i 
 
THE 
 
 MODERN GEOMETRICAL 
 
 STAIE-BUIIDEE'S GUIDE 
 
 BEING A 
 
 PLAIN PRACTICAL SYSTEM OF HAND-RAILING, 
 
 EMBRACING ALL ITS NECESSARY DETAILS, 
 
 AND 
 
 ^fomctncallg lUtiBtrateir ^Tujcntg-tao Bttd (^n^xavmp. 
 
 TOGETHER WITH THE USE 
 
 OF THE MOST IMPORTANT PRINCIPLES OF PRACTICAL GEOMETRY. 
 
 BY 
 
 SIMOlS^Cpi /-aB-AFF;, ARCHITECT, 
 
 NEW YORK: 
 
 PUBLISHED FOR THE AUTHOR. 
 1845. 
 
PREEACE. 
 
 To a work intended for those wishing to obtain a thorough knowledge of 
 the Science of Geometrical Hand-rails and Stairs much preface is unnecessary* 
 Although few subjects have of late years more employed the minds of men of 
 science and builders than that of the art and science of hand-railing, the art 
 of forming hand-rails around circular, oval, and excentric well-holes without 
 the use of a cylinder is of fecent date. The first successful method of squar- 
 ing a wreath rail upon geometrical principles was invented by Mr. Peter 
 Nicholson. Subsequent writers have contributed nothing towards the advance- 
 ment of this most useful branch of the Builder's profession ;— and have con- 
 tented themselves with methods laid down by Mr. Nicholson, which were not 
 advantageous, by not being carried out fully, and in many instances were 
 uncertain in their application owing to the workman not being able to com- 
 prehend them fully, and which consequently led to erroneous results in the 
 practice. The methods laid down by the author in this work are more simple; 
 and comprehensive than those used by Mr. Nicholson, whose diagrams are 
 so loaded with operative lines, that it has a tendency to confuse the mind of 
 the learner and give him a disgust in pursuing the study of the science. No 
 previous author seems to have had any idea of describing more than one sec- 
 tion in a quadrant of a circle. The methods herein laid down by the author 
 are entirely new and of his own invention ; being founded upon geometrical 
 principles, and carried out fully they admit of two or more sections in the 
 quadrant, as the case requires, by which principle the workman will perceive 
 that he may execute the work more expeditiously and require nearly one-half 
 less in the quantity of stuff for the rail than by the methods laid down by pre- 
 vious authors. 
 
 Since the first invention of these methods by the author, they have been 
 thoroughly tested both by theory and practice ; and,' therefore, this most essen- 
 
 2/^3 
 
tial branch as here presented is correct. The attention of the student is parti- 
 cularly directed to the arrangement of the plates, and by closely examining 
 them with the help of their explanations, he will soon discern the utility of 
 this work, and make himself master of the art and science, and also acquire a 
 thorough knowledge of the principles and use of practical geometry, so far as is 
 applicable to geometrical rails and stairs. 
 
THE 
 
 STAIR-BUILDER'S GUIDE. 
 
 INTUODUCTION. 
 
 -HAND-RAILING]. 
 
 The art and science of Hand-railing is to find the position of the 
 section of the given plan, and to form the mould according to that 
 position. 
 
 GEOMETRICAL STAIRS. 
 
 As every mansion consisting of more than one story is indebted 
 to this portion of architecture for ornament and utility, the height 
 of the Riser, and breadth and length of the Tread, should be govern- 
 ed by its location. The height of the Eiser should not exceed eight 
 inches, nor less than six and a half inches. The width of the Tread 
 should not exceed fifteen niches, nor less than ten inches. The 
 width of the Winder upon the concave end should be equal to one 
 half the width of the Flyer, which gives a graceful form to the 
 Stairs. 
 
 The Student will observe, that by applying to plate No. 22, 
 he may there obtain the use of the most important principles of 
 Geometry practically illustrated, as far as is apphcable to the art 
 and science of Hand-railing. 
 
 In conclusion— This subject having been presented to the pubUc 
 nearly three-fourths of a century, and different methods having 
 been adopted for the purpose of simplifying the matter, by subse- 
 quent writers, who were not successful by not understanding its 
 principles fully, both theoretically and practically, so as to make it 
 comprehensive and advantageous to the learner, the author has 
 laid down in this work the principles by which the moulds and 
 
6 
 
 easings are executed for the formation of the rail in practical size 
 (see Plates 6, 7, 8), and manner of producing the butt-joint at any 
 given point in the circle, also the manner of producing the formation 
 of the soUd section and spring moulds for each respective section, 
 by which method the rail may be wrought out of the least possible 
 stuff and with much less labor, and give it a more beautiful appear- 
 ance by the grain of the wood keeping the circle of the rail— subse- 
 quent writers having been governed by the termination of the 
 quadrant for the location of the joint, which necessarily required 
 nearly double the quantity of stuff for the rail than the method 
 herein adopted. The author having devoted much time in expound- 
 ing this matter, both theoretically and practically, for the benefit of 
 practical builders, and feeling confident of its correctness, beauty 
 and simplicity, and from the facilities which this work will afford, 
 and the low price at which it is offered, feels assured that it will 
 meet with the approbation and patronage of an enhghtened public. 
 
 With these considerations this work is respectfully submitted by 
 their 
 
 Humble Servant, 
 
 The Authok. 
 
PLATE 1. 
 
 Shows the plan and elevation of a cijUnder stairs, running from the 
 rake to the landing and landing to the raize. 
 
 Fig. 1 being the plan of the cyUnder. d e f the staves, being 
 ghiedand screwed together, which form the cyUnder, A and B being 
 the front strings, are glued and screwed to the cyUnder. C Cthe 
 front joist of the platform, against which the lower and upper flight 
 rest, s t being the stretch-out^ of the concave semi-circumference 
 from u to V. c c the baluster, o o the line of the nosing ; and d e 
 the rail. It will be observed that the rail D E projects, over the face 
 of the stringpieces A and B. The projection is obtained in the 
 following manner: 
 
 buppose the width of the rail to be - - - 3 inches, 
 
 from which take the size of the baluster, at the base, 
 
 which being I'i " 
 
 leaves 1\ inches, which being divided by two - 2)1} " 
 
 gives y of an inch for the projection of the rail, - % " 
 
 It is always necessary to ascertain the projection of a circular 
 part of the rail before commencing to draw the plan for executing 
 the moulds. If brackets are used, ascertain the projection from the 
 line or face of the bracket. 
 
 Fig. 2 shows the elevation of the given place, fig. 1, which 
 should be drawn upon pasteboard to obtain the length of the staves, 
 and formation of the casings ; then applying it to the concave side 
 of the cyhnder (which is then merely screwed together), trace out 
 the line of the easing. The workman will find great convenience 
 in working the bead facie or moulding upon the lower edge of the 
 cylinders before gluing the staves together, for then he may work 
 from both ways, whereas, if it was glued together before working 
 it would be more diflicult, particularly in those small openings. To 
 
 * Which is produced in the same manner as in Plate 2. 
 
8 
 
 draw the elevation and stretch-out of fig. 2 from the stretch-out s t 
 of the semi-circular part, fig. 1, erect perpendiculars to s ? or under 
 edge of the nosing of the platform ; then s t will be the length of 
 the concave semi-circumference of the cylinder. It is necessary to 
 know the width of the front joist of the platform, against which the 
 cyUnder rests, before the length of the staves can be obtained. 
 Apply the width of the joist to the stretch-out, or centre of the 
 cyhnder of the plan, fig. 2, allowing it to be | of an inch longer to 
 receive the lath and plastering ; then from the under edge of the 
 nosing at s set down the width of a riser ; and at t set up the same. 
 The depth of the stringpieces in the centre of the tread at A and B 
 being obtained from the depth of the centre of the cyUnder, then 
 draw the hypothenuse of the lower edge of the string A, upwards 
 to the centre of the stave d; also of B downwards to the centre of 
 the stave f. Then connect the points d and which allow the 
 centre of the cylinder to retain its given depth, and by intersecting 
 lines each side of the angles, the easing will be formed, and the 
 length of the staves d e and / may be obtained from the nosing to 
 the points 1, S and 3, which completes the given plan, 
 
PLATE 2. 
 
 Shows the manner of obtaining the points, and drawing the lines for the 
 formation of the face, and falling moulds, for a continued rail 
 over a small opening. 
 
 A B, fig. 1, being the given plan of the semi-circular part, having 
 a portion of straight rail attached. With the diameter ahoi the con 
 cave, and c d of the convex sides of the plan for radii, and from the 
 points ah c and d describe arcs intersecting at e and /; draw the 
 tangent lines m n and j h (parallel to the diameter a h and c d) to 
 the concave and convex sides of the given plan ; from the point f 
 draw lines through a and h intersecting the concave tangent at m 
 and n, also from the point e draw lines through c and d intersecting 
 the convex tangent aty and k ; then/ h and m n will be the stretch- 
 outs or length of the concave and convex semi-circumference from 
 a b to cdof the given plan. 
 
 To obtain the concave falling mould, fig. 2, from the given plan. Jig. 1. 
 
 Draw the horizontal line m b and make it equal in length to the 
 concave stretch-out m n, fig. 1. At the point b place the pitch 
 board brn; let m « be equal to the height of a rise, and divide 
 it into four equal parts. Draw the first and third parts parallel to 
 m b to obtain the points x and e. At the points a and b place the pitch 
 board aj I and brn; then draw the hypothenuses y e,ex and x v, set 
 up the width of the mould which is equal to the required thickness of 
 the rail, and draw said hues for the upper edge parallel to the lower 
 edge of the mould ; then apply the length of the straight parts s a 
 and yb of the given plan, fig. 1, to the base of the upper and lower 
 pitch board from nioh and a to /; erect the perpendiculars h y and 
 /5 to the upper edge of the mould, and from y \o v and s \o q 
 square across the mould ; then the fine v y and s q will be the 
 given joints for the upper and lower ends of the mould. To obtain 
 the joint in the centre of the semi-circular part, let fall the fine G H 
 perpendicular to the given plan, passing through the centre of the 
 
10 
 
 cancave and convex falling moulds ; obtain the centre of the mould 
 upon the said perpendicular from t to t, and through it square 
 across the mould from z io u and the line z u will be the required 
 joint. Draw the Mnes u u and z z parallel to said perpendicular, 
 and u t and z t will be the required overwood. Apply the overwood 
 to each side of said perpendicular upon the given plan, fig. 1. Draw 
 the lines c o through v, and z d parallel to h m, and o z and d s will 
 be the heights for the parts B and A of said mould. To form the 
 casings each side of the angle at x, divide x v and x u into six equai 
 parts each, and from the point of division next the angle x on one 
 side, draw a Hne to that point which is farthest from the angle x on 
 the other side. Do the same from all the other parts of division, 
 and by the intersection of those lines, the casings wiU be formed. 
 The same distances are applied on each side of the angle at 
 The casings being reversed to the one at x, the easing when formed 
 at the upper edge of the mould should coincide with the lower 
 edge, and then the mould will be of equal width and completed. 
 
 The convex falhng mould, fig. 3, is obtained in the same manner 
 as the concave at fig. 2. 
 
 Let j d, the base, fig. 3, be equal toj h, the stretch-out, fig. 1, of the 
 convex side of the given plan, j c is equal to the height of a rise d w, 
 and c r equal to the portions of straight rail as in fig. 2. The dis- 
 tances from each angle and point of division are obtained in the 
 same manner as at fig. 2 for the formation of the easing. Then 
 obtain the necessary overwood from the joint z u, fig. 2, and apply 
 it upon each side of said perpendicular, as is shown at x x and v 
 It will be perceived that the two falling moulds, figs. 2 and 3, have 
 different angles of incUnation, and that the lines of the overwood v 
 V and X x of the convex are shorter than u u and z z of the concave. 
 There will be no difficulty arising from the difference, as will be 
 seen by the cutting of the joint of the rail pieces. 
 
 How to obtain the face moulds, Jigs. 4 and 5, from the given plan, Jig. 1. 
 
 At the part B of the given plan, fig. 1, draw the chord line 5 9 touch- 
 ing the concave points at y z, including the overwood of the qua- 
 drantal part B. Erect perpendiculars from said chord hne at 5, y, 6, 
 7, 8, z, and q ; through the points w, d, h, and x, draw o y parallel to 
 the chord hne ; then obtain the height oz, fig. 2, and apply it upon 
 the perpendicular, from the parallel at o to z. Draw the hypothenuse 
 9 5 touching the point z, also the parallel at y ; at the angles zyoi 
 
11 
 
 tlie hypotlienuse to the perpendicular, is the pitch bevel obtained. 
 Draw lines at right angles to the hypothenuse, from the points 5, 6, 7, 
 8, 9 ; then obtain the several distances 5w, 6 d,! b,S 4 4, and 9 from 
 the chord line, fig. 1, and apply them upon the perpendicular, 5, to the 
 hypothenuse, fig. 4, from the points 5, 6, 7, S, and 9, to w, d, b, 4, 4, and 
 ^. Draw the lines x z, which will be the end of the required mould. 
 The points z 4:b and x 4 d being obtained, describe the arcs through 
 them, and b y and d to will be the length of the straight part, and y 
 to the end of the required mould. The face mould at fig. 5 is 
 ■obtained in the same manner as at fig. 4 ; the straight part being 
 thrown up, and the height from the parallel line at d to the hypothe- 
 nuse at s being obtained at d s, fig. 2. 
 
 The diagram B and A, figs. 4 and 5, gives a full description of the 
 manner in which the face moulds are applied to both sides of the 
 plank. 
 
 From the parts z and ?/, fig. 4, and u and s, fig. 5, at the upper 
 edge of the plank, the pitch bevels being obtained and applied across 
 to the lower edges z y, fig. 4, and u s, fig. 5, which give the angle of 
 inclination to the rail-pieces. Then tracing the line of the moulds 
 on both sides of the plank, and cutting away the superfluous wood 
 to the line of the moulds, the rail-pieces thus formed are ready for 
 the application of the parts B and A of the concave and convex 
 falling moulds. Take the part B, fig 2, of the concave falling mould 
 (which is supposed to be made of pasteboard), and apply it to the 
 concave edge of the rail-piece at fig. 4, having the plumblines z z, 
 of the overwood, fig. 2, applied to the lines of the pitch bevel z z 
 and y y; then bend the mould to the inner edge of said rail-piece, 
 and the plumbline y y of said mould will coincide with the line y y 
 of the pitch bevel. Then mark the joints y v and u z, and on the 
 upper edge at y, and the lower edge at z of said joints, square across 
 the ends of said rail-piece. 
 
 In the same manner apply the part B, fig. 3, of the convex mould, 
 having the plumblines of overwood x x and iv w to coincide with 
 X X and w w of the convex edge of the rail-piece ; and when the 
 lines are thus squared across the end from the concave to the con- 
 vex edge, let the points of said mould at x of the lower, and of 
 the upper edge, rest. Then trace out the Une of the moulds, and 
 cutting away the superfluous wood to them, and the butt joints 
 being cut, the rail-piece thus formed will range over its plan. 
 
 The upper wreath at A, fig. 5, being formed in the same manner 
 
12 
 
 as B, fig. 4, with the part A, fig. 2, of the concave mould being 
 apphed to the concave edge of the rail-piece, having the plumblines 
 u u and 5 5 of said mould to coincide with the pitch bevel u u and s s 
 of said rail-piece. In the same manner the convex part A, fig. 3, 
 being applied to the convex edge, with the plumbhnes v v and r r of 
 said mould, coinciding with v v and r r of said rail-piece ; the 
 butt-joints being obtained and cut, the line of the mould being 
 traced, the superfluous wood cut away, and the rail-pieces dowled 
 and well screwed together, are then ready for moulding. 
 
PLATE 3. 
 
 Shows the manner of obtaining the points, and drawing the lines for the 
 formation of the face moulds for the lower and upper wreaths of 
 Plate 2, when the edge of the plank forms an acute angle, out of 
 which the lower wreath is formed, also when the edge of the plank 
 forms an obtuse angle, out of which the upper wreath is formed, 
 which is usually termed the spring of the phnk, and the manner in 
 which the quadrantal parts, figs. 2 and 3, are thrown from the 
 quadrardal parts, B and A, fig. 1. 
 
 REFERENCES TO FIGURES AND LETTERS. 
 
 Fig. 1,AB the given plan of the cyUnder, a b the diameter of the 
 concave, and c c? of the convex sides of the plan for radii, having 
 e as centre, then e d and e I be the radius of the quadrantal part, B, and 
 zxi\ie line of overwood, b y and d w the length of the straight part, 
 z y the chord-line. To erect the perpendicular s to the chord z y, 
 from the centre e, set the point of the dividers in the centre at e, and 
 describe the arc j h upon the chord z y, then set the point of the 
 dividers at j and h, and describe the arcs at s, and at the point 
 where the arcs bisect at s draw the Une e s, then [e s will be the 
 perpendicular to the chord z y. The points in the quadrantal part 
 A are obtained in the same manner. Then draw the base line 4 
 and 5 of figs. 2 and 3 parallel to the base c d of the semi-circular 
 part A B, fig. 1 ; then erect the perpendiculars e r from the base 5 9 
 and 3 4 of figs. 2 and 3 ; then take the distance upon the perpen- 
 diculars s r, fig. 1, from the chord-fines q and p to the centre e, and 
 apply it upon the perpendiculars from the base at r, figs. 2 and 3, to 
 e, then e will be the centres for figs. 2 and 3. Then take the dis- 
 tance q t and p o upon the chord-fines from the perpendiculars s r, 
 fig. 1, to radius e d and e c of each respective quadrant, and apply 
 said distance upon the base from r, towards 4, fig. 2, also from r 
 towards 6, fig. 3 ; then through said points from r, upon the base of 
 figs. 2 and 3, draw the radius from the centre eiob d and ac ; then 
 
14 
 
 draw the radius e I of figs. 2 and 3, at right angles ioeb d and eac; 
 then take the distance in the dividers, from the centre e, fig. 1, to « 
 or b of the concave, and c or d of the convex, and apply said dis- 
 tance from the centre e, fig. 2, to b and d, and describe the arcs b z 
 of the concave, and d x of the convex ; the line z y will be the 
 required overwood. Let b y and d w be equal Xoby and d w, fig. 
 1, then z X and y w will be the ends of the quadrantal part of the 
 given plan, fig. 2 ; also apply said distance from the centre e, fig. 3, 
 to a and c, and describe the arc au of the concave, and c v of the 
 convex, the line u v will be the required overwood ; let a 5 and c r 
 be equal to a 5 and c r, fig. 1, and u v and 5 r, will be the ends of 
 the quadrantal part of the given plan, fig. 3. The object of throw- 
 ing up the quadrantal parts, figs. 2 and 3, from the parts B and A, 
 fig. 1, is because in drawing the plan for the rail full size of a circu- 
 lar stairs, the workman may find it difficult to obtain sufficient room 
 for executing the moulds for the same ; therefore, by throwing up the 
 quadrantal part or segment in a more convenient place, the mould 
 may be executed with exactness. 
 
 Fig. Q,fhg forms a right angled triangle — the distance / ^ is equal 
 to 2/ of figs. 2 and 3, from the chord-lines z y and u 5, to the paral- 
 lel lines m n and n m, and is obtained from the same, whether 
 quadrantal or segment. The distance ^ A is to give the angle of 
 inchnation to h f ; the distance g h may be greater or less as the 
 case may require ; if thin stuff the greater the distance, if thick 
 stuff the less distance is required — in figs. 7 and 8 they are equal. 
 
 Fig. 7 shows the manner of forming an acute angle from fhg, 
 fig. 6, or bevel for the edge of the plank, from which the lower 
 wreath is formed. 
 
 Fig. 8 shows the manner of forming an obtuse angle from fhg, 
 fig. 6, or bevel for the edge of the plank, from which the upper 
 wreath is formed. 
 
 Fig. 4 shows the manner of forming the face mould for the 
 spring of the plank, when the upper side and edge of the plank 
 form an acute angle as at fig. 7, for the lower wreath, from the 
 quadrantal part, fig. 2 ; first erect the perpendiculars z and y, from 
 the chord-line, fig. 2 ; then draw o y parallel to the chord z y, and 
 take the height o z, fig. 2, Plate 2, and apply it from the parallel at 
 o, to z, fig. 4,Plate 3 ; then draw the hypothenuse z y, through to 5 9. 
 
15 
 
 At the angle z of the perpendicular to the hypothenuse is the 
 pitch-bevel obtained. Draw the dotted line j k, parallel to the 
 hypothenuse, let its distance from the hypothenuse be equal to h g, 
 fig. 6, or h g, fig. 7 ; then erect the perpendicular r r, from the chord, 
 fig. 2, to the hypothenuse, fig. 4 ; draw r g, parallel to the chord, let 
 fall the perpendicular 2 2, from the hypothenuse to the chord, cutting 
 the point g of r g, then draw m n, fig. 2, parallel to the chord z y, 
 cutting the convex side of the quadrantal part of the given plan ; 
 draw the diagonal fine r / from the chord z y, to the parallel m n; 
 then draw the dotted lines parallel to / r, from the several points 
 upon the quadrantal part, to the chord commencing at w 5, dQ,b'7, 
 8 8 4, 3 3 2, 1 and x 9 ; then erect perpendiculars from the 
 several points upon the chord fine, 5, 6, 7, 4, and 9, to the hypothe- 
 nuse, 5, 6, 7, 4, and 9, of fig. 4 ; then from the angle at g, erect ghf^ 
 perpendicular to the hypothenuse 5 9 ; let the distance h fhe equal 
 to 2/ fig. 2, from the chord z y, to the parallel m n ; then from the 
 angle at r, upon the hypothenuse, draw the diagonal r f; then draw 
 the dotted lines parallel to r f from the several points, as 5, 6, 7, 4, 
 2, z, and 9^ upon the hypothenuse ; then take the several distances 
 from the chord, fig. 2, upon the dotted fines, 5 w, 6 d, 7 b, 4 8 8, 
 2 3 S, z 1, and 9 x, and apply them from the hypothenuse, fig. 4, 
 upon the dotted lines, 5 w, 6 4 7 6, 4 8 8, 2 3 3, z 1, and 9 x; draw 
 X z and w y, then x z and w y will be the ends of the required 
 mould ; draw y h and w d, which are the required straight part. 
 The points being obtained, describe the arc z h for the concave side 
 through the points 3 8, also the convex x d, through the points 13 8, 
 which completes the face moulds for the lower wreath. 
 
 Fig. 5 shows the manner of forming the face mould to the spring 
 of the plank, when the upper side and edge of the plank form an 
 obtuse angle, as at fig. 8. For the upper wreath from the quadrantal 
 part, fig. 3, first erect the perpendiculars 5 u, from the chord-line, 
 fig. 3 ; then draw d u parallel to the chord 5 w, and take the height 
 d s, fig. 2, Plate 2, and apply it from the parallel at d to 5, fig. 5, 
 Plate 3 ; then draw the hypothenuse 5 u through to 3 4, at the 
 angle 5 of the perpendicular to the hypothenuse is the pitch-bevel 
 obtained. Draw the dotted line j k parallel to the hypothenuse, let 
 the distance from the hypothenuse be equal to g h, fig. 6, or g h, fig. 8 ; 
 then erect the perpendicular r r, from the chord, fig. 3, to the 
 hypothenuse, fig. 5, draw r h parallel to the chord from the hypo- 
 
16 
 
 thenuse to the parallel or dotted liney k to the chord 5 u, let fall the per- 
 pendicular h 2, draw m 71, fig. 3, parallel to the chord 5 u, cutting the 
 concave side of the^quadrantal part of the given plan ; draw the dia- 
 gonal line r f, from the chord 5 u, to the parallel n m, then draw the 
 dotted lines parallel to/r, from the several points upon the quadrantal 
 part to the chord, commencing at r 4, c 9, a 8, 7 7 6, 3 3 2, 1 ii, and v 3; 
 then erect perpendiculars from the several points upon the chord- 
 line 4, 9, 8, 6, and 3, to the hypothenuse 3, 6, 8, 9, and 4 of fig. 5 ; 
 then through the angle h erect g f perpendicular to hypothenuse 
 3 4, let the Une g f he equal in length to 2 / fig. 3, from the chord 
 5 w to the parallel nm; then from the angle at r, upon the hypothe- 
 nuse, draw the diagonal r f, then draw the dotted fines parallel to 
 rf, from the several points 3, u, 2, 6, 8, 9, and 4, upon the hypothe- 
 nuse ; then take the several distances from the chord, fig. 3, upon 
 the dotted fines 4 r, 9 c, 8 a, 6 7 7, 2 3 3, w 1, and 3 v, and apply 
 them from the hypothenuse, fig. 5, upon the dotted fines 4 r, 9 c, 
 8 a, 6 7 7, 2 3 3, w 1, and 3 v, draw v u, and r 5 ; then v u, and r t, 
 will be the ends of the required mould. Draw 5 a, and r c, which 
 being the required straight part, the points being obtained, describe 
 the arc u a for the concave side, through the points 3, 7, also the 
 convex v c, through the points 1, 3, 7, which completes the face 
 mould for the upper wreath. 
 
Plate 4 
 
 m 
 
 R.Miller.M,,t,'r . Eng^ by JJtCAlMwod 1 
 
PLATE 4. 
 
 Shows the manner of drawing the lines for the formation of tfie 
 face and falling-moulds of a .stair-rail easing over a small open- 
 ing in the landing. 
 
 Let A B, fig. 1, be the semi-circular part of the given plan, 
 having a portion of straight rail attached at each end. A B, fig. 4, 
 shows the manner of obtaining the stretch-out of the semi-circu- 
 lar part With a b and c d, the diameters of the concave and convex 
 semi-circumference of the given plan a b c and d as centres, describe 
 the arcs intersecting at m and n; then draw the tangent lines e f 
 and j k of the concave and convex sides of the given plan. From 
 the point n draw lines through a and h, intersecting the concave 
 tangent at e and /; also from m draw lines through c and d, inter- 
 secting the convex tangent at ; and k; then j k and efwiW be the 
 stretch-outs or lengths of the convex and concave semi-circumfer- 
 ences of a b and c d. 
 
 It will be observed that A B, lig. 4, is the same as the given plan, 
 lig. 1, being drawn merely to show more clearly the maimer of 
 obtaining the different peints and ihe r stretch-outs. 
 
 To obtain the concave falhng mould, fig. 2, from the concave 
 stretch-out ef fig. 4, erect perpendiculars from the points e and / of 
 the concave stretch-out to b and o ; at the point b place the upper 
 angle of the pitch-board ab c and b d, the line of the floor ; draw 
 d e equal to half a rise at right angles to d b, then draw the hypo- 
 thenuse a e and ep parallel to the stretch-out of the given plan, fig 4. 
 Let op he equal to the straight part, dp, of the given plan, fig. I. 
 From the angle at e upon the hypothenuse measure off' eight inches 
 in length, which divide into six equal parts. Then take the length 
 of four of these parts and apply it upon the opposite side of the 
 angle at e and divide said distance into six equal parts ; then by 
 intersecting lines the easing will be formed for the Ipwer edge. Set 
 up the width of the mould, square across it at each extremity of 
 said easing, and from the points upon the upper edge of said mould 
 the upper easing may be formed. 
 3 
 
18 
 
 Take the length of the straight part c s of the given plan, fig 1, 
 and apply it upon the base of the pitch-board from c to s, fig. 2. 
 Erect the perpendicular s z, and from z square across the moulds, 
 which line will be the line of the required joint. The joint at t in 
 the centre of the semi-circular part is oblained in the same manner 
 as in the preceding plate. At p square across the mould, and this 
 is the line of the terminating joint. Draw / u, touching the lower 
 edge of the lower joint, and u t at right angles io ul ; draw t v par- 
 allel to the lower edge of said mould, and v p ai right angles to ?; ^ ; 
 then u t and v p will be the given heights for the upper and lower 
 parts of said mould. The convex falling mould, fig. 3, is obtained 
 in the same manner from j h, the convex stretch-out, fig. 4, as the 
 concave mould fig. 2 was obtained from e f, the concave stretch-out, 
 fig. 4. 
 
 To form the easing upon the convex falling mould, fig. 3, so that 
 it may coincide with the concave easing, fig. 2, take the distance 
 from r at the termination of the easing to o at fig. 2, and apply it 
 upon the concave stretch-out from /to r, fig. 4 ; then draw the line 
 r n, and through its intersection with the concave semi-circumfer- 
 ence, draw t s from the centre t to the convex semi-circumference ; 
 then draw on v through its point of intersection with the convex 
 semi-circumference, and v k will be the required distance. Apply 
 said distance upon the mould from k to v, fig. 3. Divide the dis- 
 tance from V to the angle at e into six equal parts. The distance 
 upon the hypothenuse from the angle at e is the same as upon 
 the hypothenuse of the concave mould, fig. 2. Divide this distance 
 into six equal parts, and by intersecting fines from those parts, the 
 easing will be formed; then square across the mould at each 
 extremity of the easing to the upper edge, 'where the easing is 
 formed in the same manner as at the lower. Then the convex 
 easing will coincide with the concave when applied to the rail pieces, 
 the overwood being obtained from each side of the perpendicular 
 at the joint t of the concave falhng mould, and applied to each side 
 of said perpendicular upon the convex mould, fig. 3, also at the 
 centre of the semi-circular part, fig. 1. 
 
 It will be observed that the manner of obtaining the lines and 
 heights for the formation of the face moulds, figs. 5 and 6, the appli- 
 cation of the face moulds to the plank, the cutting away of the 
 superfluous wood, the application of the falling moulds to the rail- 
 pieces, and the obtaining and cutting of the joints, are the same as 
 has been described in the preceding plates. 
 
PLATE 5. 
 
 The form of the rail being given to draw the mitre cap. 
 
 Fig. 1. Let A be the plan of the rail, and B the mitre cap. At 
 A draw the line ah \.o obtain the width of the rail, and the line o u 
 which shows the projection of the moulding of the rail. The width 
 of the cap being obtained, draw the line ah ; from the centre k 
 describe the semicircle a h, then take the distance from b to v at A, 
 being the projection of the moulding, and apply it to 5 z; at B, then 
 describe the semicircle v v and draw the ]ines h e and a f also the 
 centre line h d, and the mitre e and fa. Divide the line h h A 
 into any number of parts, and draw the lines parallel io h e from 
 the upper side of the rail through the points of division z x v t g to 
 the mitre e d. Do the same upon the side ah to the mitre f d ; 
 then place the point of the compass at the centre k and circum- 
 scribe the parallel lines from the mitre d e around to h z x v t and 
 q and draw the ordinates h c, z y, ^c. ; then prick the cap according 
 to the letters. Perform the same from the mitre dftoh a and then 
 through the points trace out the cap. 
 
 Fig. 2. Shows the manner of elevating the cap to its proper height 
 when attached to the rail, also the formation of the easing. 
 
 Let a b che the pitch-board, A the cap and B the rail. The cap 
 being raised equal to the height of half a riser, when placed upon 
 the newel, will be equal in height to the rail placed upon the long 
 balusters. 
 
lo 
 
 tiJdiUer l^raim- 
 
PLATES 6, 7, 8. 
 
 Show the manner of dravnng the lines for the formation of the face 
 and falling moulds for the rail of a semi-circular stairs, having 
 seven winders with fliers attached below, and easing over the 
 landing above. 
 
 A, B, C, fig. 1, Plate 6, being the given plan of the rail, having a 
 portion of straight rail attached to each end of the semi-circular 
 part, with the diameter a b, of the concave, and c d, of the convex 
 sides of the plan for radii, and from the points a b and c d, describe 
 arcs, intersecting at e and / Then draw the tangent lines, g h, and 
 i j, to the concave and convex sides of the given plan ; then from 
 the point e, draw lines through a and b, intersecting the concave 
 tangent at g and h ; also from the point / draw lines through c and 
 d, intersecthig the convex tangent at i and/; then g h and ij will 
 be the stretch-out, or length of the concave and convex semi-cii- 
 cumference from a c, to 6 d^ of the given plan. 
 
I'hilr 
 
PLATE 7. 
 
 To obtain the points for the formation of the concave and convex 
 falling moulds, Jigs. 2 and 3, from the given plan, Jig. 1, of 
 Plate 6. 
 
 Draw the horizontal Une t h, or base of the concave and convex 
 faUing moulds. 
 
 Fig. 2. Let e f g he the pitch-board, and ^ ^ be equal to the 
 concave stretch-out, g h, fig. 1, Plate 6. Draw h i, the height-rod, 
 perpendicular to e h, the base ; draw i j, the line of the floor, which 
 is equal in length to b k, fig. 1, Plate 6. Let j k be equal to the 
 height of half a riser, which allows the rail on the landing, when 
 placed upon the baluster, to have its proper height ; then draw 
 k I parallel to / i, draw the hypothenuse / f touching the angle at 
 i, and f e, to g ; then set up the width of the mould, which is 
 equal to the required thickness of the rail. Draw p o, o n, and n m, 
 parallel to k I, I f and / q, which gives the proper angle of incli- 
 nation to said mould. 
 
 Fig. 3. Let g hi be the pitch-board, and i j be equal to the 
 convex stretch-out, i j\ fig. 1, Plate 6 ; then draw j k the height- 
 rod (which is the same as at fig. 2) perpendicular to g j, the 
 base, and k I, the line of the floor, which length is equal to I, 
 fig. 1, Plate 6. Let I m he equal to the height of half a riser, 
 which allows the convex mould to coincide with the concave ; then 
 draw 7n e parallel to / k, draw the hypothenuse, e h, touching the 
 angle at k, and h g, to s ; then set up the width of the mould, which 
 is equal in width to fig. 2 ; draw r q, q p, and p o, parallel to m e, 
 e h, and h s, which gives the proper angle of incUnation to said 
 mould. 
 
 To form the easing upon the concave falling mould, Jig. 2. 
 
 Take the distance e g, or base of the pitch-board, and apply it 
 from the angle at / to t; let fall the perpendicular, t, to s ; then on each 
 
24 
 
 side of the angle at/ divide / t and / e into six equal parts each, 
 and from the point of di\ision next to the angle on one side, draw 
 a line to that point which is farthest from the angle, on the other 
 side ; then do the same from all the other points of division, and by 
 the intersection of those lines, the easing will be formed for the 
 lower edge of said part of the mould ; then square across the 
 mould dXe p and t u ; then from the angle at o, to w, and o, to p, 
 divide o p, and o u, into six equal parts each, and by intersecting 
 lines to each of those parts, at each side of the angle at o, the eas- 
 ing for the upper edge of said part will be formed, which completes 
 the lower easing of said mould. 
 
 To form the upper easing of said mould. 
 
 The distance from the angle at / to a being equal to eight inches, 
 divide said distance into six equal parts, and apply the distance 
 equal to four of those parts, upon the opposite side of the angle, at 
 /, to h, and divide said distance into six equal parts ; then by the 
 intersection of lines (as has been described) the easing will be 
 formed for the lower edge of said part of the mould ; then square 
 across the mould at a 6 and h 7n ; then from the angle n to m, and 
 n to h, divide n b, and n m, into six equal parts each, and by 
 intersecting lines to each of those parts, at each side of the angle 
 at n, the easing for the upper edge of said part will be formed, 
 which completes the upper and lower casings of said mould. 
 
 To form the easings upon the convex falling mould, Jig. 3. 
 
 Take the distance upon the base g s, fig. 2, and apply it upon 
 the concave stretch-out line from g to s, fig. 1, Plate 6, draw the 
 dotted line s e, and at the intersection of s e, with the concave side 
 a b, draw the dotted line from the centre o, through said point of 
 intersection of the concave, to the convex side of the given plan, 
 Draw the dotted line / u, through said point of the convex side of 
 said plan ; then take the distance, i u, of the convex stretch-out, and 
 apply it upon the base from i to u, fig. 3, Plate 7. At u erect a 
 perpendicular from the base i j, and at the point in which this 
 perpendicular intersects the lower edge of the convex mould at v 
 square across at v w. The lower point of the easing being obtained 
 at g, square across the mould 'at g r; then divide each distance on 
 each edge of the mould, from each side of the angles at h and q, 
 to g r, and w v, into six equal parts, and by the intersection of lines 
 
25 
 
 connecting those parts, the lower easing will be formed. Then 
 take the distance upon the base hne from g to the dotted line n, 
 fig. 2, and apply it upon the concave stretch-out, from g to n, fig. 1, 
 Plate 6 ; then draw the dotted line e n, and at the intersection of 
 e n, with the concave side a b, draw the dotted fine from the 
 centre o, through said point of intersection of the concave to the 
 convex side of the given plan. Draw the dotted line / y through 
 said point of the convex side of said plan ; then take the distance 
 i tf, of the convex stretch-out, and apply it upon the base hne from 
 i to y, fig. 3, Plate 7. At y erect a perpendicular from the base i y, 
 to c, square across at c d; the upper point of the easing being 
 obtained at m, square across at m o ; then divide each distance on 
 each edge of the mould, from each side of the angles at e p, to m c, 
 and o d, into six equal parts, and by the intersection of lines con- 
 necting those parts, the upper easing v^ll be formed, which com- 
 pletes the casings for the convex faUing mould. 
 
 To obtain the butt-joints upon the concave falling mouldy as at Jig. 2, 
 
 in Plate 7. 
 
 ' From the base, g h, erect as many perpendiculars as there are 
 intended to be joints, as v x, through d c to w y, on the upper edge 
 of said mould. (It should be observed that the number of pieces 
 required to form the rail should be governed by the size of the 
 opening or well-hole, that the grain of the wood may continue with 
 the circle of the rail as near as possible, which gives the rail a more 
 beautiful appearance, and is much stronger and requires much less 
 stuff in thickness.) Then obtain the centres of the mould upon the 
 part of the perpendiculars that pass through from d to w and c to y, 
 which centres are 2 5 ; then square across the mould at 1 3 and 4 6, 
 passing through the centres 2 5 ; it being squared across at Jc m, the 
 upper end, and e p, the lower end of said mould, then the lines 
 ^ m, 4 6, 1 3, and e p are the given joints. To obtain the necessary 
 overwood for each given joint, let fall the perpendiculars p q, 'S 10, 
 and 6 12 ; also erect the perpendiculars 111 and 4 13, to the base 
 r h; then each respective distance (parallel to the base) as r ^, 3 w, 
 w li, Q y, and y 12, will be the required overwood for each respec- 
 tive joint. The joint k m, at the upper end of the mould, 
 there being no overwood, for the reason that the joint is perpendicu- 
 lar to the base or floor line,y i; then take the distance ij, the floor 
 line, and r g, the base of the pitch-board, including the overwood, 
 
 4 
 
26 
 
 r e, and apply said distances from the semi-circular part of the given 
 plan, a c and b d, fig. 1, Plate 6, to 8 10 and k 1, which is the 
 required portion of straight rail attached to each end of the semi- 
 circular part. 
 
 The manner of obtaining the line of overwood upon the convex falling 
 mould, Jig. 3, from the concave, fig. 2. 
 
 Take the distances upon the base, g v and g x, fig. 2, and apply 
 them upon the concave stretch-out line from g to v and g to x, fig. 1, 
 Plate 6 ; then draw the lines e v and e x, and at the intersection of 
 e V and e x, to the concave side, a b, draw the radius lines from the 
 centre, o, through said point of intersection of the concave to the 
 convex side of the given plan, and through said points' of intersec- 
 tion to the convex side, c d; draw the lines fx and / z, then obtain 
 the required overwood for each respective joint from the concave 
 falhng mould, fig. 2, as from 2 to the line 3 10, from 2 to 1 11, from 
 5 to 6 12, from 5 to 4 13, and apply it upon each side of the radius 
 lines drawn from the centre o, fig. 1, as 7 10, 1 3, and 7 8, 4 3 ; then take 
 the distance i x and i z from the convex stretch-out and apply them 
 upon the base from i to x and i z, fig. 3, Plate 7 ; then from the base at 
 X and z erect perpendiculars through to the upper edge of said' mould 
 to b and a ; then let fall the perpendicular r s, then take the distance 
 upon the base of the convex mould from t to i, i to 7, and h to /, 
 will be equal to the convex edge of the given plan of fig. 1, Plate 6, 
 from 10 to c, c round to d, and d\o 1; also the distance upon the 
 base of the concave mould, fig. 2, from rtog,g to h, and i to j, will be 
 equal to the concave edge of the given plan of fig. 1, Plate 6, from 
 8 to a, a round to b, and b to k. * 
 
 Figs. 4, 5 and 6, of Plate 6, show the manner of obtaining the points 
 for the formation of the face moulds A B and C, from the 
 given plan, fig. 1, Plate 6, and from figs. 2 and 3, of Plate 7. 
 
 Draw the chord line 9 2 of the part A of the given plan 
 touching the concave points at 8 1; then erect perpendiculars from 
 said chord fine at 9 8 7 6 4 1 and 2 through the points 10 c a and 3 ; 
 then draw 8 7 parallel to the chord fine, then obtain the height, 7 1, of 
 fig. 2, Plate 7, and apply it upon the perpendicular from the parallel at 
 7 to 1, fig. 4, Plate 6, draw the hypothenuse 2 9, toiAihing the point at 1 
 and the parallel at 8 ; at the angle at 1 of the hypothenuse to the per- 
 pendicular, is the pitch bevel obtained ; draw fines at right angles 
 
27 
 
 to the hypothenuse from the points 9 7 6 4 and 2, then obtain the 
 several distances 9 10, 7 c, 4 5 5, and 2 3 from the chord line, fig. 1, 
 and apply them upon the perpendiculars to the hypothenuse, fig 4. 
 From the points 9 7 6 4 and 2, to 10, c, a, 5 5, and 3, draw the lines 
 
 1 '3, ac, and 8 10, then 1 3 and 8 10 will be the ends of the required 
 mould. The points 1 5 a of the concave and 3 5 c of the convex 
 part. A, being obtained, describe arcs through them, draw a 8 and 
 c 10, then a 8 and c 10 will be the length of the straight part, and 8 
 10 the end of the required mould. Draw the chord line 9 2 of the 
 part B of the given plan, touching the concave points at 7 4 ; then 
 erect perpendiculars from said chord line at 9 7 5 4 and 2, through 
 the points 10 and 3 ; then draw 7 8 parallel to the chord line, then 
 obtain the height, 8 4, fig. 2, Plate 7, and apply it upon the perpendi- 
 cular from the parallel at 8 to 4, fig. 5, Plate 6 ; draw the hypothenuse 
 
 2 9, touching the point at 4 and the parallel at 7. At the angle 4 of 
 the hypothenuse to the perpendicular is the pitch bevel obtained. 
 Draw lines at right angles to the hypothenuse from the points 9 5 
 and 2, then obtain the several distances 9 10, 5 6 6, and 2 3, from 
 the chord line, fig. 1, and apply them upon the perpendiculars to 
 the hypothenuse, fig. 0, from the points 9 5 and 2 to 10, 6 6, and 3 ; 
 draw the lines 4 3 and 7 10, which lines will be the ends of the 
 required mould. The points 4 6 7 of the concave and 3 6 10 of 
 the convex part, B, being obtained, describe arcs through them. 
 Draw the chord line 9 2 of the part C of the given plan, touching 
 the concave points at 7 k, then erect the perpendiculars from said 
 chord line at 9 7 6 4 3 A; and 2, through the points S b d and 1 ; then 
 draw 7 9 parallel to the chord line, then obtain the height, 9 k, fig. 2, 
 Plate 7, and apply it upon the perpendicular from the parallel at 9 to k, 
 
 ' fig. 6, Plate 6, draw the hypothenuse 2 9, touching the point at k, and 
 the parallel at 7. At the angle k of the hypothenuse to the perpen- 
 dicular is the pitch bevel obtained. Draw lines at right angles to 
 the hypothenuse from the points 9 6 4 3 and 2, then obtain the sev- 
 eral distances 9 8, 6 5 5, 4 5, 3 and 2 1 from the chord fine, fig. 1, 
 and apply them upon the perpendiculars, to the hypothenuse, fig. 6 ; 
 from the points 9 6 4 3 and 2 to 8, 5 5, b, d, and 1, draw the lines k 1, 
 b d and 7 8, then k 1 and 7 8 will be the ends of the required mould; 
 draw the lines k b and 1 d, then k b and 1 d will be the length of the 
 straight part required ; the points 5 5 7 of the concave and d 5 8 of 
 the convex part C being obtained, describe arcs through them, which 
 completes each respective face mould, A B and C, from the parts 
 A B and C of the given plan, fig. 1. 
 
PLATE 8. 
 
 Shows the manner of obtaining the points and drawing the lines for 
 the formation of the face moulds to the spring of the plank (when 
 the upper side and edge form an acute angle, as at A and B, and 
 when the upper side and edge form an obtuse angle, as at C) from 
 the given plan, Jig. 1. 
 
 It should be observed that the given plan, fig. 1, Plate 8, is the 
 same as the given plan, fig. 1, Plate 6 ; therefore it is unnecessary 
 to describe them again, their stretch-outs being the same, and the 
 manner of obtaining them ; the location of the joints the same, the 
 heights being obtained from fig. 2, Plate 7, for the formation of the 
 face moulds. The student will at once discern the utility of form- 
 ing the face moulds to the spring of the plank, from the same plan 
 that the solid section is of Plate 6 ; that the governing principles of 
 each may be more easily described, A B and C, fig. 1, being the 
 given plan of the rail. 
 
 To form the fojce mould A, fig. 4, to the spring of the plank {when the 
 upper side and edge form an acute angle) from the part A of the 
 given plan, Jig. 1. 
 
 Draw the chord line 9 2 (of the part A of the given 
 plan) touching the concave points at 8 1 ; then erect the per- 
 pendiculars 8 1 from the chord line, then draw 7 8 parallel to the 
 chord 9 2, and take the height, 7 1, fig. 2, Plate 7, and apply it upon 
 the perpendicular from the parallel at 7 to 1, fig. 4, Plate 8 ; then 
 draw the hypothenuse through 1 8 ; draw the dotted fine j k, paral- 
 lel to the hypothenuse, and let its distance from the hypothenuse be 
 equal to h g, fig. 8 ; then erect the perpendicular r r from the chord- 
 line to the hypothenuse, fig. 4, draw r g parallel to the chord ; let 
 fall the perpendicular 11, 11, from the hypothenuse to the chord, 
 cutting the point g of r g ; then draw n m parallel to the chord 8 1, 
 cutting the convex side of the part A of the given plan, draw the 
 diagonal fine r f from the chord to the parallel n m ; then draw 
 
30 
 
 the dotted lines parallel to fr, from the several points upon the 
 part A of the given plan to the chord, commencing at 10 9, cl, aQ>, 
 5 5 4, 11 12 12 and 3 2; then erect perpendiculars from the 
 several points upon the chord-line 9 7 6 4 and 2, to the hypothe- 
 nuse 9 7 6 4 and 2 of fig. 4 ; then from the angle at g erect g h f 
 perpendicular to the hypothenuse 9 2, let the distance k f he equal 
 to 1 1 from the chord 8 1 to the parallel n m; then from the 
 angle at r upon the hypothenuse, draw the diagonal r f; then draw 
 the dotted fines parallel to r f from the several points 9 7 6 4 11 
 and 2, upon the hypothenuse ; then take the several distances from 
 the chord upon the dotted lines 91 0, 7 c, 6 a, 45 5, 11 12 12 and 
 2 3, and apply them from the hypothenuse, fig. 4, upon the dotted 
 fines 9 10, 7 c, 6 a, 4 5 5, 11 12 12 and 2 3 draw 1 3, « c and 8 10, 
 then 1 3 and 8 10 will be the ends of the required mould ; draw 
 8 a and 10 c, which being the length of the required straight part, 
 the points being obtained, describe the arc a 1 for the concave side 
 through the points 5 12, also the convex c 3, through the points 5 12, 
 which completes the face mould A, fig. 4, from the part A of the 
 given plan, fig. 1. 
 
 To form the face mould B, fig. 5, to the spring of the plank [when 
 the upper side and edge form • an acute angle), from the part B of 
 the given plan, fig. 1. 
 
 Draw the chord-line 9 2 of the part B of the given plan, touch- 
 ing the concave points 7 4; then erect 'th4».perpendiculars 7 4 from 
 the chord-line ; then draw 8 7 parallel to the chord 9 2, and take 
 the height 8 4, fig. 2, Plate 7, and apply it upon the perpendicular 
 4 4, from the parallel at 8 to 4, fig. 5, Plate 8 ; then draw the hypo- 
 thenuse 2 9 through 4 7, draw the dotted line j k parallel to the 
 hypothenuse, and let its distance from the hypothenuse be equal to hg 
 fig. 9 ; then erect the perpendicular r r from the chord-line to the 
 hypothenuse, fig. 5, draw- r g parallel to the chord, let fall the per- 
 pendicular 14 14 from the hypothenuse to the chord-line, cutting 
 the point g oir g ; then draw n m parallel to the chord 7 4, cutting 
 the convex side of the part B of the given plan ; draw the diagonal 
 line r f from the chord to the parallel n m ; then draw the dotted 
 lines parallel to f r from the several points of the part B of the 
 given plan, to the chord, commencing at 10 9, 7 o, 1 15 15 16, 6 6 5j 
 14 13 13, 11 11 12 and 3 2; then erect perpendiculars from the 
 several points upon the chord-line, 9, e, 16, 5, 14, 12 and 2, to the 
 
31 
 
 hypothenuse 9, e, 16, 5, 14, 12 and 2 of fig. 5 ; then from the angle 
 at g, erect g h f perpendicular to the hypothenuse 9 2, let the dis- 
 tance h f he equal to 14 / from the chord 7 4 to the parallel n m ; 
 then from the angle at r upon the hypothenuse draw the diagonal 
 r f ; then draw the dotted lines parallel to r f, from the several 
 points 9, 7, e, 16, 5, 14, 12 and 2 upon the hypothenuse; then take 
 the several distances from the chord upon the dotted lines 9 10, 
 7 o, e 1, 16 15 15, 5 6 6, 14 J 3 13, 12 11 11 and 2 3, and apply them 
 from the hypothenuse, fig. 5, upon the dotted lines 9 10, 7 0, e 1, 
 16 15 15, 5 6 6, 14 13 13, 12 11 11 and 2 3 ; draw 4 3 and 7 10, then 
 4 3 and 7 10 will be the ends of the required mould. The points 
 being obtained, describe the arcs 7 4 for the concave side, through 
 the points 1, 15 6, 13 11, also the convex 10 3, through the points 
 o, 15 6, 13 11, which completes the face mould B, fig. 5, from the 
 part B, of the given plan, fig. 1. 
 
 To form the face mould C, Jig. 6, to the spring of the plank {when the 
 upper side and edge form an obtuse angle) from the part C of the 
 given plan, Jig 1. 
 
 Draw the chord-line 9 2 of the part C of the given plan, touch- 
 ing the concave points 7 k; then erect the perpendiculars* 7 ^from 
 the chord-hne ; then draw 9 7 parallel to the chord 9 2, and take 
 the height 9 k, of fig. 2, Plate 7, and apply it upon the perpendicular 
 k k, from the parallel at 9 to k, fig. 6, Plate 8 ; then draw the 
 hypothenuse 2 9 through k 7, draw the dotted line j k parallel to 
 the hypothenuse, and let its distance from the hypothenuse be equal 
 to^ ^ of fig. 10 ; then erect the perpendicular rr from the chord-fine 
 to the hypothenuse, fig. 6, draw r h parallel to the chord, let fall the 
 perpendicular h s from the hypothenuse to the chord-line ; then 
 draw m n parallel to the chord 7 k, cutting the convex side of the 
 part C of the given plan ; draw the diagonal r f from the chord to 
 the parallel m n ; then draw the dotted fines parallel to r f from 
 the several points of the part C of the given plan, to the chord 
 commencing at 8 9, ? 4 5, 5 5 6, 6 4, c? 3 and 1 2 ; then erect perpen- 
 diculars from the several points upon the chord-line 9, s, 6, 4, 3 and 
 2, to the hypothenuse 9, s, 6, 4, 3 and 2 of fig. 6 ; then through the 
 angle h erect g f perpendicular to the hypothenuse 7 k, let the line 
 ^ / be equal in length to sf from the chord 7 k to the parallel mn; 
 then from the angle at r upon the hypothenuse draw the diagonal 
 r f; then draw the dotted lines parallel to r / from the several 
 
32 
 
 points 9, s, 6, 4, 3 and 2, upon the hypothenuse ; then take the seve- 
 ral distances from the chord upon the dotted Unes 9 8, s 4 6 5 5, 
 4 b, S d and 2 1, and apply them from the hypothenuse, fig. 6, upon 
 the dotted lines 9 8, 5 4 6 5 5, 4 5, 3 and 2 1, draw 7 S, b d and 
 k 1 ; then 7 8 and k 1 will be the ends of the required mould, 
 draw k b and d 1, which being the required strait part, the 
 points being obtained, describe the arcs 7 b for the concave side, 
 through the points 4 5, also the convex 8 d, through the points t 5, 
 which completes the face mould C, fig. 6, from the part C of the 
 given plan, fig. 1, 
 
 Fig. 7 shows the manner of obtaining the proper angle of incli- 
 nation to the edge of the plank, as A, B, C, of figs. 8, 9 and 10, 
 from which the rail-pieces are formed, also from which the face 
 moulds A, B and C of figs. 4, 5 and 6, obtain their proper position 
 and form. 
 
 Ay B and C of figs. 8, 9 and 10, shows the proper position 
 of the plank, with their bevels attached for each respective 
 mould. It is necessary to obtain the position or bevel for the edge 
 of the plank, before executing the face mould, that its form may 
 coincide^with that position. 
 
 Fig. 4, Plate 7, gives a sectional view of the rail-piece ; A^ A, the face 
 mould, also showing the manner of applying the face mould A of 
 fig. 4, Plates 8 or 6, to the plank, abc d^ and e f g h being the ends of 
 the plank, the lines a e and c g being the face edge of the plank, a b 
 and e f the upper side, c d and g h the lower ; the face mould being 
 formed, apply the concave points of said mould to the face edge of 
 the plank ; then mark around the mould to obtain its form upon 
 the upper side of the plank ; then obtain the pitch-bevel from the 
 angle 1 of the perpendicular to the hypothenuse of fig. 4, Plates 8 
 or 6, and apply it to the edge of the plank, at the concave points 
 of the mould 3 8 ; then mark the line of the bevel 3 3 and 8 8 ; 
 then take the mould A, from the upper side and apply it to the 
 under side of the plank, placing the concave points of the mould 
 to the line of the bevel 3 8 ; then mark around the mould to obtain 
 its form on the lower side of the plank — as is shown in the diagram ; 
 then cutting away the superfluous wood to the line of the mould 
 of the concave side, from the bevel line 8 8 to 3 3, also to the 
 convex side from 10 10 to 1 1 ; then cutting the ends of the rail- 
 piece to the fine from 8 10, 10 8, to 8, also 3 1, 1 3 to 3 ; then said 
 
33 
 
 rail-piece is ready for Ihe application of the part A of the concave 
 and convex falling moulds ; apply the lower part of the concave 
 falling mould p q, to the line of the bevel 8 8, fig. 4, and bending it 
 to the concave edge of the rail-piece ; then the line 1 11 of said mould 
 will coincide with the line of the bevel 3 3 ; then mark the joints 
 1 3 of the upper end of the part A, also e p of the lower, and 1 of 
 the upper ends of said mould, square across the ends of the rail- 
 piece from the concave to the convex side, place the point r of the 
 lower end of the convex falling mould, fig. 3, having the line r s 
 upon the line 10 10, fig. 4, and bending it around the convex side 
 of the rail-piece ; then the line 1 11 of said mould will coincide 
 with the line 1 1 of said rail-piece, and the point 1 of the convex 
 falling mould, fig, 3, will rest vipon the line squared across the end 
 of said rail-piece, from the point 1 of the concave falhng mould? 
 fig. 2 ; then tracing the upper and lower edges of the part A of the 
 concave and convex moulds upon the rail-piece ; then cutting away 
 the superfluous wood to the line of the moulds and joints, the 
 piece thus formed will have its proper length, size and twist, and 
 will range over its plan ; also the parts B and C, figs. 5 and 6 of 
 Plates 8 and 6, are applied in the same manner. 
 
 It should be observed that, let the moulds be formed for the 
 spring of the plank, or solid section, also the pitch bevels, their 
 application to the plank is the same as has been described in the 
 part A of the given plan, fig. 4, Plate 7. 
 
 5 
 
nJifiaer.yrintir 
 
 Enp * Ay JMAlM and .ITZ 
 
PLATE 9 
 
 Shows a practical method of executing the plan of the carriage of a 
 geometrical stair, with seven winders, having fliers attached below^ 
 and resting against the landing above. 
 
 Fig. 1. Plan of the carriage from a bird's-eye view. 
 
 Fig. 2. Plan and elevation of the veneer. 
 
 Fig. 3. Plan and elevation of the concave side of the carriage. 
 
 Referetwes to letters in Jig. 1. 
 S, S, The front and middle string, or carriages of the straight part 
 W, The wall string or skirting for the straight part. 
 r r r r, Plank risers, 
 hhh, The bearings. 
 
 s s s, The staves or brackets ; those upon the concave side of the 
 carriage are formed to the circle. 
 
 V V, The veneer. 
 
 / /, Showing the projection of the concave side of the rail over the 
 concave side of the carriage. 
 
 References to letters in Jig. 2. 
 
 The stretch-out, 1 8 of the semicircular part.of the given plan, 
 fig. 1, is obtained in the same manner as has been described in the 
 preceding plates. Draw lines from the centre a through the points 
 1 2 3 4, &c., to the tangent line 1 8, cutting the face edge of the 
 plank risers r r ; then from the points upon the tangent line erect 
 perpendiculars through to the upper edge of V VL, of said veneer 
 at ^, 1 2 3 4, &c., to z, then draw the hne t u, parallel to the tangent 
 t 8. Divide u 8, the height rod, into the number of parts there 
 are risers in the given plan, fig. 1 ; draw the floor fine ^ z; then, 
 
36 
 
 at z y, set down the distance equal to the required width of the 
 trimmer joist in the landing, then draw y x, x iv, and w t, the lower 
 edge of the veneer, the form of the easing being obtained in the 
 same manner as has been described.* The line if ^ is the junction 
 of V, the veneer, to S, the string. 
 
 References to letters in fig. 3. 
 
 S. The part of the front string. 
 
 z z. Tenons being formed upon the ends of the front string, and 
 mortices formed into r, the first plank riser, for its reception. 
 
 X. The screw to connect the string to the winders. 
 
 r r r. Plank risers. 
 
 bh h. Bearers. 
 
 s s s. The staves or brackets which, being fitted, glued, and 
 screwed, to the angles of the bearers and risers, the middle and 
 back brackets are secured in the same manner, which answer for 
 lathing joist under the carriage of the stairs. 
 
 c c c. Risers. 
 
 e e e. Treads. 
 
 It should be observed that, let the well-hole have whatever form 
 it may, the carriage may be formed upon this method. 
 
 * The bead being planted to the under edge of the veneer and concave string, for the reception of 
 the lath and plastering. 
 
^h!ll<r Pritilfi 
 
 T.nq'^hyJW.Atwood .V r 
 
PLATE 10. 
 
 Needs no explanation: for it is the same as has been described in 
 Plates 6 7 and 8, its size being drawn merely to coincide with the size 
 of Plate 9. 
 
 \ 
 
M3Slla;IHraer. 
 
PLATE 11. 
 
 Exhibits the plan of forming the face and falling moulds for the rail of 
 a semi-circular stairs with six winders, having fliers attached above 
 and below. 
 
 A, B, C, fig. 1, being the given plan of the rail. 
 
 The lengths of the concave and convex semi-circumference of the 
 given plan are obtained in the same manner as in the preceding 
 Plates, a b being the diameter of the concave, and c d of the con- 
 vex, r t and e o will be their stretch-outs. The stretch-out r t oi 
 the concave being applied upon the base from r t, fig. 5, also e o 
 being applied upon the base from e to o, fig. 6 ; then at the points t 
 and o erect the height-rods, and at their upper extremities place 
 the pitch-boards C C, the base of which is equal to the straight part 
 C of the given plan, fig. 1. The pitch-board, A A, figs. 5 and 6, 
 being placed at r and e upon the base ; the distances upon the base 
 of the pitch-board A of the concave falUng moulds from r to 6 is 
 equal to the straight part A of the given plan, fig. 1. The upper 
 easing of the concave faUing mould is formed the same as the lower, 
 except its being reversed. Obtain the points from the casings of 
 the concave mould, fig. 5, and apply them to the concave stretch-out 
 of the given plan, fig. 1 ; then by the dotted lines obtain the differ- 
 ence of the concave and convex easings, and apply it to the con- 
 cave mould, fig. 6 ; then form the easings. 
 
 The manner of forming the butt-joints upon the concave falling 
 mould, fig. 5, and their application to the given plan, fig. 1, from 
 thence the formation of the face moulds. A, B and C, of figs. 2, 3 
 and 4, from the parts A, B and C of the given plan, fig. 1, are the 
 same as in the preceding Plates. 
 
Plntv I J 
 
 MiUerJ^-tnter . 
 
 K)i'/'/i.\ ' V.Atti n.ii/.A 
 
PLATE 12. 
 
 Shows the manner of obtaining the required thickness of stuff, and pro- 
 per twist of the rail-pieces, also the face moulds for each respective 
 rail-piece. 
 
 To obtain the corresponding points from the parts A B and C, 
 fig. 2, through the given plan, fig. 1, to the parts of figs. 3, 4, and 5. 
 Draw the concave stretch-out, c e, of fig. 1 (it should be observed 
 that the given plan, figs. 1 and 2, is the same as figs. 1 and 5 of 
 Plate 11, also the location of the joints). Erect the perpendiculars 
 from c e, the stretch-out, to c e, the base of fig. 2. Draw the dotted 
 line i, 4, parallel to the part of the base 1 2, 1 3, of the part A ; then 
 divide the distance i 4 into any number of parts, as g, e, c, 1 2. 
 Erect perpendiculars froni each of those points to n, r, s, t, and 
 or upper edge of said mould ; then draw the chord-line, j h, touching 
 the concave points, i, 4, of the part A, of the given plan, fig. 1, and 
 erect perpendiculars from the points /, i, and 4, h ; then draw a j 
 parallel to the chord j h ; then take the several distances, i g, g e, 
 and e c, upon the base, i 4, fig. 2, and apply them upon the parallel 
 to the chord, fig. 1, from i to g, g to e, and e to c ; then take the dis- 
 tance ij, and apply said distance from g to h, e to f and c to d ; 
 then erect the several perpendiculars from the chord-line, cutting 
 the points h gfe and d c, upon the parallel to the chord ; then take 
 the distance from c to 1, 1 to 2, and 2 to 4, and apply said distances 
 upon the stretch-out of fig. 1, from c to 1, 1 to 2, and 2 to 4 ; then 
 draw the lines 1, y, 2, and 4, «/, and at the point of intersection 
 of said lines to the concave side of the given plan, draw fines from 
 the centre z through said points of intersection to the convex side 
 of the given plan ; then erect perpendiculars from the chord and 
 convex side of the given plan, cutting the points 1 2 and b upon 
 the parallel to the chord ; then obtain the several heights from the 
 part A, fig. 2, from the base i 4 to /, n, p p, r r, s s, t t, and v «/, 
 and apply said distances upon the perpendiculars from the parallel 
 i I, g n n, e p p, c r r, 1 s s, 2 t t, and ^ v y, for the concave side ; 
 
 6 
 
42 
 
 then draw / n m, p o, r q, t ii, v w, and y x, parallel to j a, and 
 j k, h m,fo, d q,h u, and a w x, for the convex side; then trace 
 out the form of the rail-piece through the given points i, n, p, r, s, t, 
 V, for the lovi^er, and /, p, r, s, t, y, for the upper edges of the con- 
 cave side, and j, s, u, w, for the lower, and k, m, o, q, s, x, for the 
 upper edges of the convex side ; then draw the dotted line 12, 13, 
 which being the base hne of the joint, and 13, v, the height, from 12, 
 13, as 13, V, fig. 2, also the location of the joints, 12, /, and v,f, being 
 the same, which completes the form of said rail. The face-mould. A, 
 is obtained in the same manner as has been described in the pre- 
 ceding plates. The parts ^ and C, of figs. 4 and 5, are obtained 
 from the parts JB and C, fig. 2 (as will be seen by tracing the cor- 
 responding points), through the parts A and C of the given plan, 
 fig. 1, in the same manner as was the part A, fig. 3, from the part A, 
 fig. 2, through the part A of the given plan, fig. 1. 
 
PLATE 13. 
 
 Fig. 1 shows the plan of a stairs having five winders, the well- 
 hole being formed by two quadrantal parts, a larger and smaller 
 one, which may be termed a quarter space. 
 
 Fig. 2 shows the plan and elevation of the veneer for the concave 
 side of the stair. 
 
 Fig. 3 shows the plan and elevation of the veneer, or skirting for 
 the convex side of the stair. 
 
 Fig. 4 shows the plan ol' a bracket. 
 
 Fig. 5 shows the manner of diminishing the size of the given 
 bracket, fig. 4. 
 
 Fig. 1. Having c a and c i as radius, describe the quadrantal part, 
 a i, of the larger, and having e i and e f as radius, describe the 
 quadrantal part, i f, of the smaller. To obtain their stretch-out, k j\ 
 let a dhe equal to twice the given radius, c a; then having a d as 
 centres, describe the arcs d b and a b, also let fg be equal to twice 
 the given radius e f; then having f g as centres, describe the arcs 
 g c and / c, draw the tangent line h j ; then draw b through a to h, 
 and c through / to j ; then h j will be the length of the concave 
 semi-circumference from a to f, or face of v v, the veneer, which 
 commence at 2, and end at z. 
 
 S, The face side of the fronts tring, the dotted line at s showing 
 the face of the bracket, n n, the line of nosing. 
 
 C, C, the given plan of the rail. 
 
 r r rr, the risers. 
 
 b b b, the plank risers secured to the studs. The student will ob- 
 serve that the carriage is framed in the same manner as fig. 1, Plate 9. 
 
 S S, the wall-string, or skirting, having the lower end of the 
 veneer attached. 
 
 G, tlie ground for the reception of v v, the veneer. 
 
44 
 
 B B, the base upon the landing, having the upper end of the 
 veneer attached. 
 
 Fig. 2, A, the veneer being formed in the same manner as fig. 2, 
 Plate 9. 
 
 Fig. 3, A, the part of the wall-string or skirting, Ij the upper edge, 
 and u u the lower ; r r r the risers, and t t the treads. To obtain 
 the length of the part B of the veneer for the convex side, proceed 
 in the same manner as the concave ; then 5 7 9 will be the stretch- 
 out of 5 7 9, or convex end of the first and second winder ; the 
 line h j is the upper edge, and / i shows the junction of the part^ 
 of the veneer to the part A, and i g o the lower edge of the veneer 
 or skirting, which being inserted into the tread and riser by a groovf 
 being cut to the depth of half an inch, the width being cut suffic. 
 ent for the reception of the veneer or skirting, the thickness of the 
 veneer should be governed by the size of the circle, not more than 
 \ inch, nor less than \ inch. The places should not be cut out for 
 the reception of the tread and riser in the concave veneer until 
 put to its proper place and secured. Fig. 4 being the given plan 
 of the bracket, / a the height, and a h the length ; draw a h, fig. 5, 
 parallel to b, fig. 4 ; then draw the several ordinates perpendicular 
 from the base of the given bracket, fig. 4, to a b, fig. 5 ; then from 
 the point a, fig. 5, draw the diminishing line a d at any given angle ; 
 then continue the ordinates from a b to the diminishing line, a d : 
 then erect them perpendicular to a d, and obtain the several dis- 
 tances from the several ordinates of fig. 4, as a f, e c, &c., and 
 apply said distances upon the several ordinates from a d, fig. 5, as a 
 fjec, &c. Then trace out the form of the bracket. By this method 
 the bracket may be enlarged as well as diminished. 
 
I'la/, 
 
PLATE 14. 
 
 Being the plan of the rail for Plate 13. 
 
 Fig. 1, the given plan of the rail. . 
 
 Figs. 2 and 3, the concave and convex falling moulds. 
 
 Figs. 4, 5 and 6, the face moulds for each respective part of the 
 given plan, fig. 1, being formed in solid section. 
 
 Fig. 7, being the same as fig. 6, the face mould being formed for 
 the spring of the plank. 
 
 The manner of obtaining the stretch-out of the given plan, fig. 1^ 
 is the same as has been described in fig. 1, Plate 13, and its different 
 points, for the formation of the falling moulds, and casings, and 
 their application to the given plan, fig. 1, for the formation of the 
 face moulds, are the same as has been described in the preceding 
 plates ; by tracing the letters, the reader will at once perceive the 
 corresponding points. 
 
 Fig. 8 shows the plan of a rail-piece at the joint, a b, the dowel 
 or iron pins, made of thick wire to keep the rail-pieces from turn- 
 ing ; c, the hand rail screw ; d, shows the form of the groove for the 
 reception of an iron core. The groove should be formed before 
 working the moulding upon the rail. 
 
Plate 75 
 
 a Miller, JHrUa- 
 
 ■VtKoiLbk, sc M 
 
PLATE 15. 
 
 Shoves a method hy which the smith may he confined to the position of 
 the rail, so as to give the proper twist to the iron core, and obtain 
 its neat length. 
 
 First draw the ground plan of the rail or form of the well-hole, as, 
 a a, fig. 1 of the concave and c c of the convex sides of the given 
 plan ; then obtain the size intended for the core, and apply it to 
 the centre of the ground plan as is shown at the shaded part, h 6, 
 then obtain the stretch-outs of the concave side of the shaded part 
 h b, and the convex side, a a, of the given plan, the formation of 
 the falling moulds, figs. 2 and 3, and their easings, and manner of 
 obtaining the joint and their application to the given plan, fig. 1, 
 for the formation of the face moulds, figs. 4, 5 and 6, is the same 
 as has been repeatedly described. The shaded part, r s t, of the 
 concave falling, fig. 2, shows the depth of the groove for the re- 
 ception of the core. The chord lines, of each respective part of the 
 face moulds, figs. 4, 5 and 6, are drawn to the concave side of 
 the shaded part, b b, which gives the proper width to the face 
 moulds. Then by applying the face and falling moulds to the 
 plank which is of hard wood. The wreath pieces being worked 
 to the lines of the moulds and joints cut and screwed together, then 
 set a gauge to the width of the groove or shaded part, b b, or width 
 of the core, fig. 1, and apply it to the lower side of the wreath from 
 the concave edge, then set the gauge to the depth of the groove for 
 the core or shaded part, r st, fig. 2, and apply it the lower edge of the 
 concave side of the wreath ; then cutting the groove to the gauge mark 
 (which may be performed more readily by a tool usually termed a 
 router) the wreath will be complete and ready for the smith to form 
 .the core to the twist of the wreath part of the rail, which should be 
 done with great exactness that the workman may have no fitting 
 to do when placing the rail or cap upon the core. The student 
 will observe that Plate 14 is the plan of the rail for Plates 13 and 
 15 ; Plate 15 is drawn for the formation of the rail for the use of the 
 smith. 
 
PLATE 16. 
 
 To draw the scroll for a hand-rail, and curtail step for the same. 
 
 Fig. 1 shows the manner of obtaining the centres for the forma- 
 tion of the scroll and curtail step. It should be observed that the 
 width of the scroll is governed by the width of the rail, which 
 is one fifth of the extreme width of the scroll. The width of the 
 scroll being obtained, divide it into nine equal parts. Then draw 
 the line a h, equal to five of the nine parts, and draw 6 c at right 
 angles, and equal in length to a b, then a c will form the first qua- 
 drant. Take the distance equal to one of the nine parts, and apply 
 it from b to d. Draw d e at right angles to d c, then c e will form 
 the second quadrant ; then a e will be the required width of the 
 scroll. Describe the semicircle b d, and draw the diagonal line 
 from a to e, and at the point where the diagonal passes through the 
 semicircle b d, draw the line from b through the said point to f 
 and draw / A at right angles to f e ; then e h will form the third 
 quadrant. Draw dj through the point of intersection of bf to the 
 semicircle, and draw / k at right angles to / h ; then h k will form 
 the fourth quadrant. At the point of intersection of j k to bf draw 
 / m at right angles to Ik, then k m will form the fifth quadrant. At 
 the point of intersection of / m to j d, draw n o at right angles to 
 n m; then m o will form the sixth quadrant. In describing the 
 quadrant n m, the segment of the circle should terminate at p. 
 
 Fig. 2 . Shows, the manti^r, of dr.awi7ig:.tbf etfXM 'g,7\d: mriail step. 
 
 The centres o^thfe*- several <^U&di:ants"having been obtained in the 
 manner just described, jni^d:t^ir j:adii:b*e?jng*^ii^ at a, 
 
 and describe the several '€\^AmKtk'a'r,'c'e', e°K,'h k, k m, and m o, 
 which continue to p, which completes the convex side of the scroll. 
 Then set off the width of the rail, from the convex side at a to a, of 
 the same radius, and from the centres of the first and second qua- 
 drants describe the arcs from a through the radius c and e to p, which 
 completes the concave side of the scroll. Then apply the portion, 
 fa, of straight rail, to the first quadrant. (It is always necessary to 
 
 7 
 
50 
 
 have a small portion of straight rail attached to the given plan of the 
 scroll, and it should not be less in length than the required overwood 
 for obtaining the butt joint. If not attached, it will shorten the upper 
 side of the twisted part of the scroll equal to the required overwood, 
 and form an angle at the joint, when connected to the straight rail.) 
 Then square across the rail at 1 1, and obtain the stretch-outs of the 
 arcs, a c t, of the convex, also a c t, of the concave sides of the 
 given part of the scroll, which stretch-outs are obtained in the same 
 manner as the stretch-outs in the preceding plates ; then d e will be 
 the stretch-out of the concave act, and z y, the stretch-out of 
 act, of the convex sides of the given part of the scroll. 
 
 To obtain the concave falling mould at F,jig. 2>,from the part of the 
 
 given plan, jig 2. 
 
 Let ab che the pitch-board, and divide c b, the riser, into three 
 equal parts ; then at the first part from c draw e f parallel to a c. 
 Then obtain the length of the radius from the centre 6 to the convex 
 c, fig. 2, of the first quadrant, and apply it upon the base line of the 
 pitch-board, from a to v. Then obtain the point d upon the base of 
 the falling mould, which is perpendicular to v, then take the concave 
 stretch-out, d e, fig. 2, and apply it upon the base of said mould, from 
 dtoe; then from e set up the width of the mould to x ; then apply 
 the portion a f of straight rail, fig. 2, from dto f fig. 3, and erect the 
 perpendicular / h, and draw h x parallel to g e, then square across 
 the mould at g i and ex; then e x and g i will be the required joints, 
 and j i the required overwood for g i. Then form the easing, and 
 the concave falling mould will be complete. 
 
 The convex falling mould, F, fig. 4, is formed in the same manner 
 from the convex points, fig. 3, as was the concave mould, fig. 3, from 
 the concave points, fig. 2. 
 
 Todraw fie f^d-'mau1M\M}^g. 5f^{)n^tli£.^fii-t of,fho,givenplan,Jig. 2. 
 
 First draw the chord line, 1 9, touching thfe 'concave points, / t, 
 then from the choD^'Jirfe ^recf itl^ s5yemt 'perpendiculars from the 
 points 1, 2,f, 3, 4, 5, 6, t, 7, "8 *aAd*9:* ' Theh dravv^ ^ parallel to the 
 chord line, 1 9 ; then obtain the height from / g, fig. 3, or fig. 4, 
 which is the same, and apply it to gf fig. 5. Then draw the hypo- 
 thermse, 1 9, touching the point f also the parallel at t, and draw 
 fines at right angles to the hypothenuse from the points 1, 2, 3, 4, 5, 
 6, 7, 8 and 9 ; then obtain the several distances, as 1 / 2 a, 3 «, 4 4 4, 
 
5J 
 
 5 c, 6 6, 7 c, 8 8, and 9 t, from the chord Hue, fig. 2, and apply them 
 upon the perpendiculars to the hypothenuse, fig. 5, fi:om the points 
 1, 2, 3, 4, 5, 6, 7, 8 and 9, to/« a, 4 4 c, 6 8 and t ; then draw the 
 line t t, which will be the end of the required mould. The points 
 t 6 c, and t 8 c, of the segment, also c 4 «of the concave, and c i a of 
 the convex being obtained, describe the arcs through said points ; 
 then a /and « / is the straight part, and //the end of the required 
 mould. 
 
 The manner of applying the face-mould to the plank, also the 
 application of the falling moulds to the rail-piece, and the cutthig 
 away of the superfluous wood, is the same as has been described in 
 the preceding plates. 
 
 At E, fig. 6 shows the elevation of the scroll when completed, 
 and / g, the height equal to / g, of the falling moulds, and g i the 
 joint. 
 
 To draw the curtail step and block for the same, as is shown by the dotted 
 
 lines at fig. 2. 
 
 First obtain the size of the baluster i at its base, the centre of 
 which being the centre of the rail ; then draw the lines b, of the 
 bracket, from n to x, the face of which touching the base of the 
 baluster, also being the required length of the bracket, the circular 
 part a x, being drawn from the centre b of the first quadrant, which 
 produces the concave side of the Bloch Then cut a gain in the 
 Block for the reception of the String, which is glued and screwed 
 to the Block; the concave side of the Block is made sufficiently 
 smooth from the junction of the String to the Block, to x, that it may 
 not require any veneer ; Risr, the second riser. For the veneer and 
 convex side of the block, set back from the convex edge of the rail, 
 at the radius c, the distance equal that the bracket b is from the con- 
 cave, which will be the face of the lower riser, and from the centres 
 of the rail trace out the fines v, of the veneer (the face of which 
 touching the base of the baluster, the inner line of the veneer being 
 the convex side of the Block), around to G, where the end of the 
 veneer is let into the Block, a groove being made by the cut of a fine 
 saw for its reception. It will be observed that v the veneer and Riser 
 are the same piece, there being a gain cut into the Block at A, for the 
 reception of the Riser part. To obtain the length of v the veneer, 
 take a small cord, and by applying the end to the depth of the groove 
 at G, then encircle the Block with it around to A, which will be the 
 
52 
 
 length required for the veneer part, at the same time allowing for W, 
 the wedges (which are formed as w w at fig. 7), to strain the veneer 
 close and smooth to the Block The form of the Block being worked 
 out to the given fines, and gains cut for the reception of the String 
 and Biser. Then prepare a sizing of glue and size the Block upon 
 the convex side, from G around to A, also the gains, and let it 
 become dry before the veneer is appfied. The plank from which 
 the Block is formed should be perfectly seasoned. 
 
 To prepare the veneer and nser. 
 
 The length of the riser being obtained, including the length of the 
 veneer, and got to its proper width and thickness, then gauge the 
 part for the veneer about one eighth of an inch in thickness, and 
 with a rip-saw sfit out the veneer. In planing up v, the veneer, give 
 it a gradual diminish from the Biser to the end, that it may enter the 
 groove at G. Then put the veneer part into hot water, and let 
 it remain until it becomes soft and easily bent, being then ready 
 for application to the Block; give the ^/ocA; another good coat of 
 weU prepared glue from G around to A, also the gain for the recep- 
 tion of the Biser. Then place the end of the veneer square into the 
 groove at G, and carefully bend it around the Block, until it will 
 admit the Biser into the gain at A. The wedges being driven 
 properly, one from each side of the Block, will strain the veneer 
 smooth and close to the Block. It may be necessary to place a 
 smooth piece of board upon the face of the veneer, from the Biser or 
 wedges at A, to the radius c of the first quadrant, that the veneer 
 may adhere to the Bhck ; then taking a hand-screw and placing it 
 across the Block at x v, to press the board against the veneer, and 
 letting it remain until it becomes perfectly dry. 
 
 To draw the curtail step. 
 
 The diagram at N, fig. 2, shows the projection of the nosing, 
 and the outer dotted fine its continuation around the convex side to 
 the concave, to I n, at the second riser, where the return of the 
 nosing is round the end of the bracket, which gives the form and 
 size of the curtail step. 
 
PLATE 17. 
 
 Exhibits the ground plan of a stair, and rail with scroll and curtail step 
 having ten fliers resting against the landing, and five winders in the 
 semicircular part. 
 
 Fig. 1 being the given plan of the stairs and rail. 
 
 Fig. 2, the plan of the rail and face-moulds for the same. 
 
 Figs. 3 and 4, the concave and convex falling moulds. 
 
 The manner in which the different points are obtained, is the 
 same as has been repeatedly described, which will be seen by the 
 corresponding letters. 
 
PLATE 18. 
 
 Shows the manner of forming the moulds, for the rail of a circular 
 stairs, having sixteen winders, running from a landing to a rake, 
 and rake to a landing. 
 
 Fig. 1 being the given plan of the rail, the winders commencing 
 at s u and ending at t v and from t vto s u the landing from a to 6 
 and c to of figs. 2 and 3 being the length of the concave and con- 
 vex falling moulds, when placed upon the height rods 1 1 and v v. 
 The stretch-outs s t and u v, and their application to the base of 
 figs. 2 and 3 for the formation of the concave and convex falling 
 moulds, and obtaining the joints and their application to the given 
 plan, fig. 1, the heights being obtained in the same manner from the 
 several joints of fig. 2 for the formation of the face-moulds ABED 
 and E, as has been described in the preceding plates. It should be 
 observed that the lower easing at a, fig. 2, is raised half a riser 
 above its base that the balusters on the landing may be equal in 
 length to the long balusters on the rake. The perpendicular at s 
 being the first riser and 1 1 the height rod, which gives the proper 
 angle of inclination to the faUing moulds, and the same to the con- 
 cave side of the rail. The distance from a to the hypothenuse (the 
 lower edge of said mould) is twelve inches, and that upon the rake 
 eight inches, then divide the distance into six equal parts, and by 
 the intersection of lines from each of those parts the easing will be 
 formed. The upper easing is the same as has been described in 
 the preceding plates, also the manner of obtaining points for the 
 easing upon the convex falling mould. 
 
PLATE 19. 
 
 Exhibits the pkn of the rail and scroll for a stairs, the well-hole being 
 
 eUiptical 
 
 Fig. 1 being the given plan of the rail, whose centres c, d, e,f g, h, 
 i,j\ k, I, m and n, being obtained as fig. 19, Plate 22, for the formation 
 of an ellipsis by which the concave and convex sides of the given plan, ^ 
 fig. 1, are formed, the stretch-outs 1 2 (of the scroll), 3 4, 5 6, 7 8, 9 10, 
 11 12, 13 14, 15 16, 17 18, of the concave sides, and 19 20 (of the scroll), 
 21 22, 23 24, 25 26, 27 28, 29 30, 31 32, 33 34, 35 36, of the convex sides 
 of the several segments, op,pq,q r, rs,st,tu, u v, v w, of the several 
 quadrants, being obtained for the formation of the concave and con- 
 vex falling moulds, figs. 2 and 3, which being raised to the height 
 of 1 ^ of a riser above its base, also the location of each respective 
 joint and formation of the casings upon the same, and its application 
 to the given plan, fig. 1, for the formation of the face moulds, figs. 
 4, 5, 6, 7 and 8, are the same as has been described in the preceding 
 plates. It should be observed that the stretch-out must be obtained 
 from the concave and convex sides of the given plan of the rail, 
 let the well-hoJe be whatever form it may, before the concave and 
 convex faUing moulds can be formed. The carriage for an eUiptical 
 stair may be framed in the same manner as Plate 9. 
 
 8 
 
filler. Ihnisr 
 
PLATE 20. 
 
 Exhibits the plan of a rail and scroll, the well-hole being an eccentrical 
 form, the scroll being drawn as Plate 16. 
 
 Fig. 1st, having 2, 3, 4, 5 and 6, as centres, commencing at 2, hav- 
 ing 2 y and 2 r as radii, describe the quadrant c e of the concave 
 and p r of the convex, which form the first quadrant ; then from the 
 centre 3, having 3 r and 3 t as radii, describe the quadrant egoi the 
 concave and r t of the convex, which form the second quadrant; then 
 from the centre 4, having 4 t and 4 i as radii, describe the segment 
 g i of the concave and t i of the convex, which form the first seg- 
 ment ; then from the centre 5, having 5 i and 5 as radii, describe 
 the segment i j of the concave and i w of the convex, which form 
 the second segment ; then from the centre 6, having 6 and 6 2; 
 as radius, describe the quadrant j m of the concave and wzoi the 
 convex, which form the third quadrant. The manner of obtaining 
 the stretch-outs a b (of the scroll) oi a c,c d, o{ c e, e foi e g, g h, of 
 g i i y, of ij and j I, of j m of the concave side; also n o (of the 
 scroll) of np,pq,orpr,r s, of r t, t u, of ti,vw, of iw, and wyof 
 w z of the convex side of the several quadrants and segments and 
 their application to the base a, c, e, g, j, k and m of the concave, also 
 w, p, r, t, w, X and z of the convex side for the formation of the con- 
 cave and convex faUing moulds, as at figs. 2 and 3, the moulds 
 being raised the height of 1 ^ risers above its base. In obtaining 
 the joints and easings, the heights of the several joints for the for- 
 mation of the face-moulds are the same as has. been described in 
 the preceding plates. It should be observed that the several distan- 
 ces from centres to centres are thus, from 2 to 3 equal to 2 inches, 
 from 3 to 4 equal to four inches, from 3 to 5 equal to 6 inches, 
 and from 6 to 6 equal to 4 inches ; which gives the form to the 
 well-hole. 
 
PLATE 21. 
 
 To draw the plan and elevation of a staircase, the well-hole being the 
 frustum of a cone, also the mould* for the formation of the rail far 
 the same. 
 
 Fig. 1. Let abhe the diameter of the lower and c ^ of the upper 
 end of the conical frustum, divide the distance a c ox the expan- 
 sion of the frustum into four equal parts, then apply the distance 
 equal to one of the parts each side of the centre of the diameter a h 
 asstuv which form a square, then each angle of said square being 
 the centres as st uv for each respective quadrant, then from the 
 centres, s, of the first quadrant, t, of the second, u, of the third, and 
 V, of the fourth, which produces the expansion of the frustum or 
 well-hole, commencing at a and ending at c, which completes the 
 concave side of the required plan ; then set oft' the length of the 
 tread which is from a to d, then from the same centres describe the 
 convex side of the required plan, then from the centre of the diameter 
 ab describe the circle from / through ^ to / and divide the circle/ 
 h to/into the number of given parts there are intended to be treads 
 (in this there are twenty), and from the points of division upon the 
 outer circle /A to / to the centre of the given plan, draw lines from 
 the convex to the concave side of the given plan, commencing at d 
 a and c f which gives the size of each tread, and showing their 
 gradual expansion. 
 
 To draw the elevation of the given plan, fig. 1, as at fig. 2, from 
 the centre of the given plan, fig. 1, erect a perpendicular through x 
 to z, fig. 2, then draw the base line m n, which being equal in length 
 to the diameter / h, of the given plan, fig. 1 ; then erect n h, the 
 height rod, which being divided into the number of parts there are 
 intended to be risers (there being twenty treads, therefore it will 
 require twenty-one risers), then draw h f parallel to n m, which 
 being the height of the frustum for the stairs ; then from the centre 
 X, describe the semicircle a b whose diameter is equal to the 
 
62 
 
 diameter, a b, fig. 1, or base of the frustum, also from the centre, z, 
 describe the semicircle, c e, whose diameter is equal to the diameter, 
 c €, fig. 1, or expansion of the frustum; then set off the distance 
 from a b and c e, equal to the length of the tread, a d, fig. 1 ; then 
 draw the fines df,a c,b c, and h h, which give the stairs its conical 
 form. 
 
 To draw the concave string of fig. 2, and give it the spiral form. 
 
 Divide the semicircles a b and c e into half the number of parts there 
 are treads in the given plan, fig. 1, and let fall perpendiculars from 
 each point of division to the base of each of the semicircles, a b and 
 c as is shown in the diagram at 5 ^ and uv ; then from each of 
 the points upon the base, a b, draw lines to each of the points in 
 the base, c e, as from t to v, which gives the face of each riser upon 
 the concave string ; the convex string is formed in the same man- 
 ner, thus far describing semicircles from d to k, and /to h, from the 
 centres x and z, and dividing them into the same number of parts 
 as a 6 and c e, and from each of the points of division, let fall per- 
 pendiculars to the base d h and fh; then from the point of division 
 upon the base d k, draw fines to each of the points in the base fh, 
 which gives the face of each riser upon the convex string ; then 
 from each of the parts upoh the height rod, n h, draw lines parallel 
 to the base, d k, which gives the fine of each tread from the concave 
 to the convex string ; then from the upper edge of each riser upon 
 the concave and convex strings, set down the depth of each string, 
 and through those points trace out the curve or lower edge of the 
 string. The elevation of the rail is traced out in the same manner 
 which is supposed to be placed upon the balusters, that it may- 
 retain its proper height from the upper edge of the concave string. 
 
 To draw the plan of the rail, as figs. 3 and 4, for the given plan, fig. 2. 
 
 Set up the height at a, fig. 2, equal to the height of the short 
 baluster, as at o ; then draw o a to b parallel to the base d k at q, 
 erect the perpendicular q c equal in height to n h, the height rod, 
 and draw g c to e paraUel to / h, the fine of the floor ; then obtain 
 the semi-diameter of the well-hole upon the line, g c, as from 5 to ^, 
 deducfing the projection of the rail over the concave side of the 
 concave string, and apply said distance from c to z, fig. 3, at z ; let 
 fall the perpendicular, z x, then obtain the expansion of the well- 
 hole from c to a, fig. 1, and apply said distance from q to a, fig. 3 ; 
 then from the centres z and x describe the semicircles c e and a b, 
 
63 
 
 then draw the hnes a c and 6 e, which give the plan for the rail its 
 conical form (the dotted line around the given plan, fig. 3, shows 
 the projection of the rails over the concave edge of the concave 
 string) ; then divide the perpendicular y z, fig. 3, into four equal 
 parts, as 2 4 6, which lines being the diameter of the frustum at 
 each given point ; then subdivide each of these parts into two equal 
 parts each, as 5 ? w which lines being the semi-diameter of the 
 frustum at each given point, let fall a perpendicular from the base 
 of the frustum « 6 at «/, through the given plan, fig. 4 ; draw the > 
 dotted fine m through said plan parallel to the base, a 6,'of the 
 frustum, fig. 3 ; then obtain the centres stuv for each respective 
 quadrant in the same manner as has been described in fig. 1 ; then 
 take the distances equal to the semi-diameter at the several points as 
 s k for the first quadrant, t n for the second, u w for the third, and 
 V X for the fourth of fig. 3, and apply said distances from siokj to 
 n, u to w, and v to x, fig. 4, also continue skioo; then from the 
 centre s describe the quadrant k n from «, from the centre t describe 
 the quadrant n w, from the centre u describe the quadrant w x, and 
 from the centre v describe the quadrant x o, also the segment o p 
 from the centre s, for the concave side of the rail ; then set off the 
 middle of the given plan, and from the same centres describe the 
 width of the rail upon the convex side of the given plan from g 
 around to j,j to m, and mto n; then obtain the stretch-out of each 
 respective quadrant and segment. A, B, C, D, E, in the same manner 
 as has been described in the preceding plates. 
 
 To draw the concave and convex falling moulds, as Jigs. 5 and 6. 
 
 Draw the base line, g e, then obtain the concave stretch-outs, 
 fig. 4, of the several quadrants, a b of the first, c w of the second, 
 wd of the third, and e o of the fourth, and apply them upon the 
 base, fig. 5, from abed toe; ate erect the perpendicular, e o, and let 
 its height be equal to the conical side of the frustum a c, fig. 3 ; 
 then erect the perpendiculars b c and d, fig. 5, and let their height 
 be equal to « 2 « 4 and a 6, fig. 3 ; then draw the floor line, o/, 
 equal in length to the stretch-out, of, fig. 4, \etfv, fig. 5, be equal to 
 half a riser; then through those points upon the perpendiculars, 
 from the base at a b c d and e, let the curve of the lower edge of the 
 falHng mould be traced ; then set up the height of the newel, as 
 from a to b, fig. 2, and draw / /parallel to o a; then take the distance 
 a I, fig. 3, and apply it from « to /, fig. 5 ; draw / / paraUel to its base, 
 
64 
 
 ah; then set up the width of the mould, which being equal to 
 the depth of the rail, and draw the upper edge parallel to the lower 
 and form the easing at each end of said mould, the joints being 
 formed, the perpendiculars and base lines being drawn for each 
 respective piece, as 1 2, 3 4, 5 6, 7 o and fr, which completes the 
 concave falUng mould. The convex falling mould, fig. 6, is formed 
 from the same corresponding points from the convex stretch-outs of 
 fig. 4, as was fig. 5 from the concave stretch-out, fig. 4 ; also the man- 
 ner of obtaining the curvature for its formation, the manner of ob- 
 taining the casings and locating the joints on the convex mould, are 
 the same as has already been described. 
 
 To form the face moulds, AB CD2indiE, from the parts AB ( 
 D and E, fig. 7, and the manner of obtaining the heights from th'^ 
 concave falUng mould, fig. 5, for the formation of the face-moulds, 
 are the same as has already been described, which will be seen by 
 the corresponding points, as 1 2, 3 4, 5 6, 7 o and f v. It should be 
 observed that the given plans of figs. 4 and 7 are the same, and that 
 the curvature of the concave and convex falling mould, figs. 5 and 6, 
 is caused by the expansion of the well-hole. The angle at o, fig. 2, 
 shows where the bevel may be obtained, to apply upon the part of 
 the string where the tread rests, which allows the tread to lie paral- 
 lel with the base of the given plan. 
 
riah 21 
 
PLATE 22. 
 
 PRACTICAL GEOMETRY. 
 
 Definitions. 
 
 1. A right or straight line is the shortest that can be drawn 
 between two given points, a b, fig. 1. 
 
 2. A curve-line is a hne which is not straight, as a b, fig. 2. 
 
 3. Parallel lines are straight lines that are drawn equidistant from 
 each other, as a b and c d of figs. 3 and 4. 
 
 4. An angle is the space intercepted between two lines intersect- 
 ing each other, whether right or oblique. 
 
 5. Right angles is where one line meets another so as to make 
 the angles on each side equal ; then each angle is called a right 
 angle, as a c d and bed, fig. 5, and the line c d which rests on the 
 base a 6 at c is called a perpendicular. 
 
 6. Oblique angles is where one line meets another, so as not to 
 make the angles equal on each side, as a c d and bed, fig. 6. 
 
 7. An obtuse angle is greater than a right angle, as a c d, fig. 6. 
 
 8. An aeute angle is less than a right angle, as b c d, fig. 6. 
 
 Examples — a b c, fig. 7, is an acute angle, a b c, fig. 8, is a right, 
 angle, and a b c, fig. 9, is an obtuse angle. 
 
 9. A triangle is a space enclosed by three right lines, as a 6 c, 
 fig. 10. 
 
 10. A right angled triangle is that which has one right angle, 
 as a be, fig. 10, a 6 is the base, b c the perpendicular, and a c the 
 hypothenuse. 
 
 11. An aeute angled triangle is a triangle which has all its 
 
 angles acute, as a 6 c, fig. 11. 
 
 9 
 
66 
 
 12. An equilateral triangle is a triangle having all its sides 
 equal, sls ab, b c and c a, fig, 1 1. 
 
 13. A circle is a plain figure formed by one uniform curved line, 
 which is its circumference, as fig. 14. 
 
 14. The diameter of a circle is a right line drawn through the 
 centre, and terminated by the circumference, as a b, fig. 15. 
 
 15. A chord is a right line which joins the two ends of any arc 
 of a circle, sls e f, fig. 14. 
 
 16. A semicircle is one-half of a circle, being divided into two 
 equal parts by its diameter, as a b, fig. 15. 
 
 17. A segment of a circle is that portion cut off" by a chord, as e/- 
 fig. 14. 
 
 18. A quadrant i§ the fourth part of a circle, as a c, fig. 14, con- 
 tained between two radii, forming a right angle at the centre, as b c, 
 and b a. 
 
 19. An arc is any portion of the circumference of a circle, as 
 fig. 14. 
 
 20. A radius is the semi-diameter of a circle, as b a, and b c, fig. 14. 
 
 21. A sector is the portion of a circle formed by two radii, as / a h, 
 or j h i, orj b i to fig. 15. 
 
 Problem 1. 
 
 At a given distance parallel to a straight line, as a b,Jig. 8, to draw a 
 
 straight line, as c d. 
 
 In the given straight line, a b, take any two points, as e f; then with 
 the point of the dividers at e and / describe the arcs g h and i j ; 
 then draw the line c d, cutting the arcs g h and ij, then c d will be 
 parallel \o ab. 
 
 Problem 2. 
 
 ^ To bisect or divide a straight line by a perpendicular, as ah, toe d, jig, 12. 
 
 Take the distance greater than half c d, then with the point of the 
 dividers at c, describe the arc e f g, also at d describe the Bxchij ; 
 then through the bisecting points of the arcs at k and /, draw a b, 
 which will be perpendicular to c d. 
 
67 
 
 Prohlem 3. 
 
 To erect a perpendicular at tfie end of a straight line, as b c, from the 
 , base a b,fig. 13. 
 
 Take any point above the line a b, as e, and with the radius, e b, 
 describe the arc d b f ; draw the Une d e straight through to the arc 
 at/; then from the point 6, draw the hue b c through the bisecting 
 points at/; then b c will be perpendicular to the base a b. 
 
 Problem 4. 
 
 To erect a perpendicular from any given line, as d c from a b, fig. 16. 
 
 From the point d describe the arc e f; then having e and / as 
 centres, describe the arcs e g and, / h; then from the point d draw 
 the Une d c through the bisecting points g h, and d c will be per- 
 pendicular to the base a b. 
 
 To let fall a 'perpendicular from any given line, as d c from 
 a h, fig. 17. Proceed in the same manner as has been de- 
 scribed in fig. 16. 
 
 Problem 6. 
 
 In any given angle, as a b c, or a d c, fig. 18, to describe a curve by 
 
 straight lines. 
 
 Let a b c a dc he the given angle ; then divide a b ox ad 
 into six equal parts, also- 6 c or c, as is shown in the diagram ; 
 then from the point 1, nearest the angle b or d, upon one side draw 
 lines to the farthest on the other side, as 1 to 1, 2 to 2, 3 to 3, and 
 so on until the whole be completed. 
 
 This Problem is of much use in forming the curvature or easing 
 in the angles for the rail and stairs. 
 
 ^ Problem 7. 
 
 To find a right line ivhich is nearest eqml to the semi-circumference of 
 . a circle, as c d the tangent is to the semicircle a b, fig 15. 
 
 Let abhe the diameter, having a 6 as centres, describe the arcs a f 
 and b g, and at the bisecting point of ^ / at e, draw straight lines from 
 the point e through a and b to c and d; then draw the tangent 
 line c d parallel to the diameter or base a b ; then c d will be nearest 
 equal to the semicircle part, ab. To obtain the lengths of the 
 
68 
 
 several segments a h, h i and i h upon the tangent line c d, from the 
 point c draw lines through h and i to the tangent k I ; then c /v, h /, 
 / upon the tangent, will be equal to a h i and i h of the several 
 segments. By this process the student will perceive that any portion 
 of the. circle may be transferred to a straight line. 
 
 Problem 8. 
 
 To form an ellipsis hy memis of segments of circles, as fig. 19. 
 
 Let a 6 be the transverse, and c d the conjugate axis, having d as 
 centre, with d c and d h as radius, describe the quadrant cli ; then 
 from the centre d draw the radii 5 at an acute angle of 69 degrees 
 from d /?, the base ; then divide the conjugate axis c d into five 
 equal parts ; then through the point 1, draw e f parallel to the 
 transverse axis a h ; then form an equilateral triangle, as e /; then 
 having / as centre with f s and / as radii, describe the quadrant 
 s V ; then from the centre / draw / t at an angle of 53 degrees 
 from d h, the base ; then take the distance d 1, and apply said dis- 
 tance from / to i upon the line ft; then having/ as centre, with 
 j t and j X as radii, describe the quadrant t x; then from the 
 centre j draw 7 u at an angle of 30 degrees from dh, the base; 
 then at the point of intersection withy u to the transverse axis, a b, 
 will be the centre n ; then having n as centre, with n u and n 10 as 
 radii, describe the quadrant uw ; then the segments c 5 of the 
 quadrant d c h, and the segments s t oi the quadrant / s v, and the 
 segment t u of the quadrant j t x, and the segment u b of the 
 quadrant n u w, which forms one fourth part of the ellipsis from c to 
 b, the parts b d, d a and a c are formed, and the centres obtained 
 in the same manner as hds been described. 
 
 It should be observed that to form the moulds for the rail of an 
 elhptic stairs it is necessary to obtain the stretch-out of the segments 
 of each respective quadrant, in the manner as has been described 
 ^ in the preceding plates, before the moulds can be executed. 
 
 Note.— Page 62, first line from the bottom, for z x read z y. 
 
 " " third line " " '• " x " " v.