^M;»;r,»r»,t;i;t;»:( t 1,1:1 1 • «:t;^t;i:t;t:i THE AIEHICAW HOUSE-CARPENTER A TREATISE UPON ARCHITECTURE, CORNICES AND MOULDINGS FRAMING, DOORS, WINDOWS, AND STAIRS. TOCKTDCEE WITH THE MOST IMPORTANT PRINCIPLES *. *•': •'• '••.'••/• : :*•:•.*'*:*:':*: :*:*:*.* i :\ /A ^: : : : • : PRACTICAL GEOMETRY ••"*.*•• .***. \' '.V, 1 i.'i.i: •**• • I i ! t. i * *' . ?•* *-'*-' 5 III.***. BY R. G. HATFIELD, ARCHITECT. THIRD EDITION. SllustrateH hs more IJan tt)j:ec J)un»re\j Enflrabftifl** NEW YORK: * JOHN WILEY, 161 BROADWAY, AND 13 PATERNOSTER ROW, LONDON. 1849. C O/' JS Tfi H36 BnteRd aeeerding to the Aet of Congreav in the year 1844, BT m. e. HATriKLD, b flM Ctlerk'^ A0CK»/i^. tlie piatriot Court of the Southern DUtrict 9( l(ew*Y<»k. THE GETTY CENTEh ^ LIBRARY 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 authority, and subjected to the most careful examina- tion. The explanations have all been written out from the figures themselves, and not taken from any other work ; and the figures have all been drawn ex- pressly for this book. In doing this, the utmost care has been taken to make every thing as plain as the nature of the case would admit. 13 ^ -3 / 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. PREFACE TO THE SECOND EDITION. In preparing a second edition, the author regrets his want of leisure to give the work that thorough revision, which is demanded by the importance and intricacy of the subjects treated of. He has corrected all the typo- graphical errors in the references, &c., that he could discover, and added a Section on the subject of Shadows. He cannot refrain from giving expression to the satisfaction which he feels at the unexpected suc- cess of his undertaking. But it is evident that in a wide-spread, new country like ours, works of a practical character, adapted to the wants of the people, and cal- culated to instruct our operative citizens in the every- day employment of their heads and hands, cannot but meet with a favourable reception. In another edition, perhaps, opportunity will be given for additions and improvements of a still more important nature. TABLE OF CONTENTS. INTRODUCTION. Articles necessary for drawing To prepare the paper, Art. 2 5 To use the set-square, Directions for drawing. Art. 11 13 SECT. I.~PRACTICAL GEOMETRY. DEFINITIONS. Lines, - - - - 17 Angles, . - . 23 Angular point, - - - 27 Polygons, ... 28 The circle, - . . 47 The cone, ... 53 Conic sections, - - - 58 The ellipsis, - - - 61 The cylinder, - - - 68 PROBLEMS. To bisect a line, - - 71 To erect a perpendicular, - 72 To let fall a perpendicular, 73 To erect ditto on end of line, 74 Six, eight and ten rule, - 74 To square end of board, . 74 To square foundations, &c., 74 To let fall a perpendicular near the end of a line, - 75 To make equal angles, - 76 To bisect an angle, - . 77 To trisect a right angle, 78 To draw parallel lines, - 79 To divide a line into equal parts, - - - . 80 To find the centre of a circle, 81 To draw tangent to circle, 82 Do. without using centre, 83 To find the point of contact, 84 To draw a circle through three given points, . - 85 To find a fourth point in circle, 86 To describe a segnDent 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, (fee, - - - 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 viii CONTENTS. Art. To find the length of a hrace, 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- dinates, - - - 117 To trace a curve through given points, - - - 117 To describe an ellipsis by in- tersection of lines, - 118 Art. Do. from conjugate diameters, 1118 Do. by intersecting arcs, - 1119 To describe an oval by com- passes, - - - 1120 Do. in the proportion, 7 x 9> 5x7, &c., - - - 121 To draw a tangent to an el- lipsis, - - - 1122 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. HISTORY. Antiquity of its origin, - 159 Its cultivation among the an- cients, ... 160 Among the Greeks, - - 161 Among the Romans, - 162 Ruin caused by Goths and Vandals, - - - 163 Of the Gothic, - - 164 Of the Lombard, - - 165 ^ CONTENTS. IX Art. Of the Byzantine and Oriental, 166 Moorish, Arabian and Modern Gothic, - - - 167 Of the English, - - 168 Revival of the art in the sixth century, - - - 169 The art improved in the 14th and 15th centuries, - 170 Roman styles cultivated, 171 STYLES. Origin of different styles, 172 Stylobate and pedestal, - 173 Definitions of an order, - 174 Of the several parts of an order, - - 175. 185 To proportion an order, - 186 The Grecian orders, - 187 Origin of the Doric, - - 188 Intercolumniation, - - 189 Adaptation, - - - 190 Origin of the Ionic, - 191 Characteristics, - - 192 Intercolumniation, - - 193 Adaptation, - - - 194 To describe the volute, - 195 Origin of the Corinthian, - 196 Adaptation, - - - 197 Persians, - - - - 199 Caryatides, - - - 200 The Roman orders, - - 202 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 PRINCIPLKS. Origin of the art, - - 215 Arrangement and design, - 216 Ventilation and cleanliness, 217 Stability, - - . 218 Oi'naments, - - - 219 Scientific knowledge neces- sary, - - . 220 The foundation, - - 221 The column, - - - ' 222 The wall, - - - 223 The lintel, - - - 224 The arch, - . . 225 The vault, - - . 226 The dome, - - - 227 The roof, - . . 228 SECT. 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 Antee caps, - - - 240 CORNICES. Designs, - - - - 241 To proportion an eave cornice, 242 Do. from a smaller given one, . - . - 243 Do. from a larger given one, . - . . 244 To find shape of angle-bracket, 245 To find form of raking cornice, 246 mmmm 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 I, {Fig. 37,) be the given line, and 5 the number of parts. Draw a c, at any angle to a 5 ; 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,2f,3g and 4 h, parallel to cb; 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.) ^ ^ 81.— THE CIRCLE. To find the centre of a circle. Draw any chord, as « b, 22 AMERICAN HOUSE-CARPENTER. {Fig. 38,) and bisect it with the perpendicular, c d ; bisect c d with the Hne, e/, 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 lipon the same principle as Art. 85. 81, b. — A third method. Draw any chord, as a b, {F^g. 40,) PRACTICAL GEOMETRY. and from the point, a, draw a c, at right angles to a 6 ; 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 h 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.) c tig. tl. 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. d a Fig. 43. 83. — 77ie 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 From any point, as a, {Fig. 43,) in the tangent, b c, draw 2&- AMERICAN nOl'SE-CA-RPENTER. Fig. 43. 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, 6 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 h and c, with any radius greater than half h 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, a b e, and it will be the one required. Fig. 45. PRACTICAL GEOMETRY. 86. — Three points not in a straight line being given, to find a fourth that shall, with the three, lie in the circumference^ of a circle. Let a b c, {Fig. 45,) be the given points. Connect them with right lines, forming the triangle, a c b ; bisect the angle, cb a, {Art. 77,) with the line, b d ; also bisect c a in e, and erect e d, perpendicular to a c, cutting b d m. d ; then d is the fourth point required. A fifth point may be found, as at /, by assuming a, d and b, as the three given points, and proceeding as before. So, also, any number of points may be found ; simply by using any three already found. This problem will be serviceable in obtaining short pieces of very flat sweeps. (See Art. 311.) 87. — To describe a segment of a circle by a set-triangle. Let d 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 c /, by nailing them together at c, and afiixing a brace from e to /; put in pins at a and b ; move the angular point, c, in the direction, a c b ; keeping the edges of the triangle hard against the pins, a and b ; a pencil held at c will describe the arc, a c b. If the angle formed by the rulers at c be a right angle, the segment described will be a semi-circle. This problem is useful in describing centres for brick arches, when they are required to be rather flat. Also, for the head hanging- stile of a window- frame, where a brick arch, instead of a stone lintel, is to be placed over it. c e g 1 2 3 c 3 2 1 h J a 2 3 d Fig. 47. 3 2 1 b 4 AMERICAN HOUSE-CARPENTER. 88. — To describe the segment of a circle hy intersection of lines. Let a 6, [Fig. 47,) be the chord, and c d the height of the segment. Through c, draw e/, parallel to a b ; draw b f at right angles to c b ; make c e equal to c /; draw a g and 6 h, at right angles to a 6 ; divide c e, c f, d a, d b, a g and 6 A, each into a like number of equal parts, as four ; draw the lines, 1 1, 2 2, &c., and from the points, o, o and o, draw lines to c ; at the intersection of these lines, trace the curve, a cb, which will be the segment required. In very large work, or in laying out ornamented gardens, &c., this will be found useful ; and where the centre of the proposed arc of a circle is inaccessible, it will be invaluable. (To trace the curve, see note at Art. 117.) b 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, 1 1 ' ) y / e / J 6. g Fig. 85. 119.— To describe an ellipsis by intersecting arcs. Let a b 46 AMERICAN HOUSt]-CAllT>El^T*E^^ and erf, [Fig. 85,) be given axes. Between one of the foci, / and/, and the centre, e, mark any number of points, at random, as 1, 2 and 3 ; upon /and/, with b 1 for radius, describe arcs at ^, g and g ; upon/ and/, with a 1 for radius, describe arcs inter- secting the others at^,^,^ and g; then these points of intersection will be in the curve of the ellipsis. The other points, h and i, are found in like manner, viz : h is found by taking 6 2 for one radius, and a 2 for the other ; i is found by taking b 3 for one radius, and a 3 for the other, always using the foci for centres. Then by tracing a curve through the points, c, g, h, i, b, (fee, 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. f r\/ \ s Fig. 86. 120. — To describe a figure nearly in the shape of an eUip" sis, by a pair of compasses. Let a b and c d, {Fig. 86,) be given axes. From c, draw c e, parallel to a 6 ; from a, draw a e, parallel to c d; join e and d ; bisect e a in /; join /andc, inter- secting e d in i; bisect i c in o ; from o, draw og, at right angles to i c, meeting c d extended to g ; join i and g, cutting the trans- verse axis in r ; make h j equal to h g^ and h k equal to h r ; from J, through r and k, draw J m and j n; also, from g, through k, draw g I ; upon g and with g c for radius, describe the PRACTICAL GEOMETRY. 47 arcs, i I and mn; upon r and k, with r a for radius, describe the arcs, m i and In; this will complete the figure. "When the axes are proportioned to one another as 2 to 3, the exitremities, 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. a Fig. 87. 12L — 7h draw an oval in the propiirtion, seven by nine, Leit cd^ {Fig. 87,) be the given conjugate axis. Bisect c c? in o, and through o, draw a 6, at right angles locd ; 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 gl ; upon g^ with g c for radius, describe the arc, k I ; upon e, with e d(ox radius, describe the arc, J i ; upon h and /, with k k for radius, describe the arcs, j k and I i; this will complete the figure. This is a very near approximation to an ellipsis ; and perhaps no method can be found, by which a well-shaped oval can be drawn with greater facility. By a little variation in the process, ovals of different proportions may be obtained. If quarter of the trans- verse axis is taken for the radius of the circle, ef g one will be dra-wn in the proportion^ five by seven. 48 AMERICAN HOUSE-CARPENTER. 122.— To draw a tangent to an ellipsis. Let a & c d, {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 / d; bisect the angle, {Art. 77,) e d o, with the line, s r ,• then s r will be the tangent required. X23. An ellipsis with a tangent given, to detect the point of contact, hetagb /, {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 c in 0 ; through o and e, draw 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 /^ will be the diameterrequired. PRACTICAL GEOMETRY. i25.-Any diameter and its conjugate being given, to as- certain the two axes, and thence to describe the ellipsis Let a 6 and c d, {Fig. 90,) be the given diameters, conjugate to one another. Through c, draw ef, parallel to a 5 ; from c, draw c g, at right angles to ef; make c g equal to a h or h b ; join g and h; upon g, with g c for radius, describe the arc, i k cj; upon h, with the same radius, describe the arc. In; through the inltersections, I and n, draw n o, cutting the tangent, ef, in o; upon 0, with 0 g for radius, describe the semi-circle, eigf- join 0 ^ndg, also g and/, cutting the arc, i c j, in and t; from e through h, draw e also from/, through A, draw/p ; from k and ^, draw k r and ^ ^, parallel to^ h, cutting e m in r, and /p m ^ ; make A m equal to h r, and hp equal to h s ; then r and p will be the axes required, by which the ellipsis may be drawn in the usual way. 126.~-To describe an ellipsis, tahose axes shall be propor- tionate to the axes of a larger or smaller given one. Let a c b d, {Fig. 91,) be the given ellipsis and axes, and i f the trans- verse axis of a proposed smaller one. Join a and'c; from Imw i e, parallel to a c ; make o / equal to o c ; then ef will be 7 6C AMERICAN HOUSE-CARPENTER. C Fig. 91. the conjugate axis required, and will bear the same proportion to ij, use d does to a b. (See Art. 108.) 12 3 J 3 2 1 Fig. 92. 127.— To describe a parabola by intersection of lines. Let m I, {Fig. 92,) be the axis and height, (see Fig. 79,) and d d, double ordinate and base of the proposed parabola. Through /, draw a a, parallel to d d ; through d and d, draw d a and d a, parallel to ml; divide a d and d m, each into a like number of equal parts ; from each point of division in d m, draw the lines, 1 1, 2 2, &c., parallel to ml; from each point of division in d a, draw lines to I ; then a curve traced through the points of intersection, o, o and o, Avill be that of a parabola. 127, a.— Another method. Let m I, {Fig. 93,) be the axis and height, and d d the base. Extend m /, and make I a equal to m I ; join a and d, and a and d ; divide a d and a d. each into a like number of equal parts, as at 1, 2, 3, &c. ; join 1 and 1, 2 and 2, &c., and the parabola will be completed. PRACTICAL GEOMETRY. 51 Fig. 93. Fig. 9i. 128. — To describe an hyperbola by intersection of lines. Let r 0, {Fig. 94,) be the height, p p the base, and n r the trans- verse axis. (See Fig. 79.) Through r, draw a a, parallel to p p ; from p, draw a p, parallel to r o ; divide a p and p o, each into a like number of equal parts ; from each of the points of di- visions in the base, draw lines to n ; from each of the points of division in a p, draw lines to r ; then a curve traced through the points of intersection, o, o, &c., will be that of an hyperbola. The parabola and hyperbola afford handsome curves for various mouldings. • DEMONSTRATIONS. 129. — To impress more deeply upon the mind of the learner some of the more important of the preceding problems, and to indulge a very common and praiseworthy curiosity to discover the cause of things, are some of the reasons why the following exercises are introduced. In all reasoning, definitions are ne- cessary ; in order to insure, in the minds of the proponent and respondent, identity of ideas. A corollary is an inference deduced from a previous course of reasoning. An axiom is a proposition evident at first sight. In the following demonstrations, there are many axioms taken for granted; (such as, things equal to the same thing are equal to one another, &c. ;) these it was thought not necessary to introduce in form. a c li Fig. 95. d 130. — Definition. If a straight line, as a b, {Fig. 95,) stand upon another straight line, as c d, so that the two angles made at PRACTICAL GEOMETRY. 53 the point, b, are equal — a b cto a b d, (see note to Art. 27,) then each of the two angles is called a right angle. 131. — Definition. The circumference of every circle is sup- posed to be divided into 360 equal parts, called degrees ; hence a semi-circle contains 180 degrees, a quadrant 90, &c. Fig. 96. 132. — Definition. The measure of an angle is the number of degrees contained between its two sides, using the angular point as a centre upon which to describe the arc. Thus the arc, c e, {Fig. 96,) is the measure of the angle, c b e ; e a, of the angle, e b a ; and a d, of the angle, ab d. 133. — Corollary. As the two angles at b, {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 oee right angle, 90 degrees ; of half a right angle, 45 ; of one-third of a right angle, 30, (fee. 134. — Definition. In measuring an angle, {Art. 132,) no re- gard is to be had to the length of its sides, but only to the degree of their inclination. Hence equal angles are such as have the same degree of inclination, without regard to the length of their sides. a c 135. — Axiom. If two straight lines, parallel to one another, « 54 AMERICAN HOUSE-CARPENTER. as a 6 and c d, {Fig. 97,) stand upon another straight line, as e /, the angles, a bf and c d f, are equal ; and the angle, a 6 e, is equal to the angle, c d e. 136. — Definition. If a straight line, as a b, {Fig. 96,) stand obliquely upon another straight line^ as c d, then one of the an- gles, as a 6 c, is called an obtuse angle, and the other, as ab dy an acute angle. 137. — Aa:iom. The two angles, a b d and a b c, {Fig. 96,) are together equal to two right angles, {Art. 130, 133 ;) also, the three angles, a b d,e b a and cb e, are together equal to two right angles, 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. V% 98. 140. — Proposition. If to each of two equal angles a third angle be added, their sums will be equal. Let ab c and d e fy {Fig. 98,) be equal angles, and the angle, i j the one to be added. Make the angles, gba 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, if a 6 c and e/be angles of 90 degrees, and i j k, 30, then the angles, g b c and hef, will be each equal to 90 and 30 added, viz : 120 degrees. PRACTICAL GEOMETRY. 66 a d Fig. 99. 141. — Proposition. Triangles that have two of their sides and the angle contained between them respectively equal, have also their third sides and the two remaining angles equal ; and consequently one triangle will every way equal the other. Let a 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 f; the angle at b with the angle at e ; the angle at c with the angle at / ; and the triangle, 6 a c, be every way equal to the triangle, e df. a b d Fig. 100. 142. — Proposiiion. The two angles at the base of an isoceles triangle are equal. Let a b c, {Fig. 100,) be an isoceles triangle, of which the sides, a b and a c, are equal. Bisect the angle, {Art. 1$ 56 AMERICAN HOUSE-CARPENTER. 77,) 6 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 b must be equal to the angle at c. a !, b a 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 b d, and a 6 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. « J'- b n % « ^'^^ 144. — Proposition. Let abed, [Fig. 102,) be a given pa- rallelogram, and 6 c a diagonal. At any distance between a b and c rf, draw e /, parallel to a 6 ; through the point, g, the intersection of the lines, b c and e /, 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 b 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, a b c, and what remains PRACTICAL GEOMETRY. 67 in one must be equal to what remains in the other ; therefore, the parallelogram, A, is equal to the parallelogram, B. a b e f e T Fig. 101 145. — 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; 6 e being added to both a b and ef, a e equals bf; the line, ac, being equal to b d, and a e to bf, and the angle, c a e, being equal, {Art. 135,) to the angle, d bf, the triangle, a e c, must be equal, {Art. 141,) to the triangle, bfd; these two triangles being equal, take the same amount, the triangle, beg; from each, and what remains in one, a b g c, must be equal to what remains in the other, e f d g ; these two quadrangles being equal, add the same amount, the triangle, c g d,to each, and they must still be equal ; therefore, the parallelogram, ab c d, is equal to the paral- lelogram, efcd. 146. — Corollary. Hence, if a parallelogram and triangle stand upon the same base and between the same parallels, the parallelo- gram will be equal to double the triangle. Thus, the paral- lelogram, a 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 to a 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 paxal- 8 58 AMERICAN HOUSE-CARPENTER. a Fig. 104. lels, a d and h e, they are therefore equal, {Art. 145, 146 ;) and since the triangle, C, is common to both, the remaining triangles, A and By are therefore equal ; then B being equal to A, the triangle, o 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, Band C, which together amount, {Art. 137,) to two right angles ; in the same manner, the angles, A and B, form two right angles ; since the angles, A and B, are equal to B and C, take the same amount, the angle, B, from each pair, and what remains of one pair is equal to what remains of the other ; therefore, the an- gle, A, is equal to the angle, C. The same can be proved of the opposite angles, B and D. 149. — Proposition. The three angles of any triangle are equal to two right angles. Let a b c, {Fig". 106,) be a given tri- angle, with its sides extended to /, e, and d, and the line, c ^, PRACTICAL GEOMETRY. 69 b c d Fig. 106. drawn parallel ioh e. As g c is parallel to e 6, the angle, g c 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, (J.r^. 148 ;) as the angle,/ a e, is equal, (^r^. 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 nge 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. ad c Fig. 107. 156. — Proposiiio7i. 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 a b 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, a 6 c?, also, the angle at c is, for the same reason, equal to the angle, d b c ; the angle, a b c, being equal to the angles at a and c taken together, must therefore, {A7^t. 153,) be a right angle. Fig. 108. 157. — Proposition. The square of the hypothenuse of a right-angled triangle, is equal to the squares of the two remaining sides. 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. proved by showing that the parallelogram, b 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, a 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- b, being both upon the same base,/6, and between the same parallels, / b and^ c, the square, g b, is equal, {Art. 146,) to twice the triangle, f b c ; 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, /Z> c or a b d. The method of proving h c equal to c Z is exactly similar — thus proving the square, b e, equal to the squares, h c and 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- led, (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. 68' AMERICAN HOUSE-CARPENTER. The house-carpenter, especially, needs information of this kind, and were he thoroughly acquainted with the principles of geome- try, he would be much less liable to commit mistakes, and be better qualified to excel in the execution of his often difficult un- dertakings. 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 ihe construction of edifices known as dwellings, churches and other public buildings, bridges, &.C., for the accommodation of civilized man — and is the subject of the remarks which follow. 159. — This is one of the most ancient of the arts : the scrip- tures inform us of its existence at a very early period. Cain, the son of Adam, — " builded a city, and called the name of the city after the name of his son, Enoch" — but of the peculiar style or manner of building we are not informed. It is presumed that it was not remarkable for beauty, but that utility and perhaps sta- bility were its characteristics. Soon after the deluge — that me - 64 AMERICAN HOUSE-CARPENTER. morable event, which removed from existence all traces of the works of man — the Tower of Babel was commenced. This was a work of such magnitude that the gathering of the materials, according to some writers, occupied three years ; the period from its commencement until the work was abandoned, was twenty- two years ; and the bricks were like blocks of stone, being twenty feet long, fifteen broad and seven thick. Learned men have given it as their opinion, that the tower in the temple of Belus at Baby- lon was the same as that which in the scriptures is called the Tower of Babel. The tower of the temple of Belus was square at its base, each side measuring one furlong, and consequently half a mile in circumference. Its form was that of a pyramid and its height was 660 feet. It 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 Phce- nicians ; and Jerusalem, with its splendid temple, of the Israelites —show that architecture among them had made great advances. Ancient monuments of the art are found also among other nations j the subterraneous temples of the Hindoos upon the islands, Ele- phanta and Salsetta ; the ruins of Persepolis in Persia ; pyramids, obelisks, temples, palaces and sepulchres in Egypt— all prove that the architects of those early times were possessed of skill and judgment highly cultivated. The principal characteristics of their works, are gigantic dimensions, immoveable solidity, and, in some instances, harmonious splendour. The extraordinary size of some is illustrated in the pyramids of Egypt. The largest of these stands not far from the city of Cairo : its base, which is square, covers about 11| 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. §5 art, and rapidly advanced towards perfection. Dignity and grace were added to stability and magnificence. In the Doric order, their first style of building, this is fully exemplified. Phidias,' Ictinus and Callicrates, are spoken of as masters in the art at this period: the encouragement and support of Pericles stimulated them to a noble emulation. The beautiful temple of Minerva, erected upon the acropolis of Athens, the Propyleum, the Odeum and others, were lasting monuments of their success. The Ionic and Corinthian orders were added to the Doric, and many mag- nificent edifices arose. These exemplified, in their chaste propor- tions, the elegant refinement of Grecian taste. Improvement in Grecian architecture continued to advance, until perfection seems to have been attained. The specimens which have been partially preserved, exhibit a combination of elegant proportion, dignified simplicity and majestic grandeur. Architecture among the Greeks was at the height of its glory at the period immediately preceding the Peloponnesian war; after which the art declined. An excess of enrichment succeeded its former simple grandeur; yet a strict regularity was maintained amid the profusion of orna- ment. After the death of Alexander, 323 B. C, a love of gaudy splendour increased: the consequent decline of the art was visible, and the Greeks afterwards paid but little attentiwi to the science. 162.— While the Greeks were masters in architecture, which they applied mostly to their temples and other public buildings, the Romans gave their attention to the science in the construction . of the many aqueducts and sewers with which Rome abounded ; building no such splendid edifices as adorned Athens, Corinth and Ephesus, until about 200 years B. C, when their intercourse with the Greeks became more extended. Grecian architecture was introduced into Rome by Sylla ; by whom, as also by Marius and Cassar, many large edifices were erected in various cities of Italy. But under Caesar Augustus, at about the beginning of the christian era, the art arose to the greatest perfection it ever at^ 9 66 AMERICAN HOUSE-CARPENTER. tained in Italy. Under his patronage, Grecian artists were en- couraged, and many emigrated to Rome. It was at about this time that Solomon's temple at Jerusalem was rebuilt by Herod — a Roman. This was 46 years in the erection, and was most pro- bably of the Grecian style of building — perhaps of the Corin- thian order. Some of the stones of which it was built were 46 feet long, 21 feet high and 14 thick; and others were of the astonishing length of 82 feet. The porch rose to a great height ; the whole being built of white marble exquisitely polished. This is the building concerning which it was remarked — " Master, see 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 Vandals, when they overran the coun- tries of Italy, Greece, Asia and Africa, destroyed most of the works of ancient architecture. Cultivating no art but that of war, these savage hordes could not be expected to take any interest in the beautiful forms and proportions of their habitations. From this time, architecture assumed an entirely different aspect. The celebrated styles of Greece were unappreciated and forgotten; and modern architecture took its first step on the platform of existence. The Goths, in their conquering invasions, gradually extended it over Italy, France, Spain, Portugal and Germany, into England. From the reign of Gallienus may be reckoned the total extinction of the arts among, the Romans. From his time until the (ith 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 «sed 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 Verona and Ravenna. Simplicity and strength are the characteristics of the structures erected by him ; they are, however, devoid of grandeur and ele- gance, or fine proportions. These are properly of the Gothic style ; by some called the old Gothic to distinguish it from the pointed style, which is generally called modern Gothic. 165. — The Lombards, who ruled in Italy from A. D. 568, had no taste for architecture nor respect for antiquities. Accordingly, they pulled down the splendid monuments of classic architecture which they found standing, and erected in their stead huge build- ings of stone 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 7th century in Pavia, the chief city of the Lombards ; at which city, as also at many other places, a great many edifices were erected in accordance with its inelegant forms. 166.— The Byzantine architects, from Byzantium, Constantino- ple, erected many spacious edifices ; among which are included the cathedrals of Bamberg, Worms and Mentz, and the most an cient part of the minster at Strasburg ; in all of these they com- bined the Roman-Ionic order with the Gothic of the Lombards. ARCHITECTURE. 69 This style is called the Lombard-Byzantine. To the last style there were afterwards added cupolas similar to those used in the east , together with numerous slender pillars with tasteless capi- talSy and the many minarets which are the characteristics of the proper Byza7itine, or Oriental style. 167. — In the eighth century, when the Arabs and Moors de- stroyed the kingdom of the Goths, the arts and sciences were mostly in possession of the Musselmen-conquerors ; at which time there were three kinds of architecture practised ; viz : the Arabian, the Moorish and the modern-Gothic. The Arabian style was formed from Greek models, having circular arches added, and towers which terminated with globes and 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 Alhambra, or red- house. The Modern-Gothic was originated by the Visigoths in Spain by a combination of the Arabian and Moorish styles ; and introduced by Charlemagne into Germany. On account of the changes and improvements it there underwent, it was, at about the 13th or 14th century, termed the German, or romantic style. It is exhibited in great perfection in the towers of the minster of Strasburgh, the cathedral of Cologne and other edifices. The most remarkable features of this lofty and aspiring style, are the lancet or pointed arch, clustered pillars, lofty towers and flying butti-esses. 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 tJO AMERICAN HOUSE-CARPENTER. century. The Early-English is celebrated for the beauty of its edifices, the chaste simplicity and purity of design which they display, and the peculiarly graceful character of its foliage. This style is of the 13th century. The Decorated style, as its name implies, is characterized by a great profusion of enrichment, which consists principally of the crocket, or feathered-ornament, and ball-flower. It was mostly in use in the 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 Got hie, were distinguished bypeculiar characteristics as well as by different names. The first symptoms of a desire to return to a pure style in architecture, after the ruin caused by the Goths, was manifested in the character of the art as displayed in the church of St. Sophia at Constantinople, which was erected by Justinian in the 6th century. The church of St. Mark at Venice, which arose in the 10th or 11th century, was the work of Grecian archi- tects, and resembles in magnificence the forms of ancient archi- tecture. The cathedral at Pisa, a wonderful structure for the age, was erected by a Grecian architect in 1016. The marble with which the walls of this building were faced, and of which the four rows of columns that support the roof are composed, is said to be of an excellent character. The Campanile, or leaning-tower as it is usually called, was erected near the cathedral in the 12th cen- tury. Its inclination is generally supposed to have arisen from a poor foundation ; although by some it is said to have been thus constructed originally, in order to inspire in the minds of the beholder sensations of sublimity and awe. In the 13th century, the science in Italy was slowly progressing ; many fine churches were erected, the style of which displayed a decided advance in the progress towards pure classical architecture. In other parts of Europe, the Gothic, or pointed style, was prevalent. The cathedral at Strasburg, designed by Irwin Steinbeck, was erected ARCHITECTURE. 7i in the 13th and 14th centuries. In France and England during the 14th century, many very superior edifices were erected in this style. 170. — In the 14th and 15th centuries, and particularly in the latter, architecture in Italy was greatly revived. The masters began to study the remains of ancient Roman edifices ; and many splen- did buildings were erected, which displayed a purer taste in the science. Among others, St. Peter's of Rome, which was built about this time, is a lasting monument of the architectural skill of the age. Giocondo, Michael Angelo, Palladio, 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-t wisted enrichments, and the convex frieze, were unknown to pure Gre- cian architecture. Yet their efibrts were serviceable in correcting, to a good degree, the very impure taste that had prevailed since the overthrow of the Roman empire. 171. — ^At about this time, the Italian masters and numerous artists who had visited Italy for the purpose, spread the Roman style over various countries of Europe ; which was gradually re- ceived into favor in place of the modern-Gothic. This fell into disuse ; although it has of late years been again cultivated. It requires a building of great magnitude and complexity for a per- fect display of its beauties. In America at the present time, the pure Grecian style is more or less studied ; and perhaps the sim- plicity of its principles is better adapted to a republican country, than the intricacy and extent of those of the Gothic. STYLES OF ARCHITECTURE. 172. — It is generally acknowledged that the various styles in architecture, were originated in accordance with the diiferent puir 72 AMERICAN HOUSE-CARPENTER. suits of the early inhabitants of the earth ; and were brought by their descendants to their present state of perfection, through the propensity for imitation and desire of emulation which are found more or less among all nations. Those that followed agricultural pursuits, from being employed constantly upon the same piece of land, needed a permanent residence, and the wooden hut was the offspring of their wants ; while the shepherd, who followed his flocks and was compelled to traverse large tracts of country for pasture, found the tent to be the most portable habitation ; again, the man devoted to hunting and fishing — an idle and vagabond way of living — is naturally supposed to have been content with the cavern as a place of shelter. The latter is said to have been the origin of the Egyptian style ; while the curved roof of Chi- nese structures gives a strong indication of their having had the tent for their model ; and the simplicity of the original style of the Greeks, (the Doric,) shows quite conclusively, as is generally conceded, that its original was of wood. The modern-Gothic, or pointed style, which was most generally confined to ecclesiastical structures, is said by some to' have originated in an attempt to imitate the bower, or grove of trees, in which the ancients per- formed their idol- worship. 173. — There are numerous styles, or orders, in architecture ; and a knowledge of the peculiarities of each, is important to the student in the art. The Stylobate is the substructure, or base- ment, upon which the columns of an order are arranged. In Roman architecture — especially in the interior of an edifice — it frequently occurs that each column has a separate substructure ; this is called a pedestal. If possible, the pedestal should be avoided in all cases ; because it gives to the column the appear- ance of having been originally designed for a small building, and afterwards pieced-out to make it long enough for a larger one. 174. — An Order, in architecture, is composed of tw i princi- pal parts, viz : the column and the entablature. ARCHITECTURE. 73 175. The Column is composed of the base, shaft and capital. 176. The Entablature, above and supported by the columns, is horizontal ; and is composed of the architrave, frieze and cornice. These principal parts are again divided into various members and mouldings. (See Sect. 111.) 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 shafto, 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, abeam, is that part of the entablature which lies in imme- diate connection with the column. 181. — The Frieze, from ^iro?*, a fringe or border, is that part of the entablature which is immediately above the architrave and beneath the cornice. It was called by some of the ancients, zophorus, because it was usually enriched with sculptured animals. 182. — The Cornice, from corona, to crown, is the upper and projecting part of the entablature — being also the uppermost and crowning part of the whole order. 183. — The Pediment, above the entablature, is the triangu- lar portion which is formed by the inclined edges of the roof at the end of the building. In Gothic architecture, the pediment is called, a gable. 184. — The Tympanum is the perpendicular triangular surface which is enclosed by the cornice of the pediment. 185. — The Attic is a small order, consisting of pilasters and entablature, raised above a larger order, instead of a pedi- ment. An attic story is the upper story, its windows being usually square. 10 74 AMERICAN HOUSE-CARPENTER. 186. — An order, in architecture, has its several parts and mem- bers proportioned to one another by a scale of 60 equal parts, which are called minutes. If the height of buildings were al- ways the same, the scale of equal parts would be a fixed quan- tity — an exact number of feet and inches. But as buildings are erected of different heights, the column and its accompaniments are required to be of different dimensions. To ascertain the scale of equal parts, it is necessary to know the height to which the whole order is to be erected. This must be divided by the num- ber of diameters which is directed for the order under considera- tion. Tlien the quotient obtained by such division, is the length of the scale of equal parts — and is, also, the diameter of the column next above the base. For instance, in the Grecian Doric order the whole height, including column and entablature, is 8 diameters. Suppose now it were desirable to construct an exam- ple of this order, forty feet high. Then 40 feet divided by 8, gives 5 feet for the length of the scale ; and this being divided by 60, the scale is completed. The upright columns of figures, marked ^and P, by the side of the drawings illustrating the orders, designate the height and the projection of the members. The projection of each member is reckoned from a line passing through the axis of the column, and extending above it to the top of the entablature. The figures represent minutes, or 60ths, of the major diameter of the shaft of the column. 187. — Grecian Styles. The original method of building among the Greeks, was in what is called the Doric order : to this were afterwards added the loiiic 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 difier in ARCHITECTURE. 76 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, themutules; and perhaps the projection of the roof in front, to screen the entrance from the weather, gave origin to the portico. The peculiarities of the Doric order are the triglyphs — those parts of the frieze which have perpendicular channels cut in their surface ; the absence of a base to the column — as also of fillets between the flutings of the column, and the plainness of the capital. The triglyphs are to be so disposed that the width of the metopes — the spaces between the triglyphs — shall be equal to their height. 189. — The intercohimniation, or space between the columns, is regulated by placing the centres of the columns under the cen- tres of the triglyphs — except at the angle of the building ; where, as may be seen in Fig. 110, one edge of the triglyph must be over the centre of the column. Where the columns are so dis- posed that one of them stands beneath every other triglyph, the arrangement is called, mono-triglyph^ and is most common. ARCHITECTURE. 77 When a column is placed beneath every third triglyph, the ar- rangement is called diastyle ; and when beneath every fourth, araostyle. This last style is the worst, and is seldom practised. 190. — The Doric order is suitable for buildings that are des- tined for national purposes, for banking-houses, &c. Its appear- ance, though massive and grand, is nevertheless rich and grace- ful. The Custom-House and the Union Bank, in New- York city, are good specimens of this order. 191. — The Ionic Order. The Doric was for some time the only order in use among the Greeks. They gave their attention to the cultivation of it, until perfection seems to have been at- tained. Their temples were the principal objects upon which their skill in the art was displayed ; and as the Doric order seems to have been well fitted, by its massive proportions, to represent the character of their male deities rather than the female, there seems to have been a necessity for another style which should be emblematical of feminine graces, and with which they might decorate such temples as were dedicated to the goddesses. Hence the origin of the Ionic order. This was invented, according to historians, by Hermogenes of Alabanda ; and he being a native of Caria, then in the possession of the lonians, the order was called, the Ionic. 192. — The distinguishing features of this order are the volutes, or spirals of the capital ; and the dentiU among the bed-mould- ings of the cornice : although in some instances, dentils are want- ing. The volutes are said to have been designed as a represen- tation of curls of hair on the head of a matron, of whom the whole column is taken as a semblance. 193. — The intercolumniation of this and the other orders — both Roman and Grecian, with the exception of the Doric — are distinguished as follows. When the interval is one and a half diameters, it is called, pycnostyle, or columns thick-set ; when two diameters, sy style ; when two and a quarter diameters, eustyle ; when three diameters, diastyle ; and when more than ARCHITECTUflE. 79 three diameters, arccostyle, 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. Pig. irs. 80 AMERICAN HOUSK-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 s 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, r 1 1 2, with sides equal to half the diameter of the eye, viz., 2j 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, (fcc. 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 min. This is for a spiral of three revolutions ; but one of any number of revolutions, as 4 or 6, ARCHITECTURE, 81 may be drawn, by dividing o/, {Fig. 113,) into 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 o 1, and h 4, parallel to o 2 ; then the lines, ol,o2,k3,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 Ionic, though the proportions are lighter. The Corinthian displays a more airy elegance, a richer appearance ; but its distinguishing feature is its beautiful capital. This is generally supposed to have had its origin in the capitals of the columns of Egyptian temples ; which, though not approaching it in elegance, have yet a similari- ty of form with the Corinthian. The oft-repeated story of its origin which is told by Yitruvius— an architect who flourished in Rome, in the days of Augustus Caesar— though pretty generally considered to be fabulous, is nevertheless worthy of being again recited. It is this : a young lady of Corinth was sick, and finally died. Her nurse gathered into a deep basket, such trinkets and keepsakes as the lady had been fond of when alive, and placed them upon her grave ; covering the basket with a flat stone or tile, that its contents might not be disturbed. The basket was placed accidentally upon the stem of an acanthus plant, which, shooting forth, enclosed the basket with its foliage ; some of which, reaching the tile, turned gracefully over in the form of a volute! A celebrated sculptor, Calima chus, saw the basket thus decorated, and from the hint which it sug- gested, conceived and constructed a capital for a column. This was called Corinthian from the fact that it was invented and first made use of at Corinth. 197.— The Corinthian being the gayest, the richest and mdst lovely of all the orders, it is appropriate for edifices which are 11 Fig. 114. CORINTHIAN. Fig. 115 ARCHITECTURE. 83 dedicated to amusement, banqueting and festivity — for all places where delicacy, gayety and splendour are desirable. 198. — In addition to the three regular orders of architecture, it was sometimes customary among the Greeks— and afterwards among other nations— to employ representations of the human form, instead of columns, to support entablatures ; these were called Persians and Caryatides. 199. — Persians are statues of men, and are so called in com - memoration of a victory gained over the Persians by Pausanias. The Persian prisoners were brought to Athens and condemned to abject slavery ; and in order to represent them in the lowest state of servitude and degradation, the statues were loaded with the heaviest entablature, the Doric. 200. — Caryatides are statues of women dressed in long robes after the Asiatic manner. Their origin is as follows. In a war between the Greeks and the Caryans, the latter Avere 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 lelief 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. and merit, which were erected by architects from Etruria ; but Rome was principally indebted to Greece for what she acquired of the art. Although there is no such thing as an architecture of Roman invention, yet no nation, perhaps, ever was so devoted to the cultivation of the art as the Roman. Whether we consider the number and extent of their structures, or the lavish richness and splendour with which they were adorned, we are compelled to yield to them our admiration and praise. At one time, under the consuls and emperors, Rome employed 400 architects. The public works — such as theatres, circuses, baths, aqueducts, , draw v e, parallel to 6 c ; from a, e, u, p, k, t, and from as many other points as is thought ne- cessary, draw ordinates to the base-line, v e, parallel to r s ; through A, draw w x, at right angles to m I ; upon n, with r s for radius, describe an intersecting arc at x, and join 7i and x ; from the points at which the ordinates from the plan meet the base- line, V e, draw ordinates to meet the line, m I, at right angles to v 6 ; and from the points of intersection on m I, draw correspond- ing ordinates, parallel to n x ; make the ordinates which are pa- rallel to w of a length corresponding to those which are parallel to r s, and through the points thus found, trace the face-mould as required. Case 2. — When the middle height is below a line joining the other two. The lower twist in Fig. 283 is of this nature. The face-mould for this is found at Fig. 285 in a manner similar . to that at Fig. 284. The heights are all taken from the top of the falling-mould at Fig. 283 ; b j being equal to w 6 in Fig. 283, i h equal to x^ in that figure, and clXo V o^ Draw a line through J and I, and from A, draw h n, parallel to b c ; from n, draw n r, at right angles to b c, and join r and s ; then r s will be the bevil for the lower ordinates. From h, draw h x, at right an- gles toj I ; upon 71, with r 5 for radius, describe an intersecting arc at and join n and x ; then n x will be the bevil for the upper ordinates, upon which the face-mould is found as in Case 1. 388. — Elucidation of the foregoing method. — This method of finding the face-moulds for the handrailing of winding stairs, being founded on principles which govern cylindric sections, may be illustrated by the following figures. Fig. 286 and 287 repre- sent solid blocks, or prisms, standing upright on a level base, b d ; the upper surface, J a forraing oblique angles with the face, b I — 236 AMERICAN HOUSE-CARPENTER. Fig. 285. in Fig. 286 obtuse, and in Fig. 287 acute. Upon the base, de- scribe the semi-circle, b s c ; from the centre, i, draw i s, at right angles to 6 c ; from s, draw s s, at right angles to e d, and from i, draw i h, at right angles to 6 c ; make i h equal to 5 and join h and x ; then, h and x being of the same height, the line, h x, joining them, is a level line. From h, draw h n, parallel to b c, and from7t, draw n r, at right angles to 6 c; join r and s, also n STAIRS. 237 c r Fig. 286. 237. and a;; then, n and x being of the same height, n x\s a level hne ; and this line lying perpendicularly over r 5, n x and r s must be of the same length. So, all lines on the top, drawn parallel to n X, and perpendicularly over corresponding lines drawn parallel to r s on the base, must be equal to those lines on the base ; and by drawing a number of these on the semi-circle at the base and others of the same length at the top, it is evident that a curve, j X /, may be traced through the ends of those on the top, which shall lie perpendicularly over the semi-circle at the base. It is upon this principle that the process at Fi^. 284 and 28.5 is founded. The plan of the rail at the bottom of those figures is supposed to lie perpendicularly under the face-mould at the top ; and each ordinate at the top over a corresponding one at the base. The ordinates, n x and r s, in those figures, correspond to n x and r s in Fig. 286 and 287. In Fig. 288, the top, e a, forms a right angle with the face, d c ; all that is necessary, therefore, in this figure, is to find a line corresponding to A :r in the last two figures, and that will lie level and in the upper surface ; so that all ordinates at right angles to d r on the base, will correspond to those that are at right angles 238 AMERICAN HOUSE-CARPENTER, Fig. 288. r to e c on the top. This elucidates Fig. 276 ; at which the lines, h 9 and i 8, correspond to h 9 and i 8 in this figure. Fig. 289. 389.— To iindthe hevil for the edge of the plank. The plank, before the face-mould is applied, must be bevilled accord- ing to the angle which the top of the imaginary block, or prism, in the previous figures, makes with the face. This angle is de- termined in the following manner : draw w i, ( Fig. 289,) at right angles to i s, and equal to what Fig. 284 ; make i s equal to t * in that figure, and join w and s ; then sw p will be the bevil required in order to apply the face-mould at Fig. 284. In Fig. 285, the middle height being below the line joining the other two, the bevil IS therefore acute. To determine this, draw i s, {Fig. 290,) at STAIRS. 239 « to' p Fig. 290. right angles to i p, and equal to i 5 in Fig. 285 ; make s w equal Jto h w in Fig. 285, and join w and i ; then w i p will be the bevil required in order to apply the face-mould at Fig. 285. Al- though the falling-mould in these cases is curved, yet, as the plank is sprung, or bevilled on its edge, the thickness necessary to get out the twist may be ascertained according to Art. 381 — taking the vertical distance across the falling-mould at the joints, and placing it down from the two outside heights in Fig. 284 or 285. After bevilling the plank, the moulds are applied as at Art. 383 — applying the pitch-board on the bevilled instead of a square edge, and placing the tips of the mould so that they will bear the same relation to the edge of the plank, as they do to the line,^* /, in Fig. 284 or 285. 390. — To apply the moulds without bevilling the plank. Make w p, {Fig. 291,) equal to w p aX Fig. 289, and the angle, bed, equal to 6 j I in Fig. 284 ; make p a equal to the thick- ness of the plank, as a in Fig. 289, and from a draw a a, pa- rallel iowd; from c, draw c e, at right angles to w d, and join e w 0 4 Fig. 291. 240 AMERICAN HOUSE-CARPENTER. and h ; then the angle, 6 e o, on a square edge of the plank, hav- ing a line on the upper face at the distance, p a, in Fig. 289, at which to apply the tips of the mould — will answer the same pur- pose as bevilling the edge. If the bevilled edge of the plank, which reaches from p to is supposed to be in the plane of the paper, and the point, a, to be above the plane of the paper as much as a, in Fig. 289, is dis- tant from the line, wp ; and the plank to be revolved on /> 6 as an axis until the line, p w, falls below the plane of the paper, and the line, p a, arrives in it ; then, it is evident that the point, c, will fall, in the line, c e, until it lies directly behind the point, e, and the line, b c, will lie directly behind b e. k 391.— To find the bevils for splayed work. The principle employed in the last figure is one that will serve to find the bevils for splayed work— such as hoppers, bread-trays, (fcc.~and a way of applying it to that purpose had better, perhaps, be introduced in this connection. In Fig. 292, a 6 c is the angle at which the work is splayed, and b d, on the upper edge of the board, is al right angles to a b; make the angle, /^j, equal to a be, and from/, drawfh, parallel to e a; from b, draw b o, at right an- gles toab; through o, draw i e, parallel to c b, and join e and d ; then the angle, aed, will be the proper bevil for the ends from the inside, oi k d e from the outside. If a mitre-joint is re- STAIRS. 241 quired, set f the thickness of the stuff on the level, from e to m, and join m wciAd; then k d m will be the proper bevil for a mitre-joint. If the upper edges of the splayed work is to be bevilled, so as to be horizontal when the work is placed in its proper position, fsji being the same as a 6 c, will be the proper bevil for that purpose. Suppose, therefore, that a piece indicated by the lines, ^ S^S f Q-^^fh, were taken off; then a line drawn upon the bevilled surface from d, at right angles to k d, would show the true position of the joint, because it would be in the direction of the board for the other side ; but a line so drawn would pass through the point, o, — thus proving the principle correct. So, if a line were drawn upon the bevilled surface from d, at an angle of 45 degrees to k d, it would pass through the point, n. 392. — Another method for face-moulds. It will be seen by reference to Art. 388, that the principal object had in view in the preparatory process of finding a face-mould, is to ascertain upon it the direction of a horizontal line. This can be found by a method different from any previously proposed ; and as it requires fewer lines, and admits of less complication, it is probably to be preferred. It can be best introduced, perhaps, by the following explanation . In Fig-. 293, j d represents a prism standing upon a level base, b d, its upper surface forming an acute angle with the face, b I, as at Fig. 287. Extend the base line, b c, and the raking line, J /, to meet at /; also, extend e d and g a, to meet at k; from /, through k, draw fm. If we suppose the prism to stand upon a level floor, ofm, and the plane, 7^ a I, to be extended to meet that floor, then it will be obvious that the intersection between that plane and the plane of the floor would be in the line, fk; and the \'me,fk, being in the plane of the floor, and also in the inclined plane, 7^ k f, any line made in the plane, kf parallel to / k, must be a level line. By finding the position of a perpendicular plane, at right angles to the raking plane, g, we shall greatly shorten the process for obtaining ordinates. 31 m AMERICAN HOUSE-CARPENTER. This may be done thus : from /, draw / o, at right angles to/m; extend e b to o, andgj, to t ; from o, draw o t, at right angles to of, and join ^ and/; then t of will be a perpendicular plane, at right angles to the inclined plane, t g kf; becausfe the base of the former, o/, is at right angles to the base of the latter,/ A:, both these lines being in the same plane. From 6, draw h at right angles to of or parallel to/m ; fromp, draw p q, at right angles to of and from q, draw a line on the upper plane, parallel to / m, or at right angles to if/; then this line will obviously be drawn to the point, J, and the line, qj, be equal to jo b. Proceed, in the same way, from the points, ^ and c, to find a: and I. Now, to apply the principle here explained, let the curve, b s c, {Fig. 294,) be the base of a cylindric segment, and let it be re- quired to find the shape of a section of this segment, cut by a plane passing through three given points in its curved surface : one perpendicularly over 6, at the height, bj; one perpendicu- larly over 5, at the height, 5 x ; and the other over c, at the height, c I — these lines being drawn at right angles to the chord of the base, b c. Fromj, through I, draw a line to meet the chord line extended to /; from s, draw 5 parallel to 6 /, and from x, draw s k, parallel to J/; from /, through k, draw / tn ; then / m will be the intersecting line of the plane of the section with the STAIRS. m Fig. 29-t. plane of the base. This line can be proved to be the intersection of these planes in another way ; from b, through s, and from j, through ,T, draw lines meeting at m ; then the point, m, will be in the intersecting line, as is shown in the figure, and also at Fig. 293. From/, draw/p, at right angles to / m; from b and c, and from as many other points as is thought necessary, draw ordinates, parallel to fm; make p q equal to bj, and join q and/; from the points at which the ordinates meet the line, qf, draw others at right angles to qf; make each ordinate at A equal to its cor- responding ordinate at C, and trace the curve, gni, through the points thus found. Now it may be observed that A is the plane of the section, B the plane of the segment, corresponding to the plane, q p/, of Fig. 293, and C is the plane of the base. To give these planes their proper position, let A be turned on ^/ as an axis until it AMERICAN HOUSE-CARPENTER. Stands perpendicularly over the line, q f, and at right angles to the plane, B ; then, while A and B are fixed at right angles, let B be turned on the line, p f, as an axis until it stands perpendicu- larly over p /, and at right angles to the plane, C ; then the plane, A, will lie over the plane, C, with the several lines on one corres- ponding to those on the other ; the point, i, resting at I, the point, n, at X, and g at j ; and the curve, g n i, lying perpendicularly over b s c — as was required. If we suppose the cylinder to be cut by a level plane passing through the point, I, (as is done in finding a face-mould,) it will be obvious that lines corresponding to q /and pf would meet in I ; and the plane of the section, A, the plane of the segment, B, and the plane of the base, C, would all meet in that point. 393. — To find the face-mould for a hand-rail according to the principles explained in the previous article. In Fig. 295, a ecf is the plan of a hand-rail over a quarter of a dylinder ; and in Fig. 296, ab cdis the falling-mould ; / e being equal to the stretch-out of a df in Fig. 295. From c, draw c h, parallel to ef; bisect h c in i, and find a point, as b, in the arc, df, {Fig. 295,) corresponding to i in the line, h c; from i, {Fig. 296,) to the top of the falling-mould, draw i j, at right angles to he; at Fig. 295, from c, through b, draw c g, and from b and c, draw bj and c k, at right angles to ^ c ; make c k equal to h g at Fig. 296, and b j equal to ij at that figure ; from k, through j, draw k g, and from^, through a, draw gp ; then will be the intersecting line, corresponding to/m in Fig. 293 and 294 ; through e, draw p 6, at right angles to gp, and from c, draw c q. parallel to gp ; make r q equal to h g at Fig. 296 ; join;? and q, and proceed as in the previous examples to find the face-mould, A. The joint of the face-mould, u v, will be more accurately determined by finding the projection of the centre of the plan, o, as at w ; joining s and w, and drawing zi v, parallel to s w. It may be noticed that c k and 6 j are not of a length corres- ponding to the above directions : they are but J the length given. BTAIRS. 246 Fig. 295. 246 AMERICAN HOUSB-CARPENTER. Fig. 296. The object of drawing these lines is to find the point, g, and that can be done by taking any proportional parts of the lines given, as well as by taking the whole lines. For instance, supposing c k and 6 j to be the full length of the given lines, bisect one in i and the other in m ; then a line drawn from m, through i, will give the point, g, as was required. The point, g, may also be STAIRS. 247 obtained thus : at Fig: 296, make h I equal to c 6 in Fig. 295 ; from I, draw I k, at right angles to A c ; from J, draw^' k, parallel to h c ; from g, through draw ^ 7i; at Fig. 295, make b g equal to Z w in Fig. 296 ; then g will be the point required. The reason why the points, a, 6 and c, in the plan of the rail at Fig. 295, are taken for resting points instead of e, i and /, is this : the top of the rail being level, it is evident that the points, a and e, in the section a e, are of the same height ; also that the point, i, is of the same height as 6, and c as/. Now, if a is taken for a point in the inclined plane rising from the line g p, e must be below that plane ; if 6 is taken for a point in that plane, i must be below it ; and if c is in the plane,/ must be below it. The rule, then, for taking these points, is to take in each section the one that is nearest to the line, g p. Sometimes the line of intersection, g p, happens to come almost in the direction of the line, er : in such case, after finding the line, see if the points from which the heights were taken agree with the above rule ; if the heights were taken at the wrong points, take them according to the rule above, and then find the true line of intersection, which will not vary much from the one already found. ""^ ^ ^^y;;^ ^ ft % p jh \ Fig. 297. 394. — To apply the face-mould thus found to the plank. The face-mould, when obtained by this method, is to be applied to a square-edged plank, as directed at Art. 383, with this differ- ence : instead of applying both tips of the mould to the edge of 248 AMERICAN HOUSE-CARPKNTER. the plank, one of them is to be set as far from the edge of the plankj as in Fig. 295, is from the chord of the section p q — as is shown at Fig. 297. A, in this figure, is the mould applied on the upper side of the plank, B, the edge of the plank, and C, the mould applied on the under side ; a h and c d being made equal ioq x'm Fig. 295, and the angle, e a c, on the edge, equal to the angle, f qr, at Fig. 295. In order to avoid a waste of stuff, it would be advisable to apply the tips of the mould, e and h, im- mediately at the edge of the plank. To do this, suppose the moulds to be applied as shown in the figure ; then let A be re- volved upon e until the point, 6, arrives at g, causing the line, e b, to coincide with e g : the mould upon the under side of the plank must now be revolved upon a point that is perpendicularly beneath e, as /; from/, draw / h, parallel to i d, and from d, draw d h, at right angles to i d ; then revolve the mould, C, upon /, until the point, A, arrives at causing the line,/ A, to coincide with/j, and the line, i d, to coincide with k I; then the tips of the mould will be at k and I. The rule for doing this, then, will be as follows : make the an- gle, ifk, equal to the angle q v x, at Fig. 295 ; make /A: equal to/i, and through A:, draw k /, parallel to ij ; then apply the corner of the mould, t, at ^, and the other corner d, at the line, k I. The thickness of stuff is found as at Art. 381. 395. — To regulate the application of the falling-mould. Obtain, on the line, h c, {Fig. 296,) the several points, r, q,p, I and m, corresponding to the points, 6', a^, z, y, &c., at Fig. 295 ; from r qp^ (fcc, draw the lines, r t,qu,pv, &c., at right angles Xohc; make A 5, r t, q u,