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STAIR-BUILDING 
 
 IN ITS VARIOUS FORMS; 
 
 AND 
 
 The New ONE-PLA^|,E,;.]vf;ETffQi). 
 
 AS APPLIED TO 
 
 Drawing Face-Moulds, 
 Unfolding the Centre Line of Wreaths, 
 
 THEREBY 
 
 OBTAINING EXACT LENGTHS OF BALUSTERS, 
 AND ALSO UNFOLDING SIDE MOULDS. 
 
 111I10J1S BISMS 111 PLUS OF STIua IIWIIS 111 MMMhimm. 
 
 FOR THE USE OF 
 
 ARCHITECTS, 
 
 Stair-Builders, Carpenters, Iron- Workers, Pattern- Makers, and 
 
 STONE MASONS. 
 
 WOOD, mON, AND STONE STAIRS. 
 
 By JAMES H. MONCKTON, 
 
 Author of the '^American Stair-Builder," Monckton' s National Carpenter and Joiner" " Monckton s Nationa^ 
 Stair-Builder " and " Monckton' s Practical Geometry" ; Teacher for many years of the 
 Mechanical Class in the " General Society of Mechanics and Tradesmeti's 
 Free Drawing School of the City of New York." 
 
 FOURTH EDITION, REVISED AND EXTENSIVELY ENLARGED. 
 
 NEW YORK : 
 JOHN WILEY & SONS, 
 53 East Tenth Street. 
 1894. 
 
t' /S 
 
 COPVRIGHT, 1894. 
 
 By JAMES H. MONCKTON. 
 
PREFACE TO FOURTH EDITION. 
 
 NoTwi fHSTANDiNG our former belief in the completeness of this work at the conclusion of its preparation 
 and first publication, we have found since that time, from our own careful working experience and study, 
 added to the many practical questions asked by interested and intelligent correspondents in different por- 
 tions of the world, that it was important to make many changes and additions. Numerous revisions and 
 special changes have been made, all intended to perfect, widen, and maintain the usefulness and raise the 
 scientific and practical character of the work. At this time and opportunity of expression to our readers, 
 we have only to reaffirm that Geometry as applied to Stair Building and Hand-Railing throughout this book 
 is of the simplest character in its methods and perfect in its practical results. Some sixteen entirely new 
 plates have been added that have extended uses in the practice of building Wood, Iron, and Stone Stairs. 
 We will briefly mention some of the foremost useful additions that were never before published : — Plate 73. 
 Wood Panelling of the Warped Soffit of Circular Stairs. Plate 76. Side Moulds unfolded geometrically 
 correct by the use of a level line common to both planes. (The geometry of this unfoldment we have pub- 
 lished before in unfolding a central wreath-line for finding lengths of balusters, etc.) Cutting Square-top 
 Balusters to fit the Warped Surface of Wreathed Hand-rail. Plate 79. Hand-rail in One Piece over 
 Cylinders of Platform Stairs. Plate 80. Circular Stone-Work. Plate 82. Marble Wainscoting of a 
 Circular Staircase, treated with Heavy Cap and Base Mouldings. Plate 83. Unfoldment of the Warped 
 Panelled Soffit of a Stone Circular Staircase. Plate 84. Hand-rail of Circular Stone Stairs. Novel Plans 
 and Elevations of Iron Stairs. 
 
 JAS. H. MONCKTON. 
 
PREFACE. 
 
 This book presents for the first time— as applied to hand-railing— ^Z^^- 07ie-plane method of draw- 
 ing face-moulds, a method in which only one plane of projection is required; the simplicity, rapidity 
 and convenience of which in practice make it superior to all others. By the one-plane process of 
 drawing a face-mould, a centre line of wreath may also be unfolded and fixed in its relation to the 
 elevation of tread and rise; thus determining the length of each baluster on the curved plan and gain- 
 ing a knowledge and control of the wreath's exact position not before attainable. 
 
 It is intended to make this book a complete work on stair-building and hand-railing, giving a 
 large selection of plans and designs of staircases, newels and balusters, and numerous examples of 
 construction. The paper solids here introduced as an objective means of elementary and practical 
 instruction in the principles and geometrical methods of hand-railing are unequalled for the pur- 
 pose, as the construction of these solids with the drawings on their surfaces show all the positions 
 and connections required in each case, and thereby enable any fairly intelligent person to understand 
 and grasp the subject. The professional architect will find valuable suggestions in design and 
 construction ; also important considerations in planning stairs. 
 
 The experienced stair-builder will learn that this one-plane method of drawing all face-moulds — 
 and also the manner of finding the angles with which to square wreath-pieces — is simple, uniform and 
 rapid ; and no matter how skilful a stair-builder he may be, he will find that, in the extent and com- 
 pleteness of detail with which the subject is treated, it will prove a valuable work of reference. The 
 expert rail-worker will learn of the geometrical law controlling the top and bottom curves of every 
 wreath-piece, showing that a face-mould is not only a means of shaping the sides of a wreath-piece on 
 the plane of the plank, but that it carries with it a central geometrical curve (a helical line) that must 
 be observed in shaping the top and bottom surfaces of the wreath. To prove this in a practical way 
 it is only necessary to call attention to the fact that in the case of round hand-rail over any curved 
 plan, its sides hang vertically over the plan, while its top and bottom form proper curves giving its 
 own easings perfectly suited to the requirements in all cases ; and when it is considered that a 
 moulded rail over the same plan would be subject to the same centre and tangents, with the same 
 joints, then the absolute control of the curves forming the top and the bottom of a wreath by this 
 central.geometrical curve-line becomes self-evident.^ In connection with the above statements examine 
 for instance Plate 48, as giving one example among many of the correctness and practical value of un- 
 folding the centre line of a wreath. 
 
 The student or apprentice will find that the elementary study of hand-railing in the practical and 
 novel way here presented is easily acquired ; he will also see that the detail instruction given in stair- 
 building from a stepladder to expensive and difficult staircases is presented in a manner to be clearly 
 understood and quickly learned. That the drawings and descriptive page should be opposite I regard 
 as of no slight importance in a work of this character ; those who have experienced the weary task of 
 turning from reference pages to plates located at another portion of the book will appreciate the value 
 of this arrangement. The comprehensive system of hand-railing here given covers every practical re- 
 quirement, as follows: ist. By the use of tangents controlling the inclination of the plane of the plank 
 and the butt-joints ; 2d. By the one-plane method of drawing all face-moulds, simply applying to this 
 purpose a level line common to both planes ; 3d. By the further use of this last-mentioned level line, in 
 unfolding the centre line of wreath, which also unfolds correct forms of side moulds ; — all of which are based 
 upon the demonstrable laws of geometry, and point to a conclusion in the science of hand-railing as plain as 
 that in the decimal system of numbers, which, based upon the ever-truthful laws of that brancii of mathe- 
 matics, is simple in its methods and perfect in its results. 
 
 The instruction and drawings tliroughout this book apply ec[ually — except in some minor details, ivhich 
 special drawings supply — to Wood, Iron, and Stone Stairs, including Hand Railing. 
 
 Finally, it is my belief that this work is carried to an extent far beyond any publication that has 
 preceded it ; practically and scientifically covering the whole field of stair-building and hand-railing, 
 making a complete digest of the subject. 
 
 Jas. H. Monckton. 
 
CONTENTS. 
 
 HISTORY OF STAIRS 
 
 PAGE 
 
 With seven engravings, examples of ancient English stairs xxi, xxii, xxiii^ xxiv 
 
 DEFINITIONS 
 
 Of terms used in connection witli stairs and stair-building xxv and xxvi 
 
 LIST OF BOOKS 
 
 Published in England and America, treating partially or wholly on stairs or hand-railing, . . . xxvii 
 
 SUGGESTIONS. 
 
 To teachers of technical schools, xxix 
 
 To apprentices and students, xxix 
 
 Concerning the study of hand-railing, . xxix 
 
 Advantage of squaring model wreath-pieces, xxix 
 
 Fitting wreaths over circular or other curved-iron staircases, xxix 
 
 As to the fitness of close-paneled strings, ............. xxix 
 
 Self-supporting circular stairs, . xxix 
 
 Joints of wreath -pieces, . xxix 
 
 Turned newels, xxix 
 
 Connecting hand-rails with newels, xxix 
 
 Balusters with square bases, xxix 
 
 Loose newel-caps mitred to hand-rails, .............. xxix 
 
 Setting up hand-rails finished and varnished, xxix 
 
 IRON STAIRS. 
 
 Scientific and artistic treatment of iron stairs, ............. 
 
 Designs and Plans for Iron Stairs Plates 86 to 104 
 
 Circular Stone Work Plate 80 
 
 STONE STAIRS. 
 
 Circular Stone Stairs, etc Plates 81 1085 
 
PLANS AND ELEVATIONS. 
 
 PLATE 1. 
 
 Plan and elevation of a straight flight of stairs with a 7" cylinder. Plan and elevation of a half-turn 
 platform stairs with a 6" cylinder, landing with four rises above the platform. A rule to find the correct 
 proportion of tread to rise. 
 
 PLATE 2. 
 STEP-LADDERS AND STOOP. 
 
 Plans of step-ladders. Elevations of step-ladders. Elevation of stoop with platform, hand-rail, balusters 
 and newel. 
 
 PLATE 3. 
 
 T/ie old English method of stair-bitildinff. Finishing stairs on the under-side, showing the construction. 
 Ptittina- lep stairs. Covering stairs. Stair-timbering and rough-bracl<eting. Best method of joining hand- 
 rail to newel. J^/an and elevation of a common straight flight of stairs. Methods of making ordinary 
 small cylindeis and splicing them to strings. Front and wall strings laid out. Step and riser as worked 
 and glued together. 
 
 PLATE 4. 
 
 PLANNING WINDING STAIRS. 
 
 Line of travel fi.xed. Elevation of the winding flight given. Laying out the front-string and cylinders. 
 Laying out the wall-strings. 
 
 PLATE 5. 
 PLANNING STAIRS. 
 
 In the following three Plates — 5, 6 and 7 — over thirty different plans of stairs are given ; their various 
 dimensions are figured for the convenience of examination and use in the study and planning of stairs. 
 Also reference is given to each Plate, where a drawing of the plan of stairs is again projected on a large 
 scale, together with the management of the hand-rail wreath. 
 
 Fig. 1. Plan of a straight flight of stairs starting and landing with small cylinders. See Plate i, 
 Figs, i and 2; also Plates 3 and 22. 
 
 Fig. 2. Plan of a straight flight of stairs starting with a newel set off 2J", and landing with a 7" 
 cylinder, the landing riser set into the cylinder 2\". See Plate 30, FiGS- i and 2: also Plates 33 and 34. 
 
 Fig. 3. Plan of winding stairs suitable for Basement or Attic Story, with mortised — or housed — strings 
 both sides ; intended to be set between partitions or otherwise enclosed. 
 
 Fig. 4. Shows the manner of laying out the front string. A and B are continuations of the string, 5" 
 wide, and spliced to the string as shown. 
 
 Fig. 5. Plan of a qtiarter-turn of winders, with cylinder, suitable for either the starting or landing of a 
 flight of stairs. See Plate 25. 
 
 Fig. 6. A much superior plan to that of FiG. 5, turning one quarter without winders, and with the same 
 size of cylinder and regular tread. In this plan, by curving the front ends of two risers, winders are dispensed 
 with and a platform and parallel steps substituted ; this plan as shown at D C requiring 7^" more run. See 
 Plate 26. 
 
 Fig. 7. Plan the same as FiG. 6, but instead of a continued hand rail a small newel and connecting level 
 quarter-cylinder are substituted. See Plate 62. 
 
 Fig. 8. Plan for starting or landing a flight of stairs, with a single newel set diagonally. 
 
 Fig. 9. Plan of stairs combining two platforms with a parallel step between (and the front ends of its 
 two risers curved), making a half-turn. See Plate 41. 
 
 Fig. 10. Plan of a quarter-platform stairs, showing how to place the risers connecting with any quarter- 
 cylinder so that the wreath may be worked in one piece on a common inclination and in the best possible 
 shape. On these plans, from the face of the riser F and on the centre line of the rail, F E should equal half 
 a tread ; and from the face of the riser G on the centre line of rail, G E must equal half a tread. See Plate 
 31. Fig. 3, and Plate 37, Figs. 5, 6 and 7. 
 
 Fig. 11. Plan of stairs, with a quarter-cylinder turning one quarter, with winders. See Plate 36. 
 
 PLATE 6. 
 PLANNING STAIRS. 
 
 Fig. 1. Plan of half-turn platform stairs with the risers set in the cylinder as far as possible to save run. 
 The cylinder-opening is made up of two 6" quarter-circles and 5" straight between. See Plate 38. 
 
xii 
 
 CONTENTS. 
 
 Fig. 2. Plan of a flight of winding stairs with lo" cylinders, making a half-turn at the starting and at the 
 landing. See Plates 28 and 29. 
 
 Fig. 3. Plan of two connecting flights starting and landing from one 1 2'' cylinder , the flight starting 
 tiiriiing one quarter with winders, the other flight straight and landing straigiit, with its top riser set into 
 the cylinder 4". See Plate 42. 
 
 Fig. 4. Plan of a landing or starting of a flight of stairs, with a single newel set between quarter-cylinders. 
 See Plate 37, Figs, i, 2, 3 and 4. 
 
 Fig. 5. Plan of the starting of a flight of stairs, with the front-string curved out. including four treads 
 Each of the four treads are increased in width at the wall-string and diinmislied at the front-string, 
 curving the risers, — making what are called swell steps. See Plate 32. 
 
 Fig. 6. Plan of the starting or landing of a flight, turning one quarter, with platform ; finished with low- 
 down newels, or — more properly — small corner-i)ieces, and continued hand rail with ramp and goose-neck. See 
 Plate 58. 
 
 Fig. 7. Plan of half-turn plaiform open-newel stairs with wing-flights. 
 
 Fig. 8. Plan of half-turn ])latforin stairs with a large cylinder into which the front ends of the risers of 
 parallel steps are brought curved; thus ulihzing the cylinder-space anfi making a half-turn without resorting 
 to winders. See Plate 51. 
 
 Fig. 9. Plan of ship or s:''amboat stairs. At Plate 52 is a plan similar to this with angle newels. See 
 also Plate 50. 
 
 PLATE 7. 
 PLANNING STAIRS. 
 
 Fig. 1. Plan of half-turn platform-stairs with a 15" cylinder; the risers at the platform and at the starting 
 of the flight are set in the cylinders 6", tliereby saving I'.o" of run. See Plate 40. 
 
 Fig. 2. Plan of a half-turn platform stairs wiili a central riser making two platforms, and the starting 
 and landing risers each set into the cylinder 3". See Plate 39. 
 
 Fig. 3. Plan of a flight of stairs winding one quarter at the starting and one half at about the middle 
 of the flight. For the treatment of the iiand-rail at the starting see Plate 48, and for the treatment of the 
 rail at the half-turn see Plate 47. 
 
 Fig. 4. Plan of the bottom or top portion of a flight of stairs making a quarter-turn with a quarter- 
 platform, the platforvi and parallel steps secured a?id winders avoided, by curving the front ends of risers 
 to the widths of treads marked on the line of cylinder. The cylinder f)pening is formed of two quarter- 
 circles, each of 5'' radius with 8' slraiglit between. For treatment of hand-rail with larger scale-drawing 
 see Plate 46. 
 
 Fig. 5. Plan of the bottom or top portion of a flight of open-newel stairs with a 15" opening, making 
 a quarter-turn with a quarter-platform. See Plate 59. 
 
 Fig. 6. Plan of the bottom or to,) portion of a flight of stairs with a 10" cylinder winding one 
 quarter. For treatment of wreath at the bottom of a flight see Plate 27, and for the top of a flight 
 the same as at Plate 25. 
 
 Fig. 7. Plan similar to Fu;. 6, with one more winder in the plan and curved risers introduced to 
 sive room between the points B and C. 
 
 Fig. 8. Plan same as FiG. 4, with a complete semicircle for cylinder opening. For treatment of 
 wreath see Plate 44. 
 
 Fig. 9. Plan of the top or bottom portion of a flight of stairs making a quarter turn with a quarter- 
 platform and a parallel step set at the centre of a 15" cylinder. See Plate 43 for larger size scale- 
 drawing and management of wreath. 
 
 Fig. 10. Plan of a flight of circular stairs. The dotted lim s show the best method of placing the 
 carriage-timbers. The treatment of hand-rail on a larger scale of drawing will be found at Plate 53. 
 At Plate 54 full instruction is given for changing the plan of the first step to the scroll form ; the treat 
 meat of that portion of the hand-rail, also the construction of the scroll-step. 
 
 Fig. 11. Plan of an elliptic flight of stairs with the treads at the wall and front-strings properly 
 graded so as to bring the risers on lines more nearly normal to the curves and at the same time main- 
 tain an even tread on the line of travel. These stairs should be timbered in the manner shown at FiG. 10. 
 
 Self-supporting stairs, also see Plate 45. 
 
 PLATE 8. 
 
 Figs. 1 and 2. Bending wood by saw-kerfing and the construction of a circular form, with ribs of the 
 proper curve an;l narrow strips of the required length called laggings, over which to bend the saw-kerfed materiaL 
 Fig. 3. Rending wood on a curved form and keying. 
 
 Fig. 4. Bending veneer, facing and fillmg out the thickness with parallel pieces — fitted to the curve — 
 called staves. 
 
 Fig. 5. Laminated work —bending and gluing several thickness of veneer together. A rule to ascertain 
 what thickness of white pine will bear bending any given radius of cur7'ature without injuring its elasticity. 
 Fig. 6. Bending stair-strings. 
 
 Fig3. 7 and 8. Construction of a form for bending quarter-circle stair-strings, the ribs to stand on an 
 angle parallel to the inclination of string. 
 
 Figs. 9 and 10. Soffit-mouldings on the lower edge of cylinders. 
 
 Figs. 11 and 12. To find the lengths of cylinder-staves that include winder-treads. 
 
 PLATE 9. 
 
 Examples of the one-plane method in drawing face moulds. Face-moulds, their number and character. 
 
CONTENTS. 
 
 xiii 
 
 ELEMENTARY INSTRUCTION IN HAND-RAILING OVER 
 
 CURVED PLANS. 
 
 PLATE 10. 
 
 Face-mould and parallel pattern over a plan of a quarter-circle ; one tangent inclined, the other horizontal. 
 The angles with which to square the wreath-piece at the joints. 
 
 PLATE 11. 
 
 Face-mould and parallel pattern over a plan of a quarter-circle, the tangents inclined alike. The angle 
 with which to square the wreath-piece at both joints. 
 
 PLATE 12. 
 
 Face-mould and parallel pattern over a plan of a quarter-circle, the tangents dilTerently inclined. The 
 angles with which to square the wreath-piece at the joints. 
 
 PLATE 13. 
 
 Face-mould and parallel pattern over a plan less than a quarter-circle, one tangent inclined, the other 
 horizontal. The angles with which to square the wreath-piece at the joints. 
 
 PLATE 14. 
 
 Face-mould and parallel pattern over a plan of more than a quarter-circle, one tangent inclined, the other 
 horizontal. The angles with which to square the wreath-piece at the joints. 
 
 PLATE 15. 
 
 Face-mould over a plan less than a quarter-circle, the tangents inclined alike. The angle with which to 
 square the wreath-piece at both joints. 
 
 PLATE 16. 
 
 Face n-ould over a plan less than a quarter-circle, the tangents differently inclined. The angles wiili 
 which to square the wreath-piece at the joints. 
 
 PLATE 17. 
 
 Face-mould over an elliptic or eccentric curved plan, the tangents of unequal length and different inclina- 
 tions. The angles with which to square the wreath-piece at the joints. To find a coiniiion angle of inclina- 
 tion over two differ e)it lengths of pla7i tangents — when required — the total height being given. 
 
 PLATE 18. 
 
 Face mould or parallel pattern over a plan greater than a quarter-circle, the tangents mclined alike. The 
 angle with which to square the wreath-piece at both joints. 
 
 PLATE 19. 
 
 Face-mould over a plan greater than a quarter-circle, the tangents differently inclined. The angles with 
 which to square the wreath-piece at the joints. 
 
 DEVELOPMENT OR UNFOLDING OF THE CENTRE LINE OF 
 
 WREATHS. 
 
 PLATE 20. 
 
 Fig. 1. Plan of a centre line of wreath, a semicircle with three tangents inclined alike, the fourth tangent 
 level. 
 
 Fig. 2. The centre line of wreath unfolded from plan Fig. i. 
 
 Fig. 3. Plan of a centre line of wreath, a semicircle with two tangents inclined alike, the third of a less 
 inclination, and the fourth tangent level. 
 
 Fig. 4. The centre line of wreath unfolded from plan Fig. 3. 
 
 Fig. 5. Plan of a centre line of wreath-piece less than a quarter-circle, one tangent inchned, tlie other 
 level. 
 
 Fig. 6. The centre line of wreath-piece unfolded from plan Fig. 5. 
 
xiv 
 
 CONTENTS. 
 
 PLATE 21. 
 
 Pig. 1. Plan of a centre line of wreath-piece greater tlian a quarter-circle, one tangent inclined, tlie other 
 level. 
 
 Fig. 2. The centre line of wreath-piece unfolded from plan Fio. i. 
 
 Fig. 3. Plan of a centre line of wreath-piece greater than a quarter-circle, the tangents of a like 
 inclination. 
 
 Fig. 4. The centre line of wreath-piece unfolded from plan FiG. 3. 
 
 Fig. 5. Plan of centre line of wreath-piece greater than a quarter-circle, the tangents diflerently 
 inclined. 
 
 Fig. 6. The centre line of wreath-piece unfolded from plan FiG. 5. 
 
 Fig. 7. Plan of a centre line of wreath piece less than a quarter-circle, the tangents inclined alike. 
 Fig. 8. The centre line of wreath-piece unfolded from plan Fig. 7. 
 
 Fig. 9. Plan of a centre line of wreath-piece less than a quarter-circle, the tangents inclined diflerently. 
 Fig. 10. The centre line of wreath-piece unfolded from plan Fig. 9. 
 
 PLATE 22. 
 
 POSITION OF RISERS IN CONNECTION WITH CYLINDERS AT THE STARTING AND AT THE 
 
 LANDING OF STRAIGHT FLIGHTS OF STAIRS. 
 
 Fig. 1. Elevation of rises and tread to determine the face of riser at the bottom of a flight when the 
 over \vo(;d is to be all removed from the top of the wreath-piece. 
 
 Fig. 2. Elevation of tread and rise at the top of a flight to determine — 'when all the over-wood is removed 
 from I he bottom of a ivreath-piece — the relative position of the riser and cylinder. 
 
 Fig. 3. Plan of the bottom of flight with riser and cylinder placed as determined at FiG. i. 
 
 Fig. 4. Plan of the top c;f flight with cylinder and riser as determined by trial at FiG. 2. 
 
 Fig. 5. Face-mould. 
 
 Fig. 6. Perspective sketch of the wreath-piece, showing both joints prepared for squaring and the applica- 
 tion of the face-mould both sides of the plank. 
 
 Fig. 7. Elevation of tread and rises for the top and bottom of flight, showing that in some cases by a 
 change in removing the over-wood the risers may be placed at the chord-lines of cylinders of this diameter. 
 
 Laying out the joint with the pitch-board as at FiG. 5 shows how to And the thickness of wood required for 
 my size and form of hand-rail. 
 
 PLATE 23. 
 
 Half-turn Platform Stairs as given by plan and elevation at Plate i. Figs. 3 and 4. Ho'm to fix 
 the position of risers in connection with the cvliudn s, and the changes possible by varying tlie removal of 
 ;)ver-wood from the straight portion of wi cii h-[ licces. see FiGS. i and 2; also how to place the risers 
 connecting with the cylinder of a half-turn platform stairs so that the wreath may have a common incli- 
 nation, making a much superior-shaped rail and saving several inches more room than by the method 
 i^iven at FiG. 2. See FiGS. 3. 4 and 5. 
 
 PLATE 24. 
 
 Fixing the position of risers connecting with 15" cylinders at the starting and at the landing of 
 straight fliglits of stairs, so that the over-wood may be removed in the most desirable way to shape the 
 wreath-pieces, and raise the height required from the floor. — Figs, i, 2 and 3. 
 
 Plan of stairs with the top riser placed at the diameter-line of a 15" cylinder; management of hand- 
 rail, etc. — Figs. 4, 5 and 6. 
 
 PLATE 25. 
 
 Fig. 1. Plan of the top portion of a staircase winding one quarter, with a small cylinder. 
 Fig. 2. Elevation of treads and rises; development or unfolding of the centre line of wreath. Plans 
 of hand-rail and face-moulds, FiGS. 3, 4, 5 and 6. 
 
 PLATE 26. 
 
 Fig. 1. Improved plan of the top portion of a staircase winding one quarter, with a small cylinder. 
 Fig. 2. Elevation of treads and rises ; development of the centre line of wreath ; length of balusters. 
 Plans of hand-rail and face-moulds. Figs. 3, 4, 5 and 6. 
 
 PLATE 27. 
 
 Fig. 1. Plan of the bottom portion of a staircase winding one quarter, with a 10" cylinder. 
 Fig. 2. Elevation of treads and rises from plan FiG. i ; also development of the centre line of wreath, 
 giving lengths of balusters under wreath. Plans of hand-rail and face-moulds, FlGS. 3, 4, 5 and 6. 
 
 PLATE 28. 
 
 Fig. 1. Plan of the top portion of a winding staircase, making a half-turn, with a 10" cylinder. 
 Fig. 2. Elevation of treads and rises and development of the centre line of wreath, giving lengths 01 
 balusters, etc. 
 
 Figs. 3 and 4. Plan of hand rail and face mould. Fig. 4. Face-mould for both quarters of the cylinder, 
 the easement at the top finishing to a level from the upper end of wreath. 
 
CONTENTS. 
 
 XV 
 
 PLATE 29. 
 
 Plan of the bottom portion of a winding staircase maiiing a half-turn, with a lo" cylinder. 
 Fig. 1. Elevation of treads and rises as given at plan FiG. i ; also unfolding the centre line of wreath, 
 giving lengths of balusters, etc. 
 
 Fig. 2. Plan of hand-rail and face-mould. 
 Figs. 3 and 4. Face-mould for both quarters. 
 
 PLATES 30, 31, 33. 
 
 At the starting of a first flight of stairs the front-string is frequently curved out, the curve extending 
 from one to five treads. 
 
 Management of Strings and Hand-rails of Curve-outs. 
 
 Plate 30. — Fig. i. Plan and elevation. FiG. i. Parallel pattern. FiG. 3. Plan and elevation of the starting 
 of a staircase, with the front-string curved out and the newel set on top of the first step. FiG. 4. Parallel 
 pattern for curve-out of hand-rail, easing to a level at newel. 
 
 Plate 31. — Fig. i. Plan and elevation of the starting of a staircase, with the front-string curved out and 
 the newel set on top of the first step. This plan is the same as that of FiG. 3, Plate 30, but in this 
 case the hand-rail is brought straight down to the newel, not easing to a level as at Plate 30. FiG. 2. 
 Face-mould for FiG. i. FiG. 3. Plan of a quarter-platform stairs with a quarter-cylinder of 8" radius, the 
 risers improperly set at the chord-lines, thereby making it necessary to get out the quarter-wreath in two 
 pieces. Fig. 4. Parallel pattern, two of which make the quarter-wreath. 
 
 Plate 32. — Fig. i. Plan of curve-out, — with four steps, — its shape designed to make a proper connection 
 with a square newel, where the sides of the newel are required to set parallel to the side-walls of the hall- 
 way. Fig. 2. Elevation of treads and rises from plan Fig. i. Fig. 3. Face-mould from Fig. i. showing 
 the squaring of the wreath-piece at the joints. Fig. 4. Plan of starting a staircase, with the front-string 
 curved out, embracing four treads of equal vfidths at the wall-string and front-string. FiG. 5. Elevation 
 of treads and rises from plan FiG. 4. FiG. 6. Face-mould from plan FiG. 4, showing the squaring of 
 wreath-piece at the joints, 
 
 PLATE 33. 
 
 Fig. 1. Plan of the landing of a straight flight of stairs, with the top riser set in the whole deptl- 
 of a ten-inch cylinder, thereby saving five inches. 
 
 Fig. 2. Elevation of tread and rises from plan FiG. i. 
 
 Fig. 3. Face-mould from plan FiG. i ; also showing the squaring of the wreath-piece at the joints. 
 Fig. 4. Plan of the starting portion of the same flight of stairs given at FiG. i. with a ten-inch 
 cylinder, and the starting riser set in the whole radius, or depth, of cylinder, saving another five inches. 
 Fig. 5. Elevation of tread and rises from plan FiG. 4. 
 
 Fig. 6. Elevation of tread and rises same as Fig. 5, showing the unfolding of the centre line of wreath. 
 
 PLATE 34. 
 
 Hand-rail wreath in one piece over a 7" cylinder, the landing riser set in the radius— 3^"— 
 
 OF THE cylinder. 
 
 PLATE 35. 
 
 Hand-rail wreath in one piece over a 7" cylinder by a different method from that given 
 at Plate 34. In this case the landing riser is placed at the chord-line of the cylinder. Figs. 
 4 and 5. Wainscoting for stairs, its heights and construction. 
 
 PLATE 36. 
 
 Plan of winding stairs turning one quarter, with a quarter-cylinder of 8" radius. Elevation from plan. 
 Unfolding the centre line of wreath. Length of balusters under wreath. Parallel pattern for wreath-piece. 
 See Plate 5, Fig. ii. 
 
 PLATE 37. 
 
 Plan of half-turn platform stairs. Where to place the risers connecting with the cylinder when the 
 over-wood is to be removed from the wreath-pieces as shown. Drawing the face-mould for the quarter- 
 cylinders. Plan of quarter-turn platform stairs, with a quarter-cylinder of 6" radius. Where to place the 
 risers connecting with the quarter-cylinder. Elevation of plan, and unfolding the centre line of wreath. 
 Drawing the face-mould. See Plate 5, FiG. 10; also Plate 45, FiG. i. 
 
 PLATE 38. 
 
 Plan of half-turn platform stairs, with the risers next to platforn: set in the whole depth of the cyl 
 inder. Elevation from plan and unfolding the centre line of wreath. Lengths of balusters under wreath. 
 Drawing the face-mould. 
 
xvi 
 
 CONTENTS. 
 
 PLATE 39. 
 
 Plan of half-turn platform stairs, with one riser set at the centre of the cylinder dividing the half- 
 turn space in two platforms; the other two connecting risers each set in the cylinder 3". Elevation from 
 plan. Unfolding the centre line of wreath. Drawing the face-mould. 
 
 PLATE 40. 
 
 Plan of half-turn platform stairs, with two risers each set into the 15" cylintler 6". The bottom riser 
 set into the cylinder at the starting of the (light as found on trial from elevation. Drawing face-moulds. 
 See Fig. i, Plate 7. 
 
 PLATE 41. 
 
 Plan of stairs with two quarter platforms and a tread between, making a half-turn. Elevation from 
 plan. Unfolding the centre line of v/reath. Drawing the face-mould. 
 
 PLATE 43. 
 
 Plan of stairs in which two flights start and land in connection with a 12" cylinder, the upper flight 
 starting and turning one quarter with winders, the lower flight straight with the landing riser set 4" into 
 the cylinder. Elevation from plan and unfolding the centre line of wreath. Drawing the face-moulds. 
 See Plate 6, Fig. 3. * 
 
 PLATE 43. 
 
 Plan of stairs making a quarter-turn, with a 15" cylinder, a quarter platform, and a parallel step set 
 in the centre of the cylinder and at right angles to its diameter. Elevation from plan and unfolding the 
 centre line of wreath. Drawing the face-moulds. Showing that in some cases greater width of wood than 
 the width of face-mould is required for the wreath. This shows also the manner of linding the thickness required. 
 See Plate 7, Fig. 9. 
 
 PLATE 44. 
 
 Plan of stairs turning one quarter, with an 18" cylinder, a quarter- platform, and three steps with 
 their front ends curved to the cylinder, designed to take the place of wniders. Elevation from plan and 
 unfolding the centre line of the wreath, by means of which the lengths of balusters under the wreath may 
 be found. Drawing the face-moulds. See Plate 7, Fig. 8. 
 
 PLATE 45. 
 
 Pl.ui of circular stairs winding around a circular post. Plan of stairs with close string, showing how 
 to build the cylinder solid with veneered faces. Plan of quarter-platform stairs, showing how to place the 
 risers in tlie quarter-cylinder another way (when desirable), different from the one given at Plate 37, 
 Fig. 5, and Plate 5, Fig. 10. 
 
 PLATE 46. 
 
 Plan, turning one quarter, with a quarter-platform, designed to avoid winders by curving the fron^ 
 ends of the parallel steps to the cylinder. Elevation from plan and unfolding the centre line of wreath, 
 by means of which the lengths of balusters under the wreath may be found. Drawing the face-moulds. 
 See Plate 7, Fig. 4. 
 
 PLATE 47. 
 
 Plan of half- turn winding stairs with 7" cylinder. Elevation from plan, and unfolding the centre line of 
 wreath, by means of which the lengths of balusters under the wreath may be found. Drawing the face mould. 
 See Plate 6, Fig. 2. 
 
 PLATE 48. 
 
 Plan of a starting portion of a flight winding one quarter from a newel, the hand-rail wreath covering 
 nearly a semicircle on the plan, worked in one piece and no ramp used in connection. Elevation from pl.m 
 and unfoldmg the centre line of wreath, by means of which the length of each baluster under the wreath is 
 given. Drawing a parallel pattern for the wreath piece. 
 
 PLATE 49. 
 
 Plan of_ the starting portion of a flight winding more than a quarter from a newel, with a 20" cvlmder 
 containing five winders. The wreath covering nearly a semicircle on the plan, worked in one piece and 
 no ramp used in connection. Elevation from plan and unfolding the centre line of wreath, by means ol 
 which the length of each baluster under the wreath is given. Sketch, showing the wreath as squared 
 up. Drawing the face-mf)uld. See Plate 7, Fig. 3. 
 
CONTENTS. 
 
 xvii 
 
 PLATE 50. 
 
 Plan for hand-rail with parallel patterns for wreath-pieces, the plan taken Irom Plate 6. FiG. 9. 
 Steamship stairs. Elevation from plan and unfolding the centre line of wreath, showing the length ot 
 each baluster under the wreath. 
 
 PLATE 51. 
 
 Plan of half-turn platform stairs with six parallel steps, and platform placed in a large cylinder, the 
 steps at their front ends curved to the cylinder. Elevation from plan and unfolding the centre line of 
 wreath, showing the length of each baluster under the wreath. The wreath to be worked in three pieces. 
 Drawing the face-moulds. See Plate 6, Fig. 8. 
 
 PLATE 52. 
 
 Plan of open-newel steamship stairs with platform and wing flights. The newels at the starting to 
 continue above the upper deck and receive the level hand-rail and enclosing balustrade. Elevation from 
 plan of curve and unfolding the centre line of wreath. Drawing the face-mould. 
 
 PLATE 53. 
 
 Hand-rail over a circular flight of stairs starting with a newel. Plan of stairs given on PLATE 7, 
 Fig. 10. Drawing the face-moulds. 
 
 PLATE 54. 
 
 Plan for starting the circular stairs given on Plate 53, with a scroll step and scrolled hand-rail in 
 
 place of a newel. Construction of block for scroll step and riser. Scroll step complete. Face-mould for 
 
 hand-rail scroll. How to describe the scroll of a superior form and in a simple way. 
 
 PLATE 55. 
 
 Hand-rail over an elliptic flight of stairs. Drawing the face-moulds. See Plate 7, FiG. 11, for com- 
 plete plan of the stairs. 
 
 PLATE 56. 
 
 Instructions for sliding face-moulds, or for placing them in position to plumb the sides of wreath- 
 pieces. Squaring wreath-pieces. 
 
 PLATE 57. 
 
 Wholesale-store stairs. Panel-work enclosure. Construction of close front-string, paneled and capped. 
 Management of newels and hand-rail. 
 
 PLATE 58. 
 
 Plan of the landing portion of a flight of stairs turning one quarter, with a quarter-platform and square 
 corner-pieces like small low-down newels set in the angles, over which is carried a continued hand-rail, with 
 ramp and goose-neck at the landing. Construction of close front-string. Elevation showing paneled front- 
 string, corner-piece, balusters, and method of continuing the hand-rail. 
 
 PLATE 59. 
 
 Plan of quarter-platform open-newel stairs; top portion of flight. Elevation from plan, and details. 
 Laying out the angle newels. 
 
 PLATE 60. 
 
 Plan of quarter-platform stairs turning one quarter. Turned angle-newels, with hand rail ramped and 
 kneed. Elevation and details. 
 
 PLATE 61. 
 
 Design for newels, baluster, and close front-string. 
 
 PLATE 62. 
 
 Plan of stairs turning one quarter and arranged to avoid winders by curving the front end of steps, 
 making the hitter parallel, and securing a platform. Also by this plan with the small newel no twists are 
 required. Elevation from plan, showing length of newel and its connections with string, rail, balusters, etc. 
 See Plate 5, Fig. 7. 
 
xviii 
 
 CONTENTS. 
 
 PLATE C3. 
 
 Design for newel, hand-rail, and balusters; also design and construction of close front-string. 
 
 PLATE G4. 
 
 Desif^n for an open-moulded stair-string, balusters and hand-rail. The balusters to be screwed with 
 square-headed lag-screws to the side of the string, and the heads ot the screws covered with turned or 
 carved rosettes. 
 
 PLATE 65. 
 
 Design and construction of close front-string, newel and balusters. 
 PLATE 6G. 
 
 Desi"-n of a turned and carved newel, carved string, and balustrade. Design of spiral-turned newels 
 and balu"ster.s, bracketed string, ramp, and goose-neck hand-rail. Plan and elevation of a half-turn platform 
 stairs, the plan so arranged that the newels will be of equal heiglits from the platform. 
 
 PLATE 67. 
 
 Ancient staircase at Rouen, France. From the Monitcur des Architectes. 
 
 PLATE 68. 
 
 Interior view of a flight of stairs turning one quarter, witli a platform at the starting two rises up. 
 the platform ornamentally enclosed on one side with panel-work to match the hall wainscot, and above 
 which and across the hall spindle screen panels between columns. 
 
 PLATE 69. 
 
 Interior view of a grand staircase and spacious, suitably-fitted hall. 
 
 PLATES TO, 71, 72. 
 Designs for newels. 
 
 PLATE 73. 
 Paneled soffit of a circular stairs. 
 
 PLATE 74. 
 
 A third method of treating hand-rail over a large cyliiuler. Cheap way of treating the set-ofTof a newel, etc. 
 
 PLATE 75. 
 
 Another plan of stairs— starting — avoiding winders. Two ways of treating the hand-rail, etc. 
 
 PLATE 76. 
 
 The use of side-moulds in hand-railing. To cut large stone or wood square-top balusters to an exact length, 
 the tops to fit the warped bottom surface of circular rail, etc. 
 
 PLATE 77. 
 
 Plan of stairs with regular treads to chord line, 6" cylinder, quarter platform, and two rises above; the wreath 
 falling into the inclination of regular treads without the intervention of a ramp. Two ways of treating the rail. 
 
 PLATE 78. 
 
 Treatment of hand-rail for a quarter platform stairs with quarter cylinder, 8" radius, both rises at chords. 
 
 PLATE 79. 
 Hand-rail in one piece — platform stairs 7" cylinder. 
 
CONTENTS. 
 
 STONE WORK. 
 PLATE 80. 
 
 Circular slone-work. 
 
 STONE STAIRS. 
 
 PLATE 81. 
 Plan of circular stone-staircase, etc. 
 
 PLATE 82. 
 
 Management of stone steps for circular stairs of Plate 8i, stone wainscoting, etc. 
 
 PLATE 83. 
 
 Unfolding the soffit of plan at 8i, and apportioning the panels, etc. 
 PLATE 84. 
 
 Hand-railing for circular stone-staircase, from plan Plate 8i. 
 PLATE 85. 
 
 Squaring base and cap mouldings for wainscot of circular stone-staircase of Plates 8i, 82. 
 
 IRON STAIRS. 
 
 Scientific and artistic treatment of iron stairs, Plate 86. 
 A large number of plans and designs for iron staircases given through Plates 86 to 104. 
 
 PLATES 105, 106. 
 Sections of hand-rails of various forms and full size. 
 
 PLATES 107, 108, 109. 
 Newels and balusters. 
 
 Total number of plates, 112. 
 
STAIRS. 
 
 Primitive man had little use for stairs; living in a hole dug in the ground, or a hut built of 
 the branches and leaves of trees, or a log cabin, the utmost of his virants were doubtless supplied 
 with a rude ladder. 
 
 " We know little of the staircases of the Greeks and Romans, and it is remarkable that Vi- 
 truvius * makes no mention of a staircase as an important part of an edifice ; indeed, his silence 
 seems to lead to the conclusion that the staircases of antiquity were not constructed with the 
 luxury and magnificence to be seen in more recent buildings. The best-preserved ancient stair- 
 cases are those constructed in the thickness of the walls of the proraos — vestibules — of temples 
 for ascending to the roofs. According to Pausaniasf similar staircases existed in the temple of 
 the Olympian Jupiter at Elis. They were generally winding and spiral. Sometimes, as in the 
 Pantheon at Rome, instead of being circular on the plan they are triangular. Very few ves- 
 tiges of staircases are to be seen in the ruins ot Pompeii, from which it may be inferred 
 that what there were must have been of wood, and moreover that few of the houses were 
 more than one story in height. Where they exist, as in the building at the above place 
 called the country-house, and some others, they are narrow and inconvenient, with rises some- 
 times a foot in height. ... In modern architecture the magnificence of the staircase was but 
 tardily developed. The manners, too, and the customs of domestic life for a length of time 
 rendered unnecessary more than a staircase of very ordinary description." J 
 
 In England, all staircases preceding the latter part of the reign of Charles the Second 
 — previous to 1660 — were either stairs winding round a post, or the strings were framed 
 into square newels without balusters, but close-boarded, sometimes plastered between the rail 
 and steps. Fig. i is an example built about 1557 of the close-boarding, pierced as the style 
 began, and introducing afterwards a variety of designs. Later the flat-moulded baluster was 
 introduced, as shown by Figs. 2, 3 and 4, and the carved balustrade, shown at Fig. 5. As the 
 turning-lathe — a new invention of that time— came into use, turned balusters and newels were 
 adopted, an example of which is given at Fig. 6. 
 
 At Fig. 7 is given an example of an ancient English tower staircase from Naworth Castle, 
 Cumberland, England. 
 
 In other parts of Europe the state of the art in all probability had not progressed beyond 
 the examples here given. With the advancement of the arts and the enormous increase of 
 wealth many costly and magnificent staircases are now built in private dwellings of the people 
 of Europe and America. Of public buildings, a staircase in " Goldsmiths' Hall," London, Eng- 
 land, is said to have cost $150,000. At Washington, D. C, in the new hall of the House ot 
 Representatives, is a costly staircase built of stone with a highly artistic bronze balustrade. 
 There are many elaborate and expensive staircases in the capitals of Europe that are impor- 
 tant features of their public buildings. In the Grand Castle of Chapultepec, in Mexico, there is built 
 a peculiar double staircase that seems to have no supports, and is said to be the only one of its kind 
 in existence. It is built of white marble and brass. It is said that the Emperor Maximilian expressed 
 his doubts to the architect of the strength of this staircase; whereupon, as a test, a regiment of soldiers 
 was marched up and down the stairway ten abreast, thus demonstrating its strength. There has 
 recently — 1893-94 — been erected in this city a palatial residence for Collis P. Huntington, Esq. — 
 J. B. Post, architect; Ellin, Kitson Co., contractors — in which a magnificent white Vermont marble 
 and Mexican onyx staircase has been built. This staircase is constructed in a semicircle thirty-four 
 feet in diameter, of three flights — altogether ninety-eight steps, each of which weighs about a ton and 
 is eight feet long. The soffits of two of these flights are alike divided with four bands running 
 parallel to the circle and fourteen radiating or cross bands, all richly carved, that together divide the 
 sofiit into thirty-nine panels ; these panels are sunk four inches in depth, are moulded, carved, and 
 with carved rosettes in the centre of each panel. The hand-rail is of Mexican onyx, fourteen by six 
 and a half inches, moulded and carved. The balusters of onyx— one on each step— are six inches 
 
 * Vitruvius, Marcus Polii, a Roman architect, 15 B.C. \ Pausanias, an ancient Greek writer, 170 A.D, 
 
 :|: Gwilt's Encyc. of Architecture. 
 
xxii 
 
 STA/RS. 
 
 square, and turned. A marble paneled wainscot also ornaments the circular walls all the way up 
 these stairways ; this wainscoting has a heavy onyx moulding at the base and raised centre panels of 
 onyx ; the whole is capped with onyx half the width of the hand-rail, carved and moulded as before 
 described. All the onyx and marble is highly polished. This staircase is said to have cost $200,000. 
 Iron and stone stairs are built on similar principles to those of wooden ones, the difference being in 
 tlie treatment and changes required by the materials. 
 
 When the first fashion of continued hand-rail called for the skill of the workman, 
 the method adopted was that of bending and gluing together a number 
 of thin wood veneers about a convex cylinder built for the purpose. The 
 width of these veneers equaled the thickness of the rail, and the thickness 
 of the veneers altogether made up the width of the rail, as shown by the 
 accompanying sketch. The solid wreath of hand-rail formed in this way, 
 after being allowed to dry, — which a writer observes " took three weeks," 
 — was removed from the cylinder and carved into the shape of rail 
 required. As late as the year 1826 instructions in the above method — 
 although the author protested against it — were given in a book published in England by 
 M. A. Nicholson, entitled " The Carpenter and Joiner's Companion." 
 
FlO. S. 
 
 xxiv 
 
DEFINITIONS. 
 
 Stairs. — That mechanical structure in a building by which access from one story to 
 another is obtained. 
 
 Staircase. — The whole structure, consisting sometimes of a number of connecting flights of 
 stairs, and again of one flight only. 
 
 Well-hole. — The opening required for a complete staircase. 
 
 Right-hand and Left-hand Stairs. — A stair is called right-hand if, when a person is going 
 up the stairs, the hand-rail is on the right; but if in going up a stair the hand-rail is on the left, 
 then the stair is called a left-hand stair. 
 
 The Run of a flight of stairs is the horizontal distance from the first to the last riser in the 
 flight. 
 
 Tread. — The horizontal distance between two risers. One of the equal divisions into which 
 the run of a flight is divided. 
 
 Height of Story. — The vertical distance from the top of one floor to the top of the floor of 
 the story above. 
 
 Head-room. — Height required to clear the head in ascending or descending stairs. 
 
 Rise. — The vertical height between two treads — one of the equal divisions into which the 
 height of a story is divided. 
 
 Riser. — The board forming the vertical portion of the front of a step to which it is glued 
 or otherwise fastened at right angles. 
 
 Step. — The horizontal plank upon which we tread in ascending or descending a stair. 
 
 Nosing. — The outer or front edge of a step. It usually projects beyond the face of the 
 riser a distance equal to the thickness of the step, and is rounded or moulded. 
 
 Winders. — Steps of a triangular form in plan required in turning an angle or going 
 round a curve. 
 
 Scroll Step. — A bottom step with the front end shaped to receive the balusters round 
 the scroll of the hand-rail; also called a Curtail Step. 
 
 A Flight. — One continued series of steps without a landing. 
 
 A Landing. — A horizontal resting-place at the top of any flight. 
 
 A Half-turn Platform. — A landing extending across the well-hole, embracing the widths 
 of two adjoining parallel flights as they land on and start from the platform. 
 
 A Quarter Platform. — A square landing, the sides of which each equal the width of its 
 connecting flight. 
 
 Wall-string. — The string secured against the wall. A plank prepared by mortises sunk 
 into its face to receive and house the ends of steps and risers on the wall side. 
 
 Front-string. — The string on that side of the stairs over which the hand-rail hangs; a 
 plank prepared and sawed out to receive and support the front ends of steps and risers. 
 
 Open-string. — Same as front-string. 
 
 Close Front-string. — A front-string into which the ends of steps are housed the same 
 as a wall-string, the upper edge of the string capped to receive balusters ; and its outer 
 face paneled or otherwise ornamentally finished. 
 
 Pitch. — Angle of inclination. 
 
 Pitch-board. — A piece of thin board in the form of a right-angled triangle, one of the 
 sides of the right angle equal to a rise, the other side of the right angle equal to a tread 
 of a stair. The hypothenuse is the pitch or angle of incluiation of the stairs. 
 
 Cylinder. — A concave semicircle, formed b}' gluing together hollowed wooden staves, or 
 by bending over a convex cylinder. 
 
 Quarter-cylinder. — A concave quarter-circle hollowed out of a solid piece of wood, or 
 formed by being bent over a convex quarter-cylinder. 
 
 Splice. — Half the thickness of wood cut away for a few inches of its length, so that it 
 can be joined to another portion similarly treated. 
 
 Facia. — A casing, finishing the face of beams — called headers — along the floor levels. 
 
 Fillet. — A band i^" wide by \" thick, nailed to the face of a front string below the cove 
 or scotia, and extending the width of a tread ; and also a similar band mitred with the riser. 
 The fillet is also nailed to the facia under the scotia along the levels. 
 
 Curve-out. — A concave curve of the face of a front-string at its starting. 
 
 Board. — Merchantable sawed lumber of various widths and lengths, and one inch thick or less. 
 
xxvi 
 
 DEFINITIONS. 
 
 Plank. — Merchantable sawed lumber of different lengths and widths, and more than one inch 
 thick. 
 
 Timber or Beam. — Sawed lumber of large size. 
 
 Carriage-timbers. — Permanent timber supports nailed under stairs parallel to the lower 
 edge of strings 
 
 Newel. — An ornamental post or column — built or turned solid — of various sizes and 
 designs, set at the foot of stairs to receive and secure the hand-rail. In open-newel stairs, 
 posts of a small size set at the angles and into which the strings are framed. 
 
 A Straight Flight of Stairs — Is one in which all the steps are parallel and at right 
 angles to the strings. 
 
 Quarter-turn Winding Stairs. — A stairs in which the winders make a turn of one quarter 
 
 to a landing or to a continuation of the flight. 
 
 Half-turn Winding Stairs. — A stairs in which the winders make a turn of one half to a 
 landing, or to a continuation of the flight. 
 
 Circular Stairs — Are stairs with steps planned in a circle, toward the centre of which 
 they all converge, and consequently are all winders. These stairs may have a circular, a 
 square, or octagonal-shaped wall ; and on the front an open cylinder and continued hand- 
 rail, or a solid circular post with the front ends of steps and risers housed into the post. 
 
 Elliptic Stairs. — Stairs that are elliptic on the plan. The treads are spaced on the front 
 and wall strings, but being less in width on the front-string, they all converge, but not to 
 one centre like those of a circuiar stairs. 
 
 Newel Stairs. — Stairs in which newels are substituted for cylinders and continued hand- 
 rail. 
 
 Open-newel Stairs — Are so called because they have small newels arranged at the 
 angles of an opening in place of cylinders. The connecting front-strings are framed into the 
 newels. 
 
 Hand-rail.— A variously-moulded form and size of rail running parallel to the inclination 
 of a stairs, and usually kept at a vertical height of 2'.2" from the top of step to the bottom 
 of rail, at the centre of short balusters. The rail also continues around cylinders, or connect- 
 ing with newels parallel to floor levels, at a height of 2'.6" * from floor to bottom of rail. 
 
 Baluster. — A small column made of different forms and sizes, but commonly turned 
 They are set vertically on the steps of stairs, generally two on a step, and placed the same 
 distances apart on the floor levels, forming an ornamental enclosure and furnishing support 
 for the hand-rail. 
 
 Balustrade. — Balusters and hand-rai'i combined. 
 
 Wreath. — The whole of a helically-curved hand-rail, whether it makes a half-revolution 
 
 or more. 
 
 Wreath-piece. — A portion of a wreath less than the whole. 
 
 Face-mould. — A section produced on any inclined plane vertically over a curved plan of 
 hand-rail ; used on the face of plank to mark the shape of the sides of wreath-pieces. 
 
 Side-mould.— A pattern of the exact shape of the top and bottom of wreath, unfolded from the 
 centre line of rail-plan — made of thin sheet-zinc or straw-board — to be applied to the concave and 
 convex sides of wreath-pieces. See Plates 20, 21 and 76. 
 
 Ramp. — A concave or convex curve or easement of an angle, as sometimes required at 
 the end of a wreath, and the adjoining straight rail, where the two have different inclinations 
 
 Ramp and Knee. — A concave easement of hand-rail with its upper end forming an angular 
 knee. When the knee is curved convex the combined curves are called a Swan-neck. These 
 two forms — rainp and knee, and ramp and swan-neck — are used in open-newel and other 
 stairs where the newels are turned. 
 
 Butt-joint. — An end joint made at right angles to the central tangent of a wreath-piece; 
 and also an end joint made at right angles to any straight length of hand-rail. 
 
 * In figuring measurements of architectural drawings, and in specifying sizes, feet and inches are designated by accent- 
 marks— called indices — as follows : 2'. 6" meaning two feet six inches. Feet are denoted by one accent-mark over 
 the number, and a period on the right separating it from the fractions of a foot, — inches Inches have two accent- 
 marks over the number as shown. Feet and no inches are indicated thus: 25'. o" — twenty-five feet no inches ; inches 
 and no feet thus: o'.6" — no feet six inches. The latter is frequently written with the indices for inches only, as 6". 
 
BOOKS PUBLISHED TREATING ON STAIR-BUILDING. 
 
 The following is a complete list and dates, as far as ascertained, of publications in the English 
 language that either treat i)artially or wholly of stair-building and hand-railing : 
 
 Date. 
 
 1693. Moxon, " Mechanical Exercises." 
 
 1725. Halfpenny, " Art of Sound Building." 
 
 1735. Francis Price, " British Carpenter." 
 
 1738. Batty Langley, " Builder's Complete Assistant." 
 
 1741. " Workmen's Treasury." 
 
 1741. " Builder's Jewel." 
 
 1750. Abraham Swan, " The British Architect; or. The Builder's Treasury of Staircases." 
 1792. Peter Nicholson, "Carpenter's Guide." 
 1813. Peter Nicholson, New "Carpenter's Guide." 
 
 1826. M. A. Nicholson, " Carpenter, Joiner, and Builder's Companion." 
 1864. Joshua Jeays, " Orthogonal System of Hand-railing." 
 1873. Newland's " Carpenter and Joiner's Assistant." 
 
 The above are all English publications. The following are all — or have been— published in the 
 United States : 
 
 Date. 
 
 1838. Minard Lafever, " Staircase and Hand-rail Construction." 
 
 1844. R. G. Hatfield, " The American House-carpenter." 
 
 1845. Simon De Graff, " The Modern Geometrical Stair-builder's Guide." 
 1849. Cupper's " Hand-railing." 
 
 1856. Robert Riddell's "Hand-railing." 
 
 1858. Perry's " Hand-railing." 
 
 1859. Esterbrook & Monckton's "American Stair-builder." 
 
 1872. Monckton's "National Stair-builder." 
 
 1873. Wm. Forbes's " The Sectorian System of " Hand-railing." 
 1873. Louth's " Hand-railing." 
 
 1875. Gould's " Hand railing." 
 
 1886. Frank O. Cresswell's " Hand-railing and Staircasing." 
 1889. John V. H. Secor, " Nonpareil System of Hand-railing." 
 
 1888-94. Monckton's " Stair-building: Wood, Iron, Stone, and New One-Plane Method of Drawing 
 Face-Moulds; Unfolding the Centre Line of Wreaths and Side Moulds." 
 
SUGGESTIONS. 
 
 The Attention of Teachers Engaged in Giving Instruction in Architectural Drawing 
 in Technical Schools is Invited to the Examination of this Work, which is believed to be 
 well suited for taking up an important part of interior use and decoration— the planning and 
 construction of stairs, a branch of building but slightly touched, while roofs and other parts of 
 building structures are taught in much detail. 
 
 Apprentices who desire to master the contents of this book are informed that much care in 
 its preparation was given to make the whole so simple and clearly stated, that it would be easy 
 to learn ; yet if any have had no previous practice in the use of drawing-instruments, and no 
 knowledge of practical geometry, such should at once begin tiiat study. A notice of a little book of 
 Practical Geometry, treating likewise of the use of instruments and all the necessities of a beginner in the 
 study of drawing, will be found in the last pages of this work. 
 
 In the Study of Hand-railing the paper formed solids beginning with Plate No. lo should 
 be drawn as directed, cut out, and glued in shape as explained; for this purpose a little bottle 
 (with brusli) of LePage's liquid glue is best and most convenient. 
 
 The Squaring of Model Wreath-pieces, one quarter of full size — or one half size in case 
 of small cylinders — out of some soft wood brings valuable results to the apprentice, of which the 
 first is Practice; second, Experience ; third, Knowledge. 
 
 Fitting Wreaths over Circular or other Curved Iron Staircases.— Begin by chalking on 
 the iron hand-rail plate suitable lengths of wreath-pieces; and to get a parallel pattern for each 
 of these pieces, press a thick, strong sheet of paper to the top of the plate, and mark the concave 
 and convex edges of the plate as far as the length requires; parallel to the curves thus marked 
 set off each side enough to make the width and thickness equal that given by trial as at Plate No. 23, 
 Fig. 5, or Figs. 3 and 5, Plate No. 43; cut the paper joints to suit the eye, and a little long. These 
 paper patterns are used to mark the shape of the wreath-pieces on the plank; they are then sawed 
 out square through. The bottom of the wreath-piece is first fitted to the iron plate and the plate 
 let in flush ; then plumb-lines are put on the joints, the sides worked plumb and brought to a 
 width; lastly the top is cut away to the thickness, and the joints finished; when the adjoining 
 pieces are squared, their joints are fitted against the first piece, etc. 
 
 Note. — Fitting wreaths over iron staircases on iron rail plates is done as above directed because the 
 curves to which the iron is brought are often various and eccentric, consequently fitting is resorted to as 
 quickest and best. 
 
 Self-supporting Circular Stairs are rarely built. These stairs stand disconnected and away 
 from walls or any points of support, except at the top and bottom, and have hand-rails and balusters 
 over both strings. No carriage-timbers need be used if the risers are increased in thickness and 
 the strings are made thick and bent laminated as explained at Plate No. 8, Fig. 5. The strings 
 at the bottom should be run down between the floor-beams and secured with strong iron bolts ; 
 they should also be strongly bolted at the top. Only screws ought to be used on such a stair- 
 case. Jib panels should be built as high as can be permitted under the strings at the foot of 
 the stairs. The management of hand-railing for circular stairs is given at Plates Nos. 53 and 54. 
 
 Close Paneled Strings are best suited to neweled stairs, but if this construction is used 
 with cylinders they should be of large diameter, otherwise the work will appear cramped and 
 ugly. 
 
 The Joints of All Wreath-pieces with the exception of those given at Plates No. 34 
 and 35 must be made at right angles to the face of the plank. 
 Newel Caps Mitred to Hand-rails ought to be abolished. 
 
 Turned newels should be finished in one solid piece. The connection of hand-rail with newel 
 is stronger and better if run straight to the newel and bolted together. 
 
 Balusters with Square Bases insure a stronger balustrade than those with circular bases. 
 
 Hand-rails may be Finished and Varnished before being set up, if reasonable accuracy 
 be observed in the drawing and in the work. 
 
 Hand-railing. — Thorough students will give careful attention and study to each of the subjoined 
 statements and references, viz., through a knowledge and application of one simple geometrical process, that 
 of finding a level line co>nmon to both planes, the following five important operations are correctly performed: 
 ist. Measurement on one plane for drawing all Face-moulds. See Plates Nos. 9 to 19. 2d. All angles 
 found that are required for Squaring Wreath. See Plates Nos. 9 to 19. 3d. Unfolding a Central Wreath- 
 line. See Plates Nos. 20 and 21. 4th. Unfolding convex and concave Side-moulds. See Plates Nos. 20, 
 21, and 76. 5th. Finding on the horizontal plane the subtension, or opening, of any angle of tangents on a 
 cutting plane. See Plates Nos. 9 to 19. 
 
 It would be of great advantage if architects would count the height of stories by rises or at least give 
 special attention to so important a matter for their stairs. 
 
PLATE 1. 
 
 Plan and Elevation of a Straight Flight of Stairs with a Seven-inch Cylinder; 
 ALSO A Plan and Elevation of a Platform Stairs with a Six-inch Cylinder Land- 
 ing with Four Rises Above the Platform. 
 
 Fig. I. Plan of a Straight Flight of Stairs. — The plan is given to show the width of the stairs, 
 the size and position of the cylinder, the number and position of the rises and treads ; also by means of the 
 shaded lines to show the opening of the well-hole, its width and length ; its width sufficient to receive the 
 width of stairs, the diameter of cylinder and the thickness of facia ; its length sufficient for head-room. 
 
 Fig. 2. Elevation of the Straight Flight of Stairs Shown by Plan at Fig. i.— The 
 number of rises is determined by dividing the height of story— taken from the top of the floor of the 
 lower story to the top of the floor in the upper story — into any number of suitable parts ; in this case the 
 height of story is io'.4", equal to 124", which divided by 16 gives a quotient of 7^", the height of one rise. In 
 the above manner the height of any given story must be taken and divided into any number, more or less, of 
 rises. The rod E F shows the manner of taking the height and the division of rises. To obtain the tread,* 
 first find the horizontal distance that can be taken for the run of the stairs and landing room ; which in this 
 case is equal to A D, 14'. 5"; of this the landing room, B D, must never be less than the width of the stairs, and 
 is always better several inches more ; therefore take B D, 2'. 9", for landing, and C B, 5", for depth of cylinder ; 
 leaving AC, ii'.3", to be divided into treads. There is always one tread less than the number of rises in each 
 flight of stairs, because the floor itself becomes a step for the top rise ; so having sixteen rises in this flight, 
 the remaining ii'.3", equal to 135", must be divided into fifteen parts, which equals 9" for each tread, 
 as shown at plan and elevation. The line G H is the lower edge of the string-plank, which plank is sawed out 
 to receive and make a finish with the risers and steps. The dotted lines parallel to G H indicate the position 
 of the supporting carriage-timbers. Head-room is secured by constructing the well-hole of a sufficient length 
 so that the tallest person in ascending or descending the stairs would not be in danger of striking the 
 head. Head-room should not be less than ^'.o". It is not necessary to draw an elevation of steps and rises to 
 determine head-room, for that can be learned from the plan at Fig. i ; for example, count thirteen rises from 
 the top down at J ; thirteen rises, 7^ "each, equal 8'. 4^" ; subtract from this the thickness of floor, depth of 
 beam and plaster, altogether 10^", and there will remain 7'. 6" for head-room, — if the length of the well-hole 
 does not cover the step J, Fig. i. 
 
 Fig. 3. Plan of Platform Stairs. — Platform stairs ascend from one story to another in two or 
 more flights, having platforms between for resting and changing their direction. This plan has but one plat- 
 form, taking the whole width of the hall, and has four rises in the upper short flight and thirteen rises in the 
 lower starting flight. The shaded lines show the framing of the open well-hole, including the platform. 
 
 Fig. 4. Elevation of Platform Stairs from the Plan Fig. 3.—! J is the height-rod showing 
 the division and number of rises. 
 
 The head-room and tread are found as before explained at Fig. 2. Some attention must be given to 
 the position of platform K, so that the height underneath has sufficient head-room and clears the trim or fan- 
 light of doorway, if there be any ; for the platform may be one or more rises higher if space can be spared to 
 add one or more treads to the starting flight — these treads to be taken from the short landing flight. At L 
 is shown the starting of a second flight from that floor. 
 
 A Rule to Find the Correct Proportion of Tread to Rise. — To any given rise in inches add 
 a sum that together will equal twelve, double the sum added to the given rise for the tread in inches, — as 
 follows : given a 5" rise and 7 make 12, then twice 7 equals the required tread, 14"; or again, given a 7" rise 
 and 5 make 12, then twice 5 equals the tread, 10", etc. 
 
 * The tread is the distance between risers, as M N Fig. 2, without inchiding the projecting nosing ; when the projection of the 
 nosing is added the whole is called the step. The projection of the nosing is usually made equal to the thickness of the step. 
 
Plate No. 1 , 
 
 1 FT. 
 
 Fig. I. 
 
PLATE 
 
 2. 
 
 Step LADDERS and Stoop. 
 
 Fig. I. Plan of Stepladder. — This plan shows the thickness of the sides, the width of the ladder, 
 the treads and number of rises, also an extra width of tread, as at P Q, which should always be allowed at 
 the top step of a ladder. 
 
 With P Q, ^j4" deducted from the run P M, there remains Q M, equal to s'.6}4", or 42}^", to be divided 
 by 10, the quotient of which is 4%", the tread. The point of the ladder N M, if desirable, may be cut off on 
 the line N 0, and glued and nailed to the back edge of the ladder, keeping the point V to the floor. 
 
 Fig. 2. Elevation of Stepladder given at Plan Fig. i. — The height from floor to landing 
 above is S'.i}4", which divided by 10 gives a quotient of 9^", the height of each rise. The sides of step- 
 ladders having from ten to fifteen rises should be from 5" to 7" in width and not less than lya" thick. One 
 way to lay out the sides of a stepladder is as follows : let T R equal the tread and R S the rise ; connect 
 T S ; take the distance T S in the compasses and mark on the edge of the side of ladder from V to A ten spaces, 
 and with a bevel (as at X taken from T) lay out the angle and thickness of steps as shown. Another way to 
 lay out the sides of a stepladder is to use a steel square (^as at B), placing the square at the edge of the ladder 
 to the height of rise and width of tread as figured on the square and as many times as there are to be rises in 
 the ladder. The steps should be let into the sides of a ladder from 3/s" to 3^" ; 3/g" will be sufificient if the 
 sides are i" thick. 
 
 Steps are set into the sides of a ladder (as at Z Y) when the sides of ladder are 9" or 10" wide and 2" or 
 3" thick, as sometimes built in buildings used for wholesale stores. 
 
 To make a small movable stepladder strong and keep the steps from working loose a tenon should be run 
 through the sides at three points (as at i, 2, 3) and properly nailed. 
 
 Figs. 3 and 4. Plan and Isometrical Elevation of a Double Stepladder. — Where space 
 is limited and only occasional communication between stories is necessary, this ladder will answer the 
 requirements, as it can be constructed in the cheapest manner and put up in mere closet-room, as shown 
 by the plan at Fig. 3 and its perspective. Hand-rails should be put up at both sides of the laddei-, 
 hung on iron brackets well secured in the wall. At Fig. 4 there are shown fifteen rises of 8" each, mak- 
 ing a total heiffht of lo'.o"; and fourteen treads of 9" each, occupying a run of s'-3"- 
 
 Fig. 5. Elevation of Stoop with Platform. — The newel post and balusters have no turned work, 
 but are cut on the angles and chamfered. 
 
 The strings where there are so few steps may be laid out with a steel square, or can be laid out with a 
 pitchboard in the same manner as inside stairs. Fig. 6. A pitchboard, to be made of thin, well-seasoned 
 wood, N M 0, must be made perfectly square, M N the tread and M 0 the rise. The grain of the wood 
 should always run in the direction of N 0. The edge of the string C should be jointed, and a pencil-gauge 
 distance equal to C D run along from the edge C. 
 
 The distance C D is equal all together to depth of timber, thickness of riser, and thickness of ceiling 
 boards underneath. Along the line D the pitchboard is marked on the stuff as many times as there are to be 
 rises, and then sawed square through on the line of treads, and cut mitring on the line of rises. The dotted 
 lines show the correct position of the hand-rail to determine its exact length ; the platform level rail is raised 
 4" above the platform, so that when the rail at the centre of the short balusters along the flight is raised the 
 usual height, 2'.2", the level rail over the platform will be 2'. 6", the usual height for a level rail. At the 
 newel G H is raised 3>^", which added to 2'. 2" — the height at the centre of the short balusters— makes the 
 height G K at newel 2'.^}^". 
 
 Note. — The whole of the above practical details and directions relating to an ordinary stoop apply equally to inside stairs where 
 similar work is required. 
 
PLATE 3. 
 
 Plan, Elevation and Details of a Common Straight Flight of Stairs.— Stair-building 
 
 Generally. 
 
 Fig. I. Plan of a Common Straight Flight of Stairs ; showing the width of the fligiit, the 
 thickness of wall-string, the width of treads and number of treads and rises, the position of the balusters, the 
 cylinder, the width and place of hand-rail and size of newel-post. 
 
 Figs. 2 and 3. Methods of Forming Cylinders and Splicing them to Strings. 
 
 Fig. 4. Wall-string Laid Out, showing easements of angles joining floor-base at starting and landing, 
 mortises laid out with wedge-room for steps and risers which are to be let into the string 
 
 Fig. 5. Front-string Laid Out. — A B must be sufficient for depth of timber, thickness of plaster 
 and of riser. The dotletl lines at C show the wood to be left on the string for cylinder-splice. G F H D E 
 is the cylinder opened out ; F G must be the depth of floor-beam and thickness of plaster ; the line D E G is 
 the bottom edge of the cylinder, and at E the curve is raised somewhat above the direction of the line A D in 
 shaping the edge, so as to prevent a baggy appearance the cylinder would otherwise have when in place. 
 
 Fig. 6. Step and Riser as Glued Together. — The whole thickness of the riser is let into the 
 ploughed groove of the step -^/s" ; J is a glue-block, of which two or more are glued and nailed in place as shown 
 along the length of step and riser. K K are dovetail mortises cut in the end of steps to which the balusters 
 are fitted, glued and secured in position as shown at Fig. 8. The step and riser are backnailed together as at 
 R ; from two to four nails are driven in, depending on the length of steps. 
 
 Fig. 7. Stair-timbering and Rough-bracketing. — This drawing represents a vertical section 
 cut through the middle of a flight — a plan of which is given at Fig. i — showing an end view of steps and 
 risers, rough board brackets L L, the middle timber M, and piaster N. Stairs 3'.o" wide and less are usually 
 provided with two carriage, or supporting, timbers, one of which is used to strengthen the front-string, this 
 string being securely nailed to the timber ; the other timber, M, is placed at the centre of the stairs and rough 
 board brackets, L L L, fitted and nailed as shown at alternate sides of the timber. At S the nail through the 
 rough-bracket is driven into the back of the step. 
 
 Fig. 8. Side Elevation of the Starting Portion of Stairs, a Plan of which is Given at 
 Fig. I. — I'he hand-rail is brought straight to the newel at Q, as being a stronger and better connection in 
 many ways than the old plan of a loose turned cap and easement of rail mitred to the cap. P is a jib panel 
 which is usually made as a finish to the bottom of a first-story flight, and also to receive the level rail, 0, that 
 encloses the basement-stair well-hole. There is no better or stronger method of building wooden stairs than 
 what is here described in detail, where each step and riser are glued together in the manner shown at Fig. 6, 
 and housed and properly wedged with glue and hard-wood wedges in the wall-string, as shown at Fig. 4, — 
 also housed and wedged in the same manner at the front, if a close front-string is used. Carriage-timbers, 
 rough-bracketed as before described at Fig. 7, of a size from 2" by 4" to 4" by 10", and from two to five 
 timbers — never less than two — to each flight, depending on the width and extent of the stairs and the weight 
 they are expected to carry. 
 
 For good substantial work well-seasoned materials should be used throughout. So important is this con- 
 sidered in the larger-sized timber that old second-hand timber is sometimes sought for. Whole flights of 
 stairs — with the exception of circular, elliptic, or some other peculiar form — are most economically finished by 
 being wedged and nailed together, trimmed and raised to their places in the building complete ; the support- 
 ing timbers are easily put in position afterward. Generally the staircase may be put up on the dry brown 
 wall, and if made of hard wood the steps, risers, strings and newels may be completely covered with cheap 
 heavy brown paper and thin rough boards, to remain as a protection until the walls are finished with white 
 plaster, the doors hung, the mantels and grates set. 
 
 This covering when put on may be so arranged as to enable the stair-builder to easily remove some six 
 inches of it, enough to allow the hand-rail to be put up and finished, leaving the balance of the covering until 
 no longer recjuired. 
 
 A Staircase of Any Form of Plan may be Finished on the Under Side, Showing its 
 Construction with far more elegance and variety than any surface plastering or close panelling com- 
 monly done. In this finish the wedged strings will have to be cased to conceal the wedging. Both the wall 
 and front strings should have greater thickness than usual ; the front-string should be thick enough to 
 dispense with a front carriage-timber, or such timber may be used and cased. If desirable the risers can be 
 made of a thickness and strength that no middle supporting string or cased carriage-timber would be 
 recjuired : this would leave an unobstructed view of panelled steps and risers, or other ornamental finish. 
 
 The Old English Method of Stair-building — which is occasionally followed in this city, and 
 commonly in some portions of the United States — is to construct rough timber carriage-ways sawed out for 
 step and riser, with rough steps nailed on, to be used for travel during the process of building, and to be 
 plaster-finished as required on the under side, at the same time with the walls of the building. This carriage- 
 way is then cased with finished strings, steps and risers ; the wall-string and sometimes the front-string — 
 where the latter is to be close — are scribed to the grooved step and riser and set in this groove with a 3/^" 
 tongue ; the projecting step nosing is sawed to fit against the face of the scribed string. T/iis last mct/ioJ of 
 building the bodies of staircases is not as good as the more modern one previously described, and is also tnuch more 
 costly. 
 
Plate No. 4 
 
PLATE 4. 
 
 Planning Winding Stairs — Drawing Elevation of the Same— Laying Out the Strings. 
 
 Fig. I. Plan of a Staircase Winding One Quarter, Alike at the Top and at the Bottom, 
 with Cylinders lO" in Diameter. — It is important in planning winding stairs of various forms — and this 
 example will serve for all — to make the treads as nearly as possible of a uniform width on an established line of 
 travel, which is about 14" from the front-string as shown. It is well to increase the width of the stairs a few 
 inches both at the top and at the bottom, for the more convenient passage of furniture at these turns. In 
 making the plan of stairs, the first thing to be determined is the wall-lines A BCD, next the width of the body 
 of the stairs from the wall E to the front-string F. The width of the hall for this staircase will require to be 
 7' between rough walls — ^' .0" width of stairs, 10" diameter of cylinder, 3'.o passageway, and 2" thickness of 
 plaster. From the walls to the cylinder, at both the top and the bottom of the stairs, the width is made 3'. 2". 
 The line of travel is drawn in position as before mentioned ; the starting or first and top or landing riser lines 
 are now drawn as required, taking care that not less in any case than 2" level of the cylinders, as at X X, be 
 left at both top and bottom so as to make a proper finish with the facias. Between these starting and landing- 
 risers, on the line of travel, equal spaces are marked for the width and number of treads required ; next the 
 treads in the cylinders and along the line of the front-string are marked as figured ; now the lines of risers 
 are drawn from the points of division at cylinders and front-string, through the points of division first made 
 on the line of travel : and this completes the plan. 
 
 Fig. 2. Elevation of the Plan of Stairs, Fig. I. — This elevation explains itself in connection 
 with the plan beneath : The important points given by the elevation are the head-room from the top of the 
 third step, G, to the plastered ceiling, H, and the length of the well-hole as limited and shown by the line G H. 
 It is not necessary to set up an elevation of a staircase to fix the head-room. The head-room may be deter- 
 mined, and the length of well-hole, by finding how many rises down from the top — after subtracting therefrom 
 the depth of floor-beam, including floor and plaster — would equal in height y'.o", or very nearly that. 
 
 Fig. 3. Laying Out of the Front-string and Cylinder.— The distance K J, on the front-string, 
 must equal all together the depth of timber, thickness of plaster and thickness of riser. The cylinders are 
 spliced to the string on the lines 0 P and R S ; the treads, as given in the cylinders and string, agree with the 
 plan Fig. i. At the starting cylinder, N X equals 10", the width of facia, which is the depth of,floor-beam (9") 
 and thickness of plaster (i"). The top cylinder requires a straight piece of board the thickness of facia 
 glued at its upper end, of sufficient width and length (as at T X V U) to produce an easing between the lower 
 edge of the cylinder and the lower edge of the facia ; this piece is glued to the cylinder on the line T X, and 
 joins the facia on the line U V. The depth to the lower end of the cylinders is found, as shown, by describing 
 arcs of a radius equal to K J from the angles of tread and rise as shown at Q 0 and W T. It is better to 
 join the cylinder as at R, on the straight line R K, even if it has 2" or 3" more depth at that point than at K J. 
 Cylinders are sometimes — generally in the best ivork — laid out with the straight string in one flank, as here shoicn, 
 and the whole of that portion for forming the cylinders up to the lines 0 P and R S is cut away at the back, leaving 
 only a thin veneer at the face, which is bent over a convex cylinder and filled out with staves, as described at Fig. 4, 
 Plate 8. 
 
 Fig. 4. Wall-string.— This string is the starting portion, A B, at the plan Fig. i. In preparing this 
 string to join the floor-base, V S is the height of base, then S L is the easing of the angle of string to the 
 level of the base ; or the string may be mitred to the base, as at Y 0, R Y being equal to V S. The height of 
 string M N above the angle-step must be alike from the same step as M N at the next string, Fig. 5 ; also the 
 .strings connecting at these angles may be brought to a level, curved as shown, or left angular ; but they fuust 
 in all cases be brought to a level, so that the base-mouldings ivill properly connect. 
 
 Fig. 5. Wall-string.— This is the whole of that portion of wall-string marked B C of the plan Fig. 
 I. The height of the string at 0 P must be alike from the same angle-step as at 0 P of Fig. 8. This string 
 is laid out in two pieces spliced together at the centre, Z D ; it may be laid out in one piece by the use of a 
 mean tread, and in another way, each of which methods will now be given. 
 
 Fig. 6. Wall-string same as Fig. 5, Laid Out in One Piece by the Use of a Mean 
 Tread. — A mean tread is found by adding together the ten treads as figured, and dividing their sum by the 
 number of rises (9), as follows : 25 4- 1 7 -I- 13 -f 9 -f 9 -f 9 + 9 -I- 13 -|- 17 -f 23^ = 144^ 9= i6g'V -h " mean tread, 
 which is nearly 161^". Along any line, A B, beginning at A, apply the mean tread, A C, and the rise, C D, as 
 shown, marking each tread in their order of the width required, and if the work has been correctly performed 
 the last tread, 25", will come out at B of the line A B. 
 
 Fig. 7. To Lay Out Winder-strings from a Scale Drawing.— Set up to a scale 0(1}^" to i' 
 an elevation of treads and rises the same as at Fig. 6, and draw a line touching the outermost points of the 
 upper edge of string, as X E 0 ; then with a bevel set to the angle Q 0 S a string may be laid out full size 
 whose points X and E will touch the edge of plank. Begin laying out with the line 0 S, and make 0 S as 
 many inches full size as it measures on the scale. 
 
 Fig. 8. Wall-string. — This string is the landing portion of wall-string marked C D on the plan FiG. i. 
 
PLATE 5. 
 
 Through the drawings given in this plate and the two following plates, over thirty different plans of stairs 
 are presented ; they are all made to a scale and figured for convenient reference. These various plans are 
 intended to afford opportunity for the examination and study of stair plans, properly arranged for their 
 different requirements. The grading of treads next to the cylinder in the case of winders, so that the wreath 
 will make easier curves and less inclinations in its connections, is a matter of no slight importance. A little 
 more attention, a better knowledge of practical details in planning stairs, will often lead to saving valuable 
 space, or to a more comfortable passage from floor to floor. A superior plan of stairs may even prove to be 
 a question of humanity ; a cruel thing it may be to a little child or an aged and feeble person to subject them 
 to the danger and discomfort of travelling over oblique winding steps; as, for example, at Fig. 5, when a very 
 little more space, as figured, will permit a safe and easy stairway for all, as shown at Fig. 6. 
 
 Fig. I. Plan of a Straight Flight of Stairs Starting and Landing with Small Cylin- 
 ders. — I'he i)osition of cylinders with regard to starting and landing risers, as shown in this case, is 
 explained in detail at Plate No. 22, Figs, i and 2. 
 
 Fig. 2. Plan of a Straight Flight of Stairs Starting with a Newel Set-off 2>^" and 
 Landing with a 7" Cylinder ; the Landing Riser Set Into the Cylinder 2>^". — The set off of 
 a newel and its management in connection with the hand-rail is given at Plate 31, Figs, i and 2. The cylin- 
 der at the landing is treated in detail at Plate 33. 
 
 Fig. 3. Plan of Winding Staircase with Mortised Strings Both Sides. — These stairs are 
 only used where room for better cannot be spared, in such places as an attic (^r basement story. A and B 
 are plank continuations of string 5" wide and long enough when spliced to the mortised string at the top and 
 at the bottom to receive the winding steps and risers, which will be better understood by examining the 
 elevation of string set up at Fk;. 4. 
 
 Fig. 5. Plan of the Top or Landing Portion of a Quarter-turn Winding Stairs, with a 
 Small Cylinder. — The management of the hand-rail of this case is given at Plate 25. 
 
 Fig. 6. Plan of the Top Portion of a Staircase Turning One-quarter to the Landing, 
 with Diminished Steps Around the Cylinder ; Curved Risers and Platform,— This plan is an 
 improvement on that given at Fig. 5. By curving the risers, winders are avoided and a roomy platform 
 secured with the same small cylinder. But with the number and width of treads alike, this plan requires 7)^" 
 more room, as shown at C D. llw hand-rail of this case is treated in detail at Pla'i e 26. 
 
 Fig 7. Plan of the Top Portion of a Staircase Turning One Quarter with Diminished 
 Steps, Curved Risers, Newel and Level Quarter-cylinder, and Platform.— In this plan a small 
 newel is introduced with a connecting level cjuarter-cylinder ; designed to take the place of plan Fig. 6, where 
 this is preferred. By this plan no wreath or ramp will be required. A design of nnvel ; the plan, elevation and 
 management of this case in detail ivill be found at Plate 62. 
 
 Fig. 8. Plan for Starting or Landing of a Staircase. — By using a single newel and setting it 
 diagonally, as shown, it will be strong in all of its connections. In some styles of interior finish this position 
 of newel would be desiral)le. 
 
 Fig. 9. Plan of Stairs (Combining Two Platforms with Curved Risers Between) 
 Making a Half-turn. — The management of the hand-rail for this plan of stairs is given in detail at 
 Plate 41. 
 
 Fig. 10. Plan of a Platform Stairs Making a Quarter-turn with a Quarter-cylinder. — 
 
 Whatever radius is taken for the quarter-cylinder in this description of stairs, in order to make the best form 
 of wreath-piece, from E, the centre of the hand-rail, to risers F and G must be each half a tread. See Plate 
 37, Figs. 5, 6 and 7. Also Plate 45, Fig. 5. 
 
 Fig. II. Plan of Stairs Turning One Quarter with Winders and a Quarter-cylinder. — 
 
 In planning this kind of staircase experience has proved that the best shaped hand-rail is produced by 
 bringing the rail at the upper [)ortion, K, straight into the wreath-piece at the end without a ramp, for this 
 reason : one winder at K, above the quarter-cylinder, is all that should be allowed. See Plate 36. 
 
Plate No. 6 
 
 Sca'Le'4 in=Ift. 
 
PLATE 6. 
 
 Fig. I. Plan of Platform Stairs with the Risers at the Platform Set Into the Cylin- 
 der All that can be Profitable.— Placing the risers in the position given on the plan saves 6" at both 
 the landing and starting flights connected with the platform. The management of the hand-rail is given at 
 Plate No. 38. 
 
 Fig 2. Plan of a Winding Staircase with 10" Cylinders, Making a Half-turn at Each 
 Cylinder. — The hand-rail for this plan is treated in full detail for the top or landing portion at Plate No. 
 28, and for the starting at Plate No. 29. 
 
 Fig. 3. Plan of a Winding Staircase, Two Flights Connecting with a 12" Cylinder. — 
 The details and treatment of hand-rail are given at Plate No. 42. 
 
 Fig. 4. Plan of Stairs with Newel Set Between Two Quarter-cylinders. — In this case the 
 treatment of hand-rail, if at the top of a flight, will be substantially the same as that given at P'ig. 2 of 
 Plate No. 24, and if at the starting of the flight, Fig. i of Plate No. 24. 
 
 Fig. 5. Plan of the Starting of a Staircase. — Where the haU is wide enough and it is desirable 
 to make the flight broad and inviting, the front-string is curved out, embracing four or five treads. This case 
 of hand-rail is treated at Plate No. 32, Figs, i, 2 and 3. 
 
 Fig, 6. Plan of Platform Stairs with Low-down Small Corner Newels and Continued 
 Hand-rail. — A design and the management of stairs and hand-rail of this plan are given in detail at 
 Plate No. 58. 
 
 Fig. 7. Plan of Platform Newelled Stairs with Wing-flights. — This staircase, suitable for a 
 very large hall of a public building, is designed to be wainscoted and with half-newels at the walls, as shown, 
 running through and ornamentally finished at the under-side of the stairs. The best effect given to a stair- 
 case of this character is by showing the whole open construction of the under-side, tastefully finished, free from 
 plaster or close soffit panelling. 
 
 Fig. 8. Plan of Stairs Making a Half-turn, with a Large Cylinder Filled with Treads 
 of Equal Width to those of the Straight Portion of the Flights, and Curving the Ends 
 of the Risers so as to Avoid Winders and Secure an Ample Platform.— Full detail instruction 
 for the management of the hand-rail over this plan is given at Plate No. 51. 
 
 Fig. 9. Plan of Staircase Suitable for Steamboat or Ship, where Every Inch of Space 
 is Valuable. — The requirements of a hand-rail over this plan are treated in detail at Plate No. 50. See, 
 also, Plate No. 52. 
 
PLATE 7. 
 
 Fig. I. Plan of Platform Stairs. — By placing the risers six inches into both C3 linders as seen in 
 this plan, that amount of room is saved in each case — a matter of saving that is sometimes of much impor- 
 tance. The treatment of the liand-rail over this plan is given at Plate No. 40. 
 
 Fig. 2. Plan of Double Platform Stairs Made by Introducing a Riser at the Centre of the 
 Cylinder. — The treatment of the hand-rail over this plan is given at Plate No. 39. 
 
 Fig. 3. Plan of Winding Stairs Making a Three-quarter Turn.— The management of the hand- 
 rail over the centre cylinder of this flight is given at Plate No. 47, and over the starting portion at Plate 
 No. 48. 
 
 Fig. 4. Plan of a Quarter Platform Stairs.— By curving risers in the manner here shown, a good 
 roomy square stepping plan is made of what would otherwise be winders ; somewhat like those of plan at Fig. 
 6. The details and management of hand-rail over this plan will be found at Plate No. 46. 
 
 Fig. 5. Plan of a Quarter Platform Stairs with Newels Set in the Angles.— The framing of 
 these newels and their connections of hand-rail is given in complete detail at Plate No. 59. 
 
 Fig. 6. Plan of a Quarter-turn Winding Stairs at Starting. — The detailed instruction for 
 the management of hand-rail over this plan is given at Plate No. 27. 
 
 Fig. 7. Plan of the Top Portion of a Quarter-turn Winding Stairs. — The management of 
 hand-rail over a plan similar to this will be found at Plate No. 25. This plan shows a way of curving the 
 risers so as to save a sometimes much-needed space by lessening the distance from the wall B to the landing 
 riser C. 
 
 Fig. 8. Plan of a Quarter Platform Stairs Much the Same as that Given at Fig. 4, Except 
 the Shape of the Cylinder. — 'I he management of hand-rail over this plan is given at Plate No. 44. 
 
 Fig. 9. Plan of a Quarter Platform Stairs with One Tread Placed at the Centre of 
 the Cylinder. — Management and detail of hand-rail over this plan will be found at Pla'j e No. 43. 
 
 Fig. 10. Plan of a Circular Staircase. — The dotted lines show the best method of timbering a 
 staircase of this or similar form. The practical treatment of hand-rail over this plan may be found at Plate 
 No. 53 ; also at Plate No. 54 are given full instructions for changing the plan of the first step to the scroll 
 form, the management of that portion of the hand-rail, also the construction of the scroll step. 
 
 Fig. II. Plan of an Elliptic Staircase. — This plan has the treads on the line of wall and front 
 strings graded so that the risers are placed in a direction nearly normal to the curve, keeping an even tread 
 on the line of travel ; which would not be the case if the treads were made equal at the wall-string and 
 at the front-string. The hand-rail over this plan is given in detail at Plate 55. 
 
 Independent or Self-supporting Staircases. — This kind of stairs derives no support from wall 
 or partition ; they are seldom required, but when called for are mostly of a circular plan. An independent 
 straight staircase presents no difficulty ; for all that is required of it is, that it be well secured at the top and 
 bottom, and that the material and construction have ample strength to support the weight it will be liable to 
 carry. Where the plan of a self-supporting staircase is circular with cylinder opening as at Fig. 10,* the 
 timbers at the foot of the stairs R Q P should be bolted to the floor-beams, and bolted at all their connections 
 up to and including the floor-beams at the lajiding. Jib panels should be put in at the starting of both 
 strings as high up as can be allowed ; or set up a supporting column near the centre of the flight. Or again, 
 if it is convenient, let an iron bolt secured in an adjoining wall project sufficient to support the staircase at 
 about the centre L. With the supports mentioned these stairs may be finished on the under side and made of 
 sufficient strength without timbers by the use of thick laminatedf strings, the steps and risers to be well housed 
 into both strings. Iron screws only should be used — no nails. 
 
 * See Plate 45. 
 
 t See Plate 8, Fig. 5. 
 
Fig. 11. 
 
 Scale '4 in. = 1 ft. 
 
Plate No. 8 
 
PLATE 8. 
 
 Figs. I and 2. Bending Wood by Saw-kerfing. — This method of bending is the weakest 
 practised, but owing to the fact that it is thought to be least expensive is frequently adopted. To find 
 the correct distance between saw-kerfs for any required radius of curvature, select a piece of stuff of 
 suitable length and equal to the thickness of the material to be bent, as at Fig. i. Let A B equal 
 the thickness of stuff, and A C the radius of the required curve ; make a saw-kerf at B 0, leaving 
 a thin veneer A 0 uncut, nail the cut piece at S K, and move it from C to D, or just enough to close 
 the saw-kerf at B ; then C D being the distance moved will also be the exact space between each saw- 
 kerf. The same gaged thickness of veneer A 0 must be kept, and the same saw used for the work to 
 be done, as were used in the trial at Fig. i. 
 
 Fig. 2. The Construction of a Circular Form Over Which the Saw-kerfed Material 
 as Above Explained is Shown, Bent in Position. — E F G is the plank rib (made of three pieces) 
 of which two or more are required, according to the work to be done. H J L are the staves which are 
 nailed to the ribs and so complete the circular form. N M is a veneer laid over the form first, upon 
 which is bent and glued the prepared saw-kerfed material P Q R ; this must be left on the form until 
 the glue is perfectly dry. The piece of saw-kerfed work P Q R, should be drawn tight to the veneer 
 and the form by means of hand-screws, as given by one example, T U, with curved blocks, V W. 
 
 Fig. 3. Bending Wood and Keying. — This form is in plan the same as Fig. 2, except that 
 the rib £ F G is not curved at its lower edge ; — shaping the lower edge this way is done for the con- 
 venient use of hand-screws in the manner shown at Fig. 2. 
 
 By this method of bending, the wood is removed from the back of the stuff, as at X X X, etc., leaving 
 the thickness of a veneer at the face ; then after bending, the grooves XXX are filled with tightly fitted 
 strips of wood (glued in) called keys, as at S S S, etc. It greatly adds to the strength of this bent keyed 
 work to glue on three strips of veneer, — one at each edge of the keyed stuff, and one in the middle. 
 The glue should be perfectly dry before the work is removed from the form. The spaces between the 
 keys may be determined by the same method as that used to find the spaces between saw-kerfs. 
 
 Fig. 4. Bending a Veneer Facing and Filling out the Thickness with Staves. — The 
 wood is removed wholly from the back of the stuff between the points required, leaving a veneer facing 
 which is bent over the form, and then staves, Z Z Z, etc., are fitted and glued on, as shown in this drawing. 
 
 Fig. 5- Laminated Work. — Bending several thicknesses of veneer together is defined as lamina- 
 ted work. The whole of the veneers required should be heated and bent over the form together and 
 secured in place ; then releasing and applying glue to one-half, put it back in position again, and pro- 
 ceed with the other half in the same way, pressing and binding solidly the whole together and to the 
 surface of the form. 
 
 To ascertain what thickness of white pine will bear bending without injuring its elasticity, multiply the 
 radius of curvature in feet by the decimal .05 and the product will be the thickness in inches: — For example, 
 multiply a four feet radius of curvature by the decitnal given, — 4'.o" X .05=. 20, equal to two-tenths or one- 
 fifth of an inch thickness, that rvould bend without fracture. 
 
 Fig. 6. Bending Stair-strings. — This drawing shows the construction of an ordinary quarter 
 circle form with the correct position of a stair-string bent over it ; the ribs of this form are quarter 
 circles and are set parallel to each other and at right angles to the chord line R P. 
 
 Figs. 7 and 8. Construction of a Form for Bending Quarter Circle Stair-strings, the 
 Ribs to be Set on an Angle Parallel to the Inclination of Such Strings. — There are two 
 advantages claimed for forms built in this way ; one is, a saving of stuff ; the other, that the form 
 occupies less room. 
 
 Fig. 8. — Plan of a quarter turn of winders with a circular wall-string. D E, the circular wall- 
 string as laid out from the plan A B. At Fig. 7, L M and F H is the position of the ribs parallel to 
 D E the inclination of the circular string ; F G equals C A of Fig. 8 — less the thickness of stave — and 
 is the semi-minor axis, and F H becomes the semi-major axis of an ellipse ; as the shape of the rib 
 when placed on the oblique line F H, becomes a quarter ellipse. The ribs have to be beveled on the 
 edge to range with the lines L F and M H. as shown. 
 
 Figs. 9 and 10. Soffit Mouldings. — These mouldings, placed at the lower edge of stair- 
 strings, have to be carried around cylinders, and this work can be done in different ways. A cylinder 
 may be made of sufficient length and reinforced — filled out by gluing on pieces, as at 0 R S, Fig. 10 — 
 then the moulding is worked out solidly in connection with the cylinder. Another way is to shape up 
 the lower edge of the cylinder filled out as at 0 R S, then fit and shape two or more solid pieces — 
 depending on the size of cylinder — of a thickness and width sufficient to carry out a moulding similar 
 to N P, Fig. 9. 
 
 Figs. II and 12. To Find the Lengths of Cylinder Staves that Include Winder-treads. 
 
 — Set up an elevation of treads and risers sufficient to get the shape of cylinder in its connections with 
 the straight string and facia, as here shown and as before fully explained at Fig. 3, Plate No. 4. Divide the 
 opening out of the cylinder V W, into three equal parts, V Z, Z X and X W ; parallel to the risers draw 
 the lines Z L, X S and W T ; then the length of each stave and its position is given at M B, L F 
 and S X. The construction of this cylinder and the winder-treads contained in it are given at Fig. ii. 
 The manner of splicing and connecting a staved cylinder with a straight string is given at Plate No. 3, 
 Fig. 3. 
 
PLATE 9. 
 
 The method of one-plane projection is where the projection on the horizontal plane is alone required. 
 Merely illustrative examples are here i^n-e/i of the practical application of the one-plane ?nethoil in dra7cnng face- 
 moulds for hand-railing. 
 
 Fig. I. To Find the Angle of Tangents and Centre Line Over a Plan of a 
 Quarter-circle, where the Tangents are required to have a Common Inclination. — Let 
 
 A V Y be the plan, with the tangents A U and U Y ; let A W U and U T Y be the common angles of inchnation ; 
 connect U X, the level line common to both planes ; through Y and A draw the line R S indefinitely ; on U as 
 centre with U T as radius describe the arc T S and the arc at R ; connect R U and S U ; bend a flexible 
 strip and mark a curve through the points R V S: then R U S will be the length and angle of the tangents, 
 and R V S the curve-line over the plan A V Y. To find the angle ivith luhich to square the wreath-piece at both 
 joints : — On U as centre describe the arc Z B ; connect B A ; then the bevel at B will contain the angle 
 sought. 
 
 Fig. 2. To Find the Angle of Tangents and the Centre Line Over a Plan of a Quarter- 
 circle when the Tangents have Different Inclinations. — Let the plan be A F M, with the tangents 
 A B and B M ; let the two inclinations be A G B and B C M. To find the level line common to both planes : Make 
 C Q equal to B G ; parallel to M B draw Q 0 ; parallel to M C draw 0 N; connect N K : then N K is the line 
 sought. Parallel to N K draw B I ; at right angles to N K draw M D and A E; on B as centre with B C as 
 radius describe the arc C D ; again, on B as centre with A G as radius describe an arc at E ; connect E B 
 and B D ; bend a flexible strip and mark a curve through E F D ; connect E D : then E B D will be the 
 lengths and angle of tangents, and E F D the curve-line over the plan A F M. To find the angle with which 
 to square the wreath-piece at joint D : Continue B M to L ; make M L equal Q P ; connect L K : then the bevel 
 at L contains the angle required. To find the angle 7vith xvhich to square the wreath-piece at the joint E : Draw 
 I J parallel to M K ; make I J equal B H ; connect J A : then the bevel at J contains the angle sought. 
 
 Fig. 3. Plan of Hand-rail a Quarter-circle, with the Tangents to the Centre Line 
 A F and F D, the Tangents to have Common Angles of Inclination. — Let A G F and F E D be 
 the angles of a common inclination ; connect F X ; from K, parallel to F X, draw K L ; from J, parallel to 
 D E, draw J M ; through A and D draw AC; on F as centre with F E as radius describe the arc E C. To 
 find the angle with which to square the wreath-piece at both joints : Make F H equal F I ; connect H A : then the 
 bevel at H contains the angle sought. 
 
 Fig. 4. To Draw the Face-mould from Plan Fig. 3. — Make B C, B C equal B C of Fig. 3 ; 
 make B F at right angles to B C and equal to B F of Fig. 3 ; connect F C and F C ; make F M and F M each 
 equal F M of Fig. 3 ; through M and M, parallel to B F, draw K L and K L ; make M L, M K, at each side of 
 the centre equal J L and J K of Fig. 3 ; make B N 0 equal the same at Fig. 3 ; through C and C draw K P 
 and K Q ; make C P equal C K, and C Q equal C K ; let C S equal straight wood, as required ; parallel to 
 C S draw K T and Q R ; parallel to M C draw K U ; make the joints at right angles to the tangents ; 
 through Q L 0 L P of the convex and K N K of the concave trace the curved edges of the face-mould. 
 
 Fig. 5. Plan of Hand-rail a Quarter-circle, with Tangents to the Centre Line, Q Z and 
 Z X, the Tangents to have Different Inclinations. — Let Q M Z and Z G X be the required inclina- 
 tions of the tangents. To find a level line common to both planes : Make G H equal Z M ; draw H E parallel 
 to Z X, and E T parallel to G X ; connect T P : then T P is the line sought. Parallel to T P draw I J, Z A and 
 C D ; parallel to Z M draw R L ; parallel to X G draw Y F ; at right angles to P T draw X W and Q 0 ; on 
 Z as centre with Z G as radius describe the arc G W ; again, on Z as centre with Q M as radius describe an 
 arc at 0 ; connect 0 W. To find the angle with ndiich to square the 7vreath-piece at joint Q) : Draw A B parallel 
 to Q Z ; make A B equal Z N ; connect B Q : then the bevel at B contains the angle required. To find the 
 angle with which to square the wreath-piece at joint W : Prolong Z X to K ; make X K equal H 2 ; connect K P : 
 then the bevel at K will be the angle sought. 
 
 Fig. 6. Face-mould from Plan Fig. 5. — Make 0 U W equal the same at Fig. 5 ; on U as centre 
 with U Z of Fig. 5 as radius describe an arc at M ; on 0 as centre with Q M of P'ig. 5 as radius intersect 
 the arc at M ; connect 0 M, M W and M U ; make M E F equal Z E F of Fig. 5 ; make M L equal M L of 
 Fig. 5 ; parallel to M U draw I F J, 3 E V and C L D ; make L D, L C and M S 4 equal R D, R C and Z S 4 
 of Fig. 5 ; make E V, E 3 and F J, F I equal T V, T 3, Y J and Y I of Fig. 5 ; through W draw I P ; make 
 W P equal I W ; through 0 draw C R ; make 0 R equal 0 C ; make 0 T straight wood, as required ; parallel 
 to 0 T draw C N and R Q ; make the joints at right angles to the tangent ; parallel to W M draw I Z ; 
 through R D S V J P of the convex and C 4 3 I of the concave trace the curved edges of the face-mould. 
 
 Face-moulds, their Number and Character. 
 
 1. Plan : Quarter-circle — one tanfjent inclined, the other horizontal. 
 
 2. " " tanfjents " alike. 
 
 3. " " " " differently. 
 
 4. " Less than a quarter-circle — one tangent inclined, the other 
 
 horizontal. 
 
 5. " More than a quarter-circle — one tangent inclined, the other 
 
 horizontal. 
 
 6. Plan : Less than a quarter-circle — tangents inclined alike. 
 
 7. " " " " " differently in- 
 
 clined. 
 
 8. " Elliptic or eccentric — tangents inclined alike. 
 
 9. " " " " " differently. 
 
 10. " Greater than a quarter-circle — tangents inclined alike. 
 
 11. " " " '' " " differently. 
 
 It is believed that the above list of eleven face-moulds comprises all that are required in hand-railing. 
 There are two face-moulds, however, that are given in this work not on the list, one at Plate No. 34 and 
 another at Plate No. 35, each of which include the whole cylinder — a semicircle. These may be called 
 compound face-moulds, for the first is explained at Plate No. 14 and used at Plate No. 34, with a portion 
 of the curve more than the face-mould proper, and worked with the wreath, thereby completing the semi- 
 circle in one wreath-piece ; the other is double the face-mould given at Plate No. 10, the two used as 
 one and worked as directed at Plate No. 35, completing the semicircle in one wreath-piece. 
 
 Note. — These face-moulds are all given in the above order, beginning at Plate No. id. 
 
Plate No. 9 
 
Plate No. 10 
 
 V B E A B 
 
 Fig. 5. 
 
 F I G. 3 
 
PLATE 10. 
 
 This plate is the first of ten prepared for the purpose of giving instruction in a simple and practical way 
 in the scientific requirements of hand-railing, based on a few and easily-applied laws of geometry. 
 
 The object and use of the solids — rectangular, acute and obtuse angled prisms — introduced in these ele- 
 mentary plates, may be summed up briefly : as a convenient and direct means of imparting to workmen this 
 branch of geometrical knowledge ; as demonstratmg the nnportance and use of tangents as applied to hand-rail- 
 ing, for two of the vertical sides of every prism given are tangent to a curve described on the base, and tangent 
 to its trace on the cutting plane. The upper end of each prism is cut inclined on one or two angles of inclina- 
 tion in the same plane, and shows the actual relation of the inclined or cutting plane to the horizontal plane or 
 base ; * or, as maybe again stated, exhibits in every case the exact relation of a plan as given on the base and a 
 section of the plan traced vertically on the inclined plane. The cutting plane, as produced on one end of these 
 solids, is in each particular case the position of the plane or surface of plank out of which the wreath-piece 
 has to be worked. The face-mould and its tangents are found on this plane ; therefore the face-mould when 
 applied to the plank gives the shape of the convex and concave sides of the wreath-piece, which must hang 
 vertically — or plumb — over the curved plan beneath. The paper representations of solids fare to be pre- 
 ferred because they can be more easily and conveniently made than wood solids ; and in making them they 
 afford instruction in detail that wood solids do not, because in the formation of solids with paper the surfaces, 
 angles and curves all have to be found in their proper relation on one plane, which it will be seen is the 
 practice and knowledge required for drawing face-moulds correctly. 
 
 Fig. I. Represents a Solid Block or Prism Standing Vertically on a Square Base, 
 A B D C, the Upper End Cut on an Inclined Plane, A E F C, forming Oblique Angles with the 
 Sides A C and B D, and at Right Angles to the Sides ABE and CDF —Upon the base is described 
 a quarter-circle, B C, tangent to the sides of the solid C D and D B. As E F is parallel to the square base 
 C A B D, it is therefore a level line on the cutting plane C A E F ; and as B D at the base and E F on the cut- 
 ting plane are level lines, any measurement taken on level lines at the base, as G H, J K and L M, and carried 
 vertically to the cutting plane, as G R, J P and L N, and then parallel to the level line F E set off as at R S, 
 P Q and N 0, will give trace points of a curve on the cutting plane perpendicularly over the plan curve at the 
 base. 
 
 Fig. 2. Construction of a Paper Representation of a Solid with its Angles, Surfaces 
 and Curved Lines as Given in Perspective and Described at Fig. i.— Let A B C D be the square 
 base of the solid, and D F C the angle of inclmation over the base C D and A B ; make D V, B W and B E each 
 equal D F and at right angles to the sides of the base ; connect E A, B W, W V and V D ; continue D C to T, 
 and B A to U ; let A U equal A E, and C T equal C F ; connect U T. On A as centre describe the quarter- 
 circle B C, tangent to the sides of the base C D and B D ; through anv points on the curve H K M, parallel to 
 the level line B D, draw the lines H G X, K J X and M L X ; make C N P R T equal C X X X F ; draw R S, P Q 
 and N 0 parallel to U T, and equal to L M, J K and G H : then U S Q 0 C will be the trace of a curve on the 
 cutting plane lying perpendicularly over the plan curve B H K M C. With a sharp-pointed instrument scratch 
 the lines A B, B D, D C and C A ; cut out the remainder of the figure and touch the adjoining edges with a 
 little glue or thick mucilage and bring them together, leaving all lines on the outside, so that their connections 
 may be seen and understood. 
 
 Fig. 3. Plan of Hand-rail a Quarter-circle, with One Tangent to be Inclined, the 
 Other Level. — B D and D C are the plan tangents to the centre line B C ; let D F C be the angle of inclina- 
 tion over the plan tangent C D ; the tangent D B to remain level ; parallel to D B draw E Q 0, J MO and 
 R S 0. The bevel at F contains the angle with which to square the wreath- piece at joint B ; joint C is 
 squared from the face of the plank. 
 
 Fig. 4. Face-mould from Plan Fig. 3.— Make B F and F C at right angles ; let F C equal F C of FiG. 
 3, and F B equal D B of Fig. 3; through B draw V E at right angles to B F; make F 0 0 0 equal F 0 0 0 
 of Fig. 3; parallel to F B draw 01, O Y E, 0 X W and C G H; make B V equal B E; make F I equal D U of 
 Fig 3; make 0 Z equal T S of Fig 3; make 0 Y and 0 E equal P N and P E of Fig. 3, and 0 X and 0 W 
 equal L K and L J of Fig. 3, and C G and C H equal C G and C H of Fig. 3; through the points V I Z Y X G 
 of the convex and E W H of the concave trace the curved edges of the face-mould. 
 
 Fig. 5. Parallel Pattern for Round Rail, or to be Used Instead of the Face-mould for 
 Marking the Wreath-piece on the Rough Plank.— The measurements are taken from the plan at 
 Fig. 3, as indicated by the corresponding letters. Through the central points C M Q R B describe circles of 
 any radius required ; touching these circles on the convex and concave bend a flexible strip of wood and 
 mark the curved edges of the pattern. 
 
 For ordinary-sized hand-rail wreath-pieces may be worked out of stuff as thick as the width of rail, and a 
 parallel pattern about ^" wider than the required width of rail. 
 
 * It should be understood that the bases of all these solids are cut square, or at right angles to their length ; also at the upper 
 end two adjoining sides of any solid may be cut on two different angles of inclination, or a common angle of inclination ; or one side 
 may be cut at right angles to its length and the adjoining side at any inclined angle ; but in every case the opposite sides must be cut 
 parallel. \ Drawing-paper, such as Whatman's, is best to make paper representations of solids— pasteboard is too thick and clumsy. 
 
PLATE 11. 
 
 Fig. I. Represents a Solid Block or Prism Standing Vertically on a Square Base, 
 ZXWY, the Upper End Cut on the Side Z X on the Angle of Inclination, XVZ, and on the 
 Side X W on the Same Angle, K U V. — On the horizontal plane or base, Z Q W represents the plan of a 
 quarter-circle to which the sides of the solid Z X and X W represent plan tangents ; also the lines Z V and V U 
 represent the tangents on the cutting plane. The sides of this solid being cut on a common angle of inclina- 
 tion, the heights from the base X and Y to X V and YT are alike, and therefore a line drawn on the cutting 
 plane from V to T will be a level line ; and at the base a line drawn from X to Y will be a level line common 
 to both planes. At any points on the curve at the base parallel to the level line X Y, draw 0 H and P N ; 
 parallel to X V draw H R and N L : parallel to V T draw L M and R S ; make L M equal N P, and R S equal 
 HO; make V J equal XQ; then the curve Z SJ M U will lie perpendicularly over the plan-curve at the 
 base Z 0 Q P W. 
 
 Fig. 2. Construction of a Paper Representation of a Solid with its Curved Lines and 
 Angles as Given in Perspective and Explained at Fu;. i.— Let Y W X Z be the square base of the 
 solid ; on Y as centre describe the quarter-circle W Q Z ; prolong X W both ways to C and V, Z X to E, W Y to 
 3, and F, and Z Y to B. Let XVZ be the angle of inclination over XZ; make X E, W K and K F each equal 
 XV; connect K E and F E: then K F E will be the same angle of inclination over W X as X V Z is over X Z; 
 make WD, DC, Y B and Y 3 each equal XV; connect C B and 3 Z. Througli Z W draw A A; on X as centre 
 with ZV as radius describe the arcs A A; on Z as centre with A A for radius describe an arc at U; on V as 
 centre with VZ as radius intersect the arc at U; and again on Z as centre with Z3 for radius describe an 
 arc at T; and on V as centre with XY for radius intersect the arc at T; connect ZT, T U, V U and VT; at 
 right angles to Z T draw T2; from K draw K 1 at right angles to EF; at any points on the plan-cnrve 
 draw PN and OH parallel to YX; draw N G parallel to W F, and H R parallel to X V; draw RS and LM 
 parallel to VT; make L M, VJ and RS equal HO, XQ and N P; through ZSJ M U trace a curve that will 
 lie perpendicularly over the plan-curve VV P Q 0 Z. 
 
 With a sharp-pointed instrument scratch ihe lines Z V, Z X, X V\/', W Y and YZ; cut out the remainder 
 of the figure and touch the adjoining edges with a little glue or thick mucilage and bring them together, 
 leaving all lines on the outside, so that their connections may be seen and understood. 
 
 Fig. 3. Plan of Hand-rail a Quarter-circle, the Plan-tangents Z X and W X to Have the 
 Common Angle of Inclination, XVZ and WFX. — Through WZ draw A A with X F as radius; 
 on X as centre describe the arc FA and A: then A A will be the distance on the cutting plane over 
 W and Z ; and if lines be drawn from A to X and A X, then A X and A X will be the length and angle of tangents 
 on the cutting plane. From B, parallel to X Y, draw B J ; from N, parallel to W F, draw N M. To find the 
 angle for squaring the wreath-piece at both joints : Make X E equal X G ; connect E Z : then the bevel at E 
 will give the angle sought. By reference to Fig. 2 when put together as a solid it will be seen that the line 
 T 2, which is parallel to the joint required at U, is on the inclination of the cutting plane — or face of plank — in 
 that direction, and with the line I K — which is on the vertical plane — will be the angle of a plumb-line on the 
 butt joint of such a wreath-piece as this centre line Z U applies to. 2 T of Fig. 2 equals Z E of Fig. 3 ; K I 
 equals X G E of Fig. 3, and the angle T 2, I K of Fig. 2 equals the angle Z E X of Fig. 3. 
 
 Fig. 4 Face-mould over a Quarter-circle, the Tangents of a Common Inclination, as 
 Given and Explained at the Plan Fig. 3.— On a line FZ make K Z and K F each equal A K of Fig. 3 ; 
 make K V at right angles to Z F, and equal to K X of Fig. 3 ; connect Z V and FV ; make F N and Z H each 
 equal F M of Fig. 3 ; through H and N draw T D and J B, at right angles to F Z ; make V S K equal X S K of 
 Fig. 3 ; make H T and H D, and N J and N B each equal N J and N B of Fig. 3. Through F draw B I ; make 
 F I equal F B ; through Z draw D I ; made Z I equal Z D ; through I T S J I on the convex, and D K B on the 
 concave, trace the curved edges of the face-mould. The joints Z and F are made at right angles to the 
 tangents. The slide-line will be explained further along. 
 
 Fig. 5. Parallel Pattern for Round Rail, or to be Used Instead of the Face-mould (as 
 a Means of Saving Stuff) for Marking the \Vreath-piece on the Rough Plank.— On the line 
 A A make K A, K A each equal K A of Fig. 3 ; at right angles to A A draw K X, equal to K X of Fig. 3 ; connect 
 X A, X A and make the joints A, A at right angles to A X ; make A N, A N each equal F M of Fig. 3 ; make N P, 
 N P each at right angles to A A, and equal to N P of Fig. 3 ; make X Q equal X Q of Fig. 3 . describe circles 
 on the centres A P Q P A of any required radius for width of pattern. J^or ordinary-sized hand-rails — such as 
 2" thick by 3" wide, z)/^" thick by 3^" wide, 2 i^" thick by 4" wide — any wreath-piece may be worked out of stuff as 
 thick as the width of hand-rail, with a parallel pattern like Fig. 5 about voider than the widtJi of the 
 hand-rail. See Plate No. 56, Figs. 6 and 7. 
 
 Fig. 6. Exhibits the two Solids Presented— that of Fig. i, Plate No. 10, and Fig. i. 
 of This Plate — brought together, the quarter-circle of each completing the plan of a semicircle on the 
 horizontal plane and showing the vertical trace of the semicircle on the cutting planes of the two differently- 
 cut solids. 
 
 Fig. 7. This Solid is Reproduced Half the Size of Fig. i, merely to be Used for the 
 Purpose of Showing the Correctness of A A and the Angles A AX, as Described at Fig. 3. — 
 
 In this figure V T, as before explained, is a level line on the cutting plane, and X Y the position of a level line 
 on the horizontal plane common to both planes. Now if a vertical plane be conceived with Z W as base, it 
 would touch F and Z on the cutting plane, and F Z on that plane would be the distance required in position 
 on the horizontal plane. Extend the base of the vertical plane WZ to A A indefinitely. On the horizontal 
 plane, X being vertically under V of the cutting plane, and V I and X K measuring alike on both planes, set one 
 foot of the compasses on X, and with Z V or V F for radiire describe arcs at A A ; then AX A and their angles 
 on the horizontal plane will equal the angles Z V F on the cutting plane. 
 
Plate No. 11. 
 
Plate No. 12 
 
PLATE 12. 
 
 Fig. I. Represents a Solid Block or Prism Standing Vertically on a Square Base 
 A B C D, the Upper End Cut on an Inclined Plane Containing the Two Different Inclina- 
 tions A E B and E G F. — Let A N C represent a quarter-circle to which the sides of the solids A B and B C 
 are tangent. To find the direction of a level line on the cutting plane from the point H : make C I equal D 
 H, connect H I ; draw I J parallel to the base C B : then J H will be the level line sought. Draw J M parallel 
 to E B : then M D on the horizontal plane will be the direction of a level line common to both planes ; again, 
 from the point E, a level line E 0 will be found by making D Q and T 0 equal to B E ; then B T on the hori- 
 zontal plane will be the direction of a level line as before. These level lines T B, 0 E, D M and J H, being 
 all of the same length, so all level lines drawn on the base to which perpendicular lines and level lines on the 
 cutting plane are drawn, and equal measurements taken from the curve at the base (as B N and E P), will 
 give the trace of the curve on the cutting plane perpendicularly over that at the base. As the sides of the 
 solid A B and B C at the base, are tangent to the curve, so A E and E G are tangent to the curve traced on the cut- 
 ting plane. A level line eommon to both planes may be demonstrated as follows : prolong the inclination G H until 
 it meets the prolongation of the horizontal plane C D a/ R ; also continue the inclination G E until it meets the con- 
 tinuation of the base C B S ; connect S R : the/i S R is the intersecting line of the plane of the two inclinations G E 
 and E A, and the horizontal plane C R and C S ; also S R is the position of a level line common to both planes. 
 
 Fig. 2, Construction of a Paper Representation of the Solid with Its Curved Lines 
 and Angles as Given in Perspective and Described in Fig. i. — Let A B C D be the square base 
 of the solid ; on D as centre describe the quarter-circle A U V C. Prolong B C both ways to Z and N ; D C 
 to Y and I ; A D to P and A B to M. Let B Z A be the inclination of the plan tangent B A ; make B M and C K each 
 equal B Z ; let K I M be the inclination of the plan tangent C B ; let C 0, D P and D Y each equal K I ; make 0 N 
 equal B Z ; connect N P and Y A. To find the direction of a level line on the horizontal plane from the point B, 
 make D 2 equal B Z, draw 2 3 at right angles to Y D ; and 3 X parallel to Y D : then X B will be the direction of the 
 line sought ; to find the level line from the point D, make C J equal D Y, and draw J L parallel to C B ; from L 
 draw L W parallel to C I : then D W will be the direction of a level line common to both planes from the 
 point D. From A and C at right angles to the level lines X B or W D, draw A G and C H indefinitely ; with 
 Z A as radius on B as centre describe an arc at G, and with M I as radius on B describe an arc at H : then 
 H G will be the distance on the cutting plane over C A of the plan, and if lines are drawn from H to B, and 
 from G to B, the lengths and angle of tangents on the cutting plane will be given. With M I as radius set 
 one leg of the compasses on and describe an arc at E, and with H G as radius, on A intersect the arc at E, 
 connect Z E, make Z T equal M L ; on A with A Y as radius describe the arc Y F ; with Z A as radius on E 
 intersect the arc at F ; connect E F and F A ; connect FT; on A describe the arc 3 Q ; connect Q Z ; make 
 Z R equal B U, and T S equal W V : then the curve ARSE will be the trace on the cutting plane perpen- 
 dicularly over the curve at the base. From F at right angles to A F draw the line F 4, and from F at right 
 angles to F E draw F 6 ; from P at right angles to P N draw P 8 ; from J at right angles to I M draw J 5. 
 With a sharp-pointed instrument scratch the lines A B C D, and Z A ; then with a sharp knife cut through the 
 outlines of the figure, and touch the adjoining edges with a little glue or thick mucilage and bring them 
 together, leaving all lines on the outside for examination. 
 
 Fig. 3. Plan of Hand-Rail a Quarter-Circle in which the Tangents to the Centre 
 Line C B and A B Require Two Different Inclinations as B E C and A S B. — The plan of rail in 
 every case consists simply of the convex and concave curve lines embracing the width of the rail, also the 
 centre curve line and its tangents ; there has then to be added to this plan certain lines, which in their posi- 
 tion fix the kind of face-mould required, and supply points of measurement from which to draw the face- 
 mould. Let the inclination of the tangents B E C and A S B be first fixed, then find the direction of a level 
 line common to both planes as follows : Make B F equal A S ; draw F J parallel to B C ; from J draw J I 
 parallel to E B ; connect I D : then I D will be the level line sought. Parallel to D I draw L 6, R B and P X ; 
 from 8 parallel to B E draw 8 W ; from V parallel to A S draw V U. To find the distance over C A on the cut- 
 ting plane : from C and from A, at right angles to I D, draw C H and A G indefinitely ; with C E as radius 
 set one foot of the compasses on B and describe the arc at H, and with B S as radius on B describe the arc 
 S G ; connect G H : then G H will be the distance sought ; and if B H and B G are connected the lines will 
 contain the angle and length of tangents on the cutting plane. To find the angles with which to square the 
 wreath-piece : prolong B C to Z ; make C Z equal I K; connect Z D : then the bevels at Z will give the plumb 
 line to square the wreath-piece at the butt-joint over C. Continue B A to Q ; make A Q equal A T ; connect 
 Q R : then the bevel at Q will give the plumb line to square the wreath-piece at the butt-joint over A. 
 
 Fig. 4. Face-mould Over a Plan of a Quarter-circle, the Tangents of Two Different 
 Inclinations as Given at Fig. 3. — Draw the line C A and make C 0 and 0 A equal H 0 and 0 G of Fig. 3. 
 On C with the radius C E of Fig. 3 describe an arc at E ; on 0 with the radius 0 B of Fig. 3 describe an 
 intersecting arc at E, and on A with the radius B S of Fig. 3 intersect the arc at E ; connect C E, A E and 0 E; 
 make C, 8, I equal C W J of Fig. 3 ; make E V equal B U of Fig. 3. Parallel to 0 E through 8, I, V draw 
 L 6, M Y and P X; make 8, 6, 8 U I Y, I M, E 4 N and V X and V P each equal the corresponding letters of 
 Fig. 3. Through C draw L B ; make C B equal C L ; through A draw P D ; make A D equal A P. Through 
 L M N P on the concave, and B 6 Y 4 X D on the convex, trace the curves of the face-mould. The joints A 
 and C are made at right angles to the tangents A E and C E. The slide line is drawn anywhere on the face- 
 mould at right angles to the level line 0 E. 
 
 Fig. 5. Parallel Pattern for Round-rail or to be Used Instead of the Face-mould — as 
 a Means of Saving Stuff— for Marking the Wreath-piece on the rough Plank. — Make H O 
 and Q G each equal H 0 and 0 G of Fig. 3. The tangents H B and G B, and the level line B 0, are the 
 same -as Fig. 4. Make H I and B V equal C J and B U of Fig. 3 ; draw I, 5 and V 2 parallel to 0 B ; make 
 V 2, B 3 and I, 5 each equal the corresponding letters and figures at Fig. 3. The joints are made at right 
 angles to the tangents. Describe circles an the centres H 5, 3, 2 G, of any required radius for width of 
 pattern. Fig. 6. — A solid similar to Fig. i introduced to call attention to the two sections that may be cut in a 
 direction on the inclined plane, at right angles to each of the differently inclined sides or tangents, as A B and B C ; 
 and also cut down the sides of the solid in a direction at right angles to each inclination of the cutting plane as A D 
 and C E. The inclined plane of these solids should be understood as representing the surface and position of rail 
 plank J the lines A B and B C the direction of joints of face-moulds j and the lines A D and C E represent the joints 
 square through the thickmss of plank. The angle B C E tvill square the wreath-piece at the butt-joint F G ; and 
 the angle BAD squares the wreath at the joint H I. The sections here given and described are also outlined on the 
 paper solid to be formed at Fig. 2 by the lines F 7, 7. 6 and P 8, also F 4 aiid J 5. 
 
PLATE 13. 
 
 Fig. I. Represents a Solid Block or Prism Standing Vertically on a Base A B P 0, 
 the Sides of Which are Parallel, and Have Two Obtuse and Two Acute Angles. — The 
 
 upper end of this prism is cut on tlic inclination P M B, and M N at right angles to the sides, ai.d parallel to 
 the base P 0. In this solid B A, being in the horizontal plane, and also terminating the inclined plane, is a 
 level line common to both planes. On the base describe the curve A D J P tangent to the sides of the solid 
 A B and B P. To find the trace of this plan curve on the cutting plane : parallel to A B on the base and at 
 pleasure draw C D and G J ; at G and C parallel to P M draw G K and C E ; from E and K parallel to B A 
 draw K L and E F ; make E F equal C D, and K L equal G J ; through the points A F L M on the inclined 
 plane trace a curve perpendicularly over the plan curve A D J P. As the sides of the solid A B and B P at 
 the base are tangent to the plan curve, so A B and B M are tangent to the curve traced on the cutting plane. 
 
 Fig. 2. Construction of a Paper Representation of the Solid With its Curved Lines 
 and Angles Given in Perspective and Described at Fig. i.— Let A B P 0 be the form of the 
 base, the opposite sides of which are parallel and equal. From A at right angles to B A draw A X ; from P at 
 right angles to B P draw R X ; on X as centre describe the plan of curve A D J P, tangent to the sides of the 
 base B A and B P. Let P R B be the inclination — assumed or required — over the base B P ; make 0 N and 
 P M at right angles to 0 P and each equal P R ; connect N M ; make 0 U at right angles to A 0 and equal 
 to P R ; connect U A ; parallel to B A from any points on the curve D and J draw J G and D C ; parallel to 
 P R draw C E and G K ; on B with B H as radius describe the arc Q R S indefinitely ; on A with A Q as 
 radius intersect the arc at S ; connect SB; on A with A U as radius describe an arc at T ; on S with B A as 
 radius intersect the arc at T ; connect A T and T S. On B as centre describe the arcs K W and E V ; draw 
 V F and W L parallel to B A ; make V F and W L equal C D and G J ; through S L F A trace a curve on the 
 cutting plane that will lie perpendicularly over the plan curve A D J P. With a sharp-pointed instrument 
 scratch the lines A B P 0 A ; cut out the remainder of the figure and touch the adjoining edges with a little 
 glue or thick mucilage and bring them together, leaving all lines on the outside for examination and study. 
 
 Fig. 3. Plan of Hand-rail Less than a Quarter-circle, the Tangents to the Centre 
 Curve Line A P Forming the Obtuse Angle P B A. — From P draw P M and P 5 at right angles to 
 B P ; draw A 5 at right angles to B A ; on 5 as centre describe the curve A P. The position of the tangent 
 A B is horizontal, while over the tangent B P the inclination P M B is required. Draw T 0, R L, U G and X C 
 parallel to B A ; parallel to P M draw J K, F I E, S N and 0 Q ; from P at right angles to B A draw P 4 ; on 
 B with B M as radius describe the arc M 4 : then 4 A will be the distance over A and P on the cutting plane, and 
 if a line be drawn from 4 to B, then 4 B A will be the length and angle of tangents on the cutting plane. To find 
 the angle for squaring the wreath-piece at the joint over P : draw E Z parallel to B P ; from F parallel to 
 B M draw F H ; draw X Y at right angles to P M ; make X Y equal P H ; connect Y Z : then the bevel at Y 
 will give a plumb-line on the butt-joint over P, which is the angle sought. To find the angle for squaring the 
 wreath at the joint over A : make D C equal J K ; connect C A ; then the bevel at C will give a plumb-line on 
 the butt-joint and the angle sought. 
 
 Fig. 4. Face-mould Over a Plan of Less than a Quarter-circle with One Tangent 
 Fixed in the Horizontal Plane, the Other Inclined as Given at the Plan of Hand-rail, 
 Fig. 3. — Make M W equal A 4 of Fig. 3 ; with B M of Fig. 3 as radius set one foot of the compasses on M 
 and ilescribe an arc at B ; on W, with A B of Fig. 3, intersect the arc at B ; connect W B and B M. Make 
 the joints W and M at right angles to the tangents. Make M K I N equal M K I N of Fig. 3 ; through K I N 
 parallel to W B draw C L, A J and E G ; make K G equal J X, and K E equal J W of Fig. 3 ; through M draw 
 G F ; make M F equal G M ; make I J and I A ecjual F G and F U of Fig. 3 ; make N L and B 6 ecjual S L 
 and B 6 of Fig. 3 ; make W D equal W C. Through G J L 6 D on the convex and F E A C of the concave 
 trace the curved edges of the face-mould. 
 
 Fig. 5. Parallel Pattern for Round-rail or to be Used Instead of the Face-mould as a 
 Means of Saving Stuff, and for Marking the Wreath-piece on the Rough Plank.— Make 
 A M equal A 4 of Fig. 3 ; make the tangents M B and B A equal M B and B A of Fig. 3 ; make MFC ecjual 
 M I Q of Fig. 3 ; make F V and 0 T equal F V and O T of Fig. 3. The joints are at right angles to the tan- 
 gents. On M V T and A, describe circles of any required radius for width of pattern. 
 
 * Tangents to any plan curve that includes less than a quarter-circle, or a curve that measures less than ninety degrees, always 
 form obtuse angles. 
 
Plate No. 13. 
 
Plate No. 14 
 
 Fig. 4. 
 
PLATE 14. 
 
 Fig. I. Represents a Solid Block or Prism Standing Vertically on a Base A B C D the 
 Sides of which are Parallel and Have Two Acute and Two Obtuse Angles. — The upper end 
 of this prism is cut on the inclination C E B, and on the line E G at right angles to the sides and parallel to 
 the base C D. The base line B A of this solid, being in the horizontal plane and also terminating the inclined 
 plane, is a level line common to both planes. On the base draw the lines A Y and C Y at right angles to the 
 tangents ; on Y as centre draw the plan curve A L H C. At Plate No. 13, Fig. i, the solid is precisely like 
 this, but the obtuse angled tangents to the curve were required in that case, because the plan curve was less than a 
 quarter-circle ; here, however, the acute angle tnust be used because the plan curve is greater than a quarter-circle* 
 At any points on the curve as L and H parallel to A B draw L O and H J ; parallel to C E draw 0 N and J K; 
 from K and N parallel to A B draw N M and K F ; make N M equal 0 L and K F equal J H ; through the 
 points A M F E trace a curve on the cutting plane, which will lie perpendicularly over the plan curve A L H C. 
 As the sides of the solid A B and B C at the base are tangent to the plan curve, so A B and B E are tangent 
 to the curve traced on the cutting plane. 
 
 Fig. 2. Construction of a Paper Representation of the Solid with its Curved Lines, 
 Surfaces and Angles as Given in Perspective and Described at Fig. i.— Let ABC D be the 
 form of the base, the opposite sides of which are parallel and equal. Draw A Y and C Y at right angles to 
 the tangents ; on Y as centre describe the plan curve A 0 L C. Let C E B be the inclination over the base 
 C B ; make C H and D G at right angles to C D and equal to C E ; connect H G ; make D I at right angles 
 to A D and equal to D G ; connect I A. On B with B E for radius describe the arc E F and the arc K ; on A 
 as centre with A I as radius, describe an arc at J ; on A as centre with A F as radius intersect the arc at K ; 
 with B A for radius on K intersect the arc at J ; connect A J, J K and K B. Parallel to A B from any point 
 on the curve 0 and L draw 0 N and L M ; parallel to C E draw M P and N R ; make BUT equal B R P ; 
 parallel to B A draw U Q and T S ; make U Q and T S equal N 0 and M L. Through K S Q A trace a curve 
 on the cutting plane that will lie perpendicularly over the plan curve A 0 L C. With a sharp-pointed instru- 
 ment scratch the lines A B, B C, C D and D A ; cut out the remainder of the figure and touch the adjoining 
 edges with a little glue or thick mucilage and bring them together, leaving all lines on the outside for com- 
 parison and study. 
 
 Fig. 3. Plan of Hand-rail Greater Than a Quarter-circle, the Tangents to the Centre 
 Curve Line A C Forming the Acute Angle A B C. — Draw C Y, C E at right angles to C B ; draw A Y 
 at right angles to A B ; on Y as centre describe the centre curve line A N M C. The tangent A B is to remain 
 level, and over the tangent B C the inclination C E B is required. Through I and T draw I R and T V 
 parallel to A B ; at any point on the curve as X draw K G parallel to A B ; parallel to C E draw Q 0, K L and 
 V P. From C at right angles to A B draw C F indefinitely ; on B as centre with B E as radius describe the 
 arc E F : then A F will be the distance over A and C on the cutting plane ; and if a line be drawn from F to B, 
 thenV ^ k will be the length and angle of tangents on the cutting plane. To find the angle for squaring the 
 wreath-piece at the joint over C : From K draw K H parallel to B E ; make C S equal C H ; connect S J : 
 then the bevel at S will give a plumb line on the butt-joint which is the angle sought. To find the angle for 
 squaring the wreath-piece at the joint over A : make Z G equal K L ; connect G A : then the bevel at G will 
 give a plumb line on the butt-joint over A and the angle sought. In finding angles for squaring wreath-pieces 
 as much of the Joint lines as are convenient may be taken, as follows : Prolong the Joint line C Y until it meets the 
 continuation of the level line B A at 8, rnake C 6 equal C 5 ; connect 6, 8, and the same angle will be given at C, 6, 8 
 (Zi- C S J ; and again at Joint A ; from C draw the line C 7 parallel to B k ; prolong the Joint line kV to 2 ; 
 make 2, 7 equal C E : connect -j A ; then the angle 1, 1 k equals the angle Z G A. 
 
 Fig. 4. Face-mould Over a Plan of Hand-rail More Than a Quarter-circle, the Plan 
 Tangents Forming an Acute Angle, One of the Tangents to Remain Level, the Other 
 Inclined, as Given at the Plan Fig. 3. — Make A E equal A F of Fig. 3, with E B of Fig. 3 as radius ; 
 set one foot of the compasses on E and describe an arc at B ; on A with A B of Fig. 3 as radius intersect the 
 arc at B ; connect E B and A B ; make E 0 L P equal E 0 L P of Fig. 3. Parallel toA B draw P C, L D and 
 0 F. 0 G ; make OG, OF equal Ql and Q R of Fig. 3; through E draw G H; make EH equal EG; make 
 L J D equal K 3 X of Fig. 3; make P K C equal V WT of Fig. 3; make B M equal B U of Fig. 3. Tiie joints 
 E and A are at right angles to the tangents. Make AN equal AC. Through the points H FJ K M N on 
 the convex and C D G of the concave trace the curved edges of the face-mould. 
 
 Fig. 5. Parallel Pattern for Round-rail, or to be Used Instead of the Face-mould as 
 a Means of Saving Stuff, and for Marking the Wreath-piece on the Rough Plank.— A E B 
 equals A F B of Fig. 3 ; E L P equals E L P of Fig. 3 ; L M and P N equals K M and V N of Fig. 3. The 
 joints are at right angles to the tangents. On E M N A as centres describe circles of any required radius for 
 width of pattern. 
 
 * Tangents to any plan curve that includes more than a quarter-circle, or that measures more than ninety degrees, always form 
 acute angles. 
 
PLATE 15. 
 
 Fig. I. Represents a Solid Block or Prism Standing Vertically on a Base A B C D, 
 the Sides of Which are Equal and Parallel, and Have Two Acute and Two Obtuse 
 Angles. — I'he upper end of tliis prism is cut on the angle B F A on the side A B ; and on tlie side B C, on 
 the same angle of inclination G H F ; therefore, the sides of this solid have a common inclination, and a line 
 F I drawn on the cutting plane, or B D on the horizontal plane, is a level line common to both planes. On 
 the base A 0 X M C represents a plan curve less than a quarter-circle, to. which the sides A B and B C of the 
 solid represent the plan tangents ; and the lines A F and F H represent the tangents on the cutting plane. 
 To find the trace of the plan curve on the cutting plane at any points, as 0 X M,* draw 0 J and N M parallel 
 to D B ; parallel to B F draw J K and N Q ; parallel to F I draw Q P and K L ; make Q P equal N M and 
 K L equal J O ; make F R equal B X ; through the points A L R P H trace a curve on the cutting plane which 
 will lie perpendicularly over the plan curve A 0 X M C. 
 
 Fig. 2. Construction of a Paper Representation of a Solid With its Surfaces Curved 
 Lines and Angles as Given in Perspective and Described at Fig. i.— Let A B C D be the form 
 of base the opposite sides of which are parallel and equal. Draw C X and A X at right angles to the tangents ; 
 on X as centre describe the plan curve A J C. At right angles to A B and C D draw B F, C V and D W ; at 
 right angles to A D and B C draw D Y, C U and B E. Let B F A be the inclination required over the base — 
 or plan tangents— A B and B C. Make B E, C 2, 2 U, C Z, Z V, D W and D Y all equal B F ; connect U E, 
 V W and Y A. Through C and A draw S T indefinitely ; on B as centre with F A as radius describe arcs at 
 S and T. At any points on the curve, as I and 5, draw I K and 5 M parallel to D B ; parallel to B E draw 
 M N ; parallel to B F draw K L ; on A with S T as radius describe an arc at H ; on F as centre with D B the 
 level line as radius describe an arc at I ; on F with F A as radius intersect the arc at H ; on A as centre with 
 A F as radius intersect the arc at I ; connect A I, I H, H F and F I. Make F R equal E N ; parallel to F I 
 draw R Q and L 0 ; make L 0 equal I K, F P equal B J, and R Q equal M 5 ; through A 0 P Q H trace 
 a curve on the cutting plane that will lie perpendicularly over the plane curve A J C at the base. With a 
 sharp-pointed instrument scratch the lines A B C D A and A F ; cut out the remainder of the figure, and touch 
 the adjoining edges with a little glue or thick mucilage and bring them together, leaving all lines on the out- 
 side so that their connections may be seen and studied. 
 
 Fig. 3. Plan of Hand-rail Less Than a Quarter-circle, the Tangents A B and B C to 
 the Centre Curve Line A C, to Have a Common Angle of Inclination.— At right angles to B C 
 draw C H, C R ; at right angles to A B draw A R ; connect R B : then R B is the direction of a level line 
 common to both planes ; let C H B be the inclination assumed or required over the tangent C B, and let 
 B V A be the same angle of inclination over the tangent B A ; B V being at right angles to A B ; through E 
 draw E G parallel to R B ; parallel to C H draw J K. Through A C draw the line S T indefinitely ; on B as 
 centre with B H as radius describe the arc H T and S : then S T will be the distance over C and A on the 
 cutting plane ; and if lines be drawn from Tand S to B : then the lines T B and S B will be the length and 
 angle of the tangents on the cutting plane. To find the angle for squaring the wreath-piece at both joints : con- 
 tinue B C toY indefinitely ; make C Y equal C L ; connect Y R : then C Y R ivill he the angle required and the 
 bevel at Y will give a plumb-line on the butt-joints of the 7t<reath-piece over A and C. 
 
 Fig. 4. Face-mould Over a Plan of Hand-rail Less Than a Quarter-circle, the Plan 
 Tangents Forming an Obtuse Augle, and the Inclination of Both Tangents Alike, as Given 
 at the Plan Fig 3. — Let S D and D T equal S D and D T of Fig. 3 ; draw D B at right angles to S T 
 and equal to D B of Fig. 3 ; connect T B and S B. Make B X, B 0 equal B U, B Q of Fig. 3 ; make T J, 
 S J each equal H K of Fig. 3 ; through J and J draw lines at right angles to S T or parallel to D B'; make 
 J G and J E at both ends equal J G and J E of Fig. 3 ; through T draw E Z ; make T Z equal T E ; through 
 S draw E Z ; make S Z equal S E. The joints S and T are at right angles to the tangents. Through 
 Z G 0 G Z of the convex and E X E of the concave trace the curved edges of the face-mould. 
 
 * As many level lines may be drawn on the plan curve for measuring trace points on the cutting plane for face-moulds as seem 
 desirable. But in drawing face-moulds, certain points on the plan must always be taken with the level measuring lines ; as, for in- 
 stance, the angle of tangents B, and the points E, both of Fig. 3. 
 
Plate No. 15 
 
Plate No. 16- 
 
 FiG. 4. 
 
PLATE 16. 
 
 Fig. I. Represents a Solid Block or Prism Standing Vertically on a Base A B C D 
 the Sides of which are Equal and Parallel and have Two Acute and Two Obtuse 
 Angles; the Upper End of the Solid is Shown as Cut on Two Different Angles: the 
 Side A B on the Angle B M A, and the Side B C on the Lesser Angle V Q M.*— On the 
 base A E G H C represents a plan curve less than a quarter-circle, to which the sides A B and 
 B C of the solid represent the plan tangents, and the lines A M and M Q represent the 
 tangents on the cutting plane. Make B L equal D S, draw LJ parallel to A B, connect J S; then 
 J S will be a level line on the cutting plane. Make J F parallel to B M, connect F D; then FD 
 will be the direction of level lines on the horizontal plane common to both planes. At any points 
 on the curve at the base, as E G H. draw B G and I H parallel to F D; parallel to B M draw 
 I P; draw P 0 and M N parallel to J S; make P 0 equal I H, M N equal B G, and J K equal 
 F E; through A K N 0 Q trace a curve on the cutting plane which will lie perpendicularly over 
 the plan curve A E G C. 
 
 Fig. 2. Construction of a Paper Representation of a Solid with its Angles, Surfaces, 
 and Curved Lines as Given in Perspective and Described at Fig. i. — Let A B C D be the 
 
 form of base the opposite sides of which are parallel and equal. Draw A T and C T at right 
 angles to the tangents; on T as centre describe the plan curve A 0 P R C. At right angles to 
 A B and CD draw B H, C X, and D V; at right angles to B C and A D draw B G, C F, and D U; 
 let B H A be the angle of inclination required over the base or plan tangent A B; make B G 
 and C I each equal B H; connect G I; make I FG the angle of inclination over the base or plan 
 tangent B C, I F to be less in height than B H; make D U, C W and D V each equal I F; connect U A; 
 make W X equal B H; connect XV. Make B L equal D U; draw L M parallel to B A; make M N parallel 
 to B H: then N D will be the direction of level lines common to both planes. At right angles to D N 
 draw CZ and A Y indefinitely; on B as centre with A H for radius describe an arc at Y, and again on B 
 as centre with G F for radius describe an arc at Z: then Y Z will be the distance over A and C on the 
 cutting plane, and if lines are drawn from Z to B and Y to B, then Z B and Y B will be the length 
 and angle of the tangents on the cutting plane. From B parallel to N D draw B P, and at any 
 point on the curve, as R, draw R Q parallel to N D. On H as centre with G F as radius describe 
 an arc at K; on A as centre with Y Z as radius describe an arc at K; on A as centre with A U 
 as radius describe an arc at J; on M with N D as radius intersect the arc at J; connect A J, J K 
 and K H; make H E equal G 5; parallel to M J draw H 2 and E 4; make M S equal N 0, H 2 
 equal B P, and E4 equal Q R, through A S 2 4 K trace a curve on the cutting plane that will 
 lie perpendicularly over the plan curve A 0 P R C at the base. With a sharp-pointed instrument 
 scratch the lines A B, B C, C D, DA and H A; cut out the remainder of the figure and touch 
 the adjoining edges with a little glue or thick mucilage and bring them together, leaving all 
 lines on the outside so that their connections may be seen and understood. 
 
 Fig. 3. Plan of Hand-rail Less than a Quarter-circle, the Tangents to have Two 
 Different Angles of Inclination, the Angle B T A over the Tangent A B, and C S B the 
 Lesser Angle over the Tangent C B. — To find the position of a level line common to both 
 planes: draw A L parallel and equal to B C; make B W equal C S; draw W X parallel to B A; 
 make X U parallel to B T; connect U L: then U L will be the line sought. Parallel to U L 
 draw P Q, B M and G H; parallel to C S draw 0 R; parallel to U X draw F 4. At right angles 
 to L U draw C D and A E indefinitely; on B as centre with B S as radius describe the arc S D, 
 and again on B as centre with T A as radius describe an arc at E: then D E will be the distance 
 over A and C on the cutting plane; and if lines are drawn from h and D to B, D B and E B 
 will then be the length and angle of tangents on the cutting plane. 
 
 To Find the Angle for Squaring the Wreath-piece at the Joint over C:— Prolong 
 the tangent B C to N; make C N equal C R; connect N M: then the bevel at N will give a 
 plumb-line on the butt-joint over C. 
 
 To Find the Angle for Squaring the Wreath-piece at the Joint over A:— Prolong 
 the level line U L until it msets the continuation of the joint-line A K at J ; prolong the tangent 
 B A to i; make A I equal U V; connect I J: then the bevel at I will give a plumb-line on the 
 butt-joint over A. See Plate No. 74, I'lc. 5. 
 
 Fig. 4. Face-mou'd over a Plan of Hand-rad Less than a Quarter-circle, with Two 
 Different Angles of Inclination over the Plan Tangents. — Let A H equal E D of Fig. 3 
 Make H Z equal D Z of Fig. 3. On Z as centre with Z B of Fig. 3 as radius describe an arc 
 at B; on H as centre with S B of Fig. 3 as radius intersect the arc at B; connect H B and A B, 
 and if the work is correct A B will equal A T of Fig. 3. Connect B Z; make H Q equal S R ot 
 Fig. 3; make A P and A R equal A 4 and A X of Fig. 3; through Q parallel to B Z draw K G, 
 through R and P parallel to B Z draw M E and N D; make Q K and Q G equal 0 Q and 0 P 
 of Fig. 3; make B F equal B 2 of Fig. 3; make R M and R E equal U Y and U 3 of Fig. 3; 
 make P N and P D equal F H and F G of Fig. 3; through A draw D 0; make A 0 equal A D; 
 through H draw G I; make H I equal H G; through 1 K M N 0 of the convex and G F E D of 
 the concave trace the curved edges of the face-mould. The slide-line on a face-mould is always 
 drawn at right angles to the level line. 
 
 * The opposite sides of all the solids must be cut on parallel angles of inclination. 
 
PLATE 17. 
 
 Fig. I. Represents a Solid Block or Prism Standing Vertically on a Base of the 
 Given Form A B C D, the Parallel Sides of which are Equal, but Two of the Sides are 
 Larger than the Other Two. The Upper End of the Solid is Shown as Cut on Two 
 Different Angles : the Side A B on the Angle of Inclination B J A, and the Side B C on the 
 Angle of Inclination G FJ. — On ilie base AQ PC represents a plan curve elliptic or eccentric, 
 to which the sides of the solid AB and BC represent tlie plan tans^ents; and the lines A J and 
 J F represent the tangents on the cutting plane. Make D K equal B J; draw K L parallel to A D; 
 connect J L: then J L will be a level line on the cutting plane. Make L I parallel to E D; 
 connect I B: then I B will be the direction of level lines on the horizontal plane common to both 
 planes. At any point on the curve P draw PO parallel to I B; draw 0 H parallel to BJ; make 
 H M parallel to J L and equal to 0 P, and J N equal B Q; through A N M F trace a curve on 
 the cutting plane which will lie pei'pendicularly over the plan curve A Q P C. 
 
 Fig. 2. Construction of a Paper Representation of a Solid with its Angles, Surfaces, 
 and Curved Lines as Given in Perspective and Described at Fig. i.— Let A B C D be the 
 given form of b.ise, and A H G C an eccentric or elliptic curve lo which the sides A B and B C of 
 the solid are tangent. At light angles to A B draw B U, C N and D M; at right angles to B C 
 draw B E, C S and D L; let B U A be the angle of inclination required over the base or plan 
 tangent A B; make B E and C R equal B U ; l<-t R S E be the angle of inclination required over 
 the base or plan tangent B C; make D L, D M and C Q each equal R S; connect LA; make Q N 
 equal B U; connect N M; make D K equal B U, draw J I parallel to D L, connect I B. At right 
 angles to B I through C and A di'aw A P and C 0 indefinitely. On B as centre with E S as 
 radius describe an arc at 0; again on B as centre with U A as radius describe an arc at P: 
 then P 0 will be the distance over A and C on the cutting plane. On U as centre with E S as 
 radius describe an arc at X. On A as centre with P 0 as radius intersect the arc at X; on A 
 with A J as radius describe an arc at V; and again on A with A L as radius describe an arc at W. 
 On U as centre with B I as radius intersect the arc at V; connect A V W, W X, X U and U V; 
 make UY equal ET; parallel to U V draw Y2; make Y2 equal FG, and UZ equal BH; through 
 A Z 2 X trace a curve on the cutting plane that will lie perpendicularly over the plan curve A H G C 
 on the base. With a sharp-pointed instrument scratch the lines A B C D and A U; cut out the 
 remainder of the figure and touch the adjoining edges with a little glue or thick mucilage and 
 bring them together, leaving all lines on the outside so that their connections may be seen and 
 understood. 
 
 Fig. 3. Plan of Hand-rail, an Eccentric or Elliptic Curve, the Tangents of Unequal 
 Length and Two Different Angles of Inclination. — Let C P B and B Q A be the angles of 
 inclination over the plan tangents C B and B A. Make A L parallel and equal to B C; make 
 P M equal B Q; parallel to C B draw M F; parallel to C P draw F G; connect G L: then G L 
 will be the direction of a level line common to both planes. Parallel to G L draw J K, E B W 
 and U S; parallel to C P draw I N; parallel to B Q draw T 6. At right angles to G L through 
 C and A draw C D and A H indefinitely. On B as centre describe the arc P D; again on B as 
 centre with QA as radius describe an arc at H; connect H D: then H D will be the distance on 
 the cutting plane over A and C, and if lines are drawn from D and H to B, D B and H B will 
 then be the length and angle of tangents on the cutting plane. 
 
 To Find the Angle for Squaring the Wreath-piece at the Joint over C:— Prolong the 
 tangent B C to Z; prolong the joint-line C J until it meets the continuation of the level line G L 
 at Y; make C Z equal M 0; connect Z Y: then the bevel at Z will give a plumbdine on the butt- 
 joint over C. 
 
 To Find the Angle for Squaring the Wreath-piece at the Joint over A: — Prolong the 
 joint-line AU until it meets the continuation of the level line B 2 at W; prolong the tangent BA 
 to X; make A X equal B R: then the bevel at X will give a plumb-line on the butt-joint over A. 
 
 Fig. 4. Face-mould over the Plan of Hand-rail Given and Described at Fig. 3.— Make 
 A B equal H D of Fig. 3. Make BE equal D 5 of Fi;;, 3. On E as centre with 5B of FiG. 3 as 
 radius describe an arc at C; on B with B P of Fig. 3 intersect the arc at C, and if the work is 
 correct A C will equal A Q of Fig. 3. Connect CEP, B C and A C; make B F K equal P N F of 
 Fig. 3; make A M equal A 6 of Fig. 3; parallel to E C draw G F J, QK and N M L; make FJ, 
 FG equal I K, IJ of Fig. 3; make K Q --qual G 3 of Fig. 3; make C D, C P equal E B, B 2 of Fig. 3; make 
 M N, M L equal T S, T U of Fig. 3; through A draw L 0; make AO equal A L; through B draw 
 G h'; make B H equal BG; through H J D N 0 of the convex and GQPL of the concave trace 
 the curved edges of the face-moidd. The joints A and B are at right angles to the tangents. 
 The slide-line may be drawn anywhere on the face-mould, but must always be made at right 
 angles to the level lines. It is a' matter of convenience to draw the slide-line from the centre of 
 either joint of the face-mould. 
 
 Fig. 5. To Find a Common Angle of Inclination Over Two Different Lengths of 
 Plan Tangents when Required, the Total Height being Given:— The tangents are of the 
 same length and angl<; as those of FiG. 3. Let C D equal both C P and B Q of FiG. 3. Prolong 
 C B indetinitelv; on' B as centre with B A as radius describe the arc A F; connect D F; parallel 
 to F D draw B E; parallel to C D draw B H; at right angles to A B draw B G; make B G equal 
 B H; connect G A: then the angle of inclination B G A is the same as the angle C E B. In this 
 case the angle for squaring the wreath-piece will be alike for both joints. 
 
Plate No. 17 
 
 s 
 
 Fig. 4- 
 
Plate No. 18 . 
 
 Fig. 4. 
 
PLATE 18. 
 
 Fig. I. Represents a Solid Block or Prism Standing Vertically on a Base of the 
 Given Form A B C D, the Sides of which are Equal and Parallel and have Two Acute 
 and Two Obtuse Angles. — The acute angle formed by the position of the sides A B, B C of the 
 base is intended in this case to represent the angle of plan tangents, embracing in the plan 
 a curve of more than a quarter-circle. The upper end of the solid is shown as cut on the side 
 A B, on the angle of inclination B F A, and on the side B C, on the same angle of inclination 
 M G F. The sides of this solid being cut on a common angle of inclination, the heights from 
 the base D E and B F are alike, and therefore a line drawn on the cutting plane from F to 
 E will be a level line ; and at the base — or horizontal plane — a line drawn from D to B will 
 be the position of a level line common to both planes. Over the base A B and B C the lines 
 A F and F G represent the tangents on the cutting plane, — or those of a face-mould. At any 
 points on the curve at the base parallel to B D draw Q R and L K ; through R and K par- 
 allel to B F draw R I ; parallel to F E draw I H and T N ; make T N equal R Q, F J equal 
 B 0, and I H equal K L ; through A N J H G trace a curve on the cutting plane which will lie 
 perpendicularly over the plan curve A Q 0 L C. 
 
 Fig. 2. Construction of a Paper Representation of a Solid with its Angles, Surfaces, 
 and Curved Lines as Given in Perspective and Described at Fig. i. — Let A B C D be the 
 form of base the opposite sides of which are parallel and equal. Draw A M and C M at right 
 angles to B C and B A ; on M as centre describe the plan curve A C, which is greater than a 
 quarter-circle. At right angles to A B draw B U, C F and D I ; at right angles to B C draw C G, 
 B H and D X ; let D X A and B U A be the common angles of inclination ; make B H, C J, J G, 
 C 4, 4 F and D I each equal B U ; connect G H and F I. As D I and B H are of equal heights, a 
 line from D to B will be the position of a level line common to both planes. Through C A 
 draw the line P E indefinitely ; on B as centre with U A as radius describe an arc at P and 
 at E ; then P E will be the distance over A and C on the cutting plane. At any points on the 
 curve, as Q and R, parallel to M B draw Q 0 and L R ; parallel to C G draw L K ; parallel 
 to B U draw 0 S. On U as centre with P E as radius describe an arc at V ; on A as centre 
 with A X as radius intersect the arc at V ; again, on V as centre with the same radius describe 
 an arc at W ; on U with H G as radius intersect the arc at W ; connect A V, V W, W U and 
 U V ; make U 2 equal H K ; parallel to U V draw S T and 2 Z ; make S T equal Q 0, U Y 
 equal B N, and 2 Z equal L R ; through the points W Z Y T A trace a curve that will lie per- 
 pendicularly over the plan curve A Q N R C. With a sharp pointed instrument scratch the 
 lines A B C D A and A U ; cut out the remainder of the figure and touch the adjoining edges 
 with a little glue or thick mucilage and bring them together, leaving all lines on the outside, 
 so that their connections may be seen and understood. 
 
 Fig. 3. Plan of Hand-rail Greater than a Quarter-circle ; the Tangents to have a 
 Common Angle of Inclination C F B over the Tangent C B, and B K A over the Tangent 
 B A. Through A and C draw the line D E indefinitely ; on B as centre with B F as radius 
 describe the arc F E ; and again on B with the same radius as before describe an arc at D : 
 then D E will be the distance over A C on the cutting plane; and if lines are drawn from D 
 to B and from E to B, then D B and E B will be the length and angle of tangents on the 
 cutting plane. J B will be the direction of a level line common to both planes. Parallel to 
 J B draw L 0 and C P ; parallel to C F draw N G. 
 
 To Find the Angle for Squaring the Wreath-piece at Both Joints :— Prolong the tan- 
 gent B C to H ; make C H equal C G ; connect H J : then the bevel at H will give a plumb- 
 line on the butt joints over A and C. 
 
 Fig. 4. Face-mould over a Plan of Hand-rail Greater than a Quarter-circle, the Tan- 
 gents having a Common Angle of Inclination as given at the Plan Fig. 3. — Let V X U 
 equal D T E of Fig. 3. On V as centre with B F of Fig. 3 as radius describe an arc at W ; 
 and on U as centre with the same radius as before intersect the arc at W ; connect V W, U W 
 and X W : X W should equal T B of Fig. 3. Make V C and U G each equal F G of Fig. 3 ; par- 
 allel to W X draw V E, C B A, G H F and U K ; make V E and U K each equal C P of Fig. 3. 
 Make C B and C A equal N 0 and N L of Fig. 3 ; and again, make G H and G F equal N 0 
 and N L of Fig. 3. Through U draw F J indefinitely ; through V draw A D indefinitely ; make 
 VD equal AV; make UJ equal U F; make WZY equal B Q S of Fig. 3. The joints V and 
 U are at right angles to the tangents. Through DEBZHKJ of the convex and A Y F of the 
 concave trace the curved edges of the face-mould. 
 
 Fig. 5. Parallel Pattern for Round Rail, or to be Used Instead of the Face-mould 
 as a Means of Saving the Width of Stuff for Marking the Wreath-piece on the Rough 
 Plank.— Make DTE equal D T E of Fig. 3 ; make E B and D B each equal F B of Fig. 3 ; 
 connect B T, B D and B E; make E N and D N each equal F G of Fig. 3; make N M, N M 
 each equal N M of Fig. 3 ; make B R equal B R of Fig. 3. Describe circles on the points 
 E M R M D as centres of any required radius for width of pattern. Bend a flexible strip of 
 wood or other material by which to mark curve- lines touching the circles for the convex and 
 concave edges of the pattern. The joints are made at right angles to the tangents. 
 
PLATE 19. 
 
 Fig. I. Represents a Solid Block or Prism Standing Vertically on a Base of the 
 Given Form A B C D, the Sides of which are Equal and Parallel and have Two Acute 
 and Two Obtuse Angles. — The acute angle formed l)y the position of the sides A B, B C of 
 the base is intended in this case to represent the angle of plan tangents embracing in the plan 
 a curve of more than a quarter-circle. The upper end of the solid is shown as cut on the side 
 A B on the angle of inclination B F A, and on the side B C on a less angle of inclination M G T. 
 
 To Find the Position of a Level Line on the Cutting Plane :— Make B N equal D E; 
 draw N H parallel to A B; connect H E, which is the level line sought; parallel to B F draw H J : 
 then f/ie line D J on the horizontal plane will be the position of a level line coutmon to both planes. At 
 any points on the curve at the base parallel to J D draw 0 R and L K; through R and K 
 parallel to B F draw R S and K U; parallel to H E draw U T and S P; make U T equal K L, 
 S P equal R 0, and H Q equal J X; through the points A P Q T G trace a curve on the cutting 
 plane which will lie perpendicularly over the plan curve A 0 X L C. 
 
 Fig. 2. Construction of a Paper Representation of a Solid with its Angles, Surfaces, 
 and Curved Lines as Given in Perspective and Described at Fig. i. — Let A B C D be the 
 form of base the opposite sides of which are parallel and equal. C X and A X are at right 
 angles to A B and B C. On X as centre describe the plan curve A W C, which is greater than a 
 quarter-circle. At right angles to A B draw B E, C L and D M ; at right angles to B C draw B F, 
 C H and D N; let B E A be the angle of inclination over B A; make B F and CJ each equal B E; 
 let J H F be an angle of inclination — over C B — less than B E A over B A; make D N, D M and 
 C K each equal J H; make K L equal B E; connect L M and N A; make B 5 equal D N; draw 
 5 U parallel to B A, and U V parallel to E B: then a line drawn from V to D will be the position 
 of a level line common to both planes. At right angles to D V draw A Q and CP; on B as 
 centre with F H as radius describe an arc at P, and again on B as centre with E A as radius 
 describe an arc at Q: then Q P will be the distance over A C on the cutting plane. At any 
 points on the plan curve Y and J draw J I and Y 0 parallel to D V; make I G parallel to C H ; 
 make V U and 0 Z parallel to E B; on U as centre with V D as radius describe an arc at R; on 
 A with A N as radius intersect the arc at R; on E as centre with F H as mdius describe an arc 
 at S; on A as centre with P Q as radius intersect the arc at S; connect A R, R S, S E and U R; 
 make E T equal F G; parallel to U R draw Z 2 and T4; make Z 2 equal 0 Y, U 3 equal V W, and 
 T4 equal I J; through the points S, 4, 3, 2 A trace a curve on the cutting plane that will lie 
 perpendicularly over the plan curve A Y W J C. With a sharp-pointed instrument scratch the lines 
 A B C D A and A E; cut out the remainder of the figure and touch the adjoining edges with a 
 little glue or thick mucilage and bring them together, leaving all lines on the outside, so that 
 their connections may be seen and understood. 
 
 Fig. 3. Plan of Hand-rail Greater than a Quarter-circle, the Tangents to have Two 
 Different Angles of Inclination: BZA over the Plan Tangent A B, and a Less Angle 
 of Inclination C H B over the Plan Tangent C B. — Make A 4 parallel and equal to B C; 
 make B 5 equal C H; draw 5 L parallel to A B, and L V parallel to Z B: then the line V4 will be 
 the direction of a level line common to both planes. Parallel to V 4 draw S U, B F and Q X; 
 draw Y N parallel to Z B, and TJ parallel to C H. 
 
 To Find the Angle for Squaring the Wreath-piece over the Joint C: — Prolong 
 the tangent B C to G; make C G equal C D; connect G F; then the bevel at G will give a plumb- 
 line on the butt-joint over C. 
 
 To Find the Angle for Squaring the Wreath-piece over the Joint A : — Make E R 
 parallel to B A and equal to V M; connect R A: then the bevel at R will give a plumb-line on 
 the butt-joint over A. Through C and A at right angles to V 4 draw C I and A K indefinitely; 
 on B with B H as radius describe the arc H I; and again on B as centre with Z A as radius 
 describe an arc at K; connect K I: then K I will be the distance over A and C on the cutting 
 plane; and if lines are drawn from K to B, and I to B, then I B and K B will be the length and 
 angle of tangents on the cutting plane or, which is the same thing, of the face-mould. 
 
 Fig. 4. Face-mould over a Plan of Hand-rail Greater than a Quarter-circle, the 
 Tangents having Two Different Angles of Inclination as Given at the Plan Fig. 3. — 
 Make H C A equal I 0 K of Fig. 3. On C as centre with 0 B of Fig. 3 as radius describe an arc 
 at B; on H as centre with H B of Fig. 3 as radius intersect the arc at B: then if the work is 
 correct A B will equal A Z of Fig. 3. Connect A B, B H and C B; make H N equal H J of 
 Fig. 3; make B Q K equal Z L N of Fig. 3; parallel to C B draw N F, N 0, Q L and K J, K D; 
 make N 0, N F equal T U, T S of Fig. 3; make Q R L equal V W P of Fig. 3; make K J, K D 
 equal Y X, Y Q of Fig. 3; through A draw D E; make A E equal A D; through H draw F G; 
 make H G equal H F. The joints H and A are at right angles to the tangents. Through G 0 RJ E 
 of the convex and F L D of the concave trace the curved edges of the face-mould. 7'he slide- 
 line in this case and of all face-moulds of whatever character is always at right angles to the level line. 
 
Plate No. 19 
 
 G 
 
Plate No. 20 ' 
 
PLATE 2 0. 
 
 Skilful workers of hand-rail know that carefully shaping the wood next the joints of a 
 wreath-piece so that it will nicely fall ^n with its adjoining pieces, of whatever character they 
 may be, is of the first importance ; next, experience has taught them that the interval — the 
 helical surfaces — or top and bottom surfaces between joints, compel them to follow certain 
 curvatures peculiar to each kind of wreath-piece. From these facts of experience it is evident 
 that a face-mould is not only a means of shaping the sides of a wreath on the plane of the 
 plank, but that it carries with it certain geometrical curves that shape the top and bottom 
 surfaces of the wreath. This controlling curve-line of helical surfaces of wreaths is a centre 
 line,* as at A E R, Fig. 2, and is found in the development of the central cylindric line on cut- 
 ting planes as Z J U of Fig. i, Plate No. 11, and C 0 Q S E of Fig. i, Plate No. 10. See also 
 the similar cases of the two last solids referred to, with their cutting planes brought in posi- 
 tion over the plan at their bases, Fig. 6, Plate No. ii. A round hand-rail — controlled from 
 its centre as it properly must be — over a circular or curved plan affords a complete demon- 
 stration that this centre cylindric line gives shape to the top and bottom of the rail, for 
 while its curved sides hang vertically over the plan, its top and bottom also take proper 
 curves, forming its own casings perfectly suited to the requirements of every case. To meas- 
 ure and make an exact drawing of this centre line will demonstrate the peculiar form of 
 curvature in all cases of wreaths, and also show exact heights at every point over an eleva- 
 tion of treads and rises embraced within any curved plan. Thus the practical use of develop- 
 ing the centre line will be to get the length of balusters in any position as required on wind- 
 ing steps in cylinders ; also the ability to test the wreath-piece over the elevation, and 
 determine when desirable what changes to make, if any, in the inclination of tangents. 
 
 Fig. I. The Semicircle ACM is a Plan of the Centre Line of a Hand-rail with 
 Tangents to Each Quarter A B, B C and C N, N M. — In this plan the two quarters that 
 make the semicircle are of the same character as the first two solids introduced. The first 
 solid at Plate No. 10 is here repeated by D M N C ; the other, D A B C, at Plate No. ii. 
 At Plate No. ii. Fig. 6, two similar solids are brought together showing the difference in 
 their cutting planes as placed in position over the plan of a semicircle at the base. The 
 quarter-circle A C over its tangents A B, B C has a common inclination B Y A, C E B ; also the 
 quarter-circle M C over one tangent C N has the same inclination N R C, while the tangent 
 M N remains level. On the quarter A C, B D is the position of a level line common to both 
 planes, as before shown ; and on the quarter C M, N M is the level line common to both 
 planes. Divide the quarter-circle A C, and the quarter C M, each into four equal parts ; through 
 L and H draw LJ, H G parallel to the level line D B; parallel to C E draw G F; parallel to 
 B Y draw J K ; through 0, P, Q parallel to the level line M N dtaw 0 S, P T and Q U. 
 
 Fig. 2. Development or Unfolding of the Centre Line of a Semicircular Wreath of 
 Hand-rail over the Plan A C M as given at Fig. i. — Draw A X ; make A L I equal A L I of 
 Fig. I. Draw I Y at right angles to AX; make I Y and L K equal J K and B Y of Fig. t. 
 Parallel to A X draw Y C ; make Y H C equal I H C of Fig. i ; draw C E and H F at right 
 angles to A X ; make H F and C E equal G F and C E of Fig. i. Parallel to A X draw E N ; 
 make E Q P 0 N equal C Q P 0 M of Fig. i. Through N at right angles to A X draw X R ; 
 make N R, 0 S, P T and Q U at right angles to E N and equal to N R, V S, W T and X U of 
 Fig. I. Through the points AKYFEUTSR trace the centre line sought. f 
 
 Fig. 3. The Semicircle Q V S is the Plan of a Centre Line of Hand-rail with Tan- 
 gents to each Quarter Q U, U V and V W, W S. — Let U T Q be the inclination required over 
 the plan tangent Q U, and V Y U the greater inclination required over the plan tangent U V ; 
 and for the quarter S V the angle WAV must be the same as V Y U ; the plan tangent W S 
 remains level. To find the level line for the quarter Q V, let Y X equal U T ; draw X Z par- 
 allel to V U, and Z F parallel to Y V ; connect F R, which is the level line sought. Divide V J 
 in two parts ; parallel to F R draw U K ; divide K Q into three parts ; divide V S into four 
 parts ; parallel to the level line S W draw P B, 0 C and N D ; parallel to R F draw H G, L 8 
 and M 6; parallel to V Y draw G E; parallel to U T draw 8,4 and 6, 5. See Plate No. 12 
 for case of solid like R Q U V of Fig. 3. 
 
 Fig. 4. Development of the Centre Line of a Semicircular Wreath of Hand-rail 
 over the Plan Q V S as given at Fig. 3.— On the line Q F make Q M L K equal Q M L K 
 of Fig. 3. Erect the perpendiculars K T, L 4 and M 5 equal to U T, 8, 4 and 6, 5 of Fig. 3. 
 Draw T V parallel to Q F ; make T J H V equal K J H V of Fig. 3 ; perpendicular to Q F draw 
 V Y, H E and J Z ; make V Y, H E and J Z equal V Y, G E and F Z of Fig. 3 ; parallel to Q F 
 draw YW ; make Y N 0 P W equal V N 0 P S of Fig. 3 ; through W at right angles to Q F 
 draw F A ; make W A, P B, 0 C and N D equal W A, 3 B, 2 C and I D of Fig. 3. Through the 
 points Q54TZEYDCBA trace the centre line sought. 
 
 Fig. 5. This Portion of a Circle Less than a Quarter is a Plan of a Centre Line of 
 Hand-rail with its Tangents D C and C B. — Over the plan tangent DC let DEC be the 
 angle of inclination required, the tangent C B to remain level. Divide the curved line B D 
 into any number of equal parts, and from the points of division J I H G F draw lines par- 
 allel to B C and touching D C ; at right angles to D C draw 0 P, N Q, M R, L S and K T. 
 
 Fig 6. Development of the Centre Line of a Wreath-piece over a Plan of a Portion 
 of a Circle less than a Quarter as given at Fig. 5. — On the line D B make D F G H 1 J B 
 equal the spaces between the corresponding letters at Fig. 5. Make D E, F T, G S, H R, I Q 
 and J P equal D E, K T, L S, M R, N Q and OP of Fig. 5. Through the points B P Q R S T E 
 trace the centre line sought. 7'ke solid Fig. 5 is given in detail at Plate No. 13. 
 
 * The tangents are invariably placed at the centre of the width, and the heights and angles of inclination are 
 also always at the centre of the thickness of the wreath-piece. 
 
 f A practical self-imposed useful lesson would be to make the two solids composing Fig. i of wood, and glu- 
 ing them together in the manner shown at FiG. 6, Plate No. ii, remove the wood to the semicircle at the 
 base and its vertical trace on the cutting planes ; then wrap Fig. 2, the development, around the cylindric surface 
 of Fig. I, and test the curve thus obtained. 
 
PLATE 21. 
 
 Fig. I. Plan of a Centre Line of Hand-rail Greater than a Quarter-circle ; the 
 Tangent B C to have an Angle of Inclination Equal to C D B, the Other Tangent B A to 
 Remain Level. — Divide the circular line A C into any number of equal jjarts, and from each of 
 these points of division draw lines parallel to the level line A B touching the tangent B C, as 
 0 V, E F, G H, L K and M N; parallel to C D draw N P, K J, H I, F S and V W. This solid is given 
 at Plate No. 14. 
 
 Fig. 2. Development or Unfolding of the Centre Line of a Wreath-piece over a 
 Plan Greater than a Quarter-circle as given at Fig. i. — On the line AC make AOEG LMC 
 equal the spaces between the corresponding letters of Fig. 1. At right angles to A C draw C D 
 equal to C D of Fig. i. Make M P, LJ, G l. E S and OW parallel to C D and equal to N P, K J, 
 H I, F S and V W of Fig. i. Through A W S I J P D trace the centre line soiigiit. 
 
 Fig. 3. Plan of a Centre Line of Hand-rail Greater than a Quarter-circle, the 
 Tangents Q T and T X to have the Common Angle of Inclination X Y T and T Z Q — 
 Divitle the circular line Q X into any even number of equal parts, aiui fiom each of these points 
 of division draw lines parallel to the level line R T touching the tangents, as I J, G H, D E and 
 A B; make J K and H L parallel to X Y, and E M and B C parallel to T Z. This solid is given at 
 Plate No. 18. 
 
 Fig. 4. Development of the Centre Line of a Wreath-piece over a Plan Greater 
 than a Quarter-circle as given at Fig. 3. — On the line Q P let Q A D F equal the spaces 
 between the coriesponding letters of Fig. 3. At right angles to Q F draw F Z, D M and AC 
 equal to T Z, E M and B C of Fig. 3. Make Z X parallel to Q P; make Z G I X equal F G I X of 
 Fig. 3; through X draw Y P at right angles to Q P; make I K and G L parallel to X Y; make 
 X Y, I K and G L equal X Y. J K and H L of Fig. 3; through QC MZ LK Y trace the curve sought. 
 
 Fig. 5. Plan of a Centre Line of Hand-rail Greater than a Quarter-circle, the 
 Tangents to have Different Angles of Inclination; the Angle D C F over the Plan 
 Tangent D F, and a Less Angle of Inclination F E A over the Plan Tangent A F. — To find 
 a level line common to l)oth planes: draw A B ])ai-allel and equal to F D; make C L equal F E; 
 draw L J parallel to D F; make J I parallel to DC; then the line I B will be the level line soi/glit. 
 From F draw F P parallel to I B; divide P A and 0 D each in two equal parts; draw Q H and 
 N M parallel to I B; make M K parallel to D C, ami H G parallel to F E. This solid is given at 
 Plate No. 19. 
 
 Fig. 6. Development of the Centre Line of a Wreath-piece over a Plan of More 
 than a Quarter-circle, the Tangents having Different Angles of Inclination as given at 
 Fig. 5. — On the line A R make A Q P equal A Q P of Fig. 5; make P E aiui Q G perpendicidar 
 to A R and equal to F E and H G of Fig. 5; draw E D parallel to A R; make EO N D equal 
 POND of Fig. 5; through D diaw CR at right angles to A R; draw N K and OJ parallel to 
 DC; make D C, N K and 0 J equal D C, M K'and I J of Fig. 5; through the points AGE J KC 
 trace the centie-line sought. 
 
 Fig. 7. Plan of a Centre Line of Hand-rail Less than a Quarter-circle, the 
 Tangents S U and U V to have the Common Angle of Inclination V X U and U W S. — 
 Make ST parallel and equal to U V: then T U vvill be a level line common to both planes. 
 Divide the curved line SV into anv even number of equal parts SADKGYV; parallel to T U 
 draw YZ, G H. D E and A B; parallel to VX draw Z J and H I; make E F and B C parallel to 
 U W. 7'///j soliil is given at Plate No. 15. 
 
 Fig. 8. Development of the Centre Line of a Wreath-piece over a Plan of Less 
 than a Quarter-circle, the Tangents having a Common Angle of Inclination as given at 
 Fig. 7. — On the line SL make S A D K equal SA D K of Fig. 7; at right angles to S L make 
 K W, D F and A C equal U W E F and B C of Fig. 7; draw W V parallel to S L; make WG Y V 
 equal K G YV of Fig. 7; through V at right angles to S L draw XL; parallel to L X draw Y J 
 and G I; make V X, YJ and G T equal V X, Z J and H I of Fig. 7; through the points SCFWIJX 
 trace the centre line sought. 
 
 Fig. 9. Plan of a Centre Line of Hand-rail Less than a Quarter-circle, the Tangents 
 to have Different Angles of Inclination ; P R M over the Tangent, M P and N Q P a Less 
 Angle of Inclination over the Tangent P N. — Make P S equal N Q, and ST parallel to P M; 
 make TV parallel to R P; {\\)m M parallel and equal to P N draw M 0: then OV will be a level 
 line common to botii planes. Divide B M in two equal parts M A B; draw P C parallel to V 0; 
 divide C N into three equal parts C D E N; parallel to 0 V draw E F, D G and AU; parallel to N Q 
 draw F I and G H; parallel to P R diaw U J. Tin's solid is given at Plate No. 16. 
 
 Fig. 10. Development of the Centre Line of a Wreath-piece over a Plan of Less 
 than a Quarter-circle, the Tangents having Different Angles of Inclination as given at 
 Fig. 9. — On the line M F make M A B C equal M A B C of Fig. 9; at right angles to M F make 
 C R. B T and A J equal P R, V T and U J of Fig. 9; draw R N parallel to M F; make R D E N 
 equal C D E N of Fig. 9; through N (iraw FQ at right angles to M F; parallel to N Q draw El 
 and D H; make N Q, E I and D H equal N Q, F I and G H of Fig. 9; through the points 
 M J T R H I Q trace the centre line sought. 
 
 .V further important use for a knowledge of this centre line is to unfcjld side moulds. See Plate 
 No. 76, Figs, i, 2, 3, 4, 5, and 6. 
 
Plate No. 21 
 
 M A B C F 
 
 F 1 G. 10. 
 
Plate No. 22 . 
 
PLATE 22. 
 
 Position of Riser in Connection with Cylinders at the Landing and Starting of Straight 
 Flights of Stairs. — The Face-mould, and the Management of the Wreath-pieces. 
 
 Fig-. I. A Sufficient Elevation of Rises and Tread to Determine the Place of Riser 
 at the Bottom of a Flight when the Over-wood is to be all Removed from the Top of 
 the Wreath-piece, as in this Case. — Let X X be the centre of the short balusters and the 
 bottom line of the hand-rail ; make X B the thickness of rail, and X C = 3^-" the thickness of 
 plank ; draw C E parallel to X X ; make X A half the thickness of plank ; draw A D parallel 
 to X X ; make N F four inches, and F D half the thickness of the rail, being all together, from 
 the floor to S, 5I". Where the centre line A D intersects the centre level line S D, that inter- 
 section at D fixes the centre of wreath-piece as shown. From D parallel to the riser draw 
 D G indefinitely ; anywhere below the floor-line and parallel to it draw K G ; as G is the 
 centre of the rail, G 0 must be a half-inch to the face of the cylinder ; and as the diameter 
 of the cylinder is 6 ", 0 K must be 3" ; draw K L parallel to the riser ; continue the line of 
 the first riser to H ; on K with K 0 for radius describe the cylinder : then H J shows the 
 distance to be i^" between the bottom riser and the commencement of the cylinder ; and this 
 is so placed at the plan Fig. 3, as may be seen by the corresponding letters. 
 
 Fig. 2. Elevation of Tread and Rise at the Top of a Flight Sufficient to Determine, 
 when all the Over-wood is Removed from the Bottom of a Wreath-piece, the Relative 
 Position of the Riser and Cylinder. — At the bottom — Fig. i — all the over-wood, B C, is taken 
 off the top of the straight part of the wreath-piece ; not, however, because it is the top, but 
 because it is the concave face ; and in cases of this kind it makes the best shape either for 
 the top or the bottom of the flight. At the top of the flight this over-wood is taken o& at 
 the bottom side of the wreath-piece. This is so because the bottom wreath-piece is simply 
 turned the other side up at the top of the flight, but the over-wood is still taken from the 
 concave face. The position of the cylinders differ ; at the bottom of the flight the chord- 
 line of the cylinder is i^" from the face of the riser, and at the top of the flight it is 2". 
 The letters at the various points of this elevation are made to correspond with those of Fig. i, 
 so that having examined and made the drawings of that figure, a careful inspection of this 
 will be all the explanation needed. 
 
 Fig. 3. Plan of the Bottom of Flight with the Riser and Cylinder as Determined 
 at Fig. I. 
 
 Fig. 4. Plan of the Top Portion of Flight with the Cylinder and Riser Placed as 
 Determined by the Trial at Fig. 2. — Of the plan of hand-rail around the cylinder, one 
 quarter-circle has to be prepared for drawing a face-mould. Q X and X G are the plan tan- 
 gents to the centre line of rail, the pitch-board to be placed as shown ; continue G X to W ; 
 parallel to G X draw A V, Z T and K Y R. 
 
 Fig 5. Face-mould. — * Draw F W indefinitely ; make W V T R equal W V T R of Fig. 4, 
 At right angles to F W draw W B, V A, T 0 Z and R E Y ; make W B equal X G of Fig. 4. 
 Through B draw A D parallel to R W ; make B D equal B A ; make B C equal G C of Fig. 4 ; 
 make T 0, T Z, R E, R Y each equal S 0, S Z and Q Y, Q Y of Fig. 4 ; parallel to R F draw Y M 
 and E L ; through D C 0 E of the convex and A Z Y of the concave trace the curved edges 
 of the face-mould. In this connection is shown the laying out of joints B and F to square 
 the wreath-piece. 
 
 Fig. 6. Perspective Sketch of the Wreath-piece, showing both joints prepared for 
 squaring, and the application of the face-mould to both sides of the stuff. 
 
 Fig. 7. Elevation of Tread and Rises for the Top and Bottom as before Given, 
 the Object being to show that sometimes by a Change in Removing the Over-wood 
 the Wreath-piece may be Kept in its Required Position as to Height and the Chord- 
 line of the Cylinder Brought to the Face of the Riser. — G is the centre of the rail, G 0 
 is ^" ; OK is the radius of the cylinder. The height from the floor to D is fixed as before, 
 alike at the bottom and at the top. The bottom line of rail must pass through X X, the 
 centres of the short balusters ; from X X set off the thickness of rail 2^"; and from D, which 
 must be at the centre of the plank, set off both ways half the thickness of plank parallel to 
 X X : this will show how the over-wood must be removed from the straight part of each wreath- 
 piece. f T//e bottom of the rail at the centre of the wreath is kept 4" above the floor to suit the 
 required length of balusters on the level. At X X the short balusters are 2'.2" from the top of 
 the step to the bottom of the rail ; then from the floor to the bottom of the level rail the 
 height will be 4" more, equal to z' .6" , the same length that the Icmgest baluster on each step 
 has to be, because being half a tread back from the short baluster, it must therefore be a 
 half-rise longer. The merchantable lengths of ordinary balusters are 2'.4" and 2 . 8", thus allow- 
 ing one inch to go in the rail and one inch to dovetail in the step. The illustrations given 
 at Figs, i, 2 and 7 as methods of working wreath-pieces and disposing of the over-wood on the 
 straight part are not to be understood as applying only to 6'' cylinders; for after fixing the required 
 position of wreath-piece, and G and 0, then 0 K may be any radius of cylinder, more or less. 
 For instance, in the case of Fig. i, if 0 K, instead of being a radius of 3", should be 6", then 
 the riser would set i^" into the 12" cylinder. And again, if instead of 0 K being 3", it should 
 be 2" radius, then there would be 2^" straight between the riser and the chord-line of a 
 4" cylinder. 
 
 NoTK. — It must be understood that throughout this work, with the exception of the cases given at Plates 34 and 
 35, all joints of wreath-pieces are to be made at right angles to their tangents, and square through the face of the 
 plank. 
 
 * At Plate No. 10 are given perspective and geometrical drawings, and the formation of a paper representation of 
 a solid, a plan of hand-rail, and from it a drawing of a face-mould, which together give a complete practical knowl- 
 edge of the application and drawing of face-moulds of this kind. 
 
 \ At Fig. 7 about all that can be done further — if desirable — within the limits of the thickness of the plank will be 
 at the top of the flight to take all the over-wood oft the top of the wreath-piece, and at the bottom of the tli^ht take 
 all the over-wood oft the bottom of the wreath-piece; this change would brmg D at the bottom of the fligli!— ki cpiiig 
 it at the same Viflnrht about i" nearer to the riser; and D at the too. about J" nearer to the riser, at the same heiglit. 
 
PLATE 23. 
 
 Platform Stairs, Half-turn, or Such as given by Plan and Elevation at Plate No. i. Figs. 
 3 and 4. — The Position of Cylinder with the Place of Connecting Risers, and the 
 Differences Possible by Varying the Removal of Over-wood from the Straight Portion 
 OF Wreath-pieces; also How to Fix the Risers Connecting with the Cylinder, so 
 THAT the Whole Wreath may have One Common Inclination, besides Saving Several 
 Inches of Sikpimng-room. 
 
 Fig. I. Elevation of Rises and Treads Above and Below the Platform to Test the 
 Removal of Over-wood from the Wreath-pieces. — The bottom line of rail must in all cases 
 pass through the centres of the short balusters X X. As the rail is to be 2\" thick by 4" wide 
 the plank out of which the wreath-pieces are to be worked must be 4" thick. Make X B and 
 B C each 2"; draw B D and C E parallel to X X; make X N and X M 2V; connect N M; make 
 R U half a rise, 3J"; make U D half the thickness of rail, make D F and D S each 2", 
 
 half the thickness of plank; make X J 2V, the thickness of rail; draw F L, J P and S Q parallel 
 to X X. This drawing shows that keeping the rail above the platform at the height R U and 
 taking all the over-wood off at N C of the upper wreath-piece, the lower wreath-piece must 
 have j" of over-wood taken off at J L, the top, and 1" off the bottom, X Q. From D parallel 
 to the line of riser draw D G; at right .ingles to D G draw G K; make G 0, i", and 0 K, 3", the 
 radius of the cylinder; on the centre K describe the cylinder Y A; parallel to D G through 
 K draw Y V, and continue the line of risers to T Z: then A T will lje the distance between the 
 chord-line of the cylinder and the face of the risers at the platform. Another c/ianf^e can be 
 made by taking all the over-iuood off the bottom of the lower wreath-piece, and this would bring the 
 riser of that side f" further from the chord-line of the cylinder — all together 2V. 
 
 Fig. 2. Elevation the Same as that of Fig. I.— The object in repeating this drawing 
 is to call attenti'in to still another change in removing over-wood. In this case the over-wood 
 is all taken off the bottom of the upper wreath-piece, and off the top of the lower wreath- 
 piece, bringing the chord-line of the 6" cylinder to the face of the riser as shown. An inch 
 variation in the height of R U to bring the over-wood as required would be of slight importance. 
 This arrangement of wreath-pieces and over-wood is not confined to any size of cylinder; for instance, 
 if the cylinder is to be 12" diameter, then, G 0 being fixed points, 0 K would be 6", and the 
 risers would set just as they are, but would be in the cylinder 3". At FiG. i a similar change 
 would take place if the size of cylinder is altered; that is to say, that, G 0 being fixed, if 
 the radius of the cylinder is made less, the chord-line would be drawn nearer 0; and if the 
 radius is made greater, the chord-line of the cylinder advances further towards the risers and 
 into the step, as the increase is more or less. The face-mould for Figs, i and 2 is to be found 
 exactly as directed at Figs. 4 and 5 of Plate No. 22. 
 
 Fig. 3. Here again is an Elevation of the Same Tread and Rise Connected with a 
 Platform and the Same Size Cylinder as given at Figs, i and 2. — This elevation is 
 introduced for the purpose of showing how, by a wholly different treatment, the wreath in 
 this case, or indeed of any-sized cylinder — within reasonable limits — connected with a platform- 
 stairs as given by plan and elevation at Plate i. Figs. 3 and 4, may be carried around such 
 cylinder on one common inclination, saving room in the stepping, and making a superior shaped 
 wreath. After setting up the elevation, begin by drawing the bottom lines of rail through 
 the centres of short balusters at X X; set off the centre line and the thickness of rail parallel 
 to XX as shown. Draw H F indefinitely; make H N and N F each equal K G, 3^', the radius 
 of a 6" cylinder, and V' more to the centre of rail and baluster; through F parallel to the 
 riser-lines draw B S; through N draw M U; at M and J parallel to H F draw J E and M L: 
 then the four heights C E, E F, F L and L B will be equal; anywhere on the line B S at K 
 as centre with 3" = K 0 as radius describe the cylinder, and with V more radius to G describe 
 the centre line of rail R G P; draw G W, P Q and R V at right angles to PR; make the 
 heights Q T, G W, UV and RS each equal J N; connect S U, V G, W Q and TP: then the 
 four heights and inclinations are set in proper relation to their base, the plan tangents. 
 
 Fig. 4. Plan of Rail the Centre Line of which is the Quarter-circle K P G of Fig 3. — The 
 heights and inclinations Q T P and G W Q are the same as those of like lettering at Fig. 3. 
 Through Q draw S Z; parallel to Q Z draw F E and P B; through G draw P K; on Q as centre 
 draw WK; parallel to Q T draw D C. 
 
 To Find the Angle with which to Square the Wreath-piece at Both Joints :— Make 
 G Y equal G X; connect Y Z: then the angle as taken by the bevel at Y will give a plumb- 
 line — H G of Fig. 5 — on the butt-joints, by means of which the wreath-piece may he squared. 
 
 Fig. 5. Face-mould taken from the Plan Fig. 4; also Showing the Squaring of 
 the Wreath-piece at Both Joints. — On the line K K make A K, A K each equal A K of Fig. 4; 
 make A T at right angles to A K and equal to A Q of Fig. 4; connect K T, KT; make T C, T C 
 each equal T C 'of Fig. 4; parallel to A T draw KB, KB, C E, C F, C E, C F; make K B, K B 
 each equal P B of Fig. 4; make C E, C F, C E, OF equal D E and D F of Fig. 4; make T S 
 and T R equal Q S and Q R of Fig. 4; through K, K draw the lines F D, F D; make K D, 
 K D each equal F K; continue T K to L, and make K L any length desirable, or equal to A B 
 or C D of Fig. 3; parallel to K L draw D 0 and F 0; make the joints L and K at right 
 angles to the tangents. Through DBESEBD of the convex and F R F of the concave trace 
 the edges of the face-mould. 
 
 Note.— At Plate No. ii are given perspective and geometrical drawings and instruction how to form a paper 
 representation of a solid, also a quarter-circle plan of hand-rait, and Irom this plan a drawing of a face-mould; and at 
 Fig. 7 of that Plate an explanation of the line K G A P of Fic 4 of this Plate — wMch together g'lve a practical knowledge 
 of the application and drawing of face-moulds of this kind. Pirci fions in di'tail for sliding face-moulds and the correct 
 application of lievels for squaring wreath-pieces 'will lie found at Pl.A l'E No. 56. 
 
Plate No. 23 
 
 ^ Scale IV2 In.= 1 Ft. 
 
 s 
 
Plate No. 24 
 
PLATE 24 
 
 Figs. I and 2. Starting and Landing Elevations Sufficient to Show the Position in 
 the Cylinders of the Starting and Landing Risers. — The cylinders of 15" diameter; the over- 
 wood to be removed from the straight part of wreath-piece the same as at Figs, i and 2 of 
 Plate No. 22. 
 
 Fig. 3. Face-mould from Plan of Hand-rail Fig. 2. — The lettering of face-mould and 
 plan are alike; and as face-mould, plan, and elevations all have been before carefully explained 
 at Plate No. 22 (the whole being drawn to a scale of i^" to the foot), it will not be necessary 
 to repeat the same here. 
 
 Where the diameter-line of a large cylinder is placed at the face of a landing-riser it will 
 be necessary to manage the case as explained through Figs. 4, 5 and 6.* Just here attention 
 may as well be called to the important fact that whenever it is desirable the ivhole radius of cylinders 
 — such as are introduced in this Plate, and also of Figs, i, 2 and 7 of Plate No. 22, and Figs, i and 2 
 of Plate No. 23 — may be saved or used for step-room, in an entirely unobjectionable and workmanlike 
 manner, if plan and wreathpiece are treated as directed at Plate No. 33. 
 
 Fig. 4. Elevation of Tread and Rise Sufficient to Take Measurements with which 
 to Prepare the Plan of Hand-rail for the Purpose of Drawing a Face-mould. — Draw the 
 bottom line of rail through XX, the centres of short balusters; parallel to X X draw ED, the centre 
 of the thickness of rail; make R U = 4", U D = i:^"; from T draw T S parallel to the floor-line ; 
 from D at right angles to the floor-line draw D S. 
 
 Fig. 5. Plan of Hand-rail with the Top Riser Placed at the Diameter-line of a 15" 
 Cylinder. — At the centre of the width of rail T, and at right angles to T K, draw tlie tangent 
 TS indefinitely; from D of Fig. 4 parallel to the line of riser draw a line to S of Fig. 5; make 
 SD equal SD of Fig. 4; connect DT; from S draw the line SMY tangent to the centre line 
 of the plan of rail; from K at right angles to S Y draw the line K M: then M will be the joint 
 of the wreath-piece, and the remainder of the rail around the cylinder from M to I will be level. 
 Parallel to M S draw N J, 0 Q and T W; continue M S to A. From Tat right angles to S M draw 
 the line T Z indefinitely; on S with D T as radius describe an arc at Z. 
 
 To Find the Angle with which to Square the Wreath-piece at the Joint over M: — 
 Make M Y equal S D; connect Y W: then the bevel at Y will give the angle required. 
 
 To Find the Angle with which to Square the Wreath-piece over the Joint T: — Prolong 
 the tangent S T to L; make T L equal G V, and connect L J : then the bevel at L will give the 
 angle required. f 
 
 Fig. 6. Face-mould taken from the Plan Fig. 5, also Showing the Squaring of the 
 Wreath-piece at Both Joints.J — On a line E D make D T equal D T of Fig. 5. T E may be 
 any length for straight wood. On T as centre with Z M of Fig. 5 as radius describe an arc at 
 M; on D as centre with S M of Fig. 5 as radius intersect the arc at M ; connect M D and prolong 
 M D to S; make TCP equal T C F of Fig. 5; parallel to M D tlirough F, C and T draw U N, 
 R G and TB; make DS equal S A of Fig. 5; make FU and F N equal G H and G N of Fig. 5; 
 make CR and C G equal BO and B Q of Fig. 5; make TB equal T P of Fig. 5; through T draw 
 RO; make TO equal T R; parallel to T E draw R H and 0 A; make the joints E and M at right 
 angles to the tangents; make M K equal M N; through K SU R of the convex and NG B 0 of the 
 concave trace the edges of the face-mould. The bevel at Y, showing the squaring of the wreath- 
 piece at joint M, is taken from Y of Fig. 5; and the bevel at L, showing the squaring of the 
 wreath-piece at joint E, is taken from L of Fig. 5. See Plate No. 74, Figs, i and 2. 
 
 * The method of proceeding would be the same at the bottom, or starting, of a staircase. 
 
 f The angle required to square a wreath-piece at each joint is, in every case, the inclination of the plane of the plank 
 along the joint given on that plane — which is the joint of the face-mould — and a plumb-line on the butt-joint — which is a 
 joint made square from the face of the plank through its thickness. 
 
 X At Plate No. 13 are given perspective and geometrical drawings, and instructions how to form a paper representation 
 of a solid; also a plan of hand-rail less than a quarter-circle, and from this plan a face-mould — which, together, give a 
 practical knowledge of the application and drawing of face-moulds of this kind. 
 
PLATE 25. 
 
 Fig. I. Plan of the Top Portion of a Staircase Winding One Quarter, with a Small 
 Cylinder as given at Plate No. 5, Fig. 5. — Let H be the centre of hand-rail and baluster ; 
 on A as centre describe the centre line of rail H, E, N ; make H B, B M and M N tangents to 
 the centre line of hand-rail. Space the balusters around the centre line of rail and on the 
 winders and steps below as required. Parallel to H B draw A F and M L indefinitely ; parallel 
 to A B from 0, the centre of baluster, draw 0 S ; prolong M B to J indefinitely ; parallel to B J 
 draw S G indefinitely ; parallel to M L from the centre of balusters Q and C draw Q K and 
 C D indefinitely. Further heights and inclinations to complete this drawing will be obtained 
 after Fig. 2, the elevation, is set up. 
 
 Fig. 2. Elevation of Tread and Rise as Figured, and as Taken from Fig. i, the 
 Plan. — This elevation is set up for the purpose of finding heights and inclinations over the 
 plan tangents H B, B E and E M of Fig. 1 ; also through the development of the centre line 
 H G D K L to find the exact relation of the wreath-piece to the steps and rises, and by this 
 means be enabled to get the lengths of balusters wherever placed on the centre line of rail, 
 around the plan of cylinder. The elevation is necessary, too, to make the curve and fix the 
 length and joints of the ramp ; also to get the odd lengths of balusters that may occur 
 under the ramp as shown. In practice, the drawing of this elevation, and of such elevations 
 generally, may be done full size, if desired, very conveniently by the use of the pitch-board, 
 laying its hypothenuse along the edge of a drawing-board ; and for the winders transferring 
 the tread and rise lines from the pitch-board by the use of a long parallel straight-edge. 
 The treads around the cylinder must be measured on the centre line of rail as follows, at 
 the plan Fig. i : From H to 1, the first tread in the cylinder, take its measure in two equal 
 parts, and the second step from 1 to 2 into two parts : then 2 N of the remainder of the 
 centre line is on the line of the floor. After completing the elevation anywhere along the line 
 marked chord-line, set off H B, 3I", equal to H B of Fig. i. Through B draw B J indefi- 
 nitely and parallel to T H ; at any point along the line B J set off J E equal to B E of Fig. i. 
 Through E draw E F parallel to B J ; at any point along the line E F set off F M equal to 
 E M of Fig. i ; through M draw the line M L indefinitely ; from the floor-line to L set up 
 5I" ; then L becomes a fixed point from which the line L R may be drawn, R being the 
 centre line of ramp ; R may be raised or lowered to suit, but there can be no change at L. 
 Wherever the line L R cuts the lines T H, B J, E F and M L, as at H, J and F, draw the lines 
 H B, J E and F M parallel to the lines of tread. At Fig. i make B J, E F and M L each 
 equal B J of Fig. 2. As the heights are alike, connect J H, F B and L E of the last-mentioned 
 figure. Place the baluster at 0 the same distance from the chord-line as at HO of Fig. i, 
 and the other two balusters as at 1 C and 2 Q, as marked alike at Figs, i and 2. Through 
 0, C and Q draw 0 G, C D and P K parallel to the rises; make S G, C D and P K equal the 
 heights indicated by the same letters at Fig. i. Through H G D K L trace the centre line of 
 wreath ; parallel to this centre line trace the top and bottom lines of the wreath as shown 
 by the short dash-lines. Place the centres of balusters that occur under the ramp as at 
 the plan, and draw the dotted lines parallel to the risers ; then, to find the length of any of 
 the balusters around the wreath or under the ramp ; take for example C 4 of the wreath, which 
 is 2f", add this to 2'.2", the usual height of balusters at X, X, then the baluster at C will be 
 2'.4i" at its centre line from the top of step to the under side of the hand-rail or wreath. 
 
 Fig. 3. Plan of Hand-rail from the Quarter-circle H E, Fig. i. — Make the heights and 
 angles B J H and E F B agree with the corresponding letters at Fig. i ; draw A B ; from T 
 parallel to A B draw T U ; parallel to B J draw 0 G ; through E draw H Z indefinitely ; on 
 B as centre with B F as radius describe the arc F Z. To find the angle with which to square 
 the wreath-piece at both joints, prolong B E to R indefinitely ; make E R equal E K ; connect 
 R A ; then the bevel at R will give the angle sought. 
 
 Fig. 4. Face-mould from Plan Fig. 3. — On a line Z Z, make V Z, V Z each equal V Z of 
 Fig. 3. At right angles to Z Z draw V M ; make V M equal V B of Fig. 3 ; connect M Z, M Z 
 and prolong M Z to A ; make Z A equal H R of Fig. 2 ; make M G, M G each equal J G of 
 Fig. 3 ; through G and G parallel to V M draw U T, U T ; make G U, G U each equal 0 U of 
 Fig. 3 ; make G T, G T each equal 0 T of Fig. 3 ; make V S equal V S of Fig. 3 ; through 
 Z and Z draw T F, T F ; make Z F, Z F equal Z T, Z T ; make T B parallel to M Z ; make F 0 
 and T D paiallel to MA; make the joints A and Z at right angles to the tangents. Through 
 F U M U F of the convex and T S T of the concave trace the edges of the face-mould. The 
 squaring of the wreath-piece at both joints is shown through the use of the bevel R, R 
 taken from R of Fig. 3. 
 
 Fig. 5. Plan of Hand-rail from the Quarter-circle E N, the Tangents E M and N M of 
 Fig. I. — Make the angle M L E equal that given at M L E of Fig. i ; parallel to M L draw 
 U 0 and V Q. 
 
 Fig. 6. Face-mould from Plan Fig. 5, also Showing the Squaring of the Wreath- 
 piece at Both Joints. — Draw the lines M F and M X at right angles ; make M N equal M N 
 of Fig. 5 ; make the straight wood N X from 2" up, at pleasure. At right angles to X M 
 through X and N draw Z U and D E ; make M 0 Q F equal L 0 Q E of Fig. 5. Through 0, Q 
 and F parallel to X M draw W Y, V C and E S ; make F Y, F W equal E Y, E W of Fig. 5 ; 
 make Q C and Q V equal R C and R V of Fig. 5 ; make 0 S equal Z S of Fig. 5 ; make N Z 
 equal N U ; make Z D parallel to N X. Through Y C S M Z of the convex and W V U of the 
 concave trace the edges of the face-mould. The bevel L used to square the wreath-piece at 
 the joint X is from L of Fig. 5. The case of face-tnould Fig. 4 is treated in detail at Plate 
 No. II, and face-tnould Fig. 6 is likewise treated at Plate No. id. The development of the 
 centre line of the wreath H G D K L of Fig. 2 is illustrated and explained in detail at Plate 
 No. 20, Figs, i and 2. Sliding face-moulds to plumb the sides of wreath-pieces, also direc- 
 tions for the application of bevels to square wreath-pieces, is given at Plate No. 56. 
 
Plate No. 25 
 
Plate No. 26 
 
PLATE 26. 
 
 Fig. I. Plan of the Top Portion of a Staircase Turning One Quarter to the Landing 
 with Diminished Steps Around the Cylinder, Curved Rises, and Platform as given at 
 Plate No. 5, Fig. 6. — This is an improved plan of stairs turning one quarter without the 
 winders as by the old method given at Plate No. 25, but the treatment of the hand-rail is 
 precisely the same as that at the last-mentioned Plate, because the conditions around the 
 cylinder are alike ; so that if the explanation is not given in this case with quite as much 
 detail as before, it is for the reasons stated. It would be well to make a careful study of 
 the preceding Plate in connection with this. Let F C A be the centre line of hand-rail and 
 F D C, C B A the tangents. Place the balusters around the centre line of hand-rail as shown. 
 Prolong F D to G indefinitely, and prolong K C to J indefinitely ; also D B to N. Connect 
 K B ; from L parallel to K B draw L 0 ; from 0 parallel to B N draw 0 M indefinitely ; from 
 E parallel to F G draw E H indefinitely. The necessary heights and inclinations to complete 
 this drawing will be established by drawing the elevation, and proceeding as directed at 
 Fig. 2. See Plafk No. 77. 
 
 Fig. 2. Elevation of Treads and Rises as Figured and as Taken from Fig. i, the 
 Plan.— From the chord-line, which is the commencement of the cylinder, the treads are to be 
 measured on the centre line of rail, from A to Z in two parts, and from Z to Q in two 
 parts ; also on the line of floor Q F in two parts. Measuring the treads in two parts is done to get 
 more exactly the stretch-out of the centre line. After drawing the elevation set off from the chord- 
 line the length of tangent A B of Fig. i (which in this case is 3^") three times as shown, 
 drawing lines at each of these distances parallel to the chord-line, to N J and G indefinitely ; 
 make the height from the line of floor to G equal 55" : then G becomes a fixed point from 
 which the line G R may be drawn, R being the centre line of ramp it may be raised or 
 lowered to suit, but no change can be made at G without changing the usual length of bal- 
 usters. Where the line G R cuts the lines I J, B N and the chord-line as at J, and at N and A, draw 
 the lines A B, N I and J D parallel to the lines of tread. At Fig. i make B N, C J and D G 
 each equal B N of Fig. 2, and connect G C, J B and N A. Place the centre of balusters L, F 
 and E as on the plan L, C and E Fig. i, measuring from each riser on the centre line 
 except the first baluster, which is measured from A, the chord, to L, the centre of baluster ; 
 parallel to the rise lines through E, F and L draw P H, F B and L M indefinitely; make P H 
 equal P H of Fig. i ; make C B equal C J of Y\g. i ; make 0 M equal 0 M of Fig. i; then 
 through G H B M A trace the centre line of wreath ;* parallel to this centre line set off and trace 
 the top and bottom lines of the wreath as shown by the dotted lines. 
 
 To Find the Length of any Baluster around the Wreath, take for Example : — F K, 
 which is 3.y" , add this to 2'.2" the height of balusters at X X : then the baluster at F will be 
 2' .<-^V at its centre line from, the top of step to the under side of the wreath. 
 
 Fig. 3. Plan of Hand-rail from the Quarter-circle C F, the Tangents C D and F D of 
 Fig. I. — Make the angle D G C equal the angle D G C of Fig. i. From R and S parallel to 
 F G draw R Y and S X. 
 
 Fig. 4. Face-mould from Plan Fig. 3, also Showing the Squaring of the Wreath- 
 piece at Both Joints.f — Draw the lines C G and G B at right angles ; let G F equal D F of 
 Fig. 3 ; F B the straight wood added should never be less than 2"; through F and B 
 parallel to G C draw R A and D C indefinitely ; make G Y X C equal G Y X C of Fig. 3; through 
 C at right angles to G C draw Z I ; through Y and X parallel to F G draw D V and S W 
 indefinitely ; make F A equal F R ; draw A C parallel to F B ; let Y V, X W and C I equal U V, 
 T W and C I of Fig. 3 ; let X S and C Z equal T S and C Z of Fig. 3. Through A G VW I of 
 the convex and R S Z of the concave trace the edges of the face-mould. The bevel G of Fig. 3 
 is used to square the wreath-piece at the joint B as shown. The dotted lines show the least 
 wood required to form the wreath-piece. 
 
 Fig. 5. Plan of Hand-rail from the Quarter-circle A C, Tangents C B and A B of 
 Fig. I. — Let the angles C J B and B N A each equal B N A of Fig. i. Connect P B ; from S 
 draw S V parallel to P B ; from U draw U T parallel to B N ; through A C draw A L indefi- 
 nitely ; on B as centre with B J as radius describe the arc J L 
 
 To Find the Angle with which to Square the Wreath-piece at Both Joints: — Prolong 
 B C to Q indefinitely ; on C as centre with C W as radius describe the arc W Q ; connect Q P: 
 then the bevel Q will give the angle required. 
 
 Fig. 6. Face-mould from Plan J Fig. 5, also Showing the Squaring of the Wreath- 
 piece at Both Joints. — Let C J and C A each equal M L of Fig. 5. Draw C N at right 
 angles to A J ; make C N equal M B of Fig. 5 ; connect N J and N A ; prolong N A to R indefi- 
 nitely ; make A R equal A R of Fig. 2 ; make the joints R and J at right angles to the tangents ; 
 let CY equal M R of Fig. 5; make TO. TX and TO, TX both equal U V and U S of Fig. 5; through J 
 draw X Z, make J Z equal J X ; through A draw X Z ; make A Z equal A X ; parallel to A R draw 
 Z W and X S ; parallel to N J draw X F ; through Z 0 N 0 Z of the convex and X Y X of the con- 
 cave trace the edges of the face-mould. The bevel Q used to square the wreath-piece at the 
 joints J and R is taken from Q of Fig. 5. 
 
 * The development of the centre line of a wreath, as in this case, is illustrated and explained in detail at Plate 20, 
 Figs, i and 2. 
 
 Sliding face-moulds to plumb the sides of wreath-pieces, also directions for the application of bevels to square 
 wreath-pieces, are given at Plate No. 56. 
 
 f The face-mould Fig. 4 is explained in detail at Plate No. 10. 
 \ Face-mould Fig. 6 is explained in detail at Plate No. ii. 
 
PLATE 27 
 
 Fig. I. Plan of the Bottom or Starting Portion of a Staircase Winding One Quarter, 
 Similar to Plan Fig. I of Plate No. 4.— Let A C F be the centre line of rail around the 
 cylinder, and A B, B D and D F tangents. Prolong F A to Q indefinitely; prolong A B to N and 
 H C to T indefinitely. Space the balusters on the centre line as shown. Further heights and 
 inclinations necessary to complete this drawing will be obtained as directed from the elevation 
 Fig. 2, when that drawing is completed. 
 
 Fig. 2. Elevation of Tread and Rises as Figured and as Taken from Fig. i, the 
 Plan. — Besides finding heights and inclinations over tangents, this elevation is also set up to 
 develop the centre line of wreath D U L P Q in its exact relation to the step and rise, thus 
 giving the lengths of balusters under the wreath. The treads in the cylinder must be measured 
 around the centre line of rail, and each tread taken in two parts in order to get more exactly 
 the stretch-out of the circular line. After completing the elevation, anywhere along the line 
 marked chord-line, — which is the commencement of the cylinder, — set off H A or A B of Fig. i 
 (either of which is 5^') three times, drawing lines parallel to the chord-line indefinitely as shown. 
 Let X X at the centres of the short balusters be the bottom line of rail; draw R N, the centre 
 line, indefinitely and parallel to XX; at the intersection, N, draw N A at right angles to the 
 chord-line; make Q R. 3", for straight wood to be added to that end of wreath-piece. Make 
 E D, si"'^ connect D N; at T and D draw the lines D C and T B parallel to the treads. At 
 Fig. I make CT and B N each equal CT and B N of Fig. 2; let AQ equal AQ of Fig. 2; 
 connect Q B, N C and T D; make BO equal A Q; parallel to B C draw OM; parallel to B 0 
 draw M G; connect G H, the level line common to both planes; parallel to G H draw J K and 
 S R; parallel to A Q draw R P; parallel to T C draw K L and Z U. At Fig. 2 the centre of 
 balusters EZJ and S are placed on each tread as at the plan, and the lines S P, J L and VZU 
 are drawn indefinitely and parallel to the rise-lines; make V U and K L equal V U and K L of 
 Fig. i; make G P equal R P of Fig. i; through tlie points DULPQ trace the centre line of 
 wreath; the doited lines are the top and bottom of wreath set off from the centre line. D H 
 is straight wood that will be added to that end of the lower wreath-piece. 
 
 To Find the Lengths of Balusters Around the Wreath : — Take for example the baluster 
 at S; S F is 5^", which, added to the usual height of balusters at X X, 2'. 2", makes the height 
 of this baluster on its centre line from the top of step to under-side of wreath 2'.jl". 
 
 Fig. 3. Plan of Hand-rail from the Quarter-circle F C of Fig. i with the Tangents 
 F D and D C— Make CTD equal CTD of Fig. i; parallel to F D draw P B and Q A. 
 
 Fig. 4. Face-mould from Plan Fig. 3 ; also Showing the Squaring of the Wreath- 
 piece at Both Joints. — Ma.ie the lines ND and Dl at right angles; let DBAN equal DBAT 
 of Fig. 3; let D F equal D F of Fig. 3, and Fl equal DH of Fig. 2; parallel to N D through 
 F and I draw K M and J L; through B, A and N draw R L, 0 H and E G parallel to D F; make 
 D J, B R, A 0, N E and N G each equal D J, K R, X 0 and C E of Fig. 3; make A H equal XQ 
 of Fig. 3; make FK equal FM; make KJ parallel to Fl; through M H G of the concave and 
 KJ RO E of the convex trace the edges of the face-mould. TAe bevel at T used to square the 
 wreath-piece at Joint 1 is taken fromT of ¥\G. 3. The dotted lines show the width of wood required 
 to work out the wreath-piece. This face-mould is explained in detail at Plate No. 10. 
 
 Fig. 5. Plan of Hand-rail from the Quarter-circle A C of Fig. i with the Tangents 
 AB and C B. — The angles of inclination BNC and AQB are taken from Fig. i. Make BO 
 equal A Q; make 0 6 parallel to B C, and 6S parallel to 0 B; connect S H, the level line common 
 to both planes. Parallel to S H draw Y R, 5 L, B G, T U and AV; parallel to B 0 draw 4 M 
 and R P; parallel to A Q draw U W; at right angles to H S draw AE indefinitely; draw C K at 
 right angles to H S; on B as centre with N C as radius describe an arc at K; again on B as 
 centre with B Q as radius describe the arc QE; connect E K. 
 
 To Find the Angles with which to Square the Wreath-piece : — Prolong B A to F 
 indefinitely and AH to J indefinitely; make A F equal A D; connect F G: then the bevel at F 
 will square the wreath-piece over the joint A. Make H J equal SX; connect J C: then the bevel 
 at J will square the wreath-piece over the joint C. 
 
 Fig. 6. Face-mould from Plan Fig. 5; also Showing the Squaring of the Wreath- 
 piece at Both Joints. — Draw the line K Q; let ZQ and ZK equal ZE and ZK of Fig. 5. On Z 
 as centre with Z B of Fig. 5 as radius describe an arc at B; make Q B equal Q B of FiG. 5, and 
 K B equal C N of Fig. 5; connect K B, B Q and BZ; make BW equal B W of Fig. 5; make 
 B M6 P equal N M6 P of Fig. 5; parallel to Z B draw Q V, M L 5, 6, 3, 1 and PY: make Q V and 
 WT equal AV and U T of Fig. 5; make Z 2, M L and M5 equal Z 2, 4, 5 and 4 L of Fig. 5; make 
 6, 3, 6, 1, PY equal S I, S 3 and R Y of Fig. 5; through Q draw the line T C; make QC equal QT; 
 through K draw the line YA; make KA equal KY; parallel to B E dra^w TD indefinitely; make 
 Q E equal Q R of Fig. 2. The joints E and K are made at right angles to the tangents. Through 
 A3 L2VC of the convex and YI5T of the concave trace the edges of the face-mould. The slide- 
 line is drawn at right angles to B Z. The dotted lines show the least width of wood required to 
 work out the wreath-piece. This face-mould is explained in detail at Plate No. 12. The 
 development of the centre line of this case of wreath is given in detail at Plate No. 20, Figs. 
 3 and 4. Sliding face-moulds to plumb the sides of wreath-pieces, also direction for the application 
 of bevels to square wreath-pieces, is given at Plate No. 56. 
 
Plate No. 2 / 
 
Plate No. 28 
 
PLATE 28. 
 
 Fig-. I. Plan of the Top Portion of a Winding Staircase Making a Half-turn with 
 a id" Cylinder as given at Plate No. 6, Fig. 2. — Let A C B be the centre line of rail around 
 the plan of cylinder, and A E, E D and D B tangents. Prolong E D to T indefinitely ; prolong 
 F C to J, A E to L, and FA to R, all indefinitely. Space the balusters on the centre line as 
 shown. Further heights and inclination of tangents to be shown in connection with this plan, 
 including measurements that are necessary to develop the centre line of wreath and at the 
 same time fix the lengths of balusters, will be obtained afier drawing the elevation. 
 
 Fig. 2. Elevation of Treads and Rises as Figured and as Taken from Fig. i, the 
 Plan. — The treads in the cylinder must be measured around the centre line of rail, and each 
 tread taken in two parts to get a nearer stretch-out of the circular line. Draw the centre 
 line of level rail 5g" above the floor. To fix heights and inclination of tangents, try a straight- 
 edge in determining W and B, points on the upper and lower chord-lines ; the points W and B 
 are not fixed, but may be raised or lowered along the chord-lines at pleasure, taking notice, how- 
 ever, that if B is raised it will increase the length of ramp ; also if W is raised the line R 0 will 
 be shortened, a reasonable length of which is required to form the level easing as shown. At B 
 draw B Y at right angles to the chord ; set off four times 5I", the lengths of the tangents 
 B D, D C, C E and E A of Fig. i. Through each of these points of division draw lines parallel 
 to the rises ; from W parallel to the line of floor draw W R ; connect R B and prolong to 0, 
 and B to F indefinitely. Where the line R B cuts the vertical lines at T, J and L, draw the 
 lines T C, J E and L A at right angles to the rise-lines. Let B F at the ramp and the same 
 distance R E at the level easement be the allowance for straight wood to be left on those 
 ends of the wreath. At Fig. i make D T, C J, E L and A R each equal D T of Fig. 2. Con- 
 nect T B, J D, L C and R E. 
 
 To Prepare for and Develop the Centre Line of Wreath. — At Fig. i draw G N and S P 
 parallel to F E ; parallel to A B draw 2 H and P Q ; parallel to C J draw Z K and i^i M. At 
 Fig. 2 place the centres of balusters 5, 4 U G S on the treads as at the plan, and tbroi:!gh 
 eacli of these centres draw lines parallel to the rise-lines indefinitely ; at baluster 5 mr;ke 
 2 H equal 2 H of Fig. i ; at baluster 4 make 8, 6 equal D T of Fig. i ; at balurter U make 
 Z K equal Z K of Fig. i ; at baluster G make N M equal N M of Fig. i ; at bj luster S make 
 P Q equal P Q of Fig. i. Through B H 6 K M Q W trace the centre line of v.-reath. The 
 dotted lines are the top and bottom of the wreath set off from the centre li::e. 
 
 To Find the Lengths of Balusters Around the Wreath :— Take for example the baluster 
 at U. U V is which added to the usual height of baluster at X, 2'. 2", makes the height 
 
 of this baluster on its centre line from the top of step to under side of wreath 2'.3tV". 
 
 Fig. 3. Plan of Hand-rail from the Quarter-circle B C of Fig. i with the Tangents B D 
 and D C. — Let the angles D T B and C J D equal the same at Fig. i ; connect F D ; parallel 
 to F D draw N R ; parallel to C J draw V L ; through B and C draw B P indefinitely ; on D 
 as centre with D J as radius describe the arc J P. 
 
 To Find the Angle with which to Square the Wreath-piece at Both Joints :— Prolong 
 DC to Y ; make C Y equal C W ; connect Y F: then the bevel at Y will give the angle required. 
 
 Fig. 4. Face-mould from Plan Fig. 3; also Showing the Squaring of the Wreath- 
 piece at Both Joints.— Let Z P, Z P each equal Z P of Fig. 3 ; make Z D at right angles 
 to Z P and equal to Z D of Fig. 3 ; connect D P and D P ; make D L, D L each equal D L 
 of Fig. 3. Through L and L draw N R, N R parallel to ZD; make L R and L N each equal 
 V R and V N ; make Z 0 and Z S equal Z 0 and Z S of Fig. 3 ; through P and P draw N H, 
 N H ; make P H and P H each equal P N ; prolong DP to F ; make P F equal B F or RE 
 of Fig. 2. The joints F and P are made at right angles to the tangents. Make N A and N A 
 each parallel to the tangents. Through H R D R H of the convex and N S N of the concave 
 trace the edges of the face-mould. The bevel at Y and Y, used to square the wreath-piece 
 at both joints as shown, is taken from Y of Fig. 3. One face-mould (Fig. 4) in this case 
 answers for both wreath-pieces. The joint F joins the ramp at F, and the same joint joins 
 E, the level easement at the top. This face-mould is explained in detail at Plate No. ii. 
 The development of the centre line of wreath is explained in detail at Plate No. 20, Figs. 
 
 and 2, by repeating the first quarter A C, Fig. i, and the developnaent A E of FiG. 2 of that 
 f late. Sliding face-moulds and squaring wreath-pieces are given at Plate No. 56. 
 
PLATE 29 
 
 Fig. I. Plan of the Bottom or Starting Portion of a Winding Staircase making 
 a Half-turn with a lo" Cylinder as given at Fig. 2, Plate No. 6. — As this case is the same 
 as the one already given at Plate No. 28 with the exception of position, that being the top and 
 this the bottom of a flight exactly alike in plan, there seems to be no need of repeating 
 what has just been given in minute detail; but the plan, the elevation, etc., serve a most useful 
 purpose in showing at a glance such differences as are naturally caused by the changed position 
 of the same plan: such as the length of ramp (which at the bottom of a flight is shorter); 
 also the lengths of balusters if required. But there need be no change of face-mould if care 
 is taken in keeping the inclination of tangents alike. It will be seen upon examination that 
 all measurements required at Fig. 2 are taken from Fig. i, or the reverse ; those measuiements 
 taken from Fig. 2 as required at Fig. i are lettered or figured alike. The lettering likewise 
 agrees between the quarter B C of Fig. i and the plan of hand-rail Fig. 3. Also as far as 
 possible the lettering is alike between the plan of hand-rail FiG. 3 and the measurements to 
 be taken in connection with it for drawing the face-mould and squaring the wreath-piece at 
 Fig. 4. 
 
Plate No. 29 
 
Plate No. 30 
 
 Scale IVa 1n.= 1 Ft. 
 
PLATE 30. 
 
 Staircases are frequently planned of greater width at the starting by curving the front- 
 string out, embracing in the curve from one to five treads; the least curve including but one 
 step is done merely to save the width of stairs at this point by setting the newel-post a few 
 inches aside. The larger curves including more steps give the stairs an inviting and more 
 ornamental appearance. There is also a more recent practice of setting the newel on top of 
 the first step by extending this step and riser in a curve sufficient to include the curve-out 
 and the base of the newel as shown at Fig. 3. This gives in very little space a neat and 
 elegant finish at the starting of a staircase. 
 
 Fig. I. Plan of Curve-out in One Tread, together with Sufficient Elevation for 
 those Measurements Required to Fix the Height of Newel and Draw the Face-mould 
 or Parallel Pattern.— Let the bottom of rail rest on the centres of the short balusters at XX; 
 make R T equal 6", and T L half the thickness of rail; let L N be the centre line of rail; 
 draw L M at right angles to the rise-line; parallel to the rise-line draw L A: then A at the 
 plan on the centre-line of rail C A becomes a fixed point from which the tangent A B may 
 be drawn at pleasure. Make C F equal M N; connect FA; from any point on the centre curve 
 S draw a line parallel to A B touching the line C A at Z; draw Z 0 parallel to C F; at right 
 angles to A B draw C D indefinitely; on A as centre with A F as radius describe the arc F D; 
 connect D B. See Plate No. 74, Figs. 3 and 4. 
 
 To Find the Angle with which to Square the Wreath-piece at the Joint over B: — 
 Parallel to A B draw C K indefinitely; at right angles to AB draw B J; make J K equal C F; 
 connect K B: then the bevel at K will give the angle sought. 
 
 To Find the Angle with which to Square the Wreath-piece at the Joint over C: — 
 Prolong the tangent B A to G, and the tangent A C to H, indefinitely; make C H equal C E; 
 connect H G: then the bevel at H will give the angle required. 
 
 Fig 2. Parallel Pattern for Wreath-piece from the Plan Fig. i. — Make AO VF equal 
 the same letters at Fig. i. On F as centre with D B of Fig. i as radius describe an arc at B; 
 on A as centre with A B of Fig. i as radius intersect the arc at B; connect B A; make B T 
 equal B T of Fig. i; through T at right angles to A B draw Y Z; draw 0 S parallel to A B; 
 make 0 S equal Z S of Fig. i. The width of the rail being 3", the parallel pattern will be 3!". 
 Make VP and VQ each if; parallel to V F draw P G and Q E; make B L and B W each ij"; 
 parallel to B A draw L Z and W Y; on S as centre with ij" as radius describe a circle and 
 sketch the curves P Z and Q Y touching the circle. The angle for squaring the wreath-piece at 
 joint B is taken by the bevel K at Fig. i, and for the joint F the bevel H of Fig. i. 
 
 Fig. 3. Plan and Elevation of the Starting of a Staircase with the Front-string 
 Curved Out and the Newel Set on Top of the First Step ; the Elevation and Plan 
 Prepared, Fixing the Height of Hand-rail at the Newel, and for Drawing the Face- 
 mould. — Let the bottom of rail rest on X X, the centres of short balusters, and make A B, the 
 centre line of rail, parallel to X X; make D E equal 8", and E F half the thickness of rail; draw 
 F C parallel to the line of tread; draw A G parallel to the line of riser: then G becomes a 
 fixed point on the line of tangent V G from which the level tangent G J Z must be drawn, 
 but may be kept any distance from I to Z at pleasure. Parallel to Z G draw H 0 and L K; 
 parallel to V M draw 0 S and K W; from V at right angles to G J draw V N indefinitely; 
 on G as centre with G M as radius describe the arc M N; connect N J. 
 
 To Find the Angle with v/hich to Square the Wreath-piece at the Joint over J: — 
 From V parallel to G Z draw V R indefinitely; draw J T at right angles to J G; make T R 
 equal V M; connect R J: then the bevel at R will give the angle sought. 
 
 To Find the Angle with which to Square the Wreath-piece at the Joint over V: — 
 Prolong Z G to U, and G V to P, indefinitely; make V P equal V Q; connect P U: then the 
 bevel at P will give the angle required. 
 
 Fig. 4. Parallel Pattern for Wreath-piece from the Plan Fig 3 — As this rail is to be 4" 
 wide by 2V thick, the pattern will answer to get out the wreath-piece if 4J" wide. Draw the line 
 GA; make G M equal G M of Fig. 3. Let M A equal 5" more or less for straight wood; on M as 
 centre with N J of Fig. 3 as radius describe an arc at J ; on G as centre with G J of Fig. 3 
 as radius intersect the arc at J; connect J G; make G SW equal G S W of Fig. 3; parallel 
 to J G draw S H and W L; make W L equal K L of Fig. 3; make S H equal 0 H of Fig. 3. 
 With 2|" radius describe circles from the centres J, H, L, M and A; connect BF and C E. The 
 joints J and A are made at right angles to the tangents. Bend a flexible strip of wood touching 
 the circles on the convex and concave and mark the curved edges of the pattern. The angle for 
 squaring the wreath-piece at joint J is taken by the bevel at R of Fig. 3, and for the joint A by the 
 bevel at P of Fig. 3. The height of hand-rail from the top of the second step, D, to the bottom 
 of the rail, E, will equal — by adding the height of short baluster at X, which is 2'. 2", to the 8" 
 raised between D and E — 2'. 10". Face-mould for Figs, i or 3 is treated in detail by Plate No. 13. 
 One feature of this plan given at Fig. 3 which is open to objection is the increased heigJit of newels j how 
 to reduce this, if required, will be shoioti by the following Plate, No. 31. 
 
PLATE 31. 
 
 Fig. I. Plan and Elevation of the Starting- of a Staircase with the l^ront-string 
 Curved Out and the Newel Set on Top of the First Step, the Elevation and Plan Pre- 
 pared to Continue the Hand-rail on a Common Inclination to the Newel, and for a Face- 
 mould as Required. — Let the l)ottom of tlie rail rest on X X the centres of sliort balusters ; 
 draw the centre line of rail Z R parallel to X X; make the tangents C V and V P of equal 
 lengths; from V parallel to the rise-line draw V 5; parallel to the tread-line draw 5 Y; make 
 
 4 R equal V P; through R parallel to V 4 draw W 8; at R draw R 4 at right angles to V 4; 
 draw R U at right angles to R 5; make C G equal Y Z; connect G V; make V B equal 4, 5 and 
 at right angles to V P; connect B P; through C draw P D indefinitely; on V as centre with 
 V G as radius describe the arc G D; at right angles to D P, through V draw Q 0, F N M and 
 T L; at riglit angles to C V draw A K and L J. 
 
 To Find the Angle with which to Square Both Ends of the Wreath-piece : — Parallel to 
 C V draw K I; at right angles to G V draw I H; prolong C V to E indefinitely; make C E 
 equal I H ; connect E F, then the bevel at E will give the angle required. 
 
 Fig. 2. Face-mould from Plan, Fig. i; also Squaring the Wreath-piece at Both Joints. 
 — Draw the line D D; make S D, S D each equal S D of Fig. i; at right angles to D D through 
 
 5 draw Q 0; make S V equal S V of Fig. i; connect V D and V D; make V K J and V K J 
 each equal V K J of Fig. i. At right angles to D D through J K and J K draw T J, N M, 
 N M and T J; make J T and J T each equal L T of Fig. i; make K M, K N and K M, K N each 
 equal A M and A N of Fig. i; make S 0 and S Q equal S 0 and S Q of Fig. i; through D 
 draw T Z at both ends; make D Z and D Z each equal T D. The lower end of the wreath- 
 piece requires an addition of straight wood to fill out the plumb joint as shown at 6 U of 
 the elevation, Fig. i. Make D F equal 6 U of Fig. i; make D U equal 4", more or less for 
 straight wood; parallel to V U draw Z B and T A. The joints U and F are made at right 
 angles to the tangents. Through Z M 0 M Z of the convex and T N Q N T of the concave 
 trace the curved edges of the face-mould. The angle for squaring the wreath-piece at joints 
 F and U is taken by the bevel at E, Fig. i. This case of face-mould is treated in detail at 
 Plate No. 15. The difference in height between this manner of treating the hand-rail and 
 that of Fig. 3, Plate 30, is from the top of the rail at 8, where it joins the newel in that 
 case, to the top of the rail at 2, where it joins the newel in this case, just 6". When the rail 
 is set up at X X, 2'. 2", the height of rail from W to 2 will be 2'. 6^", and between those 
 points at Fig. 3, Plate 30, the height is 3'..j". Much more than 6" can be gained if desired 
 by moving the centre of the newel further towards the first riser on the line 3, 3 and increas- 
 ing the radius of the curve-out to suit the change. This might be very desirable in case of 
 designing with this style of finish a low newel. 
 
 Fig. 3. Plan of a Quarter-platform Stairs with a Quarter-cylinder of 8" Radius, the 
 Rise rs each Set at A and B, the Chord-lines. — Let A Z B be the centre line of rail on the 
 plan; divide the quarter-circle A B at Z in exactly two equal parts ; connect 6 Z ; at right angles 
 to 6 B draw B E indefinitely; through A at right angles to 6 A draw THY indefinitely; through 
 Z at right angles to 6 Z draw H E; make A Y equal 9", one tread; at right angles to A Y 
 draw Y P; make YP equal 8" — the rise; connect PA; parallel to Y P draw H K; parallel 
 to AY draw K L; make L M equal Y L; divide P M in two equal parts at N; make H J 
 equal M N; draw J W parallel to Y A; draw WX parallel to 5 A; from A draw A R parallel 
 to the tangent H Z; make A R equal the tangent H Z; connect X R, w^hich is the level 
 line. Through H draw 5 G parallel to X R; anywhere on the centre curve-line draw the line 2 U 
 parallel to X R; from A at right angles to X R draw A Q indefinitely; on the dividing radial 
 
 6 D make Z D equal M N; connect D H; from Z at right angles to H G draw Z I indefinitely; 
 on H as centre with H D as radius describe an arc cutting the line Z I at C; on H again 
 as centre with K A as radius describe an arc cutting the line A Q at Q; connect Q C. 
 
 To Find the Angle with which to Square the Wreath-piece at the Joint over Z : — 
 Make Z F equal Z 0; connect F G; then the bevel at F will give the angle sought. 
 
 To Find the Angle with which to Square the Wreath-piece at the Joint over A : 
 — Make A T equal H V; connect T 5; then the bevel at T will give the angle required. 
 
 Fig. 4. Parallel Pattern for Wreath-piece from the Plan Fig. 3. — Draw the line Q C, 
 and make Q 4 C equal the same at Fig. 3. On Q as centre with A K of Fig. 3 describe an 
 arc at K; on C as centre, with D H of Fig. 3 as radius, intersect the arc at K; also 4 K 
 will equal 4 H of Fig. 3. Connect Q K, K C and 4 K; make K 3 equal H 3 of Fig. 3; make 
 Q V equal A V of Fig. 3; draw V 2 parallel to K 4; make V 2 equal U T of Fig. 3; make Q B 
 equal 3", or more for straight wood. The size of this rail is to be 2^' thick by 3" wide, 
 therefore the parallel pattern will require to be 3!" wide. With \y as radius on C, 3, 2, 
 Q, B as centres, describe circles. Make a line touch these circles for the concave and convex 
 edges of the pattern. The joints B and C are made at right angles to the tangents. The 
 angle for squaring the wreath-piece at joint C is taken by the bevel at F of Fig. 3, and for 
 the joint B by the bevel at T of Fig. 3. The slide line is drawn at right angles to the 
 level line 4 K. This pattern serves for both wreath-pieces, C being the centre joint. A face- 
 mould of this kind is treated in detail at Plate No. 16. This quarter-cylinder with its con- 
 necting steps and risers should be planned in such a way that the quarter-wreath could be 
 got out in one piece of a common inclination as shown at Plate No. 37, Figs. 5, 6 and 7. 
 It would cost less and be a superior-shaped wreath-piece. See Plate No. 78. 
 
Plate No. 31 
 
PLATE 32. 
 
 Fig. I. Plan of Hand-rail composed of Two Curves of Different Radius as a Curve- 
 out at the Starting of a Staircase, taken from the Plan given at Fig. 5, Plate No. 6.— 
 
 This shape of curve is a necessity in order to make a proper connection with a square newel 
 where the sides of the newel are required to be set parallel to the sides of tiie hallway, as 
 siiown by the plan above mentioned. An elevation has to be set up in order to fix the length 
 of plan tangent D F, as at A C of Fig. 2. And the greater the required height of newel tiie 
 higher up the line A C must be placed, and therefore the shorter this line and the plan tangent 
 will be. 
 
 Fig. 2. Elevation of Tread and Rise as at Plan of Hand-rail Fig. i, taken from 
 Fig, 5 of Plate No. 6. — Let the bottom line of rail pass through X X, the centres of the short 
 balusters of the regular tread; make LQ 5^^", more or less, depending on what height of newel 
 is demanded; make QV half the thickness of rail; draw VGA parallel to the line of treads; 
 let C B be the centre line of rail parallel to XX. When the hand-rail is set up, the height from 
 the top of the first step, L, to the bottom of the rail, Q, will be 2'. 2" at X and 5^" more at L Q, 
 equal to 2'.'jV. At Fig. i maice D E equal A B of Fig. 2; connect E F; parallel to D E draw H T 
 I U, J V, K W and LY. 
 
 Fig. 3. Face-mould to be taken from Plan Fig. i ; also Squaring the Wreath-piece 
 at Both Joints. — Make FTU VWYE equal the same at Fig. i; draw FG at right angles to FE; 
 make FG equal FG of Fig. i; tiirough G parallel to FE draw AH; parallel to FG draw T H, 
 U I, VJ, W K, 0 L and N M; make G A equal G H. Set off all measurements from the line F E as 
 taken on the line F D of Fig. i according to the corresponding letters at the curves. Through 
 NOXXXXXA of the convex and M LKJ IH of the concave trace the curved edges of the face- 
 mould. The bevel at E of Fig. i is used to square the wreath-joint G of Fig. 3. It would be 
 well to add about 3" for straight wood to joint E. 
 
 Fig. 4. Plan of the Starting Portion of a Staircase with the Front-string Curved 
 Out, and Embracing Four Treads of Equal Widths at the Wall-string and Front- 
 string. — At the elevation. Fig. 5, the bottom line of rail rests at X X, the centres of short 
 balusters; BC and DC are the centre lines of rail. Fixing the point C controls the height 
 of rail at the newel; also the length of tangent A C at Fig. 4. FE being 9", add that to the 
 height of short baluster at X, 2'.2", and the sum 2'. 11" will be the height between F and E. 
 In a flat curve like this it is desirable to keep the point C up as high as can be allowed, for 
 it shortens the tangent A C, Fig. 4, and makes the level line C 0 rriore nearly a tangent. Let 
 the angle ABC equal the same at the elevation; connect CO; parallel to CO draw 4 W I 3 
 QU, RG, 8K and AM; parallel to A B draw 4 H, S V, U Z, 3Y and 8 X. From A draw A9 at 
 right angles to 0 C; on C as centre with C B as radius desci ibe tiie arc B9; connect 9 0. 
 
 To Find the Angle with which to Square the Wreath-piece over the Joint A:— 
 Draw S6 parallel to CB; prolong CA to L; make AL equal A6; connect LG: then the bevel 
 at L will give the angle sought. 
 
 To Find the Angle with which to Square the Wreath-piece over the Joint 0:— 
 Draw 0 2 at right angles to OC; make 2 M equal A B; connect M 0: then the bevel at M will give 
 Lue angle sought. 
 
 Fig. 6. Face-mould from Plan Fig. 4 ; also Squaring the Wreath-piece at Both 
 Joints. — Let OB equal 0 9 of Fig. 4. On B as centre with B C of Fig. 4 as radius descril)e 
 an arc at C; on 0 with 0 C of Fig. 4 as radius describe an intersecting arc at C; connect O C 
 and C B; make the spaces lettered on B C agree with those of B C at Fig. 4; parallel to 0 C 
 draw XJ, Y I, Z Q, VTR and H W: take all measurements on the tangent AC, Fig. 4, and set 
 them off from the line B C as shown by the corresponding letters; make ON equal OJ; through 
 B draw WK; make BK equal BW; prolong C B to A; make BA 4" for straigiit wood. The 
 joints A and 0 are at right angles to the tangents. Through WT5FEPN of the convex and 
 KRQIJ of the concave trace the curved edges of the face-mouUl. The angle for squaring the 
 wreath-piece at joint 0 is taken by the bevel M at Fig. 4, and that for joint A by the bevel L 
 at Fig. 4. Face-mould Fig. 3 is treated in detail at Plate No. 10, and face-mould Fig. 6 is 
 treated likewise at Plate No. 13. 
 
 Note. — The line O C of Figs. 4 and 6 is not a tangent, but is simply a level line used to fix the height, control the 
 joint, and to guide the measure, or drawing, of the face-mould. Care should be taken in laying out the squaritii; at joint O 
 of the 7ureath-pieee that the width of rail he made equal /<? K N of FiG. 4, because, although 1^ N is at right angles to O C, it 
 is oblique to the curve, and may, therefore, measure a quarter of an inch or more over the width of rail. After the wreath- 
 piece is squared at the joint O, a portion of the wood, equal to N U K of FiG. 4, is cut away to correct that joint. 
 
PLATE 33. 
 
 Fig, I. Plan of the Landing Portion of a Straight Flight of Stairs with the 
 Top Riser Set in the Whole Depth of a Ten-inch Cylinder, thereby Saving Five 
 Inches Space. — Tlie plan made as described with the addition of the plan and centre 
 line of rail, proceed to set up the elevation, Fig. 2, resting the bottom line of rail on 
 the centres XX of the short balusters, and the centre of rail A D in position parallel to 
 X X, the point D being fixed by the level line G D, and its height F G from the floor. 
 The point A is decided by the position of the chord-line as taken from the plan ; A C is 
 drawn parallel to the line of tread. This completes the preparation of the elevation, all being 
 obtained that is required in fixing the point A and D together with the height C D. Again 
 at Fig. i, parallel to the line of string, let the line F A pass through C, the centre of rail ; 
 make C D at right angles to C A, and C D A equal to C D A of Fig. 2. From A draw the 
 line A I touching the centre line of rail at Q ; the exact place of Q is determined by draw- 
 ing a line from the centre 0, at right angles to A Q; parallel to A Q draw R Y, S X, V W H J 
 and P U; parallel to C D draw T 2, N M, X 3 and YZ; from C at right angles to A Q draw 
 C E; on A as centre describe the arc D E; connect E Q. 
 
 To Find the Angle with which to Square the Wreath-piece at the Joint over Q : — 
 Anywhere along the line W J draw H I parallel to 0 Q. Make H J equal N M; connect J I; 
 then the bevel at J will give the angle sought. 
 
 To Find the Angle with which to Square the Wreath-piece at the Joint over C:— 
 From M parallel to A C draw M K; make C F equal K L; connect F G; then the bevel at F 
 will give the angle required. 
 
 Fig. 3. Face-mould from Plan Fig. i, also showing the Squaring of the Wreath- 
 piece at Both Joints. — Draw the line D Q indefinitely ; make D Q equal E Q of Fig. i. 
 On D as centre, with D A of Fig. i as radius, describe an arc at A; and on Q as centre, with 
 Q A of Fig. i as radius, intersect the arc at A. Connect A Q and A D; prolong A D to L and 
 equal to A E of Fig. 2, or at pleasure for straight wood. Make D 2 M 3 Z the same as at 
 the corresponding letters of Fig. i. Parallel to A Q draw Z R, 3 S, M V W and 2 U P; through 
 Q draw R C at right angles to A Q ; make Q C equal Q R; make A 4 equal A 4 of Fig. i ; 
 make 3, 5 S equal X 5 S of Fig. i; make M V, M W equal N V, N W of Fig. i; make 2 U, 2 P 
 equal T U,T P of Fig. i; through D draw P B; make D B equal D P; parallel to D L draw 
 P E and B F; through C 4 5 V U B of the convex and R S W P of the concave trace the edges 
 of the face-mould. The tangent Q A being a level line, the face-mould slides along the joint 
 R C. The angle with which to square the wreath-piece at joint Q is taken by the bevel at J of 
 Fig. I, and for the joint L is taken by the bevel at F of Fig. i. This face-mould is treated in 
 detail at Plate No. 14. The level poriiun of rail Q B, Fig. i, may have as much straight wood 
 attached to B as seems desirable. 
 
 Fig. 4. Plan of the Starting Portion of the Same Flight of Stairs given at 
 Fig. I with a Ten-inch Cylinder, and the Starting Riser Set in the Whole 
 Radius or Depth of Cylinder, Saving Another Five Inches. — Having made the plan 
 as described, proceed to set up the elevation. Fig. 5. Let the bottom line of rail rest 
 on the centres X X of the short balusters. Place the centre of the rail C B in position par- 
 allel to X X, the point C being fixed by the level line G C, and its height F G from the 
 floor. The point B is fixed by the position of the chord line as taken from the plan. A C G 
 is drawn parallel to the line of floor. The place of B, the height A B, and the distance A C 
 is all that is required of the elevation. Again at Fig. 4 prolong the diameter-line S A of the 
 cylinder to B indefinitely; at the centre line of the rail A, at right angles to A B, draw A C; 
 make ABC equal to A B C of Fig. 5. From C, Fig. 4, draw the line C D tangent to the centre line 
 of the rail at the point D, to be determined by drawing a line from 0 at right angles to the 
 tangent C D. The remaining portion of the rail D S is level, and at S straight wood can be 
 added at pleasure. To draw the face-mould proceed as at Figs, i and 3. It is necessary to 
 give further attention to this case of hand-rail from the fact that although the riser is set 
 in the cylinder the same at the bottom as at the top of the flight, there is a difference 
 requiring another face-mould unless it is thought worth while to make the top face-mould answer 
 also for the bottom. In the latter case let C stand exectly as it is, and make C L equal 
 C A of Fig. i and L N equal C D of Fig. i; then change the place of the chord-line — or 
 commencement of the cylinder — to L, and describe the centre line of the rail from the new 
 centre, and draw from C a new tangent, and the case will then agree with the top plan 
 tangents and face-mould. 
 
 Fig. 6. Elevation Same as Fig. 5, for the Development of the Centre Line of Wreath 
 from Plan, Fig. 4. — The corresponding letters and figures of Figs. 4 and 6 will give a 
 snfiicient explanation. This developm.ent of the centre line of the wreath-piece is introduced 
 to demonstrate the correctness of treating such cases in the manner here shown. Treating 
 cases of hand-railing like this in two quarters — the joint in the centre — makes it necessary to 
 square up both pieces and draw two face-moulds, with no better result and nearly double the 
 cost. This face-mould. Fig. 3, is treated in detail at Plate No. 14. The development of the 
 centre line of wreath is given ip detail at Plate No. 21, Figs, i and 2. 
 
Scale IV2 In. = 1 Ft. 
 
Plate No. 34 
 
PLATE 34. 
 
 A Wreath in one Piece over a Twelve-inch Cylinder. — Size of Hand-rail Four Inches Wide by Two and 
 A Half Inches Thick; the Wreath to be Got out of Plank Four Inches Thick. 
 
 Fig. I. Plan of the top portion of a straight flight of stairs with a 12'' cylinder, the landing riser set in 
 the whole depth of the cylinder. 
 
 Fig. 2. Elevation. — After drawing the plan as given and described, set up one tread and rise; rest the 
 bottom of tlie rail at X X, the centres of the sliort balusters. Tiie bottom of the level rail is set up 
 from the floor 4" as usual. Place the chord-line as at plan. From E, the intersection of the centre 
 lines of rail draw E P parallel with riser, and from J draw J P parallel with tread. 
 
 Fig. I. To Prepare the Plan for Drawing the Face-mould: — Make J P equal J P of Fig. 2. At 
 right angles to J P draw P E equal to P E of Fig. 2. At right angles to J T draw T U equal to P E ; 
 connect J U and prolong the line indefinitely ; connect T P ; parallel to T P touching the centre line of 
 rail draw Z C ; from C parallel to P E draw C D ; from Z parallel to T U draw Z 7 : then the height 
 C D touching the inclined line, J E is equal to the iieight Z 7 touching the inclined line J U ; and both 
 inclinations (whatever they may be) are in the same plane. From the centre L at right angles to Z C 
 draw L A ; parallel to Z C draw 6 Q, L N and K W ; from J at right angles to Z C draw J B indefinitely ; 
 on C as centre witii D J as radius describe an arc at B ; connect B A. To Find the Angle ivitli which to 
 Square the Wreath at the Joint over J : — Draw L 8 at right angles to 6 K ; make L 8 equal N H ; connect 
 8 J: then the level at 8 will give the angle required. To Find the Slide Distance {or JSIovement of Face-mould') 
 to Plumb the Sides of the Wreath: — From F draw K2 at right angles to DC; make AS equal 2 D; connect 
 S 5 : draw S 4 at right angles to S 5 and equal to 2" — half the thickness of plank — through 4 parallel 
 to S 5 draw 4 R : then 4 R will be the distance sought. 
 
 Fig. 3. Face-mould to Include the Whole Cylinder from Plan Fig. i. — Draw the line J A equal 
 to B A of Fig. i. On J with J D of Fig. i as radius describe an arc at D ; on A with A C of Fig. i 
 as radius intersect the arc at D ; connect D A and D J, prolonging eaeh indefinitely. Make J I G F E equal 
 the same at Fig. i ; parallel to D A draw E T, F 6, G L and I W K ; make D X Y equal C X Y of Fig. i ; 
 make E 9 V T equal P 9 V T of Fig. i ; make F 1 , 3, 6 equal Q 1 , 3, 6 of Fig. i ; make G 0 M L equal N 0 M L 
 of Fig. I, and W I K equal W 9 K of Fig. i ; connect T J and prolong the line both ways indefinitely. 
 Make J 1 1 equal J K ; make T 1 3 equal T 6 ; connect L A ; make A 1 0 equal A V ; make J 1 2 equal 2" for 
 straight wood ; make the joint 12 at right angles to J D — the joint 6, 1 3 is not changed — draw K 8 and 
 11,4 parallel to J D. Through 6 V 3 M K of the concave and 13, Y, 10, X, 9, 1 0, W, 1 1 of the convex trace 
 the curved edges of the face-mould. Over-wood has to be added to the wreath from joint T to find this. 
 From T draw T B parallel to 10, V ; make T B equal 4 R of Fig. i ; then B C will equal the required 
 over-wood. 
 
 Fig. 4. Squaring the Wreath. — This wreath with its joints is to be sawed out square as usual, 
 and the lower joint made square from the face of plank. In cutting out the wreath, if more wood 
 be allowed as indicated by the small dots, x x , it will prove an advantage in squaring up. The angle for 
 squaring the wreath at its lower joint 8 is taken by the bevel 8 at Fig. i. The line T is the joint 
 of face-mould, and T 0 — equal to B C of Fig. 3 — is the wood added to the wreath, required to make a 
 complete plumb-joint through the thickness of the plank. To Slide the Face-mould in Position to Plirmh the 
 Sides of the Wreath and the Upper Joint: — Draw from T and from 1 the lines 1, 2 and T B parallel to V 10; 
 make T B and 1, 2 each equal 4 R of Fig. i ; then place the joints of the face-mould on their centres 
 at 2 and B, marking joint B and the point B ; also the edges that can be marked on the plank as shown 
 by the short broken lines; again on the lower side of the plank (on the lines 1, 2 and T B) move the 
 face-mould as much this way from 1 and T, marking joint B and point B, and the edges of the face- 
 mould that remain on the plank as before. Cut away the wood and make the plumb-joint from B as 
 indicated on both faces of the plank ; then mark a line on the plumb-joint from the points B, B at the 
 upper and lower faces of plank ; and this will be the plumb line from which to square the wreath at this 
 joint. No straight wood is required at the upper end except that added to plumb the joint. 
 
 Fig. 5. Construction of a Paper Representation of a Solid which Presents a Practical Dem- 
 onstration in Solid Geometry of the Laws Controlling the Tangents, the Trace for Face- 
 mould, and the Position of the Plane of the Plank where the Plan, a Semicircle, is to be Cov- 
 ered in one Wreath-piece. — Let J C Z equal J C Z of Fig. i ; at right angles to J C draw C D ; at right 
 angles to J Z draw Z 7 ; at right angles to Z C draw C X and Z W ; make C D, C X, Z 7 and Z W each 
 equal C D of Fig. i ; connect D J, 7 J and W X. These form the three vertical sides of the solid, and 
 show in position the inclined and level tangents, also the inclination of the plane of the plank over the diam- 
 eter of the cylinder; showing also that the point U of the centre line of rail must invariably fall below the 
 position of the level tangent X B. To Find the Am^^les Enclosing the Cutting Plane, also the Trace of the Given Semi- 
 circle :—FdYa\\Q\ to Z C draw 8, 1 4, T P, 4 Q, 1 6, N and 2. 1 5 ; parallel to C D draw 1 4, 1 3, P E, Q F, N G and 
 15, 1 ; at right angles to Z C draw J A, 1 6, 1 8 and T V indefinitely. On X as centre witli J D as radius describe 
 an arc at A ; on W as centre with J 7 as radius intersect the arc at A; connect X A W. Make X 1 1 , 1 0, 9 Y S A 
 equal D 13 E F G IJ ; parallel to X W draw S R, Y 0, 9 M, 1 0, V and 1 1 K ; make S R equal 15, 2, and Y 0 equal 
 N 3, and 9 M equal Q 4, and 10, L equal P 5, and 11 H K equal 14, 6, 8 ; through VKBHLMORA find the 
 trace of the semicircle J 5 T. With a sharp pointed instrument scratch the lines of the baseZ J, J C and 
 CZ; also the level line W X. Cut out the remainder of the figure, touch the adjoining edges with a 
 little glue or thick mucilage, and bring them together so as to leave the shading and lines outside. 
 
 Fig 6. To Develop or Unfold the Semicircle J 5 T of Fig. 5, and its Vertical Intersection 
 with the Cutting Plane of that Figure at the Points V K B H L M 0 R A:— Draw the line J T an.i 
 mark on it tiie divisions as shown by the corresponding letters and figures on the semicircle at Fig. 5. 
 Raise perpendiculars at each of these divisions equal in height to those indicated by similar letters at 
 Fig. 5; and through these trace the points of interse'ction as shown. This curve, J1GFE13D13E is 
 the unfolded centre line of wreath the plan of which is given at Fig. i. As the point E is the centre of 
 the rail joint at the upper end, it is evident that considerable wood will have to be cut away at D in 
 shaping the wreath to a level ; but the slab that is removed from the top may be glued to the bottom. 
 
 In this treatment of a wreath in one piece, the longer the tangent J C, Fig. i, the better ; and I might also 
 add the smaller the cylinder the better. See Pla te No. 79. 
 
PLATE 35. 
 
 Another and More Simple Method of Drawing a Face-mould and Working the Whole 
 Wreath in One Piece over a Seven-inch Cylinder. — Size of Hand-rail Four Inches Wide 
 HY Two Inches and a Quarter Thick ; the Wreath to he Got out of Plank Four Inches 
 Thick. 
 
 Fig. I. Plan of the top porlion of a straight flight of stairs with a 7" cylinder, the landing- 
 riser placed at the cliord-line or commencement of the cylinder. 
 
 Fig. 2. After drawing the plan as given and described, set up this tread and rise in a con- 
 venient position to the plan as shown. Rest the bottom of the rail at X X, the centres of the short 
 balusters. The bottom of the level rail B is set up from the floor 4" as usual; draw DB parallel 
 to the floor-line; let BH equal the thickness of the rail; make HK parallel to B D; at E draw EGN 
 at right angles to XX; thiongh G parallel to the rise-line draw J K. 
 
 Fig. I. To Draw the Face-mould, also to Find the Squaring of the Wreath at its 
 Joints: — At right angles to AB draw B 2, 0 W, S Y, E 3, P T, A Z, etc., indefinitely; parallel to the 
 floor-line, FiG. 2, from 0 and C draw lines lo L and C of Fig. i, and again, parallel to LO or C C 
 draw NJ F, G Z, B D 6, and H K Y. At Fig. i make LZ and LD each equal half the thickness— 2"— 
 of plank; parallel to LC draw Z2 and D 8, showing the edge of the plank as canted in its required 
 position; touching the angles U, 6, X, and F, draw lines at right angles to Z 2. The square shaded 
 sections are the joints of the wreath as cut square through the plank, showing also on these joints 
 the lines that square the wreath. For straight wood required to make the butt joint of the lower 
 end: LetTK equal E 0 of Fig. 2; through K parallel to LC draw 5 J. At right angles to Z 2 draw 
 2 J, W M, TV, Z 5, etc., measuring all the points for tracing the edges of the face-mould from the 
 line A B, and setting them off from the line J 5; as I M equals OQ; KV equals P R, etc. Mark the 
 face-mould on the plank, and saw out square; plane the joints square from the face of the plank; 
 this done, lay out each joint for squaring the wreath with the bevel W as shown. Before making 
 the butt Joint at L, cut off at right angles to the joints the slabs UY and D F; then resting D F on 
 the saw-table, let the band-saw cut each side of the wreath plumb as far as the centre; reverse the 
 ends; this time resting the level surface U Y on the saw-table and finish sawing plumb the sides 
 of the remaining half of the wreath. 
 
 To Make the Butt Joint at L, as at G E of Fig. 2:— Gauge from the joint on the level 
 surface of F D the distance J N of Fig. 2; then cut away the superfluous wood and make the joint 
 required from the gauged line to the line X Z. The butt joint made and tested* and the sides 
 sawed plumb, finish the squaring by band-saw, cutting away for the top and bottom of the wreath, 
 rolling it on the saw-table on its convex side. To increase the thickness of plank at the point 
 most required, the slab sawed off along the line UY at the top may be glued on at Z T of the 
 lower end L. 
 
 By this method of managing a wreath in one piece, where the plan is arranged with the riser 
 set into the cylinder as at Plate 34, the only difference will be that the plank will have more cant 
 or inclination; but as a wreath it will work and answer equally as well. 
 
 Side Moulds. — For a better understanding of this case of hand-railing it will be useful to 
 unfold the side moulds, f as follows: — Let F N of Fig. i, equal S C of Fig. 2; connect N R; divide the 
 centre line P6 0 into, say six equal parts, and draw radial lines through each of these six divisions 
 as shown; again, through the same divisions draw lines parallel to 0 N as 3, 3; X, 5; 6, 6, etc. 
 
 To Unfold the Convex Side Mould: — Fig. 3. On a line FA mark the six divisions taken 
 around BA of Fig. 1; take the heights F N, 3, 3, etc., of Fig. i and place them at the corresponding 
 letters and figures of Fig. 3; through the points thus found trace the dotted lines which unfold the 
 natural form of the side mould; but this form and joint requires to be forced, — changed some to 
 make it properly connect with the hand-rail of the flight. Let A B equal the rise and B C the tread; 
 connect C A; through E make the joint line D E 0 at right angles to A C; from D draw D X parallel to 
 A B. When this convex side mould is applied to the wreath, D X agrees with the joint Z F, and N 
 with the joint U, both of Fig. i. 
 
 Concave Side Mould: — Fig. 6. The line A F shows the six divisions taken around E S, Fig. i, 
 and the heights are the same as at Fig. 3; the joints are also the same. 
 
 WAINSCOT. 
 
 Fig. 4. Vertical Cross-section of Hall Wainscot, of which an Elevation of a Portion 
 is made at Fig. 5, as it Appears along the Wall, in Connection with the Level Wainscot 
 at the Starting of a Staircase. — In the best panel-work of hard wood, the frame is put 
 together, the mouldings glued in place, and the whole finished and varnished. Afterwards the 
 varnished panels are set in from the back and fastened at XX with three-cornered pieces of 
 stout sheet-metal driven in the frame; or in place of the metal, strips of wood aie nailed into 
 the frame and against the panels at X X. As to the height wainscot should be up the flight 
 as compared with the height of the connecting level wainscot, there is no fixed rule; but as a 
 basis, let A B of Fig. 5, the level wainscot, be three feet in height as at Fig. 4. Make a vertical 
 line along any riser of the flight C D equal three feet; then if, after measuring on the line H J — 
 which is drawn at right angles to the inclined string — the width of bottom rail E, middle rail F, 
 and top rail G, as given at the section. Fig. 4, the space left for panels is unpleasantly narrow, 
 changes may be made and the height given at C D altered to suit. 
 
 * The butt joint at L may be tested with the pitch board, when the level surface U Y of the wreath is made to rest 
 on the saw-table. f See Side Moulds, Platk No. 76. 
 
Plate No. 35 
 
 Fig. 4. 
 
 Scale V4 In. = 1 Ft. 
 
PLATE 36. 
 
 Fig. I, Plan of a Winding Stairs Turning One Quarter at or about the Middle of the 
 Flight. — Tliis plan is given at Platk No. 5, Fig. ii. After making the plan, describe the 
 centre line of the rail Q H, the plan tangents Q C and C H. Place the centres of balusters as 
 required. Before the plan can be completely prepared to draw a parallel pattern or a face- 
 mould, the elevation must be drawn. 
 
 Fig. 2. Elevation of Plan as given at Fig. I.— Set up the treads and rises as figured and 
 given at the plan according to the scale. The treads in the cylinder must be measured on the 
 centre line of the rail, and each tread taken in two parts for the purpose of getting more accu- 
 rately the stretch-out of the centre line. Place the centres of balusters on each tread as shown 
 on the plan, and except at the centres of the short balusters 0, draw lines parallel to the rise- 
 lines, and indefinitely. At the upper portion of the elevation through the centres of the short 
 balusters 0 0 draw the bottom line of the rail, and place the centre line N C in position par- 
 allel to 0 0, but indefinite in length. Anywhere along the upper chord-line set of? the length 
 of plan tangent H C of Fig. i, and draw the line C M parallel to the chord-line; and where the 
 centre line N A intersects at C, draw the line C D at right angles to the chord-line. Anywhere 
 along the lower chord-line P B set off the length of the plan tangent Q C of Fig. i and draw 
 the line F E. E is a fixed point from which the line E L may be drawn to suit its position over 
 the winders and the requirements of the ramp. Wherever the line E L intersects the chord- 
 line P B as at B, draw the line B F at right angles to the chord-line. Make A J equal four 
 inches for straight wood on the upper end of the wreath-piece; and make B G also four inches 
 for straight wood on the lower end. The ramp is curved as shown. Again at Fig. 1 make 
 C E at right angles to Q C and equal to F E of Fig. 2; connect E Q; make H D at right 
 angles to C H and equal to D A of Fig. 2; connect D C. 
 
 To Find the Directing Level Line: — Make CG equal H D; make G F parallel to C Q, 
 and F 0 parallel to C E; connect 0 N, which is the level line sought. Parallel to N 0 draw 
 IZ, UO, 2T, CM and 3 W; parallel to C E draw TX and Z Y; parallel to H D draw W K; 
 from H at right angles to NO draw H A indefinitely; from Q at right angles to N 0 draw 
 Q B indefinitely; on C as centre with CD as radius describe the arc D A; again on C as 
 centre with E Q as radius describe an arc at B; connect B A. 
 
 To Find the Angle with which to Square the Wreath-piece at its Joint over H: — 
 Make H L equal H J; connect L M; then the bevel at L will give the angle required. 
 
 To Find the Angle with which to Square the Wreath-piece at its Joint over Q:— Pro- 
 long H N to R indefinitely; make N R equal 0 P; connect R Q; then the bevel at R will give the 
 angle sought. 
 
 To Develop the Centre Line of the Wreath-piece in Position over the Elevation, 
 Fig. 2. — Make Z Y, T X and W K equal the heights at the corresponding letters of Fig. i; then 
 through B Y X K A trace the centre line of the wreath. Set off half the thickness of rail each 
 side of the centre as shown by the dotted lines. 
 
 To Find the Length of Balusters: — Take for exam.ple the one marked 3; 3 V measures 
 5:^", which, added to 2'. 2", the length of short baluster at 0, equals 2'.-jY between the top of 
 the step 3 and the bottom of the rail V at the centre of the baluster. 
 
 Fig. 3. Parallel Pattern for the Wreath-piece over the Plan, Fig. i.— Make the line 
 B D equal B A of Fig. i; make D S equal A S of Fig. i; on D as centre with C D of Fig. i as 
 radius describe an arc at C; and on B as centre with Q E of Fig. i as radius intersect the arc 
 at C; also on S as centre with S C of Fig. i as radius test the intersection of the arcs at C; con- 
 nect D C, C B and S C; prolong C B to G and CD to J ; make B G equal B G of Fig. 2; make 
 D J equal A J of FiG. 2; make B Y F X equal Q Y F X of Fig. i; make C K equal C K of Fig. i; 
 parallel to the level line C S draw K 3, X 2, F U and Y I; make Y I, F U, X 2 and C V equal Z I. 
 0 U, T 2 and C V of Fig. i; make K 3 equal W 3 of Fig. i. The joints J and G are made at 
 right angles to the tangents. 
 
 At a trial — laying out the squaring of the wreath-piece at joint J with the bevel L of Fig. i, 
 also joint G with bevel R of Fig. i — it is found that the position of the form of rail at 
 joint J takes the greatest width of stuff, equal to 5", therefore, with 2^" radius describe circles 
 on the centres GB1 U2V3DJ, and trace lines touching the circles to complete the pattern. 
 
 Development of the centre line of wreath-piece in cases of this kind is given in detail at 
 Plate No, 20 by the quarter-circle Q V of Fig. 3. Face-mould and parallel pattern of this 
 character is treated in detail at Plate No. 12. 
 
PLATE 
 
 Fig. I. Plan of a Half-turn Platform Stairs with the Opening between the Strings— 
 Usually Built to Connect in the Form of a Semicircle — Composed of Two Quarter-circles 
 with Straight between. — Plans of platform stairs differently treated are given at Plates 6 
 and 7. The situation of the risers in connection with the chord lines is, in this case, determined 
 by trial through the elevations of tread and rise set up at Figs. 2 and 3. 
 
 Fig. 2. — Let the bottom line of hand-rail pass through X X, the centres of short balusters; 
 the thickness of rail is set off parallel to X X by the line E D; the line C B is the centre of 
 a four-inch plank, from which the wreath-piece is to be worked out. A J is four inches, which 
 the rail is to rise above the floor more than the height to be raised at XX; J B is half the 
 thickness of rail, and the height of B, touching the centre line C B, determines the exact 
 position of B to the riser H; and at the plan Fig. i A H is made to equal A H of Fig. 2. 
 This explanation and the corresponding letters will sen-e Fig. 3. At Fig. i, to prepare for drawing 
 the face-mould, place the pitch-board as shown, marking the line of hypothenuse W M; prolong 
 U A to M; parallel to U M draw T N and V 0. 
 
 Fig. 4. Face-mould from Plan Fig. i; also the Squaring of Wreath-piece at Both 
 Joints. — Draw U A and A K at right angles; make A I LU equal A! LU of Fig. i; make A N OW 
 equal M N 0 W of F"ig. i; make W K four inches for straight wood; through N, 0, W, K, parallel 
 to U A, draw Y Z, IV, F E and B C; make W F and W E each equal the same at Fig i; make 
 0 1 and 0 V equal PI and P V of Fig. i; make NY equal S Y of Fig. i; parallel to W K 
 draw F B and E C; make L J equal L T; parallel to L U draw J R; through U draw R Z 
 parallel to AW; through F 1 Y I J of the convex and E V T of the concave trace the edges of 
 the face-mould. The squaring of the centre joint U by the use of the pitch-board, as shown, 
 is a sufficient explanation. At joint K the sides of the rail are at right angles to the plane 
 of the plank; D H S S is the over-wood to be cut away at the bottom of the lower wreath- 
 piece, and at the top of the upper one. 
 
 Fig. 5. Plan, as given at Plate 5, Fig. 10, of a Quarter-platform Stairs turning 
 One Quarter with a Quarter-cylinder. — Draw the plan of rail, also the centre line, and upon 
 tlie latter space the balusters, as required. Make the tangents to the centre line of the rail, 
 B F and B 0, and let the distance from the angle B, both ways to each riser, equal half a 
 tr.;ad — 4^" from B to S and 4^" from B to 1 . By this arrangement there is between the two 
 risers one tread, which brings the wreath-piece on a common inclination with the flight, and 
 makes the best possible shape of it. To further prepare the plan for drawing the face-mould, 
 place the tread of the pitch-board on the tangent B 0 and mark the line B Q; prolong the 
 tangent 0 B to C; prolong A 0 to Q. A line connecting A B will be the level line. Parallel 
 to A B draw R L and H G; parallel to 0 Q draw X N and 1 M; parallel to B C draw G E; 
 from F through 0 draw F P; on B as centre with B Q for radius describe the arc Q P. 
 
 To Find the Angle with which to Square Both Joints of the Wreath-piece: — On B 
 as centre describe an arc touciiing the line C F and D; connect D F: then the bevel at D 
 will give the angle required. 
 
 Fig. 6. Elevation of Tread and Rise, including the Platform, as given at Plan and 
 as Figured. — The platform is measured at the plan on the centre line in two parts from riser 
 to riser. The heights and inclinations, the rail above and below, the joints Y and Y, are all 
 shown in position; also the development of the centre line of wreath. The letters correspond 
 wiih the plan Fig. 5 and with those at the joints of the face-mould. The length of baluster 
 is here determined as before explained. 
 
 Fig. 7. Face-mould over Plan Fig. 5 ; also Showing the Squaring of the Wreath- 
 piece at the Joints. — Draw the line P F indefinitely; make K P and K F each equal K P of 
 Fig. 5; from K at right angles to F P draw K C; make K C equal K B of Fig. 5; connect 
 C P and C F; prolong C F and C P to Y; make P Y and F Y equal the same ai Fig. 6; make 
 the joints Y, Y at right angles to the tangents; make F M and P M each equal Q M of Fig. 5; 
 through M and M parallel to C K draw L R, L R; make M L and M R equal 1 L and 1 R of 
 Fig. 5; make CZ equal B Z of Fig. 5; through F and through P draw R V; make FV equal 
 F R, and P V equal P R; parallel to C Y draw V X and R X at each end; through V L Z L V of 
 the convex and R K R of the concave trace the edges of the face-mould. The angle for 
 squaring the wreath-piece at both joints is given by the bevel D, Fig. 5. The face-mould, 
 Fig. 7, is treated in detail at Plate No. ii, and face mould, Fig. 4, is also treated in detail at 
 Plate No. 10. 
 
Plate No. 37 
 
Plate No. 38 
 
 P LATFO R M 
 
 I 
 
PLATE 38. 
 
 Fig. I. Plan of a Platform Stairs with the Risers at Platform set in the Whole 
 Depth of the Cylinder. — Draw the centre line of rail and space the balusters as required; 
 also draw O A and A Q, F G and G C tangents to the centre line of rail at each quarter- 
 circle. To prepare tlie plan for measurements that will develop the centre line of wreath, or 
 to draw the face-mould, tlie elevation must first be set up. 
 
 Fig. 2. Elevation of Treads and Rises as given at the Plan and as Figured ; also 
 Development of the Centre Line of Wreath, — Let tlie bottom line of rail above and below 
 the platform pass through X X, the centres of short balusters. Place the chord-lines — of which 
 there are four — as given on the plan. Draw the centre line of rail L B and G R parallel to 
 X X; at right angles to the chord-line D 0 draw 0 A equal to the first tangent 0 A of Fig. i. 
 Parallel to D 0 draw A B; at right angles to A B, touching B, draw Q T; make QT equal the 
 second tangent A Q of FiG. i; make C G at right angles to C M and equal to the tangent 
 G C of Fig. i; make FE equal FG of Fig. i, and draw the line EZ parallel to J F; from Z draw the 
 line ZT; where the inclined line ZT intersects the chord-line J F at F, draw F E at right ang'es to 
 J F. Place the balusters numbered 1, 2, 3, 4, 5, 6 as on the plan, and draw a line through the place 
 of each baluster parallel to the rise-line and indefinitely. 
 
 To Prepare the Plan, Fig. i, for Finding the Lengths of Balusters :— Make C M G 
 equal the same of Fig. 2; make G Z F equal E Z F of Fig. 2; make Q N A equal Q N f of 
 Fig. 2; make ABO equal the same at Fig. 2; make A J equal QN; make JY parallel to 
 A 0, and Y 8 parallel to B A; connect 8 R; parallel to 8 R draw 2 V; parallel to Q N draw 
 V W: make M H equal G Z; draw H E parallel to C G, and ET parallel to C M; connect T S; 
 parallel to S T draw 5 K and 4, 1; parallel to C M draw K U; parallel to G Z draw I P. The 
 heights for the balusters are taken as numbered and lettered, and set up at the elevation, Fig. 2, 
 as designated by the same numbers and letters; then tb'-'^ugh these letters— M U P F N W Y 0— 
 trace the centre line of the wreath-pieces. Set off each side of the centre half the thickness of 
 rail as shown by the dotted lines; next, ge( the length of balusters: take for example baluster 
 4— this measures from the platform to the bottom of the rail, 4!", which, added to 2'.2" — the 
 length of short balusters at X X — equals 2'.6f" from the top of the platform to the bottom of the 
 wreath along the centre line of baluster. 
 
 Fig. 3. Plan of the Lower Quarter Wreath-piece.— The heights A B and Q N are taken 
 from those lettered the same at Fig. 2. Make A F equal Q N; draw F E parallel to AO, and E D 
 parallel to B A; connect C D, which is the directing level line; parallel to C D draw V W, A K, R S 
 and Q Y; at right angles to the level line C D from 0 and Q draw 0 M; also Q P, each indefi- 
 nitely. On A as centre with B 0 as radius describe an arc at M; again on A as centre with A N 
 as radius describe the arc N P, connect P M. 
 
 To Find the Angle with which to Square the Wreath-piece at the Joint over 0:— 
 Prolong Q C to H; make C H equal D I; connect H 0; then the bevel at H will give the angle 
 required. 
 
 To Find the Angle with which to Square the Wreath-piece at the Joint over Q :— 
 
 Prolong A Q to L; make Q L equal Q 8; connect L K; then the bevel at L contains the angle 
 sought. 
 
 Fig. 4. Face-mould taken from Plan of Quarter-circle Fig. 3, also Showing the 
 Squaring of the Wreath-piece at Both Joints.— Make 0 N J equal M J P of Fig. 3. On N 
 
 as centre with N A of Fig. 3 as radius describe an arc at B; on 0 as centre with 0 B of Fig. 3 as 
 radius describe an intersecting arc at B; on J as centre test the intersection at B with J A of Fig. 3 
 as radius;* connect N B, 0 B and J B; prolong 0 L equal to 0 L of Fig. 2; and prolong B N 
 equal to N S of Fig 2. Make N T equal N T of Fig. 3; make 0 X E equal 0 X E of Fig. 3; 
 parallel to the level line J B draw N Y, T R , E G U and X V; make N Y equal Q Y of Fig. 3; 
 make T R and B 7 Z equal S R and A 7 Z of Fig. 3; make E G, E U equal D G, D U of Fig. 3; 
 make X V equal W V of Fig, 3; through N draw R F; make N F equal N R; make the joints S and 
 L at right angles to the tangents; from R and F draw lines to the joint parallel to N S; through 0 
 draw V D; make 0 D equal 0 V; through F Y 7 G D of the convex and R Z U V of the concave 
 trace the curved edges of the face-mould. The slide-line is made at right angles to the level 
 line J B. The angle for squaring the wreath-piece at joint L is taken by the bevel at H of 
 Fig 3, and that for the centre joint S is taken by the bevel at L of Fig. 3. This face-mould 
 is treated in detail at Plate No. 12. The development of the centre line of wreath-piece in 
 a case of this kind is shown in detail at Plate No. 20, plan of quarter-circle R Q V, and 
 Fig. 4, Q to Y. 
 
 * Instead of using the length of the level line J A of Fig. 3, as at J B of Fig. 4, as a test, it may be used in any 
 case, together with the length of either one of the tangents, to establish the angle of tangents as at B. 
 
PLATE 
 
 39 
 
 Fiff. I. Plan of Stairs with Two Connecting Platforms Divided by a Riser, set in 
 ^he Centre of the CyHnder in a Direction Parallel to the Strings; also with the Risers 
 Landing on and Starting from the Platforms, Each set into the Cylinder Three 
 Inches.* — Set off the centre line of rail and space the balusters as required; also draw the 
 ti'.norents 0 C, C 2 and A F, F H to the centre line of the rail at each quarter-circle; then to 
 find the angles of inclination for the tangents it is necessary first to set up the elevati(m. 
 
 Fig. 2. Elevation of Treads and Risers as given at the Plan and as Figured; 
 also the Development of the Centre Line of Wreath-pieces. — Let the bottom lines of rail 
 above and below the platforms pass through X X, the centres of short balusters. Place the 
 chord-lines, of which there are four, as given on the plan; draw the centre line of rail R E 
 and T G parallel to X X; at right angles to the first chord-line W D draw D C equal to the 
 first tangent D C of Fig. i; parallel to W D draw C E; touching E, at right angles to 2 B, 
 draw M Q equal to the second tangent 2 C of Fig. i: then Q becomes a fixed point. At 
 the uppermost chord draw H G at the intersection of the centre line G T, equal to the fourth 
 tangent H F of Fig. i; from the third chord-line, Y A, make AF equal F A of Fig. i; make 
 F N parallel to Y A: then N becomes the next higher fixed point. Connect the two fixed 
 points N and Q; where the inclined line N Q intersects the chord-line at A draw A F at right 
 angles to Y A; divide the straight between the two quarters B A in two equal parts at S: 
 then S will be at the centre joint of wreath-pieces. Place the balusters numbered 1, 2, 3, 4, 5 
 as on the plan, and draw a line through the place of each baluster parallel to the rise-lines 
 indefinitely. 
 
 To Prepare the Plan Fig. i for Finding the Length of Balusters : — As the angles 
 of inclination in this case happen to be all alike, the angle DEC of Fig. 2 may be set in 
 place over each tangent, as D E C, C B 2, A N F and H J F; connect Q F, also G C. From the 
 centre of baluster 5 draw 5 M parallel to H J ; from the centre of baluster 4 draw 4 K parallel 
 to Q F; parallel to F N draw K L; parallel to G C from the centre of baluster 1 draw 1, 0; 
 parallel to C E draw 0 P; at Fig. 2 make 0 P equal 0 P of Fig. i; make K L and U V of 
 Fig. 2 equal K L and 5 M of Fig. i; then through D P B, A LVJ of Fig. 2 trace the centre 
 line of wreath-pieces; set off each side of the centre line half the thickness of rail as shown 
 by the- dotted lines. 
 
 To Find the Lengths of Balusters: — Take for example No. 2 baluster, where 2Z equals 
 4^", which added to 2'.2", the length of short balusters at X X, makes the length of that 
 baluster on the line of its centre from the top of step to the bottom of rail 2'. 63". 
 
 Fig. 3. Plan of the Lower Quarter-circle with Tangents and Angles of Inclination 
 Lettered Alike and as taken from Fig. l. — From K draw K M parallel to G C; from L draw 
 LH parallel to 2 B; through D and 2 draw DA indefinitely; on C as centre with C B as radius 
 describe the arc B A. 
 
 To Find the Angle with which to Square the Wreath-piece at Both Joints :— 
 
 Prolong C 2 to F indefinitely; on 2 as centre describe an arc touching the inclined line C B and 
 at F; connect FG: then the bevel at F contains the angle required. 
 
 Fig. 4. Face-mould from Plan Fig. 3 ; also Squaring of the Wreath-piece at Both 
 Joints. — Draw the line A A indefinitely; let J A and J A each equal J A of Fig. 3; make J C at 
 right angles to J A and equal to J C of Fig. 3; connect OA and OA; prolong C A to S and CA 
 to R, each indefinitely; make CH and CH each equal C H of Fig. 3; make AR equal D R or 
 J T of Fig. 2, for straight wood; make AS equal B S or A S of Fig. 2, this being one half the 
 straight between the quarter-circles of which this cylinder is composed; through H and H 
 parallel to the level line JO draw M K, M K; make H K and H M equal LK and L M of Fig 3; 
 make J ! and J N each equal J I and J N of Fig. 3; through A at both ends draw K W; make 
 A W, A W each equal A K; the joints S and R are made at right angles to the tangents; from 
 K and W draw lines parallel to the tangents, touching the joints; through W M C M W of the 
 c onvex and K N K of the concave trace the curved edges of the face-mould The angle with 
 which to square the wreath-piece at the centre joint S and joint R is contained in the bevel 
 at F of Fig. 3. The development of the centre line is explained in detail at Plate No. 20, 
 Fig. I, quarter-circle A C, and A Y E of Fig. 2. This case of face-mould, Fig. 4, is given in 
 detail at Plate No. 11. This plan of stairs is given at Plate No. 7, Fig. 2. 
 
 * A riser is calculated as set in a cylinder three inches more or less from the chord-line to the face of the regular 
 tread, and not from the chord-line to any point of the cylinder to which the riser may be curved. 
 
Plate No. 40 
 
 Scale 1^2 U = 1 Ft. 
 
PLATE 40. 
 
 Fig. I. Plan of Half-turn Platform Stairs with 15" Cylinder, the Risers Landing on 
 and Starting from the Platform set in the Cylinder 7^", its Whole Depth. — This plan is 
 given at Plate No. 7, Fig. i. Describe the centre line of rail, and draw the tangents A E. 
 E X. Space the balusters as required. Before proceeding further in the preparation of this plan 
 for drawing the required face-mould it is necessary to set up the elevation. 
 
 Fig. 2. Elevation of Tread and Rise as given at the Plan and as Figured. — Place 
 the chord-lines, of which there are two, in position as shown on the plan Fio. i. Let the 
 bottom line of rail pass through the centres of short balusters at 0 0, and draw the centre 
 line of rail D C and G K each parallel to 0 0. From the first chord-line A make A E equal 
 A E of Fig. i; parallel to a rise-line draw EC; at A draw A E at right angles to the chord- 
 line; make J G equal A E; draw G X parallel to the rise-line; from G draw G J at right 
 angles to G X; from C draw C X parallel to the line of platform; divide G X at F in two equal 
 parts. 
 
 To Prepare the Plan Fig. i as Required to Measure for Drawing a Face-mould. — 
 
 Prolong X E to N and to C; make E C equal E C of Fig. 2; connect C A; prolong J X to F 
 and to I; make X F equal X F of Fig. 2; connect F E; make E D equal X F; draw D B parallel 
 to E A; make B K parallel to E C; connect K J, which is the level line; parallel to J K draw 
 
 Y 2; parallel to J A draw 2 W; parallel to J K draw E M, U V and X S; parallel to J X draw 
 
 V R; at right angles to J X draw A H and X Z, both indefinitely; on E as centre with E F as 
 radius describe the arc F Z; again, on E as centre with C A as radius describe an arc at H; 
 connect H Z. 
 
 To Find the Angle with which to Square the Wreath-piece at the Joint over X: — 
 
 Make X N equal XG; connect N M; then the bevel at N will contain the angle required. 
 
 To Find the Angle with which to Square the Wreath-piece at the Joint over A: — 
 Make J I equal K Q; connect I A: then the bevel at 1 will contain the angle sought. 
 
 Fig. 3. Face-mould from Plan Fig. i, also Showing the Squaring of the Wreath- 
 piece at Both Joints. — Draw the line Z H; make Z 0 and 0 H equal the same at Fig. i; 
 on Z as centre with F E of Fig. i as radius describe an arc at C; on H as centre with A C 
 of Fig. I as radius intersect the arc at C; on 0 as centre with 0 E of Fig. i as radius test 
 the intersection of the arcs at C; connect Z C, H C and 0 C; prolong C H to D; make H D 
 equal D A or H K of Fig. 2; make the joints D and Z at right angles to the tangents; make 
 H W B equal A W B of Fig. i; make C R equal E R of Fig. i; parallel to 0 C draw Z S, R U, 
 and through B, L 4 and W Y; make WY equal 2 Y of Fig. i; make B L and B4 equal K L 
 and K 4 of Fig. i; make C P equal E P of Fig. i; make R U equal V U of Fig. i; make 
 ZS equal X S of Fig. i. Through H draw Y T; make HT equal YH; through Z draw UJ; 
 make ZJ equal Z U; draw lines from T and Y parallel to the tangent touching joint D. 
 Through T L P S J of the convex and Y 4 0 U of the concave trace the curved edges of the" 
 face-mould. The slide-line is at right angles to the level line 0 C. The angle for squaring 
 the joint Z is taken by the bevel N of Fig. i, and for squaring joint D by the bevel I of 
 Fig. I. 
 
 Fig. 4. Elevation of Step and Rises at the Starting, Set up for the Purpose of Find- 
 ing what Position the Bottom Riser should take next to the Chord-line of the Cylinder 
 when the \A/'reath-piece is Treated in the Simplest Manner, with the Over-wood all to be 
 Removed from the Top. — Let the bottom line of rail pass through the centres of short bal- 
 usters X X. Set off the thickness of rail X E, and draw E H parallel to X X; let X D be the 
 thickness of plank, out of which the wreath-piece is to be worked. Make the line A F pass 
 through the centre of X D and parallel to XX; make B C four inches and C A half the thickness 
 of rail. The intersection of the line A J at its given height with the centre line F A at A fixes A 
 as the centre of the rail at the centre of the plank, and the distance from A to the chord-line J 
 must be 8|-", equal to J A of Fig. i, or D K of Fig. 5. 
 
 Fig. 5. Plan of Cylinder Connecting Step and Hand-rail at the Bottom of this Flight 
 of Stairs.— Let the chord line H of the cylinder be set at the same distance from the bottom 
 riser as shown at the elevation Fig. 4. 
 
 To Prepare the Plan for Drawing a Face-mould: — Draw the tangents H J and J K; 
 from the centre D describe the plan of rail. Set the pitch-board with the tread on the line 
 H J, and mark the pitch-line J Q; prolong D H to Q; parallel to K J draw L N, V 0 and U P 
 
 Fig. 6. Face-mould from Plan Fig. 5, also Squaring the Wreath-piece at Both 
 Joints. — Draw Y J, J K at right angles; make J N 0 PQ equal the same at Fig. 5; make J K 
 equal J K of Fig. 5. Through K parallel to J Y draw X L; through N 0 P Q Y draw lines 
 parallel to J K indefinitely; make K X equal K L; make N M, 0 W, P S and Q R equal Y M, XW, Z S and 
 HR of Fig. 5. Make 0 V,' P U and QT equal X V, Z U and H T of Fig. 5; draw lines from T and R 
 to the joint Y parallel to QY; through X I M W S R of the convex and LVUT of the concave trace the 
 curved edges of the face-mould. The wreath-piece at the joint K is squared by the use of the 
 pitch-board as shown. The joint at Y is square, and E D is the over-wood as shown at E D 
 of Fig. 4. Face-mould Fig. 6 is treated in detail at. Plate No. 10. Face-mould Fig. 3 is explained 
 in detail at Plate No. 12. llie dcveloptncnt of the centre line at FiG. 2 of ivreath-pieces is given 
 in detail at Plate No. 20, Fig. 3, quarter-circle Q V, and Fig. 4, Q Y, 
 
PLATE 41. 
 
 Pig. I, Plan of Stairs with Two Quarter Platforms and a Tread between, All 
 Connected with, and Dividing Equally, a lo" Cylinder. — This plan of stairs is given at 
 Pl.\te No. 5, Fig. 9. Describe the centre line of rail, and space the balusters as required. 
 Draw the tangents A B, BE, EG and G J to the centre line of rail A E J To find the angles 
 of inclination of these tangents, and other points of measurement by which *o get the lengths 
 of balusters, it is necessary to set up an elevation. 
 
 Fig. 2. Elevation as given at Plan Fig. i and as Figured.— Measure the three treads 
 in the cylinder on the centre hne, and, as before explained, of treads situated .^n curves. Let 
 the bottom line ot rail pass through the centres of short balusters at X, 1 below and X X 
 above; draw the centre line of rail LC parallel to X, 1, and the centre line FG paiallel to 
 X X; make A B and J G equal A B and J G of Fig. i. From A at right angks to the chord- 
 line draw A B, and from G at right angles to the chord-line draw G J; from C at right 
 angles to H E draw C E; divide E J at D in two equal parts. Place the centres of balusters 
 1, 2, 3 as at the plan, and draw lines through each parallel to the rise-lines and indefinitely. 
 
 At Fig. I, to further Prepare for the Development of the Centre Line of Wreath 
 and for the Lengths of Balusters: — Make the angles A C B and J H G equal A C B and J H G 
 of Fig. 2; prolong J G to F, and K E to D; make E D and G F each equal E D of Fig. 2; 
 connect D B; connect F E; make G U equal J H; draw U X parallel to G E; make X W parallel 
 to G F; connect W K, which is the directing level line for the quarter-circle E J; make D N 
 equal B C; draw N L parallel to E B, and LO parallel lo E D; connect 0 K: then 0 K will 
 be tlie directing level line for the quarter-circle A E; from the centre of baluster 1 draw 1 S 
 parallel to K 0; parallel to B C draw ST; from the centre of baluster 2 draw 2V parallel 
 to K 0; make V M parallel to E D; through the centre of baluster 3 draw 3 Z parallel to K W; 
 make Z Y parallel to G F. At Fig. 2 make S T equal S T of Fig. 1, and V M equal V M of 
 Fig. i; make ZY equal Z Y of Fig. i; through ATMYH trace the centre line of wreath, 
 and set off each side of the centre half the thickness of rail as shown by the dotted lines. Of 
 the three balusters around this cylinder, only one. No. 3, will require a half inch more length 
 than usual for short balusters. A L and H F are 4" straight wood to be left on the wreath- 
 pieces above and below the cylinder. 
 
 Fig. 3. Plan of Rail and the Centre Line of the First Quarter AE with the Tangent 
 A B and B E, and the Angles of Inclination ACS and B D E, from Fig. i.— Make D N equ£ 
 BC; draw N L parallel to E B, and LO paralh-l to ED; connect OK, the directing level line 
 parallel to 0 K draw 4, I, B U, W X and AH; parallel to B C draw XG; parallel to D E draw 
 Y M; at right angles to 0 K draw AQ indefinitely; and again at right angles to 0 K draw E R 
 indefinitely; on B as centre with B D as radius describe the arc D R; on B again as centre with 
 C A as radius describe an arc at Q; connect Q R. 
 
 To Find the Angle with which to Square the Wreath-piece at the Joint over E: — 
 Prohmg B E to V; make EV equal N P; connect V K: then the bevel at V will contain the angle 
 required. 
 
 To Find the Angle with which to Square the Wreai;h-piece at the Joint over A: — 
 
 At U draw U T at right angles to KA; make UT equal B F; connect T A; then the bevel at T 
 will contain the angle sought. 
 
 Fig, 4. Face-mould from Plan Fig. 3; also Squaring the Wreath-piece at Both 
 Joints. — Draw the line Q R; make RS and SQ equal the same at Fig. 3. On Q as centre with 
 A C of Fig. 3 as radius describe an arc at B; on R as centre with D B of Fig. 3 as radius intersect 
 the arc at B; from S test the intersections at B by S B of Fig. 3; connect R B, Q B and S B; 
 prolong B Q to C; make QC equal A L of Fig. 2; make the joint C and R at right angles to 
 the tangents; make QG equal A G of Fig. 3; make R M L equal D M L of Fig. 3; parallel to S B 
 draw Q H, G W, L J and 4, I; make M I and M 4 equal Y I and Y 4 of Fig. 3; make L J and B Z Y 
 equal 0 J and B Z Y of Fig. 3; make G W and Q H equal X W and A H of Fig. 3; through R 
 draw 4 P; make R P equal R 4; through Q draw W F; make Q F equal Q W; from F and W 
 draw lines to joint C parallel to B C; through F H Z I P of the convex and W Y J 4 of the 
 concave trace the curved edges of the face-mould. The slide-line is drawn at right angles to 
 the level directing line S B. Joint R of the wreath-piece is squared by the bevel at V of 
 Fig. 3, and joint C by the bevel at T of Fig. 3. The face-mould is explained in detail at Plate 
 No. 12. The development of th< centre line of the wreath-piece is explained at Plate No. 20: 
 the quarter-circle Q V of Fig. 3, 'ind Q Y of Fig. 4. 
 
Plate No. 41 
 
Plate No. 42 
 
PLATE 42. 
 
 Fig. I. Plan of Landing and Starting Two Flights, Both in Connection with a 12" 
 Cylinder. — This plan is given at Plate No, 6, Fig. 3. The centre line of tlie rail may be 
 described, the balusters spaced, and the tangents to the centre line drawn; then, to find the 
 angle of inclination of these tangents, an elevation of the plan must be made. 
 
 Fig. 2. Elevation of Plan as given at Fig. i and as Figured ; also Development 
 of the Centre Line of Wreath. — In setting up the elevation, the treads in the cylinder must 
 be measured between chord-lines on the centre line of rail, each in two parts, so as to get 
 practically near enough to the stretch-out of the circle. Draw the chord-lines in the position 
 given on the plan, and parallel to the rise-lines. Place the centre of balusters as given on 
 each step of the plan, and through these centres draw lines parallel to the rise-lines indefinitely. 
 Let the bottom line of rail above and below each pass through the centres of short balusters 
 X X; draw the centre line of rail at I G and T B each parallel to X X. Make the distance 
 from the chord-line AW to BC equal tiie tangent A C of Fig. i; draw BC parallel to AW; 
 at A draw A C at right angles to A W; make the distance from the chord-line J S to G equal 
 the tangent H G of Fig. i; draw G F parallel to J S; at G draw G H at right angles to J S; 
 from B draw B F parallel to the line of floor; divide G F in two equal parts at D; transfer 
 the angle ABC to ABC of Fig. i. The three remaining tangents all happen in this case to 
 have the same angles of inclination,* so that the angle H J G may be placed at E D C, G F E 
 and H J G, Fig. i. Make D H of Fig. i equal C B; draw H J parallel to E C; make J V parallel 
 to E D; connect V K, which is the directing level line for the quarter E A. The directing 
 level line for the quarter E H — that quarter having a common angle of inclination — is a line 
 drawn from G to K. From the centie of baluster 1 draw 1 L parallel to C B; from the centre 
 of baluster 2, parallel to K V, draw 2,0; parallel to E D draw OK and 3 M; from balusters 
 
 4 and 5 draw 4N and 5Q parallel to KG; parallel to G F draw N P; jiarallel to H J draw 
 Q R. At Fig. 2 make Z R over baluster 5 equal Q R of Fig. i. Make N P over baluster 4 
 equal N P of Fig. i; make E M over baluster 3 equal 3 M of Fig i; make B 0 equal 0 K 
 of Fig. i; parallel to B F draw 0 K; make V L over baluster 1 equal 1 L at Fig. i; through 
 J R P M K LA trace the centre line of wreath; set off each side of the centre half the thickness 
 of rail as shown by the dotted lines. 
 
 To Find the Lengths of Balusters: — For example, take baluster 3; 3 U measures 61", 
 which must be added to 2'. 2", the length of short balusters X X, and this makes the length of 
 that baluster, measured along its centre from top of step to under side of rail, 2'.8f". 
 
 Fig. 3. Plan of Rail for Quarter-circle, Centre Line A E, Fig. i, with Angles of 
 Inclination— over Tangents — A B C and C D E ; also the Directing Level Line V K as 
 Transferred from Fig. i. — Parallel to K V d raw Z 5, C Q and Y W; parallel to E D draw 4 F; 
 parallel to C B draw XN; at right angles to V K draw ET and AO indefinitely; on C as 
 centre with C D as radius describe the arc D T; again on C as centre with B A as radius 
 describe an arc at 0; connect 0 T. 
 
 To Find the Angle with which to Square the Wreath-piece at the Joint over E: — 
 Prolong C E to S; make E S equal H G; connect S K: then the bevel at S contains the angle 
 required. 
 
 To Find the Angle with which to Square the Wreath-piece at the Joint over A;— 
 
 Froin Q draw Q P at right angles to K A; make Q P equal C R; connect P A: then the bevel 
 at P contains the angle sougiit. 
 
 Fig. 4. Face-mould from Plan Fig. 3; also Showing the Squaring of Wreath-piece 
 at Both Joints. — Draw the line A T indefinitely; make T U and U A equal T U and U 0 ot 
 Fk;. 3. On T as centre with C D of Fig. 3 as radius describe an arc at B; on A as centre with 
 A B of Fig. 3 as radius intersect the arc at B; with U C of Fig. 3 test the intersection of the arcs 
 at U B; connect T B, B A and U B; make B F equal C F of Fig. 3; make B N equal B N of Fig. 3; 
 parallel to U B through F and N draw 5 Z and W Y; make F Z and F 5 equal 4 Z and 4, 5 of 
 Fig. 3; make B I U equal C I U of Fig. 3; make N Y and N W equal X Y, X W of Fig. 3; through 
 A drawYC; make AC equal AY; through T draw ZG; make TG equal TZ; make A J for straight 
 wood equal A T of Fig. 2; make the joints J and T at right angles to the tangents; from Y and C 
 draw lines to joint J parallel to B J ; through G 5 I W C of the convex and Z U Y of the concave 
 trace the curved edges of the face-mould. The wreath-piece is squared at joint J by the bevel 
 at P of Fig. 3, and joint T is squared by the bevel at S of Fig. 3. The slide-line is drawn at 
 right angles to the level line U B. A face-mould of this kind is explained at Plate No. 12, 
 and the development of the centre line of such a wreath-piece as this is given at Plate No. 20; 
 quarter-circle Q V of Fig. 3, and Q Y of Fig. 4. 
 
 Fig. 5. Plan of Rail for Quarter-circle E H of Fig. i, with Angles of Inclination over 
 Tangents E F G and G J H, as Transferred from E H of Fig. i. — Connect G K; parallel to 
 G K draw L Y; parallel to G F draw Z D; through H and E draw H W indefinitely; on G as centre 
 with E F as radius describe an arc at W. 
 
 To Find the Angle with which to Square the Wreath-piece at Both Joints : — Prolong 
 G H to N; make H N equal H R; connect N K: then the bevel at N will contain the angle 
 required. 
 
 Fig. 6. Face-mould from Plan Fig. 5 ; also Showing the Squaring of the Wreath- 
 piece at Both Joints. — Draw the line W, W; make S W, S W each equal S W of Fig. 5; draw 
 
 5 G at right angles to S W; make S G equal S G of Fig. 5; connect G W and G W; prolong 
 G W to A; make W A for straight wood equal J I of Fig. 2; make the joints A and W at right 
 angles to the tangents; make G D, G D each equal F D of Fig. 5; through D and D draw L Y and 
 L Y parallel to G S; make D Y and D L at both sides of the centre each equal Z Y and Z L 
 of Fig. 5; through W and W draw L M and L M; make W M equal W L; make G T equal 
 G T of Fig. 5; through M Y T Y M of the convex and L S L of the concave trace the curved 
 edges of the face-mould; from L and M draw lines to joint A parallel to G A. A face-mould 
 of this kind is explained in detail at Plate No. ii, and the development of the centre line 
 of wreath-piece at Plate No. 20, Fig. i, quarter-circle A C, and Fig. 2, A E. 
 
 * // the height C F had been more or less than twice the height H J, then the fuce-mtuld for the quarter-circle H E would 
 have been of tlie same kind as that of FiG. 4. 
 
PLATE 43 
 
 Fig. I. Plan of the Starting of a Staircase with One Parallel Step at the Centre of 
 a 15" Cylinder, together with a Quarter Platform. — This plan is given at Plate No. 7, 
 Fig. 9. The centre line of rail may be described, the tangents drawn, and the balusters 
 spaced as required; but to find the angle of inclination over the plan tangents, etc., the ele- 
 vation must first be set up. 
 
 Fig. 2. Elevation of Treads and Rises as given at Plan Fig. i, and as Figured. — 
 Let the bottom line of rail be drawn through the centres of short balusters X X, and draw the 
 centre line of rail A N parallel to X X. Place the chord-line H J parallel to the rise-line, and 
 in position as at the plan; make J G equal J F at Fig. i; draw F G parallel to the rise-line, 
 and at the intersection G draw G J at right angles to the rise-lines. Make F D equal F E 
 of Fig. i; draw E D parallel to the rise-line; at the intersection D draw D F at right angles 
 to the rise-line; make EC equal E C of Fig. i; make Z 1 equal Zl of Fig. i; make 1 B equal 1 B of 
 Fig. i; draw B N at right angles to the line of floor; make BK equal 4", and KN half the thickness 
 of rail; draw N E parallel to the floor-line. Place the centre of each baluster on the treads as 
 numbered, and as fixed around the centre line of rail at the plan, and draw lines through each 
 parallel to the rise-lines indefinitely. At Fig. i make the angles of inclination J H F, F G E 
 and EDO all equal J H G of Fig. 2; parallel to A D through the centre of baluster 2 draw N M; 
 from the centre of baluster 1 draw 1 L parallel to A D; draw I K parallel to A D. Connect 
 F A; from 3 parallel to A F draw 3, 0; parallel to F G draw 0 R; parallel to A F draw 4 Q; 
 parallel to J H draw Q P. Again at Fig. 2, baluster 1, make V L equal V L of FiG. i; at bal- 
 uster 2 make S M equal S M of Fig. i; at baluster 3 make 0 R equal 0 R of Fig. i; at 
 baluster 4 make Q P equal Q P of Fig. i. Through the points N L M R P H trace the centre 
 line of wi eath-piece; set off half the thickness of rail each side of the centre as shown by 
 the dotted lines. To find the length of any of these balusters proceed as before directed. 
 
 Fig. 3. Face-mould from Plan Fig. i, Quarter-circle B E, also Showing the Squaring 
 of the Wreath-piece at Both Joints. — Draw the lines D C and C H at right angles. Make 
 C D equal C D of Fig. i; make C B equal C B of Fig. i; make B H equal 3" for straight 
 wood. Make C K M equal C K M of Fig. i; draw K I parallel to C B; draw Q N through M 
 parallel to C B; draw the joint D parallel to C B; draw I T parallel to D C; make the joint 
 at H at right angles to C H; from T and I parallel to C H draw lines to the joint H; make 
 C A and K J each equal C H and T U of Fig. i; make M N and M Q each equal S N and 
 S W of Fig. i; make D Y and D X each equal E Y of Fig. i. Through T A J Q X of the con- 
 vex and I N Y of the concave trace the curved edges of the face-mould. The angle with ivhich 
 to square the jvrcath piece at Joint H is contained in the bevel at D of FiG. i. The sides of the 
 wreath-piece at joint D are at right angles to the face of plank, and the over-wood is taken 
 off both surfaces of the plank equally. Hand-rails much thicker than two thirds of their width 
 require considerably greater width and thickness of stuff to work out the wreath-piece. 
 
 Fig. 4. Plan of Quarter-circle taken from J E of Fig. i, together with Angles of 
 Inclination over the Plan Tangents Lettered Alike. — Connect F K; through E and J draw 
 E L indefinitely; on F as centre with F H as radius describe the arc H L. Parallel to F K 
 draw Y D and V B; parallel to F G draw Z X and N C. 
 
 To Find the Angle with which to Square the Wreath-piece at Both Joints : — Pro- 
 long F J to P indefinitely; make J P equal J M; connect P K: then the bevel at P will con- 
 tain the angle required. 
 
 Fig. 5. Face-mould from Plan Fig. 4; also Showing the Squaring of the Wreath- 
 piece at Both Joints, and the Additional ^Vidth Required by a Form of Hand-rail of this 
 Proportion. — Draw the line E H indefinitely; make W H and W E each equal W L of Fig. 4. At 
 right angles to E H draw WG; make W G equal W F of Fig. 4; connect G E, G H and G W; make 
 G C X, G C X equal the same of Fig. 4. Through G C X each side of the centre parallel to G W 
 draw D Y, B V; make X Y and X D at each end equal Z Y and Z D of Fig. 4; make C K V and 
 C B each side of the centre equal N K V and N B of Fig. 4; make G T S U equal F T S U of 
 Fig. 4; through H and through E draw Y 0; make E 0 equal E Y; make H 0 equal H Y; 
 make H A equal H A of Fig. 2; from 0 and Y draw lines to joint A parallel to G A; through 
 ODBTBDO of the convex and through Y V U V Y of the concave trace the curved edges 
 of the face-mould. The joints A and E of the wreath-piece are squared by the angle con- 
 tained in the bevel at P of FiG. 4. By laying out the squaring of the joint as here given, 
 the thickness and width of wood required to work out the wreath-piece are found, and with 
 half this width as radius on each of the centres E K S K H A describe arcs of circles, touch- 
 ing which the edges of a parallel pattern may be traced; or no pattern need be made, and 
 the increased width of wood required may be scribed from the edges of the face-mould on 
 the plank. The development of a centre line of wreath is given in detail at Plate No. 20, FiGS. 
 I and 2. Face-mould FiG. 3 is given in detail at Plate No. 10. Face-mould Fig. 5 is given in 
 detail at Plate No. ii. 
 
Plate No. 43 
 
 G 
 
PLATE 44. 
 
 Fig. I. A Superior Plan of Starting a Stairs making a Quarter-turn, with Parallel 
 Steps and Platform, in about the Same Space Required when Planned with Winders. — 
 
 This plan is given at Plate No. 7, Fig. 8. Describe the plan of rail and its centre line; draw 
 the tangents D B and B A, and space the balusters as required; then before proceeding further 
 the elevation must be set up. 
 
 Fig. 2. Elevation of Treads and Risers as given at Plan Fig. i and as Figured; 
 also the Development of the Centre Line of the Wreath. — Draw the treads and rises as 
 shown on the plan, taking the measure — on the centre line of the rail — of each tread in two 
 parts as before explained. Place the chord-line A F as at S on the plan Fig. i. Let the 
 bottom line of rail rest on X X, the centres of short balusters; draw the centre line of rail 
 TE indefinitely and parallel to XX; make FE equal D B of Fig. i; draw EG parallel to the 
 chord-line; make 1 S equal four inches, and S B half the thickness of rail; parallel to the floor- 
 line draw B D; make D B equal G C: then B is a fixed point, and E is also a fixed point 
 at the place where the line G E intersects the centre line of rail T E. Connect E B; and 
 where the line C D intersects the line E B at C, draw the line C G at right angles to 
 CD. 1 Z equals 1 A of Fig. i; make Z V parallel to 1 B. Place the centre of each baluster 
 on the steps in position, and number them as at the plan, drawing lines through each par- 
 allel to the rise-lines indefinitely. Let A T be three inches for straight wood to be put on 
 that end of the wreath-piece. At Fig. i make the angle D C B equal D C B of Fig. 2; 
 parallel to E C through 3 draw N G, and from H draw H F parallel to A B. 
 
 Fig. 3. Plan of Rail Quarter-circle D S of Fig. i, with Centres of Balusters 4, 5 
 and 6 in Place on the Centre Line. — Make the angle F A E equal F A E of Fig. 2; make the 
 angle E H J equal G E C of Fig. 2; make E P equal F A; draw P Q parallel to E J; draw Q T 
 parallel to H E; connect T N, the directing level line; parallel to T N draw X I, E Y, 5, 2 and 
 W V; parallel to E H draw 2 R and 4 U; parallel to F A draw 0 C. At Fig. 2 baluster 1 hap- 
 pens to be at the point B; at baluster 2 make N K equal I F of Fig. i; at baluster 3 make 
 J H equal J G of Fig. i; at baluster 4 make L U equal 4 U of Fig. 3; at baluster 5 make 
 P R equal 2 R of Fig. 3; at baluster 6 make 0 Q equal 0 C of Fig. 3; through V K H U R Q A 
 trace the centre line of the wreath. Set off each side of the centre half the thickness of rail 
 as shown by the dotted lines. To find the length of balusters proceed as before directed. 
 Again, at Fig. 3, parallel to F E draw C G; at right angles to T N draw F L and J K; on E 
 as centre with E A as radius describe the arc A L; again, on E as centre with H J as radius 
 describe an arc at K; connect K L 
 
 To Find the Angle with which to Square the Wreath-piece at the Joint over J: — 
 Prolong FN to M; make N M equal T S; connect M J; then the bevel at M contains the angle 
 sought. 
 
 To Find the Angle with which to Square the Wreath-piece over Joint F : — Prolong 
 E F to D; make F D equal G B; connect D X; then the bevel at D contains the angle required. 
 
 Fig. 4. Face-mould from- Plan Fig. 3, also Showing the Squaring of the Wreath- 
 piece at Both Joints. — Draw the line A J indefinitely; make YA and YJ equal Y L and Y K 
 of Fig. 3; on J as centre with J H of Fig. 3 as radius describe an arc at H; on A as centre 
 with A E of Fig. 3 as radius intersect the arc at H; on Y as centre with Y E of Fig. 3 as radius 
 test the intersection of the arcs at H;* connect J H, H A and Y H; make H Q U equal H Q U 
 of Fig. 3; make H C equal E C of Fig. 3; parallel to Y H through C draw X I; through Q 
 and U parallel to Y H draw Q 8 and V W; make C X and CI equal 0 X and 0 I of Fig. 3; 
 make H Z 9 equal E Z 9 of Fig. 3; make Q 8 equal T 8 of Fig. 3; make U W and U V equal 
 4 W and 4 V of Fig. 3. Through J draw W B; make J B equal J W; through A draw X D; 
 make A D equal A X; make A K equal T A of Fig. 2; make the joints K and J at right angles 
 to the tangents; from X and D draw lines to joint K parallel to H K; through B V Z I D of 
 the convex and W 8, 9, X of the concave trace the curved edges of the face-mould. The slide- 
 line is made at right angles to the level line Y H. The angle zmth which to square the ivreath-piece 
 at joint K is taken by the bevel D of Fig. 3, and for joint J the bevel M of Fig. 3. 
 
 Fig. 5. Face-mould from Plan Fig. i. Quarter A D, which Joins the Level Rail; also 
 Showing the Squaring of the Wreath-piece at the Joints. — Draw the lines B C and B Z 
 at right angles; make B F G C equal the same at Fig. i; make B A equal B A of Fig. i; 
 through A draw H E indefinitely and at right angles to B A; parallel to B A draw F H, and 
 through G and C, M N and 0 0; make F L equal I L of Fig. i; make G N and G M equal 
 J N and J M of Fig. i. Joint C is made at right angles to B C; make C 0 and C 0 each 
 equal D 0 of FiG. i; make A Z three inches for straight wood; make the joint Z at right 
 angles to B Z; make A E equal A H; draw lines from H and E to joint Z parallel to B Z. 
 Through 0 M L K E of the convex and 0 N H of the concave trace the curved edges of the 
 face-mould. The sides of the wreath at joint C are made at right angles to the face of the 
 plank, f The angle with which to square the wreath at joint Z is taken by the bevel C of 
 Fig. I. Face-mould Fig. 4 is explained in detail at Plate No. 12. Face-mould Fig. 5 is 
 explained in detail at Plate No. 10. The development of a centre line of wreath as at Fig. 2 is 
 given in detail at Plate No. 20, Figs. 3 and 4. 
 
 * In face-moulds of this kind, if the drawing is carefclly made, instead of the length of this level line, Y H being 
 applied as a test, it ?nay be used mith the length of either one of the tangents to establish the angular point H. 
 
 f The sides of wreaths are straight vertically, and can be worked correctly only in that direction with suitable hol- 
 lows and rounds. After a wreath is shaped, if a straight-edge is tried square across the side of it at any point it 
 will be found hollow on the concave side and rounding on the convex side. For this reason in cases like face-mould 
 Fig. 5, joint C — in fact, at centre joints of all face-moulds — should properly have some over-wood at O O left on in 
 marking out the stuff. 
 
PLATE 45 
 
 Fig. I. Plan of a Circular Flight of Stairs Winding Around a Circular Post. — In 
 
 planning these stairs the first tiling tu do is to fix tlie place of the starting riser. The next 
 consideration is the landing and head-room. In this case the first, or starting, rise becomes 
 the landing and seventeenth rise, making one revokition; the whole height — the rises being 8" — 
 is therefore ii'.4.". In finding the head-room at a point between the starting and landing rises, 
 from the landing deduct lo" floor-beam, and 2" for floor and plaster; deduct also the bottom 
 rise, 8" — all together 20", to be taken from 11'. 4", leaving g'.8" head-room. Then, again, the floor 
 at the landing is brought on a line of the fifth rise E B, which makes it necessary to deduct 
 from the last sum four rises, — 32", — leaving a balance of head-room at that point of y'.o". The 
 post is sometimes cased as shown. The tread around the line of travel is 7", about the least 
 that ought to be permitted, if a hand-rail * is put around the post; but if a rail is hung over 
 the outside string, then the line of travel would be further out, and give a tread of 9I", which 
 would leave the plan as it is, ample. If the staircase is to stand independent of wall or 
 partition, the string should be bent laminated — see Plate No. 8, Fig. 5 — as being stronger than 
 by any other method. To give support to the string it may be enclosed to the floor from A to C, 
 or from A to B, and then set a supporting post at C, and at D suspend the string with a small 
 iron rod from the floor-beam of the story above. If the circular post is made large enough, each 
 riser can be set into mortices in the post four or five inches, and secured with lag-screws; the 
 steps, too, sliould be let into the post about 2". 
 
 Figs. 2, 3 and 4. — Fig. 2 is a Plan of Stairs such as is given at Plate No. 7, Fig. 8, 
 in which the Cylinder, 5" Thick, is Best Built Solid and Veneered, Both Faces for a 
 Close String. — The manner of building tiie solid cylinder is shown by the two thicknesses of 
 staves between tiie veneers; the straight portion, built with the cylinder from F to N, is so worked 
 for the purpose of including the easement at the lower edge of the concave face Fig. 4. 
 
 Fig. 3. TM veneer laid out for the convex face of the cylinder, the lettering agreeing with tliat 
 face on the plan Fig. 2. Tiie vertical and irregular lines are to show the position and lengths 
 of staves required for the convex face. 
 
 Fig. 4. The veneer laid out for the concave face of the cylinder, the lettering agreeing with that 
 face on the plan Fig. 2. The vertical and irregular lines indicate the lengths and position of 
 the staves for the concave face of the cylinder. This veneer is first laid over the prepared, 
 rough-staved cylinder, and on the veneer the concave-faced staves are fitted and glued; then the 
 convex staves are fitted and glued over these again; and, finally, the convex veneer Fig. 3 is 
 glued over the whole. 
 
 Fig. 5. Plan of Quarter Platform Stairs, Showing another Way of Placing the Risers 
 Connecting with the Quarter Cylinder. — A riser may be placed at the chord A; then taking 
 A B — which is the tangent to the centre line of rail — and a portion of the other tangent, B D, 
 to equal together one tread, as follows: A B = 6^", B C = 3^", making 10" one tread to the place 
 of the next riser C. 
 
 See Plate No. 5, Fig. 10, and Plate No. 37, Fig. 5. 
 
 * For treatment of liand-rail over circular stairs see Plates 53 and 54. 
 
Plate No. 45 
 
PLATE 46. 
 
 Fig. I. A Superior Plan of Stairs Making a Quarter-turn at the Landing, with 
 Parallel Steps and Platform, in about the Same Space required when Planned with 
 Winders.— This plan is given at Plate No. 7, Fig. 4. Describe the centre Hne of rail, and 
 draw the tangents at each quarter cylinder. Space the balusters as required. To find the 
 angles of inclination over the plan tangents, and other measurements, the elevation must first 
 be set up. 
 
 Fig. 2. Elevation of Treads and Rises as given at Fig. i on the Centre Line of 
 the Hand-rail ; also the Development of the Centre Line of Wreath-piece. — Place the 
 ciiord-lines — of which there are three needed — as at the plan. Put the centres of balusters 1, 
 2, 3 as given on each tread, and draw lines indefinitely through these centres parallel to the 
 rise-lines. From chord-line A make A B equal A B of Fig. i. Through B draw B C parallel 
 to the rise-lines; on the line B C — assuming any point C that will raise the wreath a little 
 high rather than fix it low — draw C F, the centre line of ramp, and the inclined line over the 
 first plan tangent; where this line C F intersects the chord-line at A draw A B at right angles 
 to the chord-line; make E Y equal E B of Fig. i: then Y becomes a fixed point. Make H Z 
 equal P V of Fig. i; draw Z Q at right angles to HZ; make Z 0 four inches, and OQ half 
 the thickness of rail: then Q is a fixed point (unalterable if the level rail is to be kept at 
 its usual height). Connect Q Y; divide P D in two equal parts at G; make A F four inches 
 for straight wood to be left at the lower end of the wreath-piece. Again at Fig. i make the 
 angle V Q P equal V Q P of Fig. 2; draw T R and W N parallel to U Q; make the angle E D B 
 equal E D Y of Fig. 2; make the angle B C A equal B C A of Fig. 2; make B H equal E D; 
 parallel to B A draw H Z; parallel to C B draw Z G; connect G F, the directing level line; 
 parallel to G F draw 2 M and 1 K; parallel to E D draw M L; parallel to B C draw K J. 
 Again at Fig. 2, baluster 1, make K J equal K J of FiG. i; at baluster 2 make M L equal M L 
 of Fig. i; through the points D LJ A trace the centre line of wreath-piece; set off half the 
 thickness of rail each side of the centre, as shown by the dotted lines. 
 
 To Find the Length of Balusters: — Take for example baluster 2: 2 N equals which 
 added to 2'.2", the usual length of short baluster at X (between the top of step and the 
 bottom of rail), makes the length of baluster 2 between the same points 2'.^i". 
 
 Fig. 3. Face-mould from Plan of Rail at the Landing Quarter-circle Fig. i : also 
 Showing the Squaring of the Piece at the Joints. — Draw the lines G Q and Q J at right 
 angles; make Q R N P equal Q R N P of Fig. i; make P G equal P G of Fig. 2; make Q U 
 equal V U of Fig. i; make U J equal 2" for straight wood; make joint G parallel to J Q; 
 make T K and joint J parallel to Q G ; parallel to J Q draw 1 T, N W and X P X; make P X 
 and P X each equal P X of Fig. i; make R I and Q Y equal S I and V Y of Fig. i; draw lines 
 from X and X to joint G parallel to P G; make U K equal U T; from K and T draw lines 
 to joint J parallel to J Q; through K Y I X of the convex and T W X of the concave trace 
 the curved edges of the face-mould. The angle with which to square the wreath at joint J is 
 taken by the bevel Q of Fig. i. The sides of wreath-piece at joint G are made at right angles 
 to the face of the plank, and the over-wood is removed equally from both faces of the plank. 
 The dotted lines show the increased width of stuff required to work out the wreath-piece of 
 a hand-rail of so much greater depth than width. 
 
 Fig. 4. Plan of Hand-rail, Quarter-circle A E of Fig. i, with its Angles of Inclination 
 E D B and B C A.— Let B H equal E D; make H Z parallel to B A, and Z G parallel to C B; 
 connect G F, the directing level line; parallel to G F draw V S, B 5 and R I; at right angles 
 to G F draw A P and E W, each indefinitely; on B as centre with B D as radius describe the 
 arc D W; again on B as centre with C A as radius describe an arc at P; connect P W. 
 
 To Find the Angle with which to Square the Wreath-piece at the Joint over E: — 
 Prolong B E to T; make ET equal E M; connect T 5: then the bevel at T will contain the 
 angle required. 
 
 To Find the Angle with which to Square the Wreath-piece at the Joint over A: — 
 
 Prolong E F to U; make F U equal G N; connect U A: then the bevel at U will contain the 
 angle sought. 
 
 Fig. 5. Face-mould from Plan of Hand-rail Fig. 4; also Showing the Squaring ot 
 the Wreath-piece at the Joints. — Draw the line A D indefinitely; make X D and X A equal 
 X W and X P of Fig. 4. On A as centre with A C of Fig. 4 as radius describe an arc at B; 
 on X as centre with the level line X B of Fig. 4 as radius intersect the arc at B; connect 
 A B, D B and X B; prolong B D to G, and B A to F; make A F equal A F of Fig. 2; make D G 
 equal D G of Fig. 2; make the joints G and F at right angles to the tangents; make D K 
 equal D K of Fig. 4; make B Z 4 equal C Z 4 of Fig. 4; through K Z 4 parallel to B X draw 
 SV, LJ and IR indefinitely; make K V, K S equal Q S, Q V of Fig. 4; make B 0, X 0 
 equal the same at Fig. 4; make Z L and Z J, 4, I and 4 R equal G J and G L, Y I and Y R of 
 Fig. 4; through D draw V C; make D C equal D V; through A draw R H; make A H equal A R; 
 from R and H draw lines to joint F parallel to B F; from V and C draw lines to joint G 
 parallel to B G; through C S 0 L I H of the convex and V 0 J R of the concave trace the curved 
 edges of the face-mould. Four additional points on the centre line may be measured as shown, 
 and a parallel pattern made by describing arcs of circles of a radius to suit the width given 
 by the squaring of the joints; or the extra width may be scribed on the plank from the edges 
 of the face-mould. The slide-line is drawn at right angles to the directing level line B X. The 
 angle used for squaring the joint G is taken from T of FiG. 4, and for squaring the joint F the 
 angle at the bevel U of Fig. 4. J^or detailed explanation of face-mould FiG. 3 see Plate No. 10 
 and for face-mould fiG. 5 see Plate No. 12. See Plate No. 75. 
 
PLATE 47 
 
 Fig. I. Plan of a Part of a Flight of Winding Stairs of which this Portion makes a 
 Half-turn. — This plan is given complete at Plate No. 7, Fig. 3. Draw the centre line of rail 
 and the tangents. Space the balusters as required; then to find angles, measurements, etc., 
 proceed to set up the elevation. 
 
 Fig. 2. Elevation of Treads and Rises as given at Plan Fig. i; also the Develop- 
 ment of the Centre Line of the Wreath. — Place the chord-lines — of which there are two, A 
 and G — as given at the plan, and extend the upper one, J D, at an indefinite length. Put the 
 centres of the balusters on each step as at the plan, and draw lines through these parallel 
 to the rise-lines indefinitely. With a straight-edge held in the direction A J mark a point at 
 the lower chord, A, and at the upper chord, J; with the understanding that if J is placed 
 lower on the chord-line, then the upper ramp will be lengthened and the wreath brought 
 lower; and if A is raised higher on the chord-line, then the lower ramp, already about right, 
 would be increased in length and curve. So at pleasure fix A and J; then draw A D at right 
 angles to the chord-line; divide D J in four equal parts; make A B equal the tangent A B of 
 Fig. 1; draw B C parallel to the rise; make B C equal D E; connect C A, and prolong to N 
 indefinitely. At right angles to J F draw G H; let G H equal G H of FiG. i; connect H J, and 
 prolong to 0 indefinitely; make J 0 and A N each 3" for straight wood at those ends of the 
 wreath-pieces. Make the joints of ramps at N and 0 at right angles to N C and H 0. Again 
 at FiG. I, as the four heights to be raised at each tangent are alike, make all the angles of 
 inclination BCA, EFB, H KE and G H J equal B C A of Fig. 2. Draw the directing level 
 lines M B and M H; parallel to M B draw 1 R and 2 S; parallel to B C draw R L; parallel to 
 E F draw ST. Again at Fig. 2, baluster 1, make R L equal R L of Fig. i; at baluster 2, 
 make 2 T equal S T of Fig. i; at baluster 3, P K equals H K of Fig. i. Through A L T K J 
 trace the centre line of the wreath. Set off each side of the centre half the thickness of 
 rail as shown by the dotted lines. 
 
 To Find the Lengths of Balusters: — Take for example baluster 1: 1 R measures i^", which 
 added to 2'.2" — the usual length of balusters at X X between the top of step and bottom of rail — 
 makes the length of baluster 1 between the same points 2'.3^". 
 
 Fig. 3. Plan of Hand-rail, Quarter-circle A E, Fig. i, with its Tangents and Angles of 
 Inclination Lettered Alike. — Connect M B, the directing level line. Parallel to M B draw Q P; 
 parallel to E F draw 0 N; through E draw A U indefinitely; on B as centre with B F as radius 
 describe the arc F U. 
 
 To Find the Angle with which to Square the Wreath-piece at Both Joints : — Pro- 
 long BE to T; make E T equal E Y; connect T M: then the bevel at T will contain the angle 
 required. 
 
 Fig. 4. Face-mould from Plan Fig. 3, also Showing the Squaring of the Wreath- 
 piece at Both Joints. — Draw the line U U indefinitely; make V U, V U equal V U of FiG. 3. Draw 
 V B at right angles to U V; make V B equal V B of Fig. 3; connect B U and B U; prolong 
 B U to W, making U W equal A N of FiG. 2. The joints U and W are made at right angles 
 to the tangents. Make B N and B N each equal B N of Fig. 3. Through N and N parallel 
 to B Z draw P R and P R; make N R and N R each equal 0 Q of Fig. 3; make N P and 
 N P each equal 0 P of Fig. 3; make V Z equal V Z of Fig. 3. Through U and U draw R S 
 and R S, and make U S equal U R at U and U; draw R Y parallel to B U; from S and R 
 draw lines to joint W parallel to B W; through S P B P S of the convex and R Z R of the 
 concave trace the curved edges of the face-mould. The angle with which to square the wreath- 
 piece at both joints is taken by the bevel T of Fig. 3. For detailed explanation of this face mould 
 see Plate No. ii. The development of the centre line of a wreath-piece of this kind is given in 
 detail at Plate No. 20, quarter-circle A C of Fig. i, and AE of Fig. 2. 
 
Plate No. 47 
 
Plate No.48 
 
 c 
 
PLATE 48. 
 
 Fig-. I. Plan of the Bottom Part of a Flight of Winding Stairs, Turning One Quarter, 
 Starting with a Newel. — The complete plan of this flight of stairs is given at Plate No. 7, Fig. 3. 
 
 Draw the centre line of rail, and space the balusters as required ; then to find the length of plan 
 tangent A C, and other measurements, proceed to set up tiie elevation. 
 
 Fig. 2. Elevation of Treads and Rises as Given at Plan ; also the Development of the 
 Centre Line of the Wreath-piece. — Place the centre of baluster on each step as at the plan, and 
 draw lines through these centres parallel to the rise-lines indefinitely. Let the bottom line of 
 rail pass through X X, the centres of short balusters on the regular treads ; draw tlie centre line 
 of rail 0 C parallel to X X indefinitely ; make B 0 equal 2^' — or more at pleasure — for 
 straight wood at the upper end of the wreath-piece ; make M V, 6" and V N half the thickness of 
 rail ; through N at right angles to the chord-line draw A C : then A C will be the length of the plan 
 tangent A C at Fig. i. And if from C of FiG. i a line is drawn touching the centre line of rail at 
 M, it will be the level tangent. 
 
 Fig. 3. Plan of the Centre Line of Rail and Tangents A C and C M from Fig. i ; also 
 the Centres of Balusters D, G, J in Place as at Fig. i. — Through A draw T B at right angles to 
 A C ; make A B equal A B of Fig. 2; connect B C. T is the centre of the circle, and T M is at 
 right angles to C M as at Fig. i. Through the centres of balusters J, G, D, and paiallel to the 
 level tangent M C, draw J K, V H and S E; parallel to A B draw E F, H I and K L; from A draw 
 A P at right angles to the level tangent CM; on C as centre with C B as radius describe the arc 
 B P ; connect P" M. 
 
 To Find the Angle with which to Square the Wreath-piece at the Joint over A : — From 
 E parallel to C B draw E R indefinitely ; prolong C A to Q ; make A Q equal A R ; connect Q S. 
 Then the bevel at Q contains the angle required. 
 
 To Find the Angle with which to Square the Wreath-piece at the Joint over M: — Make 
 U V equal H I ; connect V M. Then the bevel at V contains the angle sought. Again at FiG. 2, 
 baluster D, make E F equal E F of Fig. 3; at baluster G, make H I equal H I of Fig. 3; at baluster 
 J, make K L equal K L of Fig. 3. Through the points B F I L N trace the centre line of the 
 wreath-piece; set ofT each side of the centre half the thickness of rail as shown by the dotted lines. 
 
 To Find the Lengths of Balusters: — Take for example baluster D; D S measuring 4^", which 
 must be added to 2'. 2" — the length of short baluster at X from the top of step to the bottom of 
 the rail — making the length of the baluster D between the same points 2'. 6^". The height of rail 
 at the newel is calculated by adding M V, 6" to the length of short baluster at X, 2'. 2"; making the 
 height from M to V 2'. 8". 
 
 Fig. 4. Parallel Pattern for Wreath-piece from Plan Fig. 3 ; also Showing the Squaring 
 of the Wreath-piece at the Joints. — Make M B equal M P of Fig. 3. On B as centre with B C 
 of Fig. 3 as radius describe an arc at C; on M as centre with M C of Fig. 3 as radius intersect 
 the arc at C: connect C M and C B; prolong C B to 0; make B 0 equal B 0 of FiG. 2. Make 
 M W equal M W of Fig. i. Make the joints 0 and W at right angles to the tangents. Make 
 B F I L equal to B F I L of Fig. 3. Make F D, I G and L J equal ED, H G and K J of Fig. 3. 
 On the centres B D G J M with a radius equal to half the required width of the pattern 
 describe circles, and touching these trace the edges of the pattern. The angle with which to 
 square the wreath-piece at joint 0 is taken by the bevel Q of Fig. 3, and the angle for squaring 
 the wreath-piece at joint W is taken by the bevel at V of Fig. 3. The development of a centre 
 line geometrically the same as tins of Fig. 2 is given at Plate No. 21, Figs, i and 2. Face-mould 
 and parallel pattern as required by this plan are treated geometrically in detail at Plate No. 14. 
 
PLATE 49 
 
 Fig. I. Plan of the Bottom Portion of a Flight of Winding Stairs, this Part of the 
 Flight Turning a httle more than a Quarter and Starting from a Newel. — Tiiis case of 
 liaiui-iailing is geometrically ihe same as lluil given at Plate No. 48. In the last-meniioned 
 Plate the cylinder is 10" in diameter, embracing three winders; but the plan here presented 
 has a cylinder 20" in diameter, containing five winders. The object of introducing this example 
 is to demonstrate the correctness and practicability of working the wreath aiound a large 
 cylinder in one piece, showing, too, by the development of the centre line of the wreath its 
 exact position and relation to step and rise. The length of plan tangent A B cannot be deter- 
 mined until the elevation is drawn, if a fixed height of rail at the newel is required. 
 
 Fig. 2. Elevation of Treads and Rises as at Plan; also the Development of the 
 Centre Line of Wreath-piece. — Let the bottom line of rail pass through X X, the centres of 
 slujrt balusters on the regular treads; draw the centre line of rail F B parallel to X X indefi- 
 nitely; make 1 E equal 8", and E G half the thickness of rail. Through G draw B A at right 
 angles to the chord-line. Again, at Fig. i, continue the centre line of rail to B, and make 
 A B equal A B of Fig. 2; make A C at right angles to A B and equal to A C of Fig. 2; from 
 B draw B Z tangent to the centre line of rail; from H, the centre of the cylinder, draw 
 H 0 at right angles to B Z; then B 0 is the level tangent. Place the balusters as required, 
 and number those that come under the wreath. Through balusters 2, 3, 4 and 5 draw S 8, 
 F7. 1.6, 5T, and from P, P R, all parallel to OB; parallel to A C draw T K, Q L, 6 N, 7 J 
 and 8, 9. 
 
 To Find the Angle with which to Square the Wreath-piece at the Joint over Z:— 
 
 Make G F equal 7 J; connect FO: then the bevel at F contains the angle required. 
 
 To Find the Angle with which to Square the Wreath-piece at the Joint over A: — 
 From N draw N M parallel to A B; make AE equal M K; connect El: then the bevel at E 
 contains the angle sought. From A draw A D indefinitely and at right angles to BO; on B 
 as centre with B C as radius describe the arc C D; connect D 0. Again, at Fig. 2, place the 
 centres of balusters on each step as at the plan, aud draw lines through these centres parallel 
 to the rise-lines. At baluster 2, make 8,9 equal 8,9 of Fig. i; at baluster 3, make 7 J equal 
 7 J of Fig. i; at baluster 4, make 6 N equal 6 N of Fig. i; at baluster 5, make T K equal T K 
 3f Fig. i; through C K N J 9 G trace the centre line of the wreath-piece; from this centre set 
 off each side half the thickness of rail as shown by the dotted lines. Proceed to find the 
 lengths of balusters and the height of rail at newel as directed at Plate No. 48. 
 
 Fig. 3. Face-mould from Plan Fig. i ; also Showing the Squaring of the Wreath- 
 piece at the Joints. — Draw the line CO equal to D 0 of Fig i; on 0 as centre with 0 B 
 of Fig. I as radius describe an arc at B; on C as centre with C B of Fig. i as radius intersect 
 the arc at B; connect 0 B and B C; prolong B 0 to Z; make 0 Z equal 0 Z of Fig. i; prolong 
 B C to F; make C F equal C F of Fig. 2; make the joints Z and F at right angles to the tan- 
 gents; make C L N J 9 equal the same at Fig. i; and through each of these divisions draw lines 
 parallel to B 0; make 0 X equal 0 X of Fig. i; make J W V equal 7 W Y of FiG. i; make N V U 
 equal 6 V U of Fig. i; make L P, L R equal Q P, Q R of Fig. i. Through C draw P E; make C F. 
 equal C P; make 0 K equal 0 S; draw lines from P and E to the joint F parallel to C B; draw 
 lines from K and S to joint Z parallel to B 0; through E R V W X K of the convex and P U Y S 
 of the concave trace the curved edges of the face-mould. Joint Z of the wreath-piece is squared 
 by the angle contained in bevel F of Fig. r, and joint F is squared by the angle contained in 
 bevel E of Fic. i. 
 
 Fig. 4. A Sketch of this Wreath-piece as it Appears when Squared up. 
 
Plate No. 50 
 
 '. , TOP or n RST^TEP. 
 
 Scale ^In.= 1 Ft ^ 
 
PLATE 50. 
 
 Hand-rail over Steamboat Stairs, from Plan given at Plate No. 6, Fig. 9. — Fia. i, 2 and 3 are 
 together the plan of the string with its different curves, including the whole number of treads. Describe the 
 centre line of rail and on this centre line place the balusters on each step numbered as shown. The 
 intention is to take two treads in the first piece of rail from A to C, and in the second piece of rail to 
 include six treads from C to N; the third piece of rail to take the two last treads with as much more as 
 it requires to bring this top wreath-piece to a level at the required height. Draw the level tangent A B 
 at right angles to A J ; through C draw B K at right angles to J D; through N at right angles to N, 43, 
 draw K 0 indefinitely; the point 0 must now be established by measurement taken from the elevation. 
 
 Fig. 4. Elevation of Treads and Rises as given at Plan ; also the Development of the 
 Centre Line of Wreath, including the Whole Flight. — On the line of the third rise C D fix D 
 distant from baluster 3, so that the bottom line of rail will pass through 3; draw D K at right angles to 
 D C; make D K equal C K of Fig, 2, draw K L parallel to the rise-lines, and equal to three rises; connect 
 L D, and prolong to B indefinitely; make C B equal C B of Fig. i where the length of tangent C B 
 intersects the inclined line D B at B; draw C A through B at right angles to CD; touching L draw 
 G N at right angles to L K; prolong the ninth rise to M and N; make N G equal N K of Fig. 2; make 
 N M equal three rises; connect M G; draw M 0 at right angles to M N; make 40, 41 equal four inches; 
 make 41, P equal half the thickness of rail. Again at Figs, i, 2 and 3, make N M of Fig. 2 at right 
 angles to K N, and equal to three rises; then at Fig. 3 make 0 P equal 0 P of Fig. 4; draw P, 39 
 parallel to ON; from N parallel to K M draw N P; from P d raw P 0 parallel to M N ; from 0 draw 
 the line 0 T, touching the centre line of rail; from the centre V draw V S at right angles to 0 T; then 
 0 S will be the level tangent; parallel to 0 S draw 12, 36; 11, 34 and Z 10, 33; parallel to 0 P draw 
 
 36, 37; 34, 35; 33, R and 9, 38. 
 
 To Find the Angle with which to Square the Wreath-piece from Fig. 3 over Joint N: — 
 Make N, 42 equal 33, 32; connect 42, Z: then the bevel at 42 contains the angle required. 
 
 To Find the Angle with which to Square the Wreath-piece from Fig. 3 at the Joint over 
 S: — From R draw R 13 parallel to N 0; make ST equal 13, P; connect T U: then the bevel at T 
 contains the angle sought. From N at right angles to S 0 draw N X indefinitely; on 0 as centre with 
 N P as radius describe an arc at X; connect X S. At Fig. 2 make K L at right angles to C K and equal 
 to three rises; connect L C; parallel to C L draw B D of Fig. i; from C through N draw C Q indefinitely; 
 on K as centre with K M as radius describe the arc M Q; connect K, 43; parallel to K W draw 8, 27; 
 7, 25; 6, 23; 5, 21 and 4, 19; parallel to N M draw 27, 28; 25, 26 and 23, 24; parallel to K L draw 21, 22; 
 19, 20 and 3, 18. 
 
 To Find the Angle with which to Square the Wreath-piece from Plan Fig. 2 at Both 
 Joints:— Draw 28,29 parallel to KN; make N, 31 equal 29,30; connect 31, Y; then the bevel at 31 
 contains the angle required. At Fig. i, parallel to A B, draw 1, 16 and 2, 14; parallel to C D draw 14,15 
 and 16,17; from C draw C E at right angles to A B; on B as centre with B D as radius describe the 
 arc D E; connect E A. 
 
 To Find the Angle with which to Square the Wreath-piece from Plan Fig. i at the Joint 
 over C: — Prolong line of joint C D, and level line A B, to G; make C F equal C, 15; connect F G: then 
 the bevel at F contains the angle required. 
 
 To Find the Angle with which to Square the Wreath-piece from Plan Fig. i at the Joint 
 over A: — Parallel to B A draw C I; make H I equal C D; connect I A: then the bevel at I contains the 
 angle sought. Again at Fig. 4: take all the heights from the plan tangents of Figs, i, 2 and 3, and 
 place them on the lines drawn through the centres of like-numbered balusters, and as shown by the 
 other corresponding numbers and letters; and through these top-numbers and letters trace the cei;tre 
 line of wreath. The governing length of baluster on this flight ought to be 2'.4" at its centre, from 
 tr>p of step to bottom of rail. The odd lengths of balusters will be found as before explained. In case 
 the bottom line of rail falls below the step or floor-line, at the centre line of baluster — as, for instance, 
 here at baluster 11 — then that distance must be subtracted from the length of the governing baluster; 
 the remainder will be the length of baluster 11. 
 
 Fig. 5. Parallel Pattern from Fig. 3 for Wreath-piece, Joining Level Rail at Top ; also 
 Showing the Squaring of the Wreath-piece at the Joints. — Make S N equal S X of Fig. 3; on N as 
 centre with N P of Fig. 3 as radius describe an arc at P; on S as centre with S 0 of Fig. 3 as radius 
 intersect the arc at P; connect S P and P N; make N 38, R 35, 37 equal the same at Fig. 3; make 
 
 37, 12 equal 36, 12 of Fig. 3; make 35, 11 equal 34, 11 of Fig. 3; make R 10 equal 33, 10 of Fig. 3; 
 make the joints N and S at right angles to the tangents. Through N, 38, 1 0, 1 1 , 1 2 and S as centres, 
 with a radius equal to half the width of the required pattern, describe circles, and touching these trace 
 the edges of the pattern. To square the wreath-piece at joint N take the bevel 42, and for squaring 
 the wreath-piece at joint S take the bevel T. 
 
 Fig. 6. Parallel Pattern for Wreath-piece over Six Treads ; also Showing the Squaring- of 
 the Wreath-piece at the Joints. — Make W M and W C each equal W Q of Fig. 2; draw W L at right 
 angles to W M; make W L equal W K of Fig. 2; connect L M and L C; make the joints C and M at 
 right angles to the tangents; make C 18, 20, 22 equal the same at Fig. 2; make L, 24, 26 and 28 equal 
 K, 24, 26 and 28 of Fig. 2; parallel to L W draw 28, 8; 26, 7; 24, 6; 22, 5; 20, 4; make 20,4 equal 19, 4 of 
 Fig. 2; make 22, 5 equal 21, 5 of Fig. 2; make 24, 6 equal 23, 6 of Fig. 2; make 26, 7 equal 25, 7 of 
 Fig. 2; make 28, 8 equal 27, 8 of Fig. 2. Through C, 18, 4, 5, 6, 7, 8, M as centres with a radius -equal to 
 one half the width of the required pattern describe circles, and, touching these, trace the curved edges of 
 the pattern. Both joints of this wreath-piece are squared by the bevel at 31 of FiG. 2. 
 
 Fig. 7. Parallel Pattern for Wreath-piece Joining the Newel at the Starting, and In- 
 cluding the Two First Treads ; also Showing the Squaring at the Joints of the Wreath-piece.— 
 Make A D equal A E of Fig. i. On D as centre with D B of Fig. i as radius describe an arc at B; on A 
 as centre with A B of Fig. i as radius intersect the arc at B; connect B D and B A; make the joints 
 A and D at right angles to the tangents; make B, 1 7, 1 5 equal the same at Fig. i; make 17,1 and 
 15, 2 equal 16, 1 and 14, 2 of Fig. i; through A, 1, 2, D as centres with a radius equal to one half 
 the required width of pattern describe circles, and, touching these, trace the edges of the pattern. The 
 angle with which to square the wreath-piece at joint D is taken by the bevel F at Fig. t, and for joint 
 A the angle is taken by the bevel I of Fig. i. Face-moulds and parallel patterns are treated in detail at the 
 following Plates: FiG. 5 at Plate No. 14; Fig. 6 at Plate No. 15; Fig. 7 at Plate No. 13. Development of 
 the centre lines of wreaths is given in detail at the following Plates and Figures: Fig. 5 at Plate No. 21, Figs, i 
 a7id 2, and of FiG. 6 at the same Plate, Figs. 7 and 8; Fig. 7 at Plate No. 20, Figs. 5 and 6. 
 
PLATE 51. 
 
 Fig. I. Plan of Stairs Showing how to Place Parallel Steps of a Uniform Width in 
 a Large Cylinder, Avoid Winders, and Make Use of the Room Afforded by Securing a Full 
 Platform ; also an Evenly-graded Hand-rail in Three Parts, Free from Abrupt Top 
 Curves or Ramps. — This pUui is given ixt Plate No. 6, Fig. 8. Describe tiie centre line of rail; 
 set off and number the balusters coming witliin the cylinder as shown. Divide the cylinder into 
 tliree equal parts by the radials R. 29, and R I; draw tangents to the centre line of rail as fol- 
 lows: At right angles to R U draw U 16; through A at right angles to R, 29 draw 16 B; through 
 10 at right angles to R 1 draw B 35; al tight angles to R 42 draw 43, 35; at right angles to A B 
 draw B D indefinitely; at right angles to B 35 draw 35, 34 indefinitely; at right angles to U 16 
 draw 16. 15 indefinitelv. I'^urthci' measuicnienls required will be obtained from the elevation. 
 
 Fig. 2. Elevation of Treads and Rises as given at Plan Fig. i ; also the Develop- 
 ment of the Centre Line of Wreath. — Place the centre of l^alusler on each step and number 
 them as al the plan. Through each of the centres draw lines parallel to the rise-lines indefinitely. 
 Let the bottom line of rail at the upper and lower ends pass through X X and I X, tlie centres of 
 short balusters; parallel to X X draw the centre line of rail B 34, parallel to 1, X draw the centre 
 line of rail A 15; at right angles to the chord-line draw U 16; make U 16 equal the tangent U 16 
 of Fig. i; parallel to the chord-line draw 16, 15; parallel to U 16 draw 15 F indefinitely; make 
 43, 34 equal 43, 35 of Fig. i; parallel to the chord-line draw 34 F; draw 34, 43 at right angles to 
 the chord-line; divide F 34 into four equal parts and draw lines tlirough each division par- 
 allel to F 15 indefinitely. Make 42 B and U A each three inches for straight wood to be added 
 to the upper end of one and the lower end of the other wreath-piece, connecting with the 
 straight rail. Again, at Fig. r, make 16, 15 equal 16. 15 of Fig. 2; connect 1 5, U ; make 
 43,42 equal the same at Fig. 2; connect 42,35. Set up the following heights: 35,34; 1 0, I ; 
 B D and A 29 — each equal one of the four equal heights at 34 F of Fig. 2. Connect 34, 10; 
 I B, D A and 29,16; make 42,44 equal 35,34; draw 44,38 parallel to 43,35; make 38,36 
 parallel to 43,42; diaw 43,37 parallel and equal to 35, 10; connect 36,37; parallel to 36,37 
 draw 1 1,30, 1 2,32 and 13.39; parallel to 43,42 draw 14,41, 39,40 and 36,38; parallel to 
 35, 34 draw 32, 33 and 30,31. Make 16,17 equal A 29; draw 17,19 parallel to 16 U; draw 
 19, 23 parallel to 16, 15; make U 25 parallel and equal to 16 A; connect 25, 23; parallel to 
 25.23 draw 4,26, 3,24 and 2,20. Parallel to A 29 draw 5,28 and 26,27; parallel to 16,15 
 draw 24, 18; 20, 21 and 1, 22. At the middle piece of hand-rail describe half its width each 
 side of the centre line. Through A and 10 draw AO indefinitely; at right angles to A 10 draw 
 BP; parallel to B P draw E F, Y T, 8 C and 7N; parallel to B D draw NM and 6 L; parallel 
 to 10, I draw X H, Z G and C K; on B as centre with B I as ladius describe the arc I, 0. 
 
 To Find the Angle with which to Square the Wreath-piece at Both Joints : — Prolong 
 T Y to W; draw GJ parallel to 10 B; make 10 S equal J H; connect S W: then the bevel at 
 S contains the angle required. Again, at Fig. 2, take all the heights from the plan tangents at 
 Fig. I and place them on the lines drawn through the centres of like-numbered balusters, and as 
 shown by the other corresponding numbers and letters; and through the top numbers and letters 
 trace the centre line of wreath. The odd lengths of balusters will be found as before explained. 
 
 Fig. 3. Face-mould for the Middle Piece of Hand-rail; also Showing the Squaring 
 of the Wreath-piece at the Joints. — Make PO and PA each equal P 0 of Fig. i; make 
 P D equal P B of Fig. i; connect A D and 0 D; make D G H and D G H equal B G H of Fig. i. 
 Parallel to P D through G, H and G, H draw E F and Y T; make D Q V equal B Q V of Fig. i; 
 make G T, G Y, G T, G Y equal Z T, Z Y of Fig. i; make H F, H E, H F, H E equal X F, X E of 
 Fig i; through A draw E S; make A S equal A E; through 0 draw E B; make 0 B equal 0 E. 
 Make the joints A and 0 at right angles to the tangents. Through S F T Q T F B of the 
 convex and E Y P Y E of the concave trace the curved edges of the face-mould. The angle for 
 squaring the wreath-piece at joints 0 and A is taken by the bevel S of Fig. i. 
 
 Fig. 4. Plan of the First Third of the Wreath, with the Tangents and Angles of 
 Inclination from U to A of Fig. I. — Make U K parallel and equal to 16 A; make 16, 0 equal 
 A 29; make 0 N parallel to U 16; make N M parallel to 15, 16; connect M K, the directing 
 level line; parallel to M K draw Q S, W L, 16 D, Z F and J I; parallel to Q 29 draw VT and 
 X Y; parallel to 16, 15 draw 2,4 and 6, 3; at right angles to K M draw A B and U H; on 16 
 as centre with 16, 29 as radius describe the arc 29 B; again, on 16 as centre with U 1 5 as 
 radius describe an arc at H; connect H B. 
 
 To Find the Angle with which to Square the Wreath at the Joint over A: — Make 
 A E ecpial A T; connect E D: then the bevel at E contains the angle required. 
 
 To Find the Angle with which to Square the Wreath at the Joint over U : — Draw 
 F G at right angles to F U; make F G equal 2, 5; connect G U: then the bevel at G contains 
 the angle sought. 
 
 Fig. 5. Face-mould over the Plan Fig. 4, also Showing the Squaring of the Wreath- 
 piece at the Joints. — Make B C H equal B C H of Fig. 4; on H as centre with U 1 5 of Fig. 4 
 as radius describe an arc at 16; on C as centre with C 16 of Fig. 4 as radius intersect 
 the arc at 16; connect B 16, C 16 and 16 H; make 16, 2, 3 equal 1 5, 4, 3 of Fig. 4; make 16, 
 Y. T equal 16, Y, T of Fig. 4; parallel to C 16, through T, Y, 2, 3, draw J I, 8 Z, W L and Q S; 
 make 3, I, 3 J, 2 Z, 2 8 equal 6 I, 6 J, 2 Z and 2, 8 of Fig. 4; make 16, P, G equal 16, P 9 of 
 Fig. 4; make Y L, Y W, T S and T Q equal X L, X W, V Q, V S of Fig. 4. Through H draw J E; 
 make H E equal H J; through B draw Q F; make B F equal B Q; make H D equal U A or 
 42, B of Fig. 2. Make the joints D and B at right angles to the tangents. Draw lines from 
 E and J to joint D parallel to 16, H; through E I Z P L S F of the convex and J 8 C W Q of 
 the concave trace the curved edges of the face-mould. Make the slide-line at right angles to 
 the level line C 16. Joint B of the wreath-piece is squared by the bevel at E of Fig. 4, and 
 joint D is squared by the bevel at G of Fig. 4. A face-mould geometrically the same as 
 Fig. 3 is given in detail at Plate No. 15, and a face-mould geometrically the same as Fig. 5 
 is also given in detail at Plate No. 16. Development of the centre line geometrically the 
 same as the centre line of this wreath-piece from U to A of Fig. i is given in detail at 
 Plate No. 21, Figs. 9 and 10; also an example of the development of a centre line of wreath 
 from A to 10 of Fig. i is given geometrically the same in the last-mentioned Plate, Figs 7 
 and « 
 
Plate No. 51 . 
 
Plate No. 52 
 
 Scale f i n. = 1 ft. 
 
PLATE 52. 
 
 Fig. I. Plan of a Platform and Double-landing Steamship Staircase with Newels at 
 the Starting, at the Angles of the Platform and at the Landings. — The posts at the starting 
 are intended to run above the upper deck a sufficient height to receive the level hand-rail and 
 balustrade of that deck as shown by the dotted lines. All the newel-posts are to be finished 
 above the hand-rails with moulded caps, the landing-post also finished with moulded drops 
 below the strings. The platform-posts are to rest on the lower deck. A plan of a staircase 
 similar to this with a continued hand-rail is given at Plate No. 6, Fig. 9, the hand-rail of 
 which is treated at Plate No. 50. 
 
 Fig. 2. Elevation of Treads and Rises between the Starting Newel and the Plat- 
 form Newel. — The treads in the curve from A to F on the plan must be measured on 
 the centre line of rail in the manner before directed, — taking each tread in two parts. Place 
 the centres of the three balusters in the curve as numbered in position on each tread, and 
 through these draw lines parallel to the rise-lines indefinitely. Make A E equal A D of Fig. i 
 through E draw E D parallel to the rise-lines; make D F equal D F of Fig. i; continue the 
 tread-line M to F and Z, from X, the centre of baluster, with a radius equal to half the 
 thickness of rail describe an arc at U; touching U draw a line to F; at E draw the line EA 
 at right angles to E D; through X parallel to U F draw the bottom line of rail X C; make B N 
 equal 3" for straight wood to be left on the upper end of the wreath-piece. At right angles to 
 F E draw F W; make F W equal half the thickness of rail; draw W Y parallel to E F. 
 
 Fig. 3. Plan of Rail with Centre Line and Tangents taken from A to F of Fig. i. — 
 Make A B at right angles to A D and equal A B of Fig. 2. Connect B D; at right angles to 
 D F draw D E equal to D E of Fig. 2; connect E F; connect D K; through A draw F C indefi- 
 nitely; on D as centre with D B as radius describe the arc B C. The numbers 1, 2 and 3 desig- 
 nate the centres of the first three balusters as placed on the treads at Fig. i. Parallel to 
 G D draw P Q, 3, 0 and 2 Z; parallel to D E draw Z T and 1 R. 
 
 To Find the Angle with which to Square the Wreath-piece at Both Joints Pro- 
 long D F to J indefinitely; make FJ equal D H; connect J K; then the bevel at J contains 
 the angle sought. Again, at Fig. 2, take the heights 0 V, Z T and 1 R from Fig. 3 and place 
 them at the like-numbered balusters as shown ; then through M R T V B trace the centre-line 
 of wreath-piece; below this centre line set off half the thickness of rail for the bottom line 
 of wreath. Find the odd lengths of balusters as before explained, first fixing the length of 
 baluster at X to suit, which should not be less than 2'. 4" from top of step to the bottom of 
 rail at the centre of baluster. 
 
 Fig. 4. Face-mould from Plan Fig. 3; also Showing the Squaring of the Wreath at 
 the Joints. — Make G C. G C each equal G C of Fig. 3. Draw G D at right angles to G C; 
 make G D equal G D of Fig. 3; connect D C, D C, and prolong each indefinitely; make C Y 
 equal W Y of Fig. 2; make C N equal B N of Fig. 2; make the joints N and Y at riglit angles 
 to the tangents; make D U, D U each equal D U of Fig. 3. Parallel to G D draw U P, U P; 
 make U P, U P each equal Q P of Fig. 3; make D X X equal D X X of Fig. 3; through C and 
 C draw P L and P L; make C L and C L equal C P; draw lines from P and L to the joints 
 parallel to the tangents; through L X L of the convex and P X P of the concave trace the 
 curved edges of the face-mould. The dotted lines show the extra width of wood required — 
 greater than the width of face-mould — to get out the wreath-piece with this proportioned form 
 of hand-rail. This wreath-piece is squared at the joints Y and N by the angle at bevel J of 
 Fig. 3. An elementary study of a face-mould geometrically the same as this is given at Plate 
 No. 15; also a like study of the development of a centre line of wreath geometrically the 
 same as at Fig. 2 is given at Plate No. 21, Figs. 7 and 8. 
 
PLATE 53. 
 
 Hand-rail for Circular Staircase from Plan given at Plate No. 7, Fig. ic— Figs, i, 2 
 and 3 are together the plan of the string with its curves, including the whole number of 
 treads. Describe the centre line of rail |'' greater radius than the front-string. The hand-rail 
 of this flight is divided into five parts: Fig. 1 from the newel-post A to D embraces three 
 treads, and three more divisions of the hand-rail will each include five treads; the fifth piece 
 of lail will take the last tread, and as much more of the curve as it requires to bring this 
 top wreath-piece to a level at its proper height. Draw the tangents to the centre line of 
 rail as follows- At right angles to the radial YE, touching D, draw B 1; at right angles to 
 the radial Y C, touching C, draw 1 F; at right angles to the radial Y L, touching L, draw 4 F; 
 at right angles to Y X, touching X, draw U 4; at right angles to D 1 draw 1,2; make 1, 2 
 equal two rises and a half, the rise being "jY; connect 2 D; make C 3 and F E each at right 
 angles to 1 F, and each equal to two and a half rises; connect 3, 1, also E C; at right angles 
 to F 4 draw 4,5; make 4,5 and LJ each equal two and a half rises; connect 5 L, also J F; 
 make X 6 equal two and a half rises; connect 6, 4. At Fig. 2, through L draw C K indefinitely; 
 on F as centre with F J as radius describe the arc J K; from M parallel to Y F draw M N, 
 Q R and H F; parallel to L J draw 0 T and P S. 
 
 To Find the Angle with which to Square the Wreath-piece at Both Joints:— 
 C and L, Fio. 2. Parallel u> F Y draw A G ; at right angles to Y C draw A B; make A B equal G D; con- 
 nect B C: then tlie bevel at B contains the angle sought. At Fig. i the curve of string which includes 
 the two first treads has a radius I Z equal to one foot; also, the limit of tangents D B and B A 
 cannot be determined until a portion of the elevation is set up; neither can the tangents X U 
 and U B of Fig. 3 be fixed, for the same reason. 
 
 Fig. 4. A Portion of an Elevation of Treads and Rises, including the Three First 
 Treads, the Top Tread, and Landing. — Let the bottom line of rail pass through XXX, the 
 centres of balusters; make W T equal 8", and T Z half the thickness of rail; draw Z D parallel 
 to line of tread; prolong the fourth line of rise to C and D; draw C U at right angles to C D 
 indefinitely; make S E equal 4", and E V half the thickness of rail. Again at Fig. i, make D C 
 equal D C of FiG. 4; parallel to 2 D draw C B; from B draw the tangent B A, touching the 
 centre line of rail; from I at right angles to B A draw I A; from D at right angles to B A 
 draw D H indefinitely; on B as centre with B C as radius describe the arc C H; connect H A; 
 parallel to B A draw P Q, R X and V W; parallel to D E draw 0 F, S M and U N. 
 
 To Find the Angle with which to Square the Wreath-piece at the Joint over D: — 
 Fig. I. Parallel to A B prolong RX to E; from S draw SL parallel to B C; make DG equal D L; 
 connect G E: then the bevel at G contains the angle sought. 
 
 To Find the Angle with which to Square the Wreath-piece at the Joint over A:— Fig. i. 
 Let D J be parallel to B A; make K J equal D C; connect J A: then the bevel at J will contain 
 the angle required. At Fio. 3, make U V equal U V of Fig. 4; draw V W parallel to U X; from 
 X parallel to 6,4 draw XV; from V at right angles to X U draw V U; from U draw U C, 
 touching the centre line; from Y at right angles to U C draw Y B; from X at right angles 
 to U B draw X H indefinitely; with X V as radius on U as centre describe an arc at H; 
 parallel to B U draw A G, E K and X D. 
 
 To Find the Angle with which to Square the Wreath-piece at the Joint over X:- 
 Make X F equal Z 0; connect F G: then the bevel at F contains the angle sought. 
 
 To Find the Angle with which to Square the Wreath-piece at the Joint over B: — 
 Fio. 3. Make B C equal U V; connect C D: then the bevel at C contains the angle required. 
 
 Fig. 5. Face-mould from Plan Fig. i ; also Showing the Squaring of the Wreath- 
 piece at the Joints. — Make C A equal A H of Fig. i. On C as centre with C B of Fig. i as 
 radius describe an arc at B; on A as centre with A B of Fig. i as radius intersect the arc 
 at B; connect C B and B A; make C F M N equal the same at Fig. i; parallel to A B through 
 F M and N draw V W, R S and Q P; make F Q and F P equal 0 Q and 0 P of Fig. i; make 
 M R and M S equal S R and S X of Fig. i; make N W equal U W of Fig. i; through C draw 
 P T; make C T equal C P; make the joints A and C at right angles to the tangents; make 
 A E equal A V; through P S W B E of the convex and T Q R V of the concave trace the curved 
 edges of the face-mould. This wreath-piece is squared at the joint C by the angle at bevel 
 G of Fig. i, and at joint A is squared by the angle at bevel J of Fig. i. 
 
 Fig. 6. Face-mould from Plan Fig. 2; also Showing the Squaring of the Wreath- 
 piece at the Joints. — Make H K, H K each equal H K of Fig. 2; make H F at right angles to H K, 
 and equal to H F of Fig. 2; connect F K and F K; make EST and EST equal the same at 
 Fig. 2; parallel to H F through ST and ST draw M N, Q R and M N, Q R; through K and K 
 draw M Z and M Z; make K Z, K Z each equal M K; make F 0 0 equal F X X of Fig. 2; make 
 S R, S Q, T N, T M each side of the centre 0 0 equal P R, P Q, 0 N and 0 M of Fig. 2; make 
 the joints K, K at right angles to the tangents; through ZNRORNZ of the convex and 
 M Q 0 Q M of the concave trace the curved edges of the face-mould. This wreath-piece is 
 squared at both joints by the angle at bevel B of Fig. 2. 
 
 Fig, 7. Face-mould from Plan Fig. 3 ; also Showing the Squaring of the Wreath- 
 piece at Both Joints. — Let B X equal B H of Fig. 3. On B as centre with B U of Fig. 3 as 
 radius describe an arc at U ; on X as centre with X V of Fig. 3 as radius intersect the arc 
 at U; connect X U and U B; make the joints B and X at right angles to the tangents; make 
 XOL equal X 0 L of Fig. 3; parallel to B U and through X, 0, L draw X J, T G and AM; 
 make U N and L M equal U P and Z S of Fig. 3; make X J, 0 T, 0 G equal X J, N K, N E of 
 Fig. 3; make B C equal B A; through X draw G F; make X F equal X G; through G M N C of 
 the convex and A T J F of the concave trace the curved edges of the face-mould. The angle 
 with which to square the wreath-piece at joint X is taken by the bevel F at Fig. 3, and for 
 joint B the angle is taken by the bevel at C. An elementary study of a face-mould geometrically 
 the same as Figs. 5 and 7 is given at Plate No. 13, and of face-mould Fig. 6 at Plate No. 
 15. A like study of the development of a centre line of wreath-piece geometrically the same 
 as required for Figs. 5 and 7 is given at Plate No. 20, Figs. 5 and 6; also the development 
 of a centre line of wreath-piece geometrically the same as required for Fig. 6 is given at 
 Plate No. 21, Figs. 7 and S. 
 
Plate No. 53 
 
LATt No. 54 
 
 SCAUC I 1 N. = 1 F T. 
 
PLATE 54. 
 
 Fig. I. P)an of Starting the Circular Staircase given at Plate No. 53, with a Scroll 
 Step and Hand-rail instead of a Newel. — The first three steps in this plan are all included in 
 the curve of the scroll, but the bottom step is properly the scroll step The radius Y D is the 
 same as that of the plan of circular string, Plate No. 53. D 2, 1 is equal to D 2, 1 at the plan, 
 Plate No. 53. Touching D, the tangents 1 A are at right angles to YD; at right angles to D A, 
 touching the centre line of rail at F, draw A F; make the joint F at right angles to F A; from 
 A parallel to D 2 draw A E; parallel to V E draw U H, I C and G B. 
 
 To Ascertain the Height of the Scrolled Hand-rail, as Regulated by the Tangent D A 
 and the Angle of Inclination D E A: — Set up an elevation of the first three treads, including the 
 fourth rise as at Fig. 2. Let X be the centre of baluster, and let the bottom line of rail pass 
 through X; also let D E A equal D E A of Fig. i. Make A B half the thickness of rail; then 
 B C, which is 4V', added to whatever height of baluster is given at X, will be the total height 
 of scroll between the top of the first step C and the bottom of tlie scroll B when the rail 
 is set up. The scroll looks best when kept at a height between C and B not exceeding 2'.6". 
 In shaping the top and bottom of the scroll it is desirable not to finish to a level at the joint 
 F, but to continue the easing an inch or two lower down, coming to a level with its ease- 
 ment at about the eye of the scroll. The scroll may also be brought lower by increasing the 
 length of tangent D A of Fig. i, forming an acute angle with the plan tangents; or it may 
 be fixed at a greater height by lessening the length of tangent D A, and forming an obtuse 
 angle with the plan tangents 
 
 Fig. 3. Face-mould from Plan of Scroll Fig. i; also Showing the Squaring of the 
 Wreath-piece at the Joints. — Let E A equal E A of Fig. i; make A F at right angles to A E 
 and equal to A F of Fig. i; make E H C B equal the same at Fig. i; through E H C B parallel to 
 F A draw V V, U M, I L and G B; make E V, E V, H U, H M, C I, C L, B K and A J equal D V, T U, 
 T M, 0 !, 0 L, S K and A J of Fig. i. Through F draw G P parallel to A E; make F P equal 
 F G; through P J K L M V of the convex and V U I G of the concave trace the curved edges 
 of the face-mould. This wreath-piece is squared at the joint F by the angle at bevel E of 
 Fig. I. At joint E the sides of the wreath are at right angles to the plane of the plank. 
 
 Fig. 4. To Draw a Scroll suitable for this Hand-rail and Staircase. — Describe a circle 
 of a diameter about sufficient to enclose the spiral line to be developed — its exact diameter 
 is unimportant. Divide the circumference of the circle into sixteen parts; make the diameter 
 of the eye of the scroll S J equal the width of the hand-rail. The spiral curve is found by 
 points on these sixteen radii, beginning at J by drawing a line at right angles to the radial 
 A J, then at right angles to the next radial on the left, and so on as shown by the position 
 of the little trying-squares, the external angles of which designate points on which as centres 
 with half the width of rail as radius describe arcs of circles. Touch ing these arcs trace 
 the curved edges of the scroll; but at the point 0 where the arc touches the eye of the 
 scroll measure on the radii from the external angles of the squares to the circle forming the 
 eye, and set off these distances outward as at 0 0, S S, etc., to J, tracing the remainder of 
 the convex curve from 0 to J through the points thus found. The spiral drawn in this way 
 may be cut off at any point where a sufficient revolution is made. In this case it is cut off 
 at D and connected with the plan at D. Fig. i. 
 
 Fig. 5. Construction of Block for Scroll Step and Riser. 
 
 Fig. 6. Scroll Step as Completed. 
 
PLATE 55. 
 
 Hand-rail over Elliptic Staircase from the Plan given at Plate No. 7, Fig. ii.— The best 
 
 division of hand-rail lor this plan is to begin at the centre, Fu;. i, taking in this first piece a portion of rail including 
 three treads each side of the minor axis; then FiG. 3, covering four more treads, FiG. 5, also taking four treads; and 
 Fig. 8, with the bottom tread and as much more of the curve as may be required to join tiie level rail at its usual 
 iieight. After deciding on the proper number of pieces in which to divide the plan of hand-rail, draw the tangents for 
 the whole as follows: Make B V tangent to the centre of rail at V, draw B N tangent to the centre line of rail at Y, 
 draw NA tangent to the centre line of rail at W ; draw AL tangent to the centre line of rail at C ; the level tangent 
 L F will be fixed further on. 
 
 Fig. I, Plan of Wreath-piece including Six Treads.— Draw B X at right angles to B Y and equal to three 
 rises; connect X Y, at right angles to B V draw V A ccjual to three rises, join A B, prolong A V to F indefinitely, from 
 Y through V draw Y D; on B as centre with B A as radius describe the arc A D; parallel to B N draw M K, S R and W T: 
 parallel to V A draw U 5, Q G and L J. 
 
 To Find the Angle with which to Square the Wreath-piece at Both Joints :— Prolong RS to F, par- 
 allel to B V draw G H; make V E ( (iiial H C; join E F. then the bevel at E coiuains the angle sought. 
 
 Fig. 2. Face-mould from Fig. i ; also Showing the Squaring of the Wreath-piece at the Joints. 
 
 — Let N A and N A each equal D N of FiG. i , make N B at right angles to N A and cqiml to N B of Fig. i ; join B A 
 and B A; make B J G 5 equal the same at FlG. i. Through J G 5 draw K M, R S and T W par.illel to N B ; make B P, 
 J K and J M equal B P, L K and L M of FiG. i ; make G R, G S, 5 T and 5 W equal Q R, Q S, U T and U W of FiG. i. 
 Apply the same measurements the other side of the centre B P, and through all tliese points trace the curved edges 
 of the face-mould. Make the joints A A at right angles to the tangents. This wreath-piece is squared at both joints 
 by the angle at bevel E f)f Fig. i. 
 
 Fig. 3. Plan of Wreath-piece including Four Treads. — Draw Y R at right angles to Y N and equal to four 
 
 rises. Parallel to Y X draw N I ; at right angles to A N draw W 5 and N, 14 ; make N. 14 equal 1 R ; join 14, W ; draw W J 
 parallel ami equal to NY; make 1 S equal 14, N ; parallel to Y N draw S G ; make G 8 parallel to 1 Y'; join 8 J parallel to 
 8 J draw T M. 2, 9 N K. 7, 22 and 10. 11 , parallel to Y I draw O F and D 3 : parallel to N, 14 draw U P and C I ; at right 
 angles to J 8 di aw Y Z; on N as centre with N I as radius describe the arc 1 Z ; at right angles to J 8 draw W B , on N as 
 centre with W 14 as radius describe an arc at B ; join B Z. 
 
 To Find the Angle with which to Square the Wreath-piece at the Joint over Y;— Draw 3 A parallel 
 
 to NY, make Y E equal A 4; join E 2 ; then the bevel at E contains the angle sought. 
 
 To Find the Angle with which to Square the Wreath-piece at the Joint over W :— Parallel to J 8 
 
 prolong 7, 22 to 5 , make W X equal U Q; join X 5; then the bevel at X contains the angle required. 
 
 Fig. 4. Face-mould from Plan Fig. 3 ; also Showing the Squaring of the Wreath-piece at the 
 
 Joints. — Make Z L B equal Z L B of FiG. 3 ; on B as centre with W, 14 of Fk;. 3 as radius describe an arc at N ; on L as 
 centre wntli L N of Fk;. 3 as radius intersect the arc at N ; join B N, N Z and L N; make B I P equal W I P ol FiG. 3; 
 make N 3 F equal N 3 F of FiG. 3; through I P. 3 F parallel to L N draw 11. 10, 7, 22, H 9 and T M ; make I 10, I 11, P 22, 
 P 7, N 12. N K equal C 10, C 11, U 22, U 7, N 12, and N K of Fig. 3; make 3 9, 3 H. F M, F T equal D 9, D H, O iVi and O T of 
 Fig. 3. Through Z dr.iw T A; make Z A equal Z T, make the jcjints B and Z at right angles to the tangents; through 
 A M. 9, 12. 22 and 10 of the convex and T H K 7, 11 o( the concave trace the curved edges of the face-mould. The angle 
 with which to square the wreath-piece at joint Z is taken by bevel E, Fig. 3, and for squaring joint B the bevel X of Fig. 3. 
 
 Fig. 5- — Make W T equal four rises. From A make A 6 parallel to W 14 ; make A B perpendicular to A C and equal 
 to 6 T , join B C. This position of the plan with its tangents and angles oi inclination is reino\ ed and completed at 
 Fig. 6. 
 
 Fig. 6. Plan of Wreath-piece over Four Treads, taken from Fig. 5.— Make c N parallel and equal to 
 
 A W ; make 6 J etjual to A B ; make J H parallel to W A and H R parallel to W 6 , join R N and prolong to IVI indefinitely; 
 prolong 6 W to M ; parallel to R N draw Z 4, K 3 and A X ; parallel to W 6 draw 2 F; parallel to A B draw O Q ; at rit;iit 
 angles to N R draw W E and C D indefinitely; on A as centre with A 6 as radius describe the arc 6 E; again, on A as 
 centre with B 0 as radius describe an arc at D; join D E. 
 
 To Find the Angle with which to Square the Wreath-piece at the Joint over w :— Make W L equal 
 
 J G ; join L IV! : then the bevel at L contains the angle sought. 
 
 To Find the Angle with which to Square the Wreath-piece at the Joint over 0 :— Draw CX 
 
 at right angles to C A ; draw X Y at right angles to X C and equal to A V ; join Y C: then the bevel at Y contains the 
 angle required. 
 
 Fig. 7. Face-mould from Plan Fig. 6, Showing also the Squaring of the Wreath-piece at the 
 Joints. — Make DUE equal D U E of FiG. 6; on D as centre with C B of Fig. 6 as radius describe an arc at A ; on U as 
 centre with U A of FlG. 6 as radius intersect the arc at A; join A E. A D and U A ; make DQ equal C Q of FiG. 6; 
 make AHF etjual A H F of FiG. 6; parallel to U A through Q draw Z4; parallel to U A through H and F draw PS 
 and K 3 : make Q Z. Q 4 and A T equal O 4, O Z and A T of Fig. 6 ; make H S, H P, F 3 and F K equal R S, R P, 2, 3 
 and 2K of FiG. 6; through D draw ZB; make DB equal DZ; through E draw KG; make EC equal EK; through 
 B 4 A S 3 C of the convex and Z T P K of the concave trace the curved edges of the face-mould. The angle with 
 which to square the wreath-piece at joint D is taken by the bevel at Y ; and for joint E by the bevel at L of FiG. 6. 
 
 Fig. 8. Plan of Wreath-piece, including the First Tread. — To find the height c O set up an eleva- 
 tion of the bottom step and two rises as at FiG. 9, let X be the centre of baluster and XO half the thickness oi 
 rail. Make HZ equal four inches and ZL half the thickness of rail; draw LC parallel to the floor-line; the anL:le 
 at O must equal the angle B of FiG. 5. Again at FiG. 8 make CO equal CO of FiG. 9; from O parallel to C B 
 draw OL; from L draw LF tangent to the centre line of rail at F; make F H at right angles to F L ; parallel to 
 FL draw C G, U T and JR; parallel to CO draw SP and RQ; at right angles to FL draw C D , on L as centre 
 with LO as radius draw the arc OD; join D F. 
 
 To Find the Angle with which to Square the Wreath-piece at the Joint over C;— Make CK 
 
 equal CP; join KE: then the bevel at K contains the angle sought. 
 
 To Find the Angle with which to Square the Wreath-piece at the Joint over F :— Make H G 
 
 equal CO; join G F: then the bevel at G contains the angle required. 
 
 Fig. 10. Face-mould from Fig. 8, Showing also the Squaring of the Wreath-piece at the 
 
 Joints. — Let F D equal F D of Fig. 8 ; on F as centre with F L of Fig. 8 as radius describe an arc at L; on D as centre 
 with O L of Fig. 8 as radius intersect the arc at L; join F L and L D; make L X X equal L Q P of Fig. 8; through X X 
 and D parallel to F L draw X J, T U and D N ; make L M. X T, X U and D N equal L M, S T, S U and C N of Fig. 8. 
 Make the joints at right angles to the tangents; make FS equal FJ; through D draw UP; make DP equal DU; 
 through S 'M Q U of the convex and J T N P of the concave trace the curved edges of the face-mould. This f:ice- 
 mould will not answer for the top of the flight, because at the top, although including but one tread as at the bottom 
 and setting up the usual height from the floor for the level rail, yet the total height is greater. This will be under 
 stood by examining FlG. 11 — which is set up for the top — and comparing it with FiG. 9. 
 
Plate No. 56 
 
 F I G 3 
 
PLATE 50. 
 
 Figs. I and 2. Wreath-piece from a Face mould, with Tangents at Right Angles, the Position 
 of one of which Tangents B L is Inclined, while that of L D is Horizontal. — Tlie sliding of the face- 
 mould along ihe joint C D and at F, the other side of the stuff, to plumb the sides of the wreath-piece, 
 is sliovvn by the dotted lines. The sides G H, J K of the wreath-piece at the centre butt-joint are not 
 straight lines. On the concave side of the wreath-piece G H is a concave curve, and J K of the convex a 
 convex curve on that side. This is true also of all wreath-pieces having butt-joints falling within a cir- 
 cular plan, but these straight sides are corrected in the hands of a skilful rail-worker, who, leaving some 
 over-wood — after the two pieces are bolted together — works the sides plumb with the proper-shaped tools 
 Fig. 2 shows the wreath-piece with the concave side cut away plumb. When this side is worked plumb, 
 a gauge, like Fig. 8, having an arm provided with a large pencil, may be used to mark the width on 
 the convex side ; next the top is shaped, and from this the thickness is gauged.* The directions here 
 given with regard to this wreath-piece apply generally to all, but particularly to the following : 
 
 Plate 24, Fig. 3, Plate 25, Fig. 6, Plate 26, Fig. 4, Plate 27, Fig. 4, 
 
 " 32, •' 3, " 37- " 4. " 40, " 6, " 43. " 3, 
 
 " 44, " 5, " 46. " 3, " 54. " 3- 
 
 Figs. 3 and 4. Squaring a Wreath-piece from a Face-mould, both Tangents of which are 
 Inclined either on a Common Inclination or on Different Inclinations. — The angle with which to square 
 a Vv'realh-piece at the butt-joint is the inclination of the face of the plank along the joint given by the 
 face-mould, in connection with a line on the joint — which is square thiough the plank — that coincides 
 with a vertical plane; hence it is commonly understood as an angle giving a plumb-line on a butt-joint 
 
 To Determine the Direction in which to Apply the Angles for Squaring a Wreath-piece at the 
 Joints : — Place the lower end of the wreath-piece towards you, turn the upper end to the right or left 
 to suit the hand of the stairs, then move the face-mould up a few inches on the slide-line as at Fig. 3, 
 and it will be seen that as the centre of the joint J is carried toward K, the plumb-line must apply in 
 the direction K M passing through the centre L, and at the upper end the other tangent will be moved 
 towards N, showing that the plumb-line on that joint must lie in the direction N P, passing through the 
 centre 0. The plank edge A D, bcvch U and V, show at once the correct position of bevels. 
 
 Another Way of Deciding the Direction in which to Apply Plumb-lines on Butt-joints is as 
 Follows: — Holding the wreath-piece as befoi'e directed, cant it u\) on the corner F — a prisition it must 
 take — when it will be at once evident that the plumb-line must apply in the direction K M. This causes 
 the stock of the bevel to lay towards the convex ; then at the upper joint reverse the stock of the 
 bevel, placing it towards the concave as shown ; also the slide-line need not be used, but the tangent 
 line K J (squared from the joint at K) and the face-mould tangent moved on this line until it reaches 
 the ptiint S and N of the upper end. 
 
 To Put the Face-mould in Position on the Planes of the Plank so that its Edges will Mark 
 the Plumb Sides of the Wreath-piece : — Hold the wreath-piece as before explained ; square a line from 
 the joint at K, K J indehnitely ; slide ihe face-mould up from the lower end along the slide-line until 
 J — tiie centre of the face-mould joint — falls on the line K J, K being a point at the face of the plank 
 of the previously-applied plumb bevel K M ; then again when tlie face-uiould is in this position its tan- 
 gent at the upper end will touch the point N of the pluuib-line N P on the upper joint ; also the con- 
 cave edge of the face-mould will touch S of the plumb-line S T. Apply the fiice-mould to the other side of 
 the wreath-piece on the slide-line, keeping the joint J of the face-mould as much below the joint of the 
 wreath-piece as it is above it on this side. 
 
 Fig. 4. The Wreath-piece Shown with the Concave Side Cut Away Plumb. — In all face-moulds 
 of this character, where a level line passes through tiie centre from which the jilan of the rail is described 
 — the minor axis — there is a place where the plane of the plank is level, and this point on the width X Y 
 and through the thickness X T is the normal place in a wreath-piece, and where the over-wood is removed 
 equally parallel to the faces of the plank. In shaping the wreath-piece the centre line of the thickness of 
 rail will correctly touch R and U, the centre at the joints, and S the centre of the plank. When the face- 
 mould is in position to plumb the sides of the wreath-piece, if the normal place is marked at V and W, the 
 plumb-line V W will pass through the centre of stuff at S, and give the direction in which to move the 
 round-faced plane in working the sides of the wreath-piece plumb. In gauging tlie wreath-piece to a width, 
 the long arm of the gauge should be held in the direction of the plumb-line W V. The instruction, g'ven under 
 the head of Figs. 3 and 4 apply particularly to the following : 
 
 Plate 25, Fig. 4, Plate 26, Fig. 6, Plate 27, Fig. 6, Plate 28, Fig. 4, Plate 29, Fig. 4, 
 
 31. " 
 
 ' 2 and 4, 
 
 " 36, ■ 
 
 ' 3, 
 
 " 37, ' 
 
 ' 7, 
 
 " 38, ' 
 
 ' 4, 
 
 " 39. ' 
 
 ' 4. 
 
 40, " 
 
 ' 3- 
 
 " 41, ' 
 
 • 4. 
 
 "42, ' 
 
 ' 4 and 6, 
 
 " 43. ' 
 
 ' 5, 
 
 " 44. ' 
 
 ' 4, 
 
 46, " 
 
 ' 5, 
 
 " 47, ' 
 
 ' 4, 
 
 " 50, ' 
 
 ' 6, 
 
 " 51, ' 
 
 ' 3 and 5, 
 
 " 52, ' 
 
 ' 4- 
 
 53, " 
 
 ' 6. 
 
 " 55, " 
 
 2, 4 and 7. 
 
 
 
 
 
 
 
 Fig. 5. Wreath-piece from a Face-mould over a Plan of Less than a Quarter-circle, the 
 Position of A K, one of the Tangents, being Level, the ether, K Z, Inclined.— The dotted lines show 
 the face-mould as placed at the joint D to plumb the sides of the wreath ;f the tangent A of the face-mould 
 is brought to C on the other side of the wreath-piece. The sliding of face-moulds of this character is always 
 along the joint, which is at right angles to the level tangent, the same as Fig. i. The above instructions 
 apply particularly to the following: 
 
 Plate 24, Fig. 6, "Plate 30, Fig. 2 and 4, Plate 32, Fig. 6, Plate 33, Fig. 3, Plate 48, Fig. 4, 
 
 " 49. " 3. " 50. " 5 and 7, " 53, " 5 and 7, " 55, " 10. 
 
 * See subject of Side Moulds as treated of at Pl.\te No. 76. 
 
 \ At Fig. 5 and the list given in tlie applieation of the angles to square wreath-pieees at the joints, tlie stock of the bevel at both joints Hcs 
 to-vards tin convex side of tlte ~'jreath-pieee as shoivn. 
 
PLATE 57. 
 
 Fig. I. Plan of Stairs Suitable for Wholesale Stores. — These stairs from the first to 
 the second story are enclosed with panel-work as shown by the elevation Fk; 4. The door 
 to shut off communication between the two stories is often placed on the platform, and in 
 that case the platform is so situated that the door trims under the end of the well-hole. 
 Side-rails are hung on strong ornamental iron brackets, sometimes on both sides of wide flights. 
 The newel-posts are never less than seven inches, and those at the top of the flights are 
 continued below the ceiling, finishing at the Invver end with turned work. 
 
 Fig. 2. Construction of Close String Paneled. — This finish of string is used in the 
 upper flights tiiat are furnished wiih hand-rail and balusters, as at Fig 5 The well-holes of 
 each story are framed shorter than the run of the flights above, so that each flight starts 
 from the floor below, resting directly on the floor-beams. 
 
 Fig. 3. Panel-work. — By this plan the middle muntins A are wider than the face-muntins 
 D, so that the mouldings may be nailed free from the panels, allowing the latter to shrink 
 without disturbing the mouldings. 
 
PLATE 58. 
 
 Fig. I. Plan of the Landing- Portion of a Staircase with Square Corner-pieces 
 like Small Low-down Newels set in the Angles with a Continued Hand-rail Over. — 
 
 This plan is given at Plate No. 6, Fig. 6 L and M are 3V' square angle-pieces that are 
 brought above the platform and floor as shown in connection with the elevations Figs 3 and 
 4. The centres of balusters A and B are equal in distance to A and Q. The square angle 
 at the corner-piece L is turned for the continued hand-rail one quarter C D V, C R S, using 
 only V radius from the face of the corner-piece E. to D, then from the joint D R of the level 
 quarter another is taken at the ramp to ease over to this level, as shown at the elevation 
 Fig 3, Z X and X K. 
 
 • Fig. 2. Design and Construction of Close Front-string. — The drawing to the scale 
 given will be a sufficient explanation 
 
 Figs. 3 and 4. Elevation of Treads and Rises, including the Square Corner-piece 
 Connecting with the Platform, the String and the Level as given at Plan Fig. i. — 
 
 0 P equals 0 P of Fig. i. The face P E of the corner-piece L, Fig, i. is along the line PEZ; 
 Z X equals D E of Fig. i; Z K equals one inch; the joint J H connects with D R of Fig i 
 The height to the bottom of rail Y from the step at the line of rise W is 2', i". The height 
 from floor to bottom of level rail SJ is 2'.6". C D, V V is the carriage-timber, showing its 
 bearing against the front platform-timber at V V. F, N and T are places of mortices to receive 
 the tenons of string. The baluster at B is intended to be set three eighths of an inch into 
 the mill-plowed hand-rail, as shown at the section A, and then pieces set between each baluster, 
 ^" thick, thus leaving a finished panel or sinkage of -J", the depth of which gives a much better 
 appearance to the bottom of the rail than when flush. 
 
PLATE 59. 
 
 F'-g. I. Plan of the Upper Portion of a Quarter Platform Staircase with Square- 
 moulded Newels set in the Angles. — Tliis plan is given at Plate No. 7, Fic. 5. The clotted 
 lines show a portion of the rough Iraming of the jilatform. 
 
 Fig. 2. Design Elevation and Details of Plan Fig. i. — Through this elevation tiie 
 lengths of the angle newels and the connections of hand-rail, balustrade work, strings, etc., are 
 obtained. The face of the newel marked A at the lower end showing its connections is face A at 
 plan Fig. i ; also the face of the newel marked E at the lower end is face E at plan Fig. i. 
 
 Fig. 3. Laying Out the Newel connected with the Platform, the Sides of which are 
 Lettered on the Plan Fig. I, A B C D. — The fom- faces of this newel are lettered at tiie top lo 
 correspond with the plan Fig. i. J K is the total length of the newel-shaft as taken from J K 
 of the elevation Fig. 2. The distances marked by the letters J L M N indicate principal points 
 of measurement taken from the coi responding letters of the A side of newel, Fig. 2 ; and the 
 same may be said of the letters 0 P Q at face D ; the sides B and C will be understood by 
 comparing them with their adjoining sides and connections, and B anrl C of Fig. i. 
 
 Fig. 4. Laying out the Landing Newel the Sides of which are Lettered on the 
 Plan Fig. I, E F G H. — The letters at the top of these sides correspond with those of the plan 
 Fig. I. R S of the side E marks the total length of this newel-shaft, and T U V the prin- 
 cipal points of measurement lettered the same at. Fig. 2. WXY of the connection at the 
 side F are also the principal points of measurement taken from the elevation of the lauding 
 newel. Fig. 2. The sides H and G will he understood by examining them in connection with 
 the adjoining sides and G and H of F'lc;. i. 
 
 Fig. 5. Balustrade Moulding as Shown in Place at the Elevation Fig. 2. 
 
 Fig. 6. Construction of Square Newel-posts. — The narrow pieces forming the sides A 
 and B of tlie newel-shaft should have blocks glued to tlie inside faces at the edges — not 
 more than one foot apart — wdien the glue is set to be jointed with the edges square from, 
 the face ; also to guard against the joints giving way, hard-wood dowels ought to be set 
 in as shown at suitable intervals. 
 
Plate No.59 
 
Plate No. 61 
 
Plate No. 62 
 
 ^3 
 
 Plan of stairs tuiniu;^ one-quarter witli 
 jiiutforin, itnd two risers curved at their 
 front ends to newels. -- The plan is designed 
 to dispense with winders and give a com- 
 fortable, easy stairs for travel, taking but a 
 few incliea more room than the usual 
 winders that are required to make a quarter 
 turn. The small angle uewcl receives the 
 straight rail of the flight uud the level 
 hand-rail at the landing, as shown, by the 
 side elevation. This mode of construction 
 requires no twists or easements ; ilv only 
 turn of the hand-rail ie the level quarter 
 cylinder. See plate 5, Fig. 7. 
 
 Scale 1 
 
Plate No. 64 
 
Plate No. 66. 
 
 Fig. 2. — Design for Spiral-turned Newels 
 and Ralustcrs, Braclceted String; Hand- 
 rail with Kainp and (3oose-neck. 
 
 Elevation. 
 
 Platform. 
 
 Plan. 
 
 Fi 
 
 G. 3. 
 
 Fig. I. 
 
 Figs. 3 and 4.— Platform Stairs with Angle Newels. The 
 bottom riser of the upper flight is set one tread from 
 the centre of newel as shown on the plan, FiG. 3, and 
 the elevation, FiG. 4; if, on the contrary, the bottom 
 riser referred to should be placed at the centre of the 
 newel, it would then be necessary to make that newel 
 one rise higher from the platform to 
 receive the hand-rail of the upper flight. 
 This difference in the height of the two 
 newels from the same platform is ob- 
 jected to by some, hence these sketchc 
 and explanations. 
 
 FiG. I. — Design for a Turned and Carved Newel, Carved 
 String and Balu.strade. 
 
 — ..^^ I 
 
Plate No. 67. 
 
 Ancient Staircase at Rouen, France.— From clie '^American Architect and Building News. 
 
Plate No. 68. 
 
Plate No. 69. 
 
Plate No. 70. 
 
PLATE No. 71 
 
Scale IxIn. ^IFt. 
 
PLATE 73. 
 
 Panelled Soffit of a Circular Stairs Showing how to Work Out the Twisted or Warped Panels, 
 Circular Bands, Radial Bands and Mouldings required. 
 
 Fig. I. A single section from the circular plan of a panelled sofilit embracing three threads. This 
 section at the front string A A is bounded by a soffit drop moulding marked S 0 F. M' G — shown at the cross- 
 section Fig. 2, marked C — and a circular front band, at cross-section Fig. 2, marked B ; then by a circular 
 band and cornice moulding at the wall, marked at the cross-section Fig. 2, E and F ; further there are two 
 straight ])arallel radial cross bands, the centres of which are placed along the line of the first and fourth risers. 
 The middle band — at cross-section Fig. 2, marked D — divides the section into two panels of equal width. 
 
 Fig. 3. Elevation of Treads and Rises at the Front and on Line of Wall.— The tread B A 
 at the Iront is taken along the convex line of string A A, and the tread C D is taken at the wall, which is the 
 convex line of the cornice moulding. A E is'the depth of string that may be required ; E F is the thickness of 
 bands. At the place of fourth rise H draw H G parallel to rise line ; prolong B 1 to J ; make J and G the 
 centres of radial bands, which as shaded show the ends of these bands in their position as given at plan Fig. 
 i; prolong the rise line C N to K indefinitely ; from M parallel to rise line draw M L indefinitely ; from G and 
 J draw the horizontal lines G L and J K ; connect L K, and parallel to L K draw the line of thickness equal 
 to E F ; then L and K are the centres of these radial bands, as shaded in position at line of wall. Surround 
 the shaded ends of bands by vertical and horizontal lines as at P 0, Q R, T S and V U, which show the width 
 and thickness of wood required to twist these radial bands to the warped surface of the soffit. 
 
 Fig. 4. Radial Band Showing End Sections and Exact Length Prepared Ready to 
 Work. — At the elevation Fig. 3 the end sections as enclosed show a slight difference in width and thick- 
 ness. This is disregarded, and the wood got out of ecjual thickness at both ends ; but care must be taken 
 that the ends are laid out relatively, just the same as at the top and sides 0 P, 0 R and T V, T S of Fig. 3. 
 0 T must be the exact length from wall line to convex face of string A A, plan Fig. i. 
 
 Fig. 5. Face-mould for Soffit Moulding under Front String and Band ; Marked at Plan 
 Fig. I, S 0 F. M'G. — Beginning at plan Fig. i make the perpendiculars from tangents C D and RG each 
 equal in height one and a half rises ; connect G C and D Z ; through Z draw R 5 indefinitely ; on C as centre 
 with C G as radius describe an arc at 5; from S at right angles to R 5 draw C S; parallel to C S draw FX 
 and J M B ; at right angles to C Z draw M N and X 0 ; prolong C R to H indefinitely ; on R as centre with 
 R E as radius describe the arc E H ; connect H W : then the bevel at H indicates the angle that will square this 
 piece 0/ soffit moulding at both ends. Now at Fig. 5 draw the line L P ; make K P, K L each equal 5 5 at plan ; 
 draw K Q at right angles to L K ; make K Q equal S C at plan ; connect Q P, Q L ; make P G I and L G I 
 each equal Z 0 N at plan ; make G D, I Y H each equal F X and J M B at plan ; trace the curves through the 
 points thus found. Make the joints at right angles to the tangents. 
 
 Fig. 6. Face-mould for Panel Marked P A N E L at Plan Fig. I. — Beginning at plan, all the lines 
 and angles are dotted to better distinguish the panel and its preparation for drawing a face-mould, as follows : Let 
 J and V i)e the centre of the width of panel at the ends ; connect J V and prolong to P indefinitely ; at right 
 angles to 2 V, also to I J, draw the tangents V K and J K ; at right angles to V K draw K S ; make K S and J 0 
 each equal one and a half rises ; connect S V and 0 K ; through K at right angles to J V draw A A ; parallel 
 to A A draw 8, 8 and 2, 6 ; at right angles to K V draw 4 Y and 3 U ; prolong K V to Q ; on K as centre with 
 K 0 as radius describe an arc at P ; prolong 8, 8 to R indefinitely ; prolong V 2 to R ; make V Q equal 3 T ; 
 connect Q R : then the bevel at Q indicates the angle required to square the panel at both ends. Now at Fig. 6 
 draw the line S S ; make E S, E S each equal L P at plan ; through E at right angles to S S draw F G ; make 
 E 0 ecpial L K at plan ; connect 0 S, 0 S ; make the ends S, S at right angles to the tangents S 0 ; make 
 S Z T each way from the centre equal V Y U at plan ; through Z and T parallel to G F draw R N and K L ; 
 make Z N, Z R equal 4, 2, 4, 6 at plan ; make T K, T L, 0 G, 0 equal 3, 8, 3, 8, K A, K A of plan ; through S 
 draw R U ; make S U equal S R ; through the ]ioints thus found trace the curves. 
 
 Fig. 7. Face-mould for Circular Middle Band. — Beginning at plan Fig i, draw the line Z Z E ; 
 make tlie tangents Z R, Z R at right angles to Z W, Z W; — W is the centre from which the plan was described ; 
 — at right angles to Z Z from R draw R W ; make the perpendiculars from tangents R J and Z 1 each one and 
 a half rises ; connect J Z and I R ; parallel to R W draw 2 F and L 0 ; at right angles to R Z draw V K and 
 OX; on R as centre witli R 1 as radius describe the arc IE; on Z as centre describe the arc M Y ; connect 
 Y W : then the bevel at Y indicates the angle that ivill square the middle band at both ends. Now at FiG. 7 draw the 
 line Z Z ; make ^1,^1 each equal W E at the plan ; draw Y 0 at right angles to Y Z ; make Y 0 equal W R 
 at plan ; connect 0 Z, 0 Z ; make the joints Z, Z at right angles to 0 Z ; make Z D C ecpial Z X K at plan ; 
 parallel to 0 Y draw C J A and C J A ; make C J, C A equal V F, V 2 at plan ; make D S equal 0 L at plan ; 
 through Z draw S F ; make Z F equal Z S, etc. 
 
 Fig. 8. Face-mould for Cornice Moulding. — Beginning at Plan Fig. i, through A and T draw 
 the line A Q ; from M at right angles Q P draw M P ; make M L and T S the vertical lines from tangents M T 
 and M A each equal one and a half rises ; connect L A and S M ; prolong M T to R ; on T as centre describe 
 the arc 0 R : connect R with W, the centre of plan : then the bevel at R indicates the angle required to square the 
 moulding at both ends. On M as centre with M S as radius describe tlie arc S Q ; parallel to M P draw H I, D F 
 and C V ; at right angles to M A draw I K, F G and V B. Now at Fig. 8 draw the line C B ; make K C, K B 
 each equal P Q at plan ; at right angles to K C draw K D ; make K D equal P M at plan ; connect D B, D C ; 
 make the ends B and C at right angles to the tangents ; make B 0 M L and C 0 M L equal A B G K at plan ; 
 parallel to D K draw L U, M T and 0 R ; measure the points as before from tangents at L M and 0, etc. The 
 shaded end sections show the squaring of the moulding at both ends. No face moulds are drawn for the 
 front and wall bands, as the proceeding is precisely the same as for Fig. 7, the middle band. All the above 
 face-moulds are alike in character, and are explained in detail at Plate 15. See Plate 84, Figs. 4, 5 and 6, 
 for the same case of face-mould and its management in squaring. See also Plate 56. 
 
PLATE 74. 
 
 Another and Third Method of Treating Hand-rail Over a Large Cylinder, Two Ways o* 
 
 WHICH are Given at Plate 24. 
 
 Fig. I. Plan of 15" Cylinder, the Hand rail and Elevation of Step and Rises to Floor at 
 Top Landing. — Let J, the centre of the hand-rail, be fixed over the phm D at its recjuired height above the 
 floor ; with J as centre describe a circle equal in diameter to the thickness of the plank out of which the 
 wreath-piece is to be worked ; draw the bottom line of rail through the centre places of short balusters X X. 
 At K, four inches below the chord, for straight wood, draw the joint line K 8 at right angles to X X ; from K 
 draw the line K A, touching the circle at A ; parallel to K A, touching the circle at M, draw 8 M ; make K L 
 equal the thickness of rail ; from L draw a line parallel to X X. The bevel at 8 indicates the angle to be used 
 for the joint through the thickness of the plank. The bevel at A — instead of the pitch-board — is used to 
 square the wreath-yjiece as shown at Fig. 2, joint D or J. 
 
 Fig. 2. Face-mould. — The measurements A 1 B 2 C, etc., correspond with and are taken from Fig. 1. 
 
 Fig. 3. Plan of Setting Off a Newel. — The rail may be shaped by using an ordinary easement cut 
 out of plank as thick as B C, so that the side curves F A and E G may be formed. Tlie stair-string need not 
 be curved or bent ; only let the end of the bottom step project sufficient to suit the ])lan curve. 
 
 Fig. 4. Elevation of Steps and Rises from Plan Fig. 3. — Let A B equal A B of Fig. 3. B C is 
 six inches, and may be more or less as recpiired. C E e(|uals C E of Fig. 3. The bottom of the rail rests at 
 X X, the centre of the places of short balusters. 
 
 Fig. 5. Case of Hand-rail Like that Given at Plate 16, Fig. 3, with this Difference. — In this 
 case the level line U L common to both planes occurs at right angles to the tangent A B, for which reason the 
 sides of this wreath-piece over joint A are at right angles to the face of plank. 
 
 Fig. 6. Face-mould. — This face-mould is measured from plan Fig. 5. The squaring of the wreath- 
 piece at the joints shown, the sliding of the face-mould as by dotted lines in position to plumb the sides of the 
 twist, are all given without further explanation than the preliminary statements and the drawings themselves 
 afford ; this being thought ample in connection with the full details given at Plate 16 and repeated at other 
 Plates. 
 
PLATE 75. 
 
 Fig-. I. Plan of Stairs Starting which Avoids the Old-fashioned Angular Winders by 
 Curving Risers, and thus Secures Parallel Steps and a Roomy Platform. (See also Plate 46.) 
 
 — Fig. 2. Elevation from plan Fig. i. Let the bottom of the rail above and below rest at the centre of 
 short balusters X X and X X ; draw a line the thickness of the rail above X X as shown. F'ix the places of 
 chord lines or commencement of cylinders measured from plan Fig. i at C, 5, J and M. Set off the length 
 of plan tangent at C B, E D, J G and M L. Let A be the centre of rail ; parallel to X X draw A B ; make 
 2 A 2!" for straight wood ; at B draw B C parallel to tread line. From 4, wliich is a fixed point, draw the 
 line 4 F, raising or lowering the point F to suit ; make E F 2^" for straight wood, and make the joint of ramp 
 F at right angles to 4 F. At H, the centre of the rail, draw H G parallel to X X ; at G draw G J parallel to 
 tread-line ; from 6, which is a fixed point, draw the line 5 M, raising or lowering the point M at pleasure ; 
 make M 0 2I" for straight wood, and make the joint of the level easement 0 at right angles to 0 6 ; from the 
 centre P describe the curve of easement as sliown. 
 
 Fig- 3- Plan of Rail and Tangents Taken from Fig. i to be Prepared for Drawing 
 the Face-mould. — Let D 4 equal the same at Fig. 2 ; connect 4 E ; make D B and Z C equal D 4 : make 
 C 2 equal the same at Fig. 2 ; make D K equal C 2 ; draw K A parallel to D E ; draw A 0 parallel to K D ; 
 from 0 draw 0 T, which is the level line common to both planes ; parallel to 0 T draw the other measuring 
 lines required ; at right angles to 0 T draw E P and Z Q ; on D as centre with 4 E as radius describe an arc 
 at P ; again on D as centre with B 2 as radius describe an arc at Q ; connect P Q. To find the angle that 
 will square the wreath over the joint Z, make Z Y equal C L ; connect Y W: then the bevel at Y will give the 
 angle required. To find the angle that will square the wreath over the joint E, make E N equal 0 X ; connect 
 N T : then the bevel at N indicates the required angle. 
 
 Fig. 4. Face-mould for Wreath-piece Over Both Quarter Cylinders. — The corresponding 
 letters with Fig. 3 show the measurements by which to draw tlie face-mould. The end sections together 
 with the dotted lines of face-mould in position show the squaring of the wreath ])iece. A face-mould like this 
 is treated in full detail at Plate 12. Also a face-mould of this character will be found at Plate 27, Fig. 6 ; 
 Plate 41, Fig. 4 ; Plate 42, Fig. 4 ; Plate 44, Fig. 4 ; Plate 46, Fig. 5. This case of hand-rail may be 
 treated differently if thought desirable, or as shown by the dotted lines, as follows : From 4, Fig. 2, draw a 
 straight line to K below ; then K is at the base of the upper height 0 0, and the lower height Z Z is made the 
 same as 0 0. This arrangement of the line 4 K and the equal heights 0 0 and Z Z would give one face- 
 mould at the first quarter of a common pitch, and one above of two different inclinations the same as Fig. 4, 
 but not quite so extreme a case as th.nt. Then the ramp is got rid of, and the rail altogether would have a very 
 easy and agreeable outline ; but at the fourth rise up it will be about 3" too high. Would this variation be as 
 objectionable as in the case of newels in place of quarter cylinders where the rail cuts straight against the 
 newel, as it does in such open-newel stairs, causing the hand to feel along the sides of the newel a foot or 
 more until able to grasp the connecting rail above ? Even if the rail connections with the newels are made 
 by ramp and knee — see Plate 60 — it is possibly even then far more awkward than such slight variations in 
 heii^hts as suggested for this other and different treatment of the continued rail. See Plate 46. 
 
Plate No. 75 
 
Plate No. 76 
 
PLATE 76 
 
 Side-moulds continued. — Also How to Cut Large Square-top Balusters, Stone or Wood, to 
 AN Exact Length ; the Tops to Fit the Warped Bottom Surface of Wreathed Hand- 
 railing. 
 
 Fig. I. Plan of Semicircular Wreath to be Prepared for the Measurement and Unfold- 
 ment of Side-moulds. — 5 K 8 6 L are plain tangents to centre line, 5 8 L ; let the three heights raised over 
 these tangents equal 6 A of a common inclination, A D 4 5 ; divide the circular centre line into any number 
 of equal parts, say eight ; draw radial lines from the centre S through each of these divisions, touching the 
 concave V T W and the convex X Y U ; from K draw the level line K S ; parallel to K S draw 0 9 and Q J ; 
 parallel to 6 A draw M 1 , N B, P 2, 8 C, and J 3 ; parallel to K 4 draw 9 E. 
 
 Fig. 2. Unfoldment of the Concave Side-mould.— Mark on the line 5, 6 tlie eight divisions at 
 V T W of Fig. i ; set up the heights 6 A, F 1, etc., taken as indicated by the corresponding letters at Fig. i ; 
 make 5 T equal K 4 and T P equal 5 K of Fig. i ; 5 Z is straight wood ; A S is also tor straight wood ; the 
 joint Z is made at right angles to 5 P ; the joint at C is the centre of the upper and lower wreath-pieces, and 
 is made at right angles to 5 P ; at A 1 B 2, etc., describe circles equal in diameter to the thickness of rail. 
 Trace lines touching the circles at opposite sides, and this completes the side-mould. 
 
 Fig. 3. Unfoldment of the Convex Side-mould. — The heights by which points for tracing the 
 unfoldment (also the joints) are as shown, the same as at Fig. 2 ; the only difference is that the eight divi- 
 sions of the convex X Y U, Fig. i, are marked along the line R W. A slight change in form as made from 
 the two points above W, I would advise wherever in a side-mould it may seem desirable. These side-moulds 
 are to be cut apart at joint C ; they apply to the ])luml)ed sides of the wreath-pieces. 
 
 Fig. 4. Plan— to be Prepared for the Unfoldment of Side-moulds— of a Wreath-piece, One 
 of its Tangents a Level Line, Producing a Twist of Double Curvature, the Upper Curve 
 Easing to a Level. — Divide the centre line A B into say four equal parts, and through these draw radial 
 lines from the centre S, touching the convex and concave P Q and N 0 ; through the points E, F, G draw lines 
 parallel to A C and touching the tangent B C ; let B D be the height raised over the tangent B C ; draw the 
 perpendiculars H M, I L and J K. This plan is taken from Fig. 2, Plate 84, it being the first wreath-piece 
 joining the newel of the stone circular stairs given at Plate 81. 
 
 Fig. 5. Unfoldment of the Convex Side-mould. — Mark on the line B Q the four parts from P to 
 Q of Fig- 4. Place the heights B D, H M, etc., taken as indicated by the corresponding letters at Fig. 4; 
 make D I W equal C B D of Fig. 4 ; make the joint D at right angles to D W ; make D 0 equal 5, 6 of Plate 
 8i ; make 0 X 6i-", or equal one rise ; then X D is the inclination of the convex side of the hand-rail, which 
 the upper and lower curves of this side-mould mast tangent as shown. In this case it happens that the angle 
 0 X D the same as the angle I W D. At D, M, etc., describe circles in diameter equal to the thickness of 
 rail, and touching these trace the upper and lower curve lines of the completed convex side-mould. 
 
 Fig. 6. Unfoldment of the Concave Side-mould from Plan Fig. 4. — Mark on the line Z Y the 
 four parts taken from N, 0, Fig. 4 ; the heights are the same as D M L, etc. ; of Fig. 5 : D I W is the same as 
 at Fig. 5 ; tlie joint D is at right angles to D W ; D 0 is equal to 2, 4, of Plate 81 ; 0 X equals one rise as at 
 P'iG. 5 ; X D is the inclination of the concave side of rail, which the upper and lower curves of this side- 
 mould must tangent as shown. 
 
 Fig. 7. Plan of Wreath-piece same as Fig. 4. — In this case, however, both tangents are inclined 
 so as to get a distance lower down equal to A B, and force an easement to the lower level in the thickness of 
 stone or wood, as shown at Figs. 8 and g. Divide the centre line C D into, say, four equal parts, and draw 
 radial lines from centre S ; make C F equal B D of Fig. 4. The face-mould for this plan and its application 
 is given at Plate ii, through Figs. 7, 8, 9 and 10. 
 
 Fig. 8. Concave Side-mould from Plan Fig. 7. — Mark on the line E G the four parts at E G of 
 Fig. 7 ; make G D equal the two heights A B and C F of Fig. 7 ; make D I W equal A C F of Fig. 7 ; let 
 D 0 X equal the same at Fig. 6 ; make the joint D at right angles to D W ; sketch the curve line E D to 
 tangent the line D X, and at the points of intersection with the perpendiculars describe circles equal in 
 diameter to the thickness of rail ; then trace the upper and lower curved edges touching the circles. 
 
 Fig. 9. Convex Side-mould from Plan Fig. 7. — Mark on the line J H the four parts J H at Fig. 7. 
 The joint D and the heights and D I W are the same as at Fig. 7. D 0 X, the inclination of the convex side 
 of the rail, is the same as at Fig. 5. These side-moulds, as stated above of the plan, apply to Figs. 7, 8, 9 
 and 10 of Plate ii. At Plates 20 and 21 examples of the unfoldment of the centre line of wreath and 
 wreath-pieces are given. Read the remarks at the beginning of Plate 20. 
 
 Fig. 10. To Cut Very Large Stone or Wood Square Top Balusters to an Exact Length : 
 the Tops to Fit the Warped Bottom Surface of Wreathed Hand-railing. — Set up treads and 
 rises, the tread to equal 3, 3 at the centre line of rail, Plate 81. Set the baluster on the step as required ; 
 make 0 K the height ; at K draw K P at right angles to 0 K ; make K S equal 0 0 at the concave face of 
 balusters, Plate 81 ; draw S U perpendicular to K P and e(iual to one rise ; through K draw the line U X, 
 which will be the angle and length of baluster facing the concave side of rail. For the side of baluster facing 
 the convex side of rail, make K P equal X X at Plate 81 ; make P T equal S U ; through K draw the line T K Z. 
 The central point K must be squared through to the opposite face of baluster, and the angle T K Z made to 
 pass through K. On the other two faces of baluster draw lines connecting those made through K. 
 
PLATE 76. 
 
 The Use of Side-moulds in Hand-railing. 
 
 Peter Nicholson, the eminent practical English mathematician, who in the year 1792, first applied 
 geometry intelligently and correctly, as far as his discoveries carried him, to the use and requirements of 
 hand-railing, gave what he called " falling-moulds " — patterns of a width equal to the thickness of rail, made 
 to fold around the convex or concave sides of wreaths, or wreath-pieces, by which to mark the shape or curves 
 of the upper and under surfaces of wreaths. I call these patterns more properly, considering their use and 
 application, side-moulds. Mr. Nicholson did not work from the centre of any form of curved hand-railing, 
 nor did he make use of tangents from which we make joints across the width and through the thickness of 
 wreath-pieces, fixing also by their adoption the exact place, practically the most useful, for the three points 
 controlling the inclination of the plank. This author's " falling-mould " is produced by the simple stretch 
 out of convex or concave, the height, straight connecting lines and curves as required at pleasure. Our side- 
 moulds are unfolded from the centre of the wreath — that is, by central points taken at the centre of the thick- 
 ness and the centre of the width; and thus we get the exact mathematical centres and resultant geometrical 
 curves, which we are at liberty to alter or not as conditions may require. In this city no stair-builders working 
 hand-rail at this time make use of " falling-moulds," or side-moulds. They seem to be content with the results 
 of trial practice, and the experience it affords in a general way of the curves required in shaping wreaths, and 
 this experience as a speciality leads to many men working only at hand-railing. Certainly no man who has 
 not had considerable practice can take up a twist and shape it correctly without being overlooked by some 
 skilful workman to prevent his spoiling it ; thus taking the time of two on what ought to be one man's work. 
 It is just so with an apprentice who is too often wholly kept from learning how to do this interesting and use- 
 ful kind of work because he requires so much time and attention. Side-moulds once well understood take 
 little time to make and apply, and by their use they insure absolutely correct and graceful curves at once 
 without the necessity of spending time eyeing the twist over and over again, or leaving overwood on each 
 piece ; so that when the complete wreath is bolted together there is more eyeing, shaping, and time taken in 
 perfecting the curve, and very likely another man invited to spend his time to decide. With the use of side- 
 moulds each separate twist-piece may be worked exactly to the lines, being sure of correct shape and thick- 
 ness — no " little too thick here," or " a little too thin there " ; so, too, when the wreath-pieces are bolted 
 together there will be no more finishing at the joints than there would be with joinings of straight pieces of rail. 
 The perfection of thickness and the shape of the upper and under surfaces of wreaths is as much under 
 the control of the draughtsman in making correct side-moulds as the shape of the sides of a wreath and the 
 width of the rails is by correct face-moulds. Side-moulds must be got out of some flexible material, such as 
 straw-board or thin sheet-zinc. 
 
PLATE 77. 
 
 The plan here presented and the treatments of hand-rail and side-moulds is given as a modified, simpler 
 form than that given at Plate 5, Fig. 6, which is treated at Plate 26. This plan takes 34" more run 
 than the plan Plate 26, and it takes ii^" more run than the plan with five winders, which is shown in 
 comparison alongside the Fig. 6 above mentioned. The hand-rail by this plan of stairs is executed without 
 the long ramp required by the plan at Plate 26. It seems desirable to avoid winders if comfortable, square 
 platforms and parallel steps can be planned to land as required with so little additional space as here shown. 
 To do this in many cases it might even be better to raise the height of rises a little, diminishing their number, 
 or contract the tread some, maybe do both, rather than amble up and down those really awkward and to 
 many people dangerously huddled steppings on tapering, angular winders. 
 
 Fig. I. Plan of Quarter Turn Platform Stairs with 6" Cylinder, One Curved Step and Two 
 Rises Above Platform. — The tangents to centre line of rail are 2 A D X 3 ; the heights, etc., raised over 
 tangents is taken from elevation, as explained further on. 
 
 Fig. 2. Elevation of Treads and Rises Shown at Plan Fig. i. — Draw the plan tangents A B, 
 C D and E F ; parallel to the rise lines draw DEL and F M ; let the bottom of the rail rest on X X, the 
 centres of short balusters, and set off the thickness of rail ; parallel to X X draw the centre line of rail S C ; 
 make A S equal the straight to be left on lower twist ; at Fig. i make A M and D Y each equal D E and F M 
 of Fig. 2 ; connect M D and Y X ; make X E equal B C of Fig. 2 ; connect E 3 ; again at Fig. i make D K 
 equal X E ; draw K N at right angles to D Y, and parallel to D Y draw N P ; connect P X ; parallel to P X from 
 the centre of baluster 0 draw 0 I, and from Q draw Q H also parallel to P X ; from R parallel to A M draw 
 R B F ; at Fig. 2 make C J equal Q D of Fig. i ; parallel to D E draw J K ; make J K equal D Y of Fig. i ; 
 make 0 Z equal I Z of Fig. 1 ; make C Q equal H L of Fig. i ; make E L equal B F of Fig. i ; through 
 A Z Q K L M trace the centre curve, and from the points A Z Q, etc., draw circles, the diameters of which equal 
 the thickness of rail, and touching these circles trace the edges of the unfolded central section of rail. M P 
 is the straight allowod on the upper wreath-piece. Side-moulds for this case may be drawn readily from 
 previous explanations; also from tliose that will follow for Figs. 8 and 9. 
 
 Fig. 3. Plan of Rail taken from Fig. i to be Prepared for Drawing the Face-moulds. — Let 
 the heights A M, D Y and 5 H ecpial F M, D E and B C of Fig. 2 ; make D E equal 5 H ; draw E P at right 
 angles to D Y ; parallel to D Y draw PZ 8 ; connect Z 5, the governing level line common to both planes ; 
 parallel to Z 5 draw F B B, R 0, U C and Q 1 ; parallel to D Y draw V 3, T 4, B I, 0 J and C K ; at right 
 angles to Z 5 draw D W and Q 9 ; on 5 as centre with 5 Y as radius describe the arc Y W ; with H Q as radius 
 on 5 as centre describe the arc at 9 ; connect 9 W. To find the angle required to square the lower wreath-piece over 
 joint D, make D 2 equal D K ; connect 2 6 : then the bevel at 2 indicates the angle sought. The angle to 
 square the wreath-piece over joint Q is found by making Z 8 equal 5 L; connect 8 Q ; then the bevel at 8 shows 
 the angle required. The angle for squaring the 11 ['per 7i<reath-piece is indicated by the bevel at M. 
 
 Fig. 4. To Drav/ the Face-mould for the Upper Wreath-piece. — Make M 3, 4 D equal M 3, 4 D 
 of Fig. 3 ; draw M P, 3 K, 4 Z T and S D R at right angles to M D ; make M G etpial A G of Fig. 3 ; make 
 G P equal the straight allowed at M P, Fig. 2 ; through G draw 0 R jjarallel to M D ; make D S, D S each 
 equal V G of FiG. 3 ; make 4 Z equal X X and 4 T equal X T of Fig. 3 ; make G 0 and G V each equal M 3 ; 
 parallel to M D draw J P K ; trace the curves ST V and S Z M 0 to complete the face-mould. The angle to 
 square the wreath-piece at joint P is taken from M, Fig. 3. 
 
 Fig. 5. To Draw the Face-mould for the Lower Wreath-piece. — Make 9 N W equal 9 N W of 
 Fig. 3 ; make 9, 5 equal G H of Fig. 3 ; make W 5 equal 5 Y of Fig. 3 ; make 9 S equal A S of Fig. 2 ; make 
 the joint at S at right angles to S 5, and the joint W at right angles to W 5 ; make W I J K equal Y I J K of 
 Fig. 3 ; connect 5 N ; parallel to 5 N draw 9 Z, K F, J R and I T B ; make K F equal C U, 9 Z equal Q 1 , J R 
 equal 0 R, and I B, I T equal B B, B F, all of Fig. 3 ; through W draw T P ; make W P equal T W ; through 
 9 draw F L ; make 9 L equal 9 F ; through TRF, PB5ZL trace the curved edges of the face-mould. The 
 slide line is at right angles to 5 N. The angle for squaring the wreath at joint W is shown by the bevel at 2, 
 Fig. 3, and for squaring the wreath at joint S, the bevel at 8, Fig. 3. The above face-mould is given in full 
 details at Plate 12. See also Fig. 6, Plate 27. There is one palpable defect in the above natural treat- 
 ment of the hand-rail over this plan, to which attention is invited, and that is at Q, as may be seen on the 
 line C Q, Fig. 2, where the rail is 3!" low. If this is thought of sufficient importance it may be corrected by 
 another and different treatment, as given at the elevation Fig. 6 and in connection Figs. 7, 8 and 9. Fig. 6. 
 Set up an elevation same as Fig. 2 ; also set off the plan tangents as before ; M, as at Fig. 2, becomes a fixed 
 point ; from M draw the line M S at pleasure, raising it at the chord A to suit ; through the centres of short 
 balusters draw the bottom line of rail X X ; draw S 0 at the centre of rail and parallel to X X ; the joint T V 
 is at right angles to the tangent A C, and will be the joint of the wreath-piece, as worked, and when squared 
 up the last thing to do is to cut away the material V Z T, leaving the required joint T Z at right angles to X X ; 
 A S is the straight allowed ; the centre joint R is at right angles to the tangent C E. 
 
 Fig. 7. Plan of Rail from which to Unfold the Central Section Fig. 6 and Figs. 8 and 9 
 Side-moulds. — Divide the centre line A L K into, say, six equal parts ; draw lines from the centre Y through 
 each of these divisions ; connect 8 Y ; parallel to 8 Y draw P W and 0 X ; make 4 W equal the three heights 
 B C, D E and F M of Fig. 6 ; parallel to 4 M draw H J, P Q, L R, W T and 8, 8 ; parallel to 4, 8 draw X G. 
 On the line A K, Fig. 6, mark the six parts at centre line A L K of Fig. 7 ; make the heights K M, 0 J, P Q, 
 etc., equal 4 M, H J, P Q, etc., of Fig. 7 ; through the points thus found describe circles equal in diameter to 
 thickness of rail ; trace the upper and lower curves, touching the circles, except the one at A, from which a 
 slight departure is made to produce the required curve at that point. 
 
 Figs. 8 and 9. Side-moulds from the Prepared Plan Fig. 7. The heights are the same in both 
 of these side-moulds, as taken from Fig. 7, and tlie same as used unfolding the central section at Fig. 6 — the 
 joints, too, are all alike. Fig. 8 is the concave side-mould, S V Z the six parts taken from Fig. 7, and Fig. 9 
 is the convex side-mould, 2, 6, 9, the six parts taken from 2, 6, 9 of Fig. 7. See Plate 46. 
 
Plate No. 77 
 
Plate No. 78 
 
 Scale 
 
 1^ I N = 1 Ft. 
 
PLATE 78. 
 
 It is desirable with this plan of stairs to get the wreath-piece out in one piece. To do this it becomes 
 necessary to force the joints and change curves to tangent the inclination of the straight hand-rail. Through 
 Figs. 3 and 4, Plate 31, this plan is treated in two pieces without tlie necessity of forcing joints or curves ; 
 but it takes more time to work two twists of this character than it does one ; besides, the two-piece wreath 
 makes a pronounced ogee-like curve that is unpleasant to the eye. The fault lies in the plan ; the risers are 
 not in a position to produce the best form of wreath. See Plate 37, Tigs. 5, 6 and 7, for the best plan and 
 wreath-piece. 
 
 Fig. I. Plan of Quarter Platform Stairs with Quarter Cylinder 8" Radius, and the Risers 
 in Connection with the Platform Placed at the Chord Lines. — M A V, N C W is the plan and centre 
 line of rail ; A B C is the plan tangents ; prolong A B to P indefinitely and C B to J indefinitely; connect 
 B F, the governing level line ; divide the centre line A C into, say, six equal parts, and through each of these 
 draw radial lines from the centre F ; again through each of these parts draw lines parallel to F B. To find 
 the heights and inclination of tangents set up the elevation of plan. 
 
 Fig. 2. Elevation of Treads and Rises as Given at Plan Fig i. — The line of platform ABC 
 must equal the plan tangents ABC, Fig. i. At X X — the centres of short balusters above and below the 
 platform— draw the bottom line of rail and set off the thickness ; also draw a centre line at D and I ; make 
 D A and S I each 2" or not more than 3" ; connect D I ; at A parallel to X C draw the line A M ; bisect M S at 
 N ; make N P parallel to M L ; from B draw B P parallel to C N ; through D draw E G at right angles to P A ; 
 and from E draw E F at right angles to X X ; through I draw H J at right angles to S P ; from H draw H K at 
 right angles to X X ; at Fig. i make CDS equal M N S of Fig. 2, and draw U P parallel to C B ; connect 
 P S ; make B J equal B P ; connect J A ; parallel to B J draw 1 , 6 and 0 K ; ])arallel to C T draw G R and 
 H Q. To find the angle that will square the ivreath-picce at both joints: — On U, as centre, describe the arc R T 
 tangent to S P ; connect T P ; then the bevel at T indicates the angle sought ; from A through C draw A E 
 indefinitely ; with P S as radius, on B as centre, describe an arc at E. 
 
 Fig- 3. To Draw the Face-mould -.—Make D E, D E each equal D E of Fig. i ; at right angles to 
 D E draw D P ; make D P equal D B of Fig. i ; connect P E, P E ; make E Q, E Q each equal D A or S I ©f 
 Fig. 2 ; make the joints at Q, Q at riglit angles to P Q ; make G L N equal G L N of Fig. i ; make H 8, 9 
 equal the same at Fig. i ; make P Y D equal B Y D, Fig. i,etc. ; through E draw N R ; make E R equal E N ; 
 trace the curved edges through the points thus found to complete the face-mould. The angle for squaring 
 the wreath-piece at both joints is found at T, Fig. i. A face-mould of this character is given in full detail 
 at Plate ii. 
 
 Figs. 4 and 5. Convex and Concave Side-moulds.— Fig. 4 is the concave ; M N is the six parts 
 taken from M N, Fig. i. The heights are taken, as the corresponding letters indicate, at Fig. i. The joints 
 are the same as shown and lettered alike at Fig. 2. Fig. 5 is the convex : VW is the six divisions taken from 
 VW of Fig. i. The heights are the same. The joints, as indicated by the corresponding letters, are alike. 
 The departure from tlie exact unfoldment, as seen by the circles, is made necessary because the joints and 
 curves are forced. When the wreath-piece is squared up by the use of angle T, Fig. i, and the side-moulds 
 to insure its correct curves, then the material F G and J K is cut away, leaving the joints E F and K H that 
 connect properly with the straight rail. 
 
 Note.— The thickness of plank required to work out a wreath-piece is invariably given by the squaring shown at the 
 ends laid out from the centre ; and the unfoldment of its natural geometrical curves are from the centre, and these curves are 
 contained within the limits of the two planes of a plank of which the two joints of a wreath-piece are only end sections. But 
 in any case of this kind where the form of a wreath-piece is forced out of its natural curve-line of geometrical unfoldment, it 
 takes more thickness, how much more will be determined by whatever, at both ends, or either end, the forced form of side 
 mould is outside of and beyond that of the natural geometrical unfolded curves. 
 
PLATE 79. 
 
 At Plates 34 and 35 two methods of treatment are given — one for a twelve-inch and the other for a 
 seven-inch cylinder — both for the toj) of straight flights of stairs ; the wreaths to be worked in one piece, 
 making a complete semicircle. The case here jjresented is also to be worked in one piece. All of this kind 
 of work has to be more or less forced.* What I mean by forced — as applied to hand-railing — is to be seen in 
 all of these cases by the changed form required from the natural geometrically unfolded centre line of wreath 
 when put in comparison, as at Figs. 3 and 4 of this Plate. I suggest that the skilled, intelligent mechanic 
 would do well, as a rule, not to decide hurriedly with a snap judgment that this or that kind of proceeding or 
 method when brought to his notice is "no good" or " unprofitable." It would be better to give a little of his 
 spare time in careful examination, and in this case of hand-railing, for instance, work a wreath up as explained 
 out of some soft wood, and so get positive knowledge one way or the other. 
 
 Fig. I. Plan of Half-turn Platform Stairs. — The treads, 10"; rise, 8"; cylinder. 7"; hand-rail, 
 4" X 2!" ; the wreath to be worked in one piece ; at right angles to 3, 3 draw 3J, DF, 4K, A8, 4H, BG and 
 3 L indefinitely ; divide the centre line D W B into six equal parts, and through these draw radial lines from 
 the centre A. 
 
 Fig. 2. Elevation of Rises and Treads from Plan Fig. i. — Draw the bottom line of rail 
 above and below the platform at X X, X X, the centres of short balusters ; set off D C, D C the centre of rail, 
 also at E and A above, the thickness of rail parallel to X X ; through E and E draw A A parallel to line of rise ; 
 parallel to tread lines draw the following : A B K, D F, E J , E L, D G and BAH; at Fig. i connect F G ; make 
 F K and F J, G L and G H each equal half the thickness of the wreath-plank ; draw the lines H J and L K ; 
 make 8, 7 equal 0 D of Fig. 2 ; make all the measurements for face-mould from the line 3 A 3 and set them 
 from line 5, 6 ; as, for example, make 7 YZ equal A 2, 2, etc. ; through the points so found trace the curved 
 edges of the face-mould ; make C C, Fig. i, eijual C C at the elevation. Fig. 2 ; connect C Q, and parallel to 
 C C through the points of divisions on centre line D W B draw lines touching C Q. 
 
 Figs. 3 and 4. Convex and Concave Side-moulds. — Fig. 3 is the convex-mould ; C Q is the six 
 parts taken from 3, 2, 3, Fig. i ; the heights C C, 0 M, V P, etc., are also taken, as indicated by corresponding 
 letters at Fig. i. The joints are the same as at Fig. 2. The broken, or dotted, lines are the natural unfolded 
 geometrical curve lines, from which a forced departure has to be made as shown. Fig. 4, the concave side- 
 mould ; 4, 2, 4, is the six parts taken from 4, 2, 4, Fig. i ; the heiglits are the same as those of the convex 
 side-mould ; the joints are the same as at Fig. i and Fig. 3. This wreath-piece is cut square through, as 
 shown by the line shading at joints. The joints G and F are made scjuare from tlie face of the plank, centred 
 and lined out, as seen in connection with Figs, i and 2. At the joints G and F the lower level squaring line 
 H, and the upper level squaring line K, the slabs should be cut off square to the joints. Then resting alternately, 
 these cut surfaces on a saw-table, the sides may be cut plumb with a band-saw, and when cleaned up apj)ly 
 and mark the lines given by the side-moulds. To cut the joints, set a gauge equal to A B, Fig. 2, and run it 
 against the squared joints G and F, gauging on the level at K, also at H. Next cut away the material from 
 gauged lines to line J and to line L, making the joints E B, E B at Fig. 2. The top and bottom of the wreath 
 may now be squared with the band-saw by rolling it on the saw-table on its convex plumbed side. 
 
 Noi K. — The thickness of plank required to work out a wreath-piece is invariably given by the squaring shown at the ends 
 or butt-joints as laid out from the centre, and the unfoklment of a wreath's natural geometrical curves are from the centre ; 
 and these curves are therefore contained within the limits of the two planes of a plank, of which the joints of a wreath-piece 
 are but end sections. In any case where the form of wreath is foiced out of its natural geometrical unfoldment it lakes more 
 thickness; how much inore will be determined by whatever distance at both ends or either end the forced form of side-mould 
 is outside of the natural geometrically unfolded curves. Set the last mentioned distance from a point touching the lower face 
 of plank, down the plumb line J or H, Fig. 2, and measure for half the thickness required at right angles to the line G F. 
 
Plate No. 79 
 
 Scale 1>&]n = I Ft 
 
Plate No. 80 
 
 S C ALE 1 I N -1 Ft. 
 
PLATE 80 
 
 Circular Stone Work. 
 
 Fig. I. Plan of Stone Work a Quarter-circle, P C F, P D G, with Projecting Capstone.— 
 
 E A, the thickness of stone wall ; F C, the width of cajjstone. The top of the stone wall with capstone is 
 recpiired to start from a level and finish to a level, rising the height J L, with geometrically reversed curva- 
 tures ; all within the quarter-circle plan. To prepare this plan for unfolding the convex and concave faces 
 of the stone and its top curvatures : — Divide the centre line I M Q into six equal parts ; throught each of 
 these divisions draw lines from the centre P ; from M at right angles to M P draw J K ; at right angles to 
 P Q draw Q K ; at right angles to P E draw I J ; parallel to P M N draw J L ; make J L any required height ; 
 connect L K ; parallel to I J draw R 0 and S 7 ; from X and from V parallel to Q K draw V 8 and X W • 
 parallel to J L draw 0 T, 7 U, 8 Z and W 9. 
 
 Figs. 2 and 3. Unfoldment and Elevation of the Convex and Concave Faces of Stone 
 Wall and Capstone on the Lines E H and A B, Fig. i.— At Fig. 2 mark on the line J K the six parts 
 of the concave line A B, Fig. i ; from each of these set up perpendicular heights J L, 0 T, 7 U, etc. ; taken 
 from the corresponding letters at Fig. i from the points LT, U N, etc., describe circles equal in diameter to 
 the thickness of capstone ; trace lines touching the circles for the top and bottom curves of capstone ; the 
 centre joint at N is made at right angles to the angle L of Fig. i — copy the angle L, as at N V V, Fig. 2 ; fix 
 the number, size and joints of stone as required. Fig. 3 is the unfolded convex face of stone and 
 curvatures of capstone on the line E H, Fig. i. The six parts on the line J K of this figure are 
 taken from the six divisions at E H, Fig. i ; also the heights are the same as indicated by the corresponding 
 letters. From the fixed joints PEG, Fig. 2, draw horizontal lines touching F E G of Fig. 3. Fig. 4. — Plan 
 of stone wall P A E, P B H from Fig. i introduced for the purpose of showing presently in plan the exact 
 sizes of each of the four stones next to capstone, as follows : — Take stone I, Fig. 2 ; apply X 2 to X 2, 
 Fig. 4, and from centre P draw 2, 2 ; then the convex 2, 0 is X 2, stone I, Fig. 3 ; connect 2 F, the joint- 
 then X 0, 2, 2 of Fig. 4 is the size in plan of stone I. To find the size in plan of stone II:— At Fig. 3 
 drop the perpendicular F Z and prolong 4, 2 to Z ; apply 2, 4 to 2, 4, Fig. 4 and draw P 4, 4 ; draw P Z ; 
 then Z4, 4 is the size of stone I I. To find the size of stotte I I I : — Apply 4,6 of Fig. 3 to 4, 6 at the 
 concave Fig. 4 ; draw P 6 6 ; make 4, 6 of Fig. 3 equal 4, 6 of convex Fig. 4 ; let fall the perpendiculars 
 G A and E B ; prolong Y 6 to A and 6, 4 to B ; make 6 A and 4 B of Fig. 4 equal the same of Fig. 3 ; draw 
 P A and P B ; then B 6, 6 is the size of stone Mi; make 6 Y W of Fig. 3 equal the same of Fig. 4 ; then 
 D AWC is the size of stone I V. 7 he drawing is now prepared to make the concave and convex templet for 
 the four stone. The unfoldment of the curvatures of the capstone is, as before stated, on the face lines of 
 the stone wall A B and E H of Fig. i, and the only use of it here is to get the exact shape of the wall at 
 X F EG, etc. ; but for the sides of the capstone as side-moulds for which it is required it must be unfolded 
 on the lines C D and F G, Fig. i. Therefore, that portion of the concave and convex in each marked M K 
 must be redrawn, using the same heights and joint at N, l)ut taking three of the divisions from the line C D, 
 Fig. I, for the concave side-mould, and three of the divisions from the line F G for the convex side-mould. 
 
 Fig. 5. Plan of Capstone taken from Fig. i, as Shown by Corresponding Letters, to be 
 Prepared for Drawing the Face-mould. — P M bisects the quarter-circle I Q ; from M draw SV at right 
 angles to M P ; draw I S at right angles to P F ; from S draw S N parallel to K P ; make S N equal M N of 
 Fig. I ; connect N M ; prolong S I to B and to K indefinitely ; make I B equal S N ; from M parallel to S B 
 draw M T ; connect B T and prolong T B fo A indefinitely ; from 0 parallel to S B draw 0 E and W LA ; 
 make M V equal S R ; connect V K ; at right angles to S I draw M U indefinitely ; on S as centre with M N 
 as radius describe an arc at U ; connect I U. 
 
 Fig. 6. Face-mould for Capstone.— Make I U etiual I U of Fig. 5 ; make I Y equal S I of Fig. 5 ; 
 make U Y equal M N of Fig. 5 ; make I Z equal the straight from L to the joint, or from K to joint Fig. 3 ; 
 through I draw V W at right angles to Z Y ; through Z parallel to P W draw B A ; make I K equal B A of Fig 
 5 ; make I R P V equal B E H T of Fig. 5 ; draw V S, R T and K D parallel to Z Y ; make V S equal T J of 
 Fig. s ; make R T, I C, K D equal C 0, I, 5 and L W of Fig. 5 ; through U draw T X ; make U X equal U T ; 
 make I W equal I P ; make joint U at right angles to U Y ; trace the curve X S P and T C D W, to complete the 
 face-mould. The angle for squaring the capstone at joint Z is found at B, Fig. 5, and the angle for squaring 
 at joint U is at V of Fig. 5. To plumb the sides of the capstone the face-mould is moved on the face of the 
 stone along the line B A, as shown by the dotted lines. After the sides E E, F F of the capstone piece is cut 
 away plumb, then the side-moulds — made as explained under the head of Figs. 2 and t, — are applied ; the 
 concave side-mould to the side E E and the convex side-mould to the side F F, marking from these the top 
 and bottom of capstone ; thus carrying the lines of squaring through from joint to joint. Two of these pieces 
 reversed complete the capstone. A face-mould of this character is given in full detail at Plate 13. For a 
 better knowledge of side-moulds study Plates 20, 21 and 3. 
 
 Note. — The treatment of the above capstone will be precisely the same for a hand-rail, if required, over this or a 
 similar plan. 
 
PLATE 81. 
 
 Plan of Circular Stone Staircase. 
 
 The shaded space around the wall indicates the face-line of the stone wainscot and the finished plaster 
 line. The outside circular line is the face of brick wall. Into this brick wall the stone steps are bedded eight 
 inches, and resting step upon step lengthwise on the horizontal checks, interlocking each other with warped 
 joints accurately fitted through the depth of each stone, form an archlike self-sup])orting structure. The 
 front ends of steps finisli in line with a circle five feet in diameter ; the nosings projecting on the line L, H, N, 
 10, which is also the concave line of the stone hand-rail. The hand-rail is seven inches wide by four inches 
 thick. The shaded squares show the position and size of balusters. The centre line of hand-rail S R U TV 
 passes through the centres of balusters. The hand-rail is divided into four pieces* — on the line G F, the 
 beginning of the quicker curve, to newel S, of which 8 is the centre, and from R to U five treads are included, 
 for the joints must be made to occur at the centres of balusters in stone-work ; from U to T five more treads 
 are taken in the third piece of rail ; the balance from T to V makes the fourth piece ; next, the tangents have 
 to be placed and the heights to be raised over each tangent, fixing their inclinations ; beginning at S, draw 
 S E at right angles to 8 S ; at R draw the tangent E Q at right angles to F G and at U draw the tangent Q W 
 at right angles to G A ; at T, at right angles to G I, draw W B ; from V, the centre of the rail, at right angles to 
 G V "draw V B ; to fix the heights, at right angles to R Q draw Q D indefinitely ; at right angles to U W 
 draw W X indefinitely ; at right angles to W B draw B C indefinitely ; make Q D two and a half rises ; con- 
 nect D R ; from E parallel to D R draw E F ; make U A, W X and T I each two and a half rises in height ; con- 
 nect A Q, X U and I W ; prolong W I to C ; from I parallel to T B draw 1 , 7. 
 
 Fig. I. Template from Plan of Circular Band to be Applied on the Soffit.— An example of 
 a portion of th& middle band will show how to ])roceed for either of the circular bands. From the centre of 
 the plan G describe the centre line of the middle band A C ; then draw a straight line from A to C ; at right 
 angles to A C draw A B ; make A B equal three rises ; connect B C ; anywhere along the line A C draw i)er- 
 pendiculars X P, Q R and V S ; at right angles to B C draw P F, R G and S T ; make S T R G and P F ecjual 
 V E, Q Z and X 0 ; on the points B F G T C as centres describe circles in diameter equal to the width of band, 
 and touching these circles trace the curved edges of band. 
 
 The hand-rail, the wainscoting, the heavy base and cap mouldings, the unfolding and paneling of the 
 soffit and the curves, angles and joinings of the stone steps have each to be treated specially in one of the 
 four more Plates following this one. 
 
 See Plate 7, Fig. 10 ; also Plates 53 and 54. 
 
 *The divisions of hand-rail, if thought desirable for any reason, may be 
 each, leaving the top piece of rail to take the odd tread. 
 
 ; — Commencing at R three pieces of three treads 
 
Plate No. 81 
 
Plate No. 82 
 
PLATE 82. 
 
 CIRCULAR STONE STAIRS. 
 
 Elevation of Sections of Steps. Their Connections, Curves, and Joints from which to Make 
 Templets. Also Sections and Elevation of Wainscot ; Its Construction, together with 
 Management of Base and Cap Mouldings. 
 
 Figs, i and 2. — Elevation of sections of the first five steps. Fig. 2 is taken at the face of wainscoting, 
 Z M K 0, Plate 81. Fig. i is taken at the line of nosing, L, H, N, 1 0, Plate 81. The check l^" A B, Fig. i, 
 is set from rise line to H at plan, Plate 81, and the dotted line G H J drawn to get the radial distance at K J, 
 the face of wainscot, which is 3+". This may be diminished to 2V , which is done at C D, Fig. 2, as it seems 
 an unnecessary amount of check, and some little stone is saved, too. The joint will not radiate truly, but if 
 it does not vary from the horizontal, as it will not when treated as directed, it will make no perceptible differ- 
 ence. Through the angle of tread and riser F and A, Fig. i, draw the line FA ; from B at right angles to 
 F A draw a line the required depth of the face end of the stone step B to E ; through E parallel to F A draw 
 G R ; on this face all the joints — except joints through curves — are parallel to B E, or at right angles 
 to F A ; at Fig. 2 draw the line XC ; at right angles to X C from D draw D P indefinitely; from E, 
 Fig. I, draw the horizontal line E P ; then P, Fig. 2, touching the joint D P fixes the depth of stone at S P 
 from V E of Fig. i ; from P at Fig. 2 parallel to X C draw 0 Q ; at Fig. i prolong the floor line W J to H ; 
 prolong the joint at G to H ; on H as centre describe the curve G J ; then to find the radial from J to the 
 face of wainscot section Fig. 2, take W J, Fig. i, in two or more parts and set them at Plate 81 ; from 9 to 
 L and from G through L draw G M ; then P M will be the radial distance to set in front of the fourth riser 
 X W at Fig. 2, Plate 82, and from W perpendicular to the floor line draw W L ; then produce a curve 0 L, 
 finishing to the floor line and tangent to 0 P ; the horizontal lines G N and K M place the joints at the curve 
 L 0. Figs. 3 and 4. — Elevation of several steps from the top of the flight down, which show in the checks, 
 the regular joints and curve joints substantially what has been before explained. At Fig. 3, at the top joint, 
 it will be preferable to cut a straight joint A B in the face of stone about three inches before commencing to 
 cut the angular joint. 
 
 Fig. 5. Cross Section of Stone Wainscot wit h Base G, Base Moulding A and Cap Mould- 
 ing B. — The line of finished plaster is at C, the face of brick-work at D. This section is given on the line 
 E F,Fig. 6. 
 
 Fig. 6. Elevation of Wainscot. — G G G, etc., is the joints of base, the face of which is worked to 
 the circle and set 25" in front of the face of wainscot, as at G, Fig. 5. The wainscot may be got out and 
 joined as at H, H, H, H. A cross-section of this piece of panel-work on the circle is shown at J J ; I is an 
 inserted raised panel ; 5 is the centre from which the lower portion of the wainscot panel is described, extend- 
 ing within the radial lines 5 V and 5 S. The cap-mouldings T M 0 and the base-mouldings S V X will have 
 to be squared up similar to hand-railing, and on this drawing it is most convenient to fix lengths of pieces — 
 material considered — fit to handle that will cover in number the total length required ; also heights, etc., must 
 be placed. At the ca])-moulding easement draw a horizontal line from the centre T to K and from M to N ; 
 draw the perpendiculars M K and ON. It is best to make M N a length on plan of which any certain number 
 of pieces will complete the circle. Divide K L into three parts, and from two of the points of division raise 
 perpendiculars. At S, the centre of the base-moulding, draw the horizontal line S P, and from V draw the 
 horizontal line V W ; draw the perpendiculars V P and X W ; divide P R into four parts, and from three points 
 of the division raise perpendiculars. Fig. 7. — Moulded lower edge at finished front end of steps. 
 
 * Sometimes two centres have to be used — one of less radius for base moulding and lower part of first and second panels 
 and base moulding — and then these panels are possibly of a greater height than the others. 
 
PLATE 83. 
 
 Unfolded Soffit of a Circular Stone Staircase. 
 
 Unfolding the Soffit of a Circular Stone Staircase, and Apportioning the Panels, etc. — 
 
 Draw a line A B indefinitely ; the centre E will not be needed in this development until later on ; make A B 
 equal I, 0 of the \Aan at Plate 8i ; make B C equal P Q of Fig. 2, Plate 82 ; make A D equal E R, Fig. i, 
 Plate 82 ; at the plan Plate 81 draw N 0, and at right angles to N 0 draw the perpendicular 0 F ; connect 
 F N ; take F N for radius, and on A — of this Plate — as centre describe an arc at C, and on B as centre describe 
 an arc at D ; at the intersection of the arcs at C and D draw the line D C ; proceed in like manner from D 
 and C, as before, from A and B for the number of six steps ; then draw the line G F ; again from A B proceed 
 and describe intersecting arcs at fK and L, and so on for five steps to M and N ; now the centre may be found 
 by drawing the line M N until it intersects the line G F, prolonged to meet at the centre E ; with E A as radius 
 describe the line of nosing J A 0 ; with E B as radius describe the face- line of wainscot ; with E W as radius 
 describe the finished plaster line ; mark at F J the two steps E B, Fig. 3, Plate 82 ; mark also at G H the 
 two steps G D, Fig. 4, Plate 82 ; connect Q R and J H ; mark at N 0 the three steps J S, Fig. i, Plate 82 ; 
 mark also at M P the three steps L 0, Fig. 2, Plate 82 ; connect S T, U V and OP; A Z is the projection of 
 nosing ; Z X is the moulded lower edge on the finished face of the front ends of steps. The drawing will 
 furnish its own explanations by measurements and observation of circular and radial bands, division of panels, 
 etc. It is not pretended that this unfoldnient of a warped surface, i.e., the soffit, is geometrically exact, but 
 is a near approach, and serves a useful purpose for at least j^lanning and viewing the paneling on the ap- 
 proximately unfolded surface. 
 
Plate No. 84 
 
PLATE 84. 
 
 Hand-railing for Circular Stone Staircase. 
 
 Fig. I. — Elevation of treads and rise sufficient in number to show the height of hand-rail, length, and 
 position of balusters, and placing the heights A B and C D at the top and bottom of the flight, giving the 
 position of D and A above the floors. The joints E and F show their place in stone-work to be at the centre 
 of baluster. At Fig. io, Plate 76, drawings and explanations are given to cut the balusters to an exact length, 
 the tops to fit the 7varped bottom of rail. Between the centres of these two balusters there are five treads, as at 
 plan Plate 81. At the top the baluster sets, as at plan, on the third tread down ; so also at the bottom the 
 baluster sets, as at plan, on the fourth tread up. The treads H J are taken on the centre line of rail 3, 3, 
 Plate 81. From F and from E draw the horizontal lines E B and F C ; CD equals 7 C and B A equals R F, 
 both of Plate 81.* 
 
 Fig. 2. Plan of First Wreath-piece 8 S R, Plate 81, to be Prepared for Drawing the Face- 
 mould. — Make H C at right angles to D H and equal A B, Fig. i ; connec^t C D ; parallel to the tangent A D 
 draw H G, L N, W X U, 0 P and S R T ; make E G equal H C ; connect G A ; then the bevel at G i^idicates 
 the angle for squaring the wreath-piece at joint over A. The angle for squaring the tvreath over joint H is found 
 as folloivs : — Prolong F H to J ; jjrolong A D to J ; on H as centre with H K as radius describe the arc K I ; 
 connect I J ; then the bevel at I indicates the angle sought. 
 
 Fig. 3. Face-mould from Plan Fig. 2 ; also Showing the Squaring of the Wreath-piece 
 at Both Joints. — Draw tlie line T N ; make A R equal A R, Fig. 2 ; make A P U N equal the same at Fig. 
 2 ; at right angles to T N draw R S, A D, U W and N L ; make A D equal A D of Fig. 2 ; also A B equal A B 
 of Fig. 2 ; make D B equal D C of Fig. 2 ; make R S equal T S of Fig. 2 ; make N Y L and U X W equal 
 M Y L and V X W of Fig. 2 ; through B draw L 0 ; make B 0 equal B L ; make A T equal A P ; make the 
 joint B at right angles to D B ; through P X Y 0 and T S D W L trace the curved edges of the face-mould. 
 Squaring the wreath-piece at joint A is done with the angle G, Fig. 2, and at joint B with the angle I, Fig. 2. 
 The dotted lines show the movement of the face-mould along joint A to plumb the sides of the wreath-piece 
 from joint to joint. Side-moulds are given for this wreath-piece at Plate 76, Figs. 4, 5 and 6. A face-mould 
 of the above character is explained in detail at Plate 13. See also Plate 32, Figs. 4, 5 and 6 ; also Note 
 appended. Side-moulds for this case are given at Plate 76, Figs. 4, 5 and 6. 
 
 Fig. 4. Plan of Second and Third Wreath-pieces R U and U T at Plate 81 to be Prepared 
 for Drawing the Face-mould. — Make the perpendiculars E D and G K each two and a half rises ; connect 
 D G and K H ; through E draw H J C ; on G as centre with G D as radius describe the arc D C ; parallel to 
 G A draw R Z, Q Z and I Z ; parallel to E D draw X P, L 0 and M N ; prolong G E to B ; on E as centre 
 describe the arc F B ; connect B A ; then the bevel at B indicates the angle with which to square the ivreath-piece 
 at both Joints. 
 
 Fig. 5. Face-mould from Plan Fig. 4, also Showing the Squaring of the Wreath-piece at 
 Both Joints. — Draw the line N C ; make J C and J N each ecpial J C of Fig. 4 ; make J M at right angles 
 to N J and equal to J G of Fig. 4 ; make C G H L and N G H L each equal D P 0 F N of Fig. 4 ; at right 
 angles to N C draw lines through each of these points ; make G F, G 0, H 0, H X, L 0 and L X equal X Z, 
 X I, L 0 and L X equal X I, X Z, L Z, L Q, M Z and M R of Fig. 4 ; through C draw F 0 ; make C 0 equal 
 C F; trace the curved edges of face-mould through the points thus found, and make the joints C and N 
 at right angles to the tangents. 
 
 Fig. 6. Squaring the Wreath-piece with Face-mould, Fig. 5, and the Angle at B, Fig. 4. — 
 The dotted lines show the position of face-mould in squaring the wreath-piece as it is moved on the slide- 
 line from Z to X. A side-mould for this wreath-piece for the convex and concave is a straight parallel strip 
 of sheet zinc of a width equal to the thickness of rail and long enough to reach from joint to joint around the 
 convex. The face-mould, Fig. 5, is explained in detail at Plate 15. The case of hand-rail, Figs 2 and 3, 
 limited by length of tangents, cannot always be made to answer the required height in this geometrically easy 
 way, and for this reason we are obliged to use other and less desirable methods, as follows : 
 
 Fig. 7. Plan of Wreath-piece same as Fig. 2 to be Prepared for Drawing a Face-mould that 
 will Bring the Rail to a Lower Level, as Much as A G, Fig. i.— Let U Y B equal H C D, Fig. 2 ; let B E 
 ecpial A G, Fig. i ; from F draw F R parallel to B U ; make F R equal B U ; make Y S equal B E ; draw S K parallel 
 to U B ; parallel to U Y draw K J ; from J draw J R, the governing level line common to both planes ; parallel 
 to J R draw V 0 6, 5 Q L, 3, 2 N, etc. ; at right angles to J R draw U W and FX indefinitely ; on B as centre, 
 with B Y as radius, describe the arc Y W ; and again on 8 as centre, with E F as radius, describe an arc at X ; 
 connect X W ; parallel to J R draw B M indefinitely ; prolong Y T to M ; prolong B U to H ; on U as centre, 
 with U G as radius, describe the arc G H ; connect H M ; then the angle at H zvill square the wrealh-piece over 
 joint U. The angle for squaring the wreath-piece over joint F will be found on the plumb face of joint 
 presently to be explained. Make D C half the thickness of stone used to get out the wreath-piece, and from 
 C draw C D at right angles to F E. D E is to be added to lower joint of face-mould for extra stone required 
 to admit of cutting a plumb joint through the centre of thickness at that end. 
 
 Fig. 8. Face-mould from Plan Fig. 7.— Draw the line X W ; make X Z W equal X Z W of Fig. 7 ; 
 make W B equal Y B of Fig. 7 ; make X B equal F E of Fig. 7 ; connect B Z ; make X R equal D E of Fig. 7 ; 
 make the joints R and W at right angles to the tangent ; make W I G K equal Y I G K of Fig. 7 ; make X P 2 
 equal F P 2 of FiG. 7 : make I, V, I, 6 equal 0 V, 0 6 of Fig. 7 ; make P 5, P L, 2 N, 2, 3 equal Q 5, Q L, 2 N 
 and 2, 3, etc., of Fig. 7 ; trace the curved edges of the face-mould through the points thus found. The 
 slide-line is at right angles to B Z. A face-mould of this character is given in detail at Plate 16. 
 
 Fig. 9. Squaring the Wreath-piece by Face-mould, Fig. 8, and the Angle at H, Fig. 7. — 
 The slide-line must be squared through to the other face of stone. The angle for squaring the wreath-piece 
 at joint 0 is found at H, Fig. 7, and controls the sliding of face-mould, which is moved up along the slide- 
 line, as shown by the dotted lines, until the tangent on the face-mould touches X, the point of plumb line, A 
 at A B on the joint ; then mark the lower point P ; the same thing is done on the other face of stone ; the 
 face-mould being moved again, but downward, along the slide-line until the point 0 — the tangent point — 
 squares from the joint to B. Fig. id is a sketch merely introduced to show that the joint P through the 
 stone is cut plumb, as at F Y, and the easement to a level is forced in the manner shown. B E and E Y is the 
 two heights B E and U Y of Fig. 7. The tangent point P, Fig. 9, if marked at both faces of the stone, as it 
 should be, is represented by P P, Fig. 9, and governs the squaring at this joint. Proper side-moulds — that is, 
 to suit this case — are given for the wreath-piece Fig. 9, at Plate 76. 
 
 * The face-mould for the wreath-piece joining at F and raising the height C D 'S not given because Fics. 2 and 3 cover it 
 the only difference being is that it i^ a little shorter piece— TV at Pi.atf, Si. See Plate 56 for sliding face-moulds, etc. 
 
PLATE 85. 
 
 Squaring Base and Cap Mouldings for Wainscot of Circular Stone Staircase. 
 
 Figs. I and 2. Plan of Base Moulding Shown at A, Fig. 5, and at SV and V X of Fig. 6, 
 Plate 82, to be Prepared for Drawing Face-moulds. — G 0 is tlic radius to face of wainscot from G 0 
 It plan Plate 81. At Fig. i on the face line of wainscot make J L etjual S R, Fig. 6, Plate 82 ; take also 
 the four divisions R P and set them from L to 0, and from 0 to N set off the four divisions V W at F"ig. 6, 
 Plate 82 ; through N draw Y G ; through 0 draw C G ; draw the tangent 0 F at right angles to 0 G ; make 
 0 B equal P V of Fig. 6, Plate 82 ; draw N S at right angles to N G ; through 0 draw S F at right angles to 
 0 G ; draw S K perpendicular to 0 S ; make S K and N Y each equal one half of W X at Fig. 6, Plate 82 ; 
 connect Y S and K 0 ; parallel to K 0 draw B F ; from F draw F J, and prolong J F to C ; on 0 as centre 
 describe the arc D E ; connect E C ; then the bevel at E indicates the angle that is required to square the 
 moulding at the joint over 0 ; from 6, 0 and M parallel to F J draw 6, 8, 0 1, M Q, and 5 R ; make H I equal 
 0 B ; connect I J ; f/ien the bevel at I indicates the angle that mill square the moulding at the joint over J ; 
 from 0 at right angles to F J draw OA; on F as centre with F B as radius describe the arc B A ; connect 
 J A. At Fig. 2 draw the radial S 9 indefinitely ; then S 9 is the governing level line ; through N draw the line 
 OX; on S as centre with S Y as radius describe the arc Y X ; prolong S N to L ; on' N as centre with N Z 
 as radius describe the arc Z L ; connect L G ; then the bevel at L shoics the angle required to square this 
 piece of mouliling at both joints ; parallel to S 9 draw U V and C C ; at right angles to S N draw V W and C P. 
 
 Fig. 3. Side-mould for Fig. 4. — Draw the line A C B ; divide 5, 6 of Fig. i into four equal parts ; 
 set these parts from B to C ; make C A equal 5 X of Fig. i ; make B D, W S, Y R, and X Z ecjual P V, 0 0, 
 
 0 0, 0 0, Fig. 6, Plate 82 ; with these points as centres describe circles equal in diameter to the height of 
 base moulding ; touching the circles, trace the curves of tlie concave side-mould. 
 
 Fig. 4. Face-mould from Plan, Fig. i ; also Showing the Squaring of this Piece of Mould- 
 ing at Both Joints. — Draw the line K L ; make J S at right angles to K L and equal to J F, F"ig. i ; make 
 J 0 equal J A of Fig. i ; make S 0 equal F B; connect 0 S ; make K J R Q I L equal K J R Q I, 3 of Fig. 
 
 1 ; parallel to J S draw L Z, 1,0, Q M, R N, K E, and make these equal N 6, H 0, Z M, X 5, J L of Fig. i ; 
 connect R N and K E ; make the joint 0 at right angles to 0 S ; trace the curves as found. 2 he angle to 
 square this piece at joint S is found at I, Fig. i, and for joint 0 at E, Fig. i. In marking the stone see that 
 there is enough width equal to X X, X X, or a little more, taking notice that more width is required from the 
 tangent one way than the other. At least one face of the stone must be perfectly true and out of wind. 
 
 Fig. 5. Face-mould from Plan Fig. 2 ; also Showing the Squaring at Both Joints.— Draw 
 the line A B ; make C A and C B each equal T X of Fig. 2 ; at right angles to A C draw D E ; make C D, C E 
 equal T S, T R of Fig. i ; connect D B and D A ; make D G 0 and D G 0 each ecjual S W P of Fig. 2 ; par- 
 allel to E D draw S 0, S 0, F G, F G ; make 0 S, 0 S each ecpial C C of Fig. i ; make G Fand G F, C D and C E 
 each equal V U, F R, and T S of F'ig. i ; make the joints B and A at right angles to the tangents ; trace the 
 curves through the points found. The angle to square this piece of moulding at both joints is given at L, Fig. 2. 
 This face-mould is the same as that given for hand-rail at Plate 84, Figs. 4, 5, and 6, and also at Pla'i e 15. 
 
 Figs. 6 and 7. Cap-moulding of Wainscot Shown at Section B, Fig. 5, and at T M 0 of 
 Fig. 6, Plate 82. — M 0 is the radius to face of wainscot ; J 0 on face of wainscot is taken in the four 
 divisions K T of Fig. 6, Plate 82 ; also 0 X is taken in four parts from M N of the last-named figure and 
 plate ; the two heights at Fig. 7, 5 S and X W, are found at N 0 of Fig. 6, Plate 82 ; draw 0 E at right 
 angles to 0 M ; make the height 0 4 ecjual K M of Fig. 6, Plate 82 ; parallel to S 0 draw 4 E ; connect E J ; 
 at right angles to E J draw J B; parallel to E J draw OH, D6, and I A; make G H equal 0 4; connect 
 H J and prolong to F and to A ; then the bevel at H indicates the angle required to square the moulding at joint over 
 J ; prolong J E to X ; make 0 R equal 0 K ; connect R X ; then the bevel at R gives the angle to square the 
 moulding at joint over 0 ; at right angles to J E X draw 0 C ; on E as centre with E 4 as radius describe the 
 arc 4 C ; connect C J ; at Fig. 7 draw 5, 9 to M indefinitely ; then 5, 9 is the governing line ; through X draw 
 OP; on 5 as centre with 5 W as radius draw the arc W P ; prolong 5 X to Y ; on X as centre with X 1 as 
 radius describe the arc I Y ; connect Y M ; then the bevel at Y indicates the angle required to square the moulding 
 at both joints over X a/id 0 ; parallel to 5, 9 draw 3 U and Q Z ; parallel to X W draw U V and Z Z. 
 
 Fig. 8. Face-mould from Plan of Moulding Fig. 6 ; also Showing the Squaring at Both 
 Joints. — Draw the line F A ; make F J H 6 A equal the same of Fig. 6 ; at right angles to F A draw A I, 6 D, 
 H 4, J E, and F S ; make F S, J E, H 4, 6 D, and A I equal J E, G 0, L D, and B i of Fig. 6 ; make the joint at 
 4 at right angles to 4 E ; trace the curve lines. The angle to square this piece of moulding at joint J is found 
 at H, Fig. 6 ; the angle for joint 4 is taken from R, FiG. 6. 
 
 Fig. 9. Side-mould for Fig. 8. — Draw the line KT; divide the concave L D I, F'ig. 6, into four 
 parts and set them from K to T ; draw K M, X X, X X perpendicular to K T ; make K M, X X, X X equal the 
 same at Fig. 6, Plate 82 ; through the points thus found describe circles equal in diameter to the thickness 
 of rail, and trace the curved lines touching the circles ; make the angle M 0 8 equal the angle 4, 0, 8, Fig. 6, 
 and make the joint at right angles to M 8. 
 
 Fig. 10. Face-mould from Plan Fig. 7; also Squaring this Piece of Cap-moulding at 
 Both Joints. — Draw the line A B ; make D B and D A each equal P T of Fig. 7 ; at right angles to B A draw 
 C D G ; make D C equal T 5, Fig. 7 ; connect B C and A C ; make the joints B and A at right angles to the 
 tangents B C and A C ; make B Z E and A Z E each equal W Z V, Fig. 7 ; parallel to C G draw Z Q, E F ; make 
 Z Q, E F, D G equal Z Q,3 U, and T N of Fig. 7 ; trace the curves. The angle to square this piece of mould- 
 ing at both joints is taken from Y at Fig. 7 Where stone is used — such as onyx — that has pronounced bed 
 lines, then in case of mouldings or hand-rail worked from a face-mould like Fig. 8, the face of stone requires 
 such extreme inclination that the bed lines are thrown on top, or obliquely, in that direction ; and when 
 joined to a piece of moulding worked from F'lG. 10, or, a similar piece of hand-railing, where the inclination 
 of the face of stone is, as in this latter case, slight, then it becomes evident that the two — the first mentioned 
 of which the bed lines are nearly vertical and the latter nearly horizontal — will not answer joined in that 
 position at alb The remedy for this mismatching is to square up such a piece as F"iG. 8 out of soft pine, and 
 copy it with patterns or templets marked on two parallel ])lane surfaces as the twisted piece lies flat. These 
 parallel plane surfaces toucli the twist at one or more points, and the space between the flat lying piece and 
 the planes is the exact thickness of stone required, and no less or greater thickness can be correctly used. 
 
Plate No. 86 
 
 POUL8ON 4 Eger, Hecla Architectuhai. Bronzf and Iron Works, 48 to 51 World Buildinq, New York. 
 
IRON STAIRS. 
 
 Iron Staircases of tlie following designs, and variously finished, have been built in New York, 
 Chicago, and most of the principal cities of the East and West. Iron balustrade work of these and 
 other elaborate designs is also combined with wood and stone stairs. Where finish is desired 
 beyond mere pai.iting, advantage is taken of several scientific and mechanical processes. 
 
 Electro-plating is done on stair work in bronze, brass, and antique brass. The railings, newels, 
 and all high-relief lines and mouldings are polished to afford artistic contrast. The electro-plate is 
 ihcrouglily substantial and durable. y\ 
 
 Bower-Barff Rustless Process. — This treatment. Imparts to the surface of the iron a blue-black 
 color on rougher castings, while on polished work the iro'rf ;rfeislfi>§; its lustrous appearance with a 
 beautiful steel-blue shade, that harmonizes well with electj-p-plalecj. v\jt>,ffc w'hsrc it is used in com- 
 bination. ;"\' ' ^ ^f,',-,^^ 
 
 Galvano-plastic. — By electrolytic process are reproducecf, /wh'en desired,' {5a'ne'i5, ,'njbuldings, and 
 ornamental detail in solid bronz-copper. These can be given a m^dl' ^fiiii<;5i" -of bronze; brjfss; an^tique 
 brass, silver, oxidized silver or Bower-Barff, by a subsequent electro-plate.'-.. Th^'-Meqla Aichi^'tejclural 
 Bronze and Iron Works reproduce by this means at a nominal cost details of des'igtl ffom expensive 
 and highly-artistic originals — of which they have a large variety — in solid metal tliat cannot be 
 equaled by any other known process. This galvano-plastic work is well adapted to panels in wain- 
 scoting and decorative interior work ; and it can be incorporated in the matter of design with electro- 
 plated cast-iron work in infinitely varying ways. 
 
< r • * « 
 
Plate No. 87 
 
Plate No. 88 
 
POULSON & EOEB, HECLA ARCHITECTURAL BRONZE AND IRON WORKS, 48 TO 51 WORLD BuJLDlNG. NEW YORK. 
 
Plate No. 90 
 
 P0UL8ON ir. Egeh, Hecla AHCHiTecruRAi Bron?f an& ikon Works, 48 to 61 World Buiidini., New York- 
 
Plate No. 91 
 
 PouLSON & Eger, Hecla Architectural Bronze and Iron Works, 48 to 51 World Building, New York. 
 
Plate No. 92 
 
Plate No. 93 
 
 POULSON & EGEH, HECL* ARCHITECTliHAL BRON7E AND IRON WORKS, 48 TO 51 WORLD BUILDING, NEW YORK 
 
Plate No. 94 
 
Plate No 95 
 
 PouLSON & Egeb, Hecla Architectural Bronze and Iron Works, 48 to 51 World Building, New York. 
 
Plate No. 96 
 
 Po jtsoN 4. Eger, Hecla ARCHiTECTiiRAi Bron;f and Iron Works, 48 to 51 World Builoinq, New York. 
 
Plate No. 97 
 
 PouLSON & Eger, Hecla Archjtectural Bronze and Iron Works, 48 to 51 World Building, New York. 
 
Plate No. 98 
 
 PouLSON & Eger, Hecla AR'^hitectuhal Bhonjf »nd Ihon Works, 48 To b1 World Building, New York 
 
Plate No. 99 
 
 PouLSON & Eger, Hecla Architectural Bronzf and Iron Works, 48 to 51 World Building, New York. 
 
Platk No. 100 
 
 J . — 
 
 PouLSON & Egeh, Hecl* AfiCMiTEcrLRAi Bronze (^^o Ikon WoftKS, 4S to b'\ WoiIi-d Bui^oino, New Yoh^ 
 
Plate 
 
 POULSON 4 EGER, HECLA ARCHITECTURAL BRONiE AND IH'JN WORKS, 48 TO 51 WORLD BuiLDING, NEW YORK. 
 
Plate No. 102 
 
 P L/\rsl 
 
 ScA LE |- I N.= 1 Ft 
 
 PLAN OF IRON STAIRS: — This plan commends itself for many situations and business purposes 
 by the very little space occupied; it being also strong, compact, simple, lasting and neat in appearance. In 
 ascending by this plan of steps it is done from side to side as numbered 1, 2, 3, etc; while a handrail 
 fixed at the middle is grasped as a guard and assistant. See Plate 103 for a section in elevation of this plan 
 
 Poulson & Eger, Hecia Architectural Bronze and Iron Works, 48 to 51 World Building, New York. 
 
POULSON & EGER, HECLA ARCHITECTURAL BRONZE AND IRON WORKS, 48 TO 51 WORLD BUILDING, NEW YORK. 
 
Plate No. 104 
 
 VIEW OF THREE DIFFERENT FORMS OF STAIRS BROUGHT TOGETHER :— The flight 
 on the right hand is a straight one; that on the left is in plan a quarter circle: the nriiddle flight is 
 also in plan a circular double flight designed for convenience of ascent and descent. They are all 
 constructed after the nnethod shown and explained in plan and section at Plates 102, 103. These 
 compact, neat and cleanly Iron Stairs supply a want more particularly in business places by their great 
 economy of space. 
 
 PouLSON & Eger, Hecla Architectural Bronze and Iron Works, 48 to 51 World Buildino, New York. 
 
Plate No. 105. 
 
Plate No. 106. 
 
Plate No. 1 07. 
 
 Newels and Balusters Manufactured by the Standard Wooo-TURNiNa Co.* 206 Greene St., Jersey City, N. J., U. S> A. 
 
 Send 4 cts. in stamps for Catalogue and Price-List. 
 
Plate No. 108. 
 
 Newels and Balusters Manufactured by the Standard Wood-Turning Co.. 206 Greene St., Jersey City, N. J., U.S.A. 
 
 Send 4 cts. in stamps for Catalogue and Price-List. 
 
Plate No. 109. 
 
 Newels and Balusters Manufactured by the Stansaro Wood-Turninq Co.. 206 Greene St.> Jersey City, 
 
 Send 4 cts. in stamps for Catalogue and Price-List. 
 
MONCKTON'S 
 
 Practical Geometry 
 
 BEING A 
 
 SERIES OF LESSONS BEGINNING WITH THE SIMPLEST PROBLEMS AND 
 IN THE COURSE EMBRACING ALL OF GEOMETRY LIKELY TO BE 
 REQUIRED FOR THE USE OF EVERY CLASS OF MECHANICS 
 OR THAT ARE NEEDED FOR INSTRUCTION 
 IN MECHANICAL SCHOOLS. 
 
 ILLUSTRATED BY 42 FULL PAGE PLATES. 
 
 BV 
 
 JAMES H. MONCKTON, 
 
 Author of Monckton's "National Carpenter and Joiner," and Monckton's "National Stair Builder." 
 Instructor for many years of the Mechanical Class in " The General Society of Mechanics 
 and Tradesmen's Free Drawing School " of the City of New York. 
 
 Price $1.00. 
 
 CONTENTS. 
 
 Introduction. Drawing instruments, tools and materials required to be- 
 gin with, and liow to use them. A 1' square. Right angle triangles. 
 Drawing boards and tacks. The compasses and its a tachments. The 
 line pen. Proper way to handle compasses. The lead pencil. Six 
 rules for using tlie drawing instrumenis. 
 
 Plate i. Geometry. A point. A line. A curve line. A composite line. 
 A mixed curve. A zigzag line. A vertical line. A level line. A per 
 pendicular. 
 
 Plate 2. A circle. The radius. The diameter. A segment of a circle. 
 Concentric circles. Eccentric circles. A tangent. Tjngent circles. 
 
 Plate 3. To erect a perpendicular from a given line. Parallel lines. 
 Oblique lines. An angle. Curvilinear angles. Mixtilinear ingles. 
 To bisect a line. 
 
 Plate 4. To bisect an angle. A diagonal. To erect a perpendicular at 
 the end of a given line. To let fall a perpendicular to a given line 
 from a given point. Angles. To trisect a quarter circle. To inscribe 
 a square in a circle. 
 
 Plate 5. A superficies. A plane figure. An equilateral triangle. An 
 isosceles. A scalene and a right angled triangle. A square. An 
 oblong. A rhombus. Aihomboid. A trapezoid. A trapezium. Quad- 
 rilaterals and tetragons. 
 
 Plate 6. Polygons and their construction. From a given side to con- 
 struct an equilateral triangle, a squ.ire, a pentagon, and a hexagon. 
 
 Plate 7. To inscribe an octagon within a square (two methods). Alti- 
 tude or height of a triangle. An angle nscribed in a semicircle. To 
 inscribe a square within a square. An equilateral triangle inscribed in 
 a circle. To circumscribe a square about a circle. 
 
 Plate 8. To place a line at right angles to a given line by the use of any 
 scale of equal parts, and its application for a mechanical purpose. To 
 circumscribe an equilateral triangle about a given circle. To find the 
 centre of a circle. To inscribe a circle in a triangle. 
 
 l^LATE 9. To copy an angle. To copy an irregular angular figure. To 
 describe a circle through any three points not in a straight line. To 
 find a right line equal to the semicircumference of a circle. 
 
 Plate 10. To find the circumference of a circle by another method. Con- 
 vex and concave. The square of the hypothenuse of a right angled 
 triangle is equal to the sum of the other two sides. 
 
 Plate ii. To construct a triangle from given sides. To divide a circle 
 into six equal parts; into eight equal parts; into twelve equal paris. 
 
 Plate 12. Measurement of angles. Rapid method of dividing a gi en 
 space into any number of equal parts. 
 
 Plate 13. To divide a line into any number of equal parts. 
 
 Plate 14. Measure of the angle at the centre and circumference of the 
 circle. To make a square equal to three given squares. To describe an 
 ellipse (two methods). 
 
 Plate 15. Three methods of describing the ellipse. 
 
 Plate 16. Two other methods of describing the ellipse. To draw a tan- 
 gent to a given point on the ellipse. To find the point of contact. 
 
 Plate 17 To find the axis of a given ellipse. To circumscribe a rectan- 
 gle by an ellipse. To find the axis of an ellipse proportional to theaxi* 
 of a given ellipse. Parallel elliptic lines impossible. Tj describe an 
 approximate ellipse with arcs of circles. 
 
 Plate 18. To describe an approximate ellipse more accurately. To de- 
 scribe an eggoid. To describe a simple spiral or scroll with arcs ol 
 circles. 
 
 Plate iq. To describe an egg-shaped oval with arcs of circles. To de 
 scribe a cycloid. To desci ibe an epicycloid. 
 
 Plate 20. To describe an epicycloid by a circle rolline; within another cir- 
 cle. To describe an involute. To describe a spiral or involute of one 
 or more revolutions. 
 
 Plate 21. Arches. A semicircular arch. A platband. The elliptic arch. 
 
 Plate 22. To describe various forms of gothic arches. 
 
 Plate 23. A rectangular prism. The cube. A cube cut by a plane pass- 
 ing through the diagonals. Cut by a plane through the centre. 
 
 Plate 24. A square prism cut at any height. To develop the surface of 
 the truncated prism. 
 
 Plate 25. A pyramid. A triangular prism. A hexagonal prism. A square 
 pyramid. A hexagonal pyramid. 
 
 Plate 26. A regular tetrahedron. A regular octahedron. The dodecahe- 
 dron. The icosahedron. 
 
 Plate 27. A right cylinder. To find a section. To find an envelope. 
 
 Plate 28. To find the section of a right cylinder cut obliquely. To find 
 the envelope. 
 
 Plate 29. A cylindroid To find the section. 
 
 Plate 30. To find the section of a cylindroid cut obliquely in two direc- 
 tions. A sphere. The covering of a sphere. 
 
 Plate 31 To describe the covering of a sphere on lines parallel to centre 
 plane. The sections of a sphere. Prolate spheroid. Sections of a pro- 
 lite spheroid 
 
 Plate 32. Covering of a prolate spheroid. An oblate spheroid. Sections 
 
 of an oblate spheroid Covering of an oblate spheroid. 
 Plate 33. The helix. 
 Plate 34. The cone and its sections. 
 
 Plate 35. The ellipse liom a cone. The parabola. The hyperbola. 
 Plate 36. To describe the parabola by another method. The ellipse from 
 a cone. 
 
 Plate 37. To describe a parabola by the intersection of tangent lines. 
 The hyperbola on the principle that sections of a cone parallel to the 
 base are circles. To develop the covering of a cone. 
 
 Plate 38. To develop the covering of a cone, with the trace of the inter- 
 section, when an hyperbola— also when a parabola— is produced by 
 cutting plane. 
 
 Plate 39. To develop the covering of an oblique cone together with the 
 
 traces of several cutting planes. 
 Plate 40. Scale of equal parts. A diagonal decimal scale. 
 Plate 41. The protractor. The straight protractor. 
 
 WILLIAM T. COMSTOCK, Publisher, 23 Warren St., New York. 
 
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