FRANKLIN INSTITUTE LIBRARY 
 
 PHILADELPHIA 
 
 Class .^.a^^. 8 Book -.R^3. Acce8sion.^4r.l3.1 
 
 comprising such works as, from their ravity or value, should not be lent 
 out, all unbound periodicals, and such text books as ought to be found in a 
 library of reference, except when required by Committees of the Institute, 
 or by Members or holders of second class stock, who have obtained the 
 sanction of the Committee. The second class shall include those books 
 intended for circulation. 
 
 Article VI. — The Secretary shall have authority to loan to Members and 
 to holders of second class stock, any work belonging to the second class, 
 subject to the following regulations : 
 
 Section 1. — No individual shall be permitted to have more than two books 
 out at one time, without a written permission, signed by at least two mem- 
 bers of the Library Committee ; nor shall a book be kept out more thdn 
 TWO WEEKS ; but if no one has applied for it, the former borrower may 
 renew the loan. Should any person have applied for it, the latter shall 
 have the preference. 
 
 Section 2. — A fine of ten cents per week shall be exacted for the 
 detention of a book beyond the limited time ; and if a book be not returned 
 within three months, it shall be deemed lost, and the borrower shall, in 
 addition to his fines, forfeit its value. 
 
 Section 3. — Should any book be returned injured, the borrower shall pay 
 for the injury, or replace the book, as the Library Committee may direct ; 
 and if one or more books, belonging to a set or sets, be lost, the borrower 
 shall replace them or make full restitution. 
 
 Article VII. — Any person removing from the Hall, without permission 
 from the proper authorities, any book, newspaper, or other property in 
 charge of the Library Committee, shall be reported to the Committee, who 
 may inflict any fine not exceeding twenty-five dollars. 
 
 Article VIII. — No Member or holder of second class stock, whose 
 annual contribution for the current year shall be unpaid, or who is in 
 arrears for fines, shall be entitled to the privileges of the Library or Read- 
 ing Room. 
 
 Article IX. — If any Member or holder of second class stock, shall refuse 
 or neglect to comply with the foregoing rules, it shall be the duty of the 
 Secretary to report him to the Committee on the Library. 
 
 Article X. — Any Member or holder of second class stock, detected in 
 mutilating the newspapers, pamphlets or books belonging to the Institute, 
 shall be deprived of his right of membership, and the name of the ofi"ender 
 shall be made public. 
 
LESSONS 01 HMD KAILIId 
 
 FOE LEAKNEKS. 
 
 ROBERT RIDDELL, 
 
 AUTHOR or "the caepenter and joiner," "practical geometry," "new elements of hand railing," etc. 
 
 PHILADELPHIA: 
 
 624, 626 & 628 :M:JLI^K:E'r STJBEIET. 
 
 187G. 
 
TH 
 
 567S- 
 
 RS3 
 
 in6 
 
 Entered, according to Act of Congress, in the year 1875, by 
 In the OflSce of the Librarian of Congress, at Washington, D. C. 
 
THE addition of another work, to those which I have already published on Hand Railing, might seem 
 a further demand upon the purchasers of former editions. This, however, is not the object ; nor do 
 I wish it to appear that this new treatise is indispensable, or that the subject shall receive a clearer elucida- 
 tion from this than from former works. It is simply a separate and distinct system of lines, the construc- 
 tive principle differing in every material point from that illustrated in my previous works ; and no similar 
 system has ever yet appeared in print. The practical results of both my systems are precisely the same^ 
 though produced by opposite means. The student, therefore, having an alternative, may adopt the one he 
 best understands. 
 
 The ideas resulting from my experience in Hand Railing have been before the pubUc for many years, and 
 the fact that their principles have been thoroughly tested and generally adopted on both sides of the water 
 is of itself sufficient evidence of their value. 
 
 Regarding the merits of a work now presented for the first time, it is eminently proper that, should they 
 be great or small, their enumeration should be made by those interested in becoming fully conversant with 
 the proper combination of geometrical lines requisite in the construction of Hand Railing. Proficiency in 
 this is justly considered the summit of the joiner's art. 
 
 R. RiDDELL, 
 
 1214 Hancock Street, 
 
 Fhiladelphia. 
 
Plate 1 
 
Plate 1. 
 
 CYLINDERS, EAMPS AND MITRE CAPS. 
 
 We here commence by giving a few of the sim- 
 plest constructions in stair building ; and, in doing so, 
 the learner is supposed to be entirely unacquainted 
 with this branch of art, but at the same time is 
 considered a competent joiner, capable of executing 
 work properly. Then, with no other guide or assist- 
 ant than the drawings and explanations given in 
 this work, he may thoroughly instruct himself in 
 the principles and practice of stair building. But 
 let it be distinctly understood, that accuracy in both 
 drawing and workmanship is essential to complete 
 success and satisfactory results. This will appear 
 evident in the construction of a simple cylinder, as 
 we will now proceed to show. 
 
 Fig. 1. Here the cylinder is formed by three hol- 
 low staves. These are glued and screwed together. 
 The usual way is to prepare a stave long enough 
 to make two or more cylinders with the edges jointed 
 to bevel K. This being cut into'proper lengths, place 
 the square ends of three pieces on the plan, bringing 
 the joints together. Now if the edges be accurately 
 jointed, they will form a perfect semicircle, agreeing 
 with the plan, and you may proceed to glue and screw 
 the joints. Nails should not be used, as they are apt 
 to make a burr or roughness on the edges and to 
 break the bond of glue. 
 
 It may however be stated that this plan of making 
 cylinders is not adopted by workmen generally. Some 
 prefer bending a veneer over a form or drum of the 
 required size, and then gluing a number of staves 
 on the back. This is a very neat method, as the 
 grain of wood may be made to run in the same di- 
 rection as in the string, which should always be done 
 when the staircase is of walnut or other hard wood. 
 But, if the work is to be painted, a cylinder made as 
 first stated answers every purpose and saves the 
 trouble and expense of making forms or drums. 
 
 Fig. 2. The proper effect of curves in a wreath 
 depends entirely on the position of risers, in either 
 platform or other stairs ; a correct and simple method 
 
 of determining this is here shown. The diameter of 
 cylinder is sixteen inches. After drawing centre 
 line of rail, set the compasses to half the width of a 
 step and space off the balusters ; three of which stand 
 on the platform as shown. Balusters A and B give a 
 direction to draw risers 2 and 3 ; from these set off 
 width of steps, which gives the position of risers land- 
 ing on and starting from the platform as shown. 
 
 Fig. 3. In this case the diameter of cylinder is 
 twelve inches : two balusters stand on the platform, and 
 position of risers regulated as in Fig. 2. But if the 
 cylinder is six or eight inches, then face of risers 
 landing on and starting from the platform should 
 stand on spring line or diameter of cylinder. 
 
 By adopting these simple directions, time will be 
 saved and the work will present a more finished ap- 
 pearance than otherwise. 
 
 Fig. 4 shows the elevation of stairs at the start- 
 ing, giving centre of newel and position of mitre cap. 
 In forming a mitre for the cap, let L P equal half the 
 width of rail, in preference to what is termed a true 
 mitre. To cut this mitre in the cap at once and cor- 
 rectly, proceed as follows : Draw the required size of 
 cap, and width of rail as shown : then from centre N, 
 draw a line parallel with one side of mitre, which de- 
 termines a distance as 2 A. Have a piece of plank, 
 shown in Fig. 5; on it draw a line as A B, and set of 
 POINTS 2 and C on each side of it, making the dis- 
 tance between, the same as between 2 A in Fig. 4. 
 From 2 C, draw lines parallel with A B. Square 
 over the end of plank, and with a panel saw cut 
 through these lines, keeping thickness of saw towards 
 A. This done fasten with a screw the centre of cap 
 over line A B, enter the saw and cut one side of mitre 
 down to 3. Set the compasses to width of rail and 
 mark point C as shown. Revolve the cap until point 
 C is opposite cut C in the block; then enter the saw 
 and complete the mitre. Should there be a hole 
 through the cap, let it revolve on a pin set in the 
 block. 
 
Plate 2. 
 
 THE QUAETEE OIEOLE WEEATH LANDING ON A LEVEL ELOOE. 
 
 Here the first lesson in hand railing begins, by showing 
 the construction of a simple wreath for stairs, having square 
 steps and a quarter circle at landing ; the wreath to form its 
 own ramp and connect with the level rail. The under side 
 of rail when in position should stand above the floor to suit 
 a long baluster. But to understand this we must commence hy 
 making an elevation of a square step, two risers and floor line, 
 as in Fig. 1. The points H H show centre of short balusters. 
 Let under side of rake rail rest on these. Make under side of 
 level rail stand half a riser above the floor line. Set off thick- 
 ness of rake and level rail intersecting at B. From this point 
 set off width of rail as D, then through half this width draw 
 a perpendicular, shown by dotted line, cutting through K, 
 and giving centre of level rail and balusters to the right of 
 Fig. 2. The position of riser landing, being already fixed by 
 the elevation, all that is necessary to complete the ground plan 
 is the quarter circle. Its radius may be assumed as L N. 
 Draw a line from N, parallel with L K, cutting rake rail at 
 A. We have now a proper direction to cut the carriage, and 
 exact points for drawing a mould for the wreath, everything 
 being obtained from the elevation. 
 
 As all inclined curves, which stand over and agree with a 
 circle are elliptical, it is evident that the curve in the rake or 
 inclined rail, to stand directly over and follow the quarter 
 circle of the stairs, must also be elliptical ; therefore this will 
 be the form of a wreath. The operation of describing an 
 ellipse, by means of a string and two pins, is familiar to most 
 joiners. 
 
 Fig. 3. This shows how to prepare a mould for giving the 
 wreath its proper form. First draw a base line, and on it set 
 off points A B C, to correspond with A B C on elevation. 
 Draw from point A a line at right angles with base. Set off 
 on this line, from A, points E D, having them correspond 
 with E B D on elevation. Take A B for radius or half 
 diameter, and with E for centre, intersect the base line ; which 
 gives points 2 and 3. Fix a pin in each of these. Tie a 
 string to pin 3, bring it around pin 2 and with a pencil 
 stretch it, touching the point E ; then strike the curve from 
 E to B. This done take A C for radius, and with D for 
 centre, intersect the base line, giving points for pins as shown; 
 from which sweep the outside curve from D to C. Let three 
 or four inches of straight wood remain on the mould below 
 B C, and any length desired on the left of E D. 
 
 The plank for a wreath should be as thick as the rail is 
 wide. Lay the mould on and mark by it. Cut the plank 
 square through. Face the top surface out of wind, and work 
 the edge D square with the surface. Run a gauge line to 
 half thickness of plank on the edge, and square' over from D, 
 
 cutting the line at A, as shown in the figure below. Now 
 with the pitch board or bevel, mark the spring line DAD. 
 Work both sides of the piece by the mould. The application 
 of mould is to keep point D opposite spring line on edge of 
 plank. 
 
 This gives lines for taking off the slabs. 
 
 The cylinder part of wreath being formed, gauge the 
 straight part on the left of E D, to thickness of rail. Cut off 
 the under slab to near the spring line, and the top slab on the 
 "wide end, square with joint. This slab is shown in elevation 
 as B C D. It will now be seen that the concave corner of 
 the piece gives a perfect direction for taking off the remaining 
 slab; thus forming the best possible ramp on top side of 
 wreath landing ; every part is parallel and the same width as 
 .rail. Let the learner remember, however, that to obtain these 
 results, he must be exact in both working and drawing. Now 
 complete the wreath by gauging to a thickness. 
 
 Fig. 4 shows the elevation of square steps and risers, and 
 gives the proper manner of taking the lengths for a rail. It 
 is very natural to suppose, that the height of riser and width 
 of step gives pitch of stairs, and it undoubtedly would, pro- 
 vided there were no errors. But it is quite possible that a 
 lauding might be fixed a little high or low, or perhaps a 
 wrong cut made in the carriage at the starting, either of which 
 mistakes would give a very different pitch. There was a 
 time when such errors would have made no difference ; then 
 joints were made and fitted on the stairs, wreaths rudely cut 
 and worked over cylinders, and little or nothing known to 
 guide or direct the workman, and yet the hand rails of olden 
 time present a finish and excellence truly wonderful, consid- 
 ering the disadvantages under which the workmen of that 
 time labored. But now hand railing is reduced to an exact 
 science, everything should be prepared on the bench, and 
 made ready for fixing. This, however, cannot be done with- 
 out having correct dimensions, the pitch of stairs being the 
 most important. To obtain this, lay a straight edge on the 
 nosings and mark a plumb line on its face as shown. This 
 gives the pitch, regardless as to how the work has been 
 put up. 
 
 The lengths of straight rails is obtained by laying a rod on 
 the nosings, and marking a plumb line for face of second 
 riser as R, and spring line of cylinder as P. The necessity 
 for marking the spring line on every piece of wreath having 
 any portion of straight wood is this : The line is first marked 
 on the edge and continued across the under side, so that on 
 fixing the rail and having it in a true position over the 
 plan, the spring line of both cylinder and wreath should stand 
 exactly opposite or be on a plumb line. 
 
Plate 2 
 
 Scale :1H. Inches, 
 
Plate 3. 
 
 THE WEEATH TOE QUAETER LANDING STAIES. 
 
 In laying down a ground plan for stairs of this description 
 the proper way to proceed is to place the risers in such a posi- 
 tion at the cylinder as will cause the surface of plank for 
 wreath to be thrown on the same inclination as that made by a 
 square step and riser ; or, in other words, the straight portion 
 of wreath and straight rail should have same pitch. 
 
 Fig. 1 shows this arrangement. Draw two sides of a square 
 as E D B, the continuation of these sides being the centre 
 line of straight rail. Set off on each side from D half the 
 width of a step. From these points draw risers landing and 
 starting ; spacing from these the width of steps. The radius 
 for centre line of rail may be assumed, as A B, or drawn so 
 that nosing on step starting will clear the baluster on the 
 landing. The ground plan being now complete, we will pro- 
 ceed to find a mould for wreath. To do this lay the pitch- 
 board on line D B and draw from D, giving pitch D H ; 
 extend line A B cutting at H. This gives B H as half the 
 height of wreath. Now draw from D through A, and from 
 E parallel with D A. Draw a line square with D A from A 
 to F. Make A N equal B H, and draw from F through N. 
 This line gives pitch of plauk. In order to strike the curves 
 on mould with a string, as in preceding example, we must 
 find half the long diameter of semi-ellipse, which is done as 
 follows : Make F K equal A B. Square up from K, cutting 
 at L. Draw through L parallel with F K, and set half width 
 of rail on each side of L, cutting the pitch line at 3 and 4. 
 This gives F 3 as half the long diameter for inside curve of 
 mould, and F 4 for outside curve. To find a bevel for joints 
 of wreath : From B as centre and a radius touching D II strike 
 an arc cutting at 2, join 2 A. This gives bevel X for the 
 joints. It will also give half the width of mould on its 
 straight part. For example, set half width of rail below B 2, 
 cutting the bevel at point 3. This gives 3, 2 for half width 
 of mould. 
 
 Fig. 2 shows the manner of finding the mould. Have a 
 piece of board sufficiently wide. Square over any line as A 
 B D. Make A D equal A D on plan. Make the distance 
 A F on right and left equal F N Fig. 1, square over the 
 lines, make F E right and left equal F E on plan, draw 
 from D through E, E. These are tangents on the mould, and 
 to be correct must equal the pitch D H on plan, make A B 
 equal A B on plan. Set half width of rail on each side of B. 
 Make A 3 4 equal F 3, 4, Fig. 1. Find points for pins by 
 the method already given, and strike the curves with a string, 
 dress to the lines and the mould is complete; Cut the plank 
 
 square through, and face one side out of wind. Lay on the 
 mould, and mark the joints, and at the same time mark on 
 surface of plank the direction of lines E K. The object of 
 having these lines is to prove the correctness of a joint at 
 once, by the application of a square which should be per- 
 fectly true and the blade nine or twelve inches long. 
 
 Place the stock against the joint, and the edge of the blade 
 along the surface of the stuff. If the joint be cut square the 
 edge and surface will agree. To prove the other direction, 
 place the stock against the joint, and lay the blade flat on 
 the surface ; when, if the joint be correct, the edge will agree 
 with line E K. By this manner of applying the square, 
 the slightest inaccuracy will be readily noticed, because 
 we have a surface, or line the entire length of the blade to 
 prove by. 
 
 The joints being made, apply bevel X, and mark square 
 section of rail on each joint, remembering to reverse the bevel, 
 as shown on the right and left of Fig. 2. Take off the out- 
 side slab on the left, indicated by the dotted line. Work it 
 by bevel X and square with joint. When this is done, apply 
 a square to the beveled edge, and from half the thickness of 
 stuff, as 2, square over 2, 3. Continue this line along the 
 beveled edge, and square with joint. Set off on this line the 
 distance between E, K on mould ; and through the point thus 
 given, draw spring line by the pitch board. The slab on the 
 right is taken off, and spring line on edge, marked in the 
 same manner. 
 
 Lay on the mould, keeping its outer edge fair with the 
 beveled edge of stuff on the left, and its inner edge fair with 
 dotted line on the right. Mark the surface of the stuff by 
 the mould, turn the piece over and mark the under side in 
 the same manner. This gives a guide for taking off the slabs, 
 and, in doing so, hold the plane in the direction indicated by 
 the spring line on the beveled edge. 
 
 The wreath having now its cylinder form, take off the 
 upper and under slabs, as directed by the square section of 
 rail on each joint. This is best done by sawing down to near 
 the spring line ; but to have a still further direction, square 
 over the line through B on both edges. Mark half thickness 
 of stuff, and on each side set off half thickness of rail ; we 
 have now clear directions for cutting off the remaining slabs, 
 which completes the wreath. 
 
 The method of applying the mould as explained in this 
 case should be adopted for all others, therefore need not be 
 repeated. 
 
Plate ^. 
 
 THE SEMI-ELLIPSE AND THEOEY OP JOINT BEVELS ILLUSTEATED. 
 
 The curves on all face moulds for wreaths being some por- 
 tion of a semi-ellipse, it is necessary for the stair builder to 
 become familiar with the manner of describing this figure. 
 While, perhaps, few workmen are entirely ignorant of this 
 beautiful problem, for its application is common in many 
 trades, there may be some readers of this work who are unac- 
 quainted with it, and to those we will briefly explain the 
 simplest and most practical manner by which it may be done. 
 The means are always at hand, being simply a string and 
 two pins. 
 
 Fig. 1. The serai-ellipse has two diameters, always at right 
 angles with each other. These are termed major and minor 
 axis, meaning long and short diameters. In this Fig. the line 
 C B represents the major axis, and AD the minor axis. We 
 desire now to describe a semi-ellipse of a given length and 
 rise. Let C B represent the given length, and A D the given 
 rise. Take one-half the major axis or long diameter (which 
 is the term we shall use hereafter), as A B or A C, for radius ; 
 and from D as centre, intersect the long diameter on the right 
 and left. This gives points H and K. Fix a pin in each of 
 these, tie a string to pin K and bring it around pin H. With 
 a pencil stretch the string until its point touches D. Strike 
 the curve from D to B, and returning to D, complete the curve 
 to C. To arrive at correct results, the hand should be steady, 
 the string fine and not allowed to stretch. Should the curve 
 appear defective from any cause, the error may be detected in 
 tlie following manner : From the end of a rod mark the short 
 diameter A D, and half the long diameter A B, by moving 
 the rod and constantly keeping these points, S and T, on the 
 long and short diameters, the end of the rod R will give any 
 number of points, through which the curve, to be correct, 
 must pass. 
 
 The most important point, and the one upon which the 
 claim of stair building as an exact science rests, is the principle 
 by which correct bevels for joints are determined from the 
 plan. This principle is not only reliable, but applicable to 
 all forms of wreaths, and when once understood, its simpli- 
 city and correctness will be self-evident. We shall now 
 illustrate the theory of the joint bevels, starting with the 
 proposition, that all inclined rails, to follow the directions of, 
 and agree with, a circular plan directly under them, must be 
 elliptical in form. This is proved by cutting a semicircle on 
 the rake, showing the section thus made to be a perfect semi- 
 ellipso. 
 
 Fig. 2. The two semicircles in this Fig. represent the base 
 of a cylinder; the line through A to the right and left its 
 diameter, and Y U its thickness. Assume A B C as the 
 rake cut, through the cylinder, and transfer points B C from 
 it to line of diameter. Then find points for pins by the 
 method already given and sweep the two semi-elliptic curves 
 as shown. This semi-ellipse is an exact section of the cylin- 
 der, and when raised to its proper position will stand directly 
 over the semicircle. Bevel X in the angle at B gives the 
 plumb cut at each end. 
 
 Let us now proceed to find tangents to a portion of the 
 semicircle. From A as centre strike a semicircle, as the 
 dotted hue passing between Y U. On this fix upon any points, 
 as 4 and 6, and draw lines from these parallel with diameter' 
 to the centre of semi-ellipse on the right and left. This gives 
 
 5 and 7. At right angles with 4 A, draw a line from 4, 
 cutting the perpendicular at T ; also draw from 6, square with 
 
 6 A, cutting at S. _ From points S and T draw lines through 
 5 and 7. These lines are tangent to elliptic curves at these 
 points, and give the direction for any straight wood that may 
 be necessary on the right and left. Draw a line from A, 
 through 5 and 7, and we will proceed to find the bevel for 
 joints on straight wood. Before going any farther, it may be 
 again stated that the plank from which a wreath is to be cut 
 should be as thick as the rail is wide. For example, if the 
 rail is four inches wide the wreath should be cut from a plank 
 four inches thick. 
 
 Fig. 2. Let Y U represent the thickness of cylinder, and 
 also the width of rail. We therefore take that for the thickness 
 of plank from which to cut the wreath. This is shown on 
 the right and left, marked edge of plank. From the 
 points where the tangents intersect the diameter, 3 on the left 
 and C on the right, square up to half the thickness, shown on 
 the edge 2, 2; then taking bevel X from the angle B, apply 
 it as shown. This gives J and N on the lower edge of plank. 
 From these points square down to the diameter, shown at F 
 on the left and L on the right. Then from F and L draw 
 lines along the straight wood and parallel with tangents, 
 cutting the lower edge of plank at P and H. Half the thick- 
 ness of plank is marked at these places as K and R. By draw- 
 ing a line from P through K, and from H through R, we 
 obtain bevels N and W for the joints of straight wood. Al- 
 though^ these bevels differ from X, yet bevel X M'as the key 
 by which they were found, and which also suggested the 
 diagrams at Fig. 3 and 4, giving the same results in a more 
 direct and practical manner, and proving them to be correct. 
 
 Fig. 3. Draw a line parallel with diameter in Fig. 2. Fix 
 a point, say B. Draw from it line B K, parallel with tan- 
 gent 3 K on the left in Fig 2. Turning to the augle in 
 which is shown bevel X, take point B for a centre and with 
 any radius draw dotted arc O O. With the same radius, and 
 with B in Fig. 3 for centre, draw arc O O ; make its length 
 equal arc in Fig. 2, and from the lower point O draw a 
 line through B. Then from any point, as V, square up a 
 line, giving D. Xow with V for centre, strike an arc, its 
 radius toucliing B K and cutting at E. From D draw 
 through E, and we have bevel N, corresponding exactly with 
 bevel X Fig. 2, and proving it correct. 
 
 Fig. 4. Draw line B K parallel with tangent 7 R in Fig. 
 2, and proceeding as in Fig. 3, and we have" bevel W; which 
 is equal to bevel W on the right. These problems have 
 never been given before, and are well worth the study of the 
 learner, as a thorough familiarity with them will materially 
 assist him in mastering the principles of hand railing. 
 
Plate 4 
 
Plate 5 
 
Plate B. 
 
 QUAETEE CIECLE WEEATH IN TWO PIECES. 
 
 Figure 1 shows ground-plan of stairs having a 
 quarter circle and quarter landing. Here the face 
 of the risers stand on the spring line, and the wreath 
 will be in two pieces. This must always be the case 
 when the arrangement of risers differs from that given 
 in Plate 3. It is, however, better to have a wreath 
 in two pieces than in one, as the grain of wood in the 
 straight part can then run with the straight rail. 
 In this form of wreath the practical value of joint 
 bevels will be demonstrated. 
 
 Divide the quarter circle which represents the 
 centre of rail in two equal parts by the point A. 
 Draw the tangents through A B and B C ; A B being 
 at right angles with a line drawn from A to the centre 
 3, and B C a continuation of centre line of straight 
 rail. 
 
 We must now make an elevation of square step, 
 riser and part of the landing as shown at Fig. 2. Set 
 off the points ABC to correspond with ABC in 
 Fig. 1 ; draw a line as E K for half height of a riser 
 above the landing and square up from A B C to cut 
 this line. Now draw pitch of stairs, cutting through 
 A and J and giving the distance E 2 ; transfer this to 
 plan, making extended tangent A 2 equal it. Draw 
 from 2 through C. This is the directing line for 
 pitch of plank. Draw from A and B, parallel with 
 
 2 C, and from centre 3, square with 2 C. Make N D 
 equal A E Fig. 2. Join D H. Draw from 3 parallel 
 with D H. This line is pitch of plank, and always 
 the major axis of a semi-ellipse; the half diameter of 
 which is determined as follows : Make 3 F equal 3 C. 
 Square up from F cutting the pitch at 0, and through 
 this point draw parallel with 3 N. Now determine 
 width of rail and set off half of it on each side of 0, 
 cutting the pitch as shown. 
 
 To draw the mould, lay the edge of a board on the 
 pitch and mark the points 3, 4, 5, 6 ; square these 
 over as shown at Fig. 3. Make 4 A, T J and 6 K cor- 
 respond with H A, T B and N C, Fig. 1. Draw from 
 J through A and K. These are tangents, and to be 
 correct must equal the pitches A, J, K, in Fig. 2. 
 Draw the joint through K square with J K. Make 
 
 3 C equal radius 3 C on plan, and set off half the 
 
 width of rail on each side of C. Now find points for 
 pins and sweep the curves with a string. This done, 
 draw from 3 through A, giving points for drawing 
 straight wood parallel with J L. This completes the 
 mould. 
 
 Fig. 4. To find bevels for joints : Draw any line as 
 P V parallel with 3, 6, Fig. 3. From any point as 
 P, draw P E and P H parallel with togents J K and 
 J A on mould ; transfer pitch of plank from Fig. 1 as 
 follows : From 6 as a centre and with any radius, draw 
 the dotted arc. With same radius, and P centre, draw 
 an arc of equal length, and from its lower corner draw 
 the pitch through P ; square over from any point as 
 E. cutting the pitch at S ; from II as centre, and a 
 radius touching P E, strike an arc cutting at V, join 
 V S. This gives bevel W for centre joint of wreath. 
 Again, take R as centre, and a radius touching P H, 
 strike an arc cutting at N, join N S. This gives 
 bevel X for joint on straight wood. The application 
 of bevels to joints is shown by the square sections of 
 plank. 
 
 Before commencing to work the wreath, see that 
 your tools are in proper order, especially the square 
 and bevels. Mark the plank full by the mould, and 
 cut square through ; face one side of each piece out 
 of wind ; lay on the mould ; mark the joints and direc- 
 tion of tangents on the surface of stuff. The reason 
 for this has been before explained. Both pieces being 
 worked to the cylinder form, take them to a thickness 
 by cutting the slabs on straight part to near the 
 spring line. Complete the piece by taking off the 
 slabs on circular part, which is easily done, as the 
 square section of rail marked on centre joint gives a 
 clear direction. It is best to leave the stuff full at all 
 joints made in the circular part of a wreath, for the 
 purpose of removing any defect that might appear 
 when the joints are bolted together. Now clean off 
 to width and thickness, and in putting together, be 
 particular to keep the lines made by bevel W on each 
 joint, exactly opposite. This throws the straight 
 wood on each piece of wreath to their proper 
 pitches. 
 
Plate 6. 
 
 THE WREATH OVER SIX WI 
 
 Figure 1 shows the ground-plan of stairs, having 
 winders and square steps. The semicircle shows 
 centre line of rail. This may be divided into any 
 number of winders, making their narrow ends equal 
 to half a square step. The pitches of this wreath 
 can be arranged so as to be equal. By this means the 
 construction becomes quite simple, as a mould can be 
 obtained from one of the pitches and the diagonal of a 
 square, the sides of which are equal to the radius 0 P. 
 This plan shows nine risers, but in making elevation 
 omit one, because in going up the stairs from square 
 step on the left, we make eight steps before arriving to 
 the top of last riser on the right. 
 
 Fig. 2, on the right, shows the height of eight 
 risers. The dotted perpendicular B B represents 
 spring line, the arrows show position of risers stand- 
 ing beyond it, the same as on plan. Let the spaces R 
 J and J B equal the radius 0 P on plan. Draw 
 through J the dotted perpendicular A A, from 4 on 
 the right square over a line, giving point B on the 
 left. Now draw pitches through square steps. This 
 shows that there must be a ramp above and below to 
 connect with wreath by gentle curves. To form 
 the ramp, draw a line from B on the left, cutting the 
 lower pitch at P. Make the upper pitch equal. The 
 whole height of wreath on spring line is shown 
 as B B. 
 
 From these pitches, a mould for the wreath is read- 
 ily obtained as follows. Form a square as shown ; its 
 sides being equal to the radius 0 P, Fig. 1. Draw 
 from R parallel with diagonal line D C. Draw 
 through C square with D C. Make height J 2 equal 
 height J A on the left. Draw from C through 2. 
 This line is pitch of plank, and on it we find half 
 
 lERS. CYLINDER 15 INCHES. 
 
 long diameter of semi-ellipse. This must always be 
 given when the curves on a mould are to be struck 
 by a string or trammel. Proceed by making C S 
 equal C R. Square up from S, cutting the pitch at V. 
 Through this point draw parallel with S C. Set off 
 half width of rail on each side of V, cutting the pitch 
 at points 2 and 3. This gives C 2 as half diameter 
 for inside curve on mould and C 3 for outside curve. 
 
 Fig. 3. From the edge of a board square over any 
 line as C A. Make C A equal the diagonal of square 
 D C, Fig. 2. Take for radius any of the pitches on 
 elevation, say A B, and from C and A as centres mark 
 intersections on right and left, giving points B B, 
 through which draw from A. These are tangents on 
 the mould. Make the joints square with them. 
 Draw from C through B on the right. Make C R 
 equal 0 P on the plan. Set half width of rail on 
 each side of R. Let C, 2, 3, equal corresponding let- 
 ters on pitch at Fig. 2. Now find points for pins and 
 sweep curves on mould with a string. 
 
 The pitches being equal, the same bevel will an- 
 swer for each joint. To obtain this bevel, find pitch 
 of plank as shown above the square. Take 2 for 
 centre and with any radius draw dotted arc. With 
 same radius return to Fig. 3 and take N for centre, 
 draw dotted arc to equal the first and from its lower 
 point E draw through N. From any point as H 
 square up a line cutting at F. Take H for centre 
 and strike an arc touching extended tangent A B 
 and cutting at K. Join K F. This gives bevel X 
 for joints on wreath. Its application is shown by the 
 square sections. 
 
 Cut the stuff square through and work the wreath 
 by the method already explained. 
 
riate 6 
 
Scale i^'^ Inches' . 
 
Plate 7. 
 
 THE WEEATH OVEE SIX WINDEES LAITDDTG Olf A LEVEL FLOOE. 
 
 Figure 1. In order to show the certainty and prac- 
 tical application of this system of hand-railing to all 
 forms of wreath, we have here given the ground-plan 
 of stairs, where the riser landing is purposely placed 
 in a most awkward position. In fact the whole 
 arrangement is badly planned, and yet we shall find 
 no difficulty in forming a wreath by the principles 
 which are laid down. 
 
 Fig. 2 shows the elevation. Here we draw a square 
 step and set off the number of risers above it, which 
 in this case is seven. The square steps and two 
 winders are shown below, and the floor line and two 
 winders above, standing in the same position as those 
 on the plan. The under side of rail at landing should 
 stand half a riser above the floor line, and the under 
 side of rail at step S should rest on the centre of short 
 balusters O O. Draw half thickness of, rake rail 
 parallel with under side ; square over the dotted line 
 from 4, cutting perpendicular on the right; the point 
 thus given represents position of riser 4 on plan. 
 But in order to have easy ramps and at the same 
 time make the rail a proper height at the landing, we 
 must raise the wreath at its centre joint above riser 
 4, say to point E. To determine the lower ramp and 
 pitch, assume any point in half thickness of rake rail 
 as J ; draw from J to E ; this is lower pitch : make 
 upper one equal to it as shown. The whole height 
 of wreath is now given on spring line as L L. 
 
 To find the mould for wreath. From D square 
 over a line cutting through H ; form the square as 
 shown, its sides being equal to radius of plan ; draw 
 a diagonal from D through A ; draw from E parallel 
 with D E; again draw through A square with D A; 
 make height A C equal height H E ; draw from B 
 through C. This is pitch of plank and long diameter 
 of semi-ellipse. To find half its diameter, make B P 
 equal one side of the square ; draw from P, cutting the 
 pitch at 0 ; draw through 0 parallel with BP; set 
 half width of rail on each side of 0, cutting the pitch 
 
 at points 2 and 3. This gives B 2 as half long 
 diameter for inside curve on mould, and B 3 for out- 
 side curve. 
 
 To find a bevel for joints. Take H for centre, and 
 with a radius touching D E, strike an arc cutting at 
 K, join K A and we have bevel X. This will apply 
 to all the joints because the pitches of wreath are 
 equal. 
 
 Fig. 4. Having obtained the points necessary, we 
 will proceed to form the mould. Upon a piece of 
 board, draw a line for the major axis. Set off" on it 
 from each side of A the distance B C Fig. 3, giving 
 B B ; square over these points ; make A D and B E 
 equal corresponding letters. Fig. 3 ; draw from D 
 through E E. These are tangents, and to be correct 
 must equal pitches on elevation. *Make A H equal 
 radius on plan, and set off" on each side of H, half 
 width of rail ; find points for pins and sweep the 
 curves ; mark the joints square with tangents. The 
 square sections show the proper application of bevels. 
 The straight wood E N of mould should equal N L 
 on elevation where upper and lower joints of wreath 
 connect with the ramps. 
 
 It may also be stated that from the bevel, half the 
 width of mould across the joint is readily found, as 
 follows : Set off half width of rail below H, Fig. 3 ; 
 draw through this parallel with H K, cutting bevel 
 at R. This gives K R as half the width of joints E N, 
 Fig. 4. The importance of these small matters will 
 be appreciated when we remember that the rail is to 
 be jointed and finished on the bench ready for fixing. 
 
 *NoTE. — Tlie radius of semi-circle on ground-plan will 
 always give centre of mould on short diameter ; as one of the 
 properties of an ellipse is, that its minor axis never changes. 
 This is made plain by the following simple experiment: 
 When a tumbler half filled with water stands level on its 
 bottom, the surface of the water presents a circle, its boundary 
 the glass. Incline the tumbler to any angle, and the water 
 immediately assumes the form of an ellipse, but still retains 
 the diameter of the tumbler as its minor axis. 
 
Plate 8. 
 
 PLATPORM STAIRS. 
 
 Figure 1 sliows a ground-plan for stairs, in which 
 convenience, simplicity and safety are combined, and 
 which should always be adopted where there is suf- 
 ficient space. As before stated, the best result de- 
 pends entirely upon the position of the risers in the 
 cylinder ; if these be placed right, the strings, wreaths, 
 and everything throughout will present a finished and 
 uniform appearance attainable in no other way. 
 Hence the necessity of starting right and keeping the 
 desired end in view from the very commencement. 
 The first thing needed is the height of risers and 
 width of step, accurately spaced off on a rod ; the 
 next a ground-plan to work by. The size of cylinder 
 is determined by the length of step and space allowed 
 for stairs. If this should be ample, the larger the 
 cylinder the better. The semicircle in this Fig. 
 shows centre line of rail, its diameter twelve inches. 
 Two balusters stand on the platform, one of them 
 opposite half diameter. Set off the balusters, making 
 the spaces between their centres equal to half a square 
 step. This at once determines the position of risers 
 A and B. The same rule applies to cylinders vary- 
 ing from ten to fifteen inches. Proceed now to set 
 off width of step on right and left, giving position of 
 risers landing on, and starting from the platform. 
 Observe that the face of riser starting should be 
 curved. This may be worked solid, the thickness of 
 riser being sufficient. 
 
 Fig. 2 shows the elevation. From this we obtain 
 the joint bevels and the points to be used in forming 
 the mould. In this case the height is three risers 
 counting from A and including B ; draw the elevation 
 of two square steps and risers A and B, placing them 
 in the same position from the spring line as on the 
 plan, indicated by arrows in both places; draw upper 
 and lower pitch through corners of square steps, cut- 
 ting spring line and giving points E and F ; divide 
 the distance between these points equally as H; square 
 over frora^ H, cutting lower pitch at L ; draw from E 
 parallel with HL; form the square as shown, its 
 sides are equal to radius of plan ; square down from L, 
 cutting at V. This is the directing point in finding 
 
 pitch of plank. Now draw from V through TJ and 
 parallel with this line, from K and E ; draw through 
 Y square with V U ; make 6, 5 equal height E H on 
 elevation; join 5, 2. This is pitch of plank and 
 major axis. In order to show the points for forming 
 the mould distinctly, the long diameter is raised above 
 the pitch and parallel with it. Make Y R equal Y U ; 
 square up from R, cutting at 0; draw through 0 
 parallel with R 2, and set off half width of rail on 
 each side of 0, giving 3, 8, and 3, 9, as half long 
 diameter of elliptic curves on mould. 
 
 To find the bevels, take P for centre, and a radius 
 touching E L ; strike an arc cutting at N ; join N H. 
 This gives bevel W for joint on straight part of 
 wreath. Now draw from H through T; and with K 
 for centre, and a radius touching H T, strike an arc 
 cutting at J ; join J E. This gives bevel X for centre 
 joint. 
 
 Lay the edge of a board on the pitch and mark 
 points 2, 3, 4, 5 ; square these over as shown in Fig. 
 3 ; make 2 E, 4 T, and 5 H equal 2 E, Z K, and 6 U, 
 Fig. 2 ; draw from T through E and II. These are 
 tangents on mould, and to be correct must equal 
 pitches E T H on elevation. Draw from 3 through 
 E; find centre of mould 0 from radius, and set off 
 half width of rail on each side of it ; find points for 
 pins and strike the curves ; mark the joint H square 
 with tangent. The termination of curves on line 
 through E gives the width for straight part of mould. 
 This may be obtained with more exactness from 
 bevel W by the method already explained. 
 
 The stuff being cut, joints made, and bevels applied 
 as shown, take off the slab on upper side, indicated 
 by dotted line ; work by bevel W and apply a square 
 to this edge and across the joint; square a line from 
 2, continuing it along the beveled edge ; set off on 
 this the distance E F, and draw the spring line by 
 the pitch board. This might be called the plumb 
 line, as it really is when in position over the plan. 
 When putting the wreath together, see that lines 
 made by bevel X on the joint of each piece come 
 exactly opposite. 
 
Plate 8 
 
Plate 9 
 
Plate 9. 
 
 PLATFOEM STATES CONVEETED 
 
 It sometimes happens tliat, either through the 
 carelessness or ignorance of a builder, stairs are put 
 hp and finished without regard to convenience or 
 L^onifort. When this is the case, it is always desir- 
 able to make an improvement, and that this may 
 sometimes be done without much trouble is shown in 
 the accompanying plate. 
 
 Fig. 1 is the plan of stairs, supposed to be finished; 
 with rail, balusters and everything complete. The 
 platform is one step below tnc floor of a room, entered 
 from it by a door on the right, which is very objec- 
 tionable, for, in leaving the room, it is necessary to 
 take one step down before ascending the next flight. 
 This fiiult, however, may he easily remedied and 
 without deranging anything, by simply throwing a 
 riser across the landing, and raising one half of it to 
 the height of the floor; converting it into two quar- 
 ter platforms. The one on the right should be wider 
 by a step than that on the left. This will throw 
 riser starting beyond the spring line and make riser 
 landing stand in the cylinder. A reference to the 
 plan and cut of carriage will clearly explain this. 
 
 Any one conversant with the preceding plates 
 would have no difticulty in constructing a wreath for 
 this plan; but, in order to make clear one or two 
 points that will assist in comprehending the plates 
 tliat follow, a brief explanation is perhaps necessary. 
 The number of risers, coimtiiig from riser A to riser 
 B, is four. 
 
 Fig. 2 shows the elevation of two steps, and the 
 height of four risers. The dotted perpendicular 
 through E represents the spring line, and from it 
 project risers A and B, the came as those on plan. 
 Draw the upper pitch cutting at J, and the lower 
 pitch cutting through H and F. This sho^vs the 
 whole height of wreath on spring-line as II J. From 
 half this height, as E, square over a line cutting the 
 lower pitch at K; join E F. Form the square F S R V, 
 its sides to equal the radius on the plan. Square 
 down from K, cutting at L; draw from F through L; 
 draw from S V R parallels with F L; again, draw 
 through R square with F L ; make T D equal height I 
 
 INTO TWO QUAETEE LANDINGS. 
 
 H E; draw from A through D. This is pitch of plank. 
 Find half the long diameter of a semi-ellipse, by 
 making R P equal . RS; square up from P, cutting 
 the pitch in 0, and through it, draw parallel with 
 R P; set half width of rail on each side of 0, giving 
 the points, as C 2 for inside curve on mould, and 
 C 3 for outside curve. 
 
 To find the mould. Lay the edge of a board on the 
 pitch, and mark the points A, B, C. D . these square 
 over, as shown at Fig. 3. Make A E, B F and D H 
 equal A V, N F and T S,. Fig. 2; draw fr om F through 
 H and E. These are tangents on the mould, and to 
 be correct, must equal the pitches H F E, Fig. 2. 
 Make C P equal radius on plan ; set half width of 
 rail on each side of P, draw from C through H. Now 
 find points for pins and strike the curves with a 
 string; add any length of straight wood desired, par- 
 allel with F H ; make the joint E square with E F. 
 This completes the mould for both pieces of wreatli. 
 
 To find joint bevels. The method here differs 
 from that given before, and will apply to any wreatli. 
 no matter what angle the tangents make on the plan, 
 while the previous method will only apply to where 
 the tangents form a right angle. Here the tangent 
 F E being extended fixes a point in 0. Draw 0 H 
 parallel with tangent F H ; transfer pitch of plank 
 from Fig. 2 by taking 0 for centre, and with any 
 radius, draw the dotted r.rc. With same radius, and 
 0, Fig. 3, as centre, draw an arc of equal length, and 
 ^rom its lower corner, draw tho pitch through 0; 
 square over from any point, as II, cutting the pitch at 
 V. With N as centre and a radius touching H 0, 
 strike an arc cutting at L ; draw from V through L. 
 This gives bevel W for joint on straight wood. With 
 same centre and a radius touching extended tangent 
 F E strike an arc cutting at T; join T V, and we 
 have bevel X for centre joint E. The square sections 
 of plank show the proper application of the bevel to 
 joints. In putting the wreath together, remember 
 to keep the line made by bevel X on each joint, 
 directly opposite. This will cause the straight wood 
 of both pieces to stand on their correct pitches. 
 
Plate 10. 
 
 THE WEEATH STAETING PEOM A LEVEL PLOOR. 
 
 Figure 1 shows the ground plan of stairs where the riser 
 landing comes within a few inches of the spring line of cylin- 
 der. When this is the case, the lower part of wreath must be 
 thrown on a pitch to meet the ramp formed on rake rail at 
 landing. The following explanation with the plate will show 
 how this may be done. We first find from the plan the num- 
 ber of risers that will stand under the wreath and ramp. 
 Counting from A and including B there are three. The posi- 
 tion of those beyond the spring line are indicated by arrows. 
 
 Fig. 2 shows the elevation; and O P H represents the 
 spring line; the steps and risers stand in the same position to 
 it as on the plan. The face of riser landing on floor projects 
 beyond the spring line as shown by an arrow. Let under 
 side of rake rail, both above and below, rest on the centre of 
 short balusters as shown. Set off half its thickness and draw 
 the upper pitch, cutting through H and D. Now draw from 
 D, say to E, or in such a manner as will make a ramp on rake 
 rail at the landing. To form this ramp, square over from E, 
 cutting perpendiculars at T and P. Make T U equal T D. 
 Draw from E through U, cutting half thickness of rake rail 
 at landing, and giving a direction to form the ramp. This 
 line also shows that the lower piece of wreath will have equal 
 pitches, and that O P will be its whole height on spring line. 
 Then P H gives the height for upper piece of wreath. We 
 have now the elevation of pitches and the form of ramp. The 
 next thing will be a mould and the joint bevels for upper 
 piece of wreath. 
 
 Fig. 3 shows the method of obtaining the bevels and the 
 points for forming the mould. First draw a square, its sides 
 equal to the radius of plan. Extend the upper line to the 
 right and left. Make T R equal height P H on elevation. 
 Square over from R, cutting at O. Make T V equal T V 
 on elevation. Draw from Y through corner of square E. 
 Now draw parallel with this line from the other three corners 
 of the square, as T S J. Draw through J square with V E. 
 Make U N equal height T R and draw from L through N. 
 This is pitch of plank nnd major axis. Make J H equal J E. 
 Square up from H, cutting at O. Draw through O parallel 
 with H L, and set oif half width of rail on each side of O, 
 cutting at 2, 3. This gives A 2 and A 3 as half diameters 
 for curves on mould. 
 
 Before drawing the mould we will find the bevels. Draw 
 from V to O. Make O F equal T C on the right. Draw 
 from R through F. -With S for centre and with a radius 
 touching F R strike an arc, cutting at K. Join K J. This 
 
 gives bevel X for centre joint. Then with T for centre and 
 a radius touching V C, strike an arc cutting at 6. Draw 
 from E through 6. This gives bevel W for joints on straight 
 part of wreath. 
 
 In drawing the mould for upper piece of Avreath lay the 
 edge of a board on the pitch and mark the points N, A, B, L. 
 Square them over as shown at Fig 4. ]\Iake N H, B D and 
 L E equal U S, 4 T and L E, Fig. 3. Draw from D through 
 H and E. These are tangents and must equal the pitches C 
 O and F R, shown above the square in Fig 3. Find centre 
 O from radius and set off on each side of it half the width of 
 rail. Draw from A through IT. Find points for pins and 
 sweep the curves. Add straight wood from II and make the 
 joints squaro with tangents. In working the piece, apply 
 the bevels as shown by the square sections. 
 
 Fio-. 5 shows how to draw the mould for lower piece of 
 wreath. Form a square, its sides equal to the radius on plan. 
 Draw the diagonal T U and parallel with it from K. Draw 
 square with the diagonal through T, cutting at B. Make T 
 D equal height T U on elevation. Draw from B through D. 
 This line is pitch of plank. Make D B on the left equal D 
 B on the right, and square over B D B. Make B O and B 
 E equal B K, and D C equal T U. Draw from C through 
 E and O. These are tangents and must equal pitches O U 
 E on elevation. Make D N equal radius on plan and set off 
 half width of rail on each side of N. The curves may now 
 be struck with a string as before ; or, in this case, with the 
 compasses, because the pitches are flat. In other words, the 
 wreath, when in position, will stand so nearly level that its 
 form will vary but slightly from a true circle. The points, 
 however, for elliptic curves are given as D, 5, 4. As the 
 pitches for this wreath are equal, one bevel will answer for 
 both joints. Take T for centre and with a radius touching 
 pitch, strike an arc, cutting at P. Join P R, which gives 
 bevel Z. Its application is shown on the square sections of 
 plank. 
 
 Fig. 6 shows cut of carriage starting. This may be deter- 
 mined at once by the intersection which under side of joist 
 make with lower edge of carriage, as the point S shows. Re- 
 member that this is a matter of some importance, if we con- 
 sider that the whole weight of the stairs is supported by the 
 carriage, and that this might be cut in such a manner as not 
 only to weaken, but make the stairs absolutely dangerous ; 
 and leave a stain on the reputation of the workman, perhaps 
 for years, and all for the want of a little consideration. 
 
Plate 11 
 
Plate 11. 
 
 PLATPOKM-STAIES. 
 
 Figure 1 shows the ground-plan. The semicircle 
 represents the centre of rail. As the cylinder here 
 is large, place three halusters on the platform. The 
 method for forming the wreath and finding the bevels 
 in this case will be the same as in the preceding 
 plates. It is said that perfection can only be obtained 
 by doing the same thing over and over again. If 
 that be true, the repetition given in this work, in de- 
 scribing the plates, if closely followed, cannot fail to 
 perfect the learner in a knowledge of everything 
 connected with stair building. 
 
 Ascertain the number of rises necessary to make 
 an elevation and proceed as usual. 
 
 Fig. 2 is the elevation showing the height of three 
 risers. The perpendicular through E F represents 
 spring line. Kisers A and B and two square steps 
 stand in the same position and project from it, the 
 same as on the plan, shown by arrows in both places. 
 Draw the pitches through corners of square steps 
 above and below, cutting spring line at F and E. 
 From half the distance between these points as K, 
 square over dotted line, cutting pitch at H. Draw" 
 from E parallel with K H. Now form the square as 
 shown, its sides equal to radius on plan. Extend the 
 upper side to the right. Square down from H, cut- 
 ting at L, and draw from L through corner of square 
 at T. • Draw from the other corners of the square 
 parallel with L T. Draw through V square with L 
 T. Make P D equal height E K. Join D A. This 
 is pitch of plank. On the dotted line above and par- 
 allel with it are points C 2, 3, which are half long 
 diameters for elliptic curves on mould. These are 
 found in the same manner as in preceding cases. 
 Next find bevels. Draw from K through J. Take 
 
 CYLINDER 17 INCHES. 
 
 R for centre and strike an arc touching line J K and 
 cutting at S. Join S T. This gives bevel X for cen- 
 tre joint of wreath. Now take 2 for centre and strike 
 an arc touching J II and cutting above 2. Draw 
 from this point to K, which gives bevel W for joint 
 on straight part of wreath. Now lay the edge of a 
 board on the pitch and mark points A, B, C, D, square 
 these over as shown Fig. 3. Make A K, B J and 
 D E equal A T, N R and P E, Fig. 2. Draw from 
 J, through E and K, giving the tangents. These 
 must equal the pitches E, J, K, Fig. 2. Make C 0 
 equal radius on plan. Set half width of rail on each 
 side of 0. Draw from C through E. Now find 
 points for pins and strike the curves. Make the 
 joints square with tangents, and the mould is com- 
 plete. The square sections on right and left show 
 the application of bevels. 
 
 Note. — It must have been noticed by this time that the 
 principles employed in the constructions aheady given are 
 simple, and may be readily comprehended and applied by 
 any one of ordinary intelligence. There is no secret or mys- 
 tery about them which will not disappear after a few moments' 
 intelligent study of any of the plates and their explanations. 
 We start by fixing the position of a single point from which 
 a line is drawn. This line determines the pitch of plank, or 
 in other words shows its true position when standing over the 
 cylinder. Upon the surface of this plank is shown two ellip- 
 tic curves representing one-half the wreath. The drawings 
 necessary for all this are simple and may be made by any one, 
 without previous experience or expensive instruments, because 
 the lines are principally right angles or parallels, and the 
 points readily determined by fixed rules suggested by common 
 sense ; the tools that are requisite being a framing square 
 and the compasses. If there be but the desire and ambition 
 to learn, there is no problem in the whole range of hand rail- 
 ing but what may be readily solved by the principles laid 
 down in this work. 
 
Plate 12. 
 
 PLATFOEM-STAIES. OYLINDEE SIX raOHES. 
 
 Figure 1 shows a plan of stairs starting from a level floor. 
 This plan should never be adopted in first-class buildings, as 
 the short steps and small space between the strings gives to 
 the stairs a narrow and contracted appearance. The riser here 
 stands on the spring line, and the wreath is of the most simple 
 description. Its whole construction is given on the elevation 
 of a square step. The plank is shown on pitch of stairs, its 
 under side resting on centre of short balusters O O. The 
 thickness of rail is gauged from under side, showing a slab on 
 top, which is taken off in working. At the centre joint is 
 given a square section of plank, and in this is drawn a section 
 of rail, a glance at which will show how to take off the slabs; 
 commence by taking off the under slab which forms the easing. 
 The lower piece of wreath is shown on the right ; its under 
 side stands half a riser above the floor. This should be cut 
 to the same thickness as the upper piece, and taken to the 
 width of the rail. The wedge-shaped slab when taken off 
 will bring the under side of both pieces at the joint.on the 
 same level. The remaining slabs on top side of straight 
 part when cut off gives a plain direction to work the easing 
 across both joints. 
 
 Fig. 2 shows the plan of platform-stairs having a six inch 
 cylinder. The risers landing on and starting from the plat- 
 form stand on the spring line. This makes the whole height 
 of wreath over spring line equal to one riser, or half a riser 
 for each piece of wreath. The method for forming the wreath 
 is the same as given before ; but in repeating here, we wish to 
 direct particular attention to the subject of joint bevels. 
 
 Fig. 3 shows how to find the points for the mould and the 
 joint bevels. Form a square, its sides equal to radius on plan ; 
 extend the upper side to right and left; set oflp on the perpen- 
 dicular above J the height of half a riser, giving K ; draw the 
 pitch of the stairs from K by the pitchboard, cutting through T, 
 and giving point E, ; from this draw through corner of square 
 P ; and from the other corners of the square draw parallel with 
 II P ; draw through E, square with R P; make F D equal 
 height J K, and join D A. This is pitch of plank. The 
 dotted line above it shows the major axis with points for half 
 diameters of curves on mould. 
 
 To find the bevels, square over K N ; make K L equal 
 
 S T ; draw from N through L ; take J for centre, and strike 
 an arc, touching L N, and cutting at U ; join U E ; this gives 
 bevel X for centre joint ; now take S for centre, and strike an 
 arc, touching T R, and cutting at a point to the left of S ; draw 
 from P through this point, which gives bevel W for joints on 
 straight part of wreath. 
 
 We are now ready to draw the mould. Lay the edge of a 
 board on the pitch, and mark points A, B, C, D ; square these 
 over as shown at Fig. 4 ; make A T, C K, and D L equal 
 A J, 2 S, and F P, Fig. 3 ; draw from K through T and L. 
 These are tangents which must equal pitches T K and N L, 
 shown above the square in Fig. 3. Complete the mould as 
 before explained. The square sections of plank and rail show 
 the application of the bevels. 
 
 Let us note here that the direction of joints across the width 
 of rail is always at right angles with the tangents on its face, 
 the bevels giving the cut through the thickness. These we 
 have obtained from the pitches of wreath, finding the bevel 
 for a joint through a given tangent, from the pitch which the 
 tangent corresponds with. This method, however, will only 
 apply when tangents on ground-plan form a right angle. But 
 by the method which follows, and which we have already used 
 in Plate 5, we can obtain correct bevels for any wreath, no 
 matter what the angle formed, finding them directly from the 
 tangents and pitch of plank. This method is therefore of 
 universal application. 
 
 Fig, 5. Take any point upon a line parallel with the long 
 diameter in Fig. 4, as 2 ; from this, draw 2 K and 2 L paral- 
 lel with tangents K T and K L on mould ; transfer pitch of 
 plank from Fig. 3, as follows : Take D for centre, and with 
 any radius draw dotted arc ; then with same radius and with 
 2, Fig. 5, for centre, draw an arc of equal length, and from its 
 lower point draw the pitch through 2 ; square up from any 
 point on long diameter, as H, cutting the pitch at S ; then 
 with H for centre, strike an arc touching 2 K, and cutting at 
 R ; draw from S through R ; this gives bevel W. Again take 
 H for centre, and strike an arc touching 2 L and cutting at 3 ; 
 join 3 S which gives bevel X. These bevels correspond 
 exactly with bevels W and X in Fig. 3, clearly proving the 
 practical value and reliability of this method. 
 
Plate 12 
 
Plate 13 
 
Plate 13. 
 
 WREATHS LANDING ON A LEVEL ELOOE. 
 
 This plate is intended to show the effect produced 
 in a wreath by the wrong position of the risers, and 
 the necessity for phicing them properly in order to 
 obtain the best results. 
 
 Fig. 1 shows plan and elevation of stairs, in which 
 the riser landing stands to the left of the spring 
 line, its position evidently fixed without any regard 
 to height of level rail at the landing. This careless 
 mode of proceeding always causes unnecessary work, 
 besides making it impossible to construct a perfect 
 wreath. The elevation above shows square step, and 
 floor line. The under side of rake rail rests on centre 
 of short balusters. The square section represents 
 centre joint of lower piece of wreath. The under 
 side of this slioidd stand the height of half a riser 
 above the floor line as indicated by the arrow on the 
 left, but does not. This error is caused by the riser 
 being in a wrong position. If the riser had been 
 placed on the right of the spring line, it would have 
 thrown the centre of wreath to its proper height. 
 The only way to get over this difficulty is as follows : 
 Make the level piece of wreath a parallel width the 
 same as rail. Cut both pieces of wreath from a plank 
 of the same thickness, and take slabs off" top and 
 bottom of upper piece as shown in elevation. This 
 brings point P to meet under side of lower piece of 
 wreath at centre joint, the slabs from top and bottom 
 of which being cut off, the ramp is easily formed. 
 This gives as good a wreath as can be formed over 
 such badly arranged stairs. 
 
 Fig. 2 shows the mould for lower piece of wreath. 
 The method for drawing similar moulds has been 
 so often explained that it need not be repeated. 
 
 Fig 3 shows plan of similar stairs but with the 
 riser In proper position. Here the face of riser stands 
 on a line with centre of rail at point S. 
 
 Fig. 4 is the elevation of a square step and floor 
 line.° Set off from riser, on floor line, two spaces each 
 equal to radius N S on plan and draw the dotted per- 
 pendiculars. Let under side of rake rail rest on 
 centre of short balusters. Show half its thickness by 
 a line through D C. Make under side of level rail 
 stand half a riser above the floor as indicated by an 
 arrow. Draw a line above this, showing its thick- 
 ness and also top of plank. Cut both pieces of wreath 
 from the same plank. Then by taking off the slabs 
 from top and bottom of upper piece of wreath as 
 shown, part of the ramp is formed. Draw thQ line 
 ABC and from centre of joint B, draw to the left 
 
 parallel with floor, and cutting perpendiculars at S 
 and L. This gives D L as the height for lower 
 piece of wreath. 
 
 To find the points for drawing the mould. Form a 
 square as usual. Extend the pitch D C, cutting at E; 
 draw from E through F, and parallel with this line 
 draw from the other corners of the square. Again 
 draw through R, square with E F, cutting at H ; 
 make height P K equal height D L ; draw from H 
 through K. This is pitch of plank. Find half long 
 diameter of semi-ellipse, by making R V equal R L ; 
 square up from Y, cutting at U ; draw through it 
 parallel with VP; set half width of rail on each side 
 of U. This gives J 2 for inside curve on mould, and 
 J 3 for outside curve. 
 
 Now lay the edge of a board on the pitch, and 
 mark the points H, N, J, K ; square these over as 
 shown at Fig. 5 ; make K D, N C, and H B equal 
 corresponding distances on the right, as P L, T S, 
 and H F; draw from C through D and B. These are 
 tangents on the mould, and to be correct must equal 
 pitches D B C, Fig. 4. 
 
 Make J 0 equal radius on plan. Set half the 
 width of rail on each side of 0 ; draw from J through 
 D. Now find points for pins and strike the curves ; 
 mark joints square with tangents and add straight 
 wood from D. This completes mould for lower piece 
 of wreath. The upper piece is shown on plan, its 
 curves are two quarter-circles struck from centre N, 
 and its width the same as rail. 
 
 Find the bevels from the tangents and pitch of 
 plank as follows: Upon any line parallel with the 
 long diameter fix upon a point as P. From this 
 draw P C and P D, parallel with the tangents on 
 mould. Now transfer pitch of plank from Fig. 
 4, as follows: Take U for centre, and with any 
 radius draw dotted arc ; with same radius, and with 
 P for centre draw an arc of equal length ; from its 
 upper point draw the pitch through P; square up 
 from any point, as J, cutting pitch at R ; with J for 
 centre strike an arc touching P C, and cutting at K ; 
 join K R, which gives bevel X for centre joint. From 
 the same centre strike an arc touching P D and cut- 
 ting at L ; draw from R through L, which gives bevel 
 W for joint on straight part of wreath. 
 
 Note.— In cutting the plank for level piece of wreath, 
 allow enough wood to make centre joint B. The direction of 
 joint on surface of plank will be parallel with line N S on 
 plan. 
 
Plate 14. 
 
 THE WEEATH STARTING PEOM A LEVEL ELOOE. 
 
 Figure 1 shows plan of stairs in which the riser 
 starting is purjoosely pLaced in a wrong position, in 
 order to show how errors and mistakes may some- 
 times be remedied by simple means. We wish to 
 impress upon the learner, however, that there will be 
 more satisfaction in starting right, and working by 
 correct principles throughout, than in correcting 
 errors and covering mistakes by means however 
 ingenious. As in the last plate, we are here required 
 to form a perfect wreath over a defective plan. The 
 face of the riser should have projected past the spring 
 line a distance equal to the radius, or, if still further, 
 the difficulty of forming the easing which the wreath 
 should have from level to rake would be lessened. 
 It is a fault conunon to many wreaths, starting from 
 a level floor, that the lower corner along the concave 
 side presents a broken or defective curve. In trying 
 to correct this, some throw the section of rail at the 
 joint out of square, but if the riser be placed aright 
 on the start, there will be no trouble in giving to the 
 wreath a uniform and graceful curve. 
 
 On account of the position of this riser, it will be 
 necessary to change the joint from what is usual 
 in order to give the wreath a regular height, when in 
 position, over floor and steps. We will assume for it 
 the line through E. This gives E F and F H as 
 tangents for upper part of wreath. 
 
 Fig. 2 shows elevation of square step and floor line. 
 The dotted perpendiculars passing through points H 
 F E show tangents on plan unfolded. Let under side 
 of lower wreath piece stand half a riser above the 
 floor. The thickness of plank will be sufficient to form 
 the easing on top and across the joint ; let under side 
 of rake rail rest on centre of short balusters ; draw 
 half its thickness through H and F; assume E as 
 half the thickness of rail on lower piece of wreath ; 
 draw from F through E ; make joint square Avith F E. 
 It will now be seen that by cutting a slab from the 
 top and bottom of the plank, an easing may be formed 
 across the joint. Square over a line from E cutting 
 at N, this gives N H as height for upper piece of 
 wreath ; extend the tangent E F on plan and make 
 E P equal N P on elevation ; draw from P through 
 H; draw from F and E parallel with P H; draw 
 through centre T square with P H ; make L D equal 
 
 height N H on elevation ; join D A, this is pitch of 
 plank. For convenience, we will find the half diam- 
 eters on the dotted line parallel with this. Make T S 
 equal T H ; square up from S, cutting at 0 ; draw 
 through 0 parallel with T S; set half width of rail on 
 each side of 0, cutting at 2 and 3, and giving B 2 and 
 B 3 as half diameter for the curves. 
 
 Lay the edge of a board on the pitch and mark the 
 points A, B, C, D ; square these over as shown at 
 Fig. 3 ; make A H, C F, and D E, equal A E, R F, 
 and L II, Fig. 1 ; draw from F through E and H. 
 These are tangents and must equal pitches on eleva- 
 tion as E F H. Make B 0 equal radius on plan ; draw 
 from B through H; set off* half width of rail on each 
 side of 0 ; find points for pins and sweep the curves ; 
 add straight wood from II, and make joints square 
 with tangents. This completes the mould for upper 
 piece of wreath. 
 
 Fig. 4. To find the joint bevels. Take any point 
 as F on long diameter, and draw lines F H and F E 
 parallel with tangents on the mould ; transfer pitch 
 of plank from Fig. 1 by taking D for centre, and with 
 any radius, draw dotted arc; with same radius, and 
 F Fig. 4 as centre, draw an arc of equal length, and 
 from its lower corner draw the pitch through F; 
 from any point as C, square up a line cutting the 
 pitch in K ; with C for centre, strike an arc touching 
 F E, and^ cutting at 3 ; join 3 K. This gives bevel 
 X for joint E. From same centre, strike an arc 
 touching F H, and cutting at 2; draw from K 
 through 2. This gives bevel W for joint on straight 
 wood. 
 
 Fig. 5 shows cut of carriage starting. J is face of 
 riser as shown on the plan. The joist stands to the 
 left of the cylinder. It will be at once seen from this 
 Fig., that if the carriage be cut to the right of L, its 
 width through the cut will be less than the depth of 
 joist, and if cut to the left, its width will be greater 
 and therefore project below the ceiling. The method 
 of ascertaining the proper cut will present itself 
 readily to any practical mind, as well as the impor- 
 tance of paying proper attention to these details. We 
 have before stated that as the carriage sustains the 
 entire weight of the stairs, it is all important to have 
 it properly fitted and well secured. 
 
Plate 14 
 
P LAT 
 
 E IB. 
 
 THE WEEATH STARTING FROM A FLOOR 0 
 
 Figure 1. There are many examples of wood, 
 metal, and stone stairs which have no cylinders, and 
 yet have regular wreaths. In such cases, the lower 
 part of wreath cannot be worked in the manner given 
 before. We here give a plan of stairs starting from 
 a passage without a cylinder, and in which the rail is 
 required to continue along the passage. 
 
 Assume P R as radius for centre line of rail, ^ila 
 shall find by making an elevation of the pitches of 
 wreath, that instead of cutting a slab from top and 
 bottom of lower piece, as in the preceding example, 
 it will be necessary to bevel the edge of straight wood 
 in order to throw the centre joint to its proper height. 
 
 Fig. 2 shows the elevation of risers, floor line and 
 one square step. By referring to the plan, we find 
 the height of three risers necessary in obtaining the 
 pitches, counting from the floor and including riser 
 B. Set off" to the right of spring line n distance equal 
 to radius P R and draw the dotted perpendicular. 
 Draw half thickness of level rail, cutting at 5. Let 
 under side of rake rail rest on centre of short balus- 
 ters. Set ofl" half its thickness, cuttino; throu2;h 2 
 and 3. Join 3, 5. This gives 2, 3, 4, as pitches for 
 upper, and 4, 5, for lower pieces of wreath. 
 
 The points for drawing the mould for upper piece 
 are obtained as follows : Draw a line to the left of 4 
 and parallel with floor, cutting through H and E. 
 Form the square and draw from E through its lower 
 corner F. From the other corners of the square 
 draw parallels with E F. Draw through N square 
 with E F. Make height 6 D equal height 4, 2. 
 Draw from A through D, This is pitch of plank. 
 Now make N J equal N 4. Square up from J, cut- 
 ting at S. Draw through S parallel with J A. Set 
 off" half width of rail on each side of S, giving the 
 half diameters for curves on the mould, C being the 
 centre. 
 
 Lay the edge of a board on the pitch and mark 
 points A, B, C, D. Square these over as shown at 
 Fig. 3. Make A 4, B 3, and D 2, equal A F, K H, 
 and 6, 4, Fig. 2. Draw from 3 throifgh 2 and 4. 
 These are tangents, and must equal pitches 2, 3, 4, 
 on elevation. Make C 0 equal radius on plan, set 
 
 PASSAGE WHERE THERE IS W CYLINDER. 
 
 half width of rail on each side of 0. Draw from C 
 through 2. Find points for pins and sweep the 
 curves. Make the joints square with the tangents. 
 This completes mould for upper piece of \Yreath. 
 
 Fig. 4. To find the bevels. From any point on a 
 line parallel with long diameter, as 3, draw lines 3, 2, 
 and 3, 4, parallel with tangents on mould. Transfer 
 pitch of plank from Fig. 2, as follows : With D for 
 centre and any radius draw dotted arc, and with same 
 radius and 3 Fig. 4 for centre, draw an arc of equal 
 length, and from its upper point draw the pitch 
 through 3. Square down a line from any point as K, 
 cutting the pitch at V. With K for centre, strike an 
 arc touching 3, 4, and cutting at J. Join J V. This 
 gives bevel X for centre joint. From same centre 
 strike an arc touching 3, 2, cutting at L. Draw 
 from V through L. This gives bevel W for joint on 
 straight wood of wreath. The sections on right and 
 left show the application of bevels. 
 
 Fig. 5 shows the mould for lower piece of wreath, 
 which is quite simple. Upon a piece of board square 
 over two lines, shown as perpendiculars through 5 and 
 6, the distance between them equal to radius on plan. 
 Make 4, 5, equal pitch 4, 5, on elevation. Draw from 
 5 parallel with edge of board, cutting through 6. Set 
 off half width of rail on each side of 4. The joint 
 through 4 is on the short diameter of semi-ellipse and 
 the line through 6 is long diameter. Find half width 
 of mould from bevel Y on elevation and set it off on 
 each side of 6; the method has already been ex- 
 plained. Find points for pins and strike the curves. 
 Add straight wood from 6. The section of plank and 
 rail on the right shows how to apply bevel Y, and the 
 section opposite 4 shows the centre joint, having 
 equal slabs cut off both sides. 
 
 When the stairs are of stone or metal and are to 
 have metal balusters, capped by a wood rail, work 
 the wreath to its proper form, fit and bolt the joints 
 together, and before moulding, rabbet the under side 
 for the iron strap that connects the top of the balus- 
 ters. Then let the smith fit this strap or flange to 
 your work. This is far preferable to the haphazard 
 way of fitting after the smith. 
 
Plate 16. 
 
 SIDE WKEATH STAETING TEOM A NEWEL. 
 
 This simple problem has often perplexed the be- 
 ginner, and sometimes the more experienced stair 
 builder, when they have had no guide in determining 
 the height for newel and its projection from string. 
 
 This plate shows for the first time, and by entirely 
 hew diagrams, how to form the wreath, no matter 
 what the height, position, or projection of the newel. 
 Or in other words, it gives plain directions for con- 
 necting the rail with the mitre cap by any desired 
 curve. The first to be considered on the ground- 
 plan is the side curve or projection, to the width of 
 passage. If there be room, make the projection as 
 much as possible so as to extend the curve, which 
 will give a much better effect to the stairs. 
 
 Fig. 1 shows elevation of square step and riser. 
 Let under side of rake rail rest on centre of short 
 balusters 0 0. Set off half its thickness, as shown. 
 Fix upon the height of newel, from the top of step 
 to under side of cap, say it shall equal the height of 
 a long baluster ; which is half a riser more than a 
 short one, when there are two on a step. Set off 
 half a riser above the step as indicated by an arrow, 
 and from this draw half thickness of rail, cutting 
 pitch at D. This is a fixed point ; and through it 
 draw the dotted perpendicular. We are now ready 
 to lay down a correct ground-plan. Draw any line 
 parallel with step, as that through A B ; cutting per- 
 pendicular through D, and fixing point A. Decide 
 upon centre of newel, giving it any desired projec- 
 tion from A B, and on a line with face of riser. 
 Assume C as the point. Draw from C through A. 
 With C for centre draw size of cap. Gut off a 
 portion equal to width of rail, by a line through 
 2, square with A C, and draw the two parallel 
 curves showing width of rail. These curves may be 
 extended to any point on the left as B. This com- 
 pletes the ground-plan. 
 
 To find the mould we proceed as follows : Square 
 up from B, or the termination of curves, cutting the 
 pitch at E. Square over from E, cutting at F. This 
 gives D F for height of wreath. Draw any number 
 of lines through width of rail, parallel with A C. 
 Extend line A C and fix a point on it as J. Square 
 over from J, cutting through D. Make D E equal 
 height D F, Fig. 1. Draw from E through J; this 
 
 is pitch of plank. Now draw from all the points 
 cutting pitch, square with E J. Make J D 2 equal 
 J A 2, and E F equal D B; draw from D through F. 
 This is tangent on the mould, and to be correct must 
 equal pitch D E, Fig. 1. Find points, through which 
 trace the curves on mould. Leave the lower end of 
 the mould square ; making length of mitre from 2 
 the same as on the plan. 
 
 We next find the bevels as follows : From any 
 point as A on the right, draw A J parallel with A J 
 on the left ; and A F parallel with tangent D F on 
 mould. From any point as K, square down, cutting 
 at J. Take K for centre, and with a radius touching 
 A F, strike an arc, cutting at N; join J N; this 
 gives bevel W for joint on upper end of wreath. 
 The line A J gives bevel X, which is to be applied 
 to square edge of plank. The line made by it will 
 stand plumb when the rail is in position, and fixes 
 the points by which to apply the mould. After the 
 stuff is cut square through and joints made, the mould 
 is to be applied to both sides of the piece ; keeping 
 the mitre point on the mould exactly over the points 
 on top and bottom made by bevel X across the joint. 
 Let the learner notice the direction which tangents 
 take when the rail is in position, and there will be no 
 difficulty in applying the mould or working the piece. 
 
 The square sections of plank and rail are given to 
 show the proper application of the bevels. Be care- 
 ful to apply bevel W properly. Observe that its 
 blade is on the same side of tangent as bevel X. 
 
 Having worked the wreath piece to its cylinder 
 form, cut off the under slab square with joint, and 
 at the same time square with line for mitre point. 
 Then cut off the under slab of straight part and 
 work it square with joint and line made by bevel W. 
 The piece will now show a curve on lower corner of 
 concave side. This is a perfect direction for working 
 off remainder of slab, thus forming a ramp from 
 level to rake. Now mark the mitre and cut it partly 
 through, finishing it after the piece is moulded. Com- 
 plete the wreath by taking off the upper slab. 
 
 Fig. 2 shows the construction of a similar wreath, 
 but reversed. Bevels, heights, and letters are the 
 same as in Fig. 1, and the explanation given there 
 will apply here. 
 
^1 
 
Plat 
 
 E 17. 
 
 DESOEIPTION or SIDE 
 
 Ojr the riglit of this plate is shown the construction of a 
 wreath for the starting of first-class stairs. The ends of three 
 steps are curved on the string, which adds greatly to the 
 , general effect and finish. 
 
 Fig. 1 shows the elevation of square step and floor line. 
 Let under side of rake rail rest on centre of short balusters, 
 O O ; set off half its thickness, which draw to any 
 length. The height for newel between top of step and 
 under side of cap should be the same as long baluster. 
 Taking that for the height, mark half a riser from top of 
 step as indicated by an arrow. Set off above this half thick- 
 ness of rail, cutting the pitch at C. This is a fixed point, and 
 from it draw the dotted perpendicular. From any point on 
 this, as B, square over a line to the left. This is one tangent 
 on ground plan ; the other is determined by fixing a point for 
 the projection of newel. This projection, in order to have an 
 easy and extended curve, should be as much as possible, but 
 not to interfere with the passage. Here we will assume S 
 for centre of newel. Then by drawing a line from B through 
 S fixes a level tangent on both plan and wreath. From 
 centre, S, draw the size of cap, and cut off a portion equal to 
 width of rail by a line through L square with B S. 
 
 The dotted perpendicular on right of C B represents face 
 of riser. From it set off the width of three square steps as 
 shown. Draw the two parallel curves giving width of rail. 
 This may be done by bending a thin lath. The form of these 
 curves is entirely a matter of taste, but as they approach the 
 mitre cap should be elliptical. They may terminate at any 
 point on the left, as line through A. The first step should 
 be wider than the others, and curved, as shown. We now 
 have a ground-plan. Square down from A, cutting the pitch 
 at D, and square over from this, cutting the perpendicular at 
 E, which gives C E for height of wreath. 
 
 Fig. 2 shows the method of forming the mould. Extend 
 the line through L to the lefl; draw from A parallel with 
 B S, cutting at J ; make height L Iv equal height C E ; draw 
 from K through J. This is pitch of plank. Draw from J 
 and K square with pitch ; make K C equal B L, and J D 
 equal J A ; draw from C through D. This is tangent on 
 mould. When the rail is in position it stands directly over 
 B A on plan, and to be correct must equal pitch D C on ele- 
 vation. The line K C falls level when in position. Make 
 the projection below K equal the length of mitre on plan. 
 Draw a few lines through the plan parallel with B S, and 
 cutting the pitch; square these over ; find points and trace the 
 curves on mould ; add straight wood from D, The angle 
 made by the pitch gives bevel, X, to be applied to the edge 
 of plank, giving a plumb-line for mitre point. The bevel 
 for joint on straight wood is found as follows : Draw J V 
 parallel with D C on mould; from any point, as N, draw 
 N P square with the pitch ; take N for centre, and strike an 
 arc touching V J and cutting at 2; draw from P through 2. 
 This gives bevel, W, as required. The application of mould 
 to the plank and working the wreath is precisely the same as 
 explained in the preceding plate. 
 
 ¥EEATH CONTINUED. 
 
 Fig. 3 shows a method by which the curves on ground-plan 
 are struck from a centre, and the elliptic curves on mould by 
 means of a string. This is certainly a quick and off-hand 
 way of describing the curves, but objectionable from the fact 
 that the radius on plan being limited, the curves described 
 are too short to give a good effect to the Avork. The question, 
 however, might be asked. Why not increase the radius, and 
 so extend the curve ? This would not help the matter, be- 
 cause the tangents would be lengthened at the same time, and 
 change the angle of their intersection on plan, and throw the 
 newel too high. This will be better understood by referring 
 to the figure and following explanation : 
 
 First draw the elevation of square step and riser. Let 
 under side of rake rail rest on centre of short balusters, O O; 
 set off half its thickness. If the newel is to be the usual 
 height, set off half a riser from top of first step, and above 
 this draw half thickness of rail, cutting the pitch, and fixing 
 point N; draw a line at any distance above the elevation and 
 parallel with step, as D C. This is centre of rail. Now, de- 
 termine upon a point for centre of newel on a line with face 
 of riser starting ; Square up from N, cutting at D, and from 
 D draw through centre of newel. Next draw size of mitre 
 cap. Cut off a portion of it to equal width of rail by drawing 
 a line through A square with D E ; make D C equal D A ; 
 draw through C square with D C, cutting the pitch at K, and 
 line through A at B. This makes B a centre, from which 
 strike width of rail on ground plan ; draw from C parallel 
 with D A ; square over from K, cutting at L. This fixes 
 height of wreath as N L; make A E equal it; join E F; 
 draw from centre B parallel with F E. This is pitch of 
 plank and long diameter of semi-ellipse. Lay the edge of a 
 board on the pitch, and mark points B, J, H ; square these 
 over as shown in Fig. 4 ; make J N and H K, equal C F, and 
 D A Fig. 3 ; draw from K through N and H. These are 
 tangents on the mould, and to be correct K N must equal 
 pitch K N on elevation ; make B C equal B C on plan ; set 
 off half width of rail on each side of C ; draw from B through 
 N. We will now find bevels for joints. Extend the tan- 
 gent N K, cutting at V; Transfer pitch of plank from Fig. 3, 
 as follows : Take 3 for centre, and with any radius draw dotted 
 arc ; with same radius and with V, Fig. 4, for centre draw au 
 arc of equal length ; from its upper point draw pitch through 
 V ; Square down a line from any point, as 2, cutting pitch at 
 R; take 2 for centre and strike an arc touching K V, and 
 cutting at P ; join P R. This gives bevel X for joint on 
 straight wood. The intersection of pitch and long diameter 
 gives bevel W to apply on square edge of plank, and when in 
 position is a plumb-line or the mitre point of wreath. The 
 exact width of mould is found from the bevels as follow.s : 
 Set off half width of rail, as 2 X, and draw parallel with 
 V P, cutting both bevels. This gives J V for half width of 
 mould on wide end, ai\d L P as half the width on the straight 
 part. Now find points for pins, and strike curves on the 
 mould with a string. The square sections of plank show the 
 proper application of bevels to joints. 
 
Plate 18. 
 
 THE WEEATH OVEE WINDEES. 
 
 In the preceding lessons we have given examples of nearly 
 every form of ordinary stairs having material points of differ- 
 ence. We now come to a class of woric that will require still 
 a little more attention. On this plate is shown the construc- 
 tion of a wreath over winders, and starting from a newel. 
 Here it will be in two pieces; the upper piece will form its 
 own ramp ; the straight part of it thrown on the same pitch 
 as rake rail over square steps ; the curves on mould may be 
 described in the usual way, or traced through points as in the 
 last two plates. 
 
 Fig. 1 shows ground plan of stairs, starting with five 
 winders ; the narrow ends on centre line of rail are equal to 
 half a square step. Eadius for centre line of rail, eight 
 inches. Neither this nor number of winders are limited, but 
 are regulated by the space allowed for the stairs. The newel 
 here stands in a position which gives the best possible effect 
 to the stairs, making them appear wider than they really are. 
 The curve on plan is extended beyond a quarter circle, making 
 tangents for lower piece of wreath form tiie right angle 
 ABC, and tangents for upper piece, the angle C D E. This 
 determines the joint as shown by the line through C. 
 
 Fig. 2 shows elevation. The height of seven risers is 
 given on the left. The tangents on ground plan are unfolded 
 or spread out, as shown by dotted perpendiculars passing 
 through points A, B, C, D, E, corresponding with A, B, C, 
 p, E, on plan. The square step and winder adjoining stand 
 in the same position as on the plan. We will now draw the 
 pitches of wreath. Let under side of rake rail rest on centre 
 of short balusters O O. Set off half its thickness, cutting 
 tlirough E D. Notice here that D being the point fixed by 
 intersection of tangents on plan must also be the point where 
 rake rail ends on the elevation. Draw from D, say through 
 B, and from B through A. The direction of these lines is 
 discretionary, but in fixing point A bear in mind the height 
 of newel, as the pitch A B starting from the mitre cap here 
 differs from that in preceding examples of side wreaths. 
 Were this made level as in those cases, the newel would be 
 too high. Hence the caution in fixing the direction of lower 
 pitch. Draw through joint C, cutting at P N. This gives 
 N E as height for ui)per piece of wreath. Draw from A, 
 cutting at V. This gives V C as height for lower piece. 
 Now set off the projection of mitre on the cap in Fig. 1 to the 
 right of A, cutting the lower pitch. We will use this when 
 we come to the mould. 
 
 To find points for drawing the mould. Extend the pitch 
 C B, cutting at H; form the square as shown, its sides equal 
 to radius A 2 on plan ; draw from H, cutting at K ; draw from 
 S tlirough K, and from the other corners of the square draw 
 l^arallels with S K ; draw through 2 square with S K ; make 
 height Z R equal height V C ; draw from 11 through Y. This 
 is pitch of plank. Make 2, 3, equal 2, 4. Set half width of 
 rail on each side of 3, and square down from these points, 
 cutting at T and F. This gives O F and O T as half diame- 
 ters for elliptic curves on mould. 
 
 To draw the mould, lay the edge of a board on the pitch, 
 and mark the points Y, O, J, R; square these over as shown 
 in Fig. 3 ; make Y C, J B, and R A, equal Y 4, K S, and 
 the spacs between Z and corner of square; draw from B 
 
 through A and C. These are tangents on the mould, and to be 
 correct must equal pitches A, B, C, on elevation. Draw 
 from O through A; make O L equal radius on plan; set 
 half width of rail on each side of L ; find points for pins and 
 strike the curves. Now set off on tangent A B projection of 
 mitre to equal that on left of A in elevation. 
 
 The bevel W shown in the pitch gives the plumb-line or 
 mitre point across the edge of stuff. The bevel for joint C 
 is found as follows : From any point on long diameter, as F, 
 draw F L parallel with tangent C B on mould ; with any 
 point on the pitch as F for centre, and with any radius, draw 
 the dotted arc ; from F on the right for centre and with same 
 radius draw an arc of equal length, and from its lower point 
 draw pitch through F ; square over from any point as U, 
 cutting pitch at V; with U for centre strike an arc touching 
 L E, and cutting below U ; draw from V through this point, 
 which gives bevel X for joint C. 
 
 To apply this mould, keep its wide end (which should be left 
 square) fiiir with edge of ])lank, the mitre point directly ov^er 
 plumb line made by bevel W on edge of plank. Mark the 
 mitre on surface by the direction shown on mould. After 
 marking the under side in the same manner, take off the slabs 
 and work the piece to its cylinder form. Cut the mitre by 
 the plumb line and the direction marked on surface. Cut off 
 a slab from the under side square with mitre point. Work 
 as near the top surface as the moulding of rail will admit of. 
 Owing to the wreath at this starting-point being on a pitch, 
 the ramp must necessarily be short and abrupt. This, how- 
 ever, is better than having the newel too high. 
 
 To draw the mould for upper piece of wreath, extend 
 centre line of rail on plan through E to the left ; make E P 
 equal N P on elevation ; draw from P through C, and from 
 D and E parallel with P C ; from any point on line through 
 P C, as R, square over a line ; make height N H equal height 
 N E on elevation ; draw from R through H. This is pitch 
 of plank. Draw from R L and H square with pitch ; make 
 H E equal N E, L D equal J D, and R C equal R C ; draw 
 from D through E and C. These are tangents on the mould, 
 and to be correct must equal pitches E, D, C, on elevation. 
 Find the junction of straight wood and curves by making R3 
 equal R 2 and drawing from 3 through E; draw any number 
 of lines through width of rail, parallel with P R, and cutting 
 the pitch ; square these over; find the points, and trace the 
 inside and outside curves on mould ; draw straight wood par- 
 allel with D E ; make the joint through C square with 
 tangent, and the mould is complete. 
 
 Fig. 4. To find joint bevels, draw a line parallel with 
 pitch in Fig. 1, as that through R D; from any point, as D, 
 draw D C and D E parallel with tangents on mould ; take 
 H, Fig. 1, for centre, and with any radius draw dotted arc ; 
 with same radius, and from D as centre, draw an arc of equal 
 length; from its uj>per corner draw the pitch through D; 
 square down from any point as U, cutting pitch at V; with 
 U for centre strike an arc, touching D E, and cutting at T; 
 join T V. This gives bevel X for joint on straight wood. 
 From same centre strike an arc, touching CD, and cutting at 
 2 ; draw from V through 2. This gives bevel W for joi'^t 
 C. The square sections show application of bevels. 
 
Plate 18 
 
Plate 19 
 
Plate 19. 
 
 THE WREATH OVER EIVE WINDERS LANDING ON A LEVEL PLOOR. 
 
 Figure 1 shows a ground plan, well adapted for 
 either large or small halls. The cylinder here is 
 nine inches radius, but may be more or less to suit 
 length of step and space allowed. The narrow ends 
 of winders on centre line of rail are equal to half a 
 square step. The mould for this wreath is easily 
 obtained. First find the number of risers over which 
 it will stand ; counting from square step we find six. 
 With this and the radius A X we proceed as follows : 
 
 Fig. 2. Here unfold twice the radius as shown by 
 perpendiculars. Then make the elevation of square 
 step and winder adjoining it, which now stand in the 
 same position as on the plan. Set off six risers from 
 top of square step, and draw floor line ; above this 
 mark half a riser, as shown by the arrow, and from 
 it set oif half thickness of rail cutting through per- 
 pendicular at 0. We are now ready to draw lower 
 pitch of wreath to connect with a ramp over square 
 step. To do this assume any point on half thickness 
 of rake rail over square step, as shown by the inter- 
 section. From this point draw through corner of 
 winder above U, cutting perpendicular at S; divide 
 S 0 equally, as P ; square over P R ; draw from R 
 through 0 and S. Pitches are now given as J, R, S, 
 for lower piece of wreath, and R O^or the upper. 
 Square over a line from J, cutting through perpendic- 
 ular on the left, giving points V C, below which form 
 the square. Now extend pitch R S, cutting at U; 
 draw from it through F; again draw from the other 
 corners of square parallel with U F; then draw 
 through K square with U F. Make height E D 
 equal height C R; join D B. This is pitch of plank 
 for lower piece of wreath. Find half long diameter 
 of semi-ellipse by making K L equal K F; square up 
 from L, cutting dotted line at 0; draw through this 
 parallel with L B; set half width of rail on each side 
 of 0, and we have A 2 for inside curve on mould 
 and A 3 for outside curve. To find joint bevels, mak'3 
 R T on the left equal V S on the right; draw from P 
 through T; take C for centre and strike an arc touch- 
 ing T P, and cutting at 6 ; join 6 K. 
 
 This gives bevel X for lower joint. Now take V for 
 
 centre and strike an arc touching U S and cutting at 
 H; join H C. This gives bevel W for centre joint. 
 The bevels being complete for lower piece of wreath. 
 The mould is formed by laying a board on the pitch 
 and marking points B, A, N, D ; square these over as 
 shown at 
 
 Fig. 3. Here make B R, N S and D J, equal B C, 
 4 V and E F, Fig. 2; draw from S through R and J. 
 These are tangents on the mould, and to be correct 
 must equal pitches J, S, R on elevation. Draw from 
 A through J ; make A X equal radius on plan ; set 
 half width of rail on each side of X ; make J Y equal 
 distance between J and joint of ramp on elevation. 
 Now find points for pins and strike the curves. Make 
 the joints square with tangents. This completes the 
 mould for lower piece of wreath. The square sections 
 of plank and rail show application of bevels ^o joints. 
 
 The mould for upper piece of wreath is found from 
 elevation at Fig. 2. Set half width of rail on each 
 side of 0; draw perpendiculars, cutting the pitch at 
 points 4 and 5. This gives R 4 for half long diameter 
 of inside curve, and R 5 for outside curve on mould. 
 The bevel Z applies to upper joint of wreath. Have 
 .a board sufficiently wide as shown at 
 
 Fig. 4. Draw any line for long diameter, as that 
 through A; Make A, 4, 0, 5, equal R, 4, 0, 5 on 
 elevation ; square over from A ; make A R equal 
 radius on plan. Set oft' half width ^f rail on each 
 side of R. Now find points for pins and strike 
 the curves; Draw tangents through the mould at 
 right angles as shown ; add four or five inches of 
 straight wood from 0. The section of plank and rail 
 shows application of bevel Z to the upper joint. The 
 centre joint, being on the short diameter, makes width 
 of mould the same as rail. Hence a square is applied 
 to joint, as shown, and equal slabs cut off" both sur- 
 faces of p^ iuk. 
 
 If it is desirable to save material, this piece may be 
 cut by a rough pattern of i parallel width the same as 
 rail. But the mould must be retained and applied 
 as drawn, as a correct mould is absolutely necessary 
 in giving this or any wreath its true cylinder form. 
 
Plate 20. 
 
 THE WEEATH OVEE WINDEKS AND LANDING ON A ELOOR. 
 
 Figure 1 is a plan similar to that on preceding 
 plate, and showing how the lower part of wreath 
 may be made to form its own ramp and still have a 
 regular height over the winders. This is a decided 
 advantage, effecting not only a saving of time but 
 also of expense, as it gives the opportunity for using 
 machine-worked straight rails. 
 
 The construction of this wreath differs from the 
 last, only in changing the joint to a position that will 
 shorten the tangents for lower piece of wreath, and 
 prevent the pitch over square steps from running too 
 far into the cylinder. This will be seen when we 
 make the elevation of pitches for the wreath. Let 
 us assume the line through C as the joint. This 
 gives A, B, C, as tangents for lower piece of wreath, 
 and C, D, F, for the upper piece. The number of 
 risers from square step to floor is six. The narrow 
 ends of winders on centre of rail are equal to half a 
 square step. We now proceed and make the elevation 
 of pitches. 
 
 Fig. 2. Here the tangents are unfolded as shown 
 by dotted perpendiculars passing through points 
 T, R, F, N. The elevation of square step and top of 
 winder are shown on left of spring line. The height 
 from top of square step to floor is shown on the right. 
 The under side of level rail is half a riser above the 
 floor. Half its thickness is drawn above, cutting the 
 perpendicular, and fixing point N. Draw under side 
 of rake rail over square step, resting on centre of 
 short balusters; set off half its thickness, cutting 
 through T and R; draw from R through N. We 
 have now the pitches for both pieces of wreath. The 
 height for lower piece is found by squaring over from 
 T, cutting the perpendicular at C, giving C F for 
 height; square over from F, cutting at D, which gives 
 D N for height of upper piece. 
 
 Now proceed and find the mould for low^er piece of 
 wreath. Extend the pitch N R, cutting at P, giving 
 C P on dotted line. Transfer this distance to plan, 
 extending tangent C B to equal it; Draw from 
 P through A; draw from B and C parallel with P A; 
 also draw from the centre, square with P A, giving 
 J ; make height V L equal height C F, Fig. 2 ; join 
 L J. This is pitch of plank. To find half long 
 diameter for semi-ellipse, draw any line parallel with 
 pitch; extend centre line of rail, cutting below J ; 
 square up from this point and from the centre, cutting 
 at H and O; set half width of rail on each side of 0, 
 giving H 7 and H 8 for half diameters of inside and 
 outside curves on mould: 
 
 Fio\ 3. Have a board of sufficient width, and draw 
 a lin? on its face ; mark points J, K, L, to correspond 
 
 with same points on pitch. Fig. 1; square these over, 
 and make J T, K R, and L F, equal J A, E B, and 
 V C, Fig. 1 ; draw the tangents from R through T and 
 F. These to be correct, must equal pitches T, R, F, 
 Fig. 2. Make L F equal radius on plan ; set half 
 width of rail on each side of F ; find points for pins, 
 and strike the curves as shown ; draw^ from centre H 
 through T; add straight wood parallel with R T; 
 Extend the long diameter, and from any point on it, 
 as H, draw line H R parallel with tangent R T on 
 mould. Transfer pitch of plank from Fig. 1 as fol- 
 lows : From L, as centre, and with any radius, draw 
 dotted arc. With same radius and from H, Fig. 3, 
 as centre, draw an arc of equal length; from its 
 lower corner draw the pitch to H ; square down from 
 any point, as V, cutting pitch at U. With V for 
 centre, and a radius touching H R, strike an arc 
 cutting at S ; join S U. This gives bevel Z for joint 
 on lower piece of .wreath. Joint F is made so near 
 the centre line that the variation from a right angle 
 is not perceptible, therefore use a square as shown. 
 
 The mould for upper piece of wreath is found as 
 follows : Extend the tangent B C, Fig. 1, through D; 
 draw from centre of semicircle and through F, par- 
 allel with C D. Square over from D, cutting at 2 ; 
 make height D 4 equal height D N on elevation ; join 
 4 2. This is pitch of plank. 
 
 Fig. 4. Upon a board of sufficient width, draw 
 any line for ^ng diameter; set ofl" 2, 3, 4, to corre- 
 spond with same points on pitch, Fig. 5 ; square these 
 over and make 2 S and 4 N, equal S F and C D, Fig. 
 1; join S N. This line is tangent on mould, and to be 
 correct must equal pitch F N on elevation ; draw the 
 tangent from N through 4, giving direction for 
 straight wood. Make 3 K equal radius on plan, and 
 set half width of rail on each side of K. Find from 
 Fig. 5, by the method before explained, half the long 
 diameter of semi-ellipse and strike the curves on 
 mould. 
 
 Bevel X in elevation is applied to upper joint as 
 shown. The bevel for joint S is found as follow^s : 
 From any point on a line parallel with the long 
 diameter, as 4 (on the line to the right), draw 4 F par- 
 allel with tangent N S on mould. 
 
 Transfer pitch of plank as follows : From 4, Fig. 5, 
 as centre, draw dotted arc of any radius. With same 
 radius return to point 4 and draw an arc of equal length. 
 From its lower corner draw the pitch through 4 ; 
 square up from any point, as N, cutting pitch at II ; 
 with N for centre, and a radius touching F 4, strike an 
 arc cutting at 3 ; join 3 H, giving bevel W for joint 
 S. Its application is showm by the section opposite. 
 
Plate 20 
 
Plate 21. 
 
 THE WEEATH OVEK WIDE WINDEES. 
 
 Figure 1 sliows the plan of stairs, in which the width of 
 winders, at their narrow ends on centre line of rail, is eight 
 inches, and the width of a square step ten inches. When this 
 is the case, or when the widths approximate these, the upper 
 part of wreath may be on the same pitch as rake rail over 
 square steps. The balusters on winders will be one and a 
 quarter inches longer than those on square step : the difference 
 is so slight as not to be noticed when the rail is in its position. 
 The radius for centre line of rail is fifteen inches, and struck 
 from centre E. The curve extends beyond a quarter circle, 
 and terminates on line through J at mitre cap. The tangent 
 J P being square with J E, makes T P equal J P. The 
 number of risers on plan is seven. 
 
 Fig. 2 shows the elevation : Draw any line for top of first 
 step, and set off from it the height of seven risers. Draw 
 any perpendicular, as that through S T. This represents the 
 spring line. Set off to the right of it the distance T P on the 
 plan, and draw the dotted line P H. Then, on the left of 
 S T, make the elevation of a square step and winder to equal 
 those on plan. Let under side of rake rail rest on centre of 
 short balusters. Set off half its thickness, cutting through 
 spring line at T, and ending at P. This is upper pitch of 
 wreath. The point P being fixed, draw the lower pitch 
 from it, say through S. This point is nearly half a riser 
 higher than in preceding example. Here we increase height 
 of newel in order to flatten the pitch from P, and make it 
 less difficult in forming the easing on lower part of wreath 
 starting from mitre cap. 
 
 The height for wreath is now fixed at S T. We find a 
 directing line for pitch of plank by drawing from S to the 
 right, and cutting extended pitch at R, giving the distance 
 R S. Transfer this to plan, extending tangent T P to equal 
 it, giving point R. Draw from R through J; draw from P, 
 T and through centre E parallels with R J; again draw 
 through centre E square with R J, cutting at D; make 
 height F A equal height S T on elevation ; draw from D 
 through A. This is pitch of plank. Draw from centre E 
 parallel with the pitch ; make E U equal E T ; square up 
 from U, cutting at Y ; set half width of rail on each side of 
 V. This gives E 4 and E 5 as half long diameters for inside 
 and outside curves on mould. 
 
 Lay the edge of a board on the pitch, and mark the points 
 A B C D ; square these over as shown at Fig. 3 ; make A T, 
 
 C P, and D J, equal F T, N P, and D J on plan ; draw the 
 tangents from P through J and T. These to be correct must 
 equal the pitches S, P, T, Fig. 2. Draw from B through T 
 and J; make B O equal radius on plan ; set off half width of 
 rail on each side of O ; find points for pins and strike the 
 curves ; add straight wood from T. 
 
 In setting off length of mitre on tangent below J, oljserve 
 that the plank here is on a pitch, and this causes the mitre on 
 surface of plank to be longer than that shown at the cap. To 
 obtain it correctly, transfer the length from J on ground plan 
 to the left of S, Fig. 2, indicated by a line cutting the pitch. 
 The exact length is now given on the pitch. After setting 
 this off below J, draw the end of mould parallel with long 
 diameter. 
 
 The angle made by pitch of plank at Fig. 1 gives bevel W, 
 which is applied to edge of plank, giving a plumb line for 
 mitre point. The bevel for joint on straight wood is found 
 as follows : From any point on a line parallel .with long 
 diameter, as 3 (on line through 3, 6), draw 3, 2, parallel with 
 tangent T P on mould. Transfer pitch of plank as follows : 
 With C, Fig. 1, for centre, and any radius, draw dotted arc ; 
 with same radius, and from 3 as centre, draw an arc of equal 
 length ; from its lower corner draw the pitch through 3 ; 
 Square up from any point, as 4, cutting the pitch at 5 ; with 
 4 as centre and a radius touching 3, 2, strike an arc, cutting 
 at 6; draw from 5 through 6. This gives bevel X as re- 
 quired. Its application is shown by the section opposite. 
 
 Work this wreath by the method explained in plate 18, in 
 which is shown a similar construction. It might be well to 
 mention, however, that when the wreath is Avorked to its cylin- 
 der form and the lower slab cut off square with the mitre, the 
 height should be tested. This may be done by standing the 
 piece on a level surface and measuring up to half thickness 
 of rail on spring line. This to be correct should equal S T, 
 Fig. 2. In case, however, of any slight variation in this, the 
 newel may be set to suit it. 
 
 To find the exact difference in length of balusters marked 
 2 and 3 : Draw from the centre of 2 parallel with R J, cutting 
 at K. This determines the distance T K on upper tangent ; 
 set this off to the right of S on dotted line. Fig. 2, giving K ; 
 square up from K to under side of rake rail. The space at 
 this point between the top of step and under side of rail is the 
 difference in length of balusters. 
 
Plate 22. 
 
 THE WEEATH STARTING PROM A NEWEL. 
 
 Figure 1 shows the plan of stairs starting with curved 
 risers. The ends of square and curved steps on centre line 
 of rail are of equal width, making balusters under the wreath 
 and straight rail the same length. The radius for centre line 
 of rail is twenty-two inches, and its curve less than a quarter 
 circle, its centre being E, and the tangents are ABC. 
 
 Fig. 2. The dotted perpendiculars through ABC show 
 the tangents unfolded. The perpendicular through C repre- 
 sents spring line E C on plan. Make the elevation of three 
 risers with square steps standing in the same position as those 
 on plan. Let under side of rake rail rest on centre of short 
 balusters O O. Set off half its thickness, cutting through 
 tangents C, B, and fixing B as a point from which to draw 
 lower pitch. It will be seen that to draw this from B parallel 
 with the steps, would throw the lower part of wreath too high. 
 Therefore assume line B A for lower pitch. This makes 
 height of newel nearly the height of a riser — more than a 
 short baluster. Now transfer length of mitre from cap, and 
 set it off from perpendicular through A, cutting the pitch to 
 the right of A. To find height of wreath and directing line 
 for pitch of plank, Draw from A to the left, cutting at 6, giving 
 6 C as height ; extend pitch B C, cutting at D, giving 6 D ; 
 transfer this distance to upper tangent on plan, making C D 
 equal it ; draw from D through A ; also draw from B and C 
 parallel with D A ; now draw through centre E square with 
 D A ; make height 6.5 equal height 6 C on elevation ; join 
 5.2. This is pitch of plank. Draw any line parallel with 
 pitch ; make E F equal E C ; square up from E and F, cut- 
 ting at 3 and O ; set half width of rail on each side of O, as 
 shown. Then 3 J and 3 L will be half the long diameters of 
 curves on the mould. 
 
 Now lay the edge of a board on the pitch, and mark the 
 
 points 2, 3, 4, 5 ; square these over as shown in Fig. 3 ; make 
 2 C, 4 B, and 5 A, equal 2 C, N B, and 6 A, Fig. 1 ; draw the 
 tangents from B through A and C. These to be correct must 
 equal pitches A, B, C, Fig. 2. Draw from 3 through A and 
 C ; make 3 E equal radius on plan ; set half width of rail on 
 each side of E; find points for pins, and strike the curves; 
 set off length of mitre, taken from pitch. Fig. 2, on tangent 
 beyond A, and draw end of mould by it, parallel with long 
 diameter. We will again state that the end of mould is to 
 be left straight, and not cut by the mitre. The lines on it 
 give a direction to mark the mitre on surface of plank. The 
 bevel X in angle of pitch, applied to the edge of the plank, 
 gives a line which will be plumb when the wreath is in posi- 
 tion ; In applying the mould to either side. Let lower end 
 be kept fair with edge of plank, and mitre point directly over 
 the line made by bevel X. 
 
 Fig. 4. To find a bevel for joint on straight wood. 
 From any point, on a line parallel with long diameter, as K, 
 draw a line K P parallel with tangent B C. Transfer pitch 
 of plank from Fig. 1 as folloM's : With 5 for centre and any 
 radius draw dotted arc ; with same radius, and from K as 
 centre, draw an arc of equal length ; from its lower corner 
 draw the pitch through K ; square up a line from any point, 
 as J, cutting the pitch at T; with J for centre and a radius 
 touching K P strike an arc, cutting at V ; join Y T, which 
 gives bevel W for upper joint. 
 
 The arrangement of this plan is such as to produce the 
 best possible effect in every part of the stairs. Particular 
 care should be taken to have the mitre accurately cut ; for if 
 there be the least variation from the exact length, either 
 wreath or newel will be thrown from its true position and the 
 balusters stand out of plumb. 
 
Plate 2 2 
 
riaie 25 
 
Plat 
 
 THE WEEATH OVER IIVE WIUDEES 
 
 Figure 1 shows a plan of quarter circle stairs, hav- 
 ing five winders, with square steps above and below. 
 The number of risers counting from square step and 
 including riser above is eight. The wreath to be in 
 two pieces, each forming its own ramp, having 
 straight wood thrown on the same pitch as rake rail 
 over square steps. To do this, divide the quarter 
 circle equally, as B ; and from centre L, draw a joint 
 through B. This makes equal tangents on the plan, 
 as 2 A B, and B S K. By unfolding two of these we 
 find height and pitches of the wreath. 
 
 Fig. 2. Here unfold tangents 2 A B of plan, and 
 through which draw perpendiculars; that through 
 2 F P represents the spring line. The winder and 
 square step above, and below, stand from the spring 
 line the same as those on plan, and the whole num- 
 ber of risers is eight. Proceed and draw the pitches 
 through corners of square steps, cutting the spring 
 line at points 2 and P. Divide this distance equally 
 as F ; square over from F, cutting at 4 ; draw from 4 
 through 3. This gives 2 3 4 as pitches for one-half 
 the wreath ; and F P is one-half the height. Find 
 the direction of a line for pitch of plank, by squaring 
 over from 2, cutting at B. This gives B T. Extend 
 tangent B S on plan to equal B T, Fig. 2, and draw 
 from T through K ; next draw from B and S parallel 
 with T K ; again draw from centre L square Avith T 
 K. Make height F P equal that of F P, Fig. 2 ; join 
 P E. Now draw from L parallel with E P, cutting 
 at V. This is pitch of pl«.nk. 
 
 Find half long diameter of semi-ellipse, by drawing 
 any line parallel with the pitch. Make L 0 equal 
 L K. Square up from this point, cutting the pitch 
 in 0, and on each side of 0 set off half width of rail. 
 This gives L 2 for inside curve on mould, and L 3 
 
 E 23. 
 
 AND AEOUND A QUAETEE CIEOLE. 
 
 for outside curve. Lay a rod on the pitch and mark 
 the points L, N, J, V, Upon a board ; draw a line for 
 the long diameter as shown on the right. Mark the 
 points L, N, J, V, from the rod and square them 
 over. Make N 4, J 3, and V 2, equal E B, C S, and 
 F K, Fig. 1. Draw the tangents from 3 through 2 
 and 4. These to be correct must equal the pitches 
 
 2, 3, 4, Fig. 2. Draw from L through 2. Make 
 L K equal the radius on plan and set half width of 
 rail on each side of K. Find points for pins and 
 strike the curves ; add straight wood parallel with 
 tangent 3, 2, and make the joint through 4 square 
 with tangent. 
 
 Fig. 3 shows the joint bevels. From any point on 
 a line parallel with the long diameter, as 3, draw 3, 
 2 and 3, 4 parallel with tangents on mould. Trans- 
 fer pitch of plank as follows : From V, Fig. 1, as 
 centre, and with any radius, draw dotted arc. With 
 same radius and 3 for centre draw an arc of equal 
 length. From its upper corner draw the pitch 
 through 3 ; square over from any point, as P, cutting 
 pitch at T. With P for centre, and a radius touching 
 
 3, 2, strike an arc, cutting at L; join L T. This 
 gives bevel X for upper and lower joints of wreath. 
 With same centre, and a radius touching 4 3, strike 
 an arc, cutting at R; draw from T through R. This 
 gives bevel W for centre joint. The application of 
 bevels are shown by the sections. 
 
 A saving of material may be effected in cutting 
 the piece to a parallel width by a rough pattern ; but 
 the mould as drawn is necessary in giving the piece 
 its cylinder form. In working it is best to tack the 
 mould to the piece, and when bolting the centre joint 
 be careful to keep the lines made by bevel W on eacli 
 piece exactly opposite. 
 
Plate 24. 
 
 THE WEEATH STAETING EEOM A NEWEL AND STANDING OTER TOUR WINDEES. 
 
 Figure 1 shows plan of stairs often adopted by country- 
 carpenters, who, as a rule, omit the rail, leaving that for 
 others to put up. Their stairs often present an awkward 
 arrangement, for in planning them they seldom pay any at- 
 tention to the rail that is to follow. In this plan the newel 
 stands in such a position as to shorten the winder starting, 
 and give to the whole stairs a contracted appearance. All 
 this could have been improved by simply extending the curve 
 as shown in previous examples ; but now the position of 
 newel is fixed, and everything complete except the rail. The 
 wreath will be in two pieces — the upper piece to form its own 
 ramp, with straight part thrown on tlie same pitch as rake 
 rail over square steps, and the lower piece forming an easing 
 starting from mitre cap. It will be necessary to place the 
 joint in such a position as will make the upper tangents 
 shorter than the lower ones. This is shown on the plan. 
 Tiie joint being drawn through C makes the upper tangents 
 form the angle C D E, and gives for lower tangents the angle 
 ABC. Tlie number of risers shown on plan is six. 
 
 Fig. 2. The perpendiculars through A, B, C, D, and E 
 show the tangents unfolded, those through A and E repre- 
 senting spring lines A and E on plan. Draw a line for top 
 of first winder as shown. Setoff from this the height of four 
 risers on line to right giving a point as E. From this draw 
 the elevation of a square step. Let under side of rake rail 
 rest on centre of short balusters O O. Set off half its thick- 
 ness, cutting through E and D. This fixes D as a point 
 from which to draw, say through B, and from B through A. 
 This gives A B as the pitch starting from mitre cap. Square 
 over from A, cutting at K. This gives K C as height for 
 lower piece of wreath. Draw through C, cutting at F and 
 H, giving H E as height for upper piece. 
 
 To find the mould for upper piece of wreath : Extend tan- 
 gent E D on plan to equal H F, Fig. 2. Draw from F 
 through C, and from D and E parallels with F C. Now 
 draw any line square with H C, as H J, Make height J L 
 equal height H E, Fig. 2. Join L H. This is pitch of 
 plank. Draw from H P L square with, pitch. Make L E, 
 
 D, and H C equal J E, S D, and H C on the left. Draw 
 tii-e tangents from D through E and C. These to be correct 
 must equal EDO, Fig. 2. To find the junction of straight 
 wood with curves on mould, make H 2 equal H 2 on the 
 left. Draw from 2 through E. Before proceeding further, 
 it is understood that this wreath is to form its own ramp. In 
 doing so, the surface of plank is thrown on two pitches, so 
 that a line across its upper edge would be almost parallel Avith 
 a step. Now if we give this mould its exact cylinder form 
 and cut the stuff by it, there would undoubtedly be a waste 
 of material, and, while we repeat that no parallel mould for 
 an inclined wreath can be correct, we may allow the use of 
 
 one in this case. The convex curve nearly forms the ramp, 
 and the concave curve, being much shorter, gives a line on 
 surface of plank which some may deem sufficiently correct. Al- 
 lowing this, we will proceed and find the curve through centre 
 of mould. Set off from each side of it half width of rail, and 
 trace the inside and outside curves. Cut the stuff full for 
 this piece of wreath. To find joint bevels : From any point 
 on a line parallel with pitch H L, as D (on the line to the 
 right), Draw D C and D E parallel with tangents on mould. 
 Transfer pitch of plank as follows : With L, [Fig. 1] for centre 
 and any radius draw dotted arc. With same radius and D 
 for centre draw an arc of equal length. From its upper 
 corner draw pitch through D. Square down a line from any 
 point, as A, cutting the pitch at B. With A for centre and 
 a radius touching D E strike an arc, cutting at L. Join 
 L B. This gives bevel X for upper joint. From same 
 centre, and with a radius touching D C, strike an arc, cutting 
 to the right of A. Draw from B through this point, which 
 gives bevel W for centre joint. The application of bevels are 
 shown by square sections of rail. 
 
 The mould for lower piece of wreath is found as follows : 
 Extend tangent C B on plan to equal N K on elevation. 
 From its extended point, N, draw through A. Then from 
 B and C draw parallel with N A. Now draw any line, as 
 that through K, square with K, A, N. Make height K C 
 equal height K C on elevation. Draw from J through C. 
 This is pitch of plank. Again draw from J R C square 
 with pitch. Make C A equal K A, R B equal T B, and J C 
 equal J C. Draw tangents from B through C and A, which 
 to be correct must equal pitches A B C on elevation. Now 
 draw any number of lines through width of rail on plan, 
 cutting the pitch. Square over from the intersections and find 
 points, through which trace the curves on mould. Make J S 
 on the right equal J S (above). Draw from S through A. 
 Set off length of mitre to the left of A on tangent, taking it 
 from lower pitch on elevation, and through the point thus 
 given draw end of mould parallel with pitch C J. The bevel 
 in the angle made by pitch is to be applied to edge of plank, 
 giving plumb line for mitre point. The bevel for joint C is 
 found as follows : Draw 2, 3, parallel with tangent B C. 
 Square over by the pitch from any point, as V, cutting at R. 
 With V for centre, and a radius touching 2, 3, strike an arc, 
 cutting at P. Draw from R through P. This gives bevel 
 Z for joint C. Its application is shown by section opposite. 
 
 The application of the mould is the same as in preceding 
 examples. If desired, the stuff may first be cut to a parallel 
 width. The plumb line for mitre cut being on square edge, 
 is a direction to cut off the slabs and form the mitre. The 
 mitre should be cut by a bevel set to the plan, and not by a 
 square, as it differs from a true mitre. 
 
Plate 2 5 
 
Plate 25. 
 
 THE WEEATH IN TWO PIECES, ONE 
 
 Figure 1 shows the plan of stairs starting from a 
 newel, having two square steps below, four winders 
 in the quarter circle, and square steps above. The 
 wreath to be in two pieces. The upper one forming 
 its own ramp, and its straight wood thrown on the 
 same pitch as rake rail over square steps. The 
 lower piece to have sufficient straight wood, to stand 
 over the two square steps at the starting, and form 
 the easing and mitre at the cap. To do this; divide 
 the quarter circle into two equal parts as E. Through 
 E and H draw tangents meeting in point F. The 
 number of risers shown on plan is eight. The quar- 
 ter circle is centre line of rail. 
 
 Now proceed, by making an elevation showing the 
 pitches and height of wreath. This is done by drop- 
 ping perpendiculars from F and H ; that on the right 
 represents spring line. From this line on lower part 
 of plate, make the elevation of the two square steps 
 starting. Now set off the number of risers same as on 
 plan, and make the elevation of square step above. 
 Draw lower pitch through corners of steps, cutting 
 the perpendiculars at V and K, draw the upper pitch 
 cutting through N and P. This gives the whole 
 height of wreath on spring as V N, which divide in 
 J. Join P J and K J. This shows both pieces of 
 wreath having equal pitches. Find the pitch of 
 plank, by squaring over from J cutting at R. Ex- 
 tend tangent H F on plan to the left. Make H T 
 equal J K on elevation. Draw from T through E, 
 draw from F and H parallel with T E. Now draw 
 from centre A square with T E. Make height K V 
 equal height V J on elevation. Join V U. Draw 
 from A parallel with U V cutting through D. This 
 is pitch of plank. 
 
 Find half long diameter of semi-ellipse, by making 
 A L equal A H. Square up from L cutting at 0, and 
 on each side of 0 set off half width of rail. This 
 gives A 2 as half diameter for inside curve on mould, 
 and A 3 for outside curve. To find the mould : Lny 
 a rod on the pitch and mark the points A B C D. 
 
 or WHICH POEMS A DOUBLE EAMP. 
 
 Upon the ftice of a board, as shown at Fig. 3, draw 
 a line for long diameter. Mark points A B C D 
 from the rod, and square them over. Make B J, C 
 K, and D Y, equal U E, K F, and L H, Fig. 1. Draw 
 from K through J and V. These are tangents on the 
 mould, and to be correct must equal pitches V K J, 
 Fig. 2. Make A II equal radius on plan, set half 
 width of rail on each side of H. Find points for 
 pins, and strike the curves on mould. Draw from A 
 through V. Add straight wood parallel with tangent 
 K V. Make V S equal V S on pitch. 
 
 Fig. 2. Here observe that the line from the step 
 giving point S, represents line S through mitre cap, 
 on plan ; both stand in the same position from riser, 
 and to the right of S on the pitch is shown length 
 of mitre. This transfer to tangent from S on mould, 
 and draw parallel with long diameter. To find the 
 mitre on surface of plank : Draw through S parallel 
 with A V, join the points and the form is shown. 
 The pitch at Fig. 1 shows bevel Y. This is applied 
 to the square edge of plank, and gives the mitre 
 point on wreath. The straight wood on upper piece 
 of wreath may be any length, as dotted line across 
 the mould shows. The bevel for this and centre 
 joint is found as follows : 
 
 Fig. 4. From any point on a line parallel with 
 long diameter, as K, draw K V and K J parallel 
 with tangents on mould. Transfer pitch of plank as 
 follows: From B, Fig. 1, as centre, and with any 
 radius draw dotted arc. With same radius and K for 
 centre, draw an arc of equal length. From its upper 
 corner draw the pitch through K. Square over from 
 any point, as P, cutting the pitch at N. With P for 
 centre, and a radius touching K V, strike an arc cut- 
 ting at L, draw from N through L. This gives bevel 
 X for joint on straight wood of upper piece of wreath. 
 With same centre, and a radius touching K J, strike 
 an arc cutting at F, draw from N through F. This 
 gives bevel W for centre joint J for both pieces of 
 wreath. 
 
Plate 26. 
 
 TEE WKEATH STANDING OVER OUEYES 01 UNEQUAL RADIUS. 
 
 In this case the rail starts from the newel with a 
 curve, which is the only difference from the plan of 
 preceding plate. But these stairs, starting in the 
 manner stated, presents a more finished and better 
 effect than the former. The wreath as usual to be 
 in two pieces. The upper forming its own ramp. 
 The lower forms mitre and easing at the cap, and 
 makes two unequal pitches from the newel. The 
 number of risers shown on plan is eight, or counting 
 from top of first step, seven. 
 
 To prevent the ramp on upper piece of wreath 
 from being too high. Let the two upper tangents on 
 plan be short. To do this, fix a joint say at H, Then 
 through H and from E, draw tangents. These lines 
 intersect at points F and J; making H J and J 7 
 equal; and E F and F H unequal. The length of 
 tangents are now given, and by unfolding three of 
 them, as E F, F H and J 7, the pitches of wreath 
 are obtained, as shown at Fig. 2, by perpendiculars 
 passing through points 7 J and F E. That through 
 7 represents the spring line, and on which set off the 
 number of risers from top of first step to equal those 
 on plan. Then draw the upper pitch, cutting 
 through 7 and J. From J as a fixed point, draw 
 say through point F ; and from F draw the pitch 
 starting from mitre cap, as F E. The pitches of 
 wreath being given, find the heights by drawing from 
 E, cutting at L. This gives L H for height of lower 
 piece of wreath. Height of upper piece being H 7, 
 Find a direction for pitch of plank by squaring over 
 from H, cutting at N ; transfer H N to upper tan- 
 gent on plan, making 7 N equal it ; draw from N 
 through H ; draw from J and 7 parallel with N H. 
 From any point as A, square over A 2 ; make height 
 A B equal height H 7, Fig. 2; join B C. This is 
 pitch of plank for upper piece of wreath. From 
 points of intersection, as C, P, B, Draw square with 
 the pitch. 
 
 Make B H, P J, and C 7 equal A H, 2 J, and C 7 
 on the left; Draw from J through 7 and H. These 
 are tangents on the mould, and to be correct must 
 equal pitches 7 J H, Fig. 2. Find the junction of 
 straight wood, by making B R equal A R ; draw from 
 R through 7. Now draw any number of lines from 
 the plan parallel with N H, cutting the pitch ; and 
 from intersections thus given, draw square with B C. 
 
 Transfer measurements from the plan, by which 
 points are given to trace the curves on mould. 
 
 Fig. 5. To find joint bevels : From any point on a 
 line parallel with the pitch, as K, draw K J and K 
 H parallel with tangents on mould. Transfer pitch 
 of plank as follows : From B, Fig. 4, as centre, and 
 with any radius, draw dotted arc. With same radius 
 and K for centre, draw an arc of equal length. From 
 its upper corner, draw the pitch through K; square 
 over from any point, as 2, cutting the pitch at 0. 
 With 2 as centre, and a radius touching K J, strike 
 an arc, cutting at 6; join 6 0. This gives bevel X 
 for joint on straight wood of upper piece of wreath. 
 With same centre, and a radius touching K H, strike 
 an arc, cutting at a point on the right of 2 ; through 
 which draw from 0. This gives bevel W for joint H. 
 
 To find the mould for lower piece of wreath : Make 
 H L, Fig. 1, equal L K, Fig. 2 ; Draw from L through 
 E ; draw from F and H parallel with L E. From 
 any point, as P, draw P V square with L E. Make 
 height P H equal height L H, Fig. 2 ; draw from V 
 through H. This is pitch of plank. Draw from 
 points y Y H, square with pitch. Make V E, Y F, 
 and H H, Fig. 3, equal V E, 2 F, and P H, Fig. 1 ; 
 draw from F, through E and II. These are tangents 
 on the mould, and to be correct must equal pitches 
 E F H, Fig. 2. Set off length of mitre from E on 
 tangent to equal that shown on pitch to the right of 
 E, Fig. 2. Make 3 T equal T S; draw from T 
 through E ; again draw through mitre point, parallel 
 with the pitch. This line represents end of mould 
 and the square edge of plank. Both are fair on the 
 application of mould. The bevel Z, shown in angle 
 of pitch, gives mitre point on edge of plank. Find 
 curves on mould, by drawing any number of lines 
 through width of rail on plan parallel with L E, and 
 cutting the pitch. From points thus given, draw 
 square with pitch. Transfer measurements from the 
 plan, and through these points trace curves on mould. 
 
 Make joint H square with H F. To find a bevel 
 for this, draw from angle of pitch H parallel with 
 tangent H F on mould. Extend T 3, giving a point 
 on line P II. With 3 as centre, and a radius touch- 
 ing S H, strike an arc, cutting on the right of 3, 
 through which point draw from point on line P H. 
 This gives bevel W for joint H. Moulds and bevels 
 for both pieces of wreath are now complete. 
 
Plat 
 
 E 2 7. 
 
 THE WEEATH OVEE SQUAEE STEPS, 
 
 Figure 1 shows a ground-plan of stairs, the wreath 
 having a joint opposite half the diameter of semi- 
 circle. Each piece of wreath to form its own ramp, 
 and have straight wood thrown on the same pitch as 
 rake rail over square steps. The numher of risers 
 shown in plan is eight. The position of risers from 
 spring line indicated by arrows. The dotted perpen- 
 dicular on the right represent the spring line; on 
 which set off the same number of risers as on plan. 
 Draw the elevation of square step above and below, 
 to stand in same position as those on plan. The pro- 
 jection of risers from spring line shown by arrows 
 the same as on plan. 
 
 Proceed by drawing lower pitch through corners 
 of square steps, cutting perpendiculars at points C 
 and Y. Draw the upper pitch cutting through H and 
 F. Divide Y F in K; square over from K; cutting in 
 J ; join F J ; draw from J through Y. This gives C 
 Y J as pitches for one-half of wreath. To find its 
 height, square over from C to the left; this gives B K 
 as height. To draw the mould, form a square as that 
 shown, its sides being equal to radius of plan. Make 
 the distance A B equal A B on the right. Draw 
 from E through B ; draw from K D A parallel with 
 E B; again draw through D square with E B. Make 
 height J T equal height B K on the right; join 
 T R. This is pitch of plank. Find half long 
 diameter by drawing any line as P 0 parallel with 
 pitch. Make D N equal D A. Square up from N 
 cutting at 0; and on each side of 0 set half width 
 of rail. This gives P 2 for inside curve on mould, 
 and P 3 for outside curve. Proceed by laying a 
 rod on the pitch. Mark the points R P S T ; have 
 a board as shown in^Fig. 3, and draw a line for long 
 diameter. Then by the rod mark the points BPS 
 T ; square these over, and make R J, S F, and T H, 
 equal R K, L E, and J A, Fig. 2. Draw from F 
 through H and J. These are tangents on the mould, 
 and to be correct must equal pitches C Y J, Fig. 2. 
 Make P 0 equal radius on plan, and on each side of 
 0, set half width of rail ; draw from P through H. 
 Find points for pins, and strike the curves on mould. 
 
 WINDEES, AND QUAETEE LANDING. 
 
 Add straight wood parallel with H F. Make the 
 joint through J square with J F. 
 
 Fig. 4 show joint bevels, which are found as 
 follows : From any point on a line parallel with long 
 diameter, as R; draw R H and R F parallel with 
 tangents on mould. Transfer pitch of plank, by 
 taking T, Fig. 2, for centre, and with any radius, 
 draw dotted arc. With same radius, and R for 
 centre, draw an arc of equal length. Through its 
 lower corner and R draw the pitch ; Square over from 
 any point, as P, cutting the pitch at N, with P as 
 centre, and a radius touching R H; strike an arc 
 cutting at T ; join T N. This gives bevel W for 
 joint on straight wood. With same centre, and a 
 radius touching R F, strike an arc cutting at S; draw 
 from N through S. This gives bevel X for centre 
 joints of both pieces of wreath. The square sec- 
 tions of plank and rail show the application of these 
 bevels to the joints. 
 
 It will be noticed, that in making the upper part 
 of this wreath form its own ramp, that the rake rail 
 over square steps has to run into the cylinder, and 
 terminate at point F. But by doing so the eleva- 
 tion shows at a glance, that the wreath over the 
 upper winder stands too high ; whilst the lower part 
 of wreath having the same pitches as the upper, its 
 height is perfectly correct. This method of forming 
 a wreath being the most simple, it is the most often 
 adopted, for, by means of it, the trouble of working 
 a ramp on the straight rail is avoided, which is al- 
 ways desirable where it can be done, and at the same 
 time have the heights uniform. This wreath when 
 finished would certainly present to the eye the best 
 possible effect, and its height everywhere regular, 
 except the point just mentioned; and this alone is 
 sufficient to condemn and make it objectionable. 
 The motive in giving this drawing is to assist and 
 instruct the learner, by pointing out such errors and 
 defects in a construction as might possibly escape the 
 notice of even practical men. Then by turning to 
 the following plate, show a more approved method 
 for producing a similar wreath. 
 
Plate 28. 
 
 SECOND EXAMPLE OF A WEEATH OVEE SQUAEE STEPS, WINDEES AND QUAETEE LANDING. 
 
 Figure 1 shows a plan similar to the preceding — the 
 risers here being placed in such a position as to give the 
 wreath a regular and uniform height throughout. To do 
 this let a riser stand on the spring line as shown on the left, 
 and make tangents 2 3 and 3 4 each to equal half a square 
 step. This at once fixes a joint at 4, and from which draw 
 riser starting from the landing. With half a square step in 
 the compasses, set off from 4 the number of winders on centre 
 line of rail. Extend tangent 3, 4, cutting at 5. This gives 
 4, 5, 6 as tangents for upper piece of wreath, and 2, 3, 4 for 
 the lower piece. To find height and pitches of this wreath, 
 let tangents, winders, and square steps be unfolded on a nar- 
 row board, and in the same manner as laying out a string. 
 This is shown at 
 
 Fig. 2. Here the dotted perpendiculars passing through 
 points 2, 3, 4, 5, 6 show the tangents of plan unfolded — the 
 square steps, landing, and winders being in the same position 
 as those on the plan. Draw the lower pitch, cutting perpen- 
 dicular at 5. Next draw the upper pitch of square step. 
 Take any point on this pitch that will form an easy ramp, as 
 K. Draw from K through 5. This determines pitches for 
 upper piece of wreath, as 4, 5, 6. To find its height. Draw 
 from points 4 and 6 parallel with steps, cutting perpendicular 
 through 5. This gives D N for the height. Extend pitch 
 6, 5, cutting at F. This gives the distance, F D, which 
 transfer to tangent on plan, making 5 F equal it; Draw 
 from F through 4 ; draw from 5 and 6 parallel with F 4 ; 
 again draw through centre O square with F 4; make 
 height P N equal height D N, Fig. 2 ; join N J. This is 
 pitch of plank. Find half long diameter of semi-ellipse by 
 drawing any line parallel with the pitch; Make O Y equal 
 radius O 6 ; square up from V, cutting at O ; on each side of 
 it set off half width of rail. This gives L 2 for inside curve 
 on mould, and L 3 for outside curve. 
 
 To find the mould, lay the edge of a board on the pitch, 
 and mark the points J K L N ; these square over as shown 
 at Fig. 3 ; Make N 6, K 5, and J 4 equal P 6, C 5, and J 4 
 on plan ; draw from 5 through 4 and 6. These are tangents 
 on the mould, and to be correct must equal pitches 4, 5, 6 on 
 elevation ; make L C equal radius on plan ; set half width of 
 rail on each side of C; draw from L through 6. Now find 
 points for pins, and strike the curves on mould ; make the 
 joints square with tangents. 
 
 Fig. 4 shows bevels for joints, and are found as follows : 
 From any point, as N on a line parallel with long diameter. 
 Draw N 4 and N 6 parallel with tangents on mould ; transfer 
 pitch from Fig. 1 ; Take N as centre, and with any radius 
 
 draw dotted arc; with same radius, and N, Fig. 4, for centre, 
 draw an arc of equal length ; from its lower corner draw the 
 pitch through N ; Square over from any point, as Y, cutting 
 the pitch at R ; with Y as centre and a radius touching 4 N, 
 strike an arc, cutting at P ; join P R. This gives bevel W 
 for joint 4. With same centre and a radius touching N 6, 
 strike an arc, cutting at Y ; join Y R. This gives bevel X 
 for joint on straight wood of wreath. The application of 
 bevels shown by square sections of plank. 
 
 To find the mould for lower wreath piece : This could be 
 done on the plan, but in order to avoid confusion, let the 
 centre line of rail be given at any convenient place, as Fig. 5. 
 Here draw tangents A B C to equal 2 3 4 on plan ; make 
 height T Y equal height B 3 on elevation. Fig. 2 ; draw 
 from H through Y. This is pitch of plank. Find half 
 diameter by squaring up from R, cutting the pitch at P. Set 
 half width of rail on each side of P. This gives Y 2 as half 
 diameter for inside curve on mould, and Y 3 for outside 
 curve. To draw the mould : From the edge of a board 
 square over a line, as Y 3, Fig. 6 ; Make Y P and Y 4 equal 
 H Y, Fig. 5 ; square over the lines ; make Y 3, 4, 2 and 
 P 4 equal T B and H A, Fig. 5 ; draw from 3 through 2 
 and 4. These are tangents on the mould, and to be correct 
 must equal pitches 2, 3, 4 on elevation. Fig. 2 ; make Y O 
 equal radius on plan ; set half width of rail on each side of O; 
 draw from Y through 2 ; Now find points for pins, and strike 
 the curves on mould ; add straight wood parallel with 2 3 ; 
 make the joints square with tangents. 
 
 The tangents on this piece of wreath being of equal lengths, 
 and having equal pitches, therefore one bevel answers for 
 both joints. To find it, draw P J parallel with tangent 3, 2 
 on mould ; Transfer pitch of plank from Fig. 5 by taking P 
 as centre, and with any radius draw the dotted arc ; with 
 same radius return to Fig. 6, and from P as centre draw an 
 arc of equal length, and from its upper corner draw the pitch 
 through P, cutting at K ; With Y as centre and a radius 
 touching P Jy strike an arc, cutting at L; draw from K 
 through L. This gives bevel X for both joints. 
 
 It is now very evident by the elevation that this wreath is 
 decidedly superior to the preceding. Here the lengths of 
 short balusters are equal on square steps and winders, the 
 pitches, heights and curves regular. We have already stated 
 that the tangents for any wreath can be unfolded on a narrow 
 board, and we may add, that the best tool for this purpose 
 is a framing square, having a fence fastened on its blades 
 to slide along the edge of a board when laying out the 
 work. 
 
Flale 29 
 
Plate 29. 
 
 THE SIDE EAIL, ALONG STRAIGHT If A 
 
 Figure 1 shows the plan of stairs where it is re- 
 quired to have a side rail. The centre line of rail in 
 the quarter-circle is divided into two equal parts as 
 point C. Through this draw a tangent intersecting 
 those from A and E in the points B and D. The 
 wreath to be in two pieces. The rake rail over 
 square steps above and below, must have a ramp to 
 meet the wreath as it stands on a different pitch, 
 owing to the winders along the wall being wider than 
 square steps. This will be better understood by 
 making the elevation of two or three square steps 
 as shown at Tig. 2. Here drop perpendiculars from 
 D and E ; take any point as S, and square over S 2 ; 
 make S B equal the height of four risers, this being 
 the same number as from E to joint C on plan ; draw 
 through B parallel with S 2 ; divide S B in L. 
 
 Now make the elevation of a few square steps and 
 risers from point 2, and through corners of steps 
 draw the pitch. Take any point as third riser on 
 the right of 2, and draw from it, cutting through L 
 and extended line 3 B. This gives the height of 
 each piece of wreath as 2, 3, and its pitches 2 L 3. 
 We have also the distance B F as direction of a line 
 for pitch of plank. Extend tangent C B to the left; 
 Make B F equal B F on elevation ; Draw from A 
 through F ; draw from B and C parallel with A F. 
 Take any point as T ; draw T 4 square with A F ; 
 make height T S equal height 2, 3 on elevation ; join 
 S 4. This is pitch of plank. 
 
 LS, AND AEOUND A QUAETER-OIEOLE. 
 
 Draw square with the pitch from points S J 4 ; 
 Make 4, 3 equal F A, and J L equal J B ; make S 2 
 equal T C ; draw from L through 2 and 3. These 
 are tangents on the mould, and to be correct must 
 equal pitches 2 L 3, Fig. 2. 
 
 Draw any number of lines through width of rail on 
 plan, cutting the pitch. From intersections thus 
 given, draw square with the pitch. Now find 
 points and trace the curves on mould. The 
 straight wood projecting from 2, make it equal 
 the space between the joint on ramp, and 2, shown 
 on elevation. 
 
 Fig. 4 shows joint bevels, and are found as follows : 
 From any point as L on a line parallel with pitch of 
 plank. Draw L 3 and L 2 parallel with tangents on 
 mould ; Transfer pitch of plank from Fig. 3, by taking 
 S as centre, and with any radius, draw the dotted 
 arc ; with same radius and L for centre, draw an arc 
 of equal length, and from its upper corner, draw the 
 pitch through L ; Square over from any point as V, 
 cutting the pitch at R ; with V for centre, and a 
 radius touching L 3 ; strike an arc cutting at P ; draw 
 from R through P. This gives bevel W for centre 
 joint of wreath. With same centre and a radius 
 touching L 2, strike an arc cutting a point below V ; 
 draw from R through the point. This gives bevel X 
 for joint on straight wood. The square sections of 
 plank show the application of both bevels. 
 
Plate 30. 
 
 THE WREATH OV 
 
 Figure 1 shows stairs where the risers are placed in 
 the most awkward position imaginable, which is no 
 unusual thing where the work is intrusted to tliose 
 who have no practical knowledge in such matters. 
 Bad and indifferent as the plan undoubtedly is, 
 there is no difficulty in producing a wreath to stand 
 correctly over it. In making the elevation it will be 
 noticed that two moulds are required, as the lower 
 piece of wreath can be made to form its own ramp, 
 and have straight wood thrown on the same pitch as 
 the rake rail over square steps; but the upper piece 
 of wreath will have to connect with a ramp worked 
 on straight rail. 
 
 Fig. 2 shows the elevation of eight risers, being 
 the same number as on plan. The perpendicular 
 passing through 2 and 6 represent the spring line. 
 The riser of square step stands on this line and is in 
 the same position as that on plan. The riser above 
 projects from spring line as indicated by an arrow, the 
 same as on plan. Now proceed by drawing the lower 
 pitch, cutting through 2 and 3; draw the upper pitch 
 through corners of square steps. Take any point on 
 this pitch that will form an easy ramp; assume for this 
 the line cutting through 6 and 5 ; divide 3, 5 equally 
 in V; square over from V, cutting at 4; draw from 4 
 through 3 and 5. This gives 2, 3, 4 as pitches for lower 
 piece of wreath, and 4, 5, 6, for upper piece. 
 
 To find height and pitch of plank for lower piece 
 of wreath, square over from 2, cutting through N and 
 P. This gives P 4 as height. Form the square as 
 shown, its sides being equal to radius of plan ; extend 
 4 3, cutting at J ; Draw from J through S; draw from 
 N P E parallel with J S. Again draw through E , 
 square with J S; make height H A equal height P 4 ; 
 draw from A through D. This gives pitch of plank. 
 Find half long diameter by making E F equal E S ; 
 set half width of rail on each side of F. This gives 
 B 7 for inside curve on mould and B 8 for outside curve. 
 
 To find joint bevels, make N 4 on the right equal 
 N 3 on the left; draw from V through N; take P as 
 centre and a radius touching V N; strike an arc, 
 cutting at R ; join R E. This gives bevel Z for joint 
 on straight wood of wreath. Take N as centre, and 
 a radius touching J 3 ; strike an arc, cutting at Y; 
 draw from S through Y. This gives bevel X for 
 centre joint of lower piece of wreath. 
 
 R riVE WINDERS. 
 
 Now proceed and find the mould. Lay a rod on 
 the pitch and mark the points A B C D. Have a 
 board, as shown at Fig. 3, and draw a line for long 
 diameter ; mark on it from the rod, points A B C D; 
 these square over, and make A 4, C 3, and D 2 equal 
 H P, K N, and D S, Fig. 5 ; draw from 3 through 2 
 and 4. These are tangents on the mould, and to be 
 correct must equal pitches 2, 3, 4, on elevation. 
 Make B 0 equal radius on plan; set half width of 
 rail on each side of 0; draw from B through 2. Now 
 find points for pins and strike the curves; add straight 
 wood parallel with tangent 3 2. This completes the 
 mould for lower piece of wreath. The square sections 
 of plank show application of bevels to joints. 
 
 To find height and pitch of plank for upper piece 
 of wreath : square over a line from 6, cutting through 
 K and R. This gives R 4 as height. Form a square 
 as shown. Extend 4, 5, cutting at J; draw from J 
 through S; draw from K R E parallel with J S. 
 Again draw through E, square with J S; make 
 height H D equal height R 4 ; draw from A through 
 D. This is pitch of plank. To find half of long 
 diameter, make E F equal E S, square up from F cut- 
 ting at 8 ; set half width of rail on each side of 8. 
 This gives C 2 as half long diameter for inside curve 
 on mould ; and C 3, for outside curve. 
 
 To find joint bevels make 4 K on the right equal 
 5 K on the left ; draw from V through K ; take R as 
 centre and a radius touching V K; strike an arc cut- 
 ting at 0 ; join 0 E. This gives bevel W for upper 
 joint of wreath on straight wood; take K as centre, 
 and a radius touching 5 J ; strike an arc cutting at 
 Y ; join Y S. This gives bevel X for centre joint of 
 wreath. Now proceed with the mould. Lay the edge 
 of a board on the pitch, and mark the points A B C D; 
 square these over as shown at Fig. 4. Make A 4, C 5, 
 and D 6 equal H R, N K, and A S, Fig. 1. Draw 
 from 5 through 4 and 6. These are tangents on the 
 mould, and to be correct must equal the pitches 4, 5, 6 
 on elevation. Make B 0 equal radius on plan; set 
 half width of rail on each side of 0 ; draw from B 
 through 6. Now find points for pins and strike the 
 curves; make the joints square with tangents. This 
 completes the m mid for upper piece of wreath. The 
 square sections of plank show application of bevels to 
 joints. 
 
Plaie 30 
 
Plate 31 
 
Plate 31. 
 
 LANDING ¥EEATHS TOE STAIES, WHICH POEM ACUTE AND OBTUSE ANGLES. 
 
 It sometimes happens that for some particular purpose the 
 landing of stairs is thrown out of square. When such is the 
 case, the wreaths must be either more or less than a quarter 
 circle. Here both positions are given, and the manner in 
 which the wreaths are produced. 
 
 Fig. 1 shows a plan where the landing makes an obtuse 
 angle with the stairs, and gives a curve less than a quarter 
 circle. The face of riser landing stands on the spring line. The 
 tangents through centre of rail meet in point C. But before 
 going further it may be well to state that the proper way to 
 proceed in fixing the position of riser landing is to drop a 
 ])erpendicular from C, and then from any point, as B, draw 
 the pitch of a square step. This makes B a point in half the 
 thickness of both rake and level rail. Then the floor line 
 should stand below B, equal to half a riser and half thickness 
 of rail. This at once determines ]iosition of riser landing. 
 To neglect this method is almost certain to have the level 
 either too high or too low for long balusters on square steps, 
 as it is usual to make these answer for the landing. But the 
 plan before us explains the point. Here the riser stands on 
 spring line, which is not the proper position for throwing the 
 level rail to just half a riser above the floor, as the arrow 
 shows a greater space. It has, however, been done in order 
 to show the ditference between a correct and incorrect method 
 for tiie position of a riser landing. 
 
 We now proceed to find pitch of plank and height of 
 this wreath, by squaring over from B, cutting at 2. This 
 gives A 2 as height; Draw from L parallel with tangent 
 C D ; make height D E equal height A 2 ; draw from E 
 through F; draw from centre N parallel with E F, cutting 
 through H. This is pitch of plank, and the points N 2 gives 
 half tiie long diameter for inside curve on mould, and N 3 
 gives the outside curve. Lay a rod on the ]>itch, and mark 
 the points N K H ; have a board as shown at Fig. 2, and draw 
 a line for long diameter; lay on the rod, and mark from it 
 the points N K H ; these square over, and make K A and 
 H B equal F L and D C, Fig. 1 ; draw from B through A. 
 This is tangent on the mould, and to be correct must 
 equal pitch A B, Fig. 1. The tangent B H when in 
 ])osition falls level, and stands over C D on plan ; make N L 
 equal radius on plan ; set half width of rail on eacl side of L; 
 Draw from N through A. Now find points for pins and 
 strike the curves ; draw the straight wood {parallel with B A, 
 and add three or four inches of straight wood below H. This 
 completes the mould. 
 
 Fig. 3. To find joint bevels. From any point, as A on a line 
 parallel with long diameter, draw A B parallel with tangent 
 B A on mould. Transfer pitch of plank from the plan as 
 follows : Take F as centre, and with any radius, draw the 
 dotted arc ; with same radius, and A as centre, draw an arc 
 of equal length, and from its lower corner draw the pitch 
 through A. This gives bevel X for upper joint on wreath ; 
 Square over from any point, as D, cutting the pitch at O ; 
 with D as centre, and a radius touching A B, strike an arc, 
 cutting at E; draw from O through E. This gives bevel W 
 for lower joint. It has already been shown that the exact 
 width of any mould can be given by having the bevels for 
 
 joints, and by the following method : From the edge of a 
 tioard draw the bevel lines E O and A O; run a gauge line 
 to half the width of rail, cutting at points F and C. This 
 gives A C as half width of mould across its wide end; and 
 E F gives half the width across the narrow end. The 
 square sections of plank show the application of bevels to 
 joints. 
 
 Fig. 4. Here the stairs and landing form an acute or sharp 
 angle. The curve for centre line of rail may be any radius, 
 as J L, touching both sides of the angle. The object now to 
 determine is the exact position of a riser on the plan, in order 
 to have the level rail stand a proper height above the floor. 
 liCt the wreath be in two pieces having a joint on any part of 
 the curve, as D, through which draw a line square with D L, 
 cutting in C; square down from J and C; Take any point, as 
 B on line from C; Draw from B the pitch of a square step; 
 then from point B set off half a riser and half thickness of 
 rail, and draw the floor; set off from the pitch half thick- 
 ness of rail, intersecting the floor. This fixes O as the 
 centre of a short baluster, on which under side of rail rests, 
 and gives a clear direction to draw face of riser, as shown ; Ex- 
 tend this, cutting through tangent at 3, and we have the 
 exact position of riser on })lan, which will cause under side 
 of level rail on landing to stand half a riser above a short 
 baluster; or make it the proper height for long balusters on 
 square steps. The same rule may be applied to all similar 
 positions. 
 
 To find the height and pitch of plank for the wreath. 
 Square over from B, cutting at 2. This gives A 2 as height ; 
 Draw from J })arallel with C D, cutting line through D L at 
 N ; make height D 2 equal height A 2 ; draw from 2 through 
 N. This is pitch of ])lank. Draw from centre L parallel 
 with C D, cutting at P. This gives P 2 as half long diameter 
 in centre of mould. Lay the edge of a board on the pitch, 
 and mark the points 2, P N ; these square over as siiown at 
 Fig. 5 ; make 2 B and N A equal C D and J N on })lan ; 
 draw from B through A. This is tangent on the mould, and 
 to be correct must equal pitch A B on the right. The tan- 
 gent 2 B when in position falls level and stands over C D on 
 plan. Draw from P through A; make P L equal radius on 
 ])lan ; set half width of rail on each side of L. Now find 
 points for pins, and strike the curves; add the straight wood 
 parallel with tangent B A. This completes the mould for 
 the wreath. 
 
 To find joint bevels : From any point, as B on a line par- 
 allel with long diameter, draw B A parallel with tangent 
 A B on mould ; Transfer pitch of plank from Fig. 4 ; Take N 
 as centre, and with any radius draw the dotted arc; With same 
 radius and B for centre draw an arc of equal length, and from 
 its lower corner draw the pitch through B. This gives bevel 
 X for joint on upper part of wreath. From any ])oint, as R, 
 square over a line, cutting the pitch in O; with E, as centre, 
 and a radius touching A B, strike an arc, cutting at K ; draw 
 from O through K. " This gives bevel W for joint on straight 
 wood. The square sections of plank show ajiplication of 
 bevels to joints; The mould for level piece shown on plan, its 
 curves are struck from centre L. 
 
Plate 32. 
 
 WEEATHS rOE OIECULAE STAIES. 
 
 Figure 1 shows the ground-plan. The construc- 
 tion here presents no difficulty whatever. The same 
 system being adopted throughout, and by the same 
 means, every form of wreath is produced, and with 
 equal simplicity, regardless of situation or plan of 
 stairs. The only points here that need most atten- 
 tion, are the starting and landing wreaths, and these 
 are regulated by. having the pitch of a single piece of 
 wreath standing over a given number of risers as five 
 or six, the joints on plan being arranged in such a 
 manner as to make one mould answer for two, three, 
 or more pieces. The steps on centre line of rail are 
 equally divided. It has been shown that all joints 
 are square with the surface of plank, and all tangents 
 across joints must be on the same pitch, and that a 
 plank can be thrown on two different pitches, and by 
 means of this, any piece of wreath can be made to 
 form its own ramp, and by the same rule we proceed 
 to find the wreaths for these stairs. 
 
 Let the joints for one piece of wreath be fixed at 
 any two points, as A C on centre of rail ; draw tan- 
 gents through A and C intersecting in point B; draw 
 the chord A G ; this divide into two equal parts as T ; 
 draw from T through B; draw from A parallel with 
 T B ; make A N equal T B ; draw from N through B ; 
 make B D equal the height of three risers ; draw from 
 D through N. Tiiis is pitch of plank. Make N D on 
 the right and left equal ; draw from N square with 
 the pitch ; make N S equal T B ; draw from D D 
 through S. These are tangents on the mould. Divide 
 A T into exact and equal parts as four; these square 
 over, cutting through width of rail ; Now divide the 
 pitch between N D on right and left, each into four 
 equal parts; and from points thus given, draw square 
 with the pitch ; make N 2 equal T 2 ; set off from 2 
 the width of rail the same as on plan ; transfer meas- 
 urement from point 4 on plan, to lines square with 
 the pitch on each side of N S ; take the next measure 
 on the left of 4 ; transfer it in like manner ; Work 
 from the right and left of line N S; until the required 
 number of points are given, and through which trace 
 the curves on mould by bending a thin lath. But to 
 be more exact in having the width of mould across 
 each joint, Find the bevel for joints as shown at Fig. 
 3 ; Draw any line parallel with the pitch ; Extend tan- 
 
 gent D S, cutting at T ; Transfer pitch of plank, by 
 taking D as centre, and with any radius, draw the 
 dotted arc ; with same radius, and T for centre, draw 
 an arc of equal length, and from its lower corner, 
 draw the pitch through T ; From any point as L, 
 square over a line cutting the pitch in 0; Take L for 
 centre, and a radius touching T S ; strike an arc cut- 
 ting at V ; join 0 V. This gives bevel X for both 
 joints on the wreath. Set half width of rail from L, 
 and draw parallel with L V, cutting the bevel. This 
 gives V 2 for half width of mould at each joint, and 
 the points through which curves on mould must pass. 
 
 Fig. 2. Find the mould for landing wreath, to stand 
 over three risers from joint C. Make B E equal tan- 
 gent B C ; Square up from E ; make E F equal the 
 height of three risers; draw from B through F. This 
 is pitch of tangent D S on mould. Draw from F the 
 floor, and above this set ofi' half a riser; draw under 
 side of level rail ; set off half its thickness, cutting the 
 pitch at H ; from H draw parallel with F E, cutting 
 at J. This gives J H for height of wreath. Make 
 C L equal B J ; Draw from L, touching centre of rail, 
 as K ; prove this line by drawing from K through 
 centre 0. Then L K to be correct must stand at a 
 right angle with K 0. Proceed by drawing from C 
 parallel with L K ; make height J P equal height 
 J H Fig. 2 ; draw from P through K. This is pitch 
 of plank. Draw from P and K square with the pitch ; 
 make P B equal J C, and make K H equal K L ; join 
 B H. This is tangent on the mould, and to be cor- 
 rect must equal the pitch B H Fig. 2. Draw any 
 number of lines through v/idth of rail on plan cutting 
 the pitch, and from intersections thus given, draw 
 square with the pitch. Now find points through 
 which trace the curves on mould as shown. The 
 bevel W in angle of pitch applies to upper joint on 
 wreath as K. To find a bevel for joint B, Draw P R 
 parallel with tangent B H on mould; From any point 
 as 2, draw square with the pitch, cutting at 3 ; take 
 2 as centre, and a radius touching P R ; strike an arc 
 cutting V ; draw from 3 through V. This gives bevel 
 X for lower joint on wreath. The square sections of 
 plank show the application of these bevels to joints. 
 To give the method for a piece of wreath starting, 
 would be a repetition of the landing wreath. 
 
?late 32 
 
-^131