Origin of the Corinthian, 196 The lintel, - 224 Adaptation, - 197 The arch, 225 Persians, . . - - 199 The vault,' - 226. Caryatides, 200 The dome, ... 227 The Roman orders, - 202 The roof, 22&- SECT. III.— MOULDINGS, CORNICES, &c. MOULDINGS, &C. Elementary forms, - - 229 Characteristics, - - 230 Grecian and Roman, - - 231 Profile, - - - 232 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 Tofind shape of angle-bracket, 245 To find form of raking cornice, 246 VIU 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 to 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 Cross-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, - 306 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. IX. 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. Scai'fing, 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, &c. 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. General directions ing doors, 347 for hanff- 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 PLATFOKM 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, . . - . 36T Elucidation of this method, 368 Two other examples, 369, 37a 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, 37T CONTENTS. Art. Timbers to support winding stairs, . - - - To find falling-mould of rail, To find face-mould of do.. Position of butt-joint, To ascertain thickness of stuff, - - - - To apply the mould to plank, 383 Elucidation of the butt-joint, 384 Quarter-circle stairs, Falling-mould for do.. Face-mould for do., Elucidation of this method, To bevil edge of plank. To apply moulds without be- villing plank, ■ - 390 378 379 380 380 381 385 386 387 388 389 To find bevils for splayed- work, - - - 391 Another method for face- moulds, - - - 392 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, 404f APPEND IX. Page. Glossary of Architectural Terms, - . . - z 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, and 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-hoard must be regulated accord- ing to the size of the drawings which are to be made upon it. Yet for ordinary practice, in learning to draw, a board about 15 1 A 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 stuff is well-seasoned, the warping of the board will be but trifling ; and by exposing the rounding side to the fire^ or to the sun, it may be brought back to its proper shape. 4. — For mere line drawings, the paper need not 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 sheet may be cut in halves, or even quarters, for practising. If the drawing is to be much used, as working drawings generally are, cartridge- paper is much better than the other kind. 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 off the rough and uneven edge. With a sponge moderately wet, rub all the surface of the paper, except a strip around the edge about half an inch wide. As soon as the glistening of the water disappears, turn the sheet over^ and place it upon the INTRODUCTION. 3 board just where you wish it ghied. Commence upon one of the longest sides, and proceed thus : lay a flat ruler upon the paper, parallel to the edge, and within a quarter of an inch of it. With a knife, or any thing similar, turn up the edge of the paper against the edge of the ruler, and put one end of the cake of mouth-glue between your lips to dampen it. Then holding it upright, rub it against and along the entire edge of the paper that 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 sufiiciency 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 stiiF 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 affijrd 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. 1. 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 inefiaceable marks. It should be used lightly, so that the extra marks that are not needed when the drawing is inked, may be easily rubbed off with the rubber. The best kind of india-ink is that which will easily rub off upon the plate ; and, when the cake is rub- bed against the teeth, will be free from grit. 9. — The drawing-instruments may be purchased of mathe- matical instrument makers at various prices : from one to one hundred dollars a set. In choosing a set, remember that the lowest price articles are not always the cheapest. A set, com- prising a sufficient number of instruments for ordinary use, well made and fitted in a mahogany box, may be purchased at Pike and Son's, (Broadway, near Maiden-lane, N. Y.,) for three or four dollars. The compasses in this set have a needle point, which is much preferable to a common point. 10. — The best scale of equal parts for carpenters' use, is one that has one-eighth, three-sixteenths, one-fourth, three-eighths, one-half, five-eighths, three-fourths, and seven-eighths of an inch, and one inch, severally divided into tivelfths, instead of being divided, as they usually are, into tenths. By this, if it be required to proportion a drawing so that every foot of the object represented will upon the paper measure one-fourth of an inch, use that part of the scale which is divided into one-fourths of an 6 AMERICAN ilOUSE-CARPENTER. inch, taking for every foot one of those divisions, and for every inch one of the subdivisions into twelfths ; and proceed in like manner in proportioning a drawing to any of the other divisions of the scale. An instrument in the form of a semi-circle, called a protractor, and used for laying down and measuring angles, is of much service to surveyors, but not much to carpenters. 11. — In drawing parallel lines, when they are to be parallel to either side of the board, use the T-square ; but when it is required to draw lines parallel to a line which is drawn in a direction oblique to either side of the board, the set-square must be used. Let a b, {Fig. 2,) be a line, parallel to which it is Fig-. 2. desired to draw one or more lines. Place any edge, as c d, 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 h. 12. — To draw a line, as k I, {Fig. 3,) perpendicular to another, as a 6, 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 discovered in using the instruments. Fig. 3. 13. — In drawing a problem, proceed, with the pencil sharpened to a point, to lay down the several lines until the whole figure is completed ; observing to let the lines cross each other at the several angles, instead of merely meeting. By this, the length of every line will be , clearly defined. With a drop or two of water, rub one end of the cake of ink upon a plate or saucer, until a sufficiency adheres to it. Be careful to dry the cake of ink ; because if it is left wet, it will crack and crumble in pieces. With an inferior camel's-hair pencil, add a little water to the ink that was rubbed on the plate, and mix it well. It should be diluted sufficiently to flow freely from the pen, and yet be thick enough to make a Mack 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, eiface the pencil-marks with the india-rubber. If 8 AMERICAN HOUSE-CARPENTER. the pencil is used lightly, they will all rub oiF, 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. {Fig. 4.) Fig. 4. Fig. 5. Fig. 6. 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 ab 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. 8. 29. — An equi-lateral triangle has its three sides equal. {Fig. 7.) 30. — ^An isoceles triangle has only two sides equal. {Fig. 8.) 31. — A scalene triangle has all its sides unequal. {Fig. 9) Fig. 10. 32. — A right-angled triangle has one right angle. {Fig. 10.) 33. — ^An acute-angled triangle has all its angles acute. {Fig. 7 and 8.) 34. — An obtuse-angled triangle has one obtuse angle. {Fig. 9.) 35. — A quadrangle has four sides and four angles. {Fig. 11 ta 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 ;) ^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, (Fi^. 19 ;) an octagon eight, (F^^. 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. Fig. 21. 49. — The diameter is a right line passing through the centre, and terminating at two opposite points in the circumference. Hence it is twice the length of the radius, (c d, Fig. 21.) 50. — An arc of a circle is a part of the circumference, (c 6, or hed, 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. {B^ Fig. 21.) 54. — A quadrant^ or quarter of a circle, is a sector having a quarter of the circumference for its arc. (C, Fig. 21.) 55. — A tangent is a right line, which in passing a curve, touches, without cutting it. {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. Fig. 23. 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 d, Fig. 23 — c d being parallel tofg.) 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 h, 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 Z, 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 ci/linder is a solid generated by the revolution of a right-angled parallelogram, or rectangle, about one of its sides ; and consequently the ends of the cylinder are equal circles. {Fig. 25.) PRACTICAL GEOMETRY. 15 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 ylane of the segment. The circular segments are called, the ends of the cylinder. {Fig. 26.) PROBLEMS. RIGHT LINES AND ANGLES. 71. — To bisect a line. Upon the ends of the line, a b, [Fig. 27,) as centres, with any distance for radius greater than half a 6, 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 Fig. 28. 72. — To erect a perpendicular. From the point, a, {Fig. 28,) PRACTICAL GEOMETRY. 17 set off any distance, as a b, 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 b, with the same radius, describe another at d ; join d and a, and the hne, 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, b c, from which the perpendicular is re- quired to fall. Upon a, with any radius greater than a d, de- scribe an arc, cutting 6 c at e and/; upon the points, e and/, with any radius greater than e c?, 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 b, above the 3 18 AMERICAN HOUSE-CARPENTER. line, c a, and with the radius, h a, describe the arc, d a e; through d and 6, draw the line, d e ; join e and «, 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.) Fig. 31. 74, a. — A second method. Let 6, {Fig. 31,) at the end of the line, a b, be the point at which it is required to erect a perpen- dicular. Upon b, with any radius less than b a, describe the 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 b, and the line,/ 6, will be the perpendicular required. Fig. 32. 74, b. — A third method. Let b, {Fig. 32,) be the given point at which it is required to erect a perpendicular. Upon &, 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 «, 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 d, {Fig. 30,) equal eight, and a e, six ; then, ii d e equals ten, the angle, e a d, is b, right angle. Because the square of six and that of eight, added together, equal the square of ten, thus : 6 X 6 = 36, and 8 X 8 = 64 ; 36 + 64 = 100, and 10 x 10 = 100. Any sizes, taken in the same proportion, as six, eight and ten, will produce the same effect : as 3, 4 and 5, or 12, 16 and 20. (See note to Art. 103.) By the process shown at Fig. 30, the end of a board may be squared without a carpenters'-square. All that is necessary is a pair of compasses and a ruler. Let c a be the edge of the board, and a the point at which it is required to be squared. Take the point, b, as near as possible at an angle of forty-five degrees, or on a mitre-line, from a, and at about the middle of the board. This is not necessary to the working of the problem, nor does it affect its accuracy, but the result is more easily obtained. Stretch the compasses from b to a, and then bring the leg at a around to d ; draw a line from d, through 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 d, obliquely to the line, c a ; bisect e d at b ; upon b, with the radius, b e, describe the arc, e a d ; join e and a ; then e a will be the perpendicular required. 76. — To make an angle, (as e df Fig. 33,) equal to a given angle, (as b a c.) From the angular point, a, with any radius, describe the arc, 6c/ and with the same radius, on the line, d e, 20 AMERICAN HOUSE-CARPENTER. and from the point, c?, describe the wcc,fg; 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 ahgle, 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 6, 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 radius, 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. 21 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. 36. 79. — Through a given point, to draw a line parallel to a given line. Let a, {Fig. 36,) be the given point, and b c the given line. Upon any point, as d, in the line, b c, with the radius, d a, describe the arc, a c; upon a, with the same radius, describe the arc, d e ; make d e equal to a c ; through e and a, 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 A, {Fig. 37,) be the given line, and 5 the number of parts. Draw a c, at any angle Xo ah ; 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 b, 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 AMERICAN HOUSE-CARPENTER. {Fig. 38,) and bisect it with the perpendicular, c d ; bisect c d with the Hne, ef, 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 b, {Fig. 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 h, and c and d ; the intersection, o, is the centre. This is upon the same principle as Art. 85. Fig. 4a 81, b. — A third method. Draw any chord, as a 6, {Fig. 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 d — which will be the centre of the circle. If a circle be not too large for the purpose, its centre may very readily be ascertained by the help of a carpenters' -square, thus : app^ y the corner of the square to any point in the circumference, as at a ; by the edges of the square, (which the lines, a b and a c, represent,) draw lines cutting the circle, as at b and c ; join b and c ; then if 6 c is bisected, as at d, the point, d, will be the centre. (See Art. 156.) b'lg. 41. 82. — At a given point in a circle^ to draw a tangent thereto. Let a, {Fig. 41,) be the given point, and b the centre of the cir- cle. Join a and b ; through the point, a, and at right angles to a b, draw c d ; c dis the tangent required. 83. — The same, without making use of the centre of the circle. Let a, {Fig. 42,) be the given point. From a, set off any distance to 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 point of con- tact. Prom any point, as a, {Fig. 43,) in the tangent, b c, draw 24 AMERICAN HOUSE-CARPENTEK. a line to the centre d ; bisect a d at e ; upon 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 Art. 156.) 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 6, 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 ^; upon the intersection, h, with the radius, h a, describe the circle, ab c, and it will be the one required. Fig. 4& PRACTICAL GEOMETRY. 25 86. — Three points not in a straight line being given, to find a fourth that shall, ivith the three, lie in the circumference of a circle. Let a b c, {Fig. 45,) be the given points. Connect them with right hnes, forming the triangle, a c h ; bisect the angle, cb a, {Art. 77,) with the line, b d ; also bisect c a in e, and erect e d, perpendicular to a c, cutting b dm. d ; then d is the fourth point required. A fifth point may be found, as at/, by assilming a, d and 6, 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-triangle. 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 affixing a brace from e to/; put in pins at a and b ; move the angular point, c, m the direction, a c b ; keeping the edges of the triangle hard against the pins, a and 6 ; 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-style of a window- frame, where a brick arch, instead of a stone lintel, is to be placed over it. 26 AMERICAN HOUSE-CARPENTER. 88. — To describe the segment of a circle hy intersection of lines. Let a b, {Fig. 47,) 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, cf 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, (fec^ 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- Fig. 49. PRACTICAL GEOMETRY. 27 heads, window-heads, &c., to rather better advantage than Art. 87. For instance, let a b, {Fig. 49,) be the width of the open- ing, and c d the height of the arc. Extend c d, and make d e equal to c d ; join a and e, also e and b ; and proceed as direct- ed at Art. 89. Fig. 50. 90. — To describe a circle within any given triangle, so that the sides of the triangle shall be tangical. Let a b c, {Fig. 50,) be the given triangle. 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 the required circle. Fig. 51. 91. — About a given circle^ to describe an equi-lateral tri- angle. Let a d b c, {Fig. 5] ,) be the given circle. Draw the diameter, c d ; upon d, with the radius of the given circle, de- scribe the arc, a e b ; join a and b ; drsiwfg, at right angles to d c ; make/c and c g, each equal to a b ; from/, through a, draw / h, also from g, through b, draw g h; then fg h will be the triangle required. 38 AMERICAN HOUSE-CARPENTER. 92. — To find a right line nearly equal to the circumference of a circle. Let abed, {Fig. 52,) be the given circle. Draw the diameter, a c ; on this erect an 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 g f will be nearly the length of the semi- circle, ad c ; and twice g f will nearly equal the circumference of the circle, ab a d,SiS was required. Lines drawn from e, through any points in the circle, as o, o and 0, to^, p and/?, will divide^ /in the same way as the semi- circle, a d c, is divided. So, any portion of a circle may be transferred to a straight line. This is a very useful problem, and should be well studied ; as it is frequently used to solve problems on stairs, domes, 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-CARPETTTER. Again : suppose it be required, in a frame building, to find the length of a brace, having a run of three feet each way from the point of the right angle. The length of the sides containing the right angle will be each 3 feet : then, as before — 3 3 9 = square of one side, 3 times 3 = 9 = square of the other side. ] 8 Product : the square-root of which is 4*2426 + ft., er 4 feet, 2 inches and fths. full. (2.) — The hypothenuse and one side being known, to find the other side. Rule. — Subtract the square of the given side from the square of the hypothenuse, and the square-root of the product will be the answer. Suppose it were required to ascertain the greatest perpendicular height a roof of a given span may have, when pieces of timber of a given length are to be used as rafters. Let the span be 20 feet, and the rafters of 3 X 4 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 -f feet, or 8 feet, 3 inches and ^ths. full. This will be the greatest perpendicular height, as required. Again : suppose that in a story of 8 feet, from floor to floor, a step-ladder is re- quired, the strings of which are to be of plank, 12 feet long ; and it is desirable to know the greatest run such a length of string will afibrd. 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 fgths. — the answer, as required. PRACTICAL GEOMETRY. 37 Many other cases might he adduced to show the utility of this prohlem, A practical and ready method of ascertaining the length of braces, rafters, &c., when not of a great length, is to apply a rule across the carpenter s'-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 f- nearly, will be the length of the rafter in feet ; viz, 13 feet and gths 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 two given circles. Let A and jB, [Fig. 71,) be the given circles. In the right-angled tri- angle, ah c, make a h equal to the diameter of the circle, B, and c b equal to the diameter of the circle, A ; then the hypothenuse, Fig. 72. 38 American house-carpenter. a c, will be the diameter of a circle, C, which will be equal in area to the two circles, A and i?, added together. Any polygonal figure, as J[, {Fig. 72,) formed on the hypo- thenuse of a right-angled triangle, will be equal to two similar figures,* as B and C, formed on the two legs of the triangle. Fig. 73. 105. — To construct a square equal to a given rectangle. Let J., {Fig. 73,) be the given rectangle. Extend the side, a 6, and make h c equal to 6 e ; bisect a c in/, and upon/, with the radius, / a, describe the semi-circle, age; extend e b, till it cuts the curve in g ; then a square, h g h d, formed on the line, h g, will be equal in area to the rectangle, A. e b A « 8 Fig. 74. 105, a. — Another method. Let J., {Fig. 74,) be the given rectangle. Extend the side, a b, and make a d equal to a c ; * Sinular figures are such as have their several angles respectively equal, and their Bides respectively proportionate. PRACTICAL GEOMETRY. 39 bisect a din e ; upon e, with the radius, e a, describe the semi- circle, afd; 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 b, {Fig. 73,) equal the base of the given triangle, and b e equal half its perpendicular height, (see Fig. 67 ;) then proceed as directed at Art. 105. Fig. 75. 107. — Two right lines being given, to find a third jtropor- tional thereto. Let A and B, [Fig. 75,) be the given lines. Make a b equal to A ; from a, draw a c, at any angle with a b ; make a c and a d each equal to B ; join c and b ; from d, draw d e, parallel to c 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. Fig. 76. 108. — Three right lines being given, to find a fourth jpro- portional thereto. Let A, B and C, {Fig. 76,) be the given lines. Make a b equal to A ; from a, draw a c, at any angle with a b; make a c equal to B, and a e equal to C ; join c and b ; from e, draw e /, parallel to c b ; then a f will be the fourth proportional required. That is, a f bears the same proportion to C, as B does to A. 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 found ; so that it will bear the same proportion to the longer axis, as the shorter of the given ellipsis does to its longer. (See also, Art. 126.) c a 1 2 3 4 5 6 Fig. 77. 109. — A line with certain divisions being given, to divide another, longer or shorter, given line i?i 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, (fee, draw lines, parallel to c b ; then tftese will divide the line, a c, in the same proportion as B is divided — as was required. This problem will be found useful in proportioning the mem- bers of a proposed cornice, in the same proportion as those of a given cornice of another size. (See Art. 243 and 244.) So of a pilaster, architrave, &c. • Fig. 78. 110. — Between two given right lines, to find a mean pro- portional. Let A and B, {Fig. 78,) be the given lines. On the line, a c, make a b equal to A, and b c equal to B ; bisect a c in e ; upon e, with e a for radius, describe the semi-circle, a d PRACTICAL GEOMETRY. 41 c ; at h, erect h 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 €one 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 Art. 56 to 60.) If the cutting planes be at right angles with the plane, a 6 c, then — 112. — To find the axis 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 A:, at right angles to h i ; then twice g k will be the conjugate axis, and e/the transverse. 6 42 AMERICAN HOUSE-CARPENTER. 113. — To find the axis and base of the parabola. Let fn I, {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 d 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 o p, at right angles to a b; then ro will be the height, nr the transverse axis, and o p half the base ; as at Fig. 94. 115. — The axis being given, to find the foci, and to describe an ellipsis with a string. Let a b, {Fig. 80,) and c d, be the given axes. Upon c, with a e or 6 e for radius, describe the arc, ff; then /and/, the jooints 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 hnes, fg and g f, show tha 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 nse 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 ia much better. PRACTICAL GEOMETRY. 43 Fig. 81 116. — The axes being given, to describe an ellipsis with a trammel. Let a b and c d, {Fig. 81,) be the given axes. Place the trammel so that a line passing through the centre of the grooves, virould 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 may be constructed thus : take two straight strips of board, and make a groove on their face, in the centre of their width ; join them together, in the middle of their length, at right angles to one another ; as is seen at Fig. 81. A rod is then to be prepared, having two moveable nuts made of wood, with a mor- tice through them of the size of the rod, and pins under them large enough to fill the grooves. Make at hole at one end of the rod, in which to place a pencil. In the absence of a regular tram- mel, a temporary one may be made, which, for any short job^ will answer every purpose. Fasten two straight-edges at right angles to 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 J.r^. 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 u 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. ■' ■^ % ^ y^ n / 3 2 1 e 1 2 ') n V ^ D 1 2 3 A i d I Fig. 82. 117. — To describe an ellipsis by ordinates. Let a b and c c?, {Fig. 82,) be given axes. With a e or e 6 for radius, de- scribe the quadrant,/^ h; divide /A, a e and e 6, each into a like number of equal parts, as at 1, 2 and 3 ; through these points, draw ordinates, parallel to c d andfg- ; take the distance, 1 *, and place it at 1 1, transfer 2 j to 2 m, and 3 kto3 n; through the points, a, n, m, 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, m, I, &.C., and a thin slip of wood bent around by them, the curve can be made quite correct. This method is mostly used in tracing face-moulds for stair hand- railing. 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 ah ; from a and 6, draw a / and h g, at right angles to ab ; divide f a, g b, ae 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 to c ; through o, o and o, draw lines from d, 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 b, instead of being at right angles to c c^ ; also, / a and g b are drawn parallel to c d, instead of being at right angles to ah. 119. — To describe an ellipsis by intersecting arcs. Let a b 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 h 1 for radius, describe arcs at g, g, g and g ; upon/ and/ with a 1 for radius, describe arcs inter- secting the others at g^ g, g and g ; then these points of intersection will be in the cm-ve of the ellipsis. The other points, h and i, are found in like manner, viz: h is found by taking b 2 for one radius, and a 2 for the other ; i is found by taldng b 3 for one radius, and a 3 for the other, always using the foci for centres. Then by tracing a curve through the points, c, g, h, i, b, &c., the ellipse will be completed. This problem is founded upon the same principle as that of the string. This is obvious, when we reflect that the length of the string is equal to the transverse axis, added to the distance between the foci. See Fig. 80 ; in which c / equals a e, the half of the transverse axis. 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 icino; from o, draw og, at right angle* 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 7, through r and k, draw j m andj n ; also, from g, through k, draw g I; upon g and jV 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, i I and m n ; 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 approximation. 121. — To draw an oval in the proportion, seven by nine. Let c d, {Pig. 87,) be the given conjugate axis. Bisect c d ino, 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 e j and e i ; also, from g, through h and/, draw g k and g I ; upon g, with g c far radius, describe the arc, k I ; upon e, with e d for radius, describe the arc, j i ; upon h and /, with h a 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 h, one will be drawn in the proportion, five by seven. 48 AMERICAN HOUSE-CARPENTER. 122. — To draw a tangent to an ellipsis. Let abed, {Fig: 88,) 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, sr; then 5 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 b, parallel to the tangent ; any where between e and/, draw c d, parallel to ab ; bisect cd in ; through o and e, drsLW f g ; 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 ]me,fg, by the last problem; then fg will be the diameter required. PRACTICAL GEOMETRY. 49 Fig. 90. d 125. — Any diameter and its conjugate being given, to as- certain the two axes, and 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 ox hb ; join g and h ; upon g, with ^ c for radius, describe the arc, i k c j ; upon h, with the same radius, describe the arc. In; through the intersections, 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 ^; from e, through h, draw e *;*, also from/, through h, draw/p ; from A; and t, draw A: r and t s, parallel to^ h., cutting e ni in r, and/^ in s ; make h m equal to h r, and A _p equal to h s ; then r 7n> and 5 p will be the axes required, by which the ellipsis may be drawn in the usual way. 126. — To describe an ellipsis, whose axes shall be propor- tionate to the axes of a larger or smaller given one. Let a cbd, {Fig. 91,) be the given ellipsis and axes, and ij the trans- verse axis of a proposed smaller one. Join a and c ; from i, draw i e, parallel to ac ; make o f equal tooe ; then e/ will be m AMERICAN HOUSE-CARPENTER. Fig. 91. the conjugate axis required, and will bear the same proportion to ij, asc d does to a h, (See Art. 108.) 1 2 3 3 2 1 o\^^ V? i ^ [^ 7 \^ a ^ e 1 n \ k \ e 1 \ d 1 2 3 m 3 Fig. 92. 2 1 {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 6, {Fig. 95,) are right angles, and as the semi-circle, cad, contains 180 degrees, {Art. 131,) the measure of two right angles, therefore, is 180 degrees ; of one right angle, 90 degrees ; of half a right angle, 45 ; of one-third of a right angle, 30, &c. 134. — Definition. In measuring an angle, {Art. 132,) no re- gard is to be had to the length of its sides, but only to the degree of their inclination. Hence eqnal angles are such as have the same degree of inclination, without regard to the length of their sides. 6 d Fig. 97. 135. — Axiom. If two straight lines, parallel to one another, 54 AMERICAN HOUSE-CARPENTER. as a 6 andc d, {Fig. 97,) stand upon another straight line, as e/, the angles, ah f and c d f^ are equal ; and the angle, a b e, is equal to the angle, c d e. 136. — Definition. If a straight line, as a h, {Fig. 96,) stand obliquely upon another straight line, as c d, then one of the an- gles, as a & c, is called an obtuse angle, and the other, as ab d, an acute angle. 137. — Axiom. The two angles, ah d and a he, {Fig. 96,) are together equal to two right angles, {Art. 130, 133 ;) also, the three angles, ah d, eh a and cb 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. b d 140. — Proposition. If to each of two equal angles a third angle be added, their sums will be equal. Let ah c and d ef, {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, gb c, will be equal to the angle, h e f; for, ii ah c and d e/be angles of 90 degrees, and i j k, 30, then the angles, gb c and h ef, will be each equal to 90 and 30 added, viz : 120 degrees. PRACTICAL GEOMETRY. a d 55 141. — 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 h 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, f, the line, b c, must coincide with the line, e /; 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, oC which the sides, a b and a c, are equal. Bisect the angle, {Art. 56 AMERICAN HOUSE-CARPENTER. 77,) b a c, by the line, a d. Then the Hne, h a, being equal to the line, a c ; the line, a d, of the triangle, A, being equal to the line, a d, of the triangle, B, being common to each ; the angle, b a d, being equal to the angle, d a c ; the line, b d, must, accord- ing to Art. 141, be equal to the line, dc; and the angle at 6 must be equal to the angle at c. Fig. 101. 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 6 d, 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 ^^^ y^^ ^^ D S B 144. — Proposition. Let abed, {Fig. 102,) be a given pa- rallelogram, and 6 c a diagonal. At any distance between a b and c d, draw e f, parallel to ab; through the point, g, the intersection of the lines, b c and e f, draw h i, parallel to b d. In every paral- lelogram thus divided, the parallelogram, A, is equal to the paral- lelogram, B. According to Art. 143, the triangle, a & c, is equal to the triangle, bed; the triangle, C, to the triangle, D ; and EtoF; this being the case, take D and F from the triangle, bed, and C and E from the triangle, ab e, and what remains PRACTICAL GEOMETRY. 5r 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 abed and efcd, {Fig-. 103,) be given parallelograms, standing upon the same base, c d, and between the same parallels, a f and c d. Then, ab and e/ being equal to c d, are equal to one another; b e being added to both a b and ef, a e equals b f; the line, a c, being equal to b d, and a e to bf, and the angle, c a e^ being equal, {Art. 135,) to the angle, db f, the triangle, a e c^ must be equal, {Art. 141,) to the triangle, b f d ; these two triangles being equal, take the same amount, the triangle, beg, from each, and what remains in one, ab g c, must be equal to what remains in the other, efdg; these two quadrangles being equal, add the same amotint, the triangle, c g d, to each, and they must still be equal ; therefore, the parallelogram, abed, 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 d, {Fig. 103,) is double, {Art. 143,) the triangle, c e d. 147. — Proposition. Let abed, {Fig. 104,) be a given quad- rangle with the diagonal, a d. From b, draw b e, parallel toa d; extend cdto e ; join a and e ; then the triangle, a ec, will be equal in area to the quadrangle, abed. Since the triangles, adb and a d e, stand upon the same base, a d, and between the same paral- 58 AMERICAN HOUSE-CARPENTER. lels, a d and b e, they are therefore equal, {Art. 145, 146 ;) and since the triangle, C, is common to both, the remaining triangles, A and B, are therefore equal ; then B being equal to A, the triangle, a e c, is equal to the quadrangle, abed. Fig. 105. 148. — Proposition. If two straight lines cut each other, as a b and c d, {Fig. 105,) the vertical, or opposite angles, A and C, are equal. Thus, a e, standing upon c d, forms the angles, B and C, which together amount, {Art. 137,) to two right angles ; in the same manner, the angles, A and B, form two right angles ; since the angles, A and B, are equal to B and C, take the same amount, the angle, B, from each pair, and what remains of one pair is equal to what remains of the other ; therefore, the an- gle, A, is equal to the angle, C. The same can be proved of the opposite angles, B and D. 149. — Proposition. The three angles of any triangle are equal to two right angles. Let a b c, {Fig. 106,) be a given tri- angle, with its sides extended to/, e, and dy and the line, egj PRACTICAL GEOMETRY. 69 Fig. 106. drawn parallel to & e. As g c is parallel to e b, the angle, g c dj is, equal, [Art. 135,) to the angle, e hd ; as the lines, /c and h e, cut one another at a, the opposite angles, f a e and b a c, are equal, {Art. 148 ;) as the angle, / a e, is equal, ( J.rf. 135,) to the angle, a eg, the angle, a c ^, 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 at c are equal, {Art. 137,) to two right angles, the three angles of the triangle, a b c, must likewise be equal to two right angles. Any triangle can be subjected to the same proof. 150. — Corollary. Hence, if one angle of a triangle be a right angle, the other two angles amount to just one right angle. 151. — Corollary. If one angle of a triangle be a right angle, and the two remaining angles 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. b Fig. 107. 156. — Proposition. If from the extremities of the diameter of a semi-circle, two straight lines be drawn to any point in the cir- cumference, the angle formed by them at that point will be a right angle. Let ah c, {Fig. 107,) be a given semi-circle ; and a b and b c, lines drawn from the extremities of the diameter, a c, to the given point, b ; the angle formed at that point by these lines, is a right angle. Join the point, 6, and the centre, d ; the lines, d a, d b and d c, being radii of the same circle, are equal ; the angle at a is therefore equal, (Art. 142,) to the angle, ab d, also, the angle at c is, for the same reason, equal to the angle, d h c ; the angle, a b c, being equal to the angles at a and c taken together, must therefore, {Art. 152,) be a right angle. Fig. 108. 157. — Proposition. The square of the hypothenuse of a right-angled triangle, is equal to the squares of the two remaining sides. Let a b c, {Fig: 108,) be a given right-angled triangle, having a square formed on each of its sides : then, the square, b e, is equal to the squares, h c and g b, taken together. This can be PRACTICAL GEOMETRY. 61 proved by showing that the parallelogram, h I, is equal to the square, gb ; and that the parallelogram, c I, is equal to the square, h c. The angle, c b d,is a. right angle, and the angle, « 6 /, is a right 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- &, being both upon the samebase,/6, and between the same parallels, / b and^ c, the square, g b, is equal, {Art. 146,) to twice the triangle, fbc; the triangle, abd, and the parallelo- gram, b Z, being both upon the same base, b d, and between the same parallels, b d and a I, the parallelogram, b I, 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, k c and g b, taken together. This problem, which is the 47th of the First Book of Euclid, is said to have been demonstrated first by Pythagoras. It is sta- ted, (but the story is of doubtful authority,) that as a thank-offer- ing for its discovery he sacrificed a hundred oxen to the gods. From this circumstance, it is sometimes called the hecatomb pro- blem. It is of great value in the exact sciences, more especially in Mensuration and Astronomy, in which many otherwise intri- cate calculations are by it made easy of solution. These demonstrations, which relate mostly to the problems pre- viously given, are introduced to satisfy the learner in regard to their mathematical accuracy. By studying and thoroughly un- derstanding them, he will soonest arrive at a knowledge of their importance, and be likely the longer to retain them in memory. Should he have a relish for such exercises, and wish to continue them farther, he may consult Euclid's Elements, in which the whole subject of theoretical geometry is treated of in a manner sufficiently intelligible to be understood by the young mechanic. 62 AMERICAN HOUSE-CARPENTER. The house-carpenterj 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. SECTION II.— ARCHITECTURE. HISTORY OP ARCHITECTURE. 158. — Architecture has been defined to be — " the art of build- ing ;" but, in its common acceptation, it is — " the art of designing and constructing buildings, in accordance with such principles as constitute stability, utility and beauty." The literal signification of the Greek word archi-tecton, from which the word architect is derived, is chief-carpenter ; but the architect has always been known as the chief designer rather than the chief builder. Of the three classes into which architecture has been divided — viz., Civil, Military, and Naval, the first is that which refers to the construction of edifices known as dwellings, churches and other public buildings, bridges, &c.j 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<;arpenteii. morable event, which removed from existence all traces of the works of man — the Tower of Babel was commenced. This was a work of such magnitude that the gathering of the materials, according to some writers, occupied three years ; the period from its commencement until the work was abandoned, was twenty- two years ; and the bricks were like blocks of stone, being twenty feet long, fifteen broad and seven thick. Learned men have given it as their opinion, that the tower in the temple of Belus at Baby- lon was the same as that which in the scriptures is called the Tower of Babel. The tower of the temple of Belus was square at its base, each side measuring one furlong, and consequently half a mile in circumference. Its form was that of a pyramid and its height was 660 feet. It had a winding passage on the outside from the base to the summit, which was wide enough for two carriages. 160. — Historical accounts of ancient cities, of which there are now but few remains — such as Babylon, Palmyra and Ninevah of the Assyrians ; Sidon, Tyre, Aradus and Serepta of the 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 5 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 Hi 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 artj 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 were added to the Doric, and many mag- nificent edifices arose. These exemplified, in their chaste propor- tions, the elegant refinement of Grecian taste. Improvement in Grecian architecture continued to advance, until perfection seems to have been attained. The specimens which have been partially preserved, exhibit a combination of elegant proportion, dignified simplicity and majestic grandeur. Architecture among the Greeks was at the height o( its glory at the period immediately preceding the Peloponnesian war} after which the art declined. An excess of enrichment succeeded its former simple grandeur ; yet a strict regularity was maintained amid the profusion of orna- ment. After the death of Alexander, 323 B. C, a love of gaudy splendour increased : the consequent decline of the art was visible, and the Greeks afterwards paid but little attention to the science. 162. — While the Greeks were masters in architecture, which they applied mostly to their temples and other public buildings, the Romans gave their attention to the science in the construction of the many aqueducts and sewers with which Rome abounded j 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 under Csesar 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 j 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 what manner of stones, and what buildings are here." For the construction of private habitations also, finished artists were em- ployed by the Romans : their dwellings being often built with the finest marble, and their villas splendidly adorned. After Augus- tus, his successors continued to beautify the city, until the reign of Constantine ; who, having removed the imperial residence to Constantinople, neglected to add to the splendour of Rome ; and the art, in consequence, soon fell from its high excellence. Thus we find that Rome was indebted to Greece for what she possessed of architecture — not only for the knowledge of its prin- ciples, but also for many of the best buildings themselves ; these having been originally erected in Greece, and stolen by the un- principled conquerors — taken down and removed to Rome. Greece was thus robbed of her best monuments of architecture. Touched by the Romans, Grecian architecture lost much of its elegance and dignity. The Romans, though justly celebrated for their scientific knowledge as displayed in the construction of their various edifices, were not capable of appreciating the simple grandeur, the refined elegance of the Grecian style ; but sought to improve upon it by the addition of luxurious enrichment, and thus deprived it of true elegance. In the days of Nero, whose palace of gold is so celebrated, buildings were lavishly adorned. Adrian did much to encourage the art ; but not satisfied with the simplicity of the Grecian style, the artists of his time aimed at ARCHITECTURE. 67 inventing new ones, and added to the already redundant embel- lishments of the previous age. Hence the origin of the pedestal, the great variety of intricate ornaments, the convex frieze, the round and the open pediments, &c. The rage for luxury continued until Alexander Severus, who made some improve- ment ; but very soon after his reign, the art began rapidly to decline, as particularly evidenced in the mean and trifling charac- ter of the ornaments. 163. — The Goths and 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 are 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 Yerona 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 which were greatly destitute of proportion, elegance or utility — their characteristics being scarcely any thing 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 kind of architecture is called, the Lombard style, and was employed in the Tth century in Pavia, the chief city of the Lombards ; at which city, as also at many other places, a great many edifices were erected in accordance with its inelegant forms, 166. — The Byzantine architects, from Byzantium, Constantino- ple, erected many spacious edifices ; among which are included the cathedrals of Bamberg, Worms and Mentz, and the most an cient part of the minster at Strasburg ; in all of these they com- bined the Eoman-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 ails and sciences were mostly in possession of the Musselmen-conquerors ; at which time there were three kinds of architecture practised ; viz : the Arabian, the Moorish and the modern-Gothic. The Arabian style was formed from Greek models, having circular arches added, and towers which terminated with globes and minarets. The 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 AlhamWa, 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 zigzag, being the most common. Buildings in this style were erected in the 12th 70 AMERICAN HOUSE-CAHPENTER. 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 13th century. The Decorated style, as its name implies, is characterized by a great profusion of enrichment, which consists principally of the crocket, or feathered-ornament, and ball-flower. It was mostly in use in the 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 termed Gothic, were distinguished by peculiar 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 wonderful structure for the age, was erected by a Grecian architect in 1016. The marble with which the walls of this building were faced, and of which the four rows of columns that support the roof are composed, is said to be of an excellent character. The Campanile, or leaning-tower as it is usually called, was erected near the cathedral in the 12th cen- tury. Its inclination is generally supposed to have arisen from a poor foundation ; although by some it is said to have been thus constructed originally, in order to inspire in the minds of the beholder sensations of sublimity and awe. In the 13th century, the science in Italy was slowly progressing ; many fine churches were erected, the style of which displayed a decided advance in the progress towards pure classical architecture. In other parts 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 11th 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, Vignola, and other celebrated architects, each in their turn, did much to restore the art to its former excellence. In the edifices which were erected under their direction, however, it is plainly to be seen that they studied not from the pure models of Greece, but from the remains of the deteriorated architecture of Rome. The high pedestal, the cou- pled columns, the rounded pediment, the many curved-and-tvvisted enrichments, and the convex frieze, were unlaiown to pure Gre- cian architecture. Yet their eflbrts 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 OP ARCHITECTURE. 172. — It is generally acknowledged that the various styles in 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 nong all nations. Those that followed agricultural being employed constantly upon the same piece of permanent residence, and the wooden hut was the leir wants ; while the shepherd, who followed his s compelled to traverse large tracts of country for the tent to be the most portable habitation ; again, ed to hunting and fishing — an idle and vagabond -is naturally supposed to have been content with i place of shelter. The latter is said to have been e Egyptian style ; while the curved roof of Chi- gives a strong indication of their having had the todel ; and the simplicity of the original style of '. Doric,) shows quite conclusively, as is generally ts original was of wood. The modern-Gothic, or rhich was most generally confined to ecclesiastical aid by some to have originated in an attempt to '■er, or grove of trees, in which the ancients per- 3l-worship. are numerous styles, or orders, in architecture ; e of the peculiarities of each, is important to the rt. The Stylobate is the substructure, or base- lich the columns of an order are arranged. In ure — especially in the interior of an edifice — it 3 that each column has a separate substructure ; pedestal. If possible, the pedestal should be jes ; because it gives to the column the appear- been originally designed for a small building, pieced-out to make it long enough for a larger pursuits, fr land, neec offspring c flocks and pasture, for the man df way of livi the cavern the origin c nese struct tent for th the Greeks conceded, pointed st structures, imitate thi formed the 173.— T and a knov student in t ment, upon Roman arch frequently this is ca^^ avoided i j ance of ^^ and aft( one. 174- pal partfc OER, in architecture, is composed of two princi- le column and the entablature. ARCHITECTURE. 7^ 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 iSect. 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 shafio, 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 column. 181. — The Frieze, from ^iroTz^ 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,- zophoruSj 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 Avhich 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. All attic story is the upper story, its windows being usually square. 10 74 AMERICAN HOUSE-CARPENTER. ' 186. — An ordery 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 samcj 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 (he 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 i?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 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 t)f 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 <;apital. 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 inter cohimniation, 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-trig-lyph, and is most common. DORIC ORDER. Fis- no. ARCHITECTURE. 11 Wlien a column is placed beneath every third triglyph, the ar- rangement is called diastyle ; and when beneath every fourth, arcBostyle. This last style is the worst, and is seldom practised. 190. — The Doric order is suitable for buildings that are des- tined for national purposes, for banking-houses, &c. Its appear- ance, though massive and grand, is nevertheless rich and grace- ful. The Custom-House and the Union Bank, in Ne\7-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 v/hich 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 v/hom 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, pyaiostyle, or columns thick-set ; when two diameters, systyle ; when two and a quarter diameters, eiistyle ; when three diameters, diastyle ; and when more than 78 IONIC. Fiff. 111. ARCHITECTURE. 79 three diameters, arceosfyle, 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 row, 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. 80 AMERICAN HOUSE-CARPENTER. Fig. 113. 195. — To describe the Ionic volute. Draw a perpendicular from a to s, {Fig. 112,) and make a s equal to 20 min. or to f of the whole height, a c ; draw 5 o, at right angles to s a, and equal to li min. ; upon o, with 2| min. for radius, describe the eye of the volute ; about o, the centre of the eye, draw the square, rt\ 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, &c., {Fig. 113,) being the centre of the first small square towards the middle of the eye from the centre for the outside arc. The breadth of the fillet at aj, is to be made equal to 2-^\ min. This is for a spiral of three revolutions j but one of any number of revolutions, as 4 or 6, ARCHITECTURE. 81 May he drawn, by dividing of, {Fig. 113,) into a corresponding number of equal parts. Then divide the part nearest the centre, o, 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, h 4, will determine the length of the heavy lines, and the place of the centres. (See Art. 396.) 196. — The Corinthian Order is in general like the lonic^ though the proportions are lighter. The Corinthian displays a more airy eleganccj 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 ; which3 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, sucll 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 forthj 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 are II 82 CORINTHIAN. 5 8 8 Fig. 115 ARCHITECTURE, 83 dedicated to amusement, banqueting and festivity — for all places where delicacy, gayety and splendour ^re 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 Persia7is and Caryatides. 199. — Persians are statues of men, and are so called in com- memoration of a victory gained over the Persians by Pausanias. The Persian prisoners were brought to Athens and condemned to abject slavery ; and in order to represent them in the lowest state of servitude and degradation, the statues were loaded with the heaviest entablature, the Doric. 200. — Caryatides are statues of women dressed in long robes after the Asiatic manner- Their origin is as follows. In a war between the Greeks and the Caryans, the latter were totally van- quished, their male population extinguished, and their females carried to Athens. To perpetuate the memory of this event, statues of females, having the form and dress of the Caryans, were erected, and crowned with the Ionic or Corinthian entablature. The caryatides were generally formed of about the human size, but the Persians much larger ; in order to produce the greater awe and astonishment in the beholder. The entablatures were pro- portioned to a statue in like manner as 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 of 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 84 AMERICAN HOUSE-CARPENTER, 9.nd merit, which were erected by architects from Etruria ; but Rome was principally indebted to Greece for what she acquired of the art. Although there is no such thing as an architecture of Roman invention, yet no nation, perhaps, ever was so devoted to the cultivation of the art as the Roman. Whether we consider the number and extent of their structures, or the lavish richness and splendour with which they were adorned, we are compelled to yield to them our admiration and praise, At one time, under the consuls and emperors, Rome employed 400 architects. The public works — such as theatres, circuses, baths, aqueducts, ^c,— ^ were, in extent and grandeur, beyond any thing attempted in modern times. Aqueducts were built to convey water from a distance of 60 miles or more. In the prosecution of this work, rocks and mountains were tunnelled, and valleys bridged. Some of the latter descended 200 feet below the level of the water ; and in passing them the canals were supported by an arcade, or sucr 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 Greepe were introduced into Rome in all the richness and elegance of their perfection. But the luxu-r rious Romans, not satisfied with the siniple elegance of their re^ fined proportions, sought to improve upon them by lavish displays of ornament. They transformed in many instances, t\\e 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 necl^ ARCHITECTURE, 85 below it 5 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 the 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 git 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 iamong ancient ruins. 205. For mills, factories, markets, barns, stables, (fcc, where utility and strength are of more importance than beauty, the im- proved modification of this order, called the modern Tuscan, {Fig. 116,) will be useful ; and its simplicity recommends i| where economy is desirable. 806, — Egyptian Styi^e, The architecture of the ancient 86 TUSCAN. Fig, 116. ARCHITECTURE. 87 Egyptians — to which that of the ancient Hindoos bears some re- semblance — is characterized by boldness of outline, solidity and grandeur. The amazing labyrinths and extensive artificial lakes, the splendid palaces and gloomy cemeteries, the gigantic pyramids and towering obelisks, of the Egyptians, were works of immen- sity and durability ; and their extensive remains are enduring proofs of the enlightened skill of this once-powerful, but long since extinct nation. The principal features of the Egyptian Style of architecture are — uniformity of plan, never deviating from right lines and angles ; thick walls, having the outer surface slightly deviating inwardly from the perpendicular ; the whole building low ; roof flat, composed of stones reaching in one piece from pier to pier, these being supported by enormous columns, very short in proportion to their height ; the shaft sometimes polygonal, having no base but with a great variety of handsome capitals, the foliage of these being of the palm, lotus and other leaves ; entablatures having simply an architrave, crowned with a huge cavetto orna- mented with sculpture ; and the intercolumniation very narrow, usually I5 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 were employed in the erection of the Pyramids of Memphis alone, — or enough to con- struct 3,000 Bunker-Hill monuments. Some of the blocks are 40 B^ Egyptian. H. p. Fij. 117. ARCHITECTURE. S3 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- lecture is that of solemn grandeur — amounting sometimes to sepulchral gloom. For this reason it is appropriate for cemete- ries, prisons, &c. ; and being adopted for these purposes, it is, gradually gaining favour. A great dissimilarity exists in the proportion, form and general features of Egyptian columns. In some instances, there is no uniformity even in those of the same building, each differing from the others either in its shaft or capital. For practical use in this country. Fig. 117 may be taken as a standard of this style. The Halls of Justice in Centre-street, New- York city, is a building in general accordance with the principles of Egyptian architecture. Buildings in General, 208. — That style of architecture is to be preferred in which utility, stability and regularity, are gracefully blended with gran- deur and elegance. But as an arrangement designed for a warm country would be inappropriate for a colder climate, it would seem that the style of building ought to be modified to suit the wants of the people for whom it is designed. High roofs to resist the pressure of heavy snows, and arrangements for artificial heat, are indispensable in norlhern climes ; while they would be regarded as entirely out of place in buildings at the equator. 209. — Among the Greeks, architecture was employed chiefly upon their temples and other large buildings ; and the proportions of the orders, as determined by them, when executed to such large dimensions, have the happiest effect. But when used for small buildings,porticos, porches, &c., especially in country-places, they are rather heavy and clumsy ; in such cases, more slender proportions will be found to produce a better effect. The 12 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 'W • ^ r d b k3 F=? f=?_ I 1 F==L^ ^ ' :^(n^g^€Ss%^^5^:::-''^'' 543 2 10 5 10 15 Sljfi Fig. 122. 96 AMERICAN HOUSE-CARPENTER. the other stories be placed perpendicularly over and under them ; and be careful to provide for head-room. To ascertain this, when it is doubtful, it is well to draw a vertical section of the whole stairs ; but in ordinary cases, this is not necessary. To dispose the windows properly, the middle window of each story should be exactly in the middle of the front ; but the pier between the two windows which light the parlour, should be in the centre of that room ; because when chandeliers or any similar ornaments, hang from the centre-pieces of the parlour ceilings, it is important, in order to give the better effect, that the pier-glasses at the front and rear, be in a range with them. If both these objects cannot be attained, an approximation to each must be attempted. The piers should in no case be less in width than the window open- ings, else the blinds or shutters when thrown open will interfere with one another ; in general practice, it is well to make the out- side piers I 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 the piers ; divide this by 10, and the product will be i- 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. 216. — In all designs for buildings of importance, utility, dura- bility and beauty, the first great principles of architecture, should be pre-eminent. In order that the edifice be useful, commodious and comfortable, the arrangement of the apartments should be such as to fit them for their several destinations ; for public as- semblies, oratory, state, visitors, retiring, eating, reading, sleeping, bathing, dressing, &c.— -^these should each have its own peculiar form and situation. To accomplish this, and at the same time to make their relative situation agreeable and pleasant, producing regularity and harmony, require in some instances much skill and sound judgment. Convenience and regularity are very import- ant, and each should have due attention ; yet when both cannot be obtained, the latter should in most 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 pullies, so as to be easily raised and lowered. Small openings near the ceiling, that may be closed at pleasure, should be made in the partitions that separate the rooms from the passages — especially for those rooms which are used for sleeping apartments. All the apartments should be so arranged as to secure their being easily kept dry and clean. In dwellings, suitable apartments should be fitted up for bathing, with all the necessary apparatus for conveying the water. 218. — To insure stability in an edifice, it should be designed upon well-known geometrical principles : such as science has de- monstrated to be necessary and sufficient for firmness and dura- 13 98 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 enrichment : an error which is carefully to be guarded against. But those who take the Grecian models for their standard, will not be liable to go to that extreme. In ornamenting a cornice, or any other assemblage of mouldings, at least every alternate member should be left plain ; and those that are near the eye should be more finished than those whichf 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 surfiice of the ground, and constitutes the foundation. Where it is intended that the super- structure shall press unequally, as at its piers, chimneys, or columns, it is sometimes of use to occupy the space between the points of pressure by an inverted arch. This distributes the pressure equally, and prevents the foundation from springing be- tween the different points. In loose or muddy situations, it is always unsafe to build, unless we can reach the solid bottom below. In salt marshes and flats, this is done by depositing tim- bers, or driving wooden piles into the earth, and raising walls upon them. The preservative quality of the salt will keep these timbers unimpaired for a great length of time, and makes the foundation equally secure with one of brick or stone. 222. — The simplest member in any building, though by no means an essential one to all, is the Column, or pillar. This is a perpendicular part, commonly of equal breadth and thickness, not intended for the purpose of enclosure, but simply for the sup- port of some part of the superstructure. The principal force which a column has to resist, is that of perpendicular pressure. In its shape, the shaft of a column should not be exactly cylin- drical, but, since the lower part must support the weight of the superior part, in addition to the weight which presses equally on the whole column, the thickness should gradually decrease from bottom to top. The outline of columns should be a little curved, spas to represent a portion of a very long spheroid, or paraboloid, lOO 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 Avas 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 diflerent 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 Kock. But many of our modern stone walls, for the sake of cheapness, have only one face of the stones squared, the inner half of the wall being completed with brick ; so that they can, 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-CARtENTER. 224. — The Lintel, or beam, extends in a right line over a vacant space, from one column or wall to another. The strength of the lintel will be greater in proportion as its transverse vertical diameter exceeds the horizontal, the strength being always as the square of the depth. The floor is the lateral continuation or connection of beams by means of a covering of boards. 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 the semi-circle. It is, however, very fre- quently a smaller arc of a circle, and, still more frequently, a por- tion of an ellipse. The simplest theory of an arch supporting itself only, is that of Dr. Hooke, The arch, when it has only its own weight to bear, may be considered as the inversion of a chain, suspended at each end. The chain hangs in such a form, that the weight of each link or portion is held in equilibrium by ARCHITECTURE. 103 the result of two forces acting at its extremities ; and these forces, or tensions, are produced, the one by the weight of the portion of the chain below the link, the other by the same weight increased by that of the link itself, both of them acting originally in a ver- tical direction. Now, supposing the chain inverted, so as to con- stitute an arch of the same form and weight, the relative situa- tions of the forces will be the same, only they will act in contrary directions, so that they are compounded in a similar manner, and balance each other on the same conditions. The arch thus formed is denominated a catenary arch. In common cases, it differs but little from a circular arch of the extent of about one third of a whole circle, and rising from the abut- ments with an obliquity of about 30 degrees from a perpendicu- lar. But though the catenary arch is the best form for support- ing its own weight, and also all additional weight which presses in a vertical direction, it is not the best form to resist lateral pressure, or pressure like that of fluids, acting equally in all direc- tions. Thus the arches of bridges and similar structures, when covered with loose stones and earth, are pressed sideways, as well as vertically, in the same manner as if they supported a weight of fluid. In this case, it is necessary that the arch should arise more perpendicularly from the abutment, and that its general figure should be that of the longitudinal segment of an ellipse. In small arches, in common buildings, where the disturbing force is not great, it is of little consequence what is the shape of the curve. The outlines may even be perfectly straight, as in the tier of bricks which we frequently see over a window. This is, strictly speaking, a real arch, provided the surfaces of the bricks tend towards a common centre. It is the weakest kind of arch, and a part of it is necessarily superfluous, since no greater portion can act in supporting a weight above it, than can be included be- tween two curved or arched lines. Besides the arches already mentioned, various others are in use. The acute or lancet arch, much used in Gothic architecture, is 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 the inner, the in- trados. The springing lines are those where the intrados meets the abutments, or supporting walls. The span is the distance from one springing line to the other. The wedge-shaped stones, which form an arch, are sometimes cdXledi .voussoirs, the upper- most being the key-stone. The part of a pier from which an arch springs is called the impost, and the curve formed by the upper side of the voussoirs, the archivolt. 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 want of lateral firmness in the walls is compensated by a bar of iron stretched across the span of the arch, and connecting the abutments, like the tie-beam of a roof. This is the case in the cathedral of Milan and some other Gothic buildings. In an arcade, or continuation of arches, it is only necessary that the outer supports of the terminal arches should be strong enough to resist horizontal pressure. In the intermediate arches, the lat- eral force of each arch is counteracted by the opposing lateral force of the one contiguous to it. In bridges, however, where individual arches are liable to be destroyed by accident, it is desi- ARCHITECTURE. 106 i'able 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 which base 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, sbnietimes called cupola, is a concave cover- ing to a building, or part of itj 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 bnly by those above and below it, but also by those on each sidej It is only necessary that the constituent pieces should have St bommon form, and that this form should be somewhat like the frustum of a pyramid, so that, when placed in its situation^ its four angles may point toward the centre, of 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 laidj 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 isecure in this respect, being the reverse of the arch. The dome of the Pantheon, at Rome, has been always open at top, and yet has stood unimpaired for nearly 2000 years. The upper circle bf stones, though apparently the weakest, is nevertheless often tnade to support the additional weight of a lantern or tower above it. In several of the largest cathedrals, there are two domes, one \vithin 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, '^his construction is called building upon pendentives, and gives open space and lOom for passage beneath the dome. The remarks which have been made in regard to the abutments of the arch, apply equally to the walls immediately supporting a dome. They must be of sufficient thickness and solidity to resist the lateral pressure of the dome, which is very great. The wails of the Roman Pantheon are of great depth and solidity. In order that a dome in itself should be perfectly secure, its lower parts must not be too nearly vertical, since, in this case, they partake of the nature of perpendicular walls, and are acted upon by the spreading force of the parts above them. The dome of St. Paul's church, in London, and some others of similar construction, are bound with chains or hoops of iron, to prevent them from spreading at bottom. Domes which are made of wood depend, -in part, for their strength, on their in- ternal carpentry. The Halle du Bled, in Paris, had originally a wooden dome more than 200 feet in diameter, and only one foot in thickness. This has since been replaced by a dome of iron. (See Art. 303.) 228. — The 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 flat, but more frequently oblique, in its shap-e. The flat or platform-roof is the least advantageous for shedding rain, and is seldom used in northern countries. The pent roof, consisting of two oblique sides meeting at top, is the most common form. These roofs are made steepest in cold cli- mates, where they are liable to be loaded with snow. Where the four sides of the roof are all oblique, it is denominated a hipped roof, and where there are two portions to the roof, of different ob- liquity, it is a curb, or mansard roof. In modern times, roofs are made almost exclusively of wood, though frequently covered with incombustible materials. The internal structure or carpen- try of rpofs is a subject of considerable mechanical contrivance. ARCHITECTURE. 107 The roof is supported by rafters, which abut on the walls on each side, like the extremities of an arch. If no other timbers existed, except the rafters, they would exert a strong lateral pres- sure on the walls, tending to separate and overthrow them. To counteract this lateral force, a tie-beam, as it is called, extends across, receiving the ends of the rafters, and protecting the wall from their horizontal thrust. To prevent the tie-beam from sagging, or bending downward with its own weight, a king- post is erected from this beam, to the upper angle of the rafters, serving to connect the whole, and to suspend the weight of the beam. This is called trussing. Queen-posts are sometimes added, parallel to the king-post, in large roofs ; also various other connecting timbers. In Gothic buildings, where the vaults do not admit of the use of a tie-beam, the rafters are prevented from spreading, as in an arch, by the strength of the buttresses. In comparing the lateral pressure of a high roof with that of a low one, the length of the tie-beam being the same, it will be seen that a high roof, from its containing most materials, may produce the greatest pressure, as far as weight is concerned. On the other hand, if the weight of both be equal, then the low roof will exert the greater pressure ; and this will increase in propor- tion to the distance of the point at which perpendiculars, drawn from the end of each rafter, would meet. In roofs, as well as in wooden domes and bridges, the materials are subjected to an in- ternal strain, to resist which, the cohesive strength of the material is relied on. On this account, beams should, when possible, be of one piece. Where this cannot be effected, two or more beams are connected together by sjilicing. 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 upon similar principles, Roofs in the United States, after being boarded, receive a ser 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 sufiicient pitch to shed the rain from slates or shingles. Yarious artificial compositions are occasionally used to cover roofs, the most com- mon of which are mixtures of tar with lime, and sometimes witlpi sand and gravel." — Enoy. Am. (See Art. 285.) iSECTION 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 : I i Annulet, band, cincture, fillet, listel or square. Fi?. 124. __) Astragal or bead. _V Torus or tore. Fig. 125. Fig. 126. L Scotia, trochilus or mouth. Ovolo, quarter-round or echinus. Fi«. 127, 110 AMERICAN HOUSE-CARPENTER. Fig. 129. CavettOj cove or hollow. Cymatiunij or cyma-recta. I J Fig. 130. J ^ Ogee. Inverted cymatium, or cyma-reversa. ) Some of the terms are derived thus : fillet, from the French word^Z, 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 which is increased by the projection of the torus above it. Ovolo, from ovum., an egg, which this member resembles, when carved, as in the Ionic capi- tal. Cavetto, from cavus, hollow. Cymatium, from kumaton, a wave. 230. — Neither of these mouldings is peculiar to any one of the orders of architecture, but each one is common to all; and al- though each has its appropriate use, yet it is by no means con- fined to any certain position in an assemblage of mouldings. The use of the fillet is to bind the parts, as also that of the astra- gal and torus, which resemble ropes. The ovolo and cyma-re- versa are strong at their upper extremities, and are therefore used to support projecting parts above them. The cyma-recta and cavetto, being weak at their upper extremities, are not used as supporters, but are placed uppermost to cover and shelter the other parts. The scotia is introduced in the base of a column, to MOULDINGS, CORNICES, &C. Ill separate the upper and lower torus, and to produce a pleasing variety and relief. The form of the bead, and that of the torus, is the same ; the reasons for givin'g distinct names to them are, that the torus, in every order, is always considerably larger than the bead, and is placed among the base mouldings, whereas the bead is never placed there, but on the capital or entablature ; the torus, also, is never carved, whereas the bead is ; and while the torus among ,the Greeks is frequently elliptical in its form, the bead retains its circular shape. While the scotia is the reverse of the torus, the cavetto is the reverse of the ovolo, and the cyma- recta and cyma-reversa are combinations of the ovolo and cavetto. 23 i. — The curves of mouldings, in Roman architecture, were most generally composed of parts of circles ; while those of the Greeks were almost always elliptical, or of some one of the conic sections, but rarely circular, except in the case of the bead, which was always, among both Greeks and Romans, of the form of a semi-circle. Sections of the cone afford a greater variety of forms than those of the sphere ; and perhaps this is one reason why the Grecian architecture so much excels the Roman. The quick turnings of the ovolo and cyma-reversa, in particular, when exposed to a bright sun, cause those narrow, well-defined streaks of light, which give life and splendour to the whole. 232. — K 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 tke Greciafi torus and scotia. Join the extremities, a and b, {Fig. 131;) and from/, the given projection of the moulding, draw/ o, at right angles to the fillets ; from b, draw b h, 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. ters of the required ellipse. To describe the curve by interset-- tion of lines, proceed as directed at Art. 118 and noie ; by a trammel, see Art^ 125 ; and to find the foci, in order to describe it with a string, see Art. 115. Fig. 132. d \ ^ a Fig-. 133 23L—Fig. 132 to 139 exhibit various modifications of the Grecian ovolo, sometimes called echinus. Fig. 132 to 136 are MOULDINGS, CORNICES, &C. 113 Fi:r. 134. ,»'"••. ' ^L — ^ Fig. 136. Fig. 137. c N a ::^ A 5=^- ^ a « c ^ ^ Fig. 13&. Fig. 139^ elliptical, a h and h c being given tangents to the curve ; parallel to which, the semi-conjugate diameters, a d and d Cj are draAVn.^ In Fig. 132 and 133, the lines, a d and'c? c, are semi^axes, the tangents, a b and b c, being at right angles to each other. To draw the curve, see Art. 118. In Fig. 137, the curve is para^ bolical, and is drawn according to Art. 127. In Fig, 138 and 139, the curve is hyperbolical, being described according to Art. 128. The length of the transverse axis, a b, being taken at pleasure, in order to flatten the curve, a b should be made short in propor- tion to a c. IS 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 \M1 IJ^ fi V c Fi?. 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 h equal to the height, and divide abed into 4 equal parallelograms ; then proceed as directed in note to Art. 118. When the projec- tion is less than the height, draw d a, [Fig. 143,) at right angles to a b ; complete the rectangle, abed; divide this into 4 equal rectangles, and proceed according to Art. 118, 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,) ^.t right angles to the fillet ; make a d equal to the projection of the moulding : then proceed as directed for Fig. 142. 238. — ^Roman mouldings are composed of parts of circles, and have, therefore, less beauty of form than the Grecian. The bead and torus are of the form of the semi-circle, and the scotia, also, in some instances ; but the latter is often composed of two quad- rants, having difierent radii, as at Fig. 146 and 147, which re- semble the elliptical curve. The ovolo and cavetto are generally a quadrant, but often less. When they are less, as at Fig. 150, the centre is found thus : join the extremities, a and 6, and bisect ahm. c ; from c, and at right angles to a b, draw c d, cutting a level line drawn from a in d ; then d v/ill 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 Fjg., 146. Fig. 148, Fig. 149, 116 AMERICAN HOUSE-CARPENTER. Fig. 150. Fig. 151. W a Fig. 152, Fig, 153. Fig, 154, Fig, 155, Fig. 156. Fig. 1«. MOULDINGS, CORNICES, &C 117 Fig. 158. Fig. 159. Fig. 160. it cuts a perpendicular drawn from a, as at d; and that will bathe centre of the curve. In a similar manner, the centres are found for the mouldings 3X Fig. 147, 151, 153, 1.56, 157, 158 and 159. The centres for the curves at Fig. 160 and 161, are found thus : bisect the line, a b, at c ; upon a, c and b, successively, with a c or c 6 for radius, describe arcs intersecting at d and d ; then those intersections will be the centres. 239. — Fig. 162 to 169 represent mouldings of modern inven- tion. They have been quite extensively and successfully used in inside finishing. Fig. 162 is appropriate for a bed-moulding under a low, projecting shelf, and is frequently used under man- tle-shelves. The tangent, i h, is found thus : bisect the line, a b, at c, and b c aX d; from d, draw d e, at right angles to e 6 ; from 6j draw b f, 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 s, the limit of the projection, to o ; make o h equal to e ; from h, through c, draw the tangent, h i; 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. 163. Fig. 164. ^>—^flrwi,^^^ MOULDINGS, CORNICES, &C 119 Fig. 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 h, nearer horizontal, and describe an elliptic curve as shown in Fig: 131, 164, 175 and 176. Fig. 163 is much used on base, or skirting of rooms, and in deep panelling. The curve is found in the same manner as that of Fig. 162. In this case, however, where the moulding has so little projection 120 AMERICAN HOUSE-CARPENTER. in comparison with its height, the point, e, being found as in the last figure, h s may be made equal to s e, instead of o e as in 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 b, 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 A, at right angles to a b ; upon 0, with o f for radius, describe the arc,/ /i, cutting bh in h ; then make o c and o d, each equal to b h* To draw the curve, see note to Art. 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. H. P. 5 15 4 12} a 11 1 9 10} 10 14} Hi 111- 10} Fig. 170. Fig. 171. 240.— F^^. 170 and 171 are designs for antae caps. The * The manner of ascertaining the length of the conjugate diameter, d c, in this figure, and also in Fig. 131, 175 and 176, is new, and is important in this application. It is founded upon well-known mathematical principles, viz: All the parallelograms that may be circumscribed about an ellipsis are equal to one another, and consequently any one is equal to the rectangle of the two axes. And again : the sum of the squares of every pair of conjugate diameters is equal to the sum of the squares of the two axes. MouLOiNGfgj 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 H and P, height and projection- The projection is measured from the middle of the antse. These will be found appropriate for porticos^ door- ways, mantle-pieces, door and window trimmingSj &c. The height of the antas for mantle-pieces, should be from 5 to 6 diameters j 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 antee, 15 diameters high, and an en- tablature of 3 diametersj will have a better appearance. CORNICES. 241. — Fig. 172, 173 and 174, are designs for eave corniceSj and Fig. 175 and 176j 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 Hand P. H. P. U 20 17i 25 m J^ Tig. 172, 18 122 AMERICAN HOUSE-CARPENTER. H. P. riiso a 3k M 25 H. P. >'44 H 2i 2} Fig. 173. Fig. 174. MOULDINGS, CORNICES, &C. 123 Fig. 176. 124 AMERICAN HOUSE-CARPENTER, d h 12 3 4c Fig. nt. 242. — To propori'w7i an save cor?iice in accordance with the height of the building. Draw the line, a c, {Fig. 177,) and make b c and b or, 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 arc,/c?; divide the curve, df c, into 7 equal parts, as at 10, 20, 30, &c., and from these points of division, draw lines to b 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, &c. If the projection of a cornice for a building 34 feet high, is required, divide the arc between 30 and 40 into 10 equal parts, and from the fourth point from 30, draw a line to the base, b c, parallel with b d ; then the distance of the point, at which that line cuts the base, from b, will be the projec- tion required. So proceed for a cornice of any height within 70 feet. The above is based on the supposition that 18 inches is the proper projection for a cornice 70 feet high. This, for general purposes, will be found correct ; still, the length of the line, b c, may be varied to suit the judgment of those who think differ- ently. Having obtained the projection of a cornice, divide it into 20 equal parts, and apportion the several members according to its destination — as is shown at Fig. 172, 173 and 174, 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 b ; join a and b ; then b 1 will be the perpendicular height of the upper jfillet 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, izd; then ec? will be the required projection for the proposed cornice, and the perpendicular lines falling upon e d will indicate the proper projection for the members. 244. — To proportion a cornice according to a larger given dne. Let A, {Fig. 179,) be the given "cornice. Extend a o to 6, 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 to/ e ; 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 AMKRICAN HOUSE-CARPENTER. c z:^ ' -^t\ ^.^^^ ^ -p e 1 1 / // / \ r K A // / A 1 / / / . / Tig. 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 nave 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, ad ; join d and b ; from/, draw / e, parallel to db ; 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 s'rnaller 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. 127 g Fig. 180. Fig. 181. lei with the wall,/c?, draw the Ime, ah ; make e c equal a /, and through c, draw c 6, parallel with e d ; join rf and 6, and from the several angular points in B^ draw ordinates to cut c? 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, , draw a line, parallel to the diag- onal, ef; 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 Cis compressed, whereas J. ^ 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 J. ^ 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 wa,ll is pressed and the tie-beam stretched. 256.— -Another example : let E A B F, [Fig. 192,) represent a gate, supported by hinges at A and K. In this casej the strain^ ing force is the weight of the materials, and the direction of course vertical. Ascertain the dividing line at the several points, G, B, I, J, H and F. It will then appear that the force at G is sustained hj A G and G E^ and the dividing line shows that the former is stretched and the latter compressed. The force atiJis supported by A Ifand HE — the former stretched and the latter compressed. The force at B is opposed hj H B and A B, one pressed — the other stretched. The force at i^is sustained by G i^and FEj G i^ being stretched and FE pressed. By this it appears that A B is in a state of tension, and E F, of compres- sion; also, that A Hand G F sue stretched, while B H and G E are compressed : which shows the necessity of having A H and G jP, each in one whole length, while B i^and G E may be, as they are shown, each in two pieces. The force at /is sus- tained by G /and J H, the former stretched and the latter com- pressed. The piece, C Z>, is neither stretched nor pressed, and could be dispensed with if the joinings at /and 1 could be made as effectually without it. In case A B should fail, then C D would be in a state of tension. 257. — The pressure of inclined beams. The centre of gravi- ty of a uniform prism or cylinder, is in its axis, at the middle of its length. In irregular bodies with plain sides, the centre of 140 AMERICAN HOUSE-CARPENTER. gravity may be found by balancing them upon the edge of a prism in two positions, making a hne 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. Fiff. 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 line, B b, cutting 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 c?, (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 B, being equal and opposite, balance one another ; and their horizontal thrusts at the tie-beam are also equal. (See Art. 250 — Fig. 186.) When the inclination of a roof, {Fig. 194,) is one-fourth of the span, or of 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 surface. The inclination, in a rafter uniformly loaded, which will produce the least oblique pressure, {b e, Fig. 193,) is 35 degrees and 16 minutes. L-v^ fig. 195. 258. — In shed, or lean-to roofs, as Fig. 193, the horizontal pressure will be entirely removed, if the bearings of the rafters, as A, B, {Fig. 195,) are made horizontal — provided, however, that the rafters and other framing do not bend between the points of support. If a beam or rafter have a natural curve, the convex or rounding edge should be laid uppermost. 259. — A beam laid horizontally, supported at each end and uniformly loaded, is subject to the greatest strain at the middle 142 AMERICAN HOUSE-CARPENTER. of its length. The amount of pressure at that point is equal to half of the whole load sustained. The greatest strain coming upon the middle of such a beam, mortices, large knots and other defects, should be kept as far as possible from that point ; and, in resting a load upon a beam, as a partition upon a floor beam, the weight should be so adjusted that it will bear at or near the ends. (See Art. 282.) 260. — The resistance of timber. When the stress that a given load exerts in any particular direction, has been ascertain- ed, before the proper size of the timber can be determined for the resistance of that pressure, the strength of the kind of timber to be used must be known. The following rules for calculating the resistance of timber, are based upon the supposition that the tim- ber used be of what is called " merchantable" quality — that is, strait-grained, seasoned, and free from large knots, splits, decay, (&C. Fig. 198. The strength of a piece of timber, is to be considered in ac- cordance with 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. 196, its strength is termed the resistance to compression. When the force tends to pull it asunder in the direction of its length, {Aj Fig. 197,) it is termed the resistance to tension. And when strained by a force tending to break it crosswise, as at Fig. 198, its strength is called the resistance to cross strains. 261. — Resistance to compression. When the height of a piece of timber exceeds about 10 times its diameter if round, or 10 times its thickness if rectangular, it will bend before crushing. The first of the following cases, therefore, refers to such posts as would be crushed if overloaded, and the other two to such as would bend before crushing. In estimating the strength of tim- ber for this kind of resistance, it is provided in the following rules that the pressure be exactly in a line with 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. Rule. — Divide the given weight in pounds by 1000 for pine and 1400 for oak, and the quotient will be the least area of the post in inches. This rule requires that the area of the abutting surface be equal to the result : should there be, there- fore, a tenon on the end of the post, this quotient will be too small. Example. — What should be the least area of a pine post that will safely sustain 48,000 pounds ? 48,000, divided by 1000, gives 48 — the required area in inches. Such a post may be 6x8 inches, and will bear to be of any length within 10 times 6 inches, its least thickness. Case 2. — To find the area of a rectangular post that will safely bear a given weight — when its height is 10 times its least thickness or more. 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-dARPENTEit. 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 foufid 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 round, or cylmdrical. post, that will safely bear 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 pitch pine and '0016 for white pine ; then multiply the square^ root of this product by the height in feet, and the square-root of the last product will be the diameter required, in inches. Exam^ j>Ze.— 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 hav^e shown that the pressure should neVerbe 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, "vvlien the strain is applied in a line with the axis of the piece, the folloAving 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. — -Mitltiply 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. Example. — What should be the breadth of a beam of oak, having a bearing of 1 6 feet and a depth of 12 inches^ m Id 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} 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, which will be the depth in inches required. The breadth is equal to the depth multiplied by the decimal, 0-6. It is obvious that any other proportion of the breadth and depth may be ob- tained by merely changing the decimal, 0'6, in the rule. Exam- ple. — What should be the depth and breadth of a beam of pitch pine, having a proportion to one another as 6 to 1, and a bearing of 22 feet, in order to sustain a ton weight, or 2240 pounds ? This, multiplied by '01 6, gives 35"84, which, divided by 0'6, gives 59-73 ; the square-root of this is T'T, which, multiplied by 22, the length, gives 169'4; the square-root of this is 13 — which is the depth required. Then 13, multiplied by 0*6, gives 7'8 inches — the required breadth. Case 4. — When the beam is inclined, as A B, Fig: 193. Rule.— Multiply together the weight in pounds, the length of the beam in feet, the horizontal distance, A c, between the supports, in feet, and the decimal, -009, for oak, "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 one 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 10-18, nearly ; and this, multiplied by 0-6, gives 6-1 ; therefore, the size of the beam should be 10*18 inches by 6-1 inches. Fig. 199. 264. — To ascertain the scantling of the stiff est beam that can he cut from a cylinder. Let d a c h, {Fig. 199,) be the sec- tion, and e the centre, of a given cylinder. Draw the diameter, ah ; upon a and 6, with the radius of the section, describe the arcs, d e and e c ; join d and a, a and c, c and 6, and h and d ; then the rectangle, d a ch^ 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 aheam out of a given quantity of timher. iSwZe.^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 deptli in inches. Example. — "What is the strongest form for a beam whose given area of section is 48 148 AMERICAN HOUSE-CARPENTER. inches, and length of bearing 20 feet ? The length in feet, 20, multiplied by the decimal, 0-6, gives 12; the given area in inches, 48, divided by 12, gives a quotient of 4, the square of v/hich is 16 — this is the depth in inches ; and the breadth must be 3 inches. A beam 16 inches by 3 vi^ould bear twice as much as a square beam of the same area of section; which shows how im- portant it is to make beams deep and thin. In many old build- ings, and even in new ones, in country places, the very reverse of this has been practised ; the principal beams being oftener laid on the broad side than on the narrower one. 266. — Systems of Framing. 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 lerst 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 slngle-joisted, double, and framed. In a single- joisted floor, the timbers, or floor-joists, are disposed as is shown in Fig. 200. Where strength is the principal object, this manner of disposing the floor-joists is far preferable ; as experiments have proved that, with the same quantity of material, single-joisted floors are much stronger than either double or framed floors. To obtain the greatest strength, the joists should be thin and deep. 268. — To find the depth of a joist, the length of hearing and thickness being given, when the distance from ceritres is FRAMING. 149 Fig. 200. 12 inches. jRz^ie.— 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 7'988 inches, the depth required ; or 8 inches will be found near enough for practice. 269. — Where chimneys, flues, stairs, &c., occur to interrupt the bearing, the joists are framed into a piece, (6, Fig. 201,) called a trimmer. The beams, a, «, into which the trimmer is framed, are called trimming-bemns, trimm,ing-joist.9, or car- riage-beams. They need to be stronger than the commion 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 anotlier trim- mer, at the other end supporting the other half,) and the carriage- ISO 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, 6, is not more dis- tant from the bearing, d, than is necessary for ordinary hearths, &c., it will be sufficient to add \ of an inch to the thickness of the carriage-beam for every joist, c, that is supported. Thus, if the thickness of c is 3 inches, and the number of joists supported be 6, add 6 eighths, or 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 accurately in extreme cases. 270. — When the bearing exceeds 8 feet, there should be struts, as a and a, {Fig. 202,) well nailed between the joists. These will prevent the turning or twisting of the floor-joists, and will greatly 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. 161 Fig. 202. upon the underside. When the length of bearing is great, struts should be inserted at about every 4 feet. 271. — Single-joisted floors may be constructed for as great a length of bearing as timber of sufficient depth can be obtained ; but, in such cases, where perfect ceilings are desirable, either double or framed floors are considered necessary. Yet the ceil- ings under a single-joisted floor may be rendered more durable by cross-furring, as it is termed — which consists of nailing a series of narrow strips of board on the under edge of the beams and at right angles to them. To these, instead of the beams, the laths are nailed. The strips should be not over 2 inches wide — enough to join the laths upon is all that is wanted in width — and not more than 12 inches apart. It is necessary that all furring for plastering be narrow, in order that the mortar may have a suffi- 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 elects 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, hiiiding-joists, b, b, and ceiling-joists, c, c. The binding-joists 152 AMERICAN HOUSE-CARPENTER. 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 sufficiently firm when well nailed. The ceiling-joists are notched into the under side of the binders, and nailed ; they are the support of the lath and plastering. 273. — Binders are laid 6 feet apart. At this distance the fol- lowing rules will give the scantling. Case 1. — To find the depth of a binding-joist, the length and breadth being given. Rule. — Divide the square of the length in feet, by the breadth in inches ; and the cube-root of the quotient, multiplied by 3-42 for pine, or by 3*53 for oak, will give the depth in inches. 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 1 The square of 12 is 144, which, di- vided by 6, gives 24 ; the cube-root of this is 2-88, which, multi- plied by 3'42, gives 9*85, the depth in inches. Case 2. — To find the breadth, when the depth and length are given. Rule. — Divide the square of the length in feet, by the 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 40y gives 5-76, the breadth in inches. 274. — Bridging-joists are laid from 12 to 20 inches apart. The scantling may be four.d by the rule at Art. 268- 275. — Ceihng-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. Example. — What should be the depth of a ceiling-joist of pine, when the length of bearing is 6 feet and the thickness 2 inches 1 The length in feet, 6, divided by the cube-root of the breadth in inches, 1-26, gives a quotient of 4*76, which, being multiplied by the decimal, 0'64, gives 3 inches, the depth re- quired. When the thickness of a ceiling-joist is 2 inches, the depth in inches will be equal to half the length of bearing in feet. Thus, if the bearing is 6 feet, the depth will be 3 inches ; bearing. 8 feet, depth 4 inches, &c. 276. — Fram,ed floors. When a good ceiling is required, and the distance of bearing is great, the binding-joists, instead of reaching from wall to wall, are framed into girders. These are heavy timbers, as d, {Fig. 204,) which reach from wall to wall, being the chief support of the floor. Such an arrangement is termed a. framed floor. The binding, the bridging and the ceil- 20 154 AMERICAN HOUSE-CARPENTKR. Fig. 201. ing-joists in these, are the same as those in double floors just described. The distinctive feature of this kind of floor is the girder. 277. — Girders should be made as deep as the timber will allow : if their being increased in size should reduce the height of a story a few inches, it would be better than to have a house suffer from defective ceilings and insecure floors. In the fallowing rules for the scantling of girders, they are supposed to be placed at 10 feet apart. Case 1. — To find the depth, when the breadth of the girder and the length of bearing are given. Rule. — Divide the square of the length in feet, by the breadth in inches ; and the cube-root of the quotient, multiplied by 4-2 for pine, or by 4-3 for oak, will give the depth required in inches. 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 3-12, 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 tiinber, 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- permost. If the floor-plank are laid down temporarily at first, and left to season a few months before they are finally driven together and secured, the joints will remain much closer. But if the edges of the plank are planed after the first laying, they will shrink again ; as 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 cosily houses, may be attributed almost solely to this negligence. A square of parti- tioning weighs about half a ton, a greater weight, when added to its customary load, such as furniture, storage, &c., than any ordinary floor is calculated to sustain. Hence the timbers bend, the ceilings and cornices crack, and the whole interior part of the house settles ; showing the necessity for providing adequate supports independent of the floor-timbers. A partition should, if practicable, be supported by the walls with which it is connected, in order, if the walls settle, that it may settle with them. This would prevent the separation of the plastering at the angles of rooms. For the same reason, a firm connection with the ceiling is an important object in the con- struction of a partition. 282. — The joists in a partition should be so placed as to dis- charge the weight upon the points of support. All oblique pieces in a partition, that tend not to this object, are much better omitted. Fig. 206 represents a partition having a door in the middle. Its 158 AMERICAN HOUSE-CARPENTER. m U Fig. 206. f)0 Fig. 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 are suspended from it. The posts, a and 6, are halved, and nailed to the tie, c d, and the sill, e /. 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, 3|x5 for 30 feet, and 4x6 for 40. The thickness of the filling-in stuff may be regulated according to what is said at Art. 271, in regard to the width of furring for plastering. The FRAMING. 159 fiUing-in pieces should be stiflened at about every three feet 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 ofl" the corners. ■■^.- ^ 3E ^ =|p= ^ ^ ^- 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 proper distances for lathing, as in Fig. 208. A partition thus constructed requires a little more space than others ; but it has the advantage of insuring greater stability to the plastering, and also of preventing to a good degree the con- versation of one room from being heard in the other. When a partition is required to support, in addition to its own weight, that of a floor or some other burden resting upon it, the dimensions of 160 AMERICAN HOUSE-CARPENTER. the timbers may be ascertained, by applying the principles which regulate the laws of pressure and those of the resistance of tim- ber, as explained at the first part of this section. The following data, however, may assist in calculating the amount of pressure upon partitions : 284. — The weight of a square, (that is, a hundred square feet,) of partitioning 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 deafening^ (as described at the latter part of Art. 271,) at about 1500 lbs. When a floor is supported at two opposite extremities, 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 e:jierted 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. S'RAMIJfG; lei in a cold climate, for the weight of snow, in addition to the weight of the materials of which the roof is composed. The force of wind may be calculated at 40 lbs. on a square foot. The weight of snow will be of course according to the depth it acquires. {See weight of materials, in Appendix.) 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 SaUARE FOOT. Tin, Rise 1 inch to a foot. ■1 to \i lbs. Copper, Lead, " 1 " " 2 inches " 1 to li " 4 to 7 " Zinc, " 3 " " li to 2 " Short pine shingles, Long cypress shingles, Slate, u 5 a u u 6 " " u Q u u lito2i '' 4 to 5 " 5 to 9 " The weight of the covering, as above estimatedj is that of the material only, added to the weight of whatever is used to fix it to the roof, such as nails, &c. ; what the material is laid on, such as plank, boards or lath, is not included. 287. — Fig. 209 to 212 give a general idea of the usual manner of constructing trusses for roofs: c, {Fig. 209,) is a common 21 162 AMERICAN HOUSE-CARPENTER. FRAMING. 163 rafter ; i2 is a principal rafter ; ^ is a king-post ; s 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 Jind 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, 012, 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. Example. — What should be the dimensions of a pine king-post, 8 feet long, for a roof having a span of 25 feet 1 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. — Tojiiid the dim,ensions 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. Example. — 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 7 12|j multiplied by 6, gives 76, ■J.64 AMERICAN HOUSE-CARPENTER. ■which, multiplied by 0:27, gives 20-52 ; which indicates a size of about 4x5?. 291. — Tojind the dimensiojis of a tie-heam, 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 tie-beam, 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 dimensions of a principal rafter when there is a king-post in the tniddle. Mule. — 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-p)ost. Rule. — The same as the last, except that the decimal, 0-155, must be used instead of 0-96. Exatnple. — What should be the dimensions of a principal rafter, having a length of 14 feet, (as in Fig. 210,) and a thickness of 6 inches, when the span of the roof is 38 feet and the wood is pine? The square of 14 is 196, which, multi- plied by 38, gives 7448 ; this, divided by 216, the cube of 6, gives FRAMING, 165 34-48, which, multiplied by 0-155, gives 5-34. The size of the rafter should, therefore, be 6x5|. 294. — To find the diniensions of a straining-heam. 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 square-root of the product. Multiply this root by 0*9 for pine, which will give the depth in inches To find the thickness, multiply the depth by the decimal, 0"7. 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 5f x7|, or 5|x8. 295. — To find the dimensions of struts and braces. Rule. — Multiply the square-root of the length supported in feet, by the length of the brace or strut in feet ; and the square-root of the product, multiplied by 0-8 for pine, will give the depth in inches ; and the depth, multiplied by the decimal, 0*6, will give the thick- ness in inches. Example. — In Fig. 210, the part supported by the brace or strut, o, 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 1 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,ensio?is 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.— yfhoX should be the dimen- sions of pine purlins, 9 feet long and 6 feet apart ? The cube of 9 is 729, which, multiplied by (>, gives 4374; the fourth root of this is 8*13 — the required depth. This, multiplied by 0*6, gives- 4'878 — the thickness. A proper size for them would be about 5x8 inches. Purlins should be long enough to extend over two, three or more trusses. 297. — To find the dimensions of coinmoji 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. Example. — What should be the depth of a pine rafter,. 7 feet long and 2 inches thick ? 7 feet, divided by 1*26, the cube- root 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 and 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. 167 290. — Fig, 213 shows a method of constructing a trass having ^ built-rib in the place of principal rafters. The proper form for the curve is that of a parabola, {Art. 127.) This curve, when as flat as is described in the figure, approximates so near to that of the circle, that the latter may be used in its stead. The height, u b, is just half of a c, the curve to pass through the middle of the rib. The rib is composed of two series of abutting pieces, bolted together. These pieces should be as long as the dimen- sions of the timber will admit, in order that there may be but few joints. The suspending pieces are in halves, notched and bolted to the tie-beam and rib, and a purlin is framed upon the upper end of each, A truss of this construction needs, for ordinary roofs, no diagonal braces between the suspending pieces, but if extra strength is required the braces may be added. The best place for the suspending pieces is at the joints of the rib. A rib of this kind will be sufiiciently 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. a } \^ t ^ ^ f / t / 1 Fig, 215. 301. — Figi 215 is a method of obtaining the proper lengths and bevils for rafters in a hip-roof, a h and h c are walls at the angle of the building ; 6 e is the seat of the hip-rafter and g f of sL 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 h^ describe the arc, h i, cutting diini; join b and tj and extend gf to meet biinj ; then gj will pkAmiijg. 160 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/j draw f k, at right angles iofg, also f I, at right angles to be; makefk equal to /^ by the arc, I k, or make g k equal to g j by the arc, j k ; then the angle at J 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 w, at right angles to be; from o, tangical to b h, describe a semi-circle, cutting 6 e in 5 ; join m and 5 and n and 5 ; then these lines will form at s the proper angle for beviling the top of the hip-rafter. DOMESi Fig. 21 6i Fig. 217. * See ako Art. 237, 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. Fig. 2ia Fij. 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 " 1^X10 " " ' 60 " 2x13 " " " 90 " 2|xl3 " " " 108 " 3x13 " 306. — Although the outer and the inner surfaces of a dome may be finished to any curve that may be desired, yet the framing should be constructed of such a form, as to insure that the curve of equilibrium will pass through the middle of the depth of the framing. The nature of this curve is such that, if an arch or dome be constructed in accordance with it, no one part of the structure will be less capable than another of resisting the strains and pressures to which the Avhole fabric may be exposed. The curve of equilibrium for an arched vault or a roof, where the load is equally diffused over the whole surface, is that of a parabola, {Art. 127 ;) for a dome, having no lantern, tower or cupola above it, a cubic parabola^ {F^S- ^^^ ?) ^^^^ ^''^^ one having a tower, ; through the points thus found, describe the section of the newel-cap, as shown in the figure. APPENDIX. GLOS SAR Y. Terms not found here can be found in the lists of definitions in other parts of this hdSk^ or in common dictionaries. Abacus. — The uppermost member of a capital. Abtatoir. — A slaughter-house. Ahiey. — The residence of ah abbot or abbess. Abutment. — That part of a pier from which the arch springs. Acanthus. — A plant called in English, bear's-breech. Its leaves are ernployed for decorating the Corinthian and the Composite capitals. Acropolis. — The highest part of a city ; generally the citadel. Acroteria. — The small pedestals placed on the extremities and apex of a pediment, originally intended as a base for sculpture. Aisle. — Passage to and from the pews of a church. In Gothic ar- chitecture, the lean-to wings on the sides of the nave. Alcove. — Part of a chamber separated by an estrade, or partition of columns. Recess with seats, &c., in gerdens. Altar. — A pedestal whereon sacrifice was offered. In modern churches, the area within the railing in front of the pulpit. Alto-relievo. — High relief; sculpture projecting from a surface so as to appear nearly isolated. Amphitheatre. — A double theatre, employed by the ancients for the exhibition of gladiatorial fights and other shows. Ancones. — Trusses employed as an apparent support to a cornice upon the flanks of the architrave. Annulet. — A small square moulding used to separate others ; the fillets in the Doric capital under the ovolo, and those which separate the flutings of columns, are known by this term. A7itce. — 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. ArchivoU. — The ceiling of a vault : the uwder surface of an arcfi. 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 Talus. Battlement. — Indentations on the top of a wall or parapet. Bay-window. — A window projecting in two or more planes, and not forming the segment of a circle. Bazaar. — A species of mart or exchange for the sale of various ar- ticles of merchandise. Bead. — A circular moulding. Bed-mouldings. — Those mouldings which are between the corona and the frieze. Belfry. — That part of a steeple in which the bells are hung : an- ciently called campanile. Belvedere.~-An ornamental turret or observatory commanding a pleasant prospect. Bow-window. — A window projecting in curved lines. Bressummer. — Abeam or iron tie supporting a wall over a gateway or other opening. Brick-nogging. — The brickwork between studs of partitions. Buttress. — A projection from a wall to give additional strength. Cable. — A cylindrical moulding placed in flutes at the lower part of the column. Camber. — To give a convexity to the upper surface of a beam. 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 Caravansera. — In the East, a large public building for the reception t>f 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 ^owers, usually encompassed with walls and moats, and having a don- jon, or keep, in the centre. Catacombs. — Subterraneous places for burying the dead. Cathedral. — The principal church of a province or diocese, wherein the throne of the archbishop or bishop is placed. Cavetto. — A concave moulding comprising the quadrant of a circle. Cemetery. — An edifice or area where the dead are interred. Cenotaph. — A monument erected to the memory of a person buried in another place. Centring. — The temporary woodwork, or framing, whereon any vaulted work is constructed. Cesspool, — A well under a drain or pavement to receive the waste- water and sediment. Chamfer, — The bevilled edge of any thing originally right-angled. Chancel. — That part of a Gothic church in which the altar is placed. 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 ©, 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. 6 APPENDIX. Column.-r-k vertical, cylindrical support under the entablature of S.n 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 ponveyance of water. Conservatory. -rnr 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.— rStqne^ or timbers fixed in a wall to sustain the timbers of 3, 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-; pnoulding and the bed-njouldings. Cornucopia. — The horn of plenty. Corridor. -T-kn open gallery or communication to the different apart- ments of a house. Cove.—r-k concave moulding. Cripple-rafters. — The short rafters which are spiked to the hip-rafter of a roof. Crockets. — In Gothic architecture, the ornaments placed along the .angles of pediments, pinnacles, &c, Crosettes. — The same as ancones, which see. Crypt. — The under or hidden part of a building. Culvert. — An arched channel of masonry or brickwork, built be? neath the bed of a canal for the purpose of conducting water under it, Any arched channel for water underground. Cupola.-^ A. small building on the top of a dome. Curtail-step. — A step with a spiral end, usually the first of the flight, Cm*P-s.— srThe pendents of a pointed arch. Cyma.—vAn ogee. There are two kinds ; the cyma-recta, having the upper part concave and the lower convex, and the cyma-reversa, with the upper part convex and the lower concave. Dado. — The die, or part between the base and cornice of a pedestal. Dairy.-r-^An apartment or building for the preservation of milk, and Jhe manufacture of it into butter, cheese, dsc. Dead-shoar. — A piece of timber or stone stood vertically in brick- Tvork, to support a superincumbent weight until the brickwork which jis to carry it has set or become hard. Decastyle. — A building having ten columns in front. Dentils. — (From the Latin, denies, teeth.) Small rectangular blockg used in the bed-mouldings of some of the orders. Diasiyle. — An intercolumniation of three, or, as some say, foup 4ian)eters. 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 of 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. Entablature. — That part of an order which is supported by the co- lumns ; consisting of the architrave, frieze, and cornice. Eustyle.-^An intercolumniation of two and a quarter diameters. Exchange. — A building in which merchants and brokers meet to transact business. Extrados. — The exterior curve of an arch. Fagade. — The principal front of any building. Face-mould — The pattern for marking the plank, out of which hand- Tailing is to be cut for stairs, &c. Facia, or Fascia. — A flat member like a band or broad fillet. Falling-mould. — The mould applied to the convex, vertical surface of the rail-piece, in order to form the back and under surface of the rail, and finish the squaring. Festoon. — An ornament representing a wreath of flowers and leaves. Fillet. — A narrow flat band, listel, or annulet, used for the separa- tion of one moulding from another, and to give breadth and firmness to the edges of mouldings. 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. -r^K building in which various metals are cast into moulds or shapes. Frieze. — That part of an entablature included between the archi- trave and the cornice. Gahle. — The vertical, triangular piece of wall at the end of a roof, from the level of the eaves to the summit. Gain. — A recess made to receive a tenon or tusk. Gallery. — A common passage to several rooms in an upper story. A long room for the reception of pictures. A platform raised on co- lumns, pilasters, or piers. Girder. — The principal beam in a floor for supporting the binding and other joists, whereby the bearing or length is lessened. Glyph. — A vertical, sunken channel. From their number, those in the Doric order are called triglyphs. •8 APPENDIX. Granary. — A building for storing grain, especially that intended to be kept for a eonsiderabie time. Groin. — The line formed by the intersection of two arches, which •cross each other at any angle. Gultce. — 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 apaitment on entering a house. The public room of a corporate body. A manor-house. Ha7n. — A house or dwelling-place. A street or village : hence Nottingham, Bucking/mm, &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. — A large inn or place of public entertainment. A large house or palace. Hot-house. — A glass building used in gardening. Hovel. — An open shed. Hvi. — 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. Inter columniation, — The distance between two columns. Intrados. — The interior and lower curve of an arch. Jack-rafters. — Rafters that fill in between the principal rafters of a roof; called also common-rafters. Jail. — A place of legal confinement. Jambs. — The vertical sides of an aperture. Joggle-piece. — A post to receive struts. 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. 9 Lacianum. — 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. L£oer-5oard5.— 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. Lohhy. — -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. Lvffer-boarding. — 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. Modillion. — A projection under the corona of the richer orders, re- sembling a bracket. Module. — The semi-diameter of a column, used by the architect as a measure by which to proportion the parts of an order. Monastery. — A building or buildings appropriated to the reception of snonks. Monopteron. — A circular coUonade supporting a dome without an enclosing walk Mosaic. — A mode of representing objects by the inlaying of small •cubes of glass, stone, marble, shells, &c. Mosque. — A Mohammedan temple, or place of worship. Mullions. — The upright posts or bars, which divide the lights in a Gothic window. Muniment-house. — A strong, fire-proof apartment for the keeping auad 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 or 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 Obelisk. — A lofty pillar of a rectangular form. Octastyle. — A building with eight columns in front. Odeum. — Among the Greeks, a species of theatre wherein the poets and musicians rehearsed their compositions previous to the public pro- duction of them. Ogee. — See Cyma. Orangery. — A gallery or building in a garden or parterre fronting the south. Oriel-window. — ^A large bay or recessed window in a hall, chapel, or other apartment. Ovolo. — A convex projecting moulding whose profile is the quad- rant of a circle. Pagoda. — A temple or place of worship in India. Palisade. — A. fence of pales or stakes driven into the ground. Parapet. — A small wall of any material for protection on the sides of bridges, quays, or high buildings. Pavilion. — A turret or small building generally insulated and com- prised under a single roof. Pedestal. — A square foundation used to elevate and sustain a co^ lumn, statue, &c. Pediment. — The triangular crowning part of a portico or aperture which terminates vertically the sloping parts of the roof; this, iir 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. Piles. — Large timbers driven into the gi'ound to make a secure- foundation in marshy places, or in the bed of a river. Pillar. — A column of irregular form, always disengaged, and aE^ APPENDIX. 11 ways deviating from the proportions of the orders ; whence the distinc- tion between a pillar and a column. Pinnacle. — A small spire used to ornament Gothic buildings. Planceer. — The same as soffit, which see. Plinth. — The lower square member of the base of a column, pedes- tal, or wall. Porch. — An exterior appendage to a building, forming a covered approach 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. Rahlet, or Relate. — 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. Begula. — The band below the tsenia in the Doric order. Riser.-^ln 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 iinion 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 cofSn made of one stone. Scantling. — The measure to which a piece of timber is to be or has been cut. Scarfing. — The joining of two pieces of timber by bolting or nailing transversely together, so that the two appear but one. Scotia. — The hollow moulding in the base of a column, between the fillets of the tori. Scroll. — A carved curvilinear ornament, somewhat resembling in profile the turnings of a ram's hom. 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. Sojit — 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. Tcenia. — The fillet which separates the Doric frieze from the archi- trave. Talus. — ^The slope or inclination of a wall, among workmen called hatter ing. Terrace. — An area raised before a building, above the level of the ground, to serve as a walk. Tesselated pavement. — A curious pavement of Mosaic work, com- posed of small square stones. Tetrastyle. — A building having four columns in front. Thatch. — A covering of straw or reeds used on the roofs of cottages, barns, &c. Theatre. — A building appropriated to the representation of drama,..c spectacles. Tile. — A thin piece or plate of baked clay or other material used for the external covering of a roof. Tomb. — A grave, or place for the interment of a human body, 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 the two vertical edges, and having two channels in the middle. Tripod. — A table or seat with three legs. Trochilus. — The same as scotia, which see. Truss. — An arrangement of timbers for increasing the resistance to cross-strains, consisting of a tie, two struts and a suspending-piece. Turret. — A small tower, often crowning the angle of a wall, &;c. Tusk — A short projection under a tenon to increase its strength. Tympanum. — The naked face of a pediment, included between the level and the raking mouldings. Underpinning. — The wall under the ground-sills of a building. University. — An assemblage of colleges under the supervision of a senate, &c. Vault. — A concave arched ceiling resting upon two opposite paral- lel walls. Venetian-door. — A door having side-lights. Venetian-window. — A window having three separate apertures. Veranda. — An awning. An open portico under the extended roof of a building. Vestibule. — An apartment which serves as the medium of commu- nication to another room or series of i-ooms. Vestry. — An apartment in a church, or attached to it, for the pre- servation of the sacred vestments and utensils. Villa. — A country-house for the residence of an opulent person. Vinery. — A house for the cultivation of vines. Volute. — A spiral scroll, which forms the principal feature of the Ionic and the Composite capitals. Voussoirs. — Arch-stones Wainscoting. — Wooden lining of walls, generally in panels. Water-table. — The stone covering to the projecting foundation or other walls of a building. 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. Zystos. — Among the ancients, a portico of unusual length, common- ly appropriated to gymnastic exercises. TABLE OF SaUARES, CUBES, AND ROOTS. (From Hutton's Mathematics.) No. Square. Cube. Sq. Root. CubeRoot. No. Square. Cube. Sq. Root. CubeRoot. 1 1 1 1-0000000 l-OOOOOO 68 4624 314432 8-2462113 4-081655 2 4 8 1-4142136 1-250921 69 4761 328509 83066239 4-101566 3 9 27 1-7320508 1-442250 70 4900 343000 8-3666003 4-121285 4 16 64 2-0000000 1-537401 71 5041 357911 8-4261498 4-140818 5 25 125 2-2360680 1-709976 72 5184 373248 8-4852814 4-160168 6 36 216 2-4494897 1-817121 73 5329 389017 85440037 4-179339 7 49 343 2-6457513 1-912931 74 5476 405224 8-6023253 4-198336 8 64 512 2-8284271 2-000000 75 5625 421875 8-6602540 4-217163 9 81 729 30000000 2-080034 76 5776 433976 8-7177979 4-235824 10 100 1000 3-1622777 2-154435 77 5929 456533 8-7749644 4-254321 11 121 1331 3-3165243 2-2-23030 78 6084 474552 8-8317609 4-272659 12 144 1728 3-4641016 2-239429 79 6241 493039 8-8881944 4-290840 13 169 2197 3-6055513 2 351335 80 6400 512000 8-9442719 4-303869 14 196 2744 3-7416574 2-410142 81 6561 531441 9-0000000 4-3i6749 15 225 3375 3-8729833 2-466212 82 6724 551358 9-0553851 4-344481 16 256 4096 4-0000000 2-519842 83 6839 571787 9-1104336 4-362071 17 289 4913 4-1231056 2-571232 84 7055 592704 9-1651514 4-379519 18 324 5832 4-2426407 2-620741 85 7225 614125 9-2195445 4-396830 19 361 6859 4-3583989 2-66 S402 86 7396 636055 9-2735185 4-414005 20 400 8000 4 -4721350 2-714418 87 7569 658503 9-3273791 4-431048 21 441 9261 4-5825757 2-758024 88 7744 681472 9-3808315 4-447960 22 484 10648 4-6904153 2-8O2O30 89 7921 704969 9-4339811 4-464745 23 529 12167 4-7953315 2-843367 90 8100 729000 9-4S68330 4-481405 24 576 13324 4-8989795 2-884499 91 8281 753571 9-5393020 4-497941 25 625 15625 5-0000000 2-924018 92 8464 773688 9-5916630 4-514357 26 676 17576 5-0990195 2-962496 93 8649 804357 9-6436508 4-530655 27 729 19683 5-1961524 3 000000 94 8836 830534 9-6953597 4-546336 28 784 21952 5 2915026 3-036589 95 9025 857375 9-7467943 4-562903 29 841 24389 5-3351648 3-072317 96 9216 884736 9-7979590 4-578857 30 900 27000 5-4772256 3107232 97 9409 912673 9-8488578 4-594701 31 961 29791 55677644 3-141331 98 9604 941192 9-8994949 4-610436 32 1024 32768 5-6568542 3-174802 99 9801 970299 9-9498744 4-625065 33 1089 35937 5-7445526 3-207531 100 10000 1000000 100000000 4-641589 34 1156 39304 5-8309519 3-230612 101 10201 1030301 10-0498755 4-657009 35 1225 42875 5-9160798 3-271066 102 10404 1061208 10-0995049 4-672329 36 1296 46656 6 0000000 3 301927 103 10609 1092727 10-1483916 4-687548 37 1369 50653 6-0327625 3-332222 104 10816 1124861 10-1980390 4-702659 38 1444 54872 6-1644140 3-361975 105 11025 1157625 10-2469508 4-717694 39 1521 59319 6-2449980 3-391211 106 11236 1191016 10-2956301 4-732623 40 1600 64000 6-3245553 3 419952 107 11449 1225043 10-3140804 4 747459 41 1681 68921 6-4031 242 3-448217 108 11664 1259712 10-3923048 4-762203 42 1764 74088 6-4807407 3-476027 109 11881 1295029 10-4403065 4-776856 43 1849 79507 6-5574335 3-503398 110 12100 1331000 10-4880885 4-791420 44 1936 85184 6-6332496 3-530318 111 12321 1357631 10-5356538 4-805895 45 2025 91125 6-708203J 3-556893 112 12544 1404928 10-5330052 4-820284 46 2116 97336 6-7823300 3-533048 113 12769 1442897 10-6301458 4-834588 47 2209 103323 6-8555546 3-608825 114 12996 1481544 10-6770783 4-848808 48 2304 110592 6-9232032 3-634241 115 13225 1520875 10 7238053 4-862944 49 2401 117649 7-ooouooo 3-659306 116 13456 1560896 10-7703296 4-876999 50 2500 125000 7-0710678 3634031 117 13689 1601613 10-8165533 4-890973 51 2601 132551 7-1414284 3-708430 118 13924 1643032 10-8627805 4-904863 52 2704 140608 7-2111026 3-732511 119 14161 1685159 10-9087121 4-918685 53 2809 148877 7-2^01099 3-756285 120 14400 1723000 10-9544512 4-932424 54 2916 157464 7-3181692 3-779763 121 14641 1771561 11-0000000 4-946087 55 3025 166375 7-4161935 3-502952 122 14884 1815848 11-0453610 4-959676 56 3136 175616 7-4833148 3-825852 123 15129 1860867 11-0905365 4-973190 57 3249 185193 7-5193344 3-843501 121 15376 1906624 11-1355-287 4-986631 58 3364 195112 7-6157731 3-870877 125 15625 1953125 11-1803399 5-000000 59 3481 205379 7-6311457 3-892996 126 15376 2000376 11-2219722 5-013298 60 3600 216000 7-7459G67 3-914853 127 16129 2048333 11-2694-277 5-026526 61 3721 226981 7-8102197 3-936497 123 16334 2097152 11-3137085 5-039684 62 3844 238328 7-8740079 3-957891 129 16641 21466S9 11-3578167 5-052774 63 3969 250047 7-9372539 3-979057 130 16900 2197000 11-4017543 5-065797 64 4U96 262144 8-0000000 4-000000 131 17161 2-248091 11-4-455231 5-078753 65 4225 274625 8-0622577 4-020726 132 17424 2299968 11-4891253 5-091643 66 4356 287496 8-1240334 4-041240 133 17689 2352637 11-5325626 5-104469 67 4489 300763 8-1853528 4-061548 134 17956 2406104 11-5758369 5-117230 APPENDIX. 15 No. Square. Cube. Sq. Root. CubeKoot. No. Square. Cube. Sq. Root. CubeRoot. 135 18225 2460375 116189500 5-129928 202 4('804 8242408 14-2126704 5-867464 136 18496 2515456 11-661903S 5-142563 203 412C9 8365427 14-2478068 5-877131 137 18769 2571353 11-7046999 5-155137 204 41616 8489654 14-23-28569 5-836765 138 19044 2628072 11-7473401 5 -16764V. 205 42025 8615125 14-3173211 5-896368 139 19321 2635611) 11-7898261 5180101 206 4213.1 8741816 14-3527001 5-905941 140 19600 2744000 11-8321596 5-192494 207 4-.i849 8369743 14-3374946 5-915482 141 19881 2803221 11-8743422 5-204828 203 43-264 8998912 14-4222051 5-924992 142 20164 2363283 11-9163753 5-217103 209 43881 9129329 14-456a323 5-934473 143 20449 2924207 11-95826 J7 5-229321 210 44100 9261000 14-4913767 5-943922 144 20736 2985334 12-0000000 5-241483 211 44521 93J3931 14-5253390 5-953342 145 21025 3048625 12-0415946 5-253533 212 44944 95-28123 14-5602198 5-962732 146 21316 3112136 12-0830460 5-265637 213 45369 9663597 14-59-15195 5-972093 147 21609 317I-.523 12-1243557 5-277632 214 45796 9800344 14-6287338 5-981424 148 21904 3241792 12-1655-251 5-289572 215 46225 9933375 14-6623783 5990726 149 22201 3307949 12-2065553 5-301459 216 46656 10077696 14-6969385 6-000000 150 22500 3375000 12-2474487 5-313293 217 47089 10218313 14-7309199 6-009245 151 22301 3442951 12-2882057 5-325074 218 475-24 10350^32 14-7648231 6-018462 152 23104 3511808 12-3238280 5-336803 219 47961 10503459 14-7986486 6-027650 153 23409 3531577 12-3693169 5-348431 220 48400 10648000 14-8323970 6-036811 154 23716 3652264 12-4096736 5-360108 221 43-(41 10793861 14-8660687 6-045943 155 24025 3723375 12-449899.. 5-371685 222 49234 loy41048 14-8996644 6-055049 156 24336 3796416 12-4399960 5-383213 223 49729 11039567 14-9331845 6-064127 157 24649 3869393 12-5299641 5-394691 224 50176 11239424 149656295 6-073178 158 24964 3944312 12-5598051 5-406120 225 50625 11390625 15-0000000 6-082202 159 25281 4019579 12-6095202 5-417501 226 51076 11543176 15-0332964 6-091199 160 25600 4096000 12-6491106 5-428835 227 51529 1 1697083 15-0665192 6-100170 161 25921 4173231 12-6335775 5-440122 228 51984 11852352 15-0996639 6-109115 162 26244 4251523 12-7279221 5-451362 229 5-2441 12008939 15-1327460 6-118033 163 26559 4330747 12-7671453 5-462556 230 52900 12167000 15-1657509 6-126925 164 26896 4410944 12-8062485 5-473704 231 53361 12323391 15-1986342 6-135792 165 27225 4492125 12-8452326 5 484807 232 538-24 12487168 15-2315462 6-144634 166 27556 4574296 12-8840987 5-495365 233 54289 12649337 15-2643375 6-153449 167 27839 4657463 12 9228480 5-506878 234 54755 1281-2904 152970585 6-162240 168 28224 4741632 12-9614814 5-517848 235 55225 12977875 15-3297097 6-171006 169 28561 4826809 13-0000000 5-528775 233 55696 13144256 15 3522915 6-179747 170 28900 4913000 13-0384048 5-539658 237 56169 13312053 15-3948043 6-188463 171 29241 5000211 13-0766968 5-550499 238 56644 13481272 15-4272486 6-197154 172 29^)84 5083448 13-1148770 5-561298 239 57121 13651919 15-4596248 6-205822 173 29929 5177717 13-1529464 5-572055 240 57600 13324000 15-4919334 6-214465 174 30276 5268024 13-1909060 5 532770 241 53081 139..7521 15-5241747 6-223084 175 30625 5359375 13-2287566 5-593445 242 58554 14172433 15-5563492 6-234630 176 30976 5451776 13-2664992 5-604079 243 59049 14348907 15-5384573 6-240251 17T 31329 5545233 13-3041347 5-614672 244 59536 145-26784 15-6204994 6-248800 178 31684 5639752 13-3416641 5-625226 245 60025 147061-25 15-65-24753 6-257325 179 32041 5735339 13-3790832 5-635741 246 60516 14836936 15-6843371 6-265327 180 32400 5832000 13-4164079 5-646216 247 6iooy 15039223 15-7162335 6-274305 181 32761 5929741 13-4536240 5-656653 248 61504 15252992 15-7480157 6-282761 182 33124 6023558 13-4907376 5-667051 249 62091 15433249 15-7797333 6-291195 183 33489 6128487 13-5277493 5-677411 250 6-2500 15325000 15-8113383 6-299605 181 33356 6229504 13-5646600 5-637734 251 63001 15313251 15-84-29795 6-397994 185 34225 6331625 13-6014705 5-693019 252 63504 16003008 15-8745079 6-316360 18n 34596 6434856 13-6331817 5-703267 253 64009 16194277 15-9059737 6-324704 187 34969 6539203 13-6747943 5-718479 254 64516 16337064 15-9373775 6-333026 188 35344 6644672 13-7113092 5-728654 255 65025 16531375 15 9687194 6-341326 189 35721 6751269 13-7477271 5-738794 256 65536 16777216 16-OOOOOJO 6-349604 190 36100 6859000 13-7840488 5-748897 257 65049 16974593 16-0312195 6 357861 191 36481 6967871 13-8202750 5-758965 253 66564 17173512 16-0623734 6-366097 192 36864 7077838 13-8564065 5-763998 259 67031 17373979 16-0934769 6374311 193 37249 7189057 13-8924440 5-778996 260 67600 17576000 16-1245155 6-382504 194 37636 7301384 13-9283383 5-783960 261 63121 17779531 16-1554944 6-390676 195 38025 7414875 13-9842400 5-798390 262 6S644 17984723 16-1854141 6-398823 196 38416 7529536 14-0000000 5-808736 263 69169 18191447 16-217-2747 6-406953 197 38809 7645373 14-0355683 5-818643 264 69696 18399744 16-2480763 6-415069 198 39204 7762392 14-0712473 5-823477 265 70225 18609625 16-2738206 6-423153 199 39601 7880599 14-1067360 5-833272 266 70756 18821096 16-3095064 6-431223 200 40000 8000000 14-1421356 5-848035 267 71289 19034163 16-3401346 6-439-277 201 40401 8120601 14-1774469 5-837766 268 71824 19248832 16-3707055 6-447306 16 APPENDIX No. Square. Cube. Sq. Root. CubeRoot. No. Square. Cube. Sq Root. CubeRoot 269 270 72351 19465109 16-4012195 6-455315 336 112896 37933056 18-3303028 6 952053 72900 19633000 16-4316767 6-463304 337 113559 38272753 18 3575598 6-953913 271 73441 19902511 16-4620776 6-471274 338 114244 38614472 18-3847763 6-965820 272 73984 20123648 16-4924225 6-479224 339 114921 38958219 18-4119526 6-972683 273 74529 20346417 16-52-27116 6-487154 340 115600 39304000 18-4390889 6-979532 274 75076 20570824 16-55-29454 6-495065 341 116281 39651821 184661853 6-986.368 275 75625 20796875 16-5331240 6-502957 342 116964 40001688 18-4932420 6-993191 276 76176 21024576 16-6132477 6-510830 313 117649 40353607 18-5202592 7-000000 277 76729 21253933 16-6433170 6-518634 344 118336 40707534 18-5472370 7-006796 278 77234 21484952 166733320 6-5-26519 345 119025 410636-25 18-5741756 7-013579 279 77841 21717639 16-7032931 6-534335 346 119716 41421736 18-6010752 7-020349 280 78400 21952000 16-7332005 6-542133 347 120409 41781923 18-6279360 7-027106 281 78961 22188041 16-7630546 6-549912 343 121104 42144192 18-6547531 7-033850 282 79524 2242576S 16-7923556 6-557672 349 121801 42508549 18-6815417 7-040581 283 80089 22665187 16-82-26033 6-565414 350 122500 42875000 18-7082869 7-047299 284 80656 22906334 16-8522995 6-573139 351 123201 43243551 18-7349940 7-054004 285 81225 23149125 16-8319430 6-580844 352 123904 43614208 18-7616630 7-060697 286 81796 23393656 16 9115345 6-538532 353 124609 4398 -.977 18-7332942 7-067377 287 82369 23639903 16-9410743 6-596202 354 125316 44361864 18-8148877 7-074044 283 82944 23387872 16-9705627 6-603S54 355 126025 44733875 18-8414437 7-080699 289 83')21 24137569 17-0000000 6-611489 356 126736 45118016 18-8679623 7-087341 290 84100 24389000 17-0293864 6-619106 357 127449 45499293 18-8944436 7-093971 291 84681 24642171 17-0537221 6-626705 353 128164 45382712 18-9208879 7-100588 292 85264 24897088 17-0880075 6-634237 359 128881 46268279 18-947-2953 7-107194 293 85849 25153757 17-1172428 6-641852 360 129600 46656000 18-9736660 7-113787 294 86136 25412184 17-1464282 6-649400 361 130321 47045381 19-0000000 7-120367 295 87025 25672375 17-1755640 6-656930 362 131044 47437928 19-026-2976 7-126936 296 87616 25934336 17-2046505 6-664444 363 131769 47832147 19-0525589 7-133492 297 8-1209 26198073 17-2336379 6-671940 384 132495 48228544 19-0787840 7-140037 298 88804 26463592 17-2626765 6-679420 365 133225 486-27125 19-1049732 7-146569 299 89401 26730899 17-2916165 6-686883 366 133956 49027396 19-1311265 7 153090 300 90000 27000000 17-3205081 6-694329 367 134639 49430863 19-1572441 7-159599 301 90601 27270901 17-3493516 6-701759 368 135424 49835032 19-1833261 7-166096 302 91204 27543608 17-3731472 6-709173 369 136161 50243409 19 2093727 7-172531 303 91809 27818127 17-4068952 6-716570 370 136900 50653000 19-2353341 7-179054 304 92416 28094464 17-4355953 6-723951 371 137641 51064311 19-2613603 7-185516 305 93025 28372625 17-4642492 6-731316 372 138384 51478848 19-2373015 7-191966 306 93636 23652616 17-4928557 6-733664 373 1391-29 51895117 19-3132079 7-198405 307 94249 28934443 17-5214155 6-745997 374 139876 52313624 19-3390796 7-204832 308 94864 29218112 17-5499288 6-753313 375 1406-25 52734375 19-3649167 7-211248 309 95481 29503529 17 5783953 6-760614 376 141376 53157376 19-3907194 7-217652 310 96100 29791000 17-6068169 6-767399 37T 142129 53582633 19-4164878 7-224045 311 96721 3(^080231 17-6351921 6-775169 3r8 142884 54010152 19-4422221 7-230427 312 97344 30371328 17-6635217 6-782423 379 143641 54439939 19-4679223 7-236797 313 97-^69 30664297 17-6918060 6-789661 330 144400 54872000 19-4935387 7-243156 311 98596 30959144 17-7200451 6-796834 331 145161 55306341 19-5192213 7-249504 315 99225 31255375 17-7482393 6-804092 332 145924 55742968 19-5448203 7-255341 31ii 99856 31554496 17-7763388 6-811235 333 146639 56181887 19-5703353 7-262167 317 100489 31855013 17-8044933 6-818462 334 147456 56623104 19-5959179 7-263482 318 101124 32157432 17-8325545 6-825624 335 148225 57066625 19-6214169 7-274786 319 101761 32461759 17-8605711 6-832771 386 148996 57512456 19-6468327 7-231079 320 102400 32768000 17-8835438 6-839904 387 149769 57960603 19-6723156 7-287362 321 103041 33076161 17-9164729 6-847021 333 150544 58411072 19-6977156 7-293633 322 103584 33336248 17-9443584 6-854124 339 151321 58863869 19-7230829 7-299894 323 104329 33698267 17-9722008 6-861212 390 152100 59319000 19-7484177 7-306144 324 104976 34012224 18-0000000 6-868285 391 152831 59776471 19-7737199 7312383 325 105625 34323125 18-0277564 6-875344 392 153664 60236288 19-7989899 7-318611 326 106276 34645976 18-0554701 6-882389 393 154449 60693457 19-8242276 7-324829 327 106929 34965783 18-0831413 6-889419 394 155236 61162984 19-8494332 7-331037 328 107584 35287552 18-1107703 6-896435 395 156025 61629875 19-8746069 7-337234 329 108241 35611239 18-1333571 6-903436 396 15-^816 62099136 19-8997487 7-343420 330 108900 35937000 18-1659021 6-910423 397 157609 62570773 19-9248588 7-349597 331 109561 36264691 18-1934054 6-917396 398 158404 63044792 19-9499373 7-355762 332 110224 36594368 18-2208672 6-924356 399 159201 63521199 19-9749844 7-361918 333 110889 36926037 18-2482376 fi-931301 400 160000 64000000 20-0000000 7-363063 334 111556 37259704 18-2756669 6-933232 401 160801 64481201 20-0249844 7-374198 335 112225 37595375 18-3030052 6-945150 402 161604 64964808 20-0499377 7-330323 APPENDIX. ir Jfo. Square. Cube. Sq. Root. CubeRoot. No. Square. Cube. Sq. Root. CubeRoot. 403 162409 65450827 20-0748599 7-3ri6437 470 2-z.mo 103323000 21-6794334 7-774980 404 16321h 65939264 20-0997512 7-392542 471 221841 104487111 21-7025344 7-7304911 405 164025 66430125 20-1246118 7-398636 472 22-2784 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 1H6464 67917312 20-1990099 7-416859 473 225625 107171875 21-7944947 7-802454 409 167281 68417929 20-2237434 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 223484 109215332 21-8632111 7-818846 412 169744 69934523 20-2977831 7-441019 479 229441 109902239 21-8860686 7-824294 413 170569 70444997 20-3224014 7-447034 430 230400 110592000 21-908J023 7-829735 414 171396 70957944 20-3469899 7-453040 431 231361 111284641 21-9317122 7-835169 415 172225 71473375 20-3715488 7-459036 432 232324 111980163 21-9544934 7-840595 41f- 173036 71991296 20-3960781 7-465022 483 233289 112678537 21-9772610 7-846013 417 173889 72511713 20-4205779 7-470999 434 234256 113379904 22-0000000 7-851424 418 174724 73034632 20-4450483 7-476966 435 235225 114084125 22-0227155 7-356823 419 175561 73560059 20-4694895 7-482924 486 236196 114791256 22 0454077 7-852224 420 176400 74088000 20-4939015 7-488872 437 237169 115501303 22-0680763 7-867613 421 177241 74618461 20-5182843 7-494311 488 238144 116214272 22-0907220 7-872994 422 178084 75151448 20-3426386 7-500741 439 239121 116930169 2-2- 11334 14 7-878368 423 178929 75636967 20-5669638 7-50666 1 490 240100 117649000 22 135943d 7-833735 424 179776 76225024 20-5912603 7-512571 491 241031 118370771 22-1535193 7-839095 425 180625 76765525 20-6155281 7-518473 492 242064 119095488 22-1810730 7-894447 426 181476 77308776 20-6397674 7-524365 493 243049 119823157 22-2036033 7-399792 427 182329 77854483 20-6633783 7-530248 494 244036 120553784 22-2261103 7-905129 428 183184 78402752 20-6881609 7-536122 495 245025 121-287375 22-2485955 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 22-2934963 7-921099 431 185761 80062991 20-7605395 7-553639 498 243004 123505992 22-315913;) 7-925408 432 186624 80621568 20-7846097 7-559526 499 249001 1'24251499 22-3333079 7-931710 433 187489 81182737 20-8086520 7-565355 300 230000 125000000 22-3605798 7-9370U3 434 188356 81746504 20-8326667 7-571174 501 251001 125751501 22-3330293 7-94-2293 435 189225 82312875 20-8566533 7-576985 502 252004 126506008 22-4053365 7-947574 436 190096 82881856 20-8806130 7-582786 503 253009 r27263527 22-42/6615 7-952848 437 190969 83453433 20-9045450 7-583579 504 254016 1280;i4064 22-4499443 7-953114 438 191844 84027672 20-9284495 7-594363 505 255025 1-28787625 22-472;i051 7-963374 439 192721 84604519 20-9523-268 7-600133 306 256036 129554216 22-4944438 7-968627 440 193600 85184000 20-9761770 7-605905 507 257049 130323343 22-5166605 7-973373 441 194481 8376)121 21-0000000 7-611663 508 253064 13109D512 225330553 7-979112 442 195364 86350388 21-0237960 7-617412 509 259081 13187^2-229 225510283 7-984344 443 196249 86938307 21-0475632 7-623152 510 260100 132651000 22-5331795 7-989570 444 197136 87528384 21-0713J75 7-6-23381 511 261121 133432^31 22-5053091 7-994788 445 198025 88121125 21-0950231 7-634607 512 262144 134217728 22-6274170 8 000000 446 198916 88716536 21-1187121 7 640321 513 263169 135003697 22-6495033 8-005205 447 199809 89314623 21-1423745 7-646027 514 264196 135796744 22-6715581 8-0LJ403 448 200704 89915392 21-1660105 7-651725 515 265223 136590873 22-6936114 8-015595 449 201601 90518849 21-1896201 7-657414 516 266256 137383096 227156334 8-020779 450 202500 91125000 21-2132034 7-663094 517 267289 13318^413 22-7376340 8-025957 451 203401 91733851 21-2367606 7-663766 518 268324 138991832 22-7395134 8-031129 452 204304 92345403 21-2802916 7-674430 519 269351 139793359 '22-7815715 8-03629:i 453 205209 92959677 21-2837967 7-680086 520 270400 140608000 22-8U35085 8-041451 454 206116 93576664 21-3072758 7-685733 521 271441 1414-20761 22-8^54244 8-046603 455 207025 941963751 21-3307290 7-691372 522 272434 14-2236648 22-84731b3 8-051748 456 207936 94818816! 21-3541555 7-697002 523 273529 143055667 22-8691933 8 056886 457 208849 95443993 21-3773383 7-702625 524 274576 143877824 ■22-8910463 8-062018 458 209764 96071912 21-4009346 7-708239 525 275625 1447031:i5 22-9128785 8-067143 459 210681 96702579 21-4242333 7-713345 526 276676 145531576 22-9346^99 8•07^^62 460 211600 97336000| 21-4476106 7-719443 527 277729 146363183 22-9554806 8077374 461 212521 97972181 21-4709106 7-725032 528 •278784 147197952 22-9732506 8 082480 462 213444 98611128 21-4941853 7-730614 529 279341 148035389 23 0000000 8-087579 463 214369 99252847 21-5174348 7-736183 530 280900 148877000 23-04l7i89 809267;i 464 215296 99897344 21-5406592 7-741733 1 531 281961 149721291 23-0434372 8-097759 465 216225 100544625; 21-5633587 7-747311 532 283024 150558768 •23-0631252 8-102839 466 217156 1011946961 21-5370331 7-732861 1 533 284089 151419437 23-0867928 8-107913 467 218089 101847563' 21-6101828! 7-758402 i 534 •285156 152-273304 23 1084400 8-ir2980 468 219024 102503232' 21-6333077 7-763.^36 1 535 286-225 153130375 23-13i>0570 8-118041 469 219961 1031617091 21-6364078 7-769462 1 536 287296 133990636 23-1516738 , 8-123090 3* IPPENDIX. No. 537 Square. Cube. Sq. Root. CubeRoot. No. Square. Cube. Sq. Root. CubeRoot. 288369 154854153 23-1732605 8-123145 604 3n4816 220348864 24-5764115 8-453028 533 239444 155720872 23-1948270 8-133187 605 3(16025 221445125 24-5967478 8-457691 539 290521 156590819 23-2163735 8-133223 606 3(i7236 222545016 24-6170673 8-462348 540 291600 157464000 23-2379001 8-143-253 607 368449 223648543 24-6373700 8-467000 541 292681 158340421 23-2594067 8-148276 608 369664 224755712 24-6576560 8-471647 542 293764 159220088 23-2308935 8-153294 609 370881 225866529 24-6779254 8-476289 543 294849 160103007 23-3023604 8-158305 610 372100 226981000 24 6981781 8-480926 544 295936 160989184 23-3233076 8-163310 611 373321 228099131 24-7184142 8-485558 545 297025 161878625 23-3452351 8-168309 612 374554 2292209-28 24-7386338 8-490185 546 298116 162771336 23-36S6429 8-173:J02 613 375769 230346397 24-7588368 8-494806 547 299209 163667323 23-3880311 8' 178289 614 376996 231475544 24-7790234 8-499423 548 300304 1^566592 23-4093998 8-183269 615 378225 232608375 24-7991935 8-504035 549 301401 165469149 23-4307490 8-188244 616 379456 233744896 24-8193473 8-508642 550 302500 166375000 23-4520788 81932-13 &17 380689 234885113 24-8394847 8-513243 551 303601 167284151 23-4733392 8-198175 618 381924 236029032 24-8596058 8-517840 552 304704 168196608 23-4946802 8-203132 619 383161 237176659 24-8797106 8-522432 553 305809 169112377 23-5159520 8-208082 &20 384400 238328U00 24-8997992 8-527019 554 306916 170031464 23-5372046 8-213027 621 385641 239483061 24-9198716 8-531601 555 308025 170953875 23 5534380 8-217966 622 386884 240641848 24-9399278 8-536178 556 309136 171879616 23-5796522 8-222898 623 38-il29 241804367 24-9599679 8-540750 557 310249 172808693 23-6008474 8-227825 624 389376 242970624 24-9799920 8-545317 558 311364 173741112 23 6220236 8-232746 625 390625 244140625 2^-0600000 8-549880 559 312431 174676879 23-6431808 8-237661 626 391876 245314376 25-0199920 8-554437 560 313600 175616000 23-6643191 8-242&71 627 393129 246491883 25-0399681 8-558990 561 314721 176558481 23-6854386 8-247474 628 394334 247673152 25-0599282 8-563538 562 315844 177504328 23-7065392 8-252371 629 395641 248858189 25-0798724 8-568081 563 316969 178453547 23-7276210 8-2572S3 630 396900 250047000 25-0998008 8-572619 564 318096 179406144 23-7486842 8-262149 631 3:98161 251239591 25-1197134 8-577152 565 319225 180362125 23-7697286 8-267029 632 399424 252435968 25^-1396102 8-581681 566 320356 181321496 23-7907545 8-271904 633 400689 253636137 25-1594913 8-586205 567 321489 182284263 23-8117618 8-276773 634 401956 254840104 25-1793566 8-590724 568 322624 1832504.32 23-8327506 8-281635 635 403225 256047875 25-1992063 8-59.5-238 569 323761 184220009 23-8537209 8-286493 636 404 »96 257259456 25-2190404 8-599748 :570 324900 185193000 23-8746728 8-291344 637 405769 258474853 25-2338539 8-604252 571 326041 186169411 23 8956063 8-296190 638 407044 259694072 25-2586619 8-608753 572 327184 187149248 23-9165215 8-301030 639 408321 260917119 25 2784493 8-613248 573 328329 188132517 23-9374184 8-3J5865 640 409600 262144000 25.2982213 8-617739 574 329476 189119224 23-9532971 8-310694 641 410881 263374721 25-3179778 8-622225 575 33062& 190109375 23-9791576 8-315517 642 412164 264609288 25-3377189 8-626706 576 331776 191102976 24-0000000 8-3203.<5 643 413449 265847707 25-3574447 8-631183 577 332929 192100033 24-0208243 8-325147 644 414736 267089984 25-3771551 8-635655 578 334034 193100552 24-0416306 8-329954 645 416025 268336125 25-3968502 8.640123 579 335241 194104539 24-0624188 8-331755 646 417316 269586136 25-4165301 8-644585 580 335400 195112000 24-0831891 8-339551 647 418609 270840023 25-4361947 8-649044 581 337561 196122941 24-1039416 8-344341 648 419904 272097792 25-4558441 8-653497 582 333724 197137368 24-1246762 8-349126 649 421201 273359449 25-4754784 8-657946 583 339839 198155287 24-1453929 8-353905 650 422500 274625000 25-4950976 8-662391 584 341056 199176704 24-1660919 8-353678 651 423801 275894451 25-5147016 8-666331 535 342225 200201625 24-1867732 8-363447 652 425104 277167808 25-5342907 8-671266 586 343396 201230056 24-2074369 8-368-209 653 426409 278445077 25-5538647 8-675697 587 344569 202262003 24-2230829 8-372967 654 427716 279726264 25-5734237 8-680124 588 345744 203297472 24 2487113 8-377719 655 429025 281011375 25-5929678 8-684546 589 346921 204336469 24-26932-22 8-382465 656 43j336 282300416 25-6124969 8-688963 590 348100 20537i)000 24-2899156 8-337206 657 431649 283593393 25-6320112 8-693376 591 349281 206425071 24-3104916 8-391942 658 432964 234890312 25-6515107 8-697784 592 350464 207474688 21-3310501 8-396673 6fr9 434281 286191179 25-6709953 8-702188 593 351649 208527857 24-3515913 8-401398 660 435600 287496000 25-6904652 8-706588 594 352836 209584584 24-3721152 8-406118 661 436921 288804781 25-7099203 8-710983 595 354025 210644875 24-3926218 8-410833 662 438244 290117528 25-7293607 8-715373 596 355216 211708736 24-4131112 8-415542 663 439569 291434247 25-7487864 8-719760 597 356409 212776173 24-4335834 8-420246 664 440896 292754944 25-7681975 8-724141 598 357604 213847192 24-4540385 8-424945 665 442225 294079625 25-7875939 8-728518 599 358801 214921799 24.4744765 8-429633 666 443556 295408296 25-8069758 8-732892 600 360000 216000000 24-4948974 8-434327 667 444889 296740963 25-8263431 8-737260 601 361201 217081801 24-5153013 8-439010 668 446224 298077632 23-8456960 8-741625 602 362404 218167208 24-5356883 8-443688 669 447561 299418309 25-8650343 8-745985 603 363609 219256227 24-5560583 8-448360 670 448900 300763000 25-8843582 8-750340 APPENDIX. 19 No. Square. Cube. Sq. Root. CubeRoot. No. Square. Cube. Sq. Root. CubeRoot. 671 450241 302111711 25-9035677 8-754691 738 544644 401947-272 27-1661554 9-036886 67a 451584 303464448 25-9229628 8-759033 739 546121 403533419 27-1845544 9-040965 673 452929 304821217 25-9422435 8-763331 740 547600 405224000 27-2029410 9-045042 674 45427b 306182024 25-9615100 8-767719 741 549081 406869021 27-2213152 9-049114 675 455625 307546875 25-9807621 8-772053 742 550564 408518488 27-2396769 9-053183 676 45697(1 308915776 26-0000000 8-776333 743 552049 410172407 27-2580263 9-057248 677 458329 310288733 26-0192237 8-78U708 744 553536 411830784 27-2763634 9-061310 678 459684 311665752 26-0384331 8-785030 745 555025 413493625 27-2946881 9-065368 679 461041 313046839 26.0576284 8-789347 746 556516 415160936 27-3130006 9-06y422 680 462401. 314432000 26-0768096 8-793659 747 558009 416832723 27-3313007 9-073473 681 463761 315821241 26-0959767 8-797968 748 539504 418508992 27-3495337 9-077520 662 465124 317214568 26-1151297 8-802272 749 561001 420189749 27-3678644 9-0ol563 683 466489 318611987 26-1342687 8-806.572 750 562500 421875000 27-3861279 9-085b03 684 467856 320013504 26-1533937 8-810868 751 564U01 423564751 27-4043792 9-08y63a 685 469225 321419125 26-17^5047 8-815160 752 565504 425^59008 27-4226184 9-Oy3672 686 470596 32282885b 26-1916017 8-819447 753 567009 426957777 27-4408455 9-097701 687 471969 3242427^3 26-2106848 8-823731 754 568516 428661064 27-4590604 9-101726 688 473344 3:^5660672 26-2297541 8-828010 755 570025 430368875 27-4772633 9-105748 6s9 474721 3;i7082769 26-2483095 8-832^85 756 571536 432081216 27-4954542 9-109767 690 476100 328509000 26-2678511 8-836550 757 573049 433798093 27-513b33j 9-113782 691 477481 329939371 26-2868789 8-840823 758 574564 435519512 27-53179y8 9-117793 692 478864 331373888 26-3053929 8-845085 759 576081 437245479 27-5499546 9-121801 693 480249 332812557 26-3248932 8-843344 760 577600 438976000 27-5680975 9-125805 i>94 48163b 334255384 26-34387y7 8-853598 761 57yl21 440711081 27-58b2284 9-12a8i/6 695 483025 335702375 26-3628527 8-857849 762 580644 442450728 27-6043475 9-133803 696 48441b 337153536 26-3818119 8-862095 763 582169 444194947 27 6224546 9-137737 697 485809 338608873 26-4007576 8.866337 764 533696 445943744 27-64u549y 9-141787 698 487204 34006839^; 26-4196896 8-870576 765 585z25 447697125 27-658o334 9-145774 699 488601 341532099 26-4386081 8-874810 766 586756 44y4550y6 276707050 9-14975o 700 49UU00 343000000 26-4575131 8-879040 767 588Z89 451217663 27-6947640 9-153737 701 491401 344472101 ii6-476404b 8-883266 768 589824 452984832 27-7128129 9-157714 702 492^04 345948408 26-4952826 8-887488 769 591361 454756609 27-730849:^ 9-16168/ 703 494209 347428927 26-514147:^ 8-891706 770 592900 456533000 27-7488739 9-165656 704 495616 3-J8913661 26 5329983 8-895920 771 594441 458314011 27-7663860 9-16902^ 705 4»7025 35040 ib25 26-5518361 8-900130 772 5y5y84 400099648 27-704^080 9-17358J 706 498436 351895816 26-5706605 8-904337 773 5975Ji9 461889917 27-8028775 9-177544 707 499849 353393243 26-5894716 8-908539 774 599076 463684824 27-8208555 9181501^ 708 501264 354894912 26-6082694 8-912737 775 600625 465484375 27-0388218 9-185453 709 502681 356400829 26-6270539 8-916931 776 602176 467288576 27-8567766 9-18940J1 710 504100 357911000 26-645825a 8-921121 777 603729 469097433 27-8747197 9-193347 711 505521 359425431 26-6645833 8-925308 778 605284 470910952 27-8926514 9-197290 712 506944 360944 12S 26-6833281 8-929490 779 606841 472729139 27-9105715 9-201229 713 50«369 362467097 26-7020598 8-933669 78u 6084oO 474552000 27-9284801 9-205164 714 50y7y6 363994344 26-7207784 8-937843 781 60y961 476379541 27-9463772 9-209096 715 511225 365525875 26-7394839 8-942014 782 611524 470211768 27-9642629 9-213025 716 512656 367061696 26-7581763 8-946181 783 613089 480048687 27-982137;: 9-2l695u 717 5140«9 368601813 26-7768557 8-950344 784 614656 481890304 28-ooooouo 9-220873 718 515524 370146232 26-7955220 8-954503 785 616^25 483736625 28 0178515 9-224791 719 516961 37i69495y 26-8141754 8-958658 786 617796 486587656 28-0356915 9-22o707 720 518400 373248000 26-8328157 8-962809 78'/ 619369 487443403 28-0535-203 9-232619 721 519841 374805361 26-8514432 8-966957 788 620944 489303372 28-0713377 9-2^6528 722 521284 370367048 26-8700577 8-971101 789 62Z5-Z1 491169069 28-0891438 9-240435 723 522729 377933067 26-8886593 8-975241 790 6^4100 4y3039000 28- 1069380 9-24433b 724 524176 3795U3424 26-907^481 8-979377 791 625681 494913671 28-12472x2 9-248234 725 5Z5625 .381078125 26-9258240 8-983509 792 627:^64 496793080 28- 142494b 9-252130 726 527076 382657176 26-9443872 8-987637 793 628849 498677:^57 28-1602557 9-25b022 727 5285ii9 384240583 26-9629375 8-991762 79* 63J436 500566184 28-1780056 9-259911 728 52i*984 38o82835:!i 26-9814751 8-995883 795 632025 502459875 28-1957444 9-26.>797 729 531441 387420489 :a7-0000000 9-000000 796 633616 504358336 28-2134720 9-2b7b8o 730 532900 389017000 27-0185122 9-004113 79/ 635209 506261573 28-2311884 9-271559 731 534361 390617891 27-0370117 9-008223 798 636804 508169592 28-2488930 9-275435 732 535824 392223168 27-0554985 9-012329 799 638401 510082399 28-2665881 9-279300 733 5a7289 39383^:837 27-0739727 9-016431 800 640000 512000000 28-2842712 9-283170 734 533756' 3;*5446904 27-0924344 9-02055i9 801 641601 513922401 28-3019434 9-5io7044 735 540225' 3t)7065375 27.1108834 y-024624 802 643204 515849608 5{8-319t.045 9-290S07 736 .54169b| 398688256 27-1293199 9-Oaa715 803 644809 51778162/ 28-337254b 9-29476/ 737 1 5431691 400315553 27-1477439 9-032802 804 646416 519718464 28-3548938 9-298624 20 APPENDIX. No. Square. Cube. Sq. Root. CubeRoot. No. Square. Cube. Sq. Root. CubeKoot, 805 806 807 808 809 810 811 81;i 813 814 815 816 817 818 819 820 821 822 823 824 825 826 827 828 829 830 831 832 833 834 835 836 837 838 839 840 841 842 843 844 845 846 847 848 849 850 851 852 853 854 855 648025 649636 651249 652864 654481 656100 657721 659314 660969 662596 52166i)125 52360C616 525557943 527514112 529475129 531441000 533411731 533387328 537367797 539353144 6642251 541343375 665856 543338496 54533^513 5473i3432 549353259 551368000 553387661 555412248 557441767 559476224 561515625 563559976 565609283 567663552 569722789 571787000 573856191 57593J368 578009537 580093704 667489 669124 670761 672400 674041 675684 677329 678976 680625 6S2276 683;»29 685584 687241 6889 lU 690561 69i!224 693889 695556 697225 582182875 698896 700569 702244 703921 705600 707281 708>j64 710649 712336 714025 715716 717409 719104 720801 722500 724201 725904 727 60y 729316 7310^5 856 732736 858 859 860 861 862 863 864 865 866 867 868 869 870 871 584277056 586376253 5a8480472 5905S9719 592704000 594823321 596947688 599077107 601211584 603351125 605495736 607645423 609300192 611960049 614125000 616295051 618470208 620650477 622835964 625026375 627222016 857 734449 629422793 736164 631628712 28 3725219 28-3901391 28-4077454 28 4233408 28-4429253 28-46'J4989 28-4780617 28-4956137 23-5131549 28-5306852 28-5482048 28-5657137 28-5332119 28-6006993 28-6181760 28-6356421 28-6530976 28-6705424 28-6879766 28-7054002 28-7228132 28-7402157 28-7576077 28-7749891 28-7923601 28-8097206 28-8270706 28-8444102 28-8617394 28-8790582 2H-8963666 28-9136B46 28-9309523 28-9432297 28-9654967 28-9827535 29-0000000 737881 739600 741321 743044 744769 746496 748225 749956 751889 753424 755161 756900 758641 633839779 636056000 638277381 640503928 642735647 29-0172363 9-442870 29-0344623 9-446607 29-0516781 9-450341 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 9-352-286 9-356095 9-359902 9-363705 9-367505 9-371302 9-375096 9-378887 9-382675 9-336460 9-390242 9-394021 9-3.7796 9-401569 9-405339 9-409105 9-412869 9-416630 9-4-20.337 9-424142 9-4278H4 9-431642 9-435383 9-439131 872 873 874 875 876 877 878 879 880 »81 832 883 834 885 886 887 888 889 890 891 892 893 894 29-0688837 29-0860791 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 29-3423015 29-3598365 29-3768616 644972544 29-3933769 647214625 29-4108823 649461896 29-4278779 651714363 29-4448637 9-535417 653972032 656234909 658503000 660776311 29-4618397 9-539082 29-4788059 29-4957624 9 546403 29-5127091 9-550059 9-45407 9-457800 9-461525 9-465247 9-463966 9-472682 9-476396 9-430106 9-483814 9-437518 9-491220 9-494919 9-498615 9-502308 9-505998 9-5U9B85 9-513370 9-517051 9-520730 9-524406 9-528079 9-531750 760384 762129 763376 765623 663J54848 665338617 6676-27624 669921875 767376 (^72221376 769129 674526133 770884 676836152 772641 679151439 774400 681472000 7761(il 683797841 777924 686128968 779639 688465387 731456 690807104 783225 693154125 78499r' 695506456 786769 697864103 733544 700227072 790321 702595369 792100 704969000 793881 707347971 ^., ot^u^oi 795664 709732288 29-8663690 797449 712121957 29-8831056 799236 714516984 29-8998328 895 801025 71691737: 896 802816 ~ """ 897 804609 898 806404 899 808201 900 81OUO0 29-5296461 29-5465734 29-5634910 29-5303989 29-5972972 29-6141858 29-6310648 29-6479342 29-6647939 29-6816442 29.6984848 29-7153159 29-7321375 29-7489496 29-7657521 29-7825452 29-8496231 9.553712 9-557363 9-561011 9-564656 9-568298 9-571938 9-575574 9-579-208 9-582840 9'C86468 9-590094 9-593717 9-597337 9-600955 9-604570 9-608182 9-542744 902 903 904 905 906 907 908 909 910 911 912 913 9H 915 916 91 918 919 92u 921 931 932 933 934 935 936 937 938 29 7993239 9-611791 29-8161030 9-615398 29-8328678 9-619002 " 9-6226U3 9-626202 9-629797 9-633S91 9-636931 9-640569 9-644154 9-647737 9-651317 9-654894 9-658463 9-662040 9-665610 9-669176 9-672740 9-6763.)2 9-679860 29-9165506 29-933-2591 29-9499583 29-9666481 29-9833287 30-0000000 30-0166620 30-0998333 30-1164407 719323136 721734273 724150792 726572699 .-.- ^.v-v.v 729000000 901 811801 73143-2701 ""■ 813604 733870808 30-0333148 815409 736314327 30-0499584 817216 738763264 30-0665928 819025 741217625 30-0832179 820836 743677416 822649 746142643 824464 748613312 826281 751089429 828100 753571000 829921 756058031 831744 758550528 30-1993377 9-697615 833569 761048497 30-2158899 9-701158 835396 763551944 30-2324329 9-704699 837225 766060875 30-2489669 9-708237 839056 768575296 840889 771095213 842724 773620632 844561 776151559 846400 778638000 -- 848241 781229961 922 850084 783777448 923 851929 786330467 924 853776 788889024 ov-a^/aooo 925 855625 791453125 30-4138127 926 857476 794022776 3J-4302481 927 859329 796597983 30-4466747 928 861184 799178752 30-4630924 929 363041 801765039 30-4795013 930 864900 804357O00 30-4959014 3-7bluw ""' 866761 806954491 30-512-2926 9-764497 868624 809557568 30-5286750 9-767992 870489 812166237 30-5450487 9-771484 872356 814780504 30-5514136 9-774974 874225 817400375 30-5777697 9-778462 876096 820025356 30-5941171 9-781947 877969 822656953 30-6104557 9-785429 879844 825293672 30-6267857 9-788909 30-1330383 9683417 30-1496269 9-686970 30-1662063 9-690521 30-1827765 9-694069 30-2654919 9-711772 30-2820079 9-715305 3J-2985148 9-718835 30-3150128 9-722363 30-3315018 9-725888 30-3479818 9-7-29411 30-3644529 9-732931 30-3809151 9-736448 30-3973683 9-739963 9-743176 9-746986 9-750493 9-753998 9-757500 9-761000 APPENDIX, 21 No. Square, j Cube. Sq. Root. CubeRoot.; No. Square. Cube. Sq. Root. CubeRoot. 939 881721 827936019 30-6431069 9-792386 970 940900 912673000 31-1448230 9-8S8933 940 883600, 8;W584000 30-6594194 9-795361 971 942841 9154J8611 31-1608729 9 -9023 -13 Ml 8854811 833237621 30-6757233 9-799334 972 94i734 918330048 31-1769145 9-905782 942 887364^ 835396888 30-6920185 9-802304 973 946729 921167317 31-1929479 9-909178 943 889249, 838561807 30-7083051 9-806271: 974 948676 924010424 31-2089731 9-912571 944 891136 8412323S4 30-7245830 9-809736 975 950825 926859375 31-2249900 9-915962 945 8930251 843908625 30-7408523 9-813199 976 952576 929714176 31-2409987 9-919351 946 894916 846590536 30-7571130 9-816659 977 954529 932574833 31-2569992 9-92273 S 947 896809 849278123 30-7733651 9-820117 978 956484 935441352 31-2729915 9-926122 948 898704 851971392 30-7896086 9-823572 979 958441 938313739 31-288^^757 9-929504 949 900601 854670349 30-8058436 9-827025 980 960400 941192000 31-3049517 9-93-2834 950 902500 857375000 30-8220700 9-830476 981 962361 944076141 31-3209195 9-936261 951 904401 860085351 30-8382879 9-833924' 982 964324 946966168 31-3368792 9-939636 95-^ 9U6304 862801408 30-8544972 9-837369 983 966289 94986-2087 31-3528308 9-9430U9 953 908209 865523177 30-8706981 9-840813 934 968256 952763904 31-3687743 9-946330 934 910116 868250664 30-8868904 9-844254 985 970225 955671625 31-3847097 9-94;)748 955 912025 870983875 30-9030743 9-847692 986 972196 958535256 31-4006369 9-953114 956 9139361 873722816 30-9192497 9-851128 987 974169 961504803 31-4165561 9956477 957 915849 876467493 30-9354166 9-854562 288 976144 964430272 31-4324673 9-959839 958 917764 879217912 30-9515751 9-857993 989 978121 967361669 31-4483704 9-963198 959 919681 881974079 30-9677251 9-861422 990 980100 970299000 31-4642654 9-966555 960 921600 884736000 30-9838668 9-864848 991 982081 973242-271 31-4801525 9-969909 9C1 923521 887503681 31-0000000 9-868272 992 984064 976191488 31-4960315 9-973262 962 925444 890277128 31-0161248 9-871694 993 986049 979146657 31-5119025 9-976612 963 927369 893056347 31-0322413 9-875113 994 988036 982107784 31-5277655 9-979960 964 929296 895841344 31-0483494 9-878530 995 990025 985074875 31-5436206 9-983305 965 931225 898632125 31-0644491 9-881945 996 992016 988047936 31-5594677 9-986649 966 933156 901428696 31-0805405 9-885357J 997 994009 991026973 31-5753068 9-989990 967 935089 904231063 31-0966236 9-888767' 998 996004 994011992 31-5911330 9-993329 968 937024 907039232 31-1126984 9-892175 999 998001 997002999 31-6069613 9-996666 969 938961 909853209 31-1287648 9-895580 1000 1000000^1000000000 31-6227766 10000000 The following rules are for finding the squares, cubes and roots, of numbers exceeding 1,000. To find the square of any numher divisible without a remainder. Rule. — Divide the given number by such a number, from the forego- ing table, as veill divide it vi^ithout 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 found in the table, gives a quotient of 2, the square of which is 4, and the square of 1,000 is 1,000,000, therefore : 4 X 1,000,000 == 4,000,000 : the Ans. Another example. — -What is the square of 1,230 ? 1,230, being di- vided by 123, the quotient will be 10, the square of which is 100, and the square of 123 is 15,129, therefore : 100 X 15,129 "= 1,512,900 : the Ans. To find the square of any numher not divisible without a remainder. Rule. — Add together the squares of such two adjoining numbers, froin the table, as shall together equal the given number, and multiply the sum by 2 ; then this product, less 1, will be the answer. Example. — What is the square of 1,487 ? The adjoining numbers 743 and 744, added together, equal the given number, 1,487, and tht. square of 743 = 552,049, the square of 744 = 553,536, and these added, = 1,105,585, therefore : 1,105,585 X 2 =- 2,211,170 — 1 = 2,211,169 : the Ans. To fold the cube of any number divisible without a remainder. Bule. — Divide the given number by such 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. Example.— What is the cube of 2,700 ? 2,700, being divided by 900, the quotient is 3, the cube of which is 27,. and the cube of 900 is 729,000,000, therefore : 27 X 729,000,000 -= 19,683,000,000 : the Ans. To find the square or cube root of numbers higher than is found in the table. Rule. — Select, in the column of squares or cubes, as the case may require, that number which is nearest the given number ; then the answer, when decimals are not of importance, will be found di- rectly opposite in the column of numbers. Example. — What is the square-root of 87,620? In the column of squares, 87,616 is nearest to the given number ; therefore, 296, im- mediately opposite in the column of numbers, is the answer, nearly. Another example. — What is the cube-root of 110,591 ? In the co- lumn of cubes, 110,592 is found to be nearest to the given number ; therefore, 48, the number opposite, is the answer, nearly. To find the cube-root more accurately. Mule. — Select, from the co- lumn of cubes, that number which is nearest the given number, and add twice the number so selected to the given number ; also, add twice the given number to the number selected from the table. Then, as the former product is to the latter, so is the root of the number selected to the root of the number given. Example. — What is the cube-root of 9,200 ? The nearest number in the column of cubes is 9,261, the root of which is 21, therefore : 9261 9200 2 2 18522 18400 9200 9261 As 27,722 is to 27,661, so is 21 to 20-953 -f- the Ans. 21 27661 55322 27722)580881(20-953 -f 55444 264410 249498 149120 138610 105100 83166 21934 APPENDIX. 23 To find the square or cube root of a whole numler 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 5000 1475 •020650 3-3166 3-33725 : the Ans. RULES FOR THE REDUCTIOxN 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 1 3 inches is /j ^^ ^ ^°°t, therefore : ySy ... 12) 3-00 •25 Ans. Another example. — ^What is the equivalent decimal of f of an inch 1 ^ .... 8) 7-000 •875 Ans. To reduce a compound fraction to its equivalent decimal. Rule. — In accordance with the preceding rule, reduce each fraction, commen- cing at the lowest, to the decimal of the next higher denomination, to which add the numerator of the next higher fraction, and reduce the sum to the decimal of the next higher denomination, and so proceed to the last ; and the final product will be the answer. Example. — What is the decimal of a foot equivalent to 5 inches, f and -J^ of an inch ? The fractions in this case are, ^ of an eighth, 4 of an inch, and -f^ of a foot, therefore : 3^ APPENDIX. i 2) 1-0 •5 3' eighths. i 8) 3-5000 •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 : 2 8 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 oif as before, and so proceed to the end ; then the seve- ral figures pointed off at the left will be the answer. Example. — What is the expression in inches of 0-390625 feet ? Feet 0-390625 12 inches in a foot. Inches 4-687500 8 eighths in an inch. Eighths 5-5000 2 sixteenths in an eighth Sixteenth 1-0 Ans., 4 inches f and ■^^. Another example. — What is the expression, in fractions of an inch, of 0-6875 inches ? Inches 0-6875 8 eighths in an inch. Eighths 5-5000 2 sixteenths in an eighth. Sixteenth 1*0 Ans., f and ^. 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, w^ill be inches ; if in feet, then feet, &c. Side of Side of Diam. Area. Circum. equal sq. Diam. Area. Circum. equal sq. •25 •04908 •78539 •22155 •75 90-76257 33-77212 9-52693 •5 •19635 1-57079 •44311 u- 95-03317 34-55751 9-74849 •75 •44178 2-35619 •66467 -25 99-40195 35-34291 9-97005 !• •78539 3-14159 •88622 •5 103-85890 36-12831 10-19160 •25 1-2^2718 3-92699 1-10778 •75 108-43403 36-91371 10-41316 ■5 1^76714 4-71-238 1-32934 12- 113-09733 37-69911 10-63472 •75 2-40528 5-49778 1-55089 •25 117^85881 38-48451 10-85627 2- 3-14159 6-23318 l-772i5 •5 122-71846 39-26990 11-07783 •25 3-97607 7-06858 1-99401 •75 127^67628 4005530 11-29939 •5 4-90873 7-85393 2-21556 13^ 132-73228 40-84070 11-52095 •75 5-93957 8-63937 2-43712 •25 137-88646 41-62810 11-74^250 3- 7-06853 9-4-2477 2-65363 •5 143^13881 42-41150 ir96406 •25 8-29576 10-21017 2-88023 -75 148-48934 43-19689 12-18562 •5 9-62112 10-99557 3-10179 14- 153-93804 43-982-29 12-40717 •75 11-04466 11-78097 3-3-2335 •25 159-48491 44-76769 12-62373 4- 12-56637 12-56637 3-54490 •5 165-12996 45-55309 12-85029 •25 14-186-25 13-35176 3-76646 •75 170-87318 46-33849 13-07184 •5 15-90431 14-13716 3-98802 15^ 176-71458 47-12338 13-29340 •75 17-72054 14^92256 4-20957 •25 182-65416 47-90928 13-51496 5- 19-63495 15-70796 4-43113 •5 188-69190 48-69468 13-73651 •25 21-64753 16-49336 4-65269 •75 194-8-2783 49-48008 13-95307 •5 23-75829 17-27875 4-87424 16^ 201-06192 50-26548 14-17963 ■75 25-96722 18-06415 5-09580 -25 207-39420 51-05088 14-40118 6^ 28-27433 18-84955 5-31736 •5 213-82464 51-83627 14-62274 •25 30-67961 19-63495 5-53891 •75 220-35327 52-62167 14-84430 •5 33-18307 20-42035 5-76047 17^ 226-98006 53-40707 15-06535 •75 35-78470 21-20575 5-98203 •25 233-70504 54-19247 15-28741 1- 33-48456 21-99114 6-20358 •5 240-52818 54-97787 15-50897 •25 41-28249 22-77654 6-4-2514 •75 247-44950 55-76326 15-73052 ■5 44-17864 23-56194 6-64670 18^ 264-46900 56-54866 15-95208 •75 47-17297 24-34734 6-86825 •25 266-58667 57-33406 16-17364 %■ 50-26548 25-13274 7-08981 •5 268-80252 58-11946 16-39519 •25 53-45616 2591813 7-31137 •75 276-11654 58-90486 16-61675 •5 58-74501 26-70353 7-53292 19-« 283-52873 59-69026 16-83831 •75 60-13204 27-48893 7-75448 •25 291-03910 60-47565 17-05986 9^ 63-61725 28-27433 7-97604 -5 298-64765 61-26105 17-28142 •25 67-20063 29-03973 8-19759 -75 306-35437 62-04645 17-59298 •5 70-83218 29-84513 8-41915 20- 314-15926 62-83185 17-72453 •75 74-66191 30-63052 8-64071 •25 322-06233 63-6 17-25 17-94609 10^ 78-53981 31-41592 8-86226 -5 330-06357 64-40264 18-16765 •25 82-51589 3-2-20132 9-03382 -75 338-16299 65-18804 18-38920 18-61076 •5 86^59014 32-98672 9-30538 21- 346-36059 65-97344 4* 26 APPENDIX. Side of Side of Diam. Area. Circum. equal sq. Diam. Area. Cireum. equal sq. '2i'i5 354^65635 66-75884 18-83232 38- 1134-]1494 119-38052 33-67662 •5 363-05030 67-54424 19-05387 -25 1149-08660 120-16591 33-89817 •75 371-54241 68-32964 19-27543 •5 1164-15642 120-95131 34-11973 22- 380-13271 69-11503 19^49699 •75 1179-32442 121-73671 34-34129 •25 388-82117 69-90043 19-71854] 39- 1194-59060 122-52211 34-56-285 •5 397-60782 70-68583 19-94010 •25 1209-95495 123-30751 34-78440 ■75 406-49263 71-47123 20-16166 •5 1225-41748 124-09290 3500596 23- 415-47562 72^25663 20-38321 •75 1210-97818 124-87830 35-22752 •25 424-55679 73-04202 20-60477 40- 1256-63704 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 1288-24933 127-23450 35-89219 24^ 452-38934 75-39822 21-26944 -75 1304-20273 128-01990 36-11374 •25 461-86320 76-18362 21-49100 41- 1320-25431 128-80529 36-33530 •5 471-43524 76-96902 21-71255 •25 1336-40406 129-59069 36-55636 •75 481-10546 77-75441 21-93411 •5 1352-65198 130-37609 36-77841 25^ 490-87385 78-53981 22-15567 -75 1368-99808 131-16149 36-99997 •25 500-74041 79-32521 22-37722 42- 1385-44236 131-94689 37-22153 •5 510-70515 80-11061 22-59878 •25 1401-98480 132-73228 37-44308 •75 520-76806 80-89601 22-82034 •5 1418-62543 133-51768 37-66464 26- 530-92915 81-68140 23-04190 •75 1435-36423 134-30308 37-88620 •25 541-18842 82-46680 23-26345 43^ 1452-20120 135-08348 38-10775 ■5 551-54586 83-25220 23-48501 •25 1469-13635 135-87383 38-3-3931 •75 562-00147 84-03760 23-70657 ■5 1486-16967 136-65928 38-55087 27^ 572-55526 84-82300 23-92812 -75 1503-30117 137-44467 38-77242 •25 583-20722 85-60839 24-14968 44- 1520-53084 138-23007 38-99398 •5 593-95736 86-39379 24-371241 -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 3965865 •25 626-79682 88-74999 25-03591 45- 1590-43128 141-37166 39-88021 •5 637-93965 89-53539 25-25746 •25 1608-15182 142-15706 40-10176 •75 649-18066 90-32078 25-47902 •5 16-25-97054 142-94246 40-32332 29^ 660-51985 91-10618 25-70058 •75 1643-88744 143-72786 40-54488 •25 671-95721 91-89153 25-92-213 46^ 1661-90-251 144-51326 40-76643 •5 683-49275 92-67698 26-14369 •25 1680-01575 145-29866 40-98799 •75 695-12646 93-46238 26-36525 •5 1698-22717 146-08405 41-20955 30^ 706-85834 94-24777 26-58680 •75 1716-53677 146-86945 41-43110 •25 71868840 95-03317 25-80836 47^ 1734-94454 147-65485 41-65266 •5 730-61664 95-81857 27-02992 •25 1753-45048 148-44025 41-874-32 •75 742-64305 96-60397 27-25147 •5 1772-05460 149-2-3565 42-09577 31- 751-76763 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 1828-45601 151-58184 42-76044 •75 791-73043 99-74556 28-13770 •5 1847-45282 152-367-24 42-98200 32- 804-24771 100-53096 28-35926 •75 1866-54782 153-15-264 43-20356 ■25 816-86317 101-31636 28-58081 49^ 1885-74099 153-93804 43 42511 •5 829-57681 102-10176 28-80237 •25 1905-83233 154-72343 43-64667 •75 842-38861 102-88715 2902393 •5 1924-42184 155-50883 43-86823 33^ 855-29859 103-67255 29-24548 •75 1943-90954 156-29423 44-08978 •25 868-30675 104-45795 29-46704 50- 1963-49540 157-07963 44-31134 •5 881-41308 105-24335 29-68860 •25 1983-17944 157-96503 44-53290 •75 894-61759 106-02875 29-91015 •5 2002-96166 158-65042 44-75445 34- 907-92027 106-81415 30-13171 •75 2022-84205 159-43582 44-97601 ■25 921-32113 107-59954 30-35327 51- 2042-82062 160-22122 45-19757 •5 934-82016 108-38494 30-57482 •25 2062-89736 161-00662 45-41912 •75 948-41736 109-17034 30-79638 •5 2083-07227 161-79202 45-64068 35^ 962-11275 109-95574 31-01794 -75 2103-34536 162-57741 45-86224 •25 975-90630 110-74114 31-23949 52- 2123-71663 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 4652691 36^ 1017-87601 113-09733 31-90416 •75 2185-41947 165-71901 46-74847 •25 1032-06227 113-88273 32-12572 53- 2206-18344 166-50441 46-97002 •5 1046-34670 114-66813 32-34728 •25 2227-04557 167-28980 47-19158 •75 1060-72930 115-45353 32-56883 •5 2248-00589 168-07520 47-41314 37^ 1075-21008 116-23892 32-79039 •75 2269-06438 168-86060 47-63469 •25 1089-78903 117-02432 33-01195 54- 2290-22104 169-64600 47-85625 •5 1104-46616 117-80972 33-23350 •25 2311-47588 170-43140 48-07781 •75 1119-24147 118^59572 33-45506 •5 2332-82889 171-21679 48-29936 APPENDIX, 27 Side of ] Side of Diam. Area. Circum. equal sq. | Diam. Area. Circum. equal sq. 54-75 2354-28008 172-00219 48-5-2092! 71-5 4015-15176 224-62337 63-36522 55- 2375-82944 172-78759 48-74248: -75 4043-27883 225 -409-27 63-58678 •25 2397-47698 173-57-299 48-964031 72- 4071-50407 2-26-19467 63-80833 •5 2419-2-2269 174-35839 49-18559 •25 4099-8-2750 226-;)8006 64-02989 •75 2441-06657 175-14379 49-40715 i •5 4128-24909 227-76546 64-35145 56- 246300864 175-92918 49-62870 -75 4156-76886 228-55086 64-47300 •25 2485-04887 176-71458 49-850-26' 73- 4185-38681 2-29-336-26 64-69456 •5 2507-18728 177-49998 50-07183 -25 4214-10293 230-12166 64-91612 •75 2520-42387 178-28538 50-29337 •5 4343-91722 230-90706 65-13767 57- 2551-75863 179-07078 50-51493 •75 4271-82969 231-69245 65-35923 •25 2574-19156 179-85617 50-73649 74- 4300-84034 333-47785 65-58079 •5 2596-7-2267 180-64157 50-95804 -25 4329-94916 333-36325 65-80234 •75 2619-35196 181-42697 51-17960 •5 4359-15615 234-04865 66-02390 58- 264207942 182 21237 51-40116 -75 4388-46132 234-83405 66-24546 •25 2664 90505 182-99777 51-6-2271 75- 4417-86466 235-61944 66-46701 •5 2687-83886 183-78317 51844-27 •25 4447-36618 236-40484 66-68857 •75 2710-85084 184-56856 52-06583 •5 4476-96588 237-19024 66-91043 59- 2733-97100 185-35396 52-28738 •75 4506-66374 237-97564 67-13168 •25 2757- 18933 186-13936 52-50894 76^ 4536-45979 238-76104 67-35334 •5 2780-30584 186-92476 52-73050 •25 4566-35400 239-54643 67-57480 •75 280392053 187-71016 52-95205 •5 4596-34640 240'33183 67-79635 60^ 2827-43338 188-49555 53-17364 ■75 4626-43696 241-117-23 68-01791 •25 2851-04442 189-28095 53-39517 77^ 4655-63571 341-90263 68-23947 •5 2874-75362 190-06635 53-61672 •25 4686-91262 243-68803 68-46103 •75 2898-56100 190-85175 53-83828 •5 4717-29771 243-47343 68-68358 61^ 2922-46656 191-63715 54-05984 ■75 4747-78098 244-25882 68-90414 •25 2946-47029 192-42255 54-28139 78- 4778-36242 245 04422 69-12570 •5 2970-57220 193-20794 54-50295 -25 4809-04204 245-82962 69-34725 •75 2994-77228 193-99334 54-72451 •5 4839-81983 246-61502 69-56881 62^ 3019-07054 194-77874 54-94606 •75 4870-79579 247-40042 69-79037 ■25 3043-46697 195-56414 55-16762 79^ 4901-66993 248-18581 70-01192 •5 3067-96157 196-34954 55-38918 •25 4932-74-225 248-97131 70-23348 •75 3092-55435 197-13493 55-61073 -5 4963-91274 249-75661 70-45504 63- 3117-24531 197-92033 55-83229 •75 4995-18140 350-34201 70-67659 •25 314203444 198-70573 56-05385 80- 5026-54824 251-32741 70-89815 •5 3166-92174 199-49113 56-27540 •25 5058-01335 252-11-281 71-11971 •75 3191-90722 200-27653 56-49696 -5 5089-57644 252-89820 7r34126 64^ 3216-99087 201-06192 56-71853 •75 5121-23781 253-88360 7r55282 •25 3242-17270 201-84732 56-94007 81^ 5152-99735 254-46900 7r78438 •5 3267-45270 202'63-272 57-16163 •25 5184-85506 255^25440 72-00593 •75 3293 83088 203-41812 57^38319 •5 5216-81095 256-03980 72-22749 65- 3318-30724 204-20352 57-60475 •75 5248-86501 256-82579 73-44905 •25 3343-88176 204-98892 57-82630 82- 5281-01725 257-61059 72-67060 •5 3369-55447 205-77431 58-04786 ■25 531326766 253-39599 72-89216 •75 3395-32534 206-55971 58-26942 -5 5345-616-24 259-18139 73-11372 66- 3421-19439 207-34511 58-49097 ■75 5378^06301 259-96679 73-335-27 •25 344716162 208^ 13051 58-71253 83- 5410-60794 260-75219 73-55683 •5 3473-22702 208-91591 58-93409 -25 5443-25105 261-53758 73-77839 •75 3499-39060 209-70130 59-15564 ■5 5475-99234 262-32298 73-99994 67- 3525-65235 210'48570 59-37720 ■75 5508-83180 263-10838 74-22150 25 3552-01228 211-27210 59-59876 84- 5541-76944 263-89378 74-44306 •5 3578-47038 213-05750 59-82031 -25 5574-80525 264-67918 74-66461 •75 360502665 212-84290 60-04187 •5 5607-93923 265-46457 74-88617 68' 3631-68110 213-62930 60-26343 •75 5641-17139 266-24997 75-30773 •25 3658-43373 214-41369 60-48498 85- 5674-50173 267-03537 75-32928 75-55084 •5 3685-28453 215-19909 60-70654 •25 5707-93023 257-82077 •75 3712-33350 215-98449 60-92810 -5 5741-45692 268-60617 75-77240 69^ 3739-28065 21676989 61-14965 •75 5775-08178 269-39157 75-99395 •25 3766-42597 217-55529 61-37121 86- 5808-80481 270-17696 76-21551 •5 3793-66947 218-34068 61-59377 •25 5842-62602 270-96236 76-43707 •75 3821-01115 219-12608 61-81432 -5 5876-54540 271-74776 76-65362 70- 3848-45100 219-91143 62-03588 •75 5910-56396 272-53316 76-88018 •25 3875-98902 220-69683 63-35744 87- 5944 67869 273-31856 77-10174 •5 3903-625-22 221-483-28 62-47899 •25 5978-89360 274-10395 77-32329 •75 3931-35959 222-26768 62-70055 -5 6013-20468 274-88935 77-54485 71- 3959-19214 223-05307 62-92311 -75 6047-61494 275-67475 77-766-il •25 3987-12386 223-83847 63-14366 88- 6082-12337 276-46015 77-98796 28 APPENDIX. Side of Side of Diam. Area. Circum. equal sq. Diam. Ai-ea. Circum. equal sq. "88^5 6116-72993 277-24555 78-20952 94-25 6976-74097 2;6-0[!510 83-52688 •5 6151^43476 278-03094 78-43103 •5 7013-80194 296-88050 83-74344 •75 6186-23772 278-81634 78-652C.3 •75 7050-36109 297-66590 83-97000 89- 6221-13885 279-60174 78-87419 95^ 7083-21842 298-45130 84-19155 •25 6256-13815 230-33714 79-09575 •25 7325-57992 299-23670 84-41311 •5 6291-23563 231-17254 79-31730 •5 7163-02759 300-0-2209 84-03467 •75 6326-43129 281-95794 79-53886 •75 7200-57944 300-80749 84-85622 90- 6361-72512 282-74333 79-76042 96^ 7238-22947 301-53239 85-07778 •25 6397-11712 233-52873 79-98193 •25 7275-97767 302-37829 85-29934 •5 6432-60730 234-31413 80-20353 •5 7313-82404 3U3- 16369 85-52089 •75 6463-1S566 285-0^953 80-42509 •75 7351-76859 303-94908 85-74245 91^ 6503-83219 285-83493 80-64669 97^ 7389-81131 304-73448 85-96401 .25 6539-66689 286-67032 80-86820 ■25 74-27-95221 305-51983 85-18556 •5 6575-54977 287-45572 81-03976 •5 7466-19129 306-30523 86-40712 •75 6511-53082 288-24112 81-31132 -75 7504-52853 307-09068 86-62868 92- 6347-61005 289-02652 81-53287 98- 7542-96396 307-87603 86-85023 •25 66S3-73745 289-31192 81-75443 •25 7581-49755 308-68147 87-07179 •5 6720-06303 290-59732 81-97599 •5 7620-12933 309-44637 87-29335 •75 6756-43678 291-33271 82-19754 -75 7653-85927 310-232-27 87-51490 93- 6792-90871 292-16811 82-41910 99- 76^7-68739 311-01767 87-73646 •25 6829-47831 292-95351 82-64066 -25 77S6-61369 311-80307 87-95802 •5 6866-14709 293-73391 82-86221 •5 7775-63816 312-58346 88-17957 •75 6902-91354 294-52431 83-08377 •75 7814-76081 313-37336 88-40113 94^ 6939-77817 295-30970 83-30533 100- 7353-98163 314-15926 83-62269 The following rules are for extending the use of the above table. To find the area, circumference., or side of equal square, of a circle having a diameter of more than 100 inches, feet, ^c. Rule. — Divide the given diameter by a number that will give a quotient equal to some one of the diameters in the table ; then the circumference or side of equal square, opposite that diameter, multiplied by that divisor, or, the area opposite that diameter, multiplied by the square of the aforesaid divisor, will give the answer. Example. — 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 diameters in the table, the area of which is 176-71458, therefore : 176-71458 100 =- 10 X 10 17,671-45800 inches. Ans. To find the area, circumference, or side of equal square, of a circle "having an intermediate diameter to those in the table. Rule. — Multiply the given diameter by a number that will give a product equal to some one of the diameters in the table ; then the circumference or side of equal square opposite that diameter, divided by that multiplier, or, the area opposite that diameter divided by the square of the aforesaid mul- tiplier, will give the answer. APPENDIX. 2d Example. — What is the circumference of a circle whose diameter is 6J, or 6-125 inches ? 6-125, multiplied by 2, gives 12-25, one of the diameters of the table, whose circumference is 38-484j 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, will give its circumference ; the square of the diameter, multiplied by -78539, will give its area ; and the diameter, multiplied by -88622, will give the side of a square equal to the area of the circle. TABLE SHOWING THE CAPACITY OF WELLS, CISTERNS, &C. The gallon of the state of New- York is required to contain 8 pounds of pure water ; and since a cubic foot of pure water weighs 62-5 pounds, the gallon contains 221-184 cubic inches. Upon these data the following table is computed. One foot in depth of a cistern of 3 feet diameter will contain H do. do. 4 do. do. 4i do. do. 5 do. do. H 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. — The area of a circle in feet, divided by the decimal, -128, will give the number of gallons per foot in depth. TABLE OF POLYGONS. (From Gregory's Mathematics.) Multipliers for Radius of cir- Factors for 12; -S areas. cum. circle. sides. 3 Trigon 0-4330127 0-5773503 1-732051 4 Tetragon, or Square 1-0000000 0-7071068 1-414214 5 Pentagon - 1-7204774 0-8506508 1-175570 6 Hexagon 2-5980762 1-0000000 1-000000 7 Heptagon - 3-6339124 1-1523824 0-867767 8 Octagon 4-8284271 1-3065628 0-765367 9 Nonagon - 6-1818242 1-4619022 0-684040 10 Decagon 7-6942088 1-6180340 0-618034 11 Undecagon 9-3656399 1-7747324 0-563465 12 Dodecagon - 11-1961524 1-9318517 0-517638 To find the area of any regular polygon, whose sides do not exceed twelve. 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 find ike radius of a circle which vjill circumscribe 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 circumscribing circle, standing opposite the name of the given poly- gon, and the product will give the answer. Example. — What is the radius of a circle which will circumscribe a regular pentagon, whose sides measure each 10 feet 1 •8506508 10 8-5065080 : Ans. To find the side of any regular polygon that may be inscribed within a given circle. Rule. — Multiply the radius of the given circle by the number in the column termed Factors for sides, standing opposite the name of the given polygon, and the product will be the answer. Ex- ample. — What is the side of a regular octagon that may be inscribed within a circle, whose radius is 5 feet ? •765367 5 3-826835: Ans. WEIGHT OF MATERIALS. Woods. His. in a cubic foot. Metals. lbs. in a cubic foot. Apple, . . - - 49 Wire-drawn brass. . 534 Ash, - 45 Cast brass, 506 Beach, ... . 40 Sheet-copper, - 549 Birch, . 45 Pure cast gold, - - 1210 Box, . 60 Bar-iron, 475 to 487 Cedar, . 28 Cast iron, - 450 to 475 Virginian red cedar, . 40 Milled lead, - - 713 Cherry, . 38 Cast lead. 709 Sweet chestnut. . 36 Pewter, - 453 Horse-chestnut, . 34 Pure platina, - 1345 Cork, . 15 Pure cast silver, - 654 Cypress, - 28 Steel, 486 to 490 Ebony, - - 83 Tin, - 456 Elder, - 43 Zinc, 439 Elm, . 34 Stone, Earths, SfC. Fir, (white spruce,) • . 29 Brick, Phila. stretchers, 105 Hickory, . 52 North river common hard Lance-wood, . 59 brick. - 107 Larch, - . - . 31 Do. salmon brick, 100 Larch, (whitewood,) . 22 Brickwork, about 95 Lignum-vitse, - - 83 Cast Roman cement, - 100 Logwood, \- - 57 Do. and sand in equal parts, 113 St. Domingo mahogany, - 45 Chalk, 144 to 166 Honduras, or ba)^maho gany, 35 Clay, - . - 119 Maple, - 47 Potter's clay, 112 to 130 White oak. 43 to 53 Common earth. 95 to 124 Canadian oak, . 54 Flint, - - 163 Red oak. . 47 Plate-glass, 172 Live oak. - 76 Crown-glass, - - 157 White pine, 23 to 30 Granite, 158 to 187 Yellow pine, 34 to 44 Quincy granite, - 166 Pitch pine, 46 to 58 Gravel, 109 Poplar, . 25 Grindstone, - - 134 Sycamore, - 36 Qvpsum, (Plaster- stone,) 142 W^alnut, - 40 Dnslaked lime, . 52 32 5 APPENDIX. Cbs. in a lbs. in a cubic foot. cubic foot. Limestone, - - 118 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 beeswax, - - 60 Do. do. including lath 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 Nyack blue stone, 171 Wood-ashes, - - 58 THE END. %\