'•*^- ^^-^^^ .e^2^ NOTICE This Book is the Property of AUG. S. BURGESS /^nd whoever borrows the same is requested not to lend at to any individual, take particular pains to keep it clean, and not lurn the corners of the leaves down, or deface it whatever. AUG. S. BUEGESS. N B. A borrowed book should be returned as good as when taken.— A. S. B. UCSB LIBRARY 1%^'^ 00 LO X O n ci ij o cc D n d < PEOPLES' POCK[T STl BUILDER -ANl Carpenters' Hand Book. CONTAINING FIFTY- ONE PLATES. AND OVER FIVE HUNDRED FIGURES, WITH A FULL DESCRIPTION FOR EVERY FIGURE. EMBRACING Carpenters' and Stalr-Bnllders' Geometry, Problems, Conic Sections, CyllndLric Keetlons, as applied in tbe construction of tbe Wreatb part of Hand-rail. Rales for tbe Measure- ment of Surfaces. Tbe Construction of I^adders, Box Stairs, Dog-legg:ed, Open Newel, Cylinder, Circular and Elliptical Staircases. ROOFING— Hip, Talley, and Jack Rafters. PurUns, Splayed Work, and Bevels for tbe same. Transverse Strengrtb of Joist and Beams, and easy formulas for tbeir safe load. ALSO, Excavators, Stone and Brick Masons, Plasterers and Carpen- ters' Memoranda ; witb a variety of miscellaneous informa- tion, useful in tbe practice of tbe Arcbltect and Builder; to^etber witb a Glossary of Arcbitectural Terms, and General Index. By WILLIAM PEOPLES. David Williams Company, Publishers, 232-238 William St., New York. PREFACE. For some jears it has been the aim of tlie author to prepare a small book for the young stair builder-, carpenter and joiner, as a handy reference, that ^vould be to them what Has well and Trautwine are to the engineer. Having now worked at the trade of carpentry and stair-bniiding 48 years, the author is free to say from experience, that a little book of this kind for the pocket is a much needed acquisition for the benefit of the apprentice and journeyman as well. So much has been sai^l on the subject that it is next to impossible to be wholly original, and no claim of that nature is preferred. It is simply an arrangement of ideas, gleaned from the various work of authorities, and modified by the author's practice, embodied in book form. If this little volume should lead the student of carpentry and stair-building to study the suljject carefully, and be induced to practice the same with pleasure and profit to himself, the author will be satisfied that his efforts liave not been in vain. The young man should provide himself with a drawing l)oard, say fi4'' l>y '.jiV^ and JX'^ thick, with two battens across the back to keep the board true; joint tlie same square and straight, and to a parallel width. Provide a .set of draughting inrtruments, T square about 3' long, and two triangles, one 45° and the other 60°, a dozen thumb tacks, a 6-H pencil and rubber. Use India ink for inking. Tlie instruments can all be bought separately, and if the young man cannot afford to ))urcliase a full set of good tools, then buy one at a time, for one good tool is worth more than a chest full of bale by means of narrow inelined passages. Stairs do not appear to have ever been used in the pyramids, unless perhaps, in some opened recently ; but in the tombs of the same date tligiits of steps are always found, i CiiAT.DKAXs. Their temples were sometimes built with oR-sets, forming a terrace every story, and readied by flights of steps or inclined planes from one story to the next, as in the case of the Tower of Babel, erected 29A7 B. C, which is described in one of the mosaics at St. Marks,- Venice, showing inclined boards for obtaining access to scaffolding, instead of ladders. ^ Herodotus* says, "numbers of houses were three and four stories high. AssYiUAXS. Owing to the scarcity of stone, sun-dried brick were used for walls, ami probably wood for rooling and interior tini.sh — hence their buildings have long since disappeared, the ruins scarcely being visible. The ruins of the largest royal palace at Nineveh, erected by Sennacherib about 704 B. C, show the remains of an inclined plane, ten feet wide; close by is also seen, the ruins of an inclined terraced temple. Pkijsiaxs. The ruins of Persipolis, the ancient capitol of Persia, show the palaces to be reached by the grandest double- flight of platform stairs in the world. = The stejjs leading to the lirst terrace are 25' 7'^ wide; the rise is 4^' high l)y 14'^ in the tread ;•' there are flfty-five rises unconcealed on one side of the platform, and forty-eight to the landing ; fiom four to six steps being cut from a single block of marble. A second double-flight leads to the next terrace ; steps 16' long, 3" rise and 14" tread, having thirty-two steps to each flight; ten to fourteen steps being cut fnmi a single block. ^ The sides of the steps are walled up and decorated with sculptures, representing processions bringing tribute to the King. These palaces formed one of the most im- posing groups of buildings ever erected, ami is said to have been partially burned bv Alexander the Great, after his conqiu'st of Persia 330 B. C. 1. N. ClilYord liii'ker. Professor of Aivliiteotuie. iu tlie liliuois University. 2. ErectedOTC;— 1071 A. r». '!. Miirwick's History of t^tair Cases. 4. Flourished 484—408 B. C. 5. Enrycloptedia liritantiiea. t>. Gwilt's ])ictionai-y of Architecture. T. Stuart's Dictionary of Arehitecluie. 10 IIjstoky UK Staik Blilding. Jeki'SALem. Solomon's Temple was erecttcl 1011 — 1004 B. C. Ezekiel B. C. 774, iii his description of the temple locates nine flights of steps. This temple was destroyed by the Chal- deans under Nebiichadnezzar, B. C. 588. We read that when the Queen of Sheba saw "the ascent by which Solomon went up to the house of the Lord, there was no more spirit in her." Among the ruins of these more eastern nations, nothing is more remarkal)le than these great flights of steps. The builders of tliose days among the Jews, Chaldeans and Persians so far as we know, were the only people who really understooil the value of this feature. The Egyptians seem wholly to have neglected it. and the Creeks to have cared little about it, but was not so at Nineveh and Persipolis, ^ for from the indistinct traces left, the stairs must have been one of the most important parts of the design. Mexico. The Pyramids of Mexico resemble those of Assyria, being terraced and having inclined planes. The Pyramid of Papantlawas seven stories high, with five flights of steps leading to the cell, which formerly existed on the top.^ This prehistoric strncture was built of stone. Baked brick was mostly used in the l)uilding of these pyramids. See Prescotfs Conquest of Mexico. Gkeece. In the Ancient Doric Temple of Concoid, at Agrigentum, Island of Sicily, [when erected unknown] a flight of forty-one steps^o leads to the roof from near the entrance; they are close, being constructed in the thickness of the wall. Similar stairs are said to be found in the temple of Jupiter at Olympia, erected about 435 B. C. The Syracuse Theatre, erected 480 — 406 B. C, had eight radiating flights of steps. The Temple of Jupiter at Aizana, has a grand flight of out- side steps 98' C wide, erected about 140 B. C. Komp:. The ancient Romans bori'owed largely from the (Greeks and Etruscans ; the temples and theatres of the Greeks served as models, from which they improved aiul built up their capitol. Vitruviiis, * ^ however, makes no mention of stairs as being an impoilant adjunct of the dwelling house; if the stair case had lieen of much consideration, he w(mld have most likely described them. He gives the numbers 3, 4 and 5 for a right angle triangle, and suggests the hypothenuse of the same as a good proportion for the inclination of a flight of stairs, liule, divide the whole height into three equal parts, and take five of those parts for the length of the inclination. The stairs in the Pantheon^'' are triangular on plan, so were they in the Baths of Caracalla.^^ jq the Baths of Diocletian^ ^ the stairs were built between walls, in the same manner as we build l»ox stairs Ijetween plastered partitions. It is said that 40,000 Christians were compelled to erect this stnicture to appease the wrath of this Roman tyrant. The summit of the historic column of Trajan^' is reached by a winding staircase, containing 185 steps ; they are carved out of solid blocks of marble twelve feet in length, and five feet thick. s. .lames Ferguson's History of Arcliitecture, Vol. 1; Pajre 192. 9. N. Clilford Kicker. 10. Josepli Gwilt's Encyclopaedia of Architecture. 11. Architect, lie nourished under Augustus Ca?sar, B. C. 15; he wrote the earliest special treatise on architectui-e extant. 12. Erected probably 20 B. C. 13. Finished about 217 A. D, 14. Erected about 302 A. D. 15. Erected 114 A. D. HlSTOKY OF bXAlli BlILUINU. 11 The country house at Pompeii has steps with a rise of twelve inches.*® The Romans, also tlie Greeks, prided in wide outside steps at tlie main entrance of tlieir temples. Tlie magnificent temple of tlie Sun God at Baall;ec, ^^ lias a flight of outside steps 164 feet wide between the buttresses. Of the thirty odd different liinds of artisans, advertised for by Constantine the Great, for the building of Constantinople, i" it appears "stair hands "i^ were required. The towers and minarets of the Mahometan Mosques, had winding inclined planes, and winding stairs to reach the upper stories ; sometimes the stairs were built winding around the outside, at other times they were built in the thickness of the walls. Italy. In this beautiful land of art, during the middle ages from 1000 to 1500 A. D.. the main stair cases were often constructed on the outside of their buildings, the climate being more favorable than in the more northern countries. In court yards they were often built alongside of the wall, and supported on pillars and rampant arches, sometimes they were protected with a raking roof, supported by columns, set on pedestals at regular intervals, and balustraded between the pedestals: these outside steps were often treated very artistically for dwellings, public buildings and palaces, as they led to the principal floor; sometimes they were double, starting from opposite sides, and landing on a spacious platform at the main enti'ance to the building; this treatment offered a good chance for architectural display. Marble was mostly used in the con- struction of these stairs. Previous to 1500 A. D., very little importance was attached to the inside stairs; they were commonly built in the thickness of the walls. The Campanile or Leaning Tower, at Pisa,^" is circular on plan; the external diameter is 53 feet, and the height is 183 feet; the walls are 13 feet thick, and the stairway is built in their thick- ness, so were the stairs in the beautiful Campanile, ^^^ near the Florence Cathedral . * - The open newel staircase is of Italian origin, and was the invention of Nicola of Pisa, -^ 1205 — 1278; the idea was after- wards utilized and improved by the renaissance architects of Italy, from the beginning of the sixteenth century. In the Belvedere of Pope Julius II, 1503 — 1518, an inclined plane or turn-pike stair case of five revolutions leads to the upper story; the stairs, or inclined plane was carried on continuous cylindric vaulting, and is supported on the side next the well by eight columns in each tier. ^* The architect, Andrew Palladio, 1518 — 1580, improved the art of stair-casing during his day.^^ 16. ytuart's Dictionary. 17. Erected it is supposed by Antoninus Pius IAS— ICl A. D. 18. Founded about 'dis A. D. 19. See London Building News, 1S90. 20. Built 1174 A. I). 21. Erected at tlie beginning of the Fourteenth Century. 23. Tlie Carupanile iu the Piazza of St. Marks, Venice; the heiglit to the loggia is about 177 feet, and is readied by a ramp or Inclined plane, making thirty-si.x rounds: the foundation was laid iu 888 A. D. 23. A noted Architect of Pisa. 24. Marwick's History and Construction of Stair-cases. 2,). He says 'stair-cases will be coniniendable if they are clear, ample and commodious to ascend; inviting, as it were, people to go up. They will be clear if they have a bright and equally diffuse light; they will be sufficiently ample if they do not seem scanty and narrow to the size and qualiiy of the fabric. But they should never be less than four feet in widtli, that two persons may pass each other. They will be convenient with respect to the whole building if the arches under them can be used for domestic purposes; and with respect to persons, if their ascent is not too steep and difficult, to avoid which, the steps should be twice as broad as high." 12 lIi.'iTOKy OF Stair BTTii.Dixa. Many circular and elliptical stair-cases were constructed through- out Italy, of a stately and monumental character. The Farnese Palace stairs is 73^ by 28^; that in Carlo Mademo's Mattei Talace, with its marble walls and rich stucco ceilings, 65' by 40'; that in Ferdinando Fugo's Palace Corsine 72' by 45'; while those in the workhouse at Genoa, and in the stupendous unfinished Royal Palace at Caserta, Naples, are no less than 115' by 63', and 163' by 85' respectively. Scamozzi,^^ 1552 — 1616, mentions a double winding staircase made by Pietro del Boyo and John Cossiu so con- trived that two persons, one ascending and the other descending, should never meet. This style of a stair- case was common tlirough France in Mediaeval times. A patent was taken out in our own country for tliis kind of a staircase, by W. J. Keim, New York, 1868. The well of the staircase in the Palace Braschi. by C. Morelli,'-^ is quadrangular, 40' by 34', having straight flights of the open newel style, similar to those of to-day. The steps arc ten feet long each in one piece, supported by the walls at one end, and resting on rampant arches, springing from columns at the outer strings; tlie columns are sixteen in number, and of red, oriental granite; they stand on pedestals, which receive a broad railing and heavy baluster, which, together with the steps, are of white marble. The wall side is decorated with pilasters opposite each column and a half balustrade corresponding to the outer railing; arches spring from the columns to the pilasters across the soffit of the stairs, forming domical vaulting which is divided into panels, and enriched with beautiful frescoes. The spandrels of the arches are paneled and ornamented with carvings; niches are formed in the walls for the display of statuary. This design of Morelli's is one of the most beautiful staircases in Rome. A fine illusti'ation of two stages of this stair case may be seen in Marwick's History of Staircases. Fraxce. During the first century A. D., the stairs in France were constructed similar to those of Italy, Outside steps were sometimes built at right angles to the building, at other times parallel to the same resting on arches, supported by piers; sometimes they were roofed over to protect them from the rain. During the first centuries the tower stairs were built in the thick- ness of the walls. The French were partial to the winding stairs, they are to l)e found in most of their buildings throughout the country, from about 1000 to 1500 A. D. On account of so many towers, mostly square or circular on plan, the winding staircase suited best, and also they re(iuircd less room in their construction. They, like the Italians first built the well for the staircase square, circular or octagonal on the plan; later on for their public buildings, the ol)long well was occupied with the open newel staircase, having straight llights and quarter pace landings. After 1540 A. D., the French Renaissance style became more classical and dignified. The leading French architects traveled extensively in Italy, from which they gathered ideas and im- proved on them to suit their taste and country. The churches, ehateaus and palaces are monuments of their genius and skill as architects and builders. 2ti. Stuart's Dictionary of Architecture. Scamozzi is said to be the inventor of tlie 1 wo-foot joint rule, also the angular Ionic Volute. 27. Flourished 1780 A. D. HlSTOUY OF StAIK BlILllINO. 13 The Chateau at Chateaudun,^^ jg chiefly noted for its mag- nificent stone winding staircase, one of the finest ever con- structed.'^* Tlie well is partly square and partly octagonal on plan; the steps wind ai'ound a solid newel built of stone; the steps are built in the newel and walls at the ends, each tread is a solid stone, rectangular in section tliroughout, giving to the soffit the same appearance as above ; the tread and rise is with- out nosing and scotia,^" as was the custom in Mediaeval times. The shaft of the newel is deeply cut away, to form a bold pro- jecting half hand-rail and string, which winds spirally around the newel ; the space between the rail and projecting base or string remains plain ; at the junction of the steps, with the newel on the underside, another set of mouldings project from the newel, forming a cornice ; the shaft between the rail and cornice is ornamental, with slender columns and panels between the columns; the panels are elaborately enriched witli carved arabesques. On the wall side, engaged columns fill each angle, extending from the steps and floors to a projecting cornice, that fills the angle made by the steps and walls ; at the upper end the newel is finished with a cap, from which spring arched semicircular groins, to connect the columns in the angles on the wall sides, this vaulting gives support to the roof, and distributes the weight of the roof and vaulting, partly on the newel and partly on the walls, and gives to the whole a beautiful and appropriate finish. Many beautiful staircases of the Renaissance period may be found throughout France, erected after the beginning of the sixteenth century. In the Chateau Chambord, ^^ * a double staircase is erected, the well is thirty feet in diameter; the stairs wind around a hollow newel ten feet in diameier, which ends at the terrace level; a smaller stairs is built in this hollow newel extending beyond the terrace, where it is terminated by an elegant circular arctade crowned by a circular tower, this being beautifully decorated with classical columns and flying buttresses. These double staircases were common throughout France, they were luiilt of wood and also of stone. Owing to the large size of newel required in this kind of stairs, to obtain a fair width of tread at the narrow ends, a third winding stairs was sometimes constructed inside the newel; at otlu>r times the space is divided into rooms, and at other times it is used as a well. Tlie French and (iermans^? often made their steps and rise from solid oak timber, quartering the log, and turning the heart side up; this made a solid and noiseless staircase. Some of these wooden stairs, built during the middle ages, are still to be seen in some of the old turrets througliout that country. Two wooden staircases in the Holy Chapel in Paris, are probably the oldest in existence, being constructed in the thirteenth century. ^s ExGLAND. In England the transition began in the reign of Henry VIII, 1509 — 1547, from the Norman conquest 1066, up to and during part of his reign, close, or blind newel stairs walled 28. A fine illu.stration of two stajres of this staircase may be seen in the American Architect and Bulldin.t? News; of September 17, lK*;i. Erected about 1408—1515 A. D. 29. N. Clifford Ricker. 30. Parkei-'s Glossary of Architecture. 31. Founded 1.526 A. 1). 32. The Stadhaus :it Leyden, Holland, has a fine outside stairs built at the beginnine of Uie seventeenth century. See illustration in the Aruerican Architect, for March 27th, 1886. 33. Marwick's History of Staircases. 14 History of Stair Building. up ou the sides, and arched over in the form of a tunnel was the prevailing style; at the angles heavy piers projected from the walls, finished with base, and cap from which spring groins; the piers were low, and took the form of newels. In the reign of Queen Elizabeth 1558 — 1603, the open newel staircase became more studied; the blind straight flights between walls were modified so as to admit light above the hand-rail, and in place of balusters the space was close and sometimes plastered under the rail. This gave place to a more ornamental construc- trou, broad short flights, with elaborate and heavy carved newels at the angles, supporting vases, baskets of flowers, minature statues, columns, lions or griffins. In this reign the newels formed one of the chief decorative and constructive elements of the design. The hand-rails were broad and massive; under the rail was filled in with fancy cut scroll work, or heavy tuined balusters set ou a close string, which was paneled and ornamented with bold carved work, and finished on the lower edge with hanging scroll, termin- ating at the newels in the form of pendants. This and the succeeding reigns of James I, I(j03— l'i;i5, also Charles 1 reign 1625 — 1649, the balustrade and newel occupied at no other period a position of such conspicuous importance. The decorative feature of the Elizabethan staircase arrived in this reign at its zenith, and the best period had passed; after this tlie style had bcome more debased. ytyle, like history, repeats itself; at the present and near the close of the nineteenth century, we are copying after the style of Elizabeth in our staircases, with short flights, close outer strings around a quadangular well hole, with massive and elaborate carved newels, sometimes extending to the ceiling, and finished with arches; at other times the newel is surmounted with statuary of some anti(iue design. (Serious objections are urged against this style of staircasing — too many quirks and projecti(jns for catching and harboring the dust, and the expense to renovate and keep them in a good sani- tary conditi(m; indeed, after a few years in use, it is impossible to avoid the disease-breeding, accumulations. P'or this reason this style of staircase should be avoided in domestic architecture. The open string, and return nosings, with circular turns in the angles in place of newels, and finished with turned balusters and a continuous hand-rail, gives a more cheerful aspect to the hall, with less expense and better health to the occupants. This style of open string and continuous hand-rail was a great improvement over the old ramn and knee hand-rail and square well hole. In England, about the begiiming of the eighteenth century, the .solid newel for winding stairs, gave place to the open circular well hole, with rail and balusters, thus admitting light and air, and a better sanitary condition of the building. The first lines i)ublislu'd for constructing a continuous hand- rail, over a circular well hole containing winders, was by HaiJ'1'k>{kv, of London, 1725. The method is very much mys- . tified^ * but shows an effort. The ingenious Mji. Fuajscis Fimci:, was tlie next in 1735. He gives a method to draw the straight ramp and knee, also a wreathed ramp and knee over a quarter circle on plan; also a falling mould for squaring the wreath part; also a method to find the length of long and short balusters under the wreath rail, laid off from the center line of rail on the stretchout. He also gives a 34. See Peter Nicholson's Dictionary of Architecture, by Lomax. Page 76. HisTUKV OF Staik Building. 15 method for building up the rail, by glueing blocks to suit the falling line of the rail; he also gave lines for the construction of the scroll, and showed how to ease off an angle, by the intersection of lines; he made use of ordinates to find the contour of raking mould iugs. Ml!. Laxglev, 17.38, was the next. He shows the face- mould for the quadrant to be one-fourth of the ellipsis, having the joints of face-mould on the major and minor axis of the ellipsis, and of course the plank would be canted only one way, without any spring ; this would require very thick plank to form the wreath rail over winders, spaced equally around the well hole. He made use of ordinates in the construction of the face-mould. Mk. William Salmon, 1748, was the next author. He fol- lows Mr. Price, and gives the lines for a face-mould over winders around a circular well hole, without springing the plank. He mentions for the first time a custom among workmen, the method of glueing up the rail with thin strips over a drum made to the concave diameter of rail, when enough strips were tlius laminated for the breadtli of the rail, and the glue had per- fectly dried, the twist was taken from the drum and afterwards moulded to the required shape, and set up all in one piece over the winders. ^'^ Mk. AiiKAiiAM SwAx. 1750 came next, and was fol- lowed by Mr. Willta:\i Paixe, 1774. In the "Practical Builder" he draws the face-mould for the turnout wreath-piece at the scroll, in the same way as shown l)y Mr. Swan ; he used ordinates at right angles to the hypothenuse of pitchboard. At this time he gave two methods how to obtain a backing for a hip rafter, either straight or curved. Later on, he issued the "Builder," illustrating various kinds of stairs, and the manner of drawing face-mould for the same, by using ordinates. For winders, in a semicircular staircase, having straight steps above and below, he shows the face-mould drawn so as to make the wreath for the semicircle all in one piece ; the transverse axis of the face mould being plumb over the chord of semicircle; the face edge of the plank would coincide with the diameter of the semicircle, and the joints connecting the straight part would be spliced joints, for the plank would be canted only the one way; hence very thick stuff would be re(iuired to avoid kinks in the twist part of the rail. Like his predecessors, he dis- covered the wreath-piece to be a portion of a cylindric section, and the curve of face-mould to be elliptical; but what portion of the ellipsis required for each correct face-mould remained a mystery. Mk. Petek Nicholson, 3« in 1793 published for the first time his "Carpenters (luide," and subsequent editions down to 1835. He cleared away some of the mist connected with the con- struction of the face-mould; he discovered how to draw the elliptic curve through three points, which he termed resting ;». In our couiiiry Mr. Asheu Bkn.jamin, 1792. superintended the erection of a ciiculiir staircase in the State House, at Hartford, Conuecticut. Tlie rail was Klued up of strips onc-eiglith of an inch thick, whifii was clainied to be the first geometrical rail, over a cir- cular well hole, constructed in this country. 30. Mh. Petek Nicholson was a native of North Britain; he was horn on tl)e 2ntV) of .Tuly, 1765, in the Parish of Preston Kirk, in the County of East Lothian; at a very early age he evinced a strong iiiechanical genius, and also a turn for drawing whatever presented Itself to liim, whether of animated nature or otlicrwisc ; three years instruction was the most he ever had at a country school, which he left at the age of twelve. Although his .scholastic instructions were very limited, he was bent on inquiry, and that decidedly of a mathe- 16 HlSTOlty OF yxAiK BriLDIXG. points; one point at each end, and tlie other at, or near the center of mould; [not certain as to tlie exact location.] This new method determined that portion of the ellipsis for the face-mould, over a (luarter circle, to contain the minor axis, and hence, the mould was wider at each end than at some intermediate point, which was eijual to the true width of the rail; this required the plank to be canted two wajs, and was termed the •'springing" of the plank. In all former autliors the narrow part of face-mould was always at one of the joints, for a quarter circle on plan, and the plank was canted but one way. Not being certain as to the exact location of the center resting point, left the system in doubt and made the subject an abstruse and difficult task for the learner, and also for the more jnactical man; the cutting plane, or plane of plank would sometimes be correct, at other times wide of the mark. In 1849 Mk. Jkakes published his orthogonal system of hand-railing. Mh. Asiipitkl published a work on hand-railing in 1S.51. In 1853, Mu Josetii (Jalimx and Mi:. Lanoi.ey Banks published a treatise on stair-casing, in which the systems practiced by several stair builders of London were published. We will notice the systems of Messrs. Clark, Foster, Weston and Perry. Mu. William Clauk's method to find the face-mould, was on the orthogonal or square cut principle; he makes use of a parallel mould, cuts the joints square to the tangents, and finds the joint bevels in the same manner as Mr. Kiddle in 1855. Mr. Clark illustrates his principle with card board. Mh. Fostei; also shows a face-mould on the orthogonal system, the mould being parallel in width. Mi:. Weston's metliod in some particulars is similar to Mr. H. Kiddell's in his 1855 and 1859 editions. Jle shows the devel- opment of tangents in elevation from the ground plan; he finds the angle of tangents for the face-mould correctly, and also the joint bevels from the tangents in elevation; he makes the joints at right angles to the tangents: he takes a point from the diagonal on plan, for the center of all face-moulds='^ at the angle of matical nfiiure. At the ajje of twelve he assisted his father in his business, but did not relish the occupation of a stone mason, and at tlie end of a year he was bound to scjve for four years as an appren- tice to a cubi'iu't maker, in the neijrliborinfr villnse of Linton. During his Hppieiiticesliip, younir Nicliolson enii)loyfd every leisure mornenb in iiiiproviii!i liis mind. He Ktudicd alfiebra assiduously, from day- lijllit in tlie summer morninjrs till six, wlieii lie went to work, as well as in the evenings when his labors were over. After serving to the full extent of liis time, he went to Edinburgh and worked at his trade, aud studied (he higlier branches of mathematics. At tlie age of t wcnt.y, he came to London, where liis uncle named Hastil, Ciiriicd on an extensive business as a builder; here lie still pursued his mathematical studies together with drawing; .soon his fame spread nmong his fellow workman, who, anxious to improve themselves, solicited to become his pupils; he opened a scliool with aViout ten, the fame of his teaching soon brought an influx of pupll.'i. This gave Mr. Nicliolson more leisure, which he devoted to the inven- tion and anangement of an original treati.se on carpentry and ioinery, whicli was published under the title of "Carpenters New Guide," in 1702. He also published severiil works on arcliitecture and the higher liranches of mathematics. He commenced his Architectural Dictionary in 1810, and finished the same in 1819." Abstract from the Memoirs of Peter NiclwUon by John Bown. S7. Thi.s point taken from the diagonal on ground plan and applying the same to the diagonal on the cutting plane for the center of rail is correct only when both tangents are of the same length. This same principle was published some time ago in the "American Builder,"^ by IMr. Langstaff. HlsTOKV OF SrAUi BL1I.D1^0. 17 taiigeuts ou the cutting plaue through which he finds tlie trace of Ihe center line of his face-mould, by using a pliable strip, froui liis center line he draws the concave and convex curves of mould to a parallel width. Mk. W. E. rERKv's method differs a little from that of Mi'. Weston's: he finds the angle of tangents for the face-mould frtmi the tangents and chord in elevation; he also finds the directing ordinate and joint bevels from the tangents in elevation; he findr, the center line of rail on the face-mould correctly by the use of ordinates, drawn from the tangents in elevation in the same way as Mr;. Johx Joxes describes in his 1888 edition of hand-railing. Mr. Perry makes use of a parallel face-mould.-*" Nkwi.and's "Carpenters and Joiners Assistant," 18C0 — Mr. Nevvland's treatment of the cylindric section for hand-railing, is similar to that of Mr. Nicholson. In 1864, Mi:. JosiirA Jeavs published his second edition "The Orthogonal .Svstera of lland-railing," in it there is a great deal of original matter. In 1871. Mk. Geouge Walton published his "New Treatise and Tractical Guide to Stair-casing and Uand-railing," the priuciplc is the same as published by Mr. Kiddle, in the 1859 edition. He shows the tangents in elevation developed from plan: he finds the angle of tangents and directing ordinate, also the joint bevels from the tangents in elevation; he describes the trace of face- mould with the trammel. In 1878, Mi:. William Twiss published his block system of hand-railing. He finds the parallelogram on the cutting plane direct from the tangents in elevation, and recommends tiie traiu- mei to give the perfect trace of face-mould: the joint bevels he finds either from the block, cut to the rake of the tangents, or from the tangents in elevation. Id 1882, Mk. Frank O. Cijesweli- published a >mall work on hand-railing and stair-casing: the principle is similar to Mr. Riddle's 1859 edition of tlie elements of hand-railing: he linds the angle of tangents, directing ordinate, and joint bevels from chord and tangents in elevation, developed from plan, The trace of face-mould he finds with the trammel. The same year, 188?, Mk. Geoi:ge C'ollings published his •* Practical Treatise on Hand-railing.'' He finds the parallel- ogram on the cutting plane, from the chord in elevation, and the hypothenuse of joint bevels from the parallelogram on the cutting |)lane : the trace of lace-mould he draws with the trammel: the position or location of the major and minor axis, he determines i»y the trammel and rod in nearlv the same way as shown at Fig. 10 Plate 1. In our own country, Mk. A.siiEK Be^.jami.n, in 18iil, pub- lisiied his line? for the construction of the face-mould, similar to that of Mr. NichoNon. In 18:38, Mk. Minakd Lafevkk issued a book of lines fur the stair builder, they were similar to Mr. Nicholson's. In 18-10. Mk. -John Hall publislied a work on Hand-railing. He claimed an improvement in the formation of the face-mould, l)y the use of the Ellipsograph, of which he claimed to be the inventor. The principle is to draw tlie ground plan of rail on ■'.■>. Till' four i;is( autlioj-s show llie fufe-iiiould diuwii to ;) l):ir:illol widtli. ciiiiul lo llie Iruc widUi of rail, from n ceutt-r liue; I his (iocs not ides of his facc-ir.ould concentric tlHnlic ciuves, his system would have been without fault ; his ccutei- line on the mould is tlie correct cllii)tic curve. 18 History of Staiu Buii.dixg. the draught board, then at tlie center of well hole erect a vertical shaft, say one incii in diameter, and one or more feet long, from which extends an arm, made to slide and turn on the shaft, the arm to Imve a pencil fixed near the end, and over from the center of shaft equal to the radius for the concave and also for the convex sides of rail. Now elevate the material from which the face- mould is to be made, over the ground plan to the required inclination of wreath piece ; then move the arm, and the pencil will describe the elliptic curve of face-mould correctly if the material has the correct inclination, lie also improved Mr. Nicholson's system. The next was Mk. Si.mox De Giiaff, 1S45. lie made use of the ordinates, but claimed tlie invention for making the wieath for the semicircle in three pieces. This work of Mr. yimon De Grafl', was the last book published exclusively on stair buildiJig, on the Nicholson system, in our country. After Mr. Peter Nicholson issued his 18.55 edition of the "Carpenter's Guide,'' the ingenious Mi;. Pi;tkk Estkrhkook, stair builder, of New York, was led to investigate the subject of hand-railing more scientifically, as to the uncertain location of the center resting point of Mr. Nicholson. He claims to have made the discovery of the tangent system, thereby locating the correct resting point, which gives the true inclination to tlie tangents, from which the cutting plane or plane of plank is determined as we have it to-day; Ln 1859, in company witli Mr. J. II. Monckton, they issued a work on stair building. In the meantime other works had appeared and gave to the world the benefit of the tangent system. In 1818, Mk. R. A. Crrj'KH, in Ohio, issued his fust book "The New Practical Stair Guilder's Guide," and later, 1851, his "Universal Stair Builder." These two books simiditied the theory of stair build- ing, The principle being correct, it gave the American stair builder a much easier task than he formerly had. Mr. Cupper claimed the discovery of the tangent system, and thereby the correct resting points; also to be the inventor of the tangent box, of which he gave several plates, clearly illustrating the principle of the tangent system, wliereby the stair builder coiild see at a glance the whole principle, as Mr. Reynolds says, in a "nut shell." lie determined \\m' correct position of the transverse axis and the priralletograui for tlie face-mould on the cutting pliine; and describes the niothod of cutting the crooks out and making the joints .scpiare to the face of plank; the use of either the plumb bevel or the two joint bevels, for squaring the wreath piece; the drawing of tlie face-mould; either with a string trannnel or by ordinates; in fact he prepared the way so plain for those having the i)ractiee, to simplify and make the subject still plainer for others to follow. In 1840 Mi:. IIkyxoi/ds issued his valuable little treatise and supplement for the stair builders library.. In 1850, Mk. David (Jaav. stair builder, of Pittsburgh, Pa., partially prepared a work on the subject, i>ut owing to a fatal accident the work was never completed. He lound the parallel- ogram or angle of tangents for the lace-mould, direct from the chord line in elevation, and described the face-mould with the trammel. In 1850, the ingenious Mn. Rir>i)i.K issued his "Scientific Stair Builder," in which he determined the parallelogram or angle of tangent on the face-mould, from the transverse axis on the cutting plane. He gave a greater variety of stair plans, explained and maile plainer the government of the tangents, showed how Ul.siXIUV ol' bTAIU Litli-DXACi. I'J tlio joint bevels may be obtained from the parallelof;iani on the cutting plane, and the general application ot tlie trammel to the construction of the face-mould, for all wreath pieces that are circular on plan. In 1S55, he issued his secoud edition of the "Scientific Stair Builder," in which he described the parallelogram or angle of tangents, and the seat of trammel, direct from the inclination of the tangents in elevation. He also described tlie underside of rail for the straight part, drawn tlirough the center of baluster, for Hie correct height of wreath rail in tlie cylinder. In 18.58 and ISGO, he published the second and third edition of liis "Elements of Hand-railing." He described the tangents, treads, anil rises in the elevation as spread out, or developed from tangents on plan. This was a decided improvement, for it gave the stair builder greater control over the inclination of tangents, and allowed the length of the odd balusters to be measured from the elevation. Mr. Kiddle also published eight or ten other books for the benefit of the stair builder, and also for the carpenter and joiner. He died March lath, 1882, at his home in Philadelphia, Pa , loved and respected by all, age 74 years. In IS.-j.T, Mr. J. K. Pekkv. of New York, issued a work on stair building; followed in the same year by Messrs. Mii-wain and Young. They found tlu^ angle of tangents direct from the chord line, and used a parallel mould. In the following year, 1850, Messrs. Vaughn and Gi.enn published a small work on the subject. They used the parallel mould. In 1850, Patrick 0"Neil, of Richmond, Va., published a book for the stair builder, on the tangent box system, much the same as described by Mr. Cupper, in his first book. Mkssus. Eastei{15Kook and Moncktox, in 1859, issued •'The American Stair Builder." They d(!scribed two ways to find the angle on tangents on the cutting plane; one from the transverse axis on the cutting plane, and the other from the tangents in elevation. ^a in the former, the trammel or rod is used to describe the curves of face-mould; and in the latter, a pliable strip is recommended in tracing the curves of mould. John Thomas, in 186a, published a book on stair building. The principle is similar to that published by Milwain and Young, and also by Mr. Langstaff. The angle of tangents is eased off by the intersection of lines for the trace of face-mould. In 1809, A. RussKTjj, of Memphis, Tenn., published a very good but brief work on hand-railing. He described the parallel- ogram or angle of tangents for face-mould, from the chord line in elevation, and used the tranunel to trace the curves of mould; he found the joint bevels from the parallelogram on the cutting plaric. In 1858, Mi:. C. E Loth issued a very good book on stair building, in which is found a great deal of interesting matter on the subject. He finds the angle of tangents from the transverse axis on the cutting i)lane, and the trace of mould through points determined by ordinalcs. In 1S7'2, also in 1888, Mu. James H. Monckton published two works on the subject; in the former he described the angle of tangents from the diagonal of the parallelogram instead of the chord, as others have done. He also described the curves of ;i9. Tliis iiietliod of fiiulitig the aii!;le of tangoiifs was piiTjlislied by Mu. K. G. IIatfiei.d, \n liis "Ameiicuri House Carpenter," he says it was invented by a Mr. Kells, an eminent stair builder of N. Y. City. 20 History of Staik Buii-DiNf^. face-mould with the trammel. In the latter work he finds the angle of tangents from the chord, and the trace of face-mould by ordinates drawn from the tangents and a flexible strip. In 1873, Mr. William Forbes published a work on hand- railing, called the " Sectonian Sj'stem." This method is similar to the tangent box system; he makes use of a parallel mould. In 1874, Mr. J. II. Reaves, of Hamilton, Canada, issued a work on stair building. He finds the parallelogram for the face- mould direct from the tangents in elevation, and the curves of mould by the trammel, similai to Mr. Riddle in his 18.5.5 edition. In 1875, Mr. L. D. Gould added his book to the stair Imilder's list. In it there are some original ideas. He finds the axis on the cutting plane from the chord and diagonal on plan, and the angle of tangents from the axis on the cutting plane. He makes use of the string, to trace the curves of face mould, similar to Mr. Riddle, in some of his later works. In 1880, Mr. R. J. Sherratt published his book on hand- railing. He finds the angle of tangents from the transverse axis on the cutting plane, and the trace of mould with a string. In 1S84, Mr. F. T. Hodgson published a treatise on stair building, the title of which is "Stair Building Made Easy.'- Another publication by an "Old Stair Buildei?," appeared in 1885, entitled "A New System of Hand-railing." The system is similar to Mr. Weston and W. E. Perry's, and later Mr. Jones. In 188',», Mr. J.V. Secor issued a work on Hand-railing. He finds the parallelogram for the face-mould direct from the chord in elevation, and the bevels from the parallelogram on the cutting plane. All the above works are not w-ithout some merit to the diligent imiuirer after a thorough knowledge in the art of stair building. PLATE I. Geometry. Figure 1. A point has position but not magnitude, as at A, Since a true point has no size, a line has no breadth. A line is the patli of a point in motion, but as we make lines they of conrse have breadth. Lines are termed right, or straight lines, curved, or mixed. Fig. 2. JBC shows a strakjht line, having length and l)readth, but no thickness; is composed of points, and if straight is the shortest distance between any two given points. Fig. 3. Shows parallel ZfJies, which may be straight as DE and FG, or curved as HXI and JXK, Fig. 5 ; and if pro- longed they would be eriually distant from eacli other throughout their extension. Fig. 4. A (turvcd line LMN, is one that does not lie in a straight line between its extremities, it may be regular or irregu- lar as OPR, Fig. 8. Fig. 5. Parallel curved lines, HXI and JXK, are such, provided they are everywhere the same distance apart; if they are circles they will be concentric. Fig. 6. Hhoyvs n. mired or compcnind Ime, STVW, being part straight and part curved. Fig. 7. A zufrmg line, AB, is composed of a series of straight lines as tiie Chevron Moulding in the Norman Archi- tecture. Fig. 8. An irrcgnlar or mixed curve, as OPR, is used in landscape gardening. Fig. 9. <^onverginci lines, CD and EF, if prolonged will converge at 0/ they are also called oblique lines. Fig. 1.0. Horizontal line, AB, indicates a level or hori- zontal line, and is at right angles to a plumb line. The surface of water at rest is always horizontal. Perpe^idlc.ulnr line, CD, is i)erpendicular to AB ; any line may be said to be perpendicular to a right line.' when the angle is a rigid angle, or an angle of 110 degrees as BCD.' Fig. 11. Acute angle : an angle that is less than a riglit angle, as ABC. Fig. 12. Obtuse angle : an angle that is greater than a right angle, as ABC. Fig. 13. CurviUnear angle. The angle OHJ, formed by the intersection of curved lines. And when the angle is formed by straight and curved lines they are termed mixtilinear angles, as KLM. t. A iMj;hf, line, is a .sffuishl liiu'. i. In cU'scril)ing ;ui angle, t lie iniddk' loller detiutes the itiiiiif , 23 Plate 1. Fig. 14. Diagonal line ; is a line joining two opposite angles, iiom corner to corner, as AB. Superficies. A surface has lous^tli and breadth, but no thii-kness, foi- instance: a shadow gives a good representation, its length and bre;idtl» can be measured, but not its depth. A .solid has length, breadtli und thick- ness. A Playie Figure, is a portion of a plane enclo.sed on all sides with lines; when the lines are straight, the figure is termed a ipolynan or rectilineal fiqure. Polygons have different names, according to the number of their sides, and means a many angled or sided figure. A Polygon of three sides, is iitrimigle; of four, a quadrilateral;^ of five, a peutagon; of six, a hexanon; of seven, a heptufjon; of eight, an octaaon; of nine, a nDiiagon; of ten, a deeagon; of twelve, a dvdecagon. A Reijidar Polygon, is a polygon whose sides and angles are equal, each to each, and its perimeter Is the sum of the bounding lines. A Parallelogram, is a quadrilateral which has its opposite sides parallel. There are four kinds, the square, rectangle, rhombus and rhomboid. Fig. 15. A sq\iare has its four sides equal, and all the angles are right angles, as at A, JB, C, D. The enclosed angles at E, F, G, H, are termed intertuU (imjles; at A, B, C and D, they are termed external angles. EoLE — To find the area, square one of the sides. Example: 16 0"X16' 0"=256 feet. Ans. Fig. 16. A rectangle, is a parallelogram whose angles are right angles, and the opposite sides only are of equal length, as ABCD. RuLK.— To find the superficial area, multiply the length, 23' 0" by the breadth, IG' 0". Example: 2:i' 0"X10' 0"=y58 feet. Ans. Fig. 17. A rhombus ABCD, is a parallelogram, having all lour sides (Kinal in length, but only the opposite angles equal. This ligiire is sometimes termed lonenge, Rule.— To find the area, multiply the side, IC 0" l)y the altitude* EA, 14' 0". Example: 16' 0",<14' 0"=224 feet. Ans. Fig. 18. A rhotnhoid, is a parallelogram liaving both opposite sides and opposite angles equal ; or two of whose sides are greater than the other two. Rule.— To find the area, multiply the length, IS' 0" by the altitude ^E. 16'6". i!/'.«-amjjie; 18'0"Al6'6"-2i)7feet. Ans. A triangle, is a polygon bounded by three straight lines, contain- ing three angles. There are four kinds, the rigid a)tgle triangle, the equilateral, the scalene, and the iso.' 4' 0"^ 24 feet. Ans. This rule answers for all triangles, for a triangle is one-half a laarallelogram, having the same base and altitude. 3. Quadrilateral, is a plane surface inclosed by four right lines. There are three classes of quadrilaterals; namely, trapezoida, trape- ziums and Parallelograms. 4. The altitude of a parallelogram is the distance between its opposite sides; of a trapezoid, it is the distan(;e l)etwoeu its parallel sides; of a triangle, it Is the distance from any vertix to the side opposite, or to tliat side prolonged as CD, Fig. 22. Plate 2. 23 Rule.— To find the length of hypothenuse AB, when the base AG and the perpendicular BC, are given. Add together the squares of the base, and perpendicular, and the square root of the sum will be the length of the hypotlienuse. Example: 6? +82 -v"' 100=10 feet. Am This rule is valuable to carpenters and builders, when estimating; for finding the length of valleys and hips, also the length of braces, and figuring for the strains in the same. Every young man should study well this problem. To find the length of base or perpendicular, when the hypoth- enuse, and either one of the other sides are given. EULE.— From the square of the hypothenuse, subtract the square of the side that is known, and the square root of the remaiiider will be the length of the unknown side. Example: lO?'— 82'=v''36'=G feet. Ans. Fig. 20. An equilateral triangle. The, three sides, AB, BC and CA, aud the three angles at A, B and C, are equal, and are equal in area to one-sixth of a hexagon whose sides are the same, and the given side BC, is equal to the radius of a circumscribed circle. The cooper to find the radius of a barrel head, sets the dividers so as to step around in the groove just six times. Rule.— To find the area multiply the length of the base BC, 16' 0" l)y half its altitude EA, 13.40 feet. Thus, 10' 0"Xi?'.»r-107.n2 feet. superH(;ial. Ans. Fig. 21. An Isosceles Triangle. ABC has two sides equal, and all the angles are acute. Rule.- To find the ai-ea, multiply the base. BC, [1.5' 0"] by half the altitude, EA, I?/)' 0"] thus; 15' 0"X20^0"=1.V 0' feet. Am. 2 Fig. 22. A Scalene Trianr/le. ABC has one obtuse and two acute angles, the angle BCA being obtuse, its altitude, AD, is found by prolojiging the base line BC to D. The area is found in the same manner as at the preceding— by multiplying Ihe base by half its altitude. Fig. 23. A Trapezoid, ABCD, is a quadrilateral which has two of its sides parallel to each other, as AB and CD. Kui.E— Tofind the area, multiply the sum of the parallel sides, AB, [20' 0") and CD, [•.'(/ 0"] by the altitude. Kn, [10' 0"1, and half the product will lie tlie aiea. Example: 2(;' 0"; 20' 0"=l(J'\tl'O"=G 4 4=322 feet. Am. * Fig. 24. A Trapezium. ABCD is a quadrilateral which has no two of its sides parallel to each other. Rule. — To find the ajfii, multiply the diagonal, CB, [2(5' 0"] by tlie sum of tlie two pcrixMiiHrnl,! is, AE, [IS' 0"] ;ind DH, [(i'O"], falling upon it from the opposli/' mi'zh-s, and half the product will be the area. Example: is' 0" i D' 0"..2(i' a"=5?i ^312 feet. Am. PLATE 2. The Cikcle. Fig. 1. A circle, ABCD, is a plane figure bounded by one line which is termed the circumference, and is equal to 360 degrees, and it is such that all straight lines from a certain point within the figure called the center, to the circuint'ereuce, are equal to one another, and the space within the whole circumference is termed the circle. 24 Plate 2. Tlie diameter of a circle is a straight line drawn through the center at O, and terminated by the circumference at AC, dividing tlie circle into two semi-circles. The rddius of a circle eiiuals lialf the diameter ; the span of the dividers, when describing the l>ouudary of any circle is termed the radius, as OC. When more than one line radiates from the center to the circumference they are term';d the radii, as OM, and ON. IJl'JjE.— To find the area of a ch'cle. Siiuare the rtiainetor AC, (2(/ 0"), and niuUiply by the decimal .7854. fi-rami^te: 20' 0"X20'0"X.7S.>t =;}4(i..36 square feet. Ans. The diaiueter being given, how to find Ihe circutnferoucc. KUI.E.— Multiply the diameter by the constant, 3.1416. E.nttitiilc: •:<)' 0">ca.l4IO-62.8:j20 feet. Ans. A taviient, to a circle, is a straight line, which touches tlie circumference but does not intersect it liow far soever tiie line be produced, as EF. An arc is part of a circumference. Fig. 2. A semicircle, ABC, is tlie figure contained by a diameter and that part of tlie circumference cut off by tlie diame- ter AC, and is equal to 180 degrees. Pig. 3. A quadrant of a circle, is the half of a semicircle bounded by the arc ABC, and the two radii, OA and OC, and is equal to 'JO degrees. Fig. 4. A chord i> a straight line joining any two points in a circumference, but not passing through the center, as C E. A segment is that ))ortiou of the circle contained between the cliord AB and the arc ACB. CD is the Versed sine. How to find the diameter of a circle when the chord and the versed sine are given. Let the chord AJ5 equal ;JO'0"aud the versed sine CD equal 6' 0". KuM-; —Divide the square of half the chord by the versed sine; to 1hi.s product add tlie length of the versed .sine, and the result will be I lie length of tlie diameter of the circle, of wliicli the arc of the circle is a pari, i'/'.rrtmjjfc ; 3 6^18x18 3|4=-54 (5 -60 ft . Aii^. The radius v.ill cqmil llie half of (»' or :jO feet. Fig. 5 shows two concentric circles, AB being the outer diameter, 20' 0'^ and CD the inner diameter, IG' 0'^. Rule— To find the area of the space included between the circuiii- fevence of the concentric circles. Multiply the sum of the two diame- ters by their difference, and this product again hy the decimal .7854. t'j. rat n'plr: 10 r20 :J6x:4/..7854 -l]:i.o;i, area rcciuired. Fig. 6. ABCD shows the convex, or outside; and EFGH shows the concave, or inside of a curved surface. IVnOHLFMS. Fig. 1. Ti) bisect a (jUien Vine. L't AB be the given straight line, from the points A and B as centers, with any div lance greater than half AB. describe arcs cutting each other at C and D. Draw the line CD, and the jioint E, wliere it cuts AB, will be the middle of the litie recpiired. CD will also be perpendicular to the given straight line AB; and the angles will all be right angles. Fig. 2. From a 'jivcn (mint ouUiidc of a 'jioen straiyhl line, to let fall a j>erT>endicvlar to the fjivcn straight line. Let A be the given point, and SC'the given straight line. WitJi any radius greater than AH, describe arcs from the center A, cutting ^Cal D and E. Again, with any radius greater than half DE, draw arcs from D and E, intersecting at F. Join FA, cutting DE^tH; then i?A will be perpendicular to BC, and ))assing Ihrough the given point A, as required. Plate 2. 25 Fig. 3. To set up a perpendicnlar at the end of a given lijic. A^. From A and B describe arcs of equal radius inter- secting in D, then with the same radius draw tlie semicircle from S throufih A to C ; draw BD produced to intersect the semi- circle at C, Join CA, then CA is the perpendicular required. Fig. 4. Another method.. I-et AB be the given line, and A the point to erect the perpendicular. Witli equal radii from A and B, draw arcs cutting at C ; through BC draw the straight line produced to D, make CD equal BC. Join DA, then DA ia the perpendicular required. Fig. 5. Another metliod to erect a perpendicular near the end of a given straight line. I^et AB be the given straight line, take any point as C, and any radius greater than AC, and des- cribe arcs at D and E; then take another point J*, and descrilx' arcs intersecting at D and E. Join D and E, then the line DE will be the perpendicular required. Fig. 6. To erect a perpendicular from a given point D, on a given curved line ABC. From the point D, describe ares with equal radius, cutting the given curved line at .Band^; again, with equal radius describe arcs from the points JBand F, intersecting at jRT. Join KD, then the line KD will bo perpen- dicular or normal to the given curved line ABC. Fig. 7. To erect a perpendicular to a given curved line ABC, from a point J, outside the given line. With any radius greater than the distance from J^ to the given line, draw the arc cutting the curved line in K and L; also draw arcs with equal radius from Kand L, intersecting at M. Join MJ, cutting; the curved line in JP, then JP will be tiie perpendicular required. Fig. 8. Three points ABC, out of a straight line being 'jiven, to find the center of a circle so that the three points mrnj he in the circumference of the circle. From C and B, witii equal radius, draw arcs, cutting at 2 and 3. Also from A and B. with equal radius, draw arcs, cutting at 4 and .5. Join 4 and .5 iiroduccd, also join 2 and o produced, intersectin:r at D; then D is the center to sweep through the three points ABC, as required. Fig. 9. Tlic span AB, and height CD, of a segment hring given, to find, the cealrr E, to drdin the cu)~ve ADB. With any radius greater tlian half AB, draw arcs from the points A and B, intersecting each other in 2 and 3. Again, with any radius draw arcs from the ))oints A and D, intersecting each other in .5 and 4; Join 5 and 4 produced, join 3 and 3 jiroduced, inter- secting at E; tlieu E is the center to sv/eep the arc through ADB. Fig. 10. Aniilhcr method. The span AB, and height CD, being given. With any radius greater than AC, and Avith A and B for centers, draw arcs intersecting at E; join ED, draw the chord nt AD. With the ))oints A ancl B for centers, draw the arc 2-3, also 4-5 indelinite. Make 4-.5 equal 2-3; join A~), prolonged to intersect DE at H, for the center required. Fig. 11. Anollicr melliod. The chord or sp%n AB, and height CD, being given, to siveep the arc ADB, uhthoul finding It center. In very llat arches, and wide span, or when the radius is very long, this method may serve a good purpose, 3 26 Pr.ATE 3. At A and B tack a lOd wire nail, then talie two strips five inches wide, and a few inches longer than the span, let them be jointed straight on one edge; now place one strip to line AD, and the other strip to line DB, have the jointed edge against the nails, cross and nail them together at D; see that the strips are close to the two nails, then near the end tack on a brace. Now hold a pencil in the angle at D, and move the triangle against the nails, and toe pencil will describe the curve required. Fig, 12. A circle ABC, and a inngent DE, to tha circle beincj given, to find the point of contact. Take any point in the tangent DE, as 2; draw 2F bisect 3 JP at H, and with tlie radius If^, describe the semicircle 2KF, cutting the tangent and circle in K, then K is the point of contact required. Fig. 13. The arc of a circle ABC, and the iangr.nt LD, heiuff given, to find ilie jmhit of contact, the center of the circle being unknown. From any point as D on the tangent, draw any line cutting the arc as DFE; bisect DE in O, then with OE for a radius, draw the semicircle EHD. At i^ erect a perpendicu- lar to ED, cutting the semicircle in K, then with D as a center, and DKior a radius draw the arc KL, cutting the given arc in L for the point of contact as required. Fig. 14. To drav) a tangent to a given circle ABC, t'"it ,f hall, pass throvgli. a given point A. From the center O, draw OA through the point A, draw EF perpendicular to OA; then EF is the tangent required. Fig. 15. A triangle ABC, hchig given to dravi a circle, tluit ivM be tangott to three given side-;. Bisect" the angles at A, B, C, and the intersection at O wiil give the center of the circle required. PLATE 3. I'KOKLEMS. Fig. 1. To bisect a given angle ABC. WiMi any radius, and JB as a center, describe arc culling AB and BC in D and E; again, with X) and J57 as centers, describ(; arcs intersect insr in F; join BF, then the angle FBA, is equal to the angle FBC. Figs. 2 and 3. To transfer an angle B, A, C, Fig, .". eiii, h<(vinii ciulit Kidcs, (octagon) from a ; divide BE into the same number of parts, connect ID, \E, 2-5, &c., as shown, and through the intersections trace the easing required. The stair-builder sometimes makes use of this problem to ease off the angle on wall strings formed by the level base and raking string; the distance BD or BE is usually taken at two- thirds of a tread, this is optional with the workman as a matter of taste. Fig. 13. Shows tJic cdshirj to br. an clliptifal <-urvt:, tlic dis- I'lnce BD bcini;i less than BE. Divide BD into any number of (•4; equals 12.5664 for tlie circumference of well ho le. Then proceed as above. 12X12=144+12.566X12.566= ■l/301,904a'56=17.35 feet; which equals 17.35 feet for the exact length of weath rail at the cylinder line. If the ground plan be elliptic(d, then find the circumfer- ence by the rule for the ellipse, and proceed aa above. Plate 5. 81 PLATE 5. The Ellipse. Plate 5. Shoivs the Ellipse, AB, Fig., 1 is the Transverse Axis, and is commonly termed the Major Axis, for short. The Conjuiiate Axis is the short axis, as CD, Fig. 1, bisect- ing the Major Axis at right angles and terminated by the curve ; it is termed for short the Minor Axis. Foci, are two points found on the major axis as J and K, Fig. 2, from which to draw the cir- cumference with a string ; the two points together are termed the Foci; singly, one is termed Focus. Vectors are the two lines that radiate from any point in the circumference to each of the foci as DJ and DK, or LJ and IjK, Fig. 2. It is a known property of the ellipse, that the sum of the two vectors is equal to the transverse diameter. Center. The center of the ellipse is at the intersection of the axis, as at F, Fig. 1. Vertix. Is the extremity of the axis, as at the mixed angles at ABC and Z), Fig. 1. Nm~mal. The axis AB and CD are always normal to the curve at their Vertices. To find a line normal to the curve at any intermediate point, bisect the vectors at the circumference, as shown by the line 1 2, Fig. 8, which is normal to the curve at the point H. In this way the mason finds the joints for the arch stones in elliptical arches at any point desired. Tcmgent. Any line at right angles to the normal at the point of contact with the curve will be tangent to the curve at that point, as the line 3 4, Fig. 8. Scmi-cllipsc. Both the axis divide the ellipse into two parts, as ADBA, or CBDC, either are termed semi-ellipsis ; the area is the same in either case. Elliptical Quadrants. The same axis divides the ellipse into four equal parts, termed elliptical quadrants, all having the same area, as CFBHC, Fig. 1. Rule.*— To ^nd the circumference of an ellipsis, the major and minor axis being known. Multiply' half the sum of the two diameters by iJ,1416, and the product will oe the circumference nearly. Example.— The major axis, Fig. 1, equals 14' 0", and the minor axis 24 5 equals Ity 6". required the circumference: W.O+Ky.S-^-^^ =12.25'X3.141C =38,484 feet, answer. Rule —To find the area of an Ellipse. Multiply tlie major axis by tlie minor axis and the product again by the decimal .7854, and the result will be the area. Example.— 14' (yx'lO.SX. 7854=115.453 square feet, answer. Fig. 1. Shows how to draw the ellipse with a trammel, the imijor axis AB, and mi)wr axis CD, being given. Pivot the trammel in the center at F, with the arms centered on the axis lines AB and CD. Make the distance from pencil to minor pin equal the semi- minor axis CF^, and the distance from pencil to major pin to equal the semimajor axis AF; now place the pins in the grooves, and trace the curve through the points ACBD, for the ellipsis required. The trammel is the most practical tool for the Stair builder to trace the curves of the face mould, especially for platform cylinders, say up to 24 inches diameter; tlie elliptic curve can be more correctly traced with the trammel than in any other way, the curve being absolutely correct. ♦Boueycastle's Mensuration. 33 Plate 5. The Stair builder snould not consider his kit complete iinless pro- vided witli a trammel. For platfoiui cylinders up to the size above mentioned, the arms of tlie trammel need not be longer tlian 7 inches each, and the rod not over 20 inches long, and 14" ^tiuare, having at the end a head drilled out to admit a pencil, and a set screw, to secure the pencil in place; also, two movable heads, with set screws and pins, the pins may be one-sixteenth in diameter, to fit neatly into the grooves of the trammel; these two Jieads are fit so as to slide easily on the rod, tlioy are V-z" by ?i" square, and the set screws should be on the opposite side from the pins, for convenience, when setting them. The first pin from the pencil is termod the mmor pin; the other is termed the major jjin. At Fig, 1, the trammel is shown having four arms; for ihe Stair builders' use, only three arms are I'equired, made U" thick ami V2" wide, with grooves at right angles and a scant eighth of an inch deep, and a strong sixteenth wide, so that tlie pins may slide freely in the grooves ; have at the center, on the underside, a small point for a pivot for centering the tramme]. This instrument sliould be made of brass. After using the tool for a short time, tlie Stair builder would not be without it. The author lias used the trammel on platform cylinders for tlie last forty years. For winders and large curves the ordinate as lierein set fortli is found to be the most convenient for economy in drawing a correct face mould. Fig. 2. A rectilineal parallelogram, 1, 2, 3, 4, being given, tn find tlie major and minor axln, (duo the foci, and draw the cUipnis UHth a string, so that tlic curve irill he tangent to the parallelogram at tlie points A, S, C and D. Bisect 1 3 and 2 4 at A and B, also bisect 1 2 and 3 4 at C and D, join A B and CD, forming right angles at the center F; then AB will be the major axis and CD the minor axis required. With the scmimajor axis AF for a radius, and J? as a center, draw the arc intersecting the major axis AB at J and K ; then the points J and K will be the foci required. Now fasten a pin in each focus J" and K, pass a thread around the pins, wrapping the two ends twice around tlie one pin, and liolding the ends under the thumb of the left hand, then with a pencil nicked slightly near the point, stretch the thread until the point D is reached ; now trace the curve through the points DBCA for the olliiisis required. Fig. 3. Around a given rectangle, ABCD, to describe an ellipsis. Bisect AC and BD at J57and F, join JE7 and J' and produced; bisect AB and CD at iifand J. Join HJ, intersecting EF &i the center O. Make HK equal HB, join KA. cutting EF at M. Then AiT will e(|ual the .semimajor axis, and AJkfthe semiminor axis. Make OiVaud OF each ecju.al KA. Let OR and OS each o(iual AM; now trace the elliptic curve through the points ABCD, as required. Fig. 4. An Ellipsis, ABCD, having been given, and. the position of the major (tnd minor ((.ris is nnknoum, to find their posUimi. Draw any two parallel lines cutting the ellipse, as AC and BD. Bisect AC and BD at 1 and 2, Join 1 2 produced to inter- sect the ellipse at J57and F; bisect EF At H; tlien the point at 3 will be the center of the ellipse. With any radius less than the semimajor axis, and 3, for a center, draw tlie arc cutting the ellipse at iJ" and J, join HJ ; tlirough the center 3, draw the minor axis KL, parallel with HJ. Bisect HJ at 4, join 4 3 and produced, cutting the ellipse at M and N, for the major axis, as required. Pl.ATK 5. 33 Fig. 5. To drwv an elUjJse or ovid, ABCD, ivith the dividers, the majnr cixii^ AC being rjivcn. Let AC be the major axis, divide the same into four equal parts, as A 3, 3 3, 3 1, 1 C. Make the two middle parts equal the diagonal of tlie square 1-6, 3-7 ; prolong the sides of the square at 1 and 3 indefinite ; theu with 3 as a center, and 3 A for radius, draw the arc FAE. Also from the center 1, draw the arc HCJ, then with 7, as a center and 7 H for a radius, draw the arc H13F. in like manner, from the center 6, trace the opposite side ED J, and the ellipse or oval is complete. Fig. 6. An Ellipse shorcn hy the dotted. Vme 3, 3, 4, .5, heluu (jiven; to draw other lives tJiat^cill he pandlcl to the fjiven ellipse: 3 4 is the major o.rf.v, and 3 5 tlic iiiitior axis. Draw eqaal arcs on each side; of the given ellipse to the required width, then trace the cur\Ts so as to tangent the arcs, for the parallel lines required as ABCD, for the convex curve; and JEHF for the concave curve. If a ]>attern be required, tra<'e eitlicr of the outer curves, theu use a gauge for the parallel width. Fig. 7. An Ellipse ABCD being given; to draw another ellipse EFGH, tJirnngh agloen point F, that shall be propor- tujnul to the given ellipse. Let ABCD be the given ellipse, AC tlie major axis and BD the minor axis, aud O tlic center, and JPiJ tlie minor axis for the jMoportlonal ellipse. Join AB, from F, aud pai-allel to the dotted hue AB, draw the proportional Iiii p;irallel to A 2, now 5 is the point t hrough whicli the proportional r-llipsc will cut on tlio radial line O ?. In the same way draw other proportional lines, as 2 D, D3, and .'{ C, &c., parallel to which draw 5H, Jlfi, and 6 G ; then trace tbo tdliptii; curve and it will be found to pass tbrouj;h the points •'«, JJ, 6', &o. Tliosii proijortional lines are some- times found useful when drawing tlio face-mould. Fig. 8. To dram a, line perpcndicidar, or normal P) the cui^e of a giv(n Ellipse; also to find a, tangent to tlie same curve. Ijct ACBD be the given ellips*; ; jiud the jioint H be selected for the line noniial 1o the curve ; liud the foci E and F, drav/ th?' vectors EH, andJ^iif pi-oduced ; bisect the iuigle formed at H, and draw the normal line 1 2, which will lie j>erpeudicular to the curve of the ellipse at tlie ]ioiut H, a.s recpiired. At the point H, and at light angles to 1 3, draw the line 3 4 ; then 3 4 will be tangent to the ellipse at the point H. Another way, to find a tangent to the curve: Let the vectors converge at the ])oint J'and piolong indeliuitely. Bisect the angle in K, join KJ, then the line KJ will be tangent to the ellipse, and the point of contact will be at J. Fig. 9. A SonieUipse ABC, bring given to find a, titngent to the enrvi' at, avg point ivitliout the foci. JiQt AB be the major axis, and O the center, and OC Ibn semiminor axis produced to H. l^rav,' the quadrant and radii OD and CF ; perpendicular to Oi^d raw .FiJ,- draw J^J" parallel with OB, join iJJ" and produced, for the tangent required, the point of contact is at J ; the tangent KL is found in the same way, K being the point of tangency. If the curve of the ellipse is not given, the point of langency may be found by taking the radius OA, and O for a center, draw 84 Ti-ATE 5. the arc cuttius OD, inixluced in M, from M draw the per]ipi)dic'- ular cuttiug KD iu K, theu the point K will be the point of contact. Fig. 10. The radius of any Quaitcv Circle, and its nccoin- panying Rhomhoidal parallelogram ABCD, being given; how to trace tlic curve of an Ellipse ivith a trammel fhrongJi the jjoints A and C, so that AB and BC shall be tangent to the, curve at the points A and C; and at the satnc time locate the position of the major and minor axis, the length of the 7najor cuyis, and, position of both, major and minor axis being roiknoini. Pivot the trammel in the point D; set from pencil to minor pin the distance eqnal to the radius DL, of quarter circle, (whidi is always etpial to the length of the semi-minor axis of its accoiii- panyins? ellipse). I'lace the pencil in the point C, and tlui minor pin iu the groove at F, and the major pin iu the groove at H; now slightly fasten the major pin, and move the rod to the point A. holding the tranuiiel firmly. If the pencil reaches beyond the point A, slide the major pin closer to the minor pin, and fastm again, also move the trannnel a little, and try again, nutil the jx-nci! will pass through the point AC. Then fasten securely the major piu, and trace the elliptic cur\e KCAJ, as required. The position of the tranmiel now gives the direction of the major axis JK, also the minor axis DL, and the pencil describes the elliptic curve. FC efpials the length of semi-minor axis, and JETC equals the length of semi-major axis, and the sides AB and BC of the Rhomboid are tangent to the cune at the points A and C, as required. Fig. 11. ThUi problem is the same as the preceding for the center ellipse, and is intended that proportional lines A L, L o, SC, CK, aivd their parallels, maybe drawn, giviny the width of concentric ellipses at the points A, L, 3 C, and K. The radius of any circle, as has been stated, always ecjuals the semi-minor axis of any accompanying ellipse. Theu DL equals tli<> length of the semi-nuuor axis : and A is anotlier fixed i>oint at the end of the parallelogram ; C, at the opposite end, is another fixed point, that the curve has to pass through: on the diagonal line DB anotlier point, o. may l>e estal)lished ; the method to establish this point is shown at Fig. 4. Tlate 24 ; also Fig. 2, Plate o2. Now joiu LA, Lo, 3 C. CK. Make 1,2, 1.2. each equal hall tho thickness of the cylindri<; section. Draw 2 4 parallel with AL, also 2 5, 5 6, 7, parallel with the renter proportionals, intersect- ing the radials DB, DC, DK aixl DA. At these intersectioiis the points are given through which the cuiwes will pass, a pliable .strip nmy l»e used to trace the curve through the points, instead of the trammel. A line that will be nonnal to the curve at the point A is drawn at right angles to the tangent AB as PS. At Can inrlefinite amount of straight wood maybe added parallel to BC prolonged fiom the points 6, 6. Fig. 12. tihows horn the Ellipse may be drawn with a straiyht^cdgc, and a pliable strip. Let AB indicate the major axis, and CD the length of the minor axis, and O the center of the ellipse, required ; let EF indi- cate the straight-edge. Make a nick at H ; make i?J equal OC ; let HK equal OA ; now move the rod at intervals, keeping the poiuts J and K directly over the axis lines, and marking points at the nick H, as 2, 3, i?' prolonged at N ; again, with J* for a center, and JF'JV as a radius, draw the curve to intersect JFV prolonged at E. Pro- ceed in like manner to draw the other quarter circles F, Q, R and S. Make ST equal the width of rail, and draw the inside curves in like manner. Draw the shaidc B Vat right angles to BA. It will be observed that any number of revolutions may be drawn in this w^ay ; the shank may be drawn at either of the quad- rants. From S around to E is termed one revolution, and from E around to B, is half a revolution. Draw the eye of scroll from tlie center H. Fig. 2. [Scale, half size.] Shows how to find the centers from whi^h to draw the Scn/U, draicn half size, and lujrees with Fig. 1. the lettering being the .sa»/u\ Fig. 3. [Scale, K^^=l foot.] Shows the ^vidth of Scroll AB, divided into seven equal parts. The centers from whifh to draw the quadrants, are found precisely the same way as at Fig. 1. Fig. 4. [%^^ scale.] Shows the curtail step, and is drawn from the same centers as Fig. 3 ; Sis the Block, and is cut out to the shape, from a thick piece of very dry stuff, the depth of the rise less the thickness t)f step. R is tlie rise, which is shouldered at W; Vis the veneer, and is reduced at Wto a scant eighth of •Ml inch, and is still fnrtlier reduced tapering towards the end at K, to a sixteenth of an inch. A kerf at K, is shown for the xeneer to enter. Folding wedges are shown at W, to be entered from l)Oth sides at the same time, to strain the veneer to place eveidy ; iS shows the outer string, and the concave side of block is shown vene(Med to match the string. This may not be necessary, unless tlie outer string be of some fancy wood ; N shows the nosings of steps. 1 Let it be observed, that by dividing the width of scroll into a greater, or less, number of spaces, the revolutions will be closer together, or open, as the case may be. The line EB will pass througli the center H, and the two lines FK and DJ will fonn right angles at H. Fig. 5. [Scale, IK '^=1 foot.] Exhibits a reciprocal Scroll, draxvn by segnients. Draw AJB and CD to fonn the four right angles at O ; divide each right angle into four equal parts, and draw the radial lines S6 Pr>ATE 7. through the center O indefinite. Draw the eye of soroll at O; from the verge of eye, draw tlie cliord 2 3 at right angles to O 2 ; from the point 3 tiraw tlie chord 3 4 at right angles to O 3 ; from the point 4 draw the chord 4 5 at riglit angles to O 4. Proceed in this way on each radial line, until a sufficient number of revolutions are laid olf. Then connect the points 2 3, 3 4, 4 5, &c., by seg- ments, in this manner. With O 31 for a radius, and 31 for a ceutei-, draw arc at JT, with the same radius and point 32 I'oi- a center; draw arc intersecting at jP. Thi'u, with F for a center, draw the arc from 31 to 32. Again, with O 30 for a radius, and point 30 for a center, draw arc at H, with the same radius and point 31 for a center, draw are intersecting at H, then, with H for a center, draw the arch from 30 to 31. In like manner find point J, and othei'S, from which to com- plete the spiral line tor the convex curve of pattern. Draw the concave side DLN, also the center line MP, parallel to the convex curve. The straight, or wreath rail, may connect the scroll at any point in the cnr\'e, at rigirt angles to the radius of any segment. This scroll has a beautiful effect at the starting of a winding flight of stairs ; a ground plan made for winders to conform to this mixed curve, the result will be a very gracefnl twist. Figs. 6, 7 and 8. [Scale, one-half full size.] Shows how to find the contour of a Mouhlcd Cap, from a given section of rail, so the timber may turn the cap to suit the profile of rail. Fig. 6. Shows a section of Hand-rail S^^^X^M^'- Draw the parallelogram ABCD, for the half section of rail. Divide BCinto any number of parts, as 1, 2, 3, 4, &c. ; draw SA indefinite ; anywhere on JBA prolonged, say O, Fig. 7. Draw OJE at right angles to OB, and equal to the radius of newel cap, say 4^''. Then with OE for a radius, and O as a center, draw the cir- cumference of cap, intersecting OA at F ; make FG equal to half width of rail BCf ; prolong CI? to intersect the -circumference of cap at JET, connect G^Jif for one side of miter. Parallel with GB, and from tlie points 1, 2, 3, S shifted on the upper and lower side so that the center of mould agrees with the center line 6 7, made from the bevel through the center of crook. Two face-moulds are shown here but only one is required ; it will be observed that by tacking the face-mould on the upper side of crook, the edge 9, 16, 11 Fig. 5, will serve for a guide to work off the concave side of wreath-piece to the required bevel. The scribe line 12, 13, 14, Fig. 5, on the opposite side of crook serves as a guide to shape the wreath piece for the lower side. For the convex side of wreath piece, if the face-mould be tacked on to the lower side of crook, the edge of pattern will give the line for the convex side of wreath piece ; the scribe line 8, F. H, Fig. 5, gives the curve of mould on the upper side of crook. The surplus wood should be removed from the concave side first, then the best way is to gauge for the convex side, using the "Cupper gauge," shown at Figs. 13 and 14, Plate 21. After the wreath piece is shaped to the bevel on the concave and convex sides, then use a pliable strip to form the twist of wreath as shown at 2, 3, B. Falling-moulds, made of paste board or tin, to the required depth of rail, were made in former days, to obtain the falling line of the wreath piece; now a thin strip }4''^ wide and one-sixteenth thick, is bent around the convex and also on the concave side of wreath piece and the twist line traced agreeably to the eje, the thin strip is easily adjusted so as to avoid any kinks or abrupt places. Next take oft" the surplus wood from the top side of wreath piece first ; when dressed oft', to please the eye and touch; gauge for the lower side. Kemember when dressing off the wreath piece use the tools so as to conform to the pilch of wreath piece as JK, Fig. 6 ; this direction is very easy found at any time by marking the end of mould on the crook, as 15, 14, Fig. 5 ; then a line from 14, through 5, Fig. G, gives the direction for the tool ; pencil lines parallel to 5 14, may be drawn with a parallel ruler from end to end of the wreath piece as a guide for the tools. A large gouge and mallet are about the best tools to rough down the surplus wood, then use the drawing-knife, cylinder-plane and spoke-shave to dress off to the concave arris of the face-mould ; also use a short straight-edge to rest on the arris of mould and the scribe line on tlie crook, in the direction of JK, Fig. 6, as a guide to true up the concave side of wreath piece. PLATE 8. Plate 8. E.rhihilfi how to draw the face-mould for a quarter circle for a wrauli-picce having an intermediate easing ; Vie tangents ivill he diffe/cnLin their inclination, hence two bevels ■will he required. Pig. 1. Shows the plan of the quarter circle. Draw OA and OB indefinite, and at right angles. From the center O diaw the quarter circle ASB to the radius of 15'^ ; draw the tau- 43 Platr 8. gents A^ and BD square to the radial lines OB and OA; on each side of A, set ofiE the half width of rail [13.^ ^'] as A 2 and A 4; from the center O draw the concave side of rail 2, 3, 2, and also the convex side 4, 4, 4, thus completing the cylindric section 3, 3, 3, and 4, 4, 4, on plan. Draw the chord AB; bisect the chord at C; bisect CB and CA at ^aud F; bisect AJF* at Gr, Fig. 2. Exhibits the dei^elopment and elevation of tangents from the plan, Fig. 1. Let XX indicate a base line. Make B£> and DA, equal the tangents BD and DA on plan Fig. 1; perpen- dicular to XX, draw BC, DE and AF. Assume BC to be the height the wreath piece has to raise; from C and square to CB, draw CE prolonged. Also assume the inclination of the lower tangent to cut DE at G. draw CG prolonged to intersect XX at H, draw the tangent AG to intersect BC at J. From D, and square to CH, draw Dk; from E, and perpendicular to AJ, draw J57I/. Make CM eciual the chord AJ5 on plan, Fig, 1. Parallel with DE. draw the half width of rail, cutting the tan- gents AG and CG at 3 and 3 respectively. Bevels. Let PiV indicate a line dravra parallel with XX: draw PQ square to XX, and equal to the radius OA, Fig. 1. Make PSequ-alDK, and Pieequal^I,; draw QR and QS, and in the angle PSQ, is found the bevel for the joint of wreath- piece at the narrow end of face-mould, and in the angle PRQ is found the bevel for the joint at the wide end of mould.* Fig. 3. Exhibits the face-mould. Make the tangent BD equal CG, Fig, 3 ; with S as a center, and MB, Fig. 3, for a radius, draw arc at A; again, with D as a center, and GA, Fig. 2, for a radius, draw arc intersecting at A, connect DA; parallel with DA and DB, draw BO and AO, for the parallelogram AODB, on the cutting plane, that will agree when in position, with the square parallelogram OADB on plan Fig. 1. Proof. The diagonal OD must equal the distance MJ, Fig. 3 ; if so, the parallelogram is correct. Make BH equal GH, Fig. 3 ; draw HO for the direction of minor axis ; make A 2 and A 4 each equal G 3, Fig. 3; make B 3 and B 4 each equal G 2, Fig. 2, draw the chord AB; bisect AB at C; bisect CA and CB at F and E; bisect AF at G; from the points A, Gr, F, C, B and B draw ordinates indefinite and parallel with the director HO. On plan. Fig. 1, make BIT equal DH, Fig. 2, join HO for the directing ordinate; from the points A, G, F, C, E and B draw ordinates parallel to HO, cutting the concave, center and convex sides of rail in the points 2, 3, 4. Now transfer the points in the ordinates Fig. 1, to the corresponding ordinates Fig. 3, using the chords as base lines, then trace the curves through the points. Make the joints at A and B square to the tangents AD and BD. The sections at M, show the joint centered by the dotted line, and the tangent line is squared over the joint intersecting the gauge line ; the bevel at R, Fig. 2, is applied through the inter- section, from the face of crook. ♦When the tangent.s happen to be nearly the same length, the difference in width of the face-mould at the jbint.swillbe slight; then apply tho steepest hevel to the joint tliat has the radial line the greatest distance from the minor axis. In that case the minor axis should always he drawn on tlie mould whenever practicable. Another sure way is to apply the steepest bevel to the joint that is made from the shortest tangent ; this rule applies to all face-moulds for wreath pieces that are circular on plan. Plate 8. 43 Observe the bevel as applied, will raise the joint at B up. The section at iV"shows the tangent squared across the joint, and the same gauge as applied at M gives the intersection through which the bevel found in the angle at S, Fig. 2, is applied. Ob- serve the bevel is applied the opposite from that at M, and will cross * the tangents, and thus pitch the joint at A down. The shaded parts at ilf and iVshow the width to saw out the crook at the joints. To find the amount of over wood required at Kon the normal line GK, to saw out the crook ; take F, Fig. 1, as a center and FC, Fig. 3, for a radius ; draw arc cutting the diagonal OD at/, join fF; draw the half depth of rail, [2^^^] parallel with Ff, cutting DO at b; from b and square to Ff, draw bJ. From J" draw a line parallel to OD, cutting the concave curve of rail at h; from h and at right angles to OD, draw he; then from c to the concave curve is the amount of over wood required on the concave side of mould at the normal line or point K, Fig. 3. On tlie convex side only enough over wood to clean off the saw marks, say J^'^, is all that is required, but the amount increases grad- ually towards the ends, to the amount shown by the shaded parts, at sections M and N. Fig. 4. Exhibits an isometHcal vie^v of the ground plan, elevation, governing ordinate, and the center line of rail on the cutting plane. It is not intended for a working drawing, but simply to show from the triangular prisms the construction of lines, to find the cutting plane and the direction of the governing ordinate on the horizontal and cutting planes. The learner will find it a great advantage to cut blocks to the inclination of tangents, whfiu drawing his face-moulds, and from them he will discern more clearly the reason for this or that line. This is termed Stereotomy, or the study of the Science of Solids. The stone cutters are noted for tliis study, as one expert in the business can tell at once if what he requires is in a particular solid. OABD shows the ground plan as described at Fig. 1, and indicates the end of two triangular blocks forming a square and is supposed to be a horizontal plane : ASB shows the center line of rail on plan; AB, the chord ; DG, on edge of solid equals, the height DG, Fig. 2; 00, on the opposite corner of solid equals G E, Fig. 2; SP indicates another corner of the solid, and equals the whole height as shown at BC, Fig. 2; draw AO, and AG, connect PG and AO, also PO for the parallelogram AGPO on the cutting plane, which indicates the parallelo- gram for the face-mould. Fig. 3. Now to find the directing ordinate on the horizontal plane, prolong BD indefinite, prolong PG to intersect BD prolonged, at Q, join QA, for the director sought; this line is termed the "intersecting line," by some; the "level " or " horizontal line," by others; the direc- tion of the minor axis is always parallel with this line. If the other side SO of solid were prolonged same as BD, and PO, also prolonged to intersect BO, then if a line be drawn from the inter- *The term "cross" or "crossing the tangents," means that the bevels are applied to the opposite sides of tangents at the joints. Ob- serve at joint A, section M, tlie bevel points to the right of the tan- gent AD. Let the eye follow the tangents around to joint B, and the sharp angle of bevel is seen to point to the opposite side of tangents at section N, thus crossing the tangents. This will be the case with all wreath pieces having two bevels, when the curve on plan is a Quarter circle or more than a quarter circle ; it is only when the plan is less than a quarter circle that the bevels do not cross ; and then only when the minor axis is not in the mould. 44 Pl.ATE 8. section to the point Q, the line would pass through the point A, thus showing the line QA to result from the intersection of the two inclinations with the horizontal plane. If the learner would dowel on the triangular prisms DQAG to the side of the solid D GA, and do the same at the adjoining side AOO, then cut them to the inclination of the cutting plane, he will then see that all lines drawn parallel to the director QA, both on the horizontal and cut- ting plane, are of the same length, those on the cutting plane being parallel to those on the horizontal plane and all parallel to the director QA. This makes a fine study for the student in stair-building. Make JBH on the horizontal plane equal DQ; join HO for the director within the parallelogram OADB, on the horizontal plane. For the director on the cutting plane AGPO, make iV equal GQ, join JO for the director on the cutting plane. It will be observed that the director HO on the horizontal plane, and also the director JO on the cutting plane, are both the same length, and parallel with the director QA. By drawing them within the parallelogram it saves room on the drawing board ; QD is repre- sented in the elevation by DH; and QG by HG, in elevation Fig. 2. The director JO on tlae cutting plane is the direction of the minor axis. If a line through O be drawn at right angles to JO, it would be the direction of the major axis; OiC equals the radius OA, Fig. 1, for the center line of rail, and is the semi-minor axis of the elliptic curve, for the center line of rail. AJB is the chord line on the horizontal i)lane ; AP is the chord line on the cutting plane, and represents BM, Fig, 2 ; the chord on the horizontal plane is fust spaced off then the cliord on tlie cutting plane is divided off proportionately with the chord on plan, as shown by the perpendicular lines CC, EE, &c. For large face-moulds less room is required and time saved in their construction by the use of ordinates in this way, as shown in the triangular prism ABD on plan, and AFG on the cutting plane. Bisect the chord AB on plan at C; bisect CB and CA at E and F, draw F 3, C 3 and E 3 parallel with the director O H; from F, C and E, on the chord AB, erect perpendiculars to intersect the chord AF at FC and E on the cutting plane ; thus dividing the chord AP in elevation proportionately with the chord AB on plan. From the points F, C and E on the chord AP, draw ordinates parallel with the director OJ; then transfer the points F3, C 3, from the ordinates on the horizontal plane to the coiTesponding ordinates on the cutting plane, and trace the elliptic curve through the points A 3, 3 P, using a pliable strip. The bevel shown at the angle G is taken from the angle AG D, Fig. 2, and the bevel shown at O is taken from the angle GC B Fig. 2. The student can test the correctness of joint bevels l>efore applying them to the crooks by squaring up a piece 2'^x 2'^x6^'' long, and cutting the same to tlie inclination that the tan- gents in elevation make with the pei"pendiculars as shown ; then by applying the bevels at M and N, square to the inclination, will pro\'e their correctness ; this iloes not require much time, and will be a satisfaction to the beginner to know he is right as he pro- ceeds with his work. Pig. 5. Sliows the manner of sliding the mould. The heavy cur\ ed line indicates the crook as sawed out from the plank, llie dotted lines at the ends show K^^ over wood to make the joints. The face of crook must be carefully taken out of wind ; then place the face-mould in the center of crook, mark the joints, also transfer the tangents from the face-mould to the crook ; Plate 8. 45 also mark the normal line LP. Now lift the mould and finish the marking of tangents AD and BD, and the normal line LP on the face of crook. Cut and dress the joints square to the face, and also S(iuare to tlie tangents ; now carry the tangents across the joints square to tlie face of crook as sho\Yn by the heavy lines S 3, at sections M and N. Then turn the crook over and mark the tangents on the opjiosite side squai'e to the joints. Novvf set a gauge to tlie half thickness of crook and mark both joints alike, as shown by the dotted line 00, drawn parallel witli the face of crook. The bevels are shown applied at sections M and N, through the intersection made by the gauge line. The block pattern is then applied at right angles to the center line G 7, made from the bevels. Observe the bevels nie applied so as to cross the tangents, and also that they cut the iip])er side at 6, and the lower side of crook at 7. Now A 4 and B 4, on the ui)per side of crook, equals 2 6, at sections Mand N; and A 5 and B 5, on the lower side of crook, eipials 7 3, at sections M and N. From 4 and 5, draw 4 D and 5 D parallel to tangents AD aiul BD; this should be carefully done as these lines 4 D and .5 D are to guide the face-mould. Now apply the face-mould so that the tangents 8 D and 9 D on the mould will agree with the tangent 4 D on the upper side of crook. Two holes are seen in the mould to aid in arranging the mould over the tangents on the cr«X)k correctly. If the mould be exact over the tangents, then the joints on the mould will be parallel with the joints on the crook. When the tangents on the mould agree with the tangents on the crook, scribe around that portion of the mould that rests ou the crook, then transfer the minor axis G^iiC from the mould to the crook on the iipper side. Now apply the moidd to the opposite side of crook, sliding the same until the tangents 1:2 D aiul 10 G on the mould agree with the tangent 5 D on the crook, then scribe around the pattern and mark the minor axis as before. At H\\\Q thickness of crook is turned up showing the i)attern applied to both sides, and the point -ffiT gives the direction to hold the tools when removing the surplus wood from tlie concave side of the wreath piece ; th(! sections M and N show the over wood removed from the concave side and ready to gauge for the convex side, or if preferred, tack the mould on the crook and use the arris of the mould for a guide where the scribe line runs out, and work off the surplus wood to the distance the mould will allow, then shift the mould to the opposite side and treat in the same manner at the opposite end of crook. At the minor axis, and for some distance beyond on either side, the mould will be in the way and must be removed, then work to the scribe line and use the judgment as to how much or little to take off. After the learner has S(inared up a few twists he will not need to lack on the face-mould for a guide. The more practical man will not be troubled in this way, but proceed at once to take off t!ie surplus wood, guided by the scribe line on the crook together with the eye and judgment, and in this way detect any abrupt places in the curve. Then by elevating the crook into its natural position by standing one end on the floor while the other is lifted until the bevels will appear plumb, and by sighting down and around the curves, he will detect any abruptness. After the sides of wreath piece have been formed, then bend a thin strip around the concave and also on the convex side of large twists, and mark the twist lines of the wreath piece, keeping the section of rail at the minor axis LP, in the center of crook, for here at the minor axis, the bevels l)lend, and the section of rail as shown by the block pattern, is at right angles or square to the face of 46 Plate 9. crook : and the center of rail is in the center of crook, as shown at Q; QS and QT being equal to the depth of rail m"]. Dbserve the curved sides of wreath piece as shown by the dotted lines at sections Jlfand JV, tangent the block pattern at the center. This is mentioned merely to warn the learner not to work down close to the block pattern at the joints, when the joints are made in the circular part of wreath piece, but leave some wood so that when the adjoining wreath piece is bolted on, they both can be dressed off together, holding the tools perpen- dicular to the ground plan. If straight wood be added on at either or both joints, let it be not less than what the mould slides, 4 9 or 5 12 ; then the block pattern will give the true contour of rail section. Comparing Fig. 5 with Fig. .5, plate 7, it will be observed that the face-mould has shifted past the joints A and B at both ends, wliile that at Fig. 5, Plate 7, the joint at A shows the pattern has shifted on the line of joint. Tlie reason of this is, because in the former case, both tangents in elevation Fig, 3, are shown inclin- ing, and the mould is shifted in two directions, while in the latter case, one tangent is inclining and the other is level, and the mould is shifted in one direction. PLATE 9. Plate 9. [Scale 114^^='^ foot]. Exhibits the construction of the face-mould for a wreath jneee over a quarter circle, v^hen tlie tantjents are both the same inclinatu>n. Hence Vie vjrcath piece u-ui have no easinq, aud only one bevel will be required for both joints ; the width of rail equals 3" by 4)4'^ deep. Fig. 1. Slutus the plan. Draw OA and OB at right angles and of indefinite lengtli. Make OA equal the radius for the center line of rail. From the center O, draw the center line of rail ASB, make A 2 and A 3 each equal the half width of rail ; draw the concave and convex sides of rail 2 7'2 and 3 H o. Perpendicular to the radial lines OA and OB, draw the tan- gents AC and BC; draw the diagonal line OC. Observe when the tangents on plan are of equal lengtii, and both the .same inclination in elevation, the diagonal OC always becomes the director. This face-mould is very simple to draw and could be laid off on the plan, to avoid a net work of lines, make a separate drawing for the face-mould. Draw the chord AB, bisect the same at D; bisect AD aud DB at B and F; bisect AB and FB at G and H. From the points A, G, E, D, F, iif and JB, draw ordinates parallel to the director OC, cutting the concave, center and convex sides of rail, at 2, 3 and 4. Fig. 2. Exhibits the development and elevation of tan- gents. Let XX iudicate the edge of drawing board ,- makeJSC and CA equal the tangents BC and CA on plan. Fig. 1. At right angles to XX draw the perfjendiculars AD, CE and BF; assume BG to be the height that the] wreatli piece is required to raise. Connect AG, cutting the perpendicular CE at H. From C and square to AH, draw Cj; make Sir equal the chord BA, iMir. 1 ; parallel with CH, draw the half width of rail, cutting A H at 2. Plate 9. 47 Bevels. In this case both tangents liave the same inclina- tion. As a result only one bevel will be required for both joints. The dotted line LM indicates a gauge line parallel with XX. Make MP equal CJ. draw MN at right angles to XX, and equal to the radius OB, Fig. 1 ; draw NP produced to the edge of board for convenience when adjusting the bevel, as sho^^'n.. Fig. 3. Exhibits the facemonld. Draw AC equal to the tangent AH in elevation ; with A for a center, and KG, in ele- vation, for a radius, draw arc at G; again with C for a center, and CA as a radius, draw arcs intersecting at G, connect CG. Parallel with CA and CG. draw GO and AO for the parallelo- gram OACG, on the cutting plane ; draw the diagonal OC tor the director. Proof. The diagonal OC nnist equal the diagonal OC on plan Fig. 1 ; if so the parallelogram is correct. This is very important, that the tangents have the correct direc- tion, for tlie joints are made square to tlie tangents, therefore be rareful to give the tangents the right angle at C, for they control the joints. Draw the chord AG; bi.sect AG at D: bisect DA and DG at E and F; bisect AE and EG at J and H. From A, J, E. F, H and G. draw ordinates, indefinite and parallel with the director OC, Now transfer the points 2, 3. 4, &c. on the ordinates on plan. Fig. 1, to the corresponding ordinates on face-mould. Make A 5 and A 6, also G 7 and G 8, each ecpial Jl2, in eleva- tion. Fig, 2. Now trace the curve through the points .5, 2, 7, &c., lor ti\e concave, and through the points 6, 4, 8. etc., for the con- vex side of face-unmld using a pliable strip. Make the joints at A and G square to the tangents AC and GC; the sections at M and N represent the joints at A and G; the tangents, after being transferred from the mould to the face side of crook. They are then squared over the joints as shov.ii at M and N; the dotted lines indicate gauge lines for centering the plank. The bevel found in the angle at P, Fig. 2, is a)ii)lied throniih the center, tlien the block pattern is applied square to the line made fronx the bevel, as shown. Observe the bevel at M as ai)])lied will iiitcli the joint at G up, and the bevel as applied at section N will pitch the joint at A down. The shadetl ])arts show the amount of over wood at the joint.s that has toints o, S, o, from the ordinates on 1 lie horizontal plane to the (•orres]»onding ordhiates on the cutting jiiane at 4, 4, 4 ; then trace the elliptic curve through the points A, 4, 4, 4. G, for the center Hue of rail. If a bevel as shown at M be applictl square to the inclination AO, both on the side of solid and over the cutting jdane, it will be found to agree with the bevel found at P, Fig. 2, and may be used by the beginner to prove the bcA'els found in the usual way before applying. The sides of the block are marked for cutting with the bevel, taken from the angle AHC, Fig. 2, and is applied as shown at H, Fig. 4, on all four sides. !^ig. 5. Exhibits the .slkliivj of the nundd to confonn to the hcvcls in the fornidtion of the wrcoth-piccv. The heavy ciuved line indicates the crook as sawed from the )>lank. The face of crook must be, carefully dressed out of wind, where there is steam, this is (luickly done on a hand jointei'; then if the face-mould b(; made of wood test the (iorrectness of joints by aj)plying the stf>ck of s(luar(^ to the joint and the blad(! to the tangent. If correct, place th(5 face-mould in Uw. center of ci'ook and tack a brad at each end to incveiit th(> mould from shifting; then mark the joints, also nuirk the position of tangents AD and BD on the crook from the. mould, also the nornial line 4 2, then lift the mould and complete the marking of tangents and the normal. Now cut and dress olf the joints square to the face of crook, and also S(iuare to the tangents, then carry the tangents across the joints sipiare lo the face of crook as shown by the lines 2 :>, at sections JW and JV; also inark the tangents AD and BD on the opposite side of crook s(|uare to the joints. Then center tht; plank with the gauge as shown by the dotted lines intersecting tiie lines 2 o, at hh ; then through the inter- section aiiply the bevel at M and the reverse way at section N, so as to have the l)evels cross the tangents. A))])ly the block ])altern square to the line (> 7, made from the bevel, showing tlie twist of wreath-piece in the rough crook ; now make AC and BE on the upper side; of crook, etiual 2 (5, on the section: draw CD and ED parallel to AD and SZ), repeat the same on the lower side of crook ; HD and FD is parallel to AD and BD. Now slide the face-moulil so as to agree with the tangents on the crook ; tangeids ad and bd on the mould must lie over tan- gents CD and ED , SRA and BDSR show three sides of tlie prism; AB is the chord lino on plan, and AD shows the chord on the cutting plane ; tlie shaded part ABD shows the block to be divided on the chord plane, thus separating the prism into two triangular prisms. Bisect the chord AB on the horizontal iilane at G; bisect GA and GB at K and J; IMsecL KA at M. From M, K, G and J" draw ordinates parallel to the director AC, to intersect the center line ANB at o, 3, 8, 3. !Now draw perpendiculars on the chord plane, dividing the cliord AD on the cutting plane ]>ropor- tiouately with the chord on the horizontal plaue at the points 2, 3, 2, 2 ; draw the ordinates 2 4. 2, 4, &c., parallel with the director AC, and equal to M 3, K 3, &c.; then trace the elliptic curve through the points D, 4, 4, 4, 4, A, using a pliable strip. To find the center of the ellipse, prolong the diagonal C/Sto intersect a perpendicular line from O at P; then PA indicates the direc- tion, and also the semi-major axis. From P and square to PA, gives the direction of the minor axis. The student in stair-building should take great interest in the study of these solids. Let him first prepare the solid to agree with the parallelogram ACBR, on plau, and 5'^ or iV^ long; divide it on the chord plane ADB, then dowel them together, and cut to the required pitch. If he adds the sections ARO and BRO, forming the trapezium ACBO, and cutting tliem all off to ihe cutting plane, he will liave the length of the semi-major axis, AP and the radial line DP, fur Ihe point of contact, and by adding anotlier triangular piece, BQO [Q is not shown], and cutting it off to tlie same plane, he will have the lengtli of the semi-minor axis PT, wliich is plund) over OQ on tlie horizontal plane, and is always equal to the radius of the circle. If the student will apply his mind to the study of these solids, he will soon master the most intricate prob- lems in hand railing. " Fig. 5. Shoics the aiipUcaiion of the Face-mould to the Crook. The heavy line shows the crook as sav/ed out from thn plank* After the crook is taken carefully out of wind on tlie face side' apply the face-mould in the center ; nuirk the joints at A and D> also the tangents AC and DC. Now lift the mould, and liiii.-,h marking the tangents, whicli should be eandully transf(;rred from the face-mould to the crook, as shown by the heavy lines AC and DC. Next, line the joints A and 23, cut and dress IJieiu square to tlie face of crook, and tlie tangents ; then square the tangents across ihe joints as shown by the ruies 2 ?>, at sections P and R, Now, center the plank with the gauge, and thnuigh the iidcrsec- tion made by the gauge, apj»ly the bevel G 7. Then set the block pattern sfpiare to the line 6 7, and scribe around llit; pattern showing the twist of ureal li-piec(! at the joints. Now the distajice, 2 and 3 7, is what the mould lias to slide at the joints, to carry the lines of twist correctly from end to end of the crook. At 5^' Plate 10. joint A, let AB, on the upper side of crook equal 2 6, at section P; and AE, on the lower side of crook equal 3 7, at section P. Then parallel with tangent AC, draw BC on iue upper side of crook; and parallel with AC, on tlie lower side of crook, draw EC. It will be observed that £C cannot be drawn on the crook, but the mould is shifted down so that the tangent ac on the face- mould will agree with the center line 6 7, as shown at section P. At joint D make DH, on the upper side of crook, equal 2 6, at section R; and DF, on the lower side of crook, equal 3 7, at section R. Now, parallel with DC, on the upper side of crook, draw HC; and parallel with tangent DC, on the lower side of crook, draw FC. The tangents are now marked on the crook, ready to apply the mould, which must lay over these lines, havinej the tangents ac and dc to agree with the tanarents JBCand HC, on the upper side of crook ; and tangents EC and FC, for the lower side of crook, in this case, the sliding of the mould will be easily done, for the line of joint A is the line of the major axis of the ellipse, and thus becomes a slide line for the joint of face- mould, if correctly made. If the face-mould is made from wood, it is well always to test the correctness of the joints before making the joint on the crook, by applying the stock of the try-square to the joint and the blade of square to the tangent, on the mould. The face-mould as shown is shifted so that the tangents ac and dc agree v/ith the tangents JSCand HC, on the upper side of crook, and while in this position scril)e around the convex side of mould. In the same manner slide the mould on the lower side of crook until the tangents ac and cd will agree with the tangents £?Cand FC, then scribe around the concave side of mould. The dotted lines at section R show the sides of rail section curved, while at section JPthe block pattern gives the true profile of rail section, because the tangent AC is level, and the joint is square to the tangent ; hence the plane of joint is parallel to (iie axis of the cylinder. The joint at D is oblique to the axis of cylinder, because the joint is made square to the tangent DC, wliich is inclining. Itwill be noticed at J" and K, on section P, that there is a great saving of material in cutting out tlie crook .square to t!io face of plank. This want at J" prevents the profile of moukl on ils concave side being transferi-ed to the face of crook, by tacking the mould to the crook in its proper ])Osition, the lower arris of mould as shown supplies this want, for the upper side of crook. On the lower side, we have the scrilje line complete for the concave side. After working off the surplus wood from the concave side of -wreath-piece, then use the Cui)per gauge, and gauge the wreath-piece to a parallel width ; then dress off to the gauge line; the bevel at joint A gives the direction to guide the tools, or a line from the edge mould through the center of plank at joint D, will give the direction. For the twist line, use a pliable strip and apjdy it to the top side of v.'reath-piece on the concave, and also on the convex sides, to agree with the block pattern at the joints. vVfter tracing carefully the twist line, remove the surplus wood from the upper side of wreath-piece, then gauge for the depth. While gauging for the depth the gauge will helj) to detect any abrui>t places in squaring the ui)per side of wreath-piece, especially when there is a want of wood on tlu; arris of wreath- piece as is often the case. If there be .several wreath-pieces, ramps or easements connected, bolt them all together before applyhig the thin strip for the falling line of rail. Plate U. PLATE II. Plate 11. [Scale, \}i'^=V[. Exhibits the construction of the face-mould for a u-rcath-piece havinrj an intermediate easing, alien the cyllndric section on plan is less than a quarter circle. Fig. 1. Sliows the cyUndric section, the radius OA, for the center line cqucds 14^\ From the center O draw the center line of rati ASB. As orclinates will be iised, draw the full width of rail, dividing equally on each side of the center line. From A and B draw the tansreuts ACand BCperpendicular to the radial linos OA and OB. Farallel with CA and CB draw BL and AL, forming the parallelogram ZiACB on plan. Prolong tangent JBC indefinite. From A, and at right angles to BC, draw AD, the seat line for bevels. Pig. 2. Exhibits the development of tangents from plan Fig. 1. As this loreath-jyiece is to have an iyitermediate easing, both tangents vnll be inclining, but each loill be different in their inclination. AjhI also two joint bevels ivill be required. Let XX indicate the edge of drawing board, and BC, CA equal the tangents JBC and CA on plan, Fig. 1; from which erect perpendiculars to XX. as shown. Assume CD to be the height that the tangent AC on plan, has to raise. Draw AD prolonged to intersect the perpendicular from JBat F. Again assume BE to be the height of both tan- gents, draw ED prolonged to intersect the base line XX at G. From E. and parallel to XX, draw EH, cutting the perpen- diculars from A and C at i? and J: malce CK and JL eacli equal CD on plan, Fig. 1. From K, and perpendicular to DG, draw KM. From L, and square to AF. draw LN. Make EP equal tlie diagonal CL on plan. Let EQ equal the chord AB on plan, Fig. 1. Bevels. Eeturning to the plan, make DF equal MK, Fig. 2, and JJi? on plan, equal LN, Fig. 3; draw J'A and HA for the bevels required, as shown. Parallel with BiJdraw the half width of rail [IK''''], cutting the bevels at Y" and X, As the face-mould in this case is traced by the use of ordi- nates, first find their direction on plan. Fig. 1. Make BJ" equal CG, Fig. 2; draw JL for the director; join AB for the chord. Bisect the chord AB at Q: bisect QA and QB at JWand N; bisect AM at P. From A, P, M, Q, iVand B draw ordinates parallel with the director LJ to cut the con- cave, center and convex sides of rail at the points 2, 3, 4, &c. Fig. 3. Exhibits the facc-moidd. Make EDG equal EDG, Fig. 2, with .E7 as a center, and BQ Fig. 2, for a radius. Draw arc at A, then with D for a center, and DA, Fig. 2. as a radius, draw arc intersecting at A. Join DA. Parallel with DA and DE, draw EL and AL, estab- lishing the parallelogram LEDA on tlie plane of plank, which will agree with the parallelogram LACB, Fig. 1, when in position. Proof. The diagonal LD must equal the distance PF, Fig. 2, join CfA for tlie director. Draw the joints at A and E square 64 Plate 11. to the tangonts AD and JSD; draw the chord AE. Bisect AE at G: l)isect AG and BG at M and N; bisect AM at P. From A, P, M, G, N and E, lJra\v ordinates indefinite, and parallel to the director AG. Now transfer points 2, o and 4, on the ordinates, Fig. 1, to corresponding ordinate on tlie face-nionkl as shown, nsing the chords for base lines. Make A .5 and A 6 each eqnal HX, Fig. 1, make E 7 and E 8 each eqnal FY, Fig. 1. Now trace throngh the points for the concave, center and convex cnrves of face- monld. Ilemember the points 5 and 6, at joint A, and 7 and 8, at joint E, are not quite correct, and would not do to set the rod or trammel by, but are near enough for the practical man, when using a strip. If the point of taugency, or the connection of straight with the curved part is required, then return to plan, Fig. 1. Prolong the chord at A, also the ordinate 4 A, and draw an ordinate from the joint at G. Now with F for a center, and AP, Fig. 3, as a radius, draw arc cutting the ordinate at 5; draw F .5 prolonged, to intersect the ordinate from 6 at 7. Return to Fig. o, and make A 1 equal 7 .5, Fig. 1, and 1 9 equal 6 *,), Fig. 1, draw 9 10 through A, for the points of contact for joint A. Tlie points for joint S'may be fouiid in the same way. If the center O be required, so as to trace the face-mould with the trannnel or rod. Then return to Fig. 2. Make ^V equal the diagonal OC, Fig. 1. From V erect the perpendicular to intersect FF pro- longed, at Y, Now prolong the diagonal DL, Fig. 3, to efpial . YF, Fig. 2, at O. From O, and parallel with the director AG, draw OB, equal to the radius OB, Fig. 1, for the semi-minor axis ; and a line at right angles to OB, through O, will be the direction of the major axis. Draw the radial lines OA and OE indefinite ; from the points 7 8 and 6 9, on the joints, draw lines parallel to the tangents, to intersect the radial lines at a 8, and 9, 10 for the points of taugency, through which the elliptic curve will pass.* The mould may now be drawn with the trammel or straight-edge, without the use of ordinates. The section at W shows tlie bevel found in the angle at H, Fig. 1, applied from the face side of crook; and the section at i? shows the bevel found in the angle at F, Fig. 1, applied through the center made by the intersection of the gauge line witli the tangent EH, that is squared over the joint ; the bevel line governs the block pattern. The shaded part shows the surplus wood that has to be removed to foj-m the twist of wreath-rail, and also the width at the joint, to saw out the crook. Fig. 4. Exlbihits an isometrical view of the prismatic solid to show the sLudent the combination of lines in perspective, (Did the paraUclocjnim on the cutilag phine in position over the p'ith tangents CA arid CB, draw BL and AL, and we have the parallelogram LACB. on i>Ian. Prolong the tangent BC io the left. From A, and at right angles to jBC prolonged, draw AD, the seat of bevel. Fig. 2. Shmos the development and, elevation of tangents for a wreath-piece loilhont an easing, the two tangents must have Die .same inciinatioti, arcd only onehcvcl wLllbe required for both joints. Let XJT indicate the edge of drawing board. Make J5C and CA each etjual tangents .BCand CA, on plan. Fig. 1. From AC aud B erect perpendiculans to xx. Assume BD to be the lieiijltt required for tli3 wreatli-pie:'e to raise, draw the inclination AD, ■cutting the perpendicular from C at E. Make BF ecpial the chord AS, on plan, Fig. 1. Make CH" ennal CD, oa plan, Fig. ]. From H, and perpendicular to AE, dravv HJ. Bevel. At Fig. 1. make DiJ equal JH, Fig. 2, connect EA; parallel with EB, draw the half width of rail, cutting EA at F. Draw the rhord AB, and diagonal CL, bisecting AB at G; bisect AG and GB at J" and H. From A, J, iJand B, draw ordinates to intersect the concave, center and convex curves of tlie cyliudric section at 3, 55, 4, «&c. Pig. 3. Exhibits the Facc-mmdd. Make DE equal the tangent DE, in elevation, Fig. 2 ; with D for a center, and DF, Fig. 2. for a radius, draw arc at A; again, with E as a center, and ED for a radius, draw arc intersecting at A, Join AE: parallel with EA and ED, draw DL and AL, for the parallelo- gram LDEA, that will agree; with the parallelogram LBCA, on plan, Fig. 1. Proof. The diagonal EL must agree with the diagonal CL o:i plan, Fig. 1. ]\Iake joints at A and D square to the tangents ASa'id DE ; make A ;i and A :'., also D 4 and D 5, each equal EF, Fig. 1 ; prolong EL to O. equal to the diagonal. CO, Fig. ]. Draw the radial lines OA and OD prolonged. From ]>oints 2 and ?,, al-o 4 and .5. draw lines parallel to the tangents AE and BE, to intersect the radial lines at the points a, b and d, h, for the points of contact. Plaxk 12. 57 As we will use ordinates, to locate points in the curves, draw chord AD, intersecting the diagonal at G. Bisect GA and GD at t7 and if. From the points A, J", jH" aad Z>, draw ordinates parallel with the director EJL indefinite.* Now transfer the points on ordinates, Fig. 1, to corresponding ordinates on face-mould, using the chords as Ijase lines; then trace the curves through the points 2, 3 and 4, for the inside, center and outside of face-mould, using a llexible strip. If a trammel or rod be preferred to trace the elliptic curve of face-mould, then pivot the trammel at O witli the arms at right angles to the semi-minor axis O 3, as shown ; set from pencil to minor pin on the rod, the distance O ;3, tor the concave side of mould, then place the pencil in point a, and drop the pins in the grooves, and fasten the major pin ; then trace the curve tluougli the points a, 2, d. For the convex side set from pencil to minor piu, the distance O 4, then place the pencil in h; drop tiie pins in the grooves, and fasten tiie major pin, and sweep the convex curve. The radial lines OA and OD give the points of contact or connection of straight with the circular part. The sections M and N show the tangents carried over the joints, sijuare from the face of crook ; the dotted lines indicate the thickness of plank centered with a gauge. The bevel is applied from the face of plank through the intersection, and the block pattern is shov/n applied siiuare to the bevel for the twist of rail at the joints ; the shaded part indicates the width at the joints to saw out the crooks, and is shown by the dotted lines 5, 7, 9, 6, 8, 10. at the face-mould. At the normal line 7 S, %" on the convex side, and %" on the concave, is all the over wood that will be required to saw out the crook at that point, as the section of rail at this point is always square to the face of plank, and the width of face-mould at this point is always equal to the true width of rail. When marking out the crooks, lay down tlie face-mould on the plank, set olf on each side of the joint the amount that llie crook has to be wider tiian the face-mould, as A G, A .5, and D y, i? 10. At the minor axis set off 4 7 to equal %", and 2 8 to equal ^i.". Now shift the mould from 3 to 5, and the point 4 move to 7, so that the mould will cut the point 7 ; after n)arking the [flank from h to 7, proceed in the same manner to mark from t) to 7 ; repeat the operation for the concave side from 10 to S and fj to 8, allowing yi" surplus wood to make the joints at A and JB, as shown. This metiiod saves making an extra mould to saw out the crook which is unnecessary. Fig. 4. Shon's an isometrical perspective of the solid, and the euttiwi plmic AEDL, in vosition over the horizontal plane, ACBL. ACBL indicates the cud of block resting on a horizontal plane and agrees with the parallelogram ACBL on plan, Fig. I ; CE is the height of the lower tangent, and BD indicates tho whole lieight of both tangents. Join DE and AE; draw DL and AL [farallel with EA and ED. This is done with the bevel set to the angle ADB, Fig. 2. Now cut the block to these lines, aiul we have the paiallelogram DLAE to indicate the parallelo- gram laid down at Fig. 3, on the face-mould. Now prolong BC * Whenever the tangents are hoth the saiuo inclination, and tlie same Icnjxtli in elevation, the diaRonal becomes tJie director, and is normal to the curve, and on it, the minor axis is set oil. i 58 I'LATE 13. indeGnite. Prolong DE to intersect BC prolonged, at G, join GA for the director of ordinates. It will be observed now that if the diagonal EL be drawn, it will be parallel with the director GA, because the doited line EG is equal io DE. This is always the case when both tangents are equal on plan and in elevation, then the diagonal LE is always the director and the minor axis. From A and square to EG draw AH; this indicates the seat for the bevel shown at AD, Fig. 1. From jffand square to EG draw JIJ; this indicates the height for the bevel as shown at HJ, Fig. 2. JA indicates the hypotlienuse of bevel at EA, Fig. 1. The same bevel is found direct from the block as shown at K, by applying the bevel scpiare to the inclination, both from the side of block and from the cutting plane. The chord lines AB, on the horizontal plane, and AD, on the cutting plane, in this case come over each other, so that the ])erpend!culars connecting the bisections arc not shown as at Fig. 4, Plate 11. To show the point for trammel, prolong the diagonal CL to O, then dowel on the triangular blocks ALO and BLO, and cut them olf to the cutting plane DLAE. We will then have the trapezium DEAN, shown correctly at DEAO, Fig. 3. The point iV answers to the point O, Fig. 3, and is the center for the trammel, which is always set at right angles to N 3, the semi- minor axis, which is equal to the radius O 'i. Fig. 5. Shows the manner of shifting the face-mould on the crouli. The heavy lines show the crook as sawed out from the rough plank, having the joints A and D cut and dressed. Tin; dotted lines at the joints show the K''^ of over wood cut away. The shaded part indicates the upper surface of crook. First dress olf the upper side of crook carefully out of wind ; place the face-mould in the center of crook ; mark the joints A and B, also the tangents and minor axis PQ on the npjier side. Now lift the mould and complete marking the tangents Ai? and DE, on the upper side of crook. Now S(iuare, cut and dress the joints carefully, by applying the stock of square to the joints and the blade to agree with the tangents. After the jointing is done, carry the tangents across the joints square to face of crook, as shown by the line 2 o, at sec- tions iVf and N; and also square the tangents from the joints on the opposite side of crook. Next center the joints at 4 with a gauge, as shown alike at both joints by the dotted line. Through the intersection at 4, ap)>ly the bevel 7, reversing them as bhown at sections ilf and N. Apply the bh>ck pattern sciuare to the line G 7, for the squared section of rail on the joints, showing the twist of wreath-piece. Now i3 G is the distan(;e the mould has to shift on the upper surface of crook at each joint, and 3 7 is the distanc(! on the lower bide. Then make A J[? and Di? equal 'J « for the ujiper side of crook, and AK, also I?J'e'reath-piece having a full easement. > Fig. 1. TJie shaded imH shotvs the cylindric section on plan. The radius OA, OC, for the center line of rail equals IS^''; the width of rail S 3, equals 4^^. At right angles to the radii OA and OC, draw the tangents AB and CB. From A, and perpendicular to tangent CB, draw A 7. for the seat of bevels. Draw the chord AC, also the diagonal BO, cutting the chord at J. Fig. 2. Exhibits the elevation of tangents. Let XX indicate a base line. Make CB and BA equal tan- gents CB and BA, Fig. 1 ; perpendicular to XX, draw CD, BG 60 Plate 13^ and AL; assiime CD to be the height that the wreath-piece has to raise ; as the wreath-piece has to have a full easing, then one tangent will be horizontal and the other tangent will incline the whole height. Then draw BD for the inclination of tangent CJB, on plan, tangent AB \YiIl be level. Make CH equal tlie diagonal HO on plan, Fig. 1 ; make CF equal the chord AC on plan, Fig. 1, Make CE equal BJ, Fig. 1 ; let B 7 equal B 7, Fig. 1. From 7, and square to BD draw 7 8. Bisect the height CD at N; draw NE produced, to intersect the perpendicular from H at K, Bevels. Returning to plan, Fig. 1, make 7 8 equal 7 8, Fig. 2, and 7 D equal CD, Fig. 2. Join A 8 and AD for the bevels at 8 and D. Fig. 3. Exhibits tlie face-mould. Make CB equal BD, Fig. 2. AVith C as a center, and FD, Fig, 2, for a radius, draw arc at A; again, with B as a center, and tangent AB, Fig. 1, for a radius, draw arc intersecting at A; Join BA. Perpendicular to BA draw AE iudefniite ; draw the chord AC. Bisect AC at D; draw BD prolonged, to iutt-rsect AE at G. Join OC and we have the trapezium ABCO, on the cutting plane, or plane of plank, that will agiee v>'hen in posi- tion, with the trapezium, or qudarilateral ABCO on plan, Fig. 1. Proof. The diagonal BD and BO nuist agree with NE and NK, Fig. 2. if so, the angle of trapezium at B is correct. From 0, and square to AE, draw OX equal to the radius OA, Fig. 1. Then OA will be the semi-major axis, and OX will be the semi-minor axis of the elliptic curve. Make X 2, X 3, each equal the half width of rail. [2'^]. Make joint at C square to tangent B C, the major axis gives the joint at A. Join XCand X A; from points 2 and 3, draw the proportional lines parallel to X A and X C to intersect the radial lines at 10, 5 and 9, 8 through which to draw the elliptic curves of face mould. Now center the trammel at O, square to the minor axis O X. Then take the rod and set from the pencil (o the minor pin, the distance O X. [The minor pin is now fastened, and the major pin is loose.] Now place the pencil at A, and drop the pins in the groove, and fasten the major pin at the point O. ■J'hen trace the curve for the center line of rail through the points A X C. For the concave side of face mould, set from pencil to minor pin, the distance O 2. Now place the pencil at .5 and drop the pins in the groove, and fasten the major pin at the point O, then trace the concave side of face mould througli the points 5, 2, 9.* For the convex side of face mould, set from the pencil to minor pin, the distance O 3, on the minor axis; then place the pencil at 10 and drop the pins in the groove, and fasten the major pin at 0. Now trace the curve througli the points 10, 3, 6, com- pleting the face-mould. If a trammel is not at hand, or the inclination of tangents be steep, requiring a large trammel, then draw the two right angles at O, to indicate the scat of trammel, and use a rod, and find points in the elliptical curve as shown at figure 12, plate 5. In this case the semi-major and semi-minor axis are given. The distance 9 (>, on the rod equals the semi- *If the elliptical curve should cut the .joint at C, making the dis- tance C 4, greater than O 12, do not conclude the work wrong, for this is natural. The difference is greater in face moulds over winders that are steep, the variation regulates itself when the wreutii yiece is worked off to the plumb bevels, showing the vertical sides of fail sec- tion at the joints to be concave and conve.x. 1'LA.TE 13. 01 minor axis O 2, and the distance 9 7 equals the semi-major axis OA, for the concave side of mould. Now move the rod at inter- vals, keeping the points 6 -and 7. over the right angles as shown, and make points at the &ad of the rod, through which trace the elliptic curve, using a pliable strip. Repeat the operation for the center and convex curves of face- mould; OA and OX arc the semi-major and minor axis for the center line, and O 10 and O 3, are the semi-major and minor axis for the convex curve of face-mould. For long and steep face- moulds, the system of ordinates as before described, will be found the most convenient and economical. At sections L and N the tangents are shown squared across the joints. The dotted line indicates a gauge line. The bevel in the angle at D, Fig. 1, is applied at section L, from the face of crook, through the intersection made by the gauge.* The block pattern is applied square to the line made from the bevel. At section N, the bevel found in the angle at 8, Fig. 1, is applied from the face of crook through the center of section; the block pattern is then applied at right angles to the bevel, thus showing the twist of wreath-piece in the crook. The shaded parts show the amount of wood required at the joints to saw out the crook; at the normal line 3 2, less wood is required, as shown at Fig, 3, riate 12. Fig. 4 shows the plan of a cylindric section greater than a quarter circle. How to construct the face-mould for a ivreatli- picce tcithout an easing. The shaded pait shoivs the cylindric section on pUtn. Let OA and OC indicate the radii. With OA as a radius, draw the center line AXC; draw the tangents AB and CB per- pendicular to OA and OC. From A, and square to tangent BC, iraw A 7, for the seat of bevel. Draw the chord AC and the liagoualSO. Fig. 5. Exhibits the development and elevation of tan- gents. Let XX indicate a base line. Make CB and BA equal tan- gents CB and BA on plan, Fig. 4. Perpendicular to XX, draw CD, BM and AL. Now as this face-mould is for a wreath- piece without an easing, consequently both tangents will have the same inclination, and but one bevel will be required for both joints. Make CD e(iual the whole height the wreath-piece has to raise. Join AD. cutting BM at E. Make CF equal the chord AC, Fig. 4, Let B 7 equal B 7, Fig. 4 ; from 7, and per- poixdicular to AD, draw 7 G. Bevels. Returning to Fig, 4, make 7 D equal FG, Fig. .5. Join DA, and in the angle at D is found the bevel for both joints. Fig. 6. Exliihits the face-mould. Make CB equal DE, Fig, r,. With C as a center, and FD Fig, .5, for a radius, draw arc at A; again, with B as a center, and AG, Fig. 5, for a radius, draw arc intersecting at A. Join BA, Draw the chord AC. Bisect AC at D. Draw the diagonal BD prolonged, to equal BO on the plan. Fig. 4, as at O. *The correctness of bevels may be proved thus, with C as a center and the hypothenuse of bevel as AD, Fig. 1, for a radius, tlic curve must tangent AD, as shown at IJ. For the other bevel, AJ must equal A 8, Fig. 1. 63 Plate 13. Draw thp radial linos AO and CO and we have tlic trapezium OABC on the cutting plane, that will agree when in position, with the trapezium OABC on plan, Fig. 4. Proof. The diagonal BD must equal BP on plan, Fig, 4. As both tangents have the same inclination, and are both the same length, then the diagonal OB becomes the director. Make OX equal OX, Fig. 4 for the semi-minor axis ; make X 2 and X 3 each equal the half width of rail [2^^]. Join XC and XA. From 2 and 3 draw the proportional lines parallel to XA and XC, cutting the radial lines OA and OC at 1 6 and 6 9, for the points through which the elliptic curve will pass. Make joints at A and C at right angles to the tangents BA and BC. Now pivot the trammel at O with the axis at right angles to the minor axis OX; then set the minor pin from pencil to equal O 2 for the concave side of face-mould. Now place the pencil in the point at 6, and drop the pins In the grooves, and fasten the major pin ; then trace the curve through the points 6 2, G for tlio concave side of face-mould. For the center line, set from pencil to minor pin the distance OX; place the pencil in the point at C and drop the minor pin in the groove, then slide the major pin until it drops into the groove, then fasten the pin and trace the center line of face-mould. For the convex side, set from pencil to minor pin the distance O 3 on the minor axis; then place the pencil in the point at 9, and (h-op tlio pins into the grooves ; then fasten the major pin and trace the convex curve of face-mould through the points 1, 3, 9. Sections L and iVshow the bevel found in the angle at D, Fig. 4, ai)plied througli the center of joints. The l)lock pattern is applied square to tlie Ijevel, and shows the twist of wreath-piece at tlie joints. For all wreath-pieces greater than a quarter circle, a bevel will I)e required at both joints, and they will always cross the tangents, because there will always be a point in the face- inoiild lliat will be equal to the true v.'idthof rail, which is termed the normal line. Fig. 7. Exh'ihits ihe (jround plan of d cyllndrie section greater Uian a quarter circle. How to construct the face-mould for a wrcath-piccc, liaving an intermediate easing. Witli OA for a radius, draw the center line of rail AXC, The shaded part shows the true width of rail on plan. Perpen- dicular to tlie radii OA and OC, draw the tangents AB and CB. Draw tiie cliord ACaiul the diagonal BO, cutting the chord at P. From A, and square to tangent BC, draw A 7. Fig. 8 shows the development and elevation of iangent from plan Fig. 7. Let XX indicate a base line. Make CB and BA equal the tangents CB and BA, Fig. 7; perpendicular to XX, draw CD, BM and AL, indefinite. Now for a wreath-piece having an int«'riiiediate casing, both tangents must be inclininir, but one must have a greater inclination than the other. Let BG be the height that tlu! lower tangent has to raise, and CD the wliole height of both tangents. ,Toin AG ami GD for the increased length of tangents in elevation; prolong tangent AG, to cut, tlje perpendic- ular CD at E; also prolong tani^ent DG to intersect the liase line XX n[, H. Make CF equal AC, Fig. 7. Let CP and CJ equal SP* and BO on plan. Fig. 7; also make B 7 and ilfiVeach equal B 7 on plan, Fig. 7. From 7 and square to HD. draw 7 8, and I'rom N and square to AE, draw N 9. Bisect ED at R. Make CS equal ER. From /S through P draw a line to intersect the perpendicular from J at K, Plate 13. 63 Bevels. As both tangents in this case have different incli- nations, two bevels will be required. Return to plan, Fig. 7, and prolong tangent CB indefinite. Make 7 8 equal 7 8 in elevation, Fig 8; also make 7 9 equal iV9 iu elevation, Fig. 8. Join 8 A and 9 A for tlie bevels required, as shown in the angles at 8 and 9 on plan. Fig. 7. Fig. 9. Exhibits the face-mould. Make CBN equal DGH, Fig. S; then with C as a center and DF, Fig. 8, lor a radius, draw arc at A; again, with B as a center, and tangent AG, for a radius, draw arc intersecting at A, join JBA, draw the chord AC; bisect AC at D: draw BD indefinite. Proof. If the diagonal BD equal PS, Fig. 8, then the angle of tangents is correct. Make BO equal KS, Fig. 8; draw the radial lines OA and OC, indefinite ; join AN for the director; from the center O, draw OX, indefinite, and parallel to director AN. Make OX equal the radius OA, Fig. 7. for the semi-minor axis; througli O, and at right angles to OX, draw 6 for the direction of the major axis. '' Make joints at A and C perpendicular to the tangents AB and CB: nuike X 2 and X o each equal the true width of rail. Connect XA and XC. From 2 and 3 draw the inoportional lines parallel to XA and XC to intersect OA and OC prolonged, at 7 7 and 8 8, for the correct points through which the elliptic cui've will pass. The points 8 8 and 7 7 on the radial lines, are the points of tangeney, or the connecting points of the straight with the circu- lar part. To trace the face mould with a trammel, pivot the trammel in O, with the arms resting on G 0. For the convex side of mould; set from pencil to minor-pin the distance O 3, then place the pencil in tlie point at 8, drop the minor-pin in tlie groove at 0, and slide the major-pin until it drops in tlie groove at 9, then fasten the major-pin, and trace the elliptic curve through the points 7, 3, 8; repeat the operation for the center and concave side of face- mould. If room be a consideration then use the ordinates for any of of these moulds, by drawing the ordinates parallel to the director AN, using tlie chords as base lines as directed in former plates. At sections L and N the tangents are shown squared across the joints, and the plank is centered as indicated, by the dotted gauge lines, and the bevel found in the angle at 9, Fig. 7, is applied at the joint A, and the bevel found in the angle at 8, Fig. 7, is applied at joint C, Tlie block pattern is then applied at right angles to the lines made from the bevels and gives the twist of wreath-piece at the joints as shown. Observe tiie bevels as ajiplied; cross the tangents, because the minor axis, O 3, is in the mould. At the points 3 and 3, the bevels blend and the section of rail at that jioint is siiuare to tlie face of crook. The sliding of the mould has been explained and applies the .same here. Observe the bevel at section L is applied, so as to throw the joint at Cup, and tiie bevel at section JVis applied so as to throw the joint at A down, corresponding to the ta.ngeuts AG and GD \n elevation. 6^ Plate 14. PLATE 14. Plate 14. [Scale, M^^=l footj. Ea:hibits the constructicni of a fcwc-inoald for a wreuthr-piece over a cylindric section, that is elliptic 071 plan, said section being less than a quarter of an ellipse. Figs. 1, 2 and 3. SJimi'showto constructthefacc-^nonld for a wrcath-yiccc vnthmit an casitKj. The plan of the twist part of rail being less than a quarter of an ellipse. Draw the center line ASB of the ellipse, also draw joints at A and B normal to (he curve.* At right angles to the joints, draw the tangents AC and BC; parallel to AC and BC draw AO and BO for the ))arallelograni OACB on plan. Par- allel to the center line ASB set off the width of rail, draw the chord AB; prolong tangent BC indefinite, from A and square to .BC draw AD Fig. 2. Exhibits the development and elevation of tan- gents from, plan, Fig. 1. Let XX indicate a base line. Make BC and CA eriual tan- gents BC and CA on plan, Fig. 1 ; perpendicular to XX draw BD and C^ indefinite. Assume BD to be the whole height the wreath-piece has to raise, connect AD, cutting CE at H, for the inclination of tangents. Make CJ" e(|ual CD, Fig. 1; make DG c(iual twice CH; let BJ' equal OC, Fig. 1, make SiT equal the chord BA, Fig, 1; from J, and perpendicular to AD, draw JL. Bevels. As both tangents have the same inclination, only one bevel will be reijuired. IJeturn to Fig. 1, make DNi:nna\ J L, Fig. '.i, join NA for the bevel at N. Parallel to .BiV draw the half width of rail to cut the hypothenuse NA at Q Fig. 3. Exhibits the face-mould. Make BC eiiiial DH, Fig. ;3; with B as a center and DK, Fig. 2, for a radius, draw arc at A; again, with C as a center, and tangent AH\n elevation. Fig. 2, for a radius, draw arc inter- secting at A; join CA. Paralh;! to AC and BC, draw BO and AO for the parallelogram OACB, on the cutting phuie, that when in position will agree with the parallelogram OACB, on plan. Fig. 1. Proof. The diagonal CO, must ecjual the distance FG, Fig. 2. If so, the angle of tangents at C, must be correct. Draw tlie chord AB, bisect AB at D; bisect BD and AD at ^ and J'. Make BH equal AH, Fig. 2; join HO for the director. From the points A, E, D, F and B, on the chord line, draw ordinates indefinite and parallel to HO. Ileturu to plan, Fig. 1; bisect the chord AB at J; bisect AcT and BJ at E and F: make SiJ equal CA in elevation. Fig. 2, connect HO tor the director on plan. From the points A, E, J, JF* and B, draw ordinates paralhil to the director iifO to cut the concave, center and convex curves at 2, S and 4. Iteturning to Fig. 3. Make joints at A and B at right angles to the tangents AC and BC. Let A 5 and A 0, also B 7 and B 8, each e(iual NQ, *To draw a lino normal to the elliptic curve at any point in the curve, sec Fig. 8, Phitc 5. Tr-ATK 14. Co Fig. 1. Now transfer tlie points 2, 3 and 4, from the orilinateson plan, Fig. 1, to corresponding ordinates on face-monld : then trace the concave side of face mould tlirough the points 7, ;3, 5; and the convex side through the points 8, 4, G, using a pliable strip. The sections at P and It show the tangents carried across the joints square to the face of crook; the joint is centered witli the gauge and tlie bevel found in the angle at N, Fig. 1. is shown applied through the center of both sections so as to pitch the joint at A down, and the joint at R, up. The block pattern is then applied square to the line made from the bevel, showing the twist of rail section at the joints; the shaded part shows the width required at the joints to saw out the crook in the rough, and also tbe amount of surplus wood to be removed in squaring up the wreath-piece. Fig's. 4, 5 and 6. Exhibits the construction of face- mould ivhen the ivrcatJi -piece has an intermediate casing, Vac jAan being Vie same as at Fig. 1. Fig 4. Shows the plan. The joints at A and B are drawn normal to the center elliptic curve ASB; and the tangents AC and BC are at riglit angles to the joints. Draw AO and BO parallel to tangents AC and BC: from B and at right angles to tangent AC prolonged, draw BT; from A, and perpendicular to tangent .BC prolonged, draw AV, connect AB f(Jv the chord. Bisect AB at D; bisect AD and B n at E and F; bisect AE at H. Fig. 5. Shows the developmcni and elevation of tangents from the plan, Fig. 4. Let XX be a base line; make .BC and CA equal tangents JBCand CA, Fig. 4. Perpendicular to XX, draw BD and CE. As the wreath-piece is to contain an intermediate easing, the tangents will liave dilferent inclinations. We will assume Cffto ))e the height that the tangent AC on plan has to raise, and BD as the lieight that both tangents are required to raise. Draw tangent DH, prolonged to intersect XX at J; draw the tangent Aiifprolonged. JIake CV equal CV, Fig. 4; also make ^r equal CTon plan. fig. 4: from Vand square to tangent DH prolonged, draw VK; from Tand at right angles to tangent Ai? prolonged, draw TM, Make BQ equal tlio diasonal CO on plan. Fig. 4; let BS equal the chord AS, Fig. 4; nu\ke DJV equal twice the height .ffC Bevels. As both tangents have different inclinations two bevels will be required. Make VB, ¥iz. 4, equal VK, in eleva- tion, Fig. V). Join BA; parallel witli BB draw the half width of rail [2^^] to cut the hypothenuse of bevel at 5. The angle at P gives the bevel as shown. For the other bevel, make BR, Fig. .5, equal S27, Fig. 4; let BZ7 equal TM; join UR, and the angle at 17" gives the bevel. I'arallel to BD draw the half width of rail [2^^] to cut the hypothenuse of bevel at G. Fig. 6. Exhibits the face-mould. Make BC equal tangent DH in elevation. Fig. 5. With B as a center, and DS, Fig. .5, for a radius, draw arc at A; again, with C as a center, ami tangent AH, Fig. 5, for a radius, draw arc intersecting at A; join CA; parallel to tangent AC and BC draw BO and AO. establishing the parallelogram OACB on the cutting plane, tiiat will, Vvhen in position, agree with the parallel- ogram OACB on plan, Fig. 4. 66 Plate 14. Proof. The diagonal OC must equal the distanre QN, Fijr. 5. Draw the chord AB. Bisect AB at JD; bisect AD and BD at E and J^' bisect AE at ^, Make BL equal JiT, Fig. .O; join LO for the director on tlie face -mould. From the points A, H, E, D, i^^and B draw ordinates indefinite and parallel to the director LO, Malie joints at A and B square to the tangents AC and BC; let A and A 7 each equal Z70, Fig. 5; make B 5 and B 8 eacli equal P 5, Fig. 4. Now transfer tiie points 2, 3 and 4 from tlie ordinates on plan, Fig. 4, to corresponding ordinates on the face- mould as shown, using the chords for base lines; at section M tlie bevel found in the angle at Z7, Fig. .5, is applied, and at sec- tion N, the bevel found in the angle at P, Fig. 4, is applied in the usual way. Observe the bevels cross the tangents because there is a point in the mould that is narrower than at either of the joints. Figs. 7, 8 and 9. Are introduced to hIiow a faee-mo^ild having both tangents inclinimj, and at the same time, the bevels will not cross the tangents. This never happens only when the eylindric section on plan is less than a quarter circle, or a quarter of an ellipsis, tor then the angle of tangents is obtuse; there are five different face- moulds for the obtuse plan as follows : FiEST. A face-mould having one tangent level, the other inclining, requiring two bevels, that in no case cross the tan- gents in their application, such as the ordinary newel wreath, or for startings and landings where one of the tangents is horizontal, as shown at Fig. 3, Plate 10. Second. When both tangents are inclining and different in their inclination, requiring two bevels, which in tlieir application, do not cross the tangents, as at Fig. 9. •, TiiiKD, When both tangents are inclining and different in their inclination, but requiring only one bevel, the square being applied at the opposite joint, as at Fig. 12. FouKxn, When both tangents are inclining and different in their inclination, requiring two bevels, but crossing the tangents, as shown at Fig. 6. Fifth, Having both tangents inclining and of the same in- clination, requiring only one bevel for both joints, and crossing the tangents in their application, as shown at Fig, 3. Fig. 7. Sliows the plan of tangents and repeats Fig. 1. Tlie joints at A and B are draAvn nonnal to the curve; the tangents AC and SC are drawn square to the joints; AO and BO are parallel to jBCand AC, creating the parallelogram AOCB on plan. From A and at right angles to tangent SCprolonged, draw AD; from B and square to tangent ACprolongeil, draw BE. Pig. 8. Exhibits the development and elevation of tangents^ Let XX indicate a base line; make BC and CA equal the tangents BC and CA, Fig. 7. At right angles to XX. draw the perpendiculars BD and CF, indefinite; assume CHio be the height for the lower tangent and BD, the height for both tangents; draw the tangent AH, prolonged indefinite; draw the tangent DH prolonged to intersect the base line XX at 3. Make CJ" equal CD on plan, Fig. 7; make EG equal CJ'on plan. Fig. 7; h^XBK equal the chord AB on plan, Fig. 7, and BL equal the diagonal CO on plan, Fig. 7. Make DM equal twice the hight of CH, Platk 14. 67 join LM for the length of diagonal on the face-mould, connect DK for the length of chord on the face-mould. From J and s(iiiare to tangent DH prolonged, draw JN; also from G and square to tangent A Jif prolonged, draw GP. Keturn to plan, Fig. 7, make BL equal C2 in elevation Fig. 8, connect LO for the liirector on plan. Bisect the chord AB at Q; bisect AQ and BQ at R and S; bisect AR at T. From the points A, T, R, Q, S, and B, draw ordiuates paral- lel to the director OL, to cut tlie concave, center and convex curves of rail at 3, .3 and 4. Bevels. Make DH on plan, Fig. 7, equal JN in elevation Fig. 8; join AiJ for the hypothenuse of bevel. Make i^J" equal GP, Fig. S; join JB for the hypothenuse of bevel at the wide end of mould. Parallel to tangent CB, draw the half width of rail [2''] to cut HA at 5; parallel to tangent AC, draw the half width of rail to cut JB at 7. Fig. 9. Erhihits the facc-moidd. Make SC equal tangent DH in elevation Fig. 8; with Sasa center and DK, Fig. 8, for a radius, draw arc at A; again, with C as a center and tangent AH, for a radius, draw arc intersecting at A; join CA, parallel to AC and BC, draw BO and AO for the parallelogram OACB on tlie cutting plane, or plane of plank tiiat will agree when elevated into position with the parallelogram OACB on plan, Fig. 7. Proof. The diagonal DC must agree with the length of LM, Fig. 8. If so, the angle of tangents at C is correct. Make BD equal ^"2, Fig. 8, connect DO for the director on the face-mould; draw the chord AB. Bisect AB at Q; bisect AQ and BQ at R and S; Insect AR at T. From tlie points A. T, R. Q, S and B, draw ordinates indefinite; make joints at A and B square to the tangents AC and BC; let A 7 and A 8 each equal J 7, Fig. 7; also make B 5 and B 6 each ecpial H 5, Fig. 8. Now transfer points 2, ." and 4 from the ordinates on plan, Fig. 7, to corresponding ordinates on the face-mould as shown; then trace the curve through the points 8, 4, 6, for the convex side, and through the points 7, 2, 5, for the concave side of face-mould. The sections at M and .ZV show the bevels applied in the usual way; observe they do not cross the tangents, because the long tangent DH, in elevation, intersects the horizontal line XX, between tlie point J" and the center perpendicular from C. This will be explained at Fig. 11. Pig's. 10, 11 and 12. Exhibit)^ a plan of tanrjcnts tluit rcpatts Fig. 1. To show the crmstruction of a face-mould on the (UvkUnfj line hclwecn one in which the bevels cross the tangents ■bi their application, and a faee-moiild in lolvicli the bevels do not cross the ianfjenls. Fig. 10. Shows the plan. The joints A and JB are made normal to the curve, and the tangents AC and BC are drawn at right angles to the joints; iiarallel to AC and BC draw BO and AO for the parallelogram OACB. At right angles to tangent BC prolonged, draw AD; also from B and at right angles to tansent AC prolonged, drav/ BF. Fig. 11. Shoics the development arul elevation of tangents. Let XX indicate a base line; make BC and CA equal the tangents BC and CA, Fig. 10. From XX erect the perpendic- 68 i^LATE 14. ulars BD ami CD; assume CF to be the height that tangent AC oil plan is required to raise; make CJ equal CD, Fig. 10. Con- nect JF, and prolonged to intersect the perpendicular SD at D; then BD is the whole height of wreath-piece, from center to cen- ter of rail, and A^and FD are the increased length of tangents AC and BC on plan Fig. 10. At right angles to BD draw DE; make EH equal CF, Fig. 10; from H, and at right angles to AF, prolonged, draw HK; make BL equal the diagonal OC. Fig. 10; make BM equal the chord .BA on plan, Fig. 10; let DN equal twice FC. Return to plan, Fig. 10. In this case DA becomes the dii'ector on plan; draw the chord AB. Bisect AB at Q; bisect AQ and BQ at R and S; bisect AR at T; from the points T, R, Q and /S, draw ordinates parallel to the director AD, to inter- sect the coucave, center and convex curves of rail at 2, 3 and 4. In this case only one Ijevel will be required, because the tan- gent DF in elevation, Fig. 11, when prolonged, intersects tiie base line XX, at the point J, and the point J coincides with the point D, Fig. 10, whieli is square from the point A to the tangent BC, Fig. 10; and DA then becomes the director, and is parallel in this case with the joint at B. Observe in elevation, Fig. 11, when the tangent DJ* is pro- longed to the bas(! line XX; if it intersects the base line between the points J" and C, as siiown at C2, Fig. S, in all such cases, two bevels will be re(juired, ))ut they will not cross the tangents in their application. But if the inclination of tangent Z)i^ should inter- sect the base line to the left of the point J, then there will be two bevels required, and also they will always cross the tangents as shown at Fig. 5, where tangent DH produced intersects the base line XX at J to the left of the jioint V. Let it be rememl^ered from this that the i>oint J, is the dividing point in the application of the bevels, as to eitlier, they cross the tangent, or tliey do not. Let the student bear tliis in mind, for it is ditficult sometimes to know which way to apply the bevels; this only liappens when the tangents on plan are obtuse at tlie anglt; C; when the tangents on ]ilan are acute, bear iu mind that the bevels will always cross the tangents iu their application. Wbcn the tangents on plan form arifcht anjile, the bevels will cross llic tiuisents on all wrealli-pieces where boMi tiuiLreuts are inclining, and wlioro one tangent is level, and the otlier inclining, the square will apply to tlie joint made on the inclining tangent. Bevel. Prolong BO, Fig. 10 indefinite; make BM equal HK \n elevation. Fin. 11; join i^ikf for the bevel at M; parallel witl\ SikTdraw the half v,'idlh of rail [2^^] to cut MF at 9. Pig. 12. Exlilhiis tli-e face-mould. Make BC equal DF, Fig. 11; with B as a center, and the distance DM, Fig. 11, for a radius, draw arc at A; again with C as a center, and tangent AF, Fig. 11, for a radius, draw arc inter- spcting at A. Connect AC; parallel to ACaud BC, drawiJO and AO, creating the parallelogram OACB on the cutting plane, that will coincide with the parallelogram OACjBoii plan when elevated into position. iVLike joints at A and B perpendicular to tiie tan- gents AC and BC; draw the cliord AB. Bisect AB at Q; Insect AQ and BQ at R and S; bisect Ai2 at T. Make JBZ) equal JF in elevnlion. Fig. 11; join DO; tlien DO becomes th(! director, wliich is at right angles to tangent BC. From the points A, T, R, Q and S, draw ordinates indefinite and parallel to the director, then transfer the points 2, 3 and 4 on the ordinates on plan, Fig. 10, to corresponding ordinates on face-mould as shown. Plate 15. 69 Make JB 7 and B S each equal B 7 and B 8 on plan. Fig. 10; let A 5 and A 6 each equal Md, Fig. 10; now through the points 5, 3, 7 trace the concave curve, and through tlio points G, 4, 8 trace the curve for tlie convex side of face moiikl. The section at N shows the bevel found in tlie angle at M, Fig. 10, applied in the usual waj'; at section M tlie siiuare is applied fiom the face of erode as sliown. If it be required to find the point of tangency, prolong DA, Fig. 10. With T as a center, and TA, Fig. 12, for a radius, intersect DA at K; draw TK ])volonged, draw G P parallel to DK. Make AP, Fig, 12, equal KF, Fig. 10; draw JPL parallel to OD and equal to L G, Fig. 10; draw LA prolonged, make AV equal AL, then Z» and V are the points of tangency. PLATE 15. Plate 15. [Scale M^'' — l foot.] Shows how to place the rUsers cuid buhisters in jiUttform (uid quarter-pace c]]U)idcrii. When winders in a semi-circle are to l>c used, Fi'fa. 15 aud la shows hoiv tlie treads tnuy he (jradnatedin ivldthattlie iKf/Twr ends so as to forma, graceful curve connectinfjtheivrcath part of string. Fig. 1. Shoxvs a C -cylinder. The tread is to be 9^^; the radius Oi^ecnials .3^^; with for a center and OF [:/'] for a radius, draw tlie semi-circle BFC; draw BA and CD to indicate tlie straight part of outer string connecting the circidar part. Witli J* as a center and FO for a radius, draw arc intersecting at H; tlirough i^'and iifdraw a line to intersect DC prolonged, and also CB prolonged, at J'aufl K; then iTis the focus and the distance JC is the stretchout of the curve from JPto C, Now make from J" to the face of No. 16 j-ise to equal half a tread Vi''\ and from face of No. IG to face of No. 17 rise, to equal 'J^^; opposite Nos. IG and 17 rise, draw the face of No. 15 and 14 rise, thus locating the face of risers iu a iV cylinder. Observe the face of Nos. 14 ;ind 17 rise are 8I/3" fi-orn the .iointof cylinder on the striii;-;ht i)art of strinjr; tliis must be carefully noled when proceeding to construct tl\e face-mould, so that the correct height of wrouth-pieco Lua,y be established. Fig's. 2 to 4. Shoivs o. 7^^, 8^^ and f/^ cijllndcr. The location of risers are found in the sanii? manner as at Fig, 1, the lettering being the same. Oljserve at Fig. 2 tlie face of No. 12 and 15 rise are opposite and equal 8''^ from the spring of cylin- der; at Fig. 3 the face of risers is 714^^ from the s])ring of cylin- der ; at Fig. 4 the tread is 10^^, and the face of Nos. i:j and 10 risers is 8^^ from the joint of cylinder. Fig. 5. Shows a cylinder 10^^ in diameter. The stretchout CJ" for the (luailer circle and location of liscrs, repeats Fig. 1, tlie treads are 10^'. Make JL vq\n\\ half a tread- [5^^J for the face of No. 18 riso and 10" more to the face of No. 19 rise; draw the face of No. 10 and 17 rise directly ojijiosite ; now locate liie position of short baluster on No. IGand 19 tieads and space off the intermediate balusters; tlien draw the face of Nos. 17 and IS rise to intersect the cylinder and to suit the balusters; No. 17 rise may continue straight into tlie cylinder, but No. 18 rise should be curved so that the nosing may return without too much peak at the miter. 70 Plate 15. The face of No. 16 and 19 rise are V from the joint or spring of cylinder. Fig. 6. Exhihits a 12" cyHrifMr. The treads are sliowu 10^^; the stretchout CJ for the quarter circle is found in the same manner as at Fig. 1. Only in this case observe the stretchout is for the center line of rail instead of the string line. I3y locating the risers around the string line in the cylinder gives the true inclination on the string line, thus allowing the veneer to be straight; having no easing in case the cylinder be veneered; by spacing off the regular treads on the center line of rail, the -veneer for the face of cylinder will have an eas-ing above and also below, connecting the circular part with the straight ; this gives to the wreath part tif rail more inclination, and thereby helps the appearance of the rail. The cylinder is 12" in diameter, and if 2"x2^^ balusters be required the center line of rail will equal Vo'}>i" diameter, or a radius of Q%". Here the face of Nos. i:5 and IG rise are opposite, and '^li" from the joint or spring of cylinder. Now locate the short baluster on No. 13 and 16, treads and space off the intermediate balusters around the cylinder, then curve No. 14 and 15 rise, to suit the balusters at M and N. The intersection of No. 14 rise is made opposite that at N; the rise No. 14 is termed a "concave rise," and No. 15 rise is termed is a " convex rise." Pig. 7. Shows the treads laid off on the center line of rail for a cuUnder 14" in dhwietcr. The dotted line shows the center line of baluster, and is struck with a radius of 1%"\ the tread is \0", JCis the stretch- out for the quarter circle PNF. Make JL equal half a tread [5'^] for the face of No. 16 rise, and to No. 17 rise to equal 10^''; draw No. 15 and 14 rise opposite. Now take LC as a radius, and the verge of baluster at N for a center, draw arc to intersect BC prolonged at F; then FN will equal the radius of curve; find point on the opposite side in same manner for No. 15 rise; the face of No. 14 and 17 rise are 2%" from tlie spring of cylinder. Fig. 8. Shous the treads laid off i)i the cylinder on the center line of rail in the same manner as at Fig. 7; the face of No. 12 ^''^ The solid line shows the string line, and the dotted line indi- cates the center of baluster, which is the center of rail. Draw the tangents AJB and CB, forming the right angle at S; from B set off half a tread [5^^] each way on the tangents for the face of No. 12 and 13 rise; set off No. 11 and 14 rise, to equal a tread \10^^\ as shown; observe the face of No. 11 and 14 rise, both cut the front string 8}4^^ from the spring of cylinder on either side. Now locate the short baluster on No. 11 and 14 treads, and space off the intervening balusters around the cylinder, then curve No. 13 rise to suit the baluster; No. 13 rise in this case may run in straight or may be curved also, as the stair-builder may desire. Pig. 11. Shows a quarter cylinder of 8'^ radius, the solid line being the face of string, and the dotted line indicates the cente)' line of rail. The radius OA equals %%'^\ AB and JSCshow the tangents forming a right angle at B. For want of room, tlie face of No. 14 rise is placed at the point B; then from B set off on the tan- gents a full tread either way, as No. 13 and 15 rise; draw them in to Intersect the outer strings, and at right angles to the same. Now space off the balusters in the cylinder and curve No. 14 rise to suit the baluster; take a tread [10'^], for a radius, and say at J? as a center, draw arc at P; then B for a center, draw the curve for No. 14 rise; the face of Nos. 13 and 15 rise is Hi'' from the joint of quarter cylinder. Fig. 12. Shows a quarter cylinder, the radius being 10". If 'y'x2," balusters be used, then the radius for the center line of rail will equal iOyi", .hs shown by the dotted Ihio. Draw the tangents AB and BC, forming a right angle at B; the spac- ing of the treads in the upper lliglit of stairs locates No. 10 rise on the tangent BC, two [■//'] inches from the angle at B; the regular tread being 10^^, then the face of No. 15 rise, from the angle at B on the tangent AB, must equal 8^^, or the difference between BD and the regular tread which eciuals [10—2=8] eight inches. Set off No. 14 and 17 rise to equal 10'^ each; the face of No. 14 rise cuts the string V from tlie spring line. Mark the posi- tion of No. 1 and baluster; then space the intervening Ijalui!,- ters Nos. 2, 3, 4 and 5 eciually, and draw the face of No. 15 and 16 rise into the cylinder to suit the balusters. Tiiefaceof No. 15 rise C(mtinucs straiglit into the cylinder, and tlie face of No. 10 rise requires to be curved. AVith the regular tread [10'^] for a radius and a point at the verge of No. 4 baluster at F, draw arc cutting No. 17 rise at H, from H draw the curve tangent to No. 16 rise to inters(!ct the face of cylinder. The curving of the risers is discretionaiy with the workman; the curve may be more or less; they should be tlrawn, however, so that they wiil miter with the return nosings without too much peak at the point of 72 Plate 15. miter. The above quarttr cylinders we have first deciderl on the radius of curve, this need not be tlie rule; for the position of risers may be first dot^ided upon, then afterward the radius of quarter circle as described at Fig. 13. Fig. 13. Shoivs how to place the risers in a quarter turn, so that one lorcatU will he rc; then niakc^ Xo. 11 and 14 rise eiiual a regular tread 'J'^ as sliown. Draw the face of No. 11 and 11 rise at right angles to BA and BC, Now decide upon the radius of quarter cylinder, say 10'^, and the balusters being 2^^x2^^ which will make the radius for t!ie center line of rail equal l()'/A; with O as a center, draw the curve i\o\n D to J' for the center line of rail. The solid line shows the face of cylinder the radius of which is %'' less or 10^^. The face of No. 11 and 11 rise ai'e 'iX^^ from the spring of cylinder. Mark the center of No. 1 and baluster, then space off the intervening balusters 2, 3, 4 and 5, equally on the center line of rail and curve No. 12 and 1.3 rise to suit the balusters as explained at Fig. 12, Fig. 14. Exhibits how to place the risers in, a quarter circle, so that two vjrcath-plcccs may be required, ami that they mny be worked from the plaidi wUh the Icnxt pi>H>^lblethl<:kncxs risers; again set olf a regular tread | ll''^] each way for the face of No. 13 and 16 risers; drav/ the face of No. 13 and 14 rise at rigiit angles to the face of string as shown; ol)serve the risers cut the face string iV from the spring of quarter circle on each side. Now locate No. 1 and 7 baluster, and space the intermediate balusters 2, 3, 4, .5 and 6, eciually; then curve No. 14 and 15 risers to suit the balusters as shown. Hoirio stuir-buildors prof(!r to yihira the risers in this way in a (|iiiiitor circle, unci iiiiikin-;- llio v/ix'!i.th in two pieces, lo relievts tliu eiising on the lower ed^e of the. ((Uintci' cylinder, paiticularly wlien llin balusters arc over 2" in dituneler; by plii.<'in^ llie risers as shown ut FiR. 14, loss abruptness is cucountered iil the johiing of tlie cylin- der with the straight string. Fig. 15. Shows hoxo to lay off a veneer lohen there arc xvindcrs Ui tlie culindcr. The plan of cylinder is 13'^ in diameter. IIow to place the winders so as to form an agreeable easing above and below at the connection of the straight willi the circular part of outer string.* *This inetliod of graduating the .steps around tlic cylinder does not govern the pitch of rail. Tlic stair-builder lias to raise the incli- nation of tangents to make the height of rail agreeable for tho.se pas- sing up or down; tliis graduating of the steps is to form a pleasing easing, connecting the straight string v/ith the cylinder. -» i'l.ATK 10. VS AB equals the diameter 13'^, AC and BD indicates the straight string; with A and B for centers, and AB as a radius, draw arcs intersecting at F; parallel with AB draw Jf J" indefi- nite, so as to tangent the curve at L. From F and through points B and A draw lines to intersect the tangent line at H and J; then JH is the stretchout for the semi circle ALB. Fig. 16. Shows the elevation of riscru and iiuinner if raduatlnfi the ividth of winders at the narrow end. At right angles to HJ draw JG equal to the height of six risers, more or less, as the case may be. Join GH. At H place the pitchboard and draw the inclination HK, also place the pitch- board at G and draw tlie true inclination GM for the straight string; bisect the angles at if and Gr. draw tlie curves 2, Sand QXR at will, the worivman using his judgment as to the amount of curve that will suit best. Now draw Nos. 13, 13, 14. 15, 1(5, IT, and 18 treads to intersect the mixed lineiTa, 3, Q, X, R. and M. From the intersections draw the riser liues parallel to JG to intersect the stretchout HJ&t 5, 6, 7, 8 and 9, from which draw lines to the focus F, intersecting the cylinder line at a, b, c, d and e for the location of the face of risers graduating them on plan. Fig. 15, agreeable to the curve in elevation Fig. 16 as shown. The face of No. 12 and 18 rise is 3^' from the spring of cylinder; No. 11 and 18 treads are each 9^^ wide, No. 10 and 19 treads are the regular width 11^^ Now locate the center of short baluster, on No. 11 and 19 treads and space of the intervening bal- usters on the center line of rail, shown by the dotted line C7D, the face of risers may be shifted a little to suit the balusters as shown at a and d, also No. 18 rise is shifted H^' to suit the baluster. From the internal angle of rise and tread. Fig. 10; set otf the width of string and draw the lower edge of string to please the eye as shown by the dotted line YXO; the veneer should be laid otf with lead pencil so as not to abraid the surface of veneer that would, when bending the veneer over a drum, cause fracture; neither should the veneer be notched out for the treads and risers until the cylinder is taken from the drum; XX shows the length of staves in the rough. These cyliuders may be constructed in different ways; there is more economy in the use of staves; !?*Jod glue and very dry mate- rial should be used: for painted work use soft pine for staves. For hard wood finish, the veneered cylinder makes the best and most finished work, but the expense is greater. For the thickness of veneer see letter press for Plate 28. PLATE 16. Plate 16. [Scale, % ^^=i foot] . Exhibits the construction of a face-mould for a wreath-piece landing on the level; also staHini) from the level to rake, the tangents on plan being either at right angles, acute or obtuse. Also shows how to place the risers in the cylinder at Vie starting or landing, so as to make one face-mould answer for both ivrcath-pieces. Figs. 1,2,3 and 4. Slwws the plan, elevation a^id face- moulds for a wreath landing on the level. The tread is 9^^ by S'^ rise. The face of No. 16 rise is B^^ from the spring of cylinder, and hence the face of No. 17 rise must be in the cylinder 6'^; the balusters are 2^^ diameter, and & 74 Plate 16. the rail double moulded, 3'' wide aud \" deep; the cylinder is \Qyi" in diameter, then the radius for the center line of rail will equal ^^^=^M"^%^^"\ six inches [6^^]. Then with ^" as a radius, and O for a center, draw the semi-circle A CJE7; en- close the semi-circle with the rectilineal parallelogram ABDE; draw OC perpendicular to AE, and we liave the tangents AB, BC, CD and DE on plan; draw the chord AC; prolong tangent BA and DE to the left for the direction of straight rail. Pig. 2. Exhibitsthedevelopmcntandelcvationof tangents. Let XX indicate the edge of drawing board; make AB, BC, CD and DE equal tangents AB, BC. CD and DE on plan, Fig. 1; from the points A, B, C, D and E draw perpendiculars to XX indefinite. Now elevate Nos. 15, 16 and 17 treads and risers, keeping tlie face of No. 16 rise 3'^ from the spring line at A; tlirough the center of baluster OO draw the under side of rail; parallel witli OO draAV the center of rail,* cutting the perpendiculars at iJand J, From the floor line set up 4^' to under side of rail, and half the depth of rail more [3^'J to the center of level rail; parallel to XX draw the center of level rail, cutting the perpendiculars at if aud L. Draw LJ prolonged to M, and cutting the per))endicular from Cat N; through N, and parallel to XX. draw WIR, cutting the perpendiculars at JPand Q; prolong HJ to intersect WR at G; parallel to XX draw HS, cutting,' BP at T; then NS is the height for the lower wreath-piece, and RKis the height for the upper wreath-piece; prolong HS to E, then EL will be the whole height. Make J?!/ equal the chord AC, Fig. 1; from IF, aud at right angles to tangent iVJ" prolonged, draw TV; from P, and at right angles to tangent HJ prolonged, draw B'i; parallel to BP draw the half width of rail [IM^'J, cutting the tangents at 3 and 4 for the increased width of mould on the radial Hues; make J? j^ equal the chord AC, Fig. 1. Bevels. Make ab parallel to XX; at right angles to XX draw dh equal to the radius OC, Fig. 1; make d 7 equal TV: aud d G equal P2, also make d 5 equal the height RK; draw h 7, ii 6 and h 5 prolonged to XX. ' Fig. 3. Exhibits the face-mould. Let AB equal HJ, Fig. 2; with A as a center, and Wt^, Fig. 2. for a radius, draw arc at C: again, with J3 as a center, and NJ. Fig. 2, for a radius, draw arc intersecting at C. Join BC. Parallel to AB and CB, draw the radial lines CO and AO for the parallelogram ABCO on the cutting plane that will coincide with the parallelogram ABCO on plan when in position. Proof. The diagonal BO must equal MU, Fig. 2, if so, thf angle of tangents at JB is correct. Prolong tangent BA 6'^ to F: make joints at F and C at right angles to tangents BA and BC. Make AD equal JG, Fig. 2; draw DO for the direction of minor axis; make CS and C 3 each equal J 4, Fig. 2, also make A 4, *Let it. he okserved right here that for all wreaths .startiug from Ihe level to rake, or those from the rake to a level, and also turn- outs at the newel, and wreaths on winders; also platform wreatb.s, when it is refiuired to find the length of odd balusters. In all such cases the center line of rail must be drawn In this way to obtain the correct bciglit for each wreath-piece. For platform wreatlis, when the length of odd balustei-s is not required, the center line of rail may be drawn at once from the external angle of .step and rise, for the inclination of tangents and height of wrcath-picce as shown at Fig. 2, Plate 29. , Plate 16 75 A 5, each equal J 3, Fig. 2. Let O 8 equal the radius OC, Fig. 1, for the length of senii-iiiinor axis; make 8 6 and 8 7 each equal the half width of rail [IH^'J. Parallel vfith AJPdraw 5 10 and 4 9 for the shauk of mould, Now pivot the trammel in O with the arms at right angles to O S; then set from pencil to minor-pin the distance O 7, place the pencil in the point at 3, drop the pins in the grooves, and fasten the major-pin; then trace the curve for the convex side of mould through the points 5, 7, 3: again, set from pencil to minor-pin the distance O 6, place the pencil in the point at 2, and drop the pins in the grooves, then fasten the major-pin and trace the curve through the points 4, 6, 2, for the concave side of face-mould; proceed in like manner to trace the center line. The section at M shows the bevel found in the angle at 7, Fig. 3, applied from the far-e of crook so as to pitch tlie joint at F down; the section at iVsliows the bevel found in the angle at 6, Fig. 2, applied so as to pitch the joint at C up; the block pat- tern shows the section of rail when squared up, the shaded parts indicate the surplus wootl that has to be removed in the formatiou of wreath-piece. -■ Fig. 4. Slunvs tlic face-iuoxild for the wvcatli-piccc on Ihc level. Draw the right angle EDC, make DC equal NL, Fig. ;J. make J) J^ equal ED on plan. Fig. 1. Parallel with DE and DC draw CO and EO for the parallelogi'am OCDE on the cutting plane, or plane of plank. Proof. The chord EC and also the diagonal OD in this case must cijual KY, Fig. 2; prolong tangent DE to i^'o". Make joint at F s(iuare to DF\ make C2 and Co each eriual the half width of rail [1H^'][ let E 4 and E 5 each eijual J 3, Fig. 2. Parallel to EF draw 4 6 and 5 7, for the shank of mould; as th<' difference between the semi-minor axis OC and the semi-major axis OE is so little, the comi)asscs will suit best to draw the concave and convex curves as shown i. The .section at i!f shows the bevel found in the angle at 5, Fig. 2, applied so as to pitch the joint at C down; (he section at iV" shows tlie square applied allowing the shank FE to have a horizontal position. The block ] atteni is applied at right angles to the line made from the bevel showing the twist of rail in the crook at the joint. Figs. 5 and 6. Sho^r how fo place the risers in the cylin- der for a tDvaith. siarling from the lei^ei, so as to iiwlce thefacc- Moidds draivn for Fig. 1 answer for Fi'j. 5. Fig. 5. ShoHS the plan. The tangents AJ5, BC, CI? and i).£7 enclosing the. semi-circl« ACE, coincide with plan, Fig. 1, Prolong tangents BA and DE to the right, for the direction of straight rail; the radius OC for the center line of rail equals 6'''. Fig. 6. Sfiows Ute elevation and development of tangents, and also the treads and risers. Let XX indicate the edge of drawing board; make AB, BC, CD and DE equal tangents AB, BC, CD and DE ou i)lau. Fig. .5. From A, jB. C, D and -E7 erect perpendiculars to XX indefi- nite; let XX also indicate the floor line; from the floor line set up 4"' to the under side of rail, and the half depth of rail [2^^] addi- tional, making 6" fnmi the floor to the center of rail as shown ou the left. Parallel with XX draw the center of rail cutting the perpendiculars in F, G, H, J" and K; make KL equal EL, Fig. 3, ft)r the wIm.u- i.:j'^^ iu the cyliuder, and as the treads are W the face of No, 2 rise will be 7>^'^ from the spring line of cylinder as shown. Thus estabiishing the location of risers so as to give the proper hcvjht to the wreath-piece as required; from iJand parallel to XX. tli-aw HJ, cutting the perpendicular from B at K; thou EK is the height that the v.'reath-piece has to raise. Make EL equal BF, Fig. 7, from L and at right angles to !£■.& prolonged, draw LM\ make JiV equal the chord AC, Fig. 7 I'l.ATK 1(>. TT Let JP eqnal the diagonal BD, Fit;-. 7; prolong JH and make JQ equal BO on plan. Fig. 7; tliiough D and P draw a line to intersect the perpeudiciilar from Q at R. Bevels. Draw the gange line /ST parallel to XX\ at right angles to XX, draw VW e(iual to AF, Fig. 7. Wake T7'4 equal LM, Fig. S, let V .5 eipial the heiijht EK in e'evation, Fig. 8; draw W 4 and IT 5 prolonged to XX, parallel to ST, draw the half width [:y'] uf rail, cutting the hypotlienuse of bevels at 6 and 7. Fig. 9. Exhibits the fncc-mouhl. Make AB equal ^iJ, Fig, 8; with A as a center and DN, Fig. 8, for a radius, draw arc at C: again, with S for a center and tangent AB, Fig. 7, for a radius, draw arc inteisecting at C, Join BC. I'aralhd to AB and BC, draw CD and AD for the parallelogram ABCD on the cutting plane or plane of plank. P7-oof. The diagonal BD must agree with DP, Fig. 8. If so, the angle of tangents at B is correct. Prolong tangent BA, G" to F; also prolong the level tangent BC, ^" to H\ make joints at J' and iT at right angles to tiie tan- gents as shown; fi'om Cand at right angles to tangent BC, draw CJ indefinite; prolong the diagonal BD to intersect CJ" at 0\ th-aw the radial line AO for the point of tangency, or the intersec- tion of straight shank with the curved part of face-monld; for I'ROok that the intersection at O is correct, the diagonal OS must equal DR, Fig. 8. Make F 2 and Fi each equal 4-6, Fig. 8; also make C 4 and C .5 each eciual .5-7, Fig. 8; parallel with FB, draw 3-!» and 2-8; parallel with CH draw .5-7 and 4-G; at right angles to OC draw OK eqnal to the radius OA, Fig. 7. Make K 1 and K 11 each equal the half width of rail [2"]; now OC is the semi- major axis and OiiC the semi-minor axis. Instead of a trammel use a straight edge to find the elliptic curves of this face-mould. Draw the two right angles at O to indicate the arms of trammel; for the concave side of mould take the straight edge and mark from the end at 8 to Li, the distance O 10, for the minor notch; again mark the major notch at P cimal to O 4; now move the rod at intervals, keeping the two notches on the riglit angle lines, then marking points at the end of rod, through whichdraw the curves, using a flexible strip; proceed in like manner to draw the center and convex curves of mould. Points on tiie diagonal OB may be found, giving three points in the cune through which to draw the curves, and thus dispensing with the trammel. To find the points return to Fig. 8: make Qyequal to the radius OA. Fig. 7; |.ara!lel to JD draw YF, parallel to YF draw the half width of rail [2^'] intersecting DR at G. At Fig 1) make B-S equal DF, Kig. 8; let S 12 and -S 13 each equal FG, Fig. 8, giving the three lioints required.* Observe bevels as applied at sections M and iV makes the wreath-piece answer for a landing on the level, instead for a start- ing off the level as shown on plan, Fig. 7; by turning the wreath- piece upside down, when squared up before moulding, it will then answer for plan. Fig. 7. The bevel applied at section Jlf is found in the angle at 5, Fig. 8; and the bevel at section iVis found in the angle at 4, Fig. 8. Observe they do not cross the tangents in their application. *The correct width for any face-mould on the diagonal line was first given by Mr. J. H. Monckton. then by Mr. Secor. The above method will suit for small-face moulds, but for long moulds more points iu the curve are required as a guide for the flexible strip. '-'8 Plate 16. Figs. 10 and 11. Shmvs hoiv to place the risers in the cylinder so 4^^ for the lloor line to agree with AD, Fig. H. Through K, and parallel to XX, draw the floor line intersecting tlie under side of rail at O; then O is the center of baluster; the baluster being 2^^ in diameter set off 1 inch more to the face of rise marked No. 15; draw No. 15 rise at right angles to XX, and set off No. 14 tread and rise, thus locating the face of No. 14 rise S%^^ from the spring line of cyl- inder. As the treads are 10''' then the face of No. 15 rise will be in the cylinder 1^'^ from the soring line. [W—^%"=\ %'^}. Observe the height LF equals the height KE, for Fig. 8. Now return to plan. Fig. 10, and make the face of No. 14 rise 8%''^ from the spring line OC; thus arranging the risers in the cylinder on plan so that the face-mould. Fig. 9, will also suit for Ihe landing wreath-piece. Figs. 12,13 and 14. Shoirs the plan, eUvation and face^ mould for a ■wrcatli-inccc, starting from the level to the ndw, the angle of tangents on plan licing acute; the curve on plan being greater than a quarter circle. TiuM-egular tread is 10^', the rise 7^', balusters 2'^ in diam- eter, and the rail 4'^ wide by 2)4^^ deep. How to place the risers in the cylinder so that the wreath- piece will raise the proper height on the level, for the long balus- ters, and at the same time be worked from the least possible thickness of plank. Fig. 12. Shows the plan. AB, BC are the tangents; at right angles to the tangents, draw AO and CO; from the center O describe the center line of rail from A to C; prolong the tangents for the direction of straiiilit rail. Draw the chord AC, from Cand at right angles to .AJS draw CD. Fig. 13. Erhlbits the development and elevation of tan- gents, and also the treads and risers. Let XX indicate the edge of drawing board and also the line of floor. iSlake AB, BC e:\\ml tangents AB, BC on plan. Fig. 12; draw perpendiculars to JfX from the points A, B, Cindelinite; from XX set up half a rise to the underside of level rail and 1 h^' additional to the center of rail at D; parallel with XX draw DF; through F draw the true inclination of the center line of rail, cutting the perpendicular from A at H; parallel with FH produced, draw the underside of rail. Now on AiT set np the height of No. 1 and 'Z risers; parallel with XX draw No. 1 and 2 ri.ATK 16. Tit treads, cutting tlie under side of rail at O, O for tlie center of bal- uster, as the balusters are 2^\ then the face of No. 1 and 2 rise will be V^ to the right, or the face of No. 3 rise will be 4K^^ from the spring line AH; and as the treads are 10'^ the face of No. 1 rise will be 53'o^^ in the cylinder. Now return to plan. Fig. 12; make the face of No. 2 rise4K'''' from the spring line; set off the face of No. 1 and 3 rise and draw them at right angles to JBA prolonged; locate the balusters at No 3 tread and space them off around the cylinder, and curve No. 1 rise to suit the baluster. Returning to Fig. 13 draw iJJ" paral- lel to XX, cutting the perpendicular from S at if, then JTfis the height the wreath-piece is required to raise; bisect DJ" at P. Make JL equal HB on plan. Fig. 13; let J!2\f equal the chord AC. Fig. 13; draw PL prolonged to intersect the perpendicular from R&tN. Make FQ equal DB, Fig. 13; from Q and at right angles to i?P produced, draw QS.^ Bevels. Parallel with XX draw TV; draw TW perpen- dicular to XX, and equal to DC, Fig. 12; make Ti equal SO, Fig, 13, make T 5 equal the height FK, Fig. 13. Draw W4 and TVs produced to XX for the inclination of bevels. Parallel with rVdraw the Iralf width of rail, cutting the hypothenuse of bevels at 6 and 7. Fig. 14. Exhibits the eonstructicni of face-mould. Make AB equal HF, Fig. 13, with A as a center, and DM, Fig. 13, for a radius, draw arc at C; again, witli .5 as a center, and the tangent CB on plan. Fig. 13, for a radius, draw arc inter- secting at C, draw BC, draw the chord AC, bisect AC at H, draw the diagonal .Bi? prolonged. From C draw a perpendicular to tangent BC, and inter- secting BH produced at 0; draw the radial line OA for tlio trapezium OABC on the cutting plane, that will agree, when in position, with the parallelogram OAJBCon plan, Fig. 13. Proof. The diagonal BHO must equal PLN, Fig. 13. Prolong tangent BA ^" to J" for shank; also prolong BC to Z», 1", for straight wood to help the easing at the connection of straight rail. Make joints at J and i at right angles to the tan- gents. Let C3 and C3 each equal 5 7, Fig. 13; make J" 4 and Jo, each equal 4 6, Fig. 13. Draw the semi-minor axis OF in- definite, and at right angles to OC, make OF equal the radius OA, Fig. 13. f Let F 10 and P 11 each equal the half width of rail [3^'] parallel with JB draw 4-6 and .5-7; also parallel to LB diaw 3-l» and 3-8; now place the trammel in the point at O with the arms on the semi-major axis OC. Set from the pencil to minor pin on tiie rod the distance O 11, then place the pencil iu the point 3, drop the pins iu the grooves and fasten the major pin; now trace the convex curve through the points 7, 11, 3. Eepeat the opera- tion for the center and concave curves of face-mould. The section at M shows the bevel found in the angle at 4. Fig, 13, and the bevel applied at section JVis shown in the angle at 5, Fig. 13. *The point Q should be opposite the point D, Fig;. 12, or to the left of the perpendicular BK, as shown at Fig. 2, Plate 1.3; the results how- ever are the same either way, for a mould of this kind ha\ing a full easement. - . so Plate 17. Pigs. 15 and 16. Sh9ws ike plan and elevation of a cylin- der landinfi to accompany Figs. 12, and 13, which is for a starting, Jlovf to jjlace the risers so that one face-mould will answer for both to, eath-pieces. Fig. 15. Shows the plan. OC is the radius, CB and BA are the tangents. Fig. 16. Shows the develojnne)it and elevation of tangents and the trends and risers. Let XX indicate a base line; malce AB and BC equal tan- gents AB. BC. Fig. 15. From the points A, B, C erect perpen- diculars to XX indefinite; parallel to XX draw DF, from F draw the true inclination of the center line of rail, cutting the perpendicular from A at H; parallel to i^iif produced, draw t\w underside of rail shown Ity the dotted line. INIake DK equal FB, Fig. 13; [4M^^] from IT and parallel to XX. draw the floor line, cutting the underside of rail at 0, then is in the center of baluster, the baluster is 2'', then the face of landing rise. No. 15, will be V^ to the right of 0. Now set off No. 14 tread and rise ■which locates the face of No. 14 rise h%'' from the spring line AH of the cylinder. Parallel to XX draw HJ; it will be observed that the height JF is equal to the heiyht KF, Fig. 13, and.that they both are the same height [4%'^] from the floor line to the center of rail. Now return to plan. Fig. 15, draw the face of No. 14 rise h%" from the spring line OA, then as the regular tread is W\ the face of No. 15 rise will be in the cylinder 4^'^; now space oft" the balusters from No. 14 rise around the cylinder, then cun'e No. 15 rise to suit the baluster as shown. PLATE 17. Plate 17. [Scale K'"'=l foot.] Exhibits the construction of a face-mould, for a quaHer cylinder having flyers above and below the quaiter pace. Also the face-mould for a wreafh-plece staHing from a newel, the cylinder being less than a quarter circle on plan. Figs. 1, 2 and 3. Shows the plan, elevation and face- mould. For a quarter circle, tread is W^ and rise 1^', the balusters are S^^xS''^ and rail 4^'x2X^^. Fig. 1. Slwws plan similar to Fig. 10, Plate 15; the risers are placed, in the quaHer cylinder so the ivreath-piece nmy he worhed out in one piece, and may luwe the same inclination on tlie line of tangents as the straight part of rail. The radius OA for the center line of rail equals 65^^'', draw the radii OA and OC at right angles; parallel to OA and OC draw the tangents CB and AB, forming the square parallelogram OABC on plan; prolong tangents BA and BC for the direction of straight rail, the face of No. 12 and 13 rise is 5'^ from the angle at B; No. 11 and 14 rise is spaced off a regular tread lO^'', locating the face of No, 11 and 14 rise 8K^^ from the spring of cylinder at A and C* f ■ Often the stair builder has to furnish rails to order from the country. In all such cases he must know the diameter of cylinder, height of rise, width of tread, the location of risers at the cylinder, if at the spring, or out from the same, size of balustera, also rail; if the rail is on the right or left ascending the stairs. Plate 17. 81 Fig. 2. Sliowfi the elevation and development of tangents , also the treads and risers from plan. Let XX imlicate the edge of drawing board. Make AB and JBC equal tangents AB and .BC on plan. Fig. 1; from the points A, JSand Cdraw lines perpendicular to XX indefinite; now elevate the treads and risers, keeping the face of No. 11 and 14 rise 81^''^ from the spring lines A and C; draw the true incli- nation of rail from the external angle of No. 11 and 13 tread anattern is applied at right angles to the line made from the bevel, giving llie section of rail and twist of wreath-piece at the joints. Figs. 4, 5 and 6. Shows the construction, of face-mould for a quHrtcr cylinder on pUin c(n'rcs)}onding t/} Fig. 12, Plate t'), "■/'crc the learner is instruvicd how to place Vie risers in the cyl- inder. Fig. 4. Shoirs the jdan. The face of No. 14 rise is located 7^' from the sjuingof cylin- der and the face of No. 17 rise is placed l'^'' from the spring of cylhider; the radius for tlie center line of rail eqnals lOyi'^- The tangents AB and CB are at right angles to the radial lines OA and OC, forming the square parallelogram OABC, ou plan. 83 Plate 11. Fig. 5. Slujws the development and elevation of tangents, ulsn the treads and risers. Let XX indicate tiie ed^e of drawing board; make AB and BC equal the tangents AB and BC on plan. Fig. 4; from the points A, B and C draw perpendiculars to XX indefinite; now elevate the risers and treads, keeping the face of No. 14 rise 7^^ from the spring line at A, and the face of No. 17 rise lU^^ to the right of the spring line at C; from the external angle of No. 14 rise, draw the true inclination of the rail, cutting the perpendic- ulars at JD, JE7 and F; from D, and parallel to XX, draw DH, cutting BE &t J; then HF equals the height the wreath-piece is required to raise; from J, and at right angles to DE, draw JK; parallel with BE draw the half width of rail ['i''^] cutting the inclination of rail at N; make HL equal the chord AC, Fig. 4. Bevels. Draw PQ parallel to XX; at right angles to XX draw PR e(iual to the radius OC, Fig. 4; make P 5 equal JK, draw R 5 prolonged to XX for the bevel as shown. Pig. 6. Shmvs the fnce-monld. Make AB equal DE, Fig. 5; with A as a center, and LF, Fig. 5, for a radius, draw arc at C: again, with S as a center, and BA for a radius, draw arc iritersecting at C, join BC; par- allel to tangent BC nuA BA, draw tlie radial lines OA ami OC, and we have tl»«liarallelogram OABC on the cutting plane, that when in position, '.vill agree with the parallelogram OA.BC on plan, Fig. 4. Proof. The diagonal BO must agree with the diagonal BO on plan, Fig. 4, if so, the angle of tangents at B is correct. Prolong tangents BA and BC to D and E 6'^ for straight wood on shank; make joints at JD and E at right angles to BD and BE; draw the diagonal BO for the direction of minor axis. Malte OJ" e(iual the radius OC, Fig. 4; let J'C and J 7 each equal tlie half widtli of rail [-y^]; let A 2 and A 3, also C 4 and C 5, (!ach e(iual EN, Fig. 5. Parallel to BD antl BE draw ii 10 and o 11, also 5 9 and 4 8, for the width of shank. if it be preferred to find tlie ijeveral points in the elliptic curve and use a pliable strip in tracing the curves, then return to j)lau, Fig. 4; draw the true width of rail [4'^j from the center O equally on each side of the center line. Draw the director OB and the chord AC, forming the angle at V; bisect A V and CV at Tand W; from the points A, T, V, Wand Cdrawordiuates parallel to the director OB. cutting the convex, center and con- cave curves of rail at a, d, h, &c. Now return to Fig. G; draw the diagonal OB and the chord AC, forming the ang'e at V: bisect A V and CVal Wand T. From the points A, T, Wand C draw ordinates indefinite and parallel to the director BO; now transfer the points on the ordinates from plan, Fig. 1 to corre- sponding ordinates on face-mould, using the chord AC as a base line, then trace the curve through the points as shown, using a flexible strip. The sections at M and N show the bevel found in the angle at ."), Fig. 5, applieii-through the center of plauk, so as to pitch the joint at E up, and the joint at D down. Figs. 7, 8, 9 and 10. Shoivs how to construct the face- inoidd for a quarter cylinder haviwj two wreath -pieces; the stv dent is shown hoiv to place the risers in (he cylinder at Fig. 14, Plate 15. Plate 17. S:.: Fig. 7. SJiows the plan. Draw OA and OB to equal 12%'''' and forming the right angle at O; from the center O draw the center liu« of rail ACB; bisect the curve ACB at C; draw the ladial line CO; at right angles to the radial lines CO, AO and BO. draw lines tangent to the curve and intersecting at D and F, for the length of tan- gents AF, FC, CD and DB on plan. Draw the tangents FA and DB prolonged, for the direction of straight rail; from C, and at right angles to tangent AF pro- longed draw CH parallel to tangents CD and BD, draw BL and CL for the parallelogram LCBD on plan. Now from C on the line of tangents CF, FA and CD, BD, set otf half a tread each way, then a whole tread more, thus locating the face of No. 13 and 10 risers Q'^ from the spring line of cylinder. Fig. 8. Shoivs the development and elevation of tangents, also the treads and risers. Let JCXindicate the edge of drawing board. Make AF, FC, CD and DB equal the tangents AF FC, CD and DB, Fig. 7. From the points A, F, C, D and B draw perpendiculars to XX indefinite; elevate the treads and risers, keeping the face of No. 13 and 1(> rise 6'^ from the spring lines A and B; from the exter- nal angles of No. 13 and 16 rise, draw the inclination of rail, cut- ting the perpendiculars at H, J, K, L and M: from H and parallel to XX, draw HN, cutting FJ-dt P; then iCiVis the heiuht the wreath-piece has to raise from A to C on plan, Fig. 7. From if and parallel to XX, draw KQ, cutting the perpendicular from F at R; make RS equal FH, Fig. 7; from S and at right angles to HK, draw SV; make HT equal the chord BC on plan, Fig 7. Bevels. As the tangents all have the same inclination, only one bevel will be required for all joints. Draw ab parallel with XX; perpendicular to XX" draw di equal to HC, Fig. 7. Make dh equal SV, Fig. 8; draw fh pro- longed to m; parallel with XX draw the half width of rail cutting mf at y. Fig. 9. Shows the facc-iaotdd, the trammel or straiQht edgehcing used to draio the cUlptie curves. Make AJP equal HJ"in elevation, Fig. 8; with A as a center, and TQ, Fig. 8, for a radius, draw arc at C; again with JPas a center and FA for a radius, draw arc intersecting at C. .Join FC parallel with FC and FA, draw AL and CL for the paral- lelogram LAFC on the cutting plane. Proof. The diagonal J/J" must equal the diagonal LD, Fig. 7. If so, the angle at F is correct. Trolong the diagonal FL to O equal to DO, Fig. 7, draw the radial lines OC and OA. Make OP equal the radius OA on plan, Fig. 7; prolong tangent JPA, 6'^ to T for length of shank; make joints at rand Cat right angles to FT and FC. Let C2, and C3, also Ti and T5 equal yni. Fig. S; from 4 and 5 draw lines parallel to TA, intersecting the radial line OA produced at 7 and (5; let P 8 and P 9 each equal the half >vidth of rail [2^^]. Now pivot the trammel in the point at O with the arms at right angles to OP; for the convex curve, set from pencil to minor-pin on rod, the distance 9, [14M^^]. tlien place the pencil in the point at 7. and shift the major-pin until l)Oth drop into the grooves, then fasten the major-pin and trace the curve through the points 7, 9, 3, for the convex curve of mould; proceed in like manner to trace the concave curve through the points 6, 8, 2, for which set the minor-pin to O 8, or 10%^^ for the semi-minor axis, and proceed as above. ' r- 84 PI.ATK 17. The sectious Jif and iVshow the bevels applied from the face of crook through the center of plank, so as to pitch the shank down, and the center joint up. Fig. 10. Shows the face-mould drawn by the use of ordinates; less space being retiuired than by the trammel; in many cases the ordi- nates are to be preferred, as in large sweeps where the rooru required for the trammel is limited; it will be seen in this way any number of ordinates can be used. On plan, Fig. 7, set off the half width of rail [2^^] on each side of the center line of rail ACB. Draw Cy indefinite ; draw the chord BC prolonged ; perpendicular to BC as a center, and TQ, Fig. 8, for a radius, draw arc cutting Cy at 2; draw B 2 prolonged; draw any number of ordinates parallel with the director LD to cut the curves of rail, and also the chord .BC on plan, and prolonged to intersect J3 2 at 3, 4, 5 and 6, &c. Fig. 10. Shoivs the fncc^noiild. The parallelogram LCDB is laid off in the same manner as at Fig. 9, and need not be repeated. Prolong tangent DB. ^'' to T; make joints at T and C at right angles to -DT and DC; draw the chord BC prolonged, nuike BCequal B 2, Fig. T. Now transfer the spaces 6 5, .5 4, 4 2 and 2 3, on plan, Fig. 7, to the chord BC, Fig. 10, now draw ordinates through the points 3, B, 4, 5. 6, 5, 4, C and 3, parallel to the director BL indefluite; then transfer points en the ordinates on plan. Fig. 7. to corresponding ordinates on face-mould, using the chords BC as a base line; now through the points trace the curves for the face-mould. The sections at Jif and N show the bevels applied reverse to those at Fig. 9; at section iVthe bevel is applied so as to pitch the shank up, and at M the bevel is shown applied so as to pitch the joint at Cdown. Figs. 11, 12 and 13. Shoirs the elevation, ground plan and face-mould for a *• turnout" starting from a newel, ihc V08t is 8^^X8^^, balusters 2"y:2", rail 4"yC2}i'', the rise and tread is l"y^lO", Fig. 11. Shows tlie elevation. Begin by elevating two or three risers and treads from the base line XX, as No. 1, 2 and 3 rise. From the top of No. 1 step set up to the underside of rail the difference [.5^^] that the newel post is longer than a short baluster and IH^^ more, making dH^^ from the top of No. 1 step to the ceuter of rail as shown. At this height draw the center of level rail DH parallel with XX; from the face of No. 2 and 3 rise set off the ceuter of short balus- ter 00, through 00, draw the inclination of the underside of rail; parallel with 00 draw the center of inclining rail to intersect the center of level rail at F; from F let fall the perpendicular inter- secting the base line XX at B. Now commence the plan, Fig. 12. Draw YT indefinite, to indicate the center line of rail or center of baluster on plan. Set off No, 1, 2, 3 rise and the point S to correspond with Fig. 11. Set the ceuter of newel at H on line with the face of No. 1 rise. and in this case let the side of newel be on line with the center of baluster, and as the newel is 8'^ in diameter, the center of newel will be out 4^'' from the center of baluster. For small halls this is the usual way to locate the newel when a turnout wreath-piece is required; the newel may be set on an angle, or square with the rise, or may be set further out or in, as the stairbuilder may desire; the location is a matter of taste and couvenieuce; the piinciple in getting the face-mould is the same. Pi- ATE 17. sr> Fi-om the center of uewel at H ^ll•a^^• HB, draw the verge of cap cutting HB at 2; from 3 set oil' the halt width of rail [3^^] to the point of miter at A; make BC equal BA for the tangents on plan. From the points A and C draw the radial lines at right angles to the tangents AB and CB; from the converging point (not shown on plan) draw the curves of rail on plan. The face of Xo. o rise is ^}i^' from the spring of cylinder; draw AO and CO jiarallel with the tangents BC and BA for the parallelogram ABC on plan; from A and square to YT draw Ai). Now return to elevation, Fig. 11, make BA and BC each equal tangents BA and SC on plan, Fig. 13, from the points A and C, draw perpendiculars to XX, cutting the center of level rail at D and the inclining rail at G; prolong jDJ'to intersect CG at H, then HG is the hcifjlit the wreath-piece is required to raise in the curve. Make HJ equal the chord AC, Fig. 12; lat HK equal the diagonal BO, Fig. 12. jNIake JX equal JBJD on plan, ¥\g. 12; from L and at right angles to G^ J* prolonged, draw LM. Bevels. Draw dP parallel to XX: draw Pq perpendicular to XX and equal to AD, Fig. 12; make jjr equal to LM, Fig. 11, also make ps equal the height HG, Fig. 11, draw qr and qs prolonged to edge of board for convenience when setting the bevels. Parallel with np draw the half width of rail [2^'], cut- ling rq at o, and also sq at 4. ^ Fig. 13. Shows the face-mould. Let CB equal FG in elevation. Fig. 11. With C as a center and JG, Fig. 11, for a radius, draw arc at A; again, with .B as a center and tangent BA, Fig. 12, for a radius, draw arc intersect- ing at A, join BA; draw AO and CO parallel with tangents BC and BA for the parallelogram OABC on the cutting plane. Proof. The diagonal OB must equal the distance KG, Fig- 11. If so, the angle of tangents at B is correct. I'rolong tangent BC, T'^ to 25 for length of shank, make joints at A and D at right angles to AB and BD; make A 2 and A o each e(iual S 4, Fig. 11, make D4 and 1? 5 each equal rC, Fig. 1 1 : prolong the joint line at A indelinite, also prolong the diagonal BO to intersect the joint line at L, from L, through C, draw tJie radial line for the point of tangency or connection of straight with the circular part. From the points 4 and 5, draw lines par- allel with BD to cut the radial line LC produced at 6 and 7; now connect the points 2, 6 and 3, 7 with the curves as shown, using a flexible strip, being careful to draw the curves to tangent the straight part at points 6 and 7; also see that the curves connect tlie joint A about square to the joint. The curves may be drawn with the traunnel centered at L, with the arms resting on the semi-major axis LA ; but to use the trammel in this case would be tdo unhandy and attended with too much loss of time. Face- moulds of tliis kind the practical stairbuilder can draw the curves l>y using a pliable strip thinned down at one end. 'i'he sections of crook at M and N show the bevels applied from the face of plank; the bevel found in the angle at S, Fig. 11, is sliown applied at section Jf and the bevel shoMii at the angle r. Fig. 1 1, is applied at section N. Observe they do not cross the tangents in their application. Fig. 14. Shows the same face-mould draivn ^vith less lines, hut not so correct as shonm at Fig. 13. 'i"o find the angle of tangents at B lay down the steel square and draw the right angle ASD indefinite; make jS.B, BCand CD equal MF, FG and GV, Fig. 11. AYith B as a center ami tan- 86 Plate 18. gent AB, Fig. 12, for a radms, draw arc intersecting SA at A, connect BA, thus establisliing the angle of tangents at B. Proof. The distance AC must equal JG, Fig. 11. Make joints at A and D at right angles to BA and BD; let A 2 and A 3 each equal S 4, Fig. 11, also let D 4 and D .5 equal rZ, Fig. 11; from 4 and 5 draw lines indefinite, and parallel to SD; now draw the concave and convex curves of mould, using a pliable strip tapered at one end. Connect A 4 on plan. Fig. 12, for the miter on plan; make 3 6 equal 3 4, Fig. 12, draw A 6 for the miter on face-mould. Any face-mould may be dra^vn in this way, if the stairbiiilder prefers; by increasing the number of lines a third point in tlie curve may be found on the diagonal OB prolonged, Fig. 13, to which Ave will refer at another time. The bevels as applied at sections M and JV are the same and applied in lilce manner as those shown at Fig. 13. At Fig. 14, if the points of tangency, or connection of the straight with the circular part is required, then draw the proportional line AC; from 2 and 3 and parallel to AC, draw lines to intersect the par- allel lines from 4 and 5 at 6 and 7; then the points at G and 7 show the connection of the straight with the circular part without extending the long lines as at Fig. 13. PLATE 18. Fig. 1. Exhibits the plan of n hrtlf pace irinding '^ doq- IcrirjccV stair-case. Hall b.S" wide, the center of rail being the center of liall, and so of all succeeding fliijhts composing the sta ircase. The rail starts at the first newel and terminates at the second, and from second to the newel at the landing finishing with a ramp and knee against the newel; if instead of winders there be a jilatfonn, then the first length would intersect the lower edge of outer string on the return flight, miter and finish down the lower edge of string and the triangular space filled with balusters of odd lengths or ornamental panel work. On account of room winders occupy the half pace in this case instead of a platfonn. The newel at the winders is placed in the center of ball 2'' 7)4^^ from wall to center of post. The space is divided off on the semi-circle into six winders; the risers and tread part of steps should be housed into the post, also the nosings that join the post at right angles to its face. The manner of getting out winders for tlais kind of stairs is explained at Plate 20. Fig. 2. Shoxvs 2>lan of quarter pace winding stairs of 13 risers. The flyers to the winders is open on one side and the flyers above the winders is closed between partitions; in the quarter pace four winders are used divided off equally on the quarter circle. The corner is allowed to be cased up so as to receive the risers and steps at the narrow end. The flyers or straight steps, Nos. 1,2, 3, 9, 10, 11, 12 and 1.3 are housed into wall strings on the right and left hand side. I'lie first three steps are sided up on the right with narrow flooring and cut off hand-rail high and capped, the siding is ploughed into the post }4''\ In first-class houses No. 4 and 9 steps may be diminished at A, so as to increase the width of wind- ers at the narrow eiuls. the corner may be rounded and the string carried around, and the steps and risers housed into the string, Plate 18 87 thus taking away the sharp angle of the winders at the narrow ends, and giving a better finish to the string and wall rail, if used. Fig. 3. Shows iJlun of a quarter pace booc stairs containing 12 rl'^ers. [Scale >4''^=1 foot]. The stairs are planned so as to admit light at the quarter pace, a window should not be neglected in these close stairs, either on the side, or as a skylight in the roof, for both light aud ventilation. The construction is the same as explained for Fig. 1, Plate 19. Fig. 4. Sfwws plan of an outside steps at the front door. [Scale ^'"=1 foot]. The height at a point 3' S^^ from the building to the level witli the top of the front door sill equals 3^ 9, '.<'", which being divided into seven equal parts, will allow 6K^^ for the height of each rise; and as we are not limited to the run, we may use Blondel's form- ula for the relative width of tread to the height of rise. Rule. From the constant 24''' take twice the height of rise, l-j4''_6>^'^X2=ll''], for the tread, equals 11^' for the breadth of tread; this gives an easy grade to the steps, 6)^'^ rise, by ll'" tread. The breadth of steps from wall to outer edge of nosing is :'>' 5", the landing should drop down one rise below the door sill, then there will be 6 risers as shown to the platform. A stone at jB is bedded in the pavement to start the first rise off, sometimes the first step is of stone, which is better, as the wood work is kept higher off the ground. Fig. 5. Shows the elevation. The steps are of white pine, Ui"' thick, the nosings are returned on the front string over plain brackets; tbe balusters are shown dovetailed into the steps; the newel post is shown anchored into the stone, having an "upset" bolt leaded into the stone, and extended up into the post with a nut at the end as shown at A. For the full width of steps, take the tread If^ plus the pro- it'Ctiou of nosing [IK''^] equals l'33'i'^ plus ^s^^ for tongue into tlie rise at the internal angle, equals, [ll^'+l>a'^-K^|''''=ia%^'] fur the full width of steps Vi}^^^. For all outside steps, each step should have a wash of ^i^^ in every 12''^, so as to allow the water to run off; this is not too much, for the step is liable to turn up at the nosing, and the middle of .>tep to become dished and hold the water after a rain; the reason that the steps become caped, arises from the dampness under- neath, aud for this reason the steps and risers should have a heavy coat of paint on the underside, before fixing in place, so as to prevent the wood from taking the moisture. Then after the pitchboard is made in the usual way GJ^" by If; take olf on the tread side of pitchboard J4'^ at the right angle and nothing at the point, this will give each tread the re- quired wash, and reduce the rise to 5%'''' by 11^'' on the cut out of "horse" or string. How to take the lengths of rail so as to cut aud make all joints at the l)ench. Plumb down the face of No* 2 and No. 6 rise on the face of string as 2 J3 and 6 C, then the distance BC, [I'.Sj-.;''] taken parallel with the lower edge of string, is the length for the inclining rail; then measure from the face of land- ing rise, [No. 6] to the center of baluster on the corner [.5''.2'''] then take the distance HJ on plan, from center of baluster to wall. 88 Plate 18 Rnviiuj the lengths of rail, how to make the easinri patterns, and mark the points B and C, on the patterns, so that the rail will he the required height at the newel, and also on tlie level, and cut all joints at the bench. First witli the pitchboard lay off a few steps and risers as 1, 2 and 3, from the edge of a board, say 12'^ wide; draw the under side of rail through the center of baluster kk; parallel to kk draw 2121 for the center of rail. Now make ma equal half a rise, [3Ji^^] and draw the under side of rail up; then transfer the angle apk to the board from which the pattern is to be made, and ease off the angle with the easing pattern for the under side of rail; now gauge for the width of pattern; then lay the pattern down on the drawing so the lines kp and ap will agree witli same lines on the pattern; then mark the face of No. 6 rise across the pattern to intersect the center line nn at r. For the easement at the newel, decide upon tiie length of short baluster from top of steps to the under side of rail at the center of baluster, say, I'.ll'^. Also decide upon how much higher the newel is than a short baluster measuring from tiie top of step to the under side of rail, say !'/\ Then make FS equal 5'', and draw the dotted line ts at right angles to the rise; then transfer the angle tsk to tlie board from which the pattern is to be made, and ease the angle to suit; then dress off to the curve, and gauge to the width required for the depth of rail; now lay the pattern over the drawing to the lints ts and sk so tiiat they will agree with same lines on pattern, then mark the face of No. 2 rise cutting the center line n2l at x; also mark the line of newel post. Then the distance x^r eriuals the distance BCou the string, and is etiual to the length to cut the straight rail on the rake, le^s the straight wood on the patterns, as shown on rail in elevation. Now the post is to be "/'longer than a short Itahistcrl 1'. ll"] equals [V. W"^^ h"^^l' . \"\ frinu the t^cnle ^i''^=l foot]. "j The opening is 4'. 0'' wide, and the run for the steps is 4'. 6'". The height from cellar floor to kvel with the top of stone sill at pavement is 6'. 3". Fig. 7. Shoujx tiic clenaliiin. The height (j'. S^'beingdividiwl iuto nine risers [G.' 3^^X12'''=^ 7:, H9=:8/e full] equals 8v'„'' full fen- the height of rise, the njn being divided by i). [4'. 0'';^ 12''=.54^-y ^-G''J equals 0'^ for each tread. Right here let it be observed we have divided tiie run into the same ituuilier of parts as tlie lieij;lit foi- tiie risers, and iir>t madu tlie spac^^^ for each tread, and also 3}4^^ more for the foot of ladder at A ; also the sides of ladder should be at least IJi^'' thick and 8'^ wide. In this case it will not be necessary to make a pitchboard take the steel square and with 12'''' on the blade and SK ^'' on the tongue, draw the tread J)F from the tongue ; and the rise FJ from the blade, continue the line DF to H, and with tht^ compasses take the hypothenuse DJ, and space off eleven spaces on the edge of horse AX as J 2, 2 3. Then place the two horses together and transfer the divisions; then set the bevel to the angle JDH, and with it lay off the steps, also their thickness and gain in the horses )4."' At two or three intervals tenon the step through the horses and glue and wedge the tenons F and K. The ladder may be 26^'' wide. At i a trap door is shown leading into the garret and the scuttle M in the roof; the door in the roof should be carefully covered with tin and the curb around raised above the roof with flashings carried up and turned over the upper edge of curb; also the tinning on the door extended down to the roof to answer for cap flashing; if the tinning be done in this way there will be no leaking; two hooks and steeples are required to fasten and keep the door in place. At C is shown a light rail- ing for hand-holt. For outside steps, sap ■wood should not be used, a sound knot is to be preferred: if tlie heart side be turned downtlie stop will wear lontrer for exposed work; if inside then the heart side will be more durable, taut is more difticult to work smooth than the side nearest to the bark. Oak is to be preferred for outside steps; white pine is better than yellow pine; all outside steps sliould 1)0 constructed so as to admit a free circulation of air, and painted luiderneath as well as above to protect them from the dampness. The following is a very good pro- portion for the thickness to length of steps. Steps 3'. 0" long, l^^" thick. Steps C. 0" long, IV^" thick. Steps 3'. fi" long, 1 3-16" thick. Steps 6'. 6" long, \%" thick. Steps 4'. 0" long, l>.i" thick. Steps 7'. 0" long, \\" thick. Steps 4'. G" long, 1 5-16" thick. Steps 7. 6" long, I's" thick. Steps .V. 0" long, l^g" thick. Steps 8'. 0" long, T thick. Steps 5', 6" long. 1 7-16" thick. PLATE 19. Plate 19. [Scale, \i"=\ foot]. S/ioics various 29?«7is of close or box stairs, built either between two walls or a wall on the one side and lined up on the other with flooring. In first-class box stairs a wall rail is secured to the loall by "■holdr-fasts," and is alloived to project clear of the ivall not less than one and a half [IM'^] inches. 90 PLATE 19. Fig. 1. Shoivs the plan of a flir/ht of box stairs having a straight run of 16 risers, landing on a level with the flow in the next story. How to proceed and construct the same. First take the height of story from top to top of joist, [10''.2''''] as shown in elevation, Fig. 2; then tlie widtli of opening, [3^.0^^], then the hoj'izontal distance from the face of first rise to plumb Avith the last rise. [ll^S^^J. Having taken the measures, then proceed to layoff the risers. Provide a story rod made from pine IH^^ square, and 13 or 10 feet long, dressed np square, and for convenience, mark off one side into feet and number the same; have the rod kept in a rack for this special purpose. Now set oif on the rod the height of story,* Fig. 2, equal to lO'', 3^^, then with a large pair of dividers, space off the height into 16 parts, which will equal "7%^^ each, and are termed the "risers"; the number of full steps will always be one less; there- fore in this case there will be 15 full steps required. Now the horizontal distance including the landing equals 14'', 3^^. We must allow for the quarter pace at the landing at least 3'', C^, leaving for the run the horizontal distance 11^, 3^^. which equals 135''^ to be divided into 1.5 equal parts. fl35-=-15^^ =9^''] equals 9^'' for each division termed the " tread"; if including the projecting nosing, each division is then termed the " step." Now we have the pitch or cut out on horse 7^ rise by 9'^ tread. The next will be to make a pitchboard, have the grain to run parallel with the hypothenuse or long side; see that it is exact to the rise on the rod as shown at A in elevation, also to the tread as shown at B on plan. Fig. l;if there are several flights of stairs to the same pitch then make the pitchboard from some hard wood. Two wall strings will be required, the one on the left ascending will require an easing at the starting and also at the landing, the string on the right will not require any easing, as the room will not admit of any. Now select .the planks for the wall strings about 13'^ wide; allow the external angle Cof rise and tread to extend down from the top edge of string from ?/^ to 4'^, tlien the breadth of pitch- board more, say to the internal angle of tread and rise marked D; through the point marked D, run a gauge line 3 3 from the edge of string; then take the pitchboard and turn the hypothe- nuse or long side down on a flat surface, and with a well pointed knife-blade mark the extreme points of pitchboard, (see Fig. 7, Plate 30) take the space thus marked otf, with a large pair of dividers carefully; and step off on the gauge line the required number of risers and treads. Then apply the pitchboard as at Jf, with the long edge of board to the gauge line, and with a knife blade mark the rise and tread as indicated in elevation. After all the risers and treads have been marked ofl; apply the J lousing pattern and mark the honsings for the rise, tread and wedge-roomf with the pencil. Now with a square mark all the in- ternal angles of risers and treads out on the edge of string as at 3, 3, 3; then turn the face side of No. 3 string doAvn on No. 1 and transfer the divisions. Carry the points over to the gauge line that has already bten run on No. 3 board and apply the pitch- board and housing pattern as before; both strings will now agree. " *The stair-builder's height of .story is from top to top of floors; the carpenter's height of story is in the clear from floor to ceiling. ^ +The wedges aie u.sually % at the thick end and tapered to a U" at the thin end in a distance of 10". For all lengths the taper should be the same. Platk 19. 91 Glue on at 4 the triangular piece to form the easing at the lower end. Measure up from the floor line the height of base [7^^], draw lines with pencil parallel with the floor line as shown. Then take the easing pattern, Fig. 3, and apply the shank CD, to the upper edge of string and tangent the level line 4 5, in elevation, and draw the full easing as shown. Do the same at the lower end, clean off the surplus wood and kerf in at each end K''''; so as to make a clean joint at the joining of the base. The rule for height of base, inclvddng the mouldinfj relative to the height of story is one inch for every foot high the story is in the clear. The easing pattern shown at Fig. 3 is drawn M^^ to the foot, the arc ABC has for its radius 19^^, and is made from X^^ stufl", For all full easements starting and landing, the stair-builder will find this pattern very usefnl. Housing in the Wall String. Where the stair-builder has steam power, there is nothing to equal Mr. Parry's Router for gaining in the Avail strings. Much time is saved over the old method, for with its use, the gauge line nor any laying out with the pitchboard is required on the wall string to be housed, all that is necessary to be done is to set the dividers to the hypothenuse of pitchboard, and mark the divisions 3, 3, 3, &c., along the edge of string, then each division serves as a guide to shift the string on the machine. If the work is to be done by hand labor, the speediest way is to bore two or three holes in each tread and rise at the external angle C, in elevation. Then mortice and clean out to the required depth, and cut down to the depth of housing with a "back-saw," having a guide fastened to each side of the saw to gauge the depth (if the housing; then clean out with a chisel, or a "hand-roiiter," that is sold in most of the hardware stores for that purpose.* Stairs of this description having a straight run of 15 or 10 risers may be put together in the shop and set up in the building; the stair-builder in that case must be careful in taking the dimen- sions, also see that the walls are built straight; if between stud partitions there is not so much risk; if the walls are crooked and long wide flights, the best way is to put them together in the build- ing. When proceeding to cut out the steps and risers, consider the thickness of lath and plaster, the base and depth of housings; as the walls in this case are stud partitions, f allow X^^ on each side for lath and plaster; also allow the strings to project one inch [l^^J on each side beyond the plaster for the thickness of base; allow three-quarters of an inch [H^^] on each side for the housings. Now take a strip a little longer than the width of well, setoff from left to right the width of opening [3^. 0^^], then measure to the left for the plastering and base, [Ji^''~'r]4^^-{- 2'^=r:-3%'^J equals SH'^' Now ?/. C minus 3%^^ equals 2' 8H^^ to which add the two housings [M'^-rM'^4-3^ 8}4^^= V. W4."\ equals 1'. 9%'^ for the total length of steps and risers; they should be cut half an inch longer to allow for squar- ing the ends. *A very good method to house out the nosings and scotias, is to take a step, shoulder the nosing \" from the end down to the face of rise, then stand the step in the housing tliat is already made in the string; tlieu with a fine scribe awl mark around the nosing earelully, then house out and the nosing will be sure to come up close. The same method is applied when the steps, risers and scotias are all glued up. tFor brick walls %" is allowed for plaster, and for stud partitions li" is allowed for lath and plaster. 1)2 Pirate 19. The width of steps equals the tread [9''''] plu9 the projecting nosing* [lU^^] plus the tongue [94^^] into the rise at the internal angle D, of step and rise; thus Q''-\'iyi^'-^%^'=lQ%'' equals 10>8 ^^ for the total width of step. The width to get out the risers equals the height of rise [iH^ I unless the risers betongued into the steps, then allow the tongue [h^^] niore; the steps should be ploughed }4^^ deep to receive the scotia unless in very common work. Now square and cut the treads and risers to the exact length 2^, 9M^': take two trestles, set them out of wind and place tlie strinjis on edge with the nosings down, enter a step at each end, and at the center, wedge them slightlj' and tack a nail in each end of step, then square the flight with a rod; brace and nail the steps linn; enter the balance of steps and risers, glue, wedge, and nail them firm; the landing step leave out until set in place at tlie building. If it be preferred to build them up in the building, place the two strings in place, and if brick walls, mark and plug the walls every tenth course in the horizontal beds, then cut and enter the first rise, and also the two upper steps. See that the strings are square to each other then fix in place. Now cut and fix the steps and risers by wedges and glue; also nailing the steps and risers from above and also below for good stairs; also glue and nail blocks 00 firmly in the angle; cTZiC shows a scantling underneath to which rough brackets may l>e nailed in case the steps are long. Fig. 4. Shoivs the plan of a close stairs havinfj 13 1'iscrs landing on a quarter pace, with one rise off on ejich side, this rise off the quarter pace, should be avoided if possible in all cases as it may cause stumbling at the head of stairs. 'I'lie construction is the same as explained for Fig. 1. Fig. 5. Slwics the plan of a close ulairs havhig 16 risers landing on a quarter pace level with the floor above; also having a quarter pace at the starting one rise from the floor, tin; rise is allowed to project to receive the door. Fig. 6. Exhibits the plan of a box stairs havvnrj 1?> risers with one winder and two risers on each side of the triangular pace. This form of a stair case would be bad in a dwelling house where there are children, and should be avoided if possible. The plan is introduced here to show how the difficulty may be overcome in a hampered place; the treads are 8^'. Fig. 7. Shows the plan of a iloiible cpiartcr pace, liox stair case, having 20 risers. The lauding rise is kept away from the door; at the starting the first rise projects out to receive the door. The treads are T%^^; in this case where the tread is narrow the nosing of the step may be increased a little over the general rule, but not over a half inch; some would prefer to divide each quarter pace into two steps and take one rise from the short branch landing, so as to give more room at the door, and also take one rise from the middle branch, and thereby increase the treads to 8>4. At the starting the first rise could project out beyond the door far enough to receive the finish, this would improve the stairs, although some will not allow winders of any kind in a house. Fig. 8. Shows the plan of a dovble quarter pace ivindlng box stairs cnniainlixg IG risers. Part of the first step to gain ''run*' *As a rule the nosing is allowed to project past the rise equal to the thickness of steps. Plate 30. 93 is allowed to stand out in tlie room, tlie step is made long enough to receive the finish around the door. For want of space three winders are placed at each quarter pace divided off equally on the et (.ff the newel to equal b"yy\ the center O will be on line with the center of wainscoting. A, A, A, A, shows the face < f wall siring, stt out from siuds IX^^; for the lath and plastering \H^^] and for the thickness of base [1^^]. Hi3" shows the string [If.'"] on the right. 94 Plate 20. From the center O sweep the arc BC to any radius; then divide the arc into three equal divisions JBD, DE and EC; from the center O draw the rise OS produced, cutting the wall string at F. Again from O, draw the rise OD produced, cutting the wall string at G. Draw the straight treads 4, 5, 6, t^c, parallel with the rise O 3, the dotted lines show the projecting nosing. It will be noticed that if the face side of No. 1 winder be turned over it will answer for No. 3 winder, but the grain of wood will run the wrong way. The stair builder will find it convenient in cutting out tlie winders to make two patterns as shown at Figs. 2 and 3. They should be made substantial from strips and braced, that they may keep their shape make them a little large at the nosing; also allow for the tongue on the back edge; they should be longer than shown on plan so they may be made to answer for longer steps. For shorter steps an additional strip tacked on at the wide end will save an extra pattern. For four winders in a quarter pace two other patterns made in the same way will be found to save time. String them up for future use. Laying out the icall struKjs for loiudcrs. Pig. 4. Shoivs the irall string starling. The rise equals 't}4^^, No. 1 winder measures 20>2^^ at the wall string, then take the steel Sfjuare with 20K^' on the blade and 7)^'''' on the tongue, and apply it to the edge AB of string; draw the face of rise CD from the tongue. Now set a bevel to tiie line CD and draw No. 1 rise and winder. Set up at D the height of rise [73^^'J and draw No. 3 steji, then transfer the distance fnmi No. 3 rise to the corner [HM^^] on plan, Fig. 1, and allow ^^^^ more for the tongue shown in the corner; make the easing at Fto suit; glue on a triangular piece at JGTto form the easing connecting the regular base; draw the curves lo please the eye and suit the height of base and string in the corner. Proceed in the same manner to lay off the wall string at Fig. 5. The bevel is shown, also the; divisions are laid off to corre- spond with the divisions on the plan; the joint at i? connecting the straight string is 4K'' ftom tiie face of No. 4 rise; instead of a tongue in the corner there will be a groove, and the string cut off 1 >8 ^'' from tlie face edge of groove. Make the height of easing in the corner to agree with that at Fig. 4. in this case 8'^ At i? a portion of the straight string is drawn to give the direction of the straiglit wall string; now with a flexible strip, draw the curve tangent to the straight string to suit the eye; it will be noticed that the distance from the external angle of No. 3 rise to the curve is less than the regular distanc(! [4^'!, while at No. 4 rise the distance is greater; this will often happen and must not be considered a serious fault; of course, it should be the aim to have the margin regular, but a graceful curve must be considered, for that will be the first to catch the eye. The splice at R is made with a tongue and groove; this is very nicely done on the shaper and afterwards fit up and glued: when dried and smoothed off, tack or screw a piece of board 2' long on temporarily to keep the joint firm while hauling to the building; this can be made a neat and substantial joint; the newel l>ost is ploughed to receive the wainscoting. Pig. 6. Shoivs the manner of boring and cutting in the housings. AB shows the gauge line run on from the face of string far enough down to allow 3X'^ to 4^' from the external angle of step Plate ai. 95 and rise to the edge of string; at Cfour or five holes are bored ';{" deep; and at D they are shown mortised out ready to apply tlie back saw as explained for Fig, 3, Plate 19. Fig. 7. Shows Iww to find the stretch of pitchboard for setting the compasses. Kun a light gauge line AB on a flat surface, turn the liypoth- euuse of pitchboard down on the gauge line, and with a thin pointed knife blade held along the slope of rise and tread, mark lightly on the gauge line the points 1 and 2; then set the com- passes carefully to the distance 1 2, for the stretch of pitchboard. Fig. 8. Shows Vie housing pattern. In a hand shop this pattern will be found very handy when laying off the housings for the tread, rit-e and wedge room. Fig. 9. Shows a templet for savnng ou^ the wedges with Vie circular saw. Where the stair-builder is favored with steam power, very little material in a stair shop need be wasted. A shows the saw guide; JE3 a piece of board about 2', 0'^ long, and 5 wide; the thickness is [!"] shown in elevation at C, having a handle D, and ]iotehf d out at JE, tiie size of wedge, the saw is shown at F. In a stair shop very littleslcK'k need kg to waste; all the cuttings may be collected und plied away until a slack tiuio comes, then sawed up into wedges and triangular blocks to glue in the internal angle of step and rise; also into circular nosings, scotias and easing on wall, and front strings, and many other little things for the turn- ing lathe. Mind economy is the secret of success in all branches of liusiness. PLATE 21. Plate 21. Exhibits the construction of palter as for a waU rail on unnders for a box stairs; the plan of risers and treads corresponds to Fig. 1, Plate 20. Therailis 2y/' wide by :i):i^' deep.* Figs. 1 and 2. [Scale ?i'"=^l foot]. Shows the plan and elevation; 1, 2, 3, indicate the position of risers for the three wind- ing treads on plan; 4. 5, 6, show the risers for tlie straight treads. A, A, A, shows the lino of i)laster from which the center of rail is set out 3}.<^^. The wall rail is 2X ''' wide, thus allowing a space for the hand between the wall and rail of 2}^^^^. This is optional with the stair-builder, the space, however, should not be less than Ij-y^ between the wall and rail. The rail is shown starting out of a rosette at JHTand couliuu- ing up and terminating in the architrave or finish around door at the landing. At XXX are shown iron ' ' hold-fasts," or brackets. To support the rail they should screw into the studs or into pieces cut between the studs as shown at S; this should be altcMided to before the lathing is done; if brick walls, then blocks should be placed in the walls every 4^. C for the hoid-fasts. Now lay off the center line of rail DDD on plan 3>./' from the line of plaster, make quarter turn in the corner to whatever radius desired, say 6", then make DE and D^each euual ()" and at right angles to jD J' and .£72?, draw J70 and J'O, establishing jthe parallelogram EDFO; then with O as a center and OE for a *In specifying timber the horizontal measure is first mentioned, and the vertical measure last; as for a joist 3"X10" means that the 10" way is vertical and the 2" is horizoutal, UC Platk 31. tadius draw the curve for the center line of rail from E to F. Join FE for the chord. Now marli the treads on the center line of rail; from the joint at D to face of No. I rise is \i''\ from No. 1 rise to No. 2 equals 18^^^. From No. 2 rise to E, the spring of quarter turn, 1)4^'. Then from F to No. 3 rise 7K, and from No. 3 rise to No. 4 rise 18X, and from No. 4 rise to No. 5 rise 9". For the inclination of the straight rail the common pitch is 7,'i'' rise by 9'^ tread. Fig. 2. Shows the deration and development of the treads, risers and tangents corresponding to the center line of rail on plan. First take the average pitch of 5 risers around the winders on the center line of rail, thus: From D to face of No. I rise on plan, equals 12^^ to No. 2 rise 18>^'', to No. 3 rise around the angle of tangents equals 27'', to No. 4 rise equals 18 ^.j", and one straight tread 9", makes a total (12"4-183?/'4-27"-l-18K"+9" =85") equal to 85", which being divided by the number of lisers (.5-f-85^'=17") equals 17" for the average tread. Now let XX indicate the edge of drawing board, then with the steel square take 17" on the blade and the height of rise (73^") on the tongue, apply to the edge of board and draw the perpendiculars EG, DN and FJ from the tongue to agree with tangents ED and DF, on plan, Fig, 1. Elevate the treads and risers, keeping the face of No. 3 and 3 rise 7'^" from the spring lines as shown, corresponding to plan, Fig. 1. At the external angle of the treads and risers set off the half depth [!%"] of rail as shown by the arcs, then draw lines AB and CK" indefinite, to tangent the arcs for the center of rail. At Qset up 4''' to underside of rail and l?.i"more to Z» the cen- ter of rail. Dx&w LA parallel to the floor line, cutting AB at A. Prolong No. 2 rise to cut AB at B, also prolong No. 3 ris'i. Make 3 ilf equal 2 B, join BM jnoduced, rutting CK at C. also cutting the perpendiculars EG, DH and JVat 6, 7 and 8; then 6 7 and 7 8 show the increased length of tangents in elevation. Now ease off the angles at A, B and C to please the eye. using a flexible strip, being careful to draw the curves tangent to the center lines at the joints. Next set off the half depth of rail [1%"] on each side of center line for the ramp patterns. The pattern for the level quarter turn at L is shown at JEf on plan. At right angles to EG draw 6 N, cutting DH ^t R, then iVS is the height the wreath-piece will rise from E io F on plan. Make iVJP equal the cliord EF, Fig. 1 ; from JR and at right angles to 6 8, draw BS. Taraliel with DH draw the half width of rail, cutting the tangent 6 7 at 10 for the increased width of face-mould on the radial lines. Fig. 3. Shines the facc-monld. The tangents6 7 and 7 8 in elevation are the same length, hence there will be but one bevel required. Select a piece of heavy paper, draw the right line 7 8 produced to t: make 7 8 t equal 7 8 Tin elevation; then with 8 P in elevation as a radius, and 8 for a center, draw are at 6, then with 8 7 as a radius and 7 for a center, draw arc intersecting at 6; also with the same radius and the points 6 and 8 as centeis draw arcs intersecting at 0; join 6 7, 6 and O 8, and we have the parallelogram 8 7 6. Proof. If the diagonal 7 equals the diagonal OD on plan. Fig, 1, the parallelogram is correct. On the radial lines 6 O and 8 0, make 6 1,6 2 and S 1. 8 2, each equal 7 10 in elevation. On the diagonal 7, make 3 equal the radius O 3 on plan. On each side of 3 set off the half width of rail {IVi). _ , . . Plate 31 ,«7 Now pivot the trammel iu and set the arms at right angles to the diagoual O 7, for in tliis case the diagonal is the direction of the minor- axis. Set from pencil to minor- piu the distance O 3 for the center line, leaving the uiajor-pin loose. Then plaoe the pen- cil iu 8, drop the pins iu the grooves, hold the trammel firai. Now fasten the niajor-piu and draw the center line 6, 3, 8. Next draw the convex curve, set from pencil to minor-piu the distance O 4. (the trammel remaining as before) place the pencil in 2 and drop the pins in the grooves, then fasten the major-pin and trace the curve 3, 4, 2, for the outside of mould. For the inside or concave side set from pencil to minor-pin the distance 5 0. Now place tlie pencil in 1 and drop the pins in the grooves, then fasten themajor- piu aud trace the concave side 1, 5, 1, of mould. Make the joint at 6 perpeudicular to tangent 7 6; from 1 aud 2 draw the width of shank parallel to tangent 7 t: at t make joint at right angles to tangent 1 7. Bevel. Make OS on plan equal RS in elevation; join SE, then in the angle OSE is found the bevel for both joints; they will cross the tangents. The application of bevels are shown at sections a and b; the dotted line shows the gauge line and tlie tangent line squared across the joint, gives the center through which to apply the bevel; the block pattern is applied square to the bevel, and gives the twit^t of rail at the joints; the shaded parts show the amount of ov'r wood that has to come off. The sliding of the mould is shown at I'late 24. If a round rail is required, as shown at Fig. 4, then draw the l)ara!lelogram S 7 (5, and draw the center line of rail 8 3 6, with the trammel, make joints at t and 6 as at Fig. 3, Then set off the half width of rail [U4 ''j on each sid« of the center line as shown. For a round rail the thickness of plank required, need be only a slight increase over the diameter of rail, just enough to dress, because the elliptic curve, passing through the center of wreath-piece, gives the correct "falling line," and as the center of a round rail is eiiui-distant from the circumference in all its parts, the wreath when iu position, must be correct. At sections a and b, the shaded part shows the amount of over wood that lias to come oft"; if the bevel be applieil through the center, as at Fig. 3, it will give the direction to bore for the bolt nuts. Fig. 5. Shows the wreath-piece landing. This face-mould is the most simple to draw in stair building, as the plank is not sprung; the lines can all be taken from the pitchboard, (except the trace of mould), including the bevel shown at 8. First draw the parallelogram OAbD for the center of rail on plan; elevate rise No. 16 perpendicular to cb, and at any point, draw the pitchboard e/g* at right angles to No. 16 rise; prolong the hypothenuse eg* indefinite. From points c. A, 3, D, and 4, on plan, draw lines parallel with No. 1(3 rise, culling the pitch line eH at 5, 6. 7, 8 and H. Then again, draw lines indefinite from the same points perpendicular to the inclination eH. Face-mould. Now parallel with the inclination draw cab in elevation, make ao and bd equal AO and bD on plan; join od and prolonged, establishing the parallelogram oabd on the cutting plaiie; the points 3 and h show the increased width of mould at joint; add on straight wood parallel with 3 h sufficient to let iu and fasten to the architrave. 98 Plate 21 The line od is the semi-major axis, and the line oa is the Bemi-minor axis, the trammel centers in 0; make aj and ak each equal the half width of rail; now trace the curve for the inside and outside of mould, as has been explained. From j and k draw lines parallel with the tangents cb for the shank, the joint at c is at right angles to the tangent cb, and is drawn to agree with joint c. on plan. The section at b shows the bevel applied in the usual way; at joint A the square is shown applied from the face of plauk because there is no spring in this case; the shaded part shows the required thickness and breadth of plank to contain the twist of wreath-piece. In this case the pitehboard gives the correct Bevel. Fig. 6. Slioivs a section of round rail H full size. The dark shade sliows the iron bolt, the light shading shows the hollow dowel through which the rail bolt is to pass freely; this hollow dowel answers a good purpose; J^ '^ is a good size, they enter the rail about y^'' on each side of the joint, and are glued in on one side at the bench. Then bore through the dowel for the bolt. For round rails and center joints of moulded rails, they suit well. Figs. 7 and 8. Exhibits a handled scraper one-half full size. This is found a good tool for moulding the twist part of a hand rail; they may be made of all sizes and patterns, the draw- ing explains itself. Fig, 7 shows the length of scraper. Fig. 8 shows a cross section of the same; iifJ^T shows the handles, JB, the steel bit; C, the cap held down by the set screws iSfif In cross section; T shows the throat; apple tree is a very good wood from which to make this tool; the cap at C may be made from some harder wood. Figs. 9, 10 and 11. Sliows a dovetail patt<:ni, (Scale X full size). Every stairbuilder has his own way of la> ing olf dovetails. This pattern is introduced here as a handy means to lay oil" the work in a hand shop where something of the kind is needed. Mr. Parry has a dovetailer he furnishes with his router that saves all the trouble of laying off the dovetails. Fig. 9. Shows the front of tool. Fig. 10, the side and Fig. 11, a cross section. A is the stem 2}i''yji^^ thick, and is Including the guard B 1"^' long. A slot %''y,W is shown at C rebated on one side as shown at J, Fig. 11, the guard is 11^'' long and %"yiH" in cross section, having a projecting lip D, Fig. 10. E, E, E, E, shows tlie size of mortise on top of the step. F, F, F, shows the size of dove- tails and are made fast to the pieces EE, as shown at H, Fig. li. i is the binder. MM are set screws, having washers JVJV'and nut in the binder to adjust the pattern for the different widths of steps. The binder keeps the dovetail pattern square on the stem. ' This pattern is intended to apply on the stops before tlie ends of steps are cut off for the return nosings.* Some stair-build- ers prefer this method. It will be observed the pattern is revers- able for either right or left hand stairs, and should be nuide of cherry or some hard wood. *The ends of steps are cut olT when the nosings and brackets are to bo tit on iu tlio building wbeu putting up the rail. Plate 22 99 Figs. 12 and 13. Exhibits the face and end view of the " Cupper gauge"* [Scale U full size]. A is the stock, B is tlie stem adjusted by the set screw D, and holdiug a long tooth C made of steel, and held in place by a key E. Pig. 13. Sliows an end view of the same. The stair-builder should have different sizes of this tool, some holding a pencil and others steel points. They are very handy In squaring the wreath part of rail. After workinar tlie concave side of wreath-piece to the plurab, the convex or outside may be easily ;iauged; also Tchen moulding the wreath part of rail the gauge will be found handy in tracing the different members around the twist, espe- cially in double rails. PLATE 22. Plate 22. Exhibits the plan and elevation of a. platform or half -pace stair case; and also Jww to construct the cyliyider and strings for the same. Figs. 1 and 2. Shows the plan and elevation. [Scale, ^//=1 loot]. f The height of story is 10^ 6'^; the width of hall equals 6', 3'^; width of joist in the second story equals 10^^; the door under the platform is 6^, S^'' high; the run from the face of No. 1 rise, including the platfonn and cylinder, is 14^. 2'^; the steps are to be 1>4^' thick, and returned ou the outer string. The rail is 3%'" wide by 2}i^^ deep; balusters are 2'''X-''^; the newel post, ^z/y^//. tjie cylinder 7'^ diameter; the rail is on the right ascending, therefore is termed a ''right hand rail" There are to be 18 risers, and the height is 10^ G'^ then 10^ 6^' reduced to inches (10^ 6''^X12^^— 12'j''^) equals 126''', which being divided by the number of risers, IS. (126''-^18=7'') equals 7''' for each rise, as shown on story rod XX in elevation, Fig. 3. The height of door under the platform is 6^ 8", and we should have at least C/^ over the door for the casing or finish; the platform joist is 10^'' deep; the lath and plaster is X''' and the floor on the platform is %^^ thick. Then the whole height from floor (6^, 8'M-6'''-r>8''-r;-s'^n-lO''^97M") equals 97%'^; now 14 risers at 7^'' each equals 98'^ making 8', 3'^ from floor to top of platform, thus allowing the space required under the platform lor door ami finish. For the relative width of tread to the height of rise for dwel lings if practicable, use Bloudel's constant of 24 inches, from which take twice the rise (24^'— i^'— 7''r=l0'0 equals ICfor t e width of each tread. Now we have for the run of stairs including the platform an cylinder, 14', 2''; there is always one full step less than there ar? risers in the run of a flight of stairs, and as we have a run of 14 risers to the platform, there will be i;j full treads and one landing. ♦Termed this because 5Ir. Cupper was the first to introduce the tool in print for the stairbuilder tW'hcn takinpr the height of story the stair-builder will do well to try his lod at ditlerent points, as sometimes the joist are out of level and mia,ht cause troul:)le; therefoi'e, care in this particular shouid not bo neglected; the best point to take the height is at the starting of the first rise. Also let tlie young man think twice before setting down one Item in his dimension book, for a mistake of a few inches may cause the whole flight to be got out the second time. 100 Plate 22. Then 13 treads at 10''' each, equals 130'', or 10^ IC for the run, and 4'^ more for tlie cylinder, equals 11'', 2'^; then 14^ 2'''' minus 11', 1" leaves S', 0'' lor to frame the platform iu the rough, as shown at Fig. 1; the width of platform should equal the half width of hall, more is better and less crowds the passage between the rail and wall. The next will be the flight off the platform, there are 4 ri.sers and three full treads, at W each, equals 2', (}'', and one lauding, for which allow 6'' for the rough bearer to catch well on to thv joist at the lauding; also allow 3'. 0'', for tlie platform, and 4" for the cylinder, (2', 6''H-6'' ;-3', C-t-4=6', 4''). thus making the whole distance from the wall to the joist at the landing 0', \" to allow for the flight off the platl'orni. The next is to determine the " headway;" that should be T'. C' at least, more i-s better, but in cramped places the .stair-builder has often to do with less. AVhen the lu-adway is cut short it destroys the good effect that the stairs wouid otherwij;e have if there was more headroom. A good allowance is 8'. C, when- e\er it is convenient to have it. Now we will count down 14 ri.sers from the landing at 7" each, minus the joist {W), plastering (1''), flooring {\"), (14X7" =98''— iO'''+l''-fr'"=8G''--12''=;7'.2'0 equals 7'. 2" for the headroom. Xow place the face of headway joist plumb over Xo. 15 rise, numbering down from above as .shown in elevation; or the hori- zontal distance from the wall to the headway joist will equal 3'. 0" for the platform, 4" for cylinder, and 10 tj^'eadsat 10" each, and l"for fascia {?>'. 0''+4''+l6xiO'^^l^''''^8'-4''+l'':^;]l'. 9'') eciuais 11'. U", as .shown on plan. The next is to decide the width of landing in the second floor. The width of hall is G'. 2" in the rough, ami we have to provide for a 7" cylinder, and the center of cylinder will be the center of hall. The face string is oiie (I") inch thick. The center of hall IS 3'. 1" from wall. (;3'. l"i;]j.^"-l"^^^3'. .'>'.'"). Hence the \s idth of well equals 3'. .5} ,'". and the width of framing will equal ('/. S" minus 3'. hM", or 2'. 83^" for the width of framing at the landins. The length will Ijc 11'. 9" minus 6'. 4", the space for the flight oif the platform; equals {11'. 9"— 6'. 4":= 5'. 5"), for the length of trimming :^' . 5" by 2'. 8K" in width as shown on plan, Fig. 1. Fig. 3. (.Scale 1)4" ^^\ foot). Sluncs how to place the risers in the cyUiider so o-s to qivc the same inclination .to the wreath of cylinfJcr tluit the fftraUjht, part of slrimj has. Draw AJB indefinite and parallel to the edge of drawing board ICX, set the compasses to :>',;", and with O as a center, draw the semi-circle ACB, draw OC at riglit angles to AJ3, also from A andS draw the clirection of the straight part f)f outc.-r string per- pendicular to XX; With A for a center and AB as a radius, draw arc at Y; again with the same radius and B as a center draw the intersection at Y", draw KL parallel to XX and to tangent the point C. From Y amA through A and J5 draw lines to intersect KL at the poi^its K and L; then KL is the stretchout of the semi-circle ACB. The treads are 10", make CM" and CiVeach The stair-buiir'c;" should mnke tiie above calculation in the build- ing when takinii tie dm.pn-ions lor tli-- stan-s. He wiil save time by calling the atiOEt'on tf the f reman oarpentev to llio .startinss and liiidin^s, raavkir.K ■^■hf'vp the jjist must be trimmed to and floors laid for all straight stairs. Thl.<5 can easily le done in the buildnig. For winders the be.st way is to make a plan to a scale. And all trininiinjr laid off fi-om the dia^svin?. All measurcaionts should ba cai-efuily noted In the book for dimensiona. Flatk 23. 101 e(iual to half a tieaD i)arallel witli AB, and we have the position of risers in the ry Under. The manner of placing risers in the cylinder is explainetl at Plate 15. Fig. 4. Shows the same plan as at Fig. 3 on the string line. NN shows the nosing, C the string. The risers are .shoul- dered to receive the thickness (1^^) of string, and a U'^ is allowed to ]>roject and miter with the bracket, the nosings project over the rise equal to their thickness {l)i"), and are shown mitered atthe ends to return over the brackets. The treads are W\ hence the balusters will be spaced half a tread (5''^) from center to center as shown at A and X. Observe the two face sides of baluster at A line with the face of rise, and also with the face of bracket line. The mortise for dovetail is shown in the center of baluster. Fig. 5. Shows how to lay off the staves foi' a 1" cylinder. AC indicates a gauge line run on the draft board a conveni- ent distance from the edge. tSet the compasses toa radius of syi^^ and with O as a center draw the semi-circle IBD. With the same ra- shown applied; GH shows the joint of cylinder at the turn- out. The fii"st rise is sliown reduced the thickness of a step. (XH^^) so that the height of the rise will be the same as the regu- lar height when the .step is in place. Fig. 8. Shoxvs the shott front string off the platform. This string is laid off from the wall string in the same man- ner as Fig. 7. Make the distance from No. 10 rise to the joint of cylinder AD to equal 9%^^ as sliown on plan, Fig. 4, and draw A2? perpendicular to No. 15 tread. Parallel with the joint AZ) add on %^^ for the tongue. At KE is shown the end view of same. At ilf the string is notched }i^^, because the steps are that much thicker than the flooring, and tliis allows the string at that point to come close under the flooring. For the width of string at the easing allow 10^'' for the joist, and for lath andplastei- %^^ equals lOK. Now make MG equal lOX, aud draw the line at G parallel with the floor line EM. Then u.se the easing pattern for the curve of easment. A saw kerf is shown at the end for join- ing the level fascia or string. Fig. 9. [Scale, %^^=1 foot]. ExMbiis the elevation of staves, to obtain their different lenghts. This drawing may be made on the back of cylinder board, and used for subsequent like cylinders; apply the hypothenuse of pitehboard to the edge of draft board, and along the tread draw the line AB indefinite. Make 1 2, 2 3. 3 4, eacli equal the back of stave 3 3, Fig. 10; perpendicular to AB draw 1 8, 2 7, 3 6, 4 5, indefinite. Make 1 8 equal DE. Fig. 7, plus the height of a rise [15'^]; also make 4 5 equal i?Z>, Fig. 8. \1"]. Draw 8 .5 for the length of staves, they should be cut longer for trimming, as shown. For ordinary work that Is painled, this raethotl of staving up a cylinder can be made a good job, but for hard wood tiaish with large cylinders they should be built di&'ereutiy : that is, the grain of wood, instead of beinjj perpendicular to the treads, should run in the direc- tion of the straight string lor all fli'st-class work; this requires more labor, and consequentlythecostwill be greater; the manner of doing this will ije explained when describing a 12" cylinder. To prepare the staves for glueing, first see that they are out of wind on the face 1 2, Fig. .5; then dress oif on one edge 2 3, to the Ijevel at O, then gauge for the width 2 1, and reduce the other edge to the required bevel. After the staves have been brought to their width, make a cylinder pattern DBIl, of thin stuff, and mark the ends to the curve 1, 2, 3, then with the cylinder plane dress down to tlie curve. Where the stair-builder is favored with steam, take a circular saw havinj' a dia,nieter not over the diameter of cylinder, to use as a wab- ble; tnen place a guide across the table, held down with two hand screws, now run the staves over the saw several times until leduced sufliciently n-^ar tlie curve, leaving a small margin for a guide to keep the stuves pirallel wh'^n glueing them up. Also if having a hand .iointci , witli an adjustable guide, the staves may be reduced to the required spiay, thus saving much time. Glueing. For good work the glue must be of the best grade, and carefully prepared; have the glue so that it will run smooth from the brush without any lumps or being stringy, and the hotter the better. Plate 22. ^-^^ Rub the joint \Yell after applying the glue, to lessen the thick- ness of glue in the joint; now set the "dogs," Fig. 8, Plate 23, along the joint, on the outside, tapping them lightly with the hammer, then set thom away to dry; afterwards to make the joints stronger, screws or nails may be used. If there is .luy sap wood on the joints, rub a little chalk over the SHp part before applyiusr the glue to preveut the sap wood froni jib-iorbing the glue too fast. For all light work the doa: will be found convenient; with their use, the staves for large cylinders may all be glued up with one heating of the glue. Dress out the cylinder to the pattern with the "cylinder plane," then see that the two joints are out of wind, using the winding strips, or over the hand jointer, if one is at hand. Fig. 10. Shows plan of cylinder and the strings Joining. Fig. 11. Slwivs the same in elevation. The joints AB and CD are reduced to the thickness of strings [f ]; take the cylinder square (that is, a right angle triangle, similar to the pitch-board. Fig. 6, made from pasteboard), and square the top line BD, from the etli^e BA; make BF equal the heighth of a rise ["''l, make J'A equal DE, Fig 7 [S^^j; then BA will equal [T^'-h8^'=l5'^] 15'^; makeUCequal DH. Fig. S [T'H; now take a thin pliable strip, bend it into the cylinder to cbe points A and C. let the ends of strip extend out from the joint, tack oxie end, and hokl the other inider the thumb, then apply the pitch-board to the joint of cylinder, and regulate the ends of strip to the pitch by raising or lowering that part of strip in the cyMader; when satis- factory, trace the falling line of cylinder with a lead peacil, and trim off the surplus wood as shown at Fig. 1 1. The shaded part shows the cut and also the plough on the face of joint to receive the tongues on the strings; the dotted lines show the staves as glued up. Now, after the cylinder is fit on the sti'ings, and the lower edge moulded, glue and nail the cylinder to the lower string, square to the face, tack a brace on, to hold the cylinder firm; cover ail ex- posed joints so as to protect them. Dimensions for jointing the rail on the bench. Next, take the lengths off the strings, for jointing the rail, and enter them in the order book, to be used when getting out the rail. At Fig. 7 the first length is shown, taken from the joint GH, along the guage line to the cylinder joint at the point 2) [11'', 8''''], At Fig. 8 the length is sliown taken from the joint of cylinder at H, along the guage line BC, to p umb with the face of No, 18 rise [2^ 11%''], The level lengths must be taken at the building and after' wards entered in the order book, , Figs. 12, 13, 14. Shoivs plan of cylinder and elevation of stHngs. In this case the strings are IK thick, and halved out to suit the sides of cylinder; the cylinder is made fast by screws driven from the back of string, sufficiently slanting to draw the joint close after being glued. This method makes a good firm job, but requires more time. f^~. The other method described above can all be done on the shaping machine, where steam power is used. Preparing the steps and risers. The steps and risers will be next in order; take a rod, say half the width of hall [3', 1''] in length, set off from right to left 3}i^^ for the radius 104 Plate 23. of eyHud<>r: then measure to the right '4''^ for bracket: then l^^^ for the uosiDg [3', V^—S';4^^-i-}4^^~]-^i''-='y, 11"J equals 2^ IV for the length to cut the steps in the rough; for the risers they may he cut the thickness of nosing [li"^'^] less, or 2^, 9yi^^'i no allowance has been made for squarhig the risers; as the length of steps has been taken between the rough walls, and the face of wall string projects out \J4'^ for base, lath and plastering, the steps are housed %^^ into the wall strings, leaving then l}o^^ for squaring the ends and for \nievcn walls. Now, there are 18 risers, the rise No. 1 and No. 1.5 starting both flights, will be the thickness of step {IK''\ narrower than the rest; there will be two landing steps 8'^ wide; the one at the ])latform will extend across the platform having the well hole cut outi to Q]4'^ diameter to receive the thickness of bracket; the other will be the regu ar length. Then again, Nos. 1, 2 and 15 steps will be 2''^ longer than the regular length; that will leave 13 steps that will cut off to the regidar length. And the two landings deducted otf the whole number w^ill leave Itt steps that will re(iuire to be the full width. Now after the steps and risers are cut otf to their proper lengths, size thera to width; the width for the regular risers will be the true height of rise 1'^ plus %'^ for tongue equals T^g'''^. see Fig. .3, Plate 23. The width of steps will be equal to the tread [10^^] plus %'^ for tongue, at the internal angle B, and plus the thickue.ss of s-tep 1 )i^' for the projection of nosing, tinis : W -\-%'' ^l}i'' =^ n%'\ equals U% for the full width of treads, see Fig. 3, Plate 23. After the steps and risers are grooved and tongued, cut the miter on the steps for the return nosings; for a handy means, use the miter jack, Fig. 7, Plate 23. The risers should be grooved so as to allow the rise to be a little full in width, so that when keying up they may be forced up tight; nothing but dry stuff diould be used in the manufacture of stair work, as unseasoned material will be sure to give trouble. Next square the ends of the risers for a right hand rail, then shoulder back }i^^ for the thickness of bracket [k'^'J pius V for th-Mhickness of string, [}i''-+-V—\'^4:"\ equals Di^', as showu at A, Fig. 4, then miter the ends of risers to receive the brackets. For the "platform step" use the platform pattern for a 1'^ cylinder, see Fig, 10, Plate 23. With this pattern mark off the miter and the circular part to be sawed out, also the notch for the string off the platform. The stair builder will find the above pattern answers a good purpose; the 'circular end steps" at the turnout and how to find their location is shown at Figs. 1 and 3, Plate 24. T^ig. 4. The dovetails may now be laid off on the ends of steps, and also ou the top at the same time; the balusters are 2^'' square, the center of baluster should be the center of dovetail; then take the dovetail pattern and make from the guide to center of first dovetail lli'^ for the projection of nosing, and V more to center of baluster, equals 2 Vj^^', thence to the center of next, the dist.ince of half a tread, 5^^ in this case. To facilitate the oiieratiou of cutting in the dovetail on the steps, string them alon:;side of llie bench, all face out, mark and rip them in to the proper distance. Now stock them on the trestles and mortise tlit-ui half through ou the biick, leaving the core remaining to keep the mortise clean so the glue will adhere when glueing in the balusters. Plate 33. luo Where there Is steam power the dovetails may be cut in uicely on the band saw by liaving a beveled blocli to cant tlie step to the splay of mortise. Plaoe the short baluster with the two faces on line with the fai-.i of rise, and also the face of bracket, or end of step, as shown at A and X. For the circular end steps have a single pattern, with the size of mortise on top of the step tacked on, so that the mortise can be marked on top of the step, and also on the end, at the same time. Glueing up the step and rise. Have the glue not loo thin, place the step on the bench face down, glue the rise in phice. beiug careful to keep the miter on rise in line with the miter on step; glue and nail tin- triangular blocks in the internal angle of step and rise. .See .B, Fig. 3, Plate 28. Xow glue and nail scotiu D into place; use the pitch-board in the angle of step and rise to keep them square; sponge olf all surplus glue from the face side of step anil rise, and lay away to dry. Sometimes, as in oak, the step may be bowed and difficult to glue up. They may be mamiged in this way: Have a stout Ijench top, and, with two hand scrcv>s, draw liiy lise and step down to the bench top, then block and nail (he ri:^e to place. Nosing the steps. After the glue is dry the nosings and scotias may be Avorked, being careful to have them all aliice. In using tlie ''nosing plane"' work down to the round, being par- ticular of the last shaving. This is done that they may fit snug in the housings. Panel under the first flight, Fig. 1. The triangular panel termiiiiiles in tliis case at the joint of cylinder, and extends up back of the string J<"; find the starting point A, say plumb witli the face of second rise, then count the uu)nber of full treads from A t(i joint of cylinder. There are 11 full treads at 10'^ each, and the last tread e.xteuds into the cylinder I", leaving 9>" to be counted into the pane!; the total [11><10'^H'9'^=119^'--^^ W'—'i'. \\"\ equals 9', li'^ from A to C, along the floor. Now make a scale drawing V ','' to the foot; make AC equal 9'. 11'^; draw a i)erpendicu!ar from C, then place the point of pitch-ljoard at A, with the tread along the liue AC, and draw the hypothcnuse cutting the ijcrpendicular from C at D; divide the right angle triangle ACD into panels to suit the fancy. Brackets. For ornamenting, the front strings are usually yi'^ thick, made to fancy; they are mitered to t he rise for good work, and the nosing returns at the end; those for the cylinder may be cut having the grain of wood perpendicular to the tread, then steaming and bending them over a heated stove pipe to the required curvature; or they may be kerfed on the back and sprung into place. Circular Nosings. Are worked from the solid including the scotia: the scotia is sometimes worked separate, and each nailed to place in its turn; they also may be lurued in the lathe for small cylinders. Return Nosings. Are cut a shade longer between the heels of miters than the length of brackets,* to allow for smooth- *Thc lii-ackets are usually cut '^" lonucr .£^0; then from H circle to wall (3^ V) on plan Fig. 1. These lengths enter in the order book when returned to the shop. PLATE 23. Plate 23. (Scale %'''— 1 foot). Exhibits the steps and rf-sers mid vianner of constructing the scone; aUo, how the steps are sxipported underneath. Fig. 1. Shows the elevation of Oirce steps at the slatting of Fi2^' t6 the under side of rail for the difference that the newel is longer than a short baluster; for example say the short baluster is •?/, 2^^ from the top of first step to the underside of rail at the center of the baluster, then the newel post will be ry-^-^plus 4)4^', or 2^, 63^''^ high from top of step to the under side of rail. Then make JE?!^ equal 4K^' and FG eqnallJi''^ to the center of rail: from G, and parallel to AB, draw CiT indefinite, cutting the inclination of the center line of rail DC prolonged, at iif. From H, and perpendicular to AB, draw HB, to Fig. 1. Draw BC indefinite and parallel to AB, to indicate the center of rail on plan, Fig. 1; draw No. 1, 2 and 3 rise opposite those at Fig. 2: place the center of newel post on line with the face of No. 1 rise, and out any distance from the center of rail that may be desired, say in this case, as the hall is narrow, we place the side of newel to line with the back of balusters; then the center of newel at O will be out from the center of baluster 2i4''^; draw OB, draw the size of cap, cutting OB at H. Make HD equal half width of rail [2"] for the point of miter,* make BC equal BD, then BD and BC are the tangents on plan. From C, and perpendicular to tangent BC, draw CE indefinite; also from D, and at right angles to tangent BD, draw DF prolonged to intersect CE, (for want of room ncit shown). Make CK etiual %^^ for the face of string; make KL equal K'''' for the face of bracket; also make JjJH" equal IJ^''' for the projection of iiosiug. From the intersec- tion (not shown), draw the curve of turnout through the point K, for the cylinder, shown by the solid line; also draw the bracket and nosing lines. Now draw the risers to intersect the bracket line, and the projection of the nosings to intersect the nosing line; then through the intersections draw the miters, thus locating the risers aud treads and the radius lor the turnout at newel post. This drawing should be male on paper to prepare the steps and risers, also the cylinder; afterwards rolled up until the rail is wanted, then the drawing may be completed for the conslnictiou of the face-mould. Pigs. 1, 2, 3 and 4. Show the construction of face mould for the turnout at the neioel post. The position of newel post, location of risers in the cylinder, tangents, and location of the first rise [No. 3] outside the spring of cylinder are shown [r>^^] on plan Fig. 1. This, together with so much of the elevation. Fig. 2. as the base Hue, the elevation of treads and risers, and the inclinati(jn of the center line of rail, *Some prefer to extend the curve of turnout to the point of miter as here described, instead of termiuatliig the curve at the intersec- tion of cap, and adding on straight wood for the miter as shown at Figs. 12 and 14, Plate 29. By carrying the curve to the point of miter, and moulding th« wreath-piece to suit the cap, gives the best results. 110 Plate 24. has been explained. The drawing, it is supposed, has been made on paper and laid away until the rail is wanted. Now, to prevent a confusion of lines on plan Fig. 1, we will make another plan. Fig. 3. Draw the tangents DJ3 and JSCto correspond in length with tangents on plan Fig. 1, and also in the angle at J3; D 2 and D 3 show the miter into the cap; DF and CJ57 indicate the radial lines. [For want of room their intersection is not shown]. Now, parallel with DB and BC draw DH and CH for the parallelogram HDBC on plan; prolong the tangent BC towards K for the direction of siraiglit rail; from D, and at right angles to CB prolonged, draw DJ; join CD for the chord on plan; draw the diagonal BH; cutting Ihe short chord at 4, draw the curve CD for the center line of rail on plan. Draw the width of rail as shown. Now, at Fig. 1, draw the radial line EC on plan perpendicit- lar to AB, cutting the inclination of the center of rail at N\n elevation; prolong GH to cut CiV" at K, then KN will equal the height that the wreath-piece has to raise, and UN will be the length of tangent in elevation. Make K 2 equal the chord DC, Fig. 3; make K 6 equal the diagonal BH, on plan Fig. 3; join N 2: make H 4 equal BJ, Fig. 3; from 4 and perpendicular to NH prolonged draw 4 5. Make joint at D plumb over No. 3 rise and perpendicular to the inclination DH. Fig. 4. SJiorrs the face-mould for the turnout. At any convenient place on the paper for face-mould, draw BC, with Cas a center, and iV2, Fig, 2, for a radius; draw arc at D; then with BD, Fig. 3. as a radius, and B for a center, draw arc intersecting at D. Join BD, draw DO and CO parallel with BC and BD for the parallelogram ODBC on the cutting plane, or plane of plank. Proof. The diagonal HO must equal the distance N 6, Fig. 2; if so, the angle of tangents at B is correct. Prolong tangent BC 6^' to E; make joints at D and JE7 per- pendicular to the tangents BD and BE. Bevels. To find the bevels return to Fig. 2. Suppose the base line AB, Fig. 2, to be the edge of draught-board; at any convenient distance over draw a gauge-line 7 8; perpendicular to AB draw PR equal to DJ, Fig. 3; make P 9 equal 4 5; make P 10 equal the height KN; draw R 9 and JE 10 prolonged to edge of board, then the angles at 9 and 10 give the bevels required. Parallel witli 7 8 draw the half width of rail as 1 12, cutting the hypothenuse of bevels at 13 and 14. PiCturn to Fig. 4. Make DM and D 13 each equal 10 13, Fis. 2; also, make JEiV^and E 14 each equal 9 14, Fig. 2. Make C T, Fig. 2, equal B 4, Fig. 3; from T, and parallel to KN, draw TS, Fig. 2: prolong the joint DM, and also the diagonal BO, to in- tersect each other (not shown on plan), and from the intersection draw KJ indi'finite. Make BF equal 6 S, Fig. 2, join FD. From 13, and parallel with DF, draw the proportional line 13 H, cutting OB produced at H; make FL equal FH; from N and 14 draw lines parallel to BE to intersect the radial line at K and J. Now, with a pliable strip applied to the points DFC, draw the curve for the center of rail; also, through the points 13, H, J, trace the curve for the convex side of mould, and through the pointsMLiT trace the curve for the concave side of mould. At sections P and R the tangents are shown carried across the joint intersecting the g-auge-line. The hevel found in the angle atlO, Fig. 2, is shown applied from the face of crook, through the intersec- Plate 24. Ill tlon. In like manner, the bevel found In the angle at 9, Fl?r. 2, Is ap- plied at section R, and the block pattern shows the twist of wreath- piece. The shaded part shows the amount of wood required to saw out the crooks; and the arcs, shown laid ofif from the center line on face mould, gives the lines to follow when sawing out the crook square through the plank. Observe the bevels do not cross the tangents in their application In this case because the minor axis is not in the mould, or there is no point in the width of mould that is equal to the true width of rail [4"1. Hence the bevels both apply the same way; also, observe the steepest bevel is applied to the widest end of mould. For sliding this kind of a mould on the crook see Fig. 5, Plate 10. Sight holes are shown on the face-mould to aid in adjusting the mould over the tangent on the crook. Fig. 5. SJimvs plan of the center line of rail for a 1^' cyl- inder, and 2"ys2" balusters. The radius for a 7^^ cylinder will equal S14^^, and the bracket equals >4 ^'' in thickness; the face of baluster will be flush with the line of bracket, and hence the center of baluster will project %^'' beyond the line of cylinder, making the radius for the center line of rail r3K^^+K''=4M''J equal 4^'^ Draw AJB indefinite; then with O as a center, and OB [4.^^^] for a radius, draw the semi-circle BCA,* enclose the semi- circle with the rectilineal parallelogram ABDE; draw the radius OC. and we have the two square parallelograms OBDC and OCEA, on plan. Prolong tangents DB and EA towards M and N for the direction of straight rail; make the face of No. 13 and 14 rise 93s'''' from the spring of cylinder to correspond with Fig. 4, Plate 22, then the face of No. 14 and 15 rise will be in the cylinder )i^^. Fig. 6. Shoivs the development of tangents. Let XX indicate tlie edge of drawing board; make BD, DC, CE and EA equal tangents BD, DC, CE and EA on plan, Fig. .5. Through B, D, C, E and A draw perpendiculars to XX indefinite; now elevate the risers and treads, keeping the face of No. 13 and 16 rise 9)^^'' from the spring lines at A and B: (No. 16 rise not .sliown), through tlie center of baluster JJ" and (S-S, draw the inclination of the under side of rail parallel with the under side of rail, draw the center line of rail, cutting the perpen- diculars B atid D at G and H; and also cutting the perpendicu- lars A and E at IT" and T: loin Ti?, cutting the perpendicular from C at K. Parallel with XX, draw GL, cutting DH at 3; parallel ^vith XX diaw KM, cutting Z>jFf prolonged at 3. Now. LK will be the height the wreatli-piece is required to raise, and as LK is one-lialf the whole height around the semi-circle, only one face-mould will be required. Prol<)iig tangent KH to intersect BM at JV; prolong tangent GH U) intersect MK at P; fnnu 3. and at right angles to GP- draw 3 4; fnuQ 2, and perpendicular io NK, draw 2 5; parallel with DH. draw the half width of rail, cutting GH nt. 6, and KH at 7. Make GO equal the clwrd JSCou plan. Fig. 5. A squared section of rail is shown at Q, Joint Bevels. Parallel with XX draw the dotted line 8 9: draw 9 10 pvrpendicular to XX and equal to the radius OC. Fig. .5. Make 9 11 equal 3 4, Fig. 6; also, make 9 13 equal 2 5, Fig. 6; draw 10 12 and 10 11 prolonged to edge of board. The bevel.> are shown in tiie angles at 11 and 12. Fig. 7. Shows the face-mould. Make GH equal GH, Fig. 6; with G as a center, and CM, Fig. 6, for a radius, draw arc atK; again, with fi" as a center, and HK, Fig. 6, for a radius, draw arc intersecting at K; draw HK; parallel with HG and JEfJTdraw KG and GO for the parallelo gram OGHK on the cutting plane, or plane of plank. 113 Platk 24, Pioof. The diagonal OH must equal ON, Fig. 6; if So, tlip angle of tangents at H is correct. Prolong tangent HG, 6" to F for length of shank; make joints at J' and iiC at right angles to tlie tangents; prolong OG and Oif indefinite; make G 5 and G 6 p:ieh tqual H 7. Fig. 0: also, make if 7 and K S each equal H C>, V\g. C; make GA equal HP, Fig. 6; draw OA for the direction of minor axis; let O -.1 equal OC, Fig. 1. for the semi-minor axis; make 2 3 and 2 4 each equal the half width of rail [2"]. Draw fi 10 and .5 parallel to GF for the shank of mould. Now pivot the trammel at O with the arms at right angles to the minor axis as shown. Then set from pencil on rod to minor pin the distance O i, ])lace the pencil at 7, drop the pins in the grooves, and fasten the major pin; then trace the concave side of mould through tlu> jioints «, 4, 7. Again, set from pencil to minor pin the distance O .3. then place the pencil in ^, drop the pins in tiie grooves, fasten the major pin, and trace tlie convex sMc of mould through the points r>, a, S. At section .B the bevel found in the angle at 11. Fig. G, is shown applied through the center of plank; at secllou C the bevel found in the angle at 12, Fig. 6, is applied in the same manner. The block pattern shows the squ.ii'e serlion of rail; the shaded part indicates tlio amount of over wood IIki; Ins to be removed in the formation of the wreath-picco, also the llii'-kiiess of plank required for the twist, [4"1. In many cases less willr! o; this is owing to Ihu style of rail, as the corners may work olf or I'-inain full If a trammel is not at liand, draw llie two right angles at O. and use a rod iis previously described, or find anotlier point on the diag- onal OH, and use a flexible strip. T)ie poi.'it on the diai^onal OH "is found on the din yronal OA' in elevation from O to tlio intersection at. the pcrpendicubir. For a correct face-mould the trammel gives the curves absolutely correct foi all cylindric soirtions that are circular on plan. Figs. 8 and 9. Show how to nppl]/ the tang'iids to the crook ojid slide the mould. The face-mould at Fig. 8. is made from '■.,''' stuff, and has three holes bored through on the tangents for siu'iit holes to aid in slid- ing the mould over the tangents on the crook. The heavj' line XXXX, Fig, 8, shows the crook sawed out \:> a parallel width and square through the jdank. First see that the crook is out of wind on the face and near a uniform thickness: now place tlie face-mould in the center of crook, and transfer the tangents from the mould to the crook, as shown by the heavy dotted lines 2 8 and :'. 4. Mark the joints X2 X and X i X; then cut and dress off the joints square to the tangents, and also to the face of plank. Now cany the tangents across the joints square to the face; then mark the tangents on the opposite side of crook. Where there is a shank rt^' or 6'''' long, as XA, a good plan is to joint the shank XA square to the face of crook, then run a gauge-line on for the tangent 2 .3; then place the mould to this line and mark the tangent 3 4. Then tlie shank joint cati be squared from the face and side of shank. Xow center the joints as shown at sections P and R with a gauge .shown by the dotted lines on the face of joints, making both centers the same distance from the face of crook. Next apply the bevels through the points P and R; then, from where the bevels cut the upper and lower faces of crook, as 5 6 and 7 8, draw another set of tangents FH and HK parallel to 2 3 oud 3 4; repeat the operation on the opposite side of crook: these last lines are the lines on which the mould has to slide. Now slide the face mould so that the tangents FH and HK on fl,ATE 34. 113 the moiild will exactly foincide with the tangents J'J3' and iifiiC on tlie crook. The sitfht holes will aid in locatins? the mould, as shown. Now mark around the mould for the lines to dress oft' the overwood; before lifting the monld transfer the minor axis 10 to the face of crook; then apply the mould to the opposite side in the same way. It will be observed that at B an the monld extends beyond the crook, and the lim; for wiv.kini? off to the plnmb will be lost, while on the opposite side the line will show. A good plan for the learner in this case will l>e to tiick the mould in its place, and the arris of mould will be a guide to work off the surplus wood to the plumb. Fig. 9. Shows the crook turned up, and the concave aide is shown worked off to the plumb lines. Two patterns are here shown to give a better idea how the monld is applied. One pattern is all that is required iu practice, lly dressiug down the coucave side, as shown, and using the (.'upper gauge shown at Fig. \:l, Plate 21. the convex side of wreath-piece may be gauged to the reijuired width without the aid of face-mould. As the bevels are applied they show the wreath- piece intended fora right hand rail and laniling on the platform; for the wreath-piece off the platform apply the bevels the reverse way of this; or it will be seen that if the wreath-piece be turned face down it will answer for the wreath-piece off the platform. One thing the beginner must remember; that is, when work- ing off the surplus wood to the plumb he must hold his tools near as he can to the plumb. The minor axis that is marked on both sides of the crook will be the direction to hold the tools, as 9 0. After the crooks have been dressed to the phnnb bevels on the concave side, then bolt them together at the center joint, keeping the center lines on the joints opposite eacli other. Tlien stand the twist on the floor and cast the eye clown on the concave side and see if the curve i.s correct, having no kinks or abrupt places, when all is satisfactory. Then use the Cupper gauge and gauge from the concave side for the width of rail, then unbolt and dress otf the convex side. Then bolt them together again and dress otf at the center joint carefully that the two wreath-pieces may join each other without kinks. Just here a good way to prove the correct- ness of center joint is, first, see that the center lines made from the bevel are exactly opposite, or parallel to, each other when bolted together. Then place the wreath dov\-n on a true surface on the side, as shown at Fig. 9, and see if the shank of upper wreath- piece is parallel with the true surface; if so, the joint is correct. Again, just here, the wreath for a platform may be tested, to see if the shanks have the right direction for the pitch to and off the platform, by laying down the Inclination of the pitch to the plat- form, and also otf tiie platform on the drawing board, forming the acute angle, as CGE, Fig. 2, Plate 29. Then place the wreath on its side, same as above, with the shank to agree with one in- clination; then sight the other shank to see if it agrees with the other inclination; if so, the wreath is correct. To find the twist lines. Carry the minor axis on the plumb across the wreath-piece, as 9 9; bisect 9 9, Fig. 9; center 9 9 at 13 with the gauge used at section P, for the center of plank; set off half the thickness of rail on each side of 13; do the same on the convex side of wreath. Now we have three points, 11, 14, 12, through which to bend a pliable strip. Be careful when bending the strip to keep it square with the joint at the shank. 114 1*I,ATE 24. Use the try square oflen at the shank joint when working off the over wood, applying the stock to the Joint and the blade to the straiglit part of shank at times. After the surplus wood is removed from the top side of wreath-piece, then gauge for the thickness using the Cupper gauge. On small wreaths, the learner will discover at the center joint, the gauge will cut below the block pattern at 1.5, as shown by the dotted line, while at the opposite side the gauge will cut at the corner of block pattern. This arises partly from the direction of the joint on the concave side; in small cylinders the joint is more of a plumb joint than a perfect butt joint; on the convex side the joint will be found to be, or nearly so, a perfect butt joint; in large cylinders and narrow rails the center joist is nearly a perfect butt joint both on the inside and also on the outside of rail. The learner must be careful to allow plenty of over wood at the center joint until he has the two pieces bolted together, then dress down to please the eye and touch, so as to avoid all abrupt places. Fig. 10. Shoivs hoiv to place the easement pattern for mark- ing the points to cut the straight rail. Draw a tread [10''''] and rise [7^''] and the level landing: through the center of short baluster XX draw the under side of rail, place the lower edge of pattern along the line XX, and slide up until it will measure a half rise [3>^''^J from the floor line to the point A. Now draw the face of landing rise across the pattern, marking the center at B for the point from which to measure for the length off the platform. Then from Cto joint A equals \Z]4'^, to allow for the easement on the level. At F the length of a regular long baluster is shown Z^i'^ longer than a regular short baluster, and at H the baluster is shown 2)4'' longer than a regular short one, and at J" the baluster is shown Z% longer than a regular short one. Fig. 11. Exhibits a straight easement starting from a neiocl post. Elevate 2 or more risers and treads; through the center of baluster XX, draw the under side of rail, parallel with XX draw the center line of rail AB; from the top of first steps set up to the under side of rail the difference that the newel is longer than a short baluster, say 4X'", and the half thickness of rail [l¥'''l more to the center of rail at O. From O, draw a line parallel to the first step to intersect the inclination of the center of rail at A; plumb with No. 1 rise, draw the center O of rap; from the center O, draw the verge of cap \Z}4^^], cutting OA at F. Make jPC equal the half width of rail [2^'], then C is the point of miter: parallel with CA, draw the half width of rail to intersect the verge of cap at D and E. Through JE and D, and at right angles to CA, draw ED indefinite, cuttinar CA at J. Make AL equal AJ; from L draw The radial line LH, at right angles to AL, then H is the center to draw the curves for the easing pattern. Prolong the face of No, 2 rise to intersect the center line of rail at K, then KB shows T}4'' to joint, to be deducted when cutting the straight rail. Fig. 12. Shows a square section of rail. The dark shade indicates the rail-bolt at the center, a dowel on each side of the bolt keeps the rail from turning at the joints, and should be used in all good work, so that when the joints are dressed off they remain substantial. The hollow dowel suits best Plate 24. 115 at the center Joint, for then the wreath-piece can be turned around on the joints, the dowel keeping them from sliding off the center. Cutting and jointing the rail at the bench. The first length from spring of turnout to the spring line of cylinder at platform is 11^ 8'^ See Fig. 2, Plate 22. Now we have 6''' of straight wood on shank of face-mould. Fig. 4, and 6^^ on shank of face mould. Fig. 7. Then 11^ 8^^—Q^^-j-Q^'^lO^ 8^' for the measure to cut the straight piece for the first length. Again, the second length from spring of cylinder at the plat- form to the face of rise landing equals 2^ 11%^^, and we have to allow 6^^ for shank at face-mould. Fig. 7, and ti};^^^ for straight wood on lower end of easement, Fig. 10; then 2^ 11%'''' — 6'^+ 6X''=i'' IIM^' for the length to cut the straight rail off the plat- form. The third length from face of landing rise to spring of quarter cylinder equals 5^ 6>^'^ from which deduct 13^4^''' for the easement landing. Fig. 10, js' G}.i^^—V 134^^=4^ o}4^q equals V o}i^^ for the length to cut No. 3. From the spring of quarter cylinder to wall equals S'' V, Fig. 1, Plate 23, from which deduct fi" for shank on quarter turn, see Fig. 5. [3' 1^'— 6'':==2'' 7^^J equals 2^ 7^^ for to cut the straight rail for No. 4 length. Hanging Rail. This part of the work every stair-builder has a way of his own; a very good plan is to first have the mor- tise for the balusters cleaned out, the brackets fit and nailed on, and the nosings fitted and dressed off. Now number the nosings and lay them aside, then mark the center of baluster V^ from the end of step, and 2'^ more to the convex side of rail, [l^^~\-2^^=y^] equals 3^^ from the end of step or the bracket Hue, to the convex or outside of rail. Then mark a few steps 3'''' from the ends, also around the cylinder on the platform and along the level, for points to plumb the rail. Next place the first length of straight rail on the nos- ings, keeping the convex side to the points marked on the steps; now try the plumb bob at the end of wreath-piece at the platform, and down the side of rail when correct, plumb through the center of cap for the center of newel post on the floor. While the rail is in this position, measure from the floor to underside of cap, say that it equals 13''^; then at the center of a short baluster measure from the top of step to the under side of rail, say that it measures 13 j''''; then 13^'' minus 1}4'^ equals IIK^'' that the newel post is longer than a short baluster at its center when in position. For example, 2^.1}4^^, say the height of a short baluster when in position measures 2. 1 3^ at its center, from the top of step to the underside of rail; now the newel from floor to under side of cap equals [2' iy/'-^n}4^^='y 1^^] three feet one inch. Now set and fix the newel to the center on floor and height iust found* [3'' 1^^]; then hang the rail firmly on stanchions; glue and drive up the nuts on the rail bolts, plug the holes made for the nuts, see that the rail is straight between the crooks, then plumb and bore for the balusters; now dovetail and glue in the balusters. Before glueing be careful to see that the rail is straight, without any bends. Often the straight rail is bowed, and can be straightened nicely when putting in the balusters. Next nail on the nosings. Trim down the wall string and spandril under the stairs, leaving the hand rail for the last job to dress down, so the finisher may apply a coat of "filler" immediately after the stair- builder has completed his work. *The post may be cut to length in the shop and set in place, and the rail hung at once in the building, but the method described above is considered the most economical iu the end. Plate S5 PLATE 25. Plate 25. [Scale, H''=l foot]. Exhibits the plan and elevation of a tivo-story stair case. In the first story the stair- case is ilivided into two Jlights; the first flight lands on a lyUitform lercl with the floor in the hack hnildimj, thc7i braiiches ofl, and lands on the floor in the second story. And in the second stonj, leading to the attic, the stair-case is composed of one continiutu-i fiiiflit. The height of the first story is 11'' 1''^ from top to top of joist; the height of the first story back equals 9^ 4'^; the width of joist in the second story is 10^'; ilie width of hall iu the rough is 8^ 2^^. The lieight of second story, from top to top of joist, equals 10' 0"; widtli of joist in third floor is 10^'; steps are 1)4^' thick; nosings are to be returned ovei- lirackets; size of rail 4>^''x2?i''', fiat moulded. Now we have 11'' V for the height of the first story, which, reduced to inches, [11^ l'^X12''=i;!3''] equals 1:53'^ to be «li- vided by ?'', equals 19, for the number of risers in tlie first story. Now the height for tlie first story back equals 9' A^\ which, re- duced to inches [9' 4"X12''~112''J equals 112'^; that being di- vided by 1'' [112'''-J-T'''':=16] equals 16, for the number of risers to the platform. That leaves three [3] risers for the short flight landing in the second story. For the tread take Blondel's formula, but use for a constant 25'^ for an easy step [2.5'^— 7''+7'^=ll''] equals 11^' for the tread or cut on horse. '' For the flight in the second story leading to attic we have IC C for tlie height, whicli, reduced to inches, [10' 0''X12''=^120''] equals 120"; that being divided by 8 [120''^ 8''=1.5] equals 15 risers to the attic; then for the relative width of tread to the lieight of rise, by the above formula and constant [2r/' — s^'-j-S''' -—W\ we have 9'' for the width of tread by S'' rise for the cut ttut of horse. Fig. 1. Shows Hie plan for the first story. The Vv'ell hole is 12'' ))elween the outer strings; the cylinder uiil then have a radius of 6". The width of half pace from the luce of cylinder to wall should equal the half width of hall [4' 1"]. But this cannot always be, for in cramped places the room will not allow more than the length of a step, and should never be less Now in a 12" cylinder with the steps laid out on the center line of rail, there will be about 2" from the face of the landing rise [No. ItjJ to the face of platform joist, see Fig. 1, Flate 26. Then if we allow 4' 0" from tlie wall to the face of the platform, and 2" more to face of the landing rise, and 1.5 treads at 11" each, we will have [15Xll'^=lt35"H-12"=l'j' 9"+2"-4' 0"= 17' 11"] for the run of first flight, including the platform 17' 11" as shown on plan, Fig. 1. And for the flight of[ the platform we have 4' 0" for the width of platform, and 2" from the face of platform to face of the first rise off the platform, and two treads at 11" each; also allow 6" beyond the face of last rise to the face of trimming joist to allow "the bearers a good bearing against the joist for nailing; tliis then will give us [2Xll^'-!-2"-^G"-f 4' 0"=6' 6"] from the wall to the face of joint landing including the platform C 6". as shown on plan. Plate 25. 117 The next to decide !s the distance out from the back wall the second flight will come, as that will determine the head room for tlie first flight. The cylinder landing in the third stor^^ we will keep plumb over the cylinder at the platform in the flrst flight; Iben we will have 4' 0^' for the width of landing, and 1'^ more to face of rise, then 14 treads at K)" each, and %" more from the face of No. 1 rise to the ronuh ioist to admit the cvlinder. Therefore [14X9^'=12«''-^1-'=10' li^'+4' id"^\"^yi,"~\\' 1%"\ we will have for the whole distance from the back wall to the trimmer at the starting of the flight in the second story 14' ">%", as f^howu, Fig. 3. Headway for the First Flight. The horizontal dis- tance from wall to face of trimmer in the second story equals 14'' VJg", from which deduct 4' Q'^ for platform, and 2'^ from plat- form to face of No. 16 rise [14' 1% "—A' (i''~^"^W h% '■'] eqiial.-i say 10' '', which being rednecd to inches and divided by the width of a tread [11''] in the flrst flight (10' 6"X12"=l:i6"---ll"=--ll 4"] gives eleven full treads and four inches more to count down from the platform to plumb under the face of the trimniing joist in the second story, which brings us 4" over on Xo. 4 .--te)). This then leaves the height of four risers (28") at the starting plus the width of joist [10"J in the second story, also the thickness of floor [1"]. and plastering [1"], to be deducted from the whole height [11' 1"]. Then [2S"-i-10"+l"+l"H-12=.r 4"] the height of the ^tory 11' 1" minus 3' 4" equals 7' 9" for the headway as shown Fig. 2, the joist will round off at D, and allow perhaps 2" more which will be ample headway, for the first flight. Width of Landing Pace. For the width of framing for the landing pace, take the Jialf width of hall [4' 1"], and deduct [6"J for half the cylinder, and 1" for thickness of front string, [4' 1"— 6"-rl"=3' 6"] and we have 3' 6" for the width of land- ing pace in the rough, which if deducted from the whole width of liail [8' 2"—:'/ %"=\' S"] equals 4' 8" for the width to trim the well in the rongh. Length of Landing Pace. Now we have 14' 7?4"from the wall to face of trimmer in the second story, and 6' 6" from the wall to the fare of joist at the landing, then 14' 1%" minus 'i' G" equals 8' 2" nearly, for the length of landing pace as shown. The next will be the head room for the tiiyht in the second -tory; the height of rise is 8", and the tread 9"; we will count ■ iown 1:5 risers at 8" each. [13X8"-^12"— 8' 8''J that will equal ' 8". Then the depth of joist in the third floor is 10". and the plastering and flooring may be counted at 2" more, [S' 8" — 1(7S^2"=::7' 8"] which equals 7' 8" from the top of No. 2 step to the ceiling. We will then place the faseia on headway trimmer plumb over the center of No. 2 step and have very good head-room tor the second story flight going to the attic. The distance from back wall to the face of trimmer at head- way will be equal to 12 treads at 0" each, one-half tread 4><". .ind 4' 0" from the wall to the face of trimmer at the landing, and ?.'' more from the triunner to face of No. ].5 ri^e and the thick- ness of fascia at headway, [12Xa'^-r^K'^-r'l' 0"-r2'^=13' G^"] equals for the whole distance 13' ti'^ ". from the wall to the face of trimmer. Then 13' 6M"— 4' 0"=9' ^W, for the length to frame the well: and, as the cylinder in the attic has the same radius as iu the first story, the well will have the same width of lis Plate 25. opening, i^ &^^\ then the trimming of the well in the attic will be 4^ 8^^ by 9' 6^^^ in the rough. Fig. 1. At P is shown the manner of building the plat- form. The flooring will run in the same direction as the steps wherever this can be done; more satisfaction will be the result. A, A, A, are the bearers, 3''' bj' 4^^ scantling. They are doubled at the outer string on the long flight. At JB, B, B, the dotted lines show the rough brackets, the center bearer having two rows, one on each side. The center of newel post is placed in the cen- ter of hall. The arc and radius for the turnout is determined fi'om the difference that the newel post is longer than a siiort baluster, and the location of newel see explained Plate 26. Fig. 1 Fig. 2. Slwws the elevation of tirst flight of 16 rinens landing on tlie platform, ttienee, icith 3 risers more, hranchiiuj off and landing in tlie second story. SS shows the front string; WW the wall string; J, J. J. shows the joints of cylinder connecting tl.e front string; AB shows the story rod divided off to the required number of risers [19]. At C is shown the pitch-board, made to suit the height of rise. At ^Pthe run of treads is spaced off on a rod. H shows the pitch-board made to suit the tread. The correct lise and tread should be entered in the order book, as the pitch-board is liable to shrink. At Q is shown the panel work underneath the first flight continued to the joint of cylinder, having an arch-way for a wash-stand or coat closet. At K is shown the rough bracketing underneath the steps. Fig. 3. Slioios elcvaliun of the flight leading to the attic, having a utruight run of 15 risers. MN shows the story rod spaced olf ; AC shows the run spaced o, draw AS to indicate the center line of rail. Now ^et the center of newel O on Une with the face of No, 1 rise, and Oyi''^ out from the center line of rail; from D drop the perpendicular to intersect AB at F. Join OF; from tho center O draw the verge of cap 7^' in diameter, and cutting OJPat H. Make H2 equal half the width of rail (SJ^'O also set off on each side of OJPhalf the width of rail to cut the cap at 3 and 4; join 3 4, intersecting OF at 5; then make jPJ" equal F 5, Then FJ and F 5 are the It^xsrth of tangents on the plan. From J, and perpendicular to FJ, draw JR indeiiuite; from 0. and at right angles to 5 F, draw 5 JFJ, intersecting JR at R; then il 5 is the radius (2^ 2'") for the turnout at the center of rail, and %^^ less, or 2'' 1^4^^ is the radius for the curve of front string 6 7; draw 7 8 parallel with AJB. Now carry down the face of No. 1. 2, 3 and 4 risers from the elevation to plan. Nos. 1, 2, 3 and 4 show their position on plan. The face of No. 3 riser is shown on plan 4>^^^ out from the joint of the turnout; the radius and position of risers for the turnout are now shown as required for prejjariug the turnout, steps and risers for the same. This drawing should he luado on paper, then after tlie steps are prepared, the drawing can be rolled up until the rail is required. Fig. 5. Shows the semi-circle divided off into an odd num- ber (5) of staves. Then the back surface of the center stave will lay solid against the face of platform. AJ3 shows the edge of drawing board; CD is a gauge-line run on for the diameter of cylinder. From draw the semi-circle to a radius of 6^^; space off the semi- circle into the required number of staves, as D 2, 2 3; join 2 D prolonged to edge of board at M for a handy means to set the bevel from the edge of board as shown. SS shows the edge of face string and manner of connecting the cylinder by splicing and drawing the joint up with screws from the back. Fig. 6. Shows the quarter circle divided into two staves. The face of one stave is prolonged to cut the edge of draw- ing-board to set the bevel as shown. Fig. 7. Shows the wpiycr and lower ends of wall string for the first flight, and is laid off as has been described at Fig. 2, Plate 19. AB is the gaug« e, run on 10%''' from the upper edge; one inch is allowed for e grounds, which is worked on the solid as shown at C; the compasses are set to the hypothenuse of pitchboard, and the required numb r of risers are spaced off on the gauge line as 00; DD shows the wedge's; the wedge for riser being entered first and those for the step last. At EE the string is gained half through to join with the base; FF shows the base moulding, it is rebated on the lower edge to drop down on the base H^^ as shown at C Plate 26, 121 Pig. 8. Shows t?e vpper and loiver ends of the front or outer strUig. AB shows a gauge line run on lightly as a guide for the pitchboard as showu at P, Fig. 10; the points O, O, 0, &c., along the lower edge are transferred from O, 0, O, &c., on the wall string. Fig. 7, then squared over to intersect the gauge line AB at XXX, Ac. ; if the pitchboard is applied to these points care- fully, the two strings cannot fail to be both of the same length. Cylinder Joints. At Fig. l the face of No. 15 rise is showu 5%^' from the joint of cj Under. Now as the end of rise is reduced to H^^ thick for to miter with the brackets, the distance from the cut out of string to joint of cylinder will be H^^ less, making 5^'^ as shown; the joint CD is 10%'^ long. Then the joint of turnout at Fig. 4 is shown from the face of No. 3 rise to joint 4>^^^ in this case, the }i'^ will be added, making 4^^' from the cut out of No. 8 rise to the spring FE of turnout. No, 1 rise is shown reduced to 5^4", allowing the thickness of a step 1H'\ that when the step is in place, the risers will all be uniform in height. The points from which to take the length, to cut the rail are showu from joint to joint of cylinders, [14'' OH'']. Fig. 9. Shows f}i€ outer stHng having three risers landing in the second story. The wall string is not showu, but the points O, O, on the lower edge of string are transferred from corresponding ]!oints on the wall string as above; the points are squared over to Intersect the gauge line, from which to apply the pitchboard. At Fig. 1 the face of No. 18 rise is shown 5Ji^^ from the joint of cylinder, then H^^ more for the reduced thickness of rise will equal 63^^'' from the joint of cylinder AB to the cut out of the face string as shown. The joint is 11>4^'' long. At Cthe string is notched K^' for the difference between the thickness of step {l}4'') and the flooring (1") in the second story. For the full easement landing allow 10'^ for the width of joist, and X^^ more for the lath and plaster equals lOX^'- Then draw FG parallel with CD, and ease off the angle OFG with the easement pattern. HH shows two balusters on the level. At No. 19 rise the landing step and rise are shown returned over the bracket. The lengths to joint the rail are shown, V ^14" for th^ short string and 8'' 0%'^ for the level length, which is taken from the face of No. 19 rise to joint of cylinder starling second flight. Fig. 10. Shows the vpper and lower end of the wall string in the second story. The lining off is the same as has been described for Fig. 7. The lower end is shown lined off to the pitch-board, and after- wards with the housing pattern. At the upper end the pitch- board Pis shown applied along the gauge-line and ready to apply the housing pattern. The upper end is notched out to fit up against the joist at the lauding. At A is shown a triangular piece glued on to form the easement. The string is shown rebated on the upper edge forming the grounds. Fig. 11. Shows the upper and lower ends of the outer string. It is lined out from the wall string in the same manner as Fig. 8. The joint AB is showu on plan Fig. 2 to be 2^''' from the face of No. 2 rise. The rise is shouldered, allowing ^4^^ to connect the bracket. Then from the joint to cut-out for No. 3 rise will equal 3>8 ^''. The joint is 10^'"' long and parallel to the risers. 123 Plate 27. At the upper end the face of No. 14 rise is shown on plan Fig. 3 to be 2X^^ from the joint of cylinder. In this case the ^," for rise must be taken off, leaving from the cut-out of No. 14 rise to the joint CD equal to 2^^^ as shown. Make the joints per- pendicular to the treads; the length of joint equals \i%" . The length, 12' %%''■, for jointing tlie straight rail is taken from joint to joint of cylinders, and parallel to the lower edge of string. The first level length iu the attic (8'' ^M") is taken from joint of cyl- inder landing to joint of quarter cylinder; and, again, from joint of quarter cylinder to wall (4' 1'^) for the last length. PLATE 27. Plate 27. [Scale Yx^'—X foot]. Exhibits a method how to obtain the length of staves for the cylinders at Plate 26. Fig* 1. Shows tlw development of staves for the cylinder, Fig. 1, Plate 26. Let AB indicate the edge of drawing board; draw Nos. 15. 16, 17 and 18 risers and treads 7^'Xll'^. Make CD equal 6}i^^ perpendicular to AC draw DF; then with the breadth of a stave as D 2, Fig, 5, Plate 26, space off five staves and draw them par- allel to DF; make LO [lO^^^J equal DC, Fig. 8, Plate 26- also makeX>J'(llJ^^O equal AB, Fig. 9. Plate 26, join OF for the length of staves as shown. Observe the staves extend beyond the steps at the upper ends and also below the inclination OF at the lower end for over wood. MN and CP show the width of string (6''') at the internal angle of step and rise. Fig. 2. Shears the dci'clopmcnt of sta^'cs from the concave side of cylinder, Fig. 2, Plate 26, starting from the level to rake. Let AB indicate the edge of drawing board; from AB draw Nos. 1 and 2 treads and risers; also line off joist; No. 1 rise is re- duced }i^^ for the difference in thickness of the steps and fiooring boards. From the cut out of No. 2 rise at C, make CD equal o}^^^ as shown at Fig. 11, Plate 26. Draw ZJii* perpendicular to No. 1 step; from F space off five staves equal to D 2, Fig. 5, Plate 26, draw the staves parallel to DF; from C, set off 6^' to lower edge of string FG, which is parallel to AB; the joist is 10'''', and lath and plastering is X'^- Then drop down from the joist 10J<^' to K; draw ifif jiarallel to the joist line, cutting G^ prolonged, at H^ for the lengtli of staves as shown; the staves are shown to be cut longer to allow for trimming. Pig. 3. Shoivs the development and length of staves for Fig. 3, Plate 26, landing on the Itrcl in tlic third story. Let AB indicate the edge of drawing-board. Draw Nos. 14 and 15 treads and risers; draw BC for line of joist. As the steps are U^' thicker than the flooring. No. 15 rise is drawn a U^^ higher. The cylinder is notched out, as shown, to receive tho step. From the face of No. 14 rise set off 2%^^ to D. Draw DF perpendicular to No. 14 tread; from D^ space off five staves, each equal to jD 2, Fig. 5, Plate 20. Make DF {Vi}^'') equal DC, Fig. 11, Plate 26. Make CiT equal tlie width of joist (10''''), phis the thiokness of lath and plaster (X^O- equals 10%^^ Draw KH parallel to the joist line BC; also, draw FH parallel to AB for the length of staves as shown. Cut the staves long enough to al- low for trimming. Plaxe 27. 123 Fig. 4. Shows the sUives for Fvj. 1 jnintcd and glued ■up, formiiKj the cylinder, ready to trim off and i^plice to the strings. At A and JB the strings are shown gained in to receive the joints C and D of ej Under. Tlie joint at C is 10%''^ Jong, and the joint at JD is llj^^^ long. Tlic twist line 6, 5, 4, 3, 2, 1, is obtained by bending a pliable strip in tlie cylinder from 1 to 6, governing its direction at the joints with the pitch-board by the use of a tack at points 1, 6, 4, 5 and '2; tiien by shifting the tacks at 5 and 2 the direction of strip at the joints is easily adjusted to the inclination of pitch-board. The jointing and preparing of the staves for glueing is ex- plained for Plate 22. ( .- Fig. 5. Shows the cylinder, for the starling from the level to the rake, for the flight in the second story, and repeats Fig. 4 only in this case: A full easing is required to connect the level with the rake. The width of face string along the level is 10^-' for the joist and K'^ for the lath and plaster; equals lOX'^. Then the length of joint at D will be lOX^'; the length of joint at C is shown lO?!^^ At A and B are shown the splice johits on the ends of straight strings to receive the joints of cylinder sliown at D and C, which are to be glued and screwed from the back of strings. The twist line is found as described at Fig. 4. Only the direction of strip at joint D is regnlated by tlie square and at joint C the pitch-board g^^■es the direction of the twist line to agree with the straight string. At the intermediate points 5, 4, 3, 2, the learner mnst use his judgment as to the direction so as to avoid any abruptness in the curve. Also the learner will dis- cover from experience that if the direction of twist lines at the joints be a little steeper than the pitch-board it will ini])rove the curve at the joints. The steps arc 1}^^^ thick and the tlo2 full size.) Fig. 10. Shoios the upper edije of icall string. S shows the string, G the grounds, P the plaster, B tlie base moulding, the rebate of base, H the housing for step. The grounds to receive the plaster is sliown worked on the solid; this strhig is usually !}.<'' thick, and the steps are housed in %^' deep; on brick walls allow %^^ for plaster,* and on partions X^^ for lath and plaster. Preparing the steps are the .saine as has been described in Plate '22 tor a T" cylinder. This being a vy^ cylinder, there will be more circular end steps. I'atterns may be made the same as shown for a 7^^ cjlinder, or they may be laid off from the draw- ing. If the drawing be made on thick paper the cuiTes may be pricked through on to the stuff and afterward traced and dressed to the required shape. For the construction of the concave and convex risers, see Fig. 5 and 0, Plate 31. *If adamantine or Keen's cement be used for the -walls, then allow %" for brick and ^i" for lath and plaster. Plate 38. 126 PLATE 28. Plate 28. [Scale %=:1^J. Exhlhits the manner of veneer- imi the cylinder. Ttie true pitch of stairs is laid off on the cyl- inder line instead of o^i the center line of rail, as at Fig. 1, Plate Fig. 1. Shows the semi-circle 12'' in diameter and the stretchout C 3, for the quarter circle is drawn in the same way as at Fig. 1, Plate 26. On the stretchout 3 Cset off half a tread (5H^^) to 4, or the face of Xo. 17 rise; from the face of No. 17 rise to the face of No. 18 will equal a tread or 11^', draw No. 15 and 16 rise oppo- site to No. 18 and 17 rise ; now locate the short baluster on No. 15 and 18 treads, then space off the intervening baluster equal and curve No. IG and 17 rise to suit the baluster. The face of No, 15 and 18 rise is shown 1" from the spring of cjiinder. Fig. 2. SJimvs the thiclincss of veneer. The shaded part shows the staves. The string AA is !}{'' thick and is shown reduced at the circular part sufflcientlj to bend around the drum, the reduced part is extended beyond the spring line from V to 2'^, so as to relieve the spring at the junc- tion of the straight with the circular part and thus avoid a possible fracture. To avoid making the recess on the back of string a better waj- is to make the veneer of an even thickness throughout, then after the veneer is lapped over the drum, lag out for the full thickness of string, thus avoid a tediotis job in reducing the string to the required thickness. How to Determine the Thickness of Veneer : Barlow in his experiments on the curvature of different woods has formulated a rule for the safe elastic limit before fracture for three kinds of wood, oak. th- and larch. Rule. Multiply the radius in feet or a decimal of a foot by the constant for the kind of wood used and the result will be the thickness in inches, or the decimal of an inch. Thus the form^ ula reads; For Oak— the radius in feet X 0.05=thickness in inches. " Fir— the radius in feet X 0.035=thickness in inches. " Larch— the radius in feet X 0.077=thickness in inches. For Example, suppose the radius of curve to be 33^', which is equal to 2' V. or the inches reduced to the decimal of a foot (9'^-^12=r.75) would equal 2.75^ Now 3.73^X0.05 for oak equals .1375 of inch. This reduceil to sixteenths (.1375X16=3.3000) equals 3.200. or say two-six- teenths strong. For white pine the constant 0.0625 will be about right and would not injure the elasticity of the wood. Of coiu-se straight grained lumber should be selected for veneering in all cases. Suppose then we use the decimal 0.0625 for white pine and the radius of curvature is 6^', or the decimal of a foot equals .5. Then 0,0625X. 5=0. 03125 of an inch. Now the decimal 0.03125 reduced to thirty seconds (.03125X.32=1. 00000) equals one-thirty-second (3*2) of an inch for the thickness of pine veneer, and for every additional 6'' that the radius increases the thickness of veneer will increase one-thirty-second of an inch. By steaming the veneer, this thickness may be increased Ijy 2, but in steaming hard wood, the grain becomes discolored and Is not to be recommended for fine work unless the wood be pine and then painted; or of walnut the discoloring does not show so much when varnished. 156 : Plate 58. Fig. 3. Shoios the Vcnccr. Tlie treads and risers shoulil be lined off with load pencil and not cut out until removed from the drum. Alter the treads and risers Nos. 14, 15, 16, 17, 18 and 19 are lined off, make from the cut out of No. IS rise to ?pring of cylinder equal 734 ^^; and from the cut out of No. 15 rise to spring of cylinder 6%''^; draw AB and CD, each perpendicular to the treads for the spring lines of cylinder; the points from which to take the lengths for jointing the rail at the bench are showti taken from the spring of cylinder and parallel witli the lower edge of string. The length off the platform is 1^. 10>^^^ and the level length is 8'' OX'''' to the joint of cylinder; at F the string is notched M^^ to fit up close to the floor at the l)ack of landing step; at H the string is gained in for joining the level fascia; at K is shown a splice joint connecting the straight string. Pig. 4. Shoics irhat is termed a drum, made to the size of well hole in diameter, having straight sides; AB indicates the spring line. Fig. 5. Shrms the string with the veneered part bent over the form; fasten the straight part at one end firmly to the side of drum with two hand screws, keeping the spring line AB on the string, over the spring line AB on the drum, if the spring line on the string agrees with the spring line on the drum at the opposite side, then clamp the string down firmly, pressing the veneered part down close to the "form." As a precaution on small cylinders when bending the veneer over the drum, put a few staves across tlie veneer with a screw at each end temporarily every few inches apart, this may save a fracture of the veneer; DD shows staves cut out a little longer than required, when putting them oil com- mence at one end and fit one down, put a screw at each end into the drum, thus drawuig them closely to the veneer; then fit, glue and screw the next, finishing up at the other end. After the glue is dry, dress off the high places on the back of staves, then tack and glue on two courses of hard wood strips 3^^^ thick and 2^^ wide, in two thicknesses. Let the lower course be 1^^ above the lower edge of string, and the other course far enough down from the upper edge not to interfere with the cut out of string. After the glue is dry remove the string from the form, and dress off the over wood to the edge of veneer. If there are mould- ings to be worked on the lower edge of string, the staves can be extended and rebated to form grounds for the mouldings. The woll-liole finished in this way makes flrst-class work, and in liard wood finish sliould always be done; for tlie character of finish in a liousc is often deteruiined by tlie appearance and workmanship of the stairs, thus adding to or diminishing the value of the building. Fig. 6. Shows (I method of curvini^^ for length of the regular steps. Step No, 1, is .5^'' longer, No. 2 is !)<<'''' longer, and No. 17 is '1)4'^ longer than the regular length above. The platform step may be 8^ 2''' long to extend the full length of platfonn. In the llight to attic No. 1 step is 3K''' longer than the regular length and the landing step may be long enough to include the diameter of cylinder and 2^^ more to bed on the joist, or 13^^ longer than a regular straight step. The risers may be cut the thickness of step less or 3'' 7K^'' long. These lengths are taken between the walls, the steps on 128 Pt-ate 28. that accoiLit will be long enough for fitting into the housings. However, when taking the dimensions preparatory to getting out the stairs, see that the walls are plumb, and note the ' ' set offs" of brick work, if anj% at the height of the different s'^orys. Width of S'^'OpS. The tread or cut out c-^ front string is 11^^ the projection o'; "osing is usually equal to the thickness of steps in this case IJ^'''. The tongue at the back of step is ^^^ then the width [ll'^+l H^^+ %"=Vi.%"\ of steps equals \%%^' for the first story. And the steps in the second story will equal ^" for the tread, plus 1% for the projection of nosing, and %^^inore for the tongue at the back of step, {^"+\)i"-^r%"=\^%") equals 10%'''' for the width of steps in the second story. Risers. The risers for the first story are to be dressed up neatly to 1", and in the flight to the attic they are to be dressed 8''' wide. If the risers are to be tongued }>i," into the step as shown at B, Fig. 7, then the width of rise will be increased 3^^^, or 7J^ wide for the first flight and %%'' for the second flight. Should they be joined as at the external angle F and internal angle iif tongued into the step above and below, then the width of rise would be minus the thickness of step, plus the depth of tongue above M" , and also the tongue below J^'''' equals (1" — IM''^ =Wi."-VW^W=-'o}i") for the net width of rise ^%", and the step would be increased in width to the back of rise as shown. The Circular End Steps can be laid off on the ground plan. From the center with the dividers strike the curves and lay off the miters. Or lay off the plan on thick paper and prick through on to the stuff for steps and trace the curves on the steps. For a guide, cut sight holes through the paper at different points ••\ong the edge of steps so the drawing may l)e easily regulated on the step. The system of patterns shown and explained for Fig. 10, Plate 23, will be found the most convenient for a large stair shop. The mitering of steps, risers and glueing the same has been explained in connection with Plate 22. After the panel work under the first flight is prepared, steps nosed and together with the scotia cleaned off, the nosings let into the wall strings, cylinders glued on to the long front strings, and joinis dressed off, the work is ready for the building. If the work has to be handled a good deal in shipping, then cover the miters on the steps and risers; also protect the exposed joints of cylinders and strings. The cylindrvs that are glued to the string should be well braced before leavia.^ ^o shop for the building. The height of the platform may be markcu uu the cylinder before leaving the shop. When the stairs are stepped up take the level length and enter them in the order boolc when returned to the shop. Also have blocks well fastened in the walls at the termination of rail so that the rail and rosette may be securely fastened to the wall. Also at the. starting and landing of all flights, see that blocks are cut between the studs and the walls plugged that solid railing may be had for the 6 3iS6 111 O t^ S Bracket well, underneath the stairs, always have the grain of wood perpendicular to the step. Keep the ends of brackets V^" above the lower edge of carriage or bearer so that in case of shrinkage the ends of brackets would not break the plaster. In the first flight to the platform two 3"X4" scantling, AA, Fig. 1, Plate 25, are shown placed against the front string, and rough brackets on the opposite side from the string, the center bearer Is bracketed on both sides to distribute the weight equally, and also deafen the walking line. , In the flight to the attic, there Is no support from a panel, as in the first flight; therefore, double the center and also the outer bearer and bracket the same as shown for the first flight to the platform. Scantling 3"X4", doubled in this case, will give ample strength, but will deflect in time, and cause the plaster to crack. A good and economical way to prevent the bearer deflecting whenever it can be done, is to put a Vi' rod through the outer bearer parallel to the risers, and anchored well Into or through the wall; Plate 38. 129 sive the rod all the inclination possible from the lower edge of hearer to the internal angle of step and rise on the under side. The rods may be used every four [4' 0"] feet. Also, brace well between the outer bearer and the wall horizontallj'. Strength of hearers underneath the stairs to prevent tlw crackiim of plaster. At Fig. 9, Plate 23 is shown the outer bearer for the Hisht to the attic, Fig. 3, Plate 35. For stiffues.s of joist to prevent tiie cracliing of plaster, Tred- gokl allows a flexure of ^g of an inch for every foot in leugth. Mr. R. G. Hatfield, in his valuable treatise on Tmpi^ueJ'se Strains, recommends ^ of an inch, or the decimal .03'^ per foot in leugth for a full load. D, Copet Berg, in his article on safe buildiug, in the Americati Archilcrt, makes use of the same constant, stating the deflection should not exceed 0.03 of an inch per foot of span, or else the plastering will be apt to crack. The stair-builder should know how to calculate the dimen- sions of his beams, to suit the weights likelj" to come upon the stairs. Mr. R. G. Hatfield, in his work on Transverse Strains, has given us the benefit of his experiments iu the weight of crowds per superficial foot. He states the greatest load to be provided for is 70 lbs. per superficial foot for a croud. We will now see what load likely to come on the bearers iu the flight to the attic in this case. For the dead load we have : 14 yellow pine steps 4j^ cubic ft. @ 33 lbs. per ft. =15 1 lbs. 15 white pine risers and front string, say 4 cubic ft. For bearers and rough brackets, . .13 " " 1? Equals 17 cubic ft. of white pine @ 28 lbs., cubic ft.=^476 lbs. Plastering. The soffit measures 14' 6^'X3' 6''i-=50%'^ square ft. 50,75 square feet of plastering @ 9 lb. per foot -456.75 Walnut rail and balusters, say 3>g cubic ft. @ 33 lbs.— 110.00 Total, 1193.75 This gives us 1,193%' lbs for the dead load. Should iron be used as flitches for the bearers, or rods for trusses or hangers, their weight should be added. For the live load we will use Mr. Hatfield's constant of 7(» lbs per superficial foot. The horizontal distance between the trim- mois is m' (5'^, and width of stairway between the plastering and hand rail eciuals 3' 3'^ then 10' 6'^ by 3'' 3^'' equals say 34 square feet at 70 lbs per foot [34 X'' 0=2, 380] equals 3.380 lbs plus the dead load, [2.3S0 ^1.193?^=3, 573%] equals 3,574 lbs for the whole load likely to come tipon the stairs uniformly distril>uted. To distribute this load on the bearer underneath the stairs, two methods may be considered. First, by ]ilacing one l>earer at the outer string sufficiently strong to take up one-half the whole load, allowing the wall to take up the other half. Second, or by using a center bearer which wotild divide the load into four equal parts, thus giving to the center bearer one-half the whole load, and the outer bearer and the wall each one-fourth the whole load. We will consider the former method in this case as the most eco- nomical. Rule to find the size of bearers allowing .03 of an inch for every foot in length for deflection. Tredgold is quoted as good authority among engineers; he gives a formula for inclining beams, and is formulated there by Mr. F. E. Kidder, author of the Arch- itects' and Builders' Pocket Book. 130 Plate 28. Formula no. 1. BreadthXcube of the depthXe. Safe load at the center= Length Xhorizontal distance between supports Formula No 2. Load xlongthXhorizontal distance between supports. Breadlh= — Cube of the depth Xe. e is a constant and equals for white pine, 83. ' e is a constant and equals for hemlock, 80. e is a constant and equals for white oak, 95. e is a constant and equals for wrought iron, 2,000. The deflection from a weight uniformly distributed, is to the deflection caused bj' the same weight placed in the center as 5 is to 8; or in other words, % of the uniformly distributed load will deflect the same beam to the same extent if the load be placed at the center.* Now we have a uniformly distributed load of 3,574 lbs. to be divided into 2 equal parts [3, .574-^-2=1, 787 Ibs.J equals 1,787 lbs; allowing^ of this for acenter load, equals [( 1,787 x;5)-=-8=l, 1169 j or say 1,118 lbs. for the center load on the outer bearer. Example. fWhat must be the breadth of a white pine bearer to carry a load at the center of 1.118 lbs,? The length of bearer on the rake being 14^5, and the horizontal distance between supports being 10^5, and the depth of bearer to equal G'^. Breadth equals (1.118Xl■4.;yX10.5^-f-6.3//x«-=9.6^^ equals 9.6'''' by 6^'' deep, or say lO^^'X^^^ malting 60^'' for the cross sec- tion of timber. Then by placing 2 bearers 5^^X6^'' side by side at the outer string, and by bolting and spiking them together, will give the area of cross section and be amply .stiif enough to carry the load and not deflect a sufficient amount to injure the plaster- ing. In this case, a center bearer will be rr^iuired to nail up the brackets, and afford nailing for t!ie lath; lighter material as two jj//<^4// scantling spiked together will be ample. If brackets be nailed on both sides of the bearer, they will deafen the steps on the treading line. In this case the length of bearer is I4..y long, and at .0:; of an inch per foot for deflection, would deflect fl4.5^X.03=.4o.'5] the bearer equal to .43.5 of an inch; this decimal reduced to sixteenths would equal [,435X 16=0 960], very near -/^ t>f ^" iucli. if tlie bearers be kerfcd in from the upper side | their depth, and oak wedges driven in tlie kerfs, so as to give tliem a camber, this will add jL to their stiffness for flexure.:]: Very dry material shouhl be used in this case to make the work substantial. Stuff for bearers should be kept on hand perfectly seasoned, for it is impossible to make substantial work without good, dry material. Where two scantlings are thus spiked together, reduce th(! two inner edges to form a V shape so the plastering nuiy have a chance to key, , In long and heavy flights iron flitches or angle irons should be bolted to the bearers, for to depend all together on Avood for bearers would increase the width of front string too much, and give to the stair-case a clumsy appearance. *See Nicholson's Dictionary, Vol 2, Page 530. +NoTE.— The student will notice tiie length of bearer, 14'. C", and also tlie iioiizontal dislam-c, or run, K/.O", has been substituted for 15.0", and 11' 2", ^howu at Fig. 9, Plate 23. ■, iMicholbou dictionary, Vol. ~, Page 530. Pt-ATK 9.S. isi A stimip iron or oast iron shoe slionltl he nsecT at tiie lower end to support tlie bearer and its load. Also the floor joist at the lower end of bearers should be well 1>ridged, for whei'e cracks show in the plastering nine cases out of ten they will appear there first. In case the outer bearer be flitched with wrought iron at its center the scantling may be reduced in size. The proportion for the thickness of iron to the wooil being equal to one-twelfth. Suppose we try two white pine bearers 4^^ by 5^' flitched at the center, the thickness of flitch will equal one-twelfth the thick- ness of the two bearers (S^''), which equals ^^ of an inch, and for the depth we will allow 43^^^, or half an inch less than the depth of scantling to allow for shrinkage of timber and for the plaster to key. AYe will first find the safe load at the center for the wrought iron flitcli by Formula Xo. 1. Safe load at the center='*^'^^-^^^'^^"=748.1 pounds. 8X14.5X10,5 which equals say 749 lbs as the safe load for the iron flitch. We will next find the safe load at the center for one of the white pine bearers 4''''X5''', hy Formula No. 1. Safe load at the centers ^ — =2(59.3 pounds. 14.5X10.5 which equals 2G9 lbs as the safe load for one bearer. Then the two ))earers (2094-269=538 lbs), will equal 538 lbs, plus the safe load for the iron flitch (749 lbs), which (538+749= 1,287 lbs) equals 1,287 lbs and not deflect .03 of an inch per lineal foot. This over-runs the whole load [1,287—1,118=169 lbs] at the center of outer bearer 169 lbs, which will make up for the iron flitch, the weight of same being 139 lbs. The two bearers and flitch are to be bolted together, let the bolts pass through the center of the pine bearers, for the stress in the fibers of wood are less on that line, it being the neutral axis between the tensile and compressive stresses. At the upper ends the bearers should be well spiked to the trimmer, but on account of the extra weight the outer bearer must be supported l)y a stirrup iron, or a cast iron shoe bolted to the trinnner at the lower end, as shown at Fig. 9, Plate 23. The resistance of wrought iron to a tensile strain equals 60,000 lbs to the square inch, a factor of 6 is usually allowed for safety, hence 10.000 lbs per square inch is allowed as a safe working load for wrought iron. In this case then a stirrup made from a light iron bar, say k'^^Xli "will be more than enough to support the load at the lower end. The area of cross section for one side [V4'''X1''''=0.25] equals 0.25 square inches, plus the area of the other side (0.'i5-|- 0.25=0.50) equals for both sides 0.5 of a siiuare inch. Then 10,000 ll)s by 0.5 equals 5,000 lbs as the safe load for the stirrup iron. In using the light iron the ends that lap over the trimmer had better bo upset, thus increasing their stiffness. In addition to the stirrup iron the bearer can be well spiked to the trimmer, thus insuring good and substantial work. 133 Pi.ATK 29. PLATE 29. Plato 29. [Scale. M^^=l foot]. Exhibits a simple method h(/w to construct the face^mould for the stcvlr-case, Plate 25. Fig. 1. Shows the 23?a''>i of the center line of rail for the cyllncicr, Fig. 1, Plate 26. The rise is 7''^X'^^^^ tread, with 6^'' for a radius, and O as a center, draw the semi-circle as shown by the dotted line for the face of cyli'nder; the balusters are2^^X2^', and the bracket is \4^'' thick; the face of baluster is flush with the face of bracket, that will locate the center of baluster back from the face of front string %''''; the radius for the center line of rail will then equal (i%^^. Now with 6H^^ for a radius, and O as a center, draw the semi-circle ACE; through O draw the diameter AE. Draw the rectilineal parallelogram ABDE to tangent the curve at the points A, C and E. Prolong tangents JBA and DE indefinite towards 3 for direction of straight rail; draw OC parallel with AB for the two square parallelograms ABCO and EDCO on plan. The face of No. 15 and 18 rise is 5%^' from tlie spring of cylinder as shown. Pig. 2. Shows the length of tangents in elevation; the tj. 3, Plate 26. The dotted line indicates the face of cylinder, the radius being G'''. The solid line shows the center of rail, and is struck with a radius of (3%^^ from the center O. Through the center 0, draw AE indefinite, enclose the semi- circle ACE with the rectilineal parallelogram ABDE; draw OC at right angles to AE, and we have the two square parallelograms OABC m\A. OCDE on plan; now AB and ^Care the tangents for tlie inclining wreath-piece, and ED, DC are the tangents for the wreath-i)iece on the level; prolong BA and DE to llie left for the direction of straight rail. *Tlie face of first rise outside the cylinder is No. 14, being 3'^ from the spring line; the face of No. 15 rise will then extend into the cylinder 6^^. Fig. 9. Shous the elevation of tan(j€iits; in tliis case Oiey (trc fiihled similar to tliosc at Fig. 2, the shaded part shows the twist of raH. Let XX indicate the edge of drawing hoard. Make AB e(|ii:il AB, i''ig. S, ft)%^^]; draw AC and BD perpenJV equal J'C, Fig. 12. From iVand at right angles to QD prolonged draw NK; draw HP parallel to FD. Make J* 2 equal iif 2 on plan for the point of miter. Make Z/Tt qual FO, Fig. 12. Bevels. Let iS 5 indicate a gauge line parallel to XX. Perpendicular to XX draw SR and equal to HC, Fig. 12. Make (S 5 equal the hc'ujht LQ. Let iS 6 equal NK, draw R 5 and R t> prolonged to edge of board XX. Parallel to /S 5, draw the hall' width of rail (2J4^0« cutting the hypotheuuse of bevels at T and 8. Fig. 14. Shoivs the face-mould. At any convenient place on the paper draw J'P indefinite. Make FJ eciual DQ, Fig. 13. With J" as a center and MQ, Fig. 13, for a radius, draw arc at H. Again with JF'as a center and FH, Fig. 12, for a radius, draw arc intersecting at .H"; draw i^iiZ" pro- longed to equal H2 on plan. Fig. 12. Draw HO and JO parallel to JVand FH for the parallelogram OHFJ on the cutting plane, which will coincide when in position with the parallelogram OHFJ on pM\, Fig. 12. Proof. The diagonal FO must equal the distance TQ, Fig. 13. If so, the angle of tangents at F must be correct. Make JTPetiual 6" for length of shank, make Joints at 2 and P at right angles to the tangents F 2 and FP. From iif and at right angles to FH draw a line indefinite. Prolong the diagonal FO to intersect the line from H\ from the intersection (not shown) draw the radial line through J^ prolonged for the points of contact, or the connection of straight with the curved part of mould. Make P C and P 13 each equal 6 7, Fig. 13. Make H4 and H 3 each equal 5 8, Fig. 13; draw 13 7 and 6 8 parallel to JP; draw 3 1 1 and 4 12 parallel to H 2. Now the intersection (not shown) is the point to pivot the trammel, and from the inter- section to the point 3 is the semi-major axis for the concave side of mould, and the minor axis would be at right :inglcs. but on account of the extreme length of the axis as shown at J, Fig. 13, Plate 17, it would be attended with too much trouble. A better way to use the ordinates as jireviously explained, and thus find all the points in the curve that may be desired. Or points may be found on the diagonal FO", through wliich the curves may be drawn by using the pliable 'Tlit'fie points on the diagonal sive another point in each curve u \\(\ :iiiswcrs very well for small cylinders. But for tlie large cylinders and for face-moulds over winders more points in the curves are retjuiied for a correct trace of the face-mould. These three points iiiiiy be found on the proof diagonal in elevation for any wreath- l)iece as sliown by the dots on the line YZ, Fig. 2, Plate 32, for all wreatli-picces standing over a quarter circle on plan; and for all wreatli pieces less than a quarter circle the points are found on the proof diagonal as shown at Fig. 4, Plate 24. 1!^8 I*TATR 29. strip. Tlius return to the elevation Fig. 13, draw TQ, make L 3 equal 5, Fig. 13, draw ?, 4 parallel to LQ. Now return to Fig 14, nialve 5 equal Q 4, Fig. 13, and draw tlie proportional line H 5, draw 4 10 parallel to JffS, make 5 9 equal 5 10. Xow williaplia- l)le strip, draw the curve 7, 9, .", for tlie concave side of mould and through the points 8, 10, 4, draw the curve for the convex sidp of mould, and tlirough the points H 5, J, draw the center line of mould. At section N the bevel foinid in the angle at 5, Fig, 13, is shown applied from the face of crook through the center of plank. At section D the bevel found in the angle at 6, Fig. 13, is shown applied tlnough the center of planlc, and the block pat- tern is applied square to the line made from the bevel, thus show- ing the twist of wreath-piece in the crook. Observe the bevels do not cross the tangents in their application. The shading shows tlie surplus wood to be removed, and also the thickness and width of plank required to saw out the crook. Fig. 15. Shows the easement pattern for the landing of tlie ftrst flight. IIov) to marTi the point Fan tlie pattern so as to cut and joint the rail at the bench. Draw a tread and rise and floor line anywhere on the draft- board; through the center of baluster XX, draw the under side of rail. Then take the easement pattern already made, and apply the lower edge to the inclination through XX, as shown; now slide the pattern until the lower edge at AB will equal the distance XQ, [4^^J Fig. 5 ; then mark the pattern opposite the landing rise at center of rail, as shown at F, and we have 5'''' to the joint at Con the inclination to allow when cutting the straight rail. And from F to J3, parallel with the floor line, we have 'M}i^^ to allow for the easement when cutting the straight rail for the level. At 1 is shown how much longer the long baluster is tlian the regular short baluster for the first flight; 4 5 shows how much longer the balusters for the level are to be over the regular short balusters; at 2 and 3 the difference over the regular short balusten-s is shown for these two odd balusters. Figs. 8, 9 and 10, Plate 28. Shoivs how to construct the face-mould over a quaHer pace xvindinrj, so as to form two e'lsinfjs in the wreath-piece, thus avoidimj the short ramp usually at the lov)er end of wreath- piece. Fig. 8. Shows the plan of quarter cylinder having three winders in the quarter pace; AB and BC are the tangents en- closing the center line of rail; AD and CB indicate the direction of straight rail; OABC shows the parallelogram on plan. Fig. 9. tiliows the elevation of ta7igents. Let XX indicate the edge of drawing board. Make AB '^ equal AB, Fig. 8. Draw AC and BD perpendicular to XX and of indelinite length. Now elevate the treads and risers, keeping the face of No. 9 rise 2^^ and also the face of No. 13 rise 10^^ from the spring line AC; through the center of baluster 00 and 00 draw the under side of rail; parallel with the under side of rail, draw the cenler of rail, intersecting AC at E, and at the upper end prolong the inclination indefinite, cutting BD at H: draw HE prolonged indefinite; perpendicular to AF draw FD to cut EH prolonged at Gr; make EJ perpendicular to AC; make EK equal the chord AC, Fig. 8; make FL equal twice DH: parallel with BD, draw the half width of rail, cutting EH and FH at 2 and 3; ease the angle at E; at right angles to ME, and at the spring of easing, draw the joint line 4 5, cutting the center of straight rail at 6, and also £rJ7 prolonged at 7; parallel Pt.aTR 29. isy with HE prolonged, draw the half thickness of plank, cutting the joint line 4 7 at 5; from 5, and perpendicular to i?^ prolonged, draw 5 8, cutting HE prolonged at N. Now NE is the length required for the shank of face-mould at the lower end; the shank of mould at the upper end may be any length, as the wreath- piece contains a natural easing. Bevels. From the center D, draw a line to tangent HG, and intersect HB at P; join PF; the angle at P gives the bevel for the lower end of wreath-piece; again, from t!ie center J draw a line to tangent FH prolonged, and intersect BD at Q; join QE for the bevel required at the upper end of wreath-piece. The bevel shown in the angle at 5, applies from the joint after the sides of wreath-piece are worked off to the plumb. Pig. 10. Sliovs the face-mould. Make NEH eqiml NEH Fig, 9; with S as a center, and the distance FK. Fig. 9. for a radius, draw arc at J"; again, with if for a center, and FH, Fig. 9, for a radius, draw arc, cutting at F; chaw JTH" prolonged; parallel with EH and HF, draw FO and EO for the parallelogram OFHE on the cutting plane. Proof. The diagonal HO must equal the distance LK, Fig. 9, if so, the angle of tangents at JFfmust be correct. Make EG equal HG, Fig. 9; draw GO indefinite for the direction of minor axis; make OC equal OC on plan, Fig. 9; make C2 and C3 each equal the half width of rail \l}4^^]\ make E 4 and E 5 each equal H 3, Fig. 9; make F 6 and F 7 each equal H 2, Fig. 9; make joints at N and S perpendic- ular to the tangents HE and HF; from the points 6, 7, and 5, 4. draw lines for the straight wood parallel to the tangents HS and HN to intersect the joints at 8, 9, and 10, 11; now draw the curves of face-mould in the usual way. The bevel found in the angle at Q, Fig. 9, is shown applied at section B; and the bevel found in the angle at P, Fig. 9, is shown applied at section A; the block pattern is shown applied in the usual way. Now work off the sides of wreath- piece to the plumb, then through the center of rail section draw ab as shown at section A; from a and b draw lines on the sides of wreath-piece square to the joint, then mark on these lines from tlie joint the distance N 7, Fig. 9. Now apply the bevel shown in the angle at .5, Fig. 9, from the joint and through the points just found on the sides of wreath-piece which will give the cripple joint 5 4, Fig. 9: a section of this joint is shown at D. the points 7 and 6 cor- respond to the two centers 7 and 6, Fig. 9; after the cripple joint is cut, then carry the line ab across the joint, and also the plumb line made from the i)evel; raise up the distance 7 G, Fig. 9, for tlie center of block pattetn, and shape the easing square from the joirit last made; the points 7 and correspond to the two centers 7 and 6, Fig. 9. Observe a piece may be glued on the top of wreath-piece to accommodate the easing; the crook must be sawn out wider on the concave side to allow the block pattern to raise up the required height, 7 6, Fig. 9. This method of forming the unnatural or forced easing in the wreath-piece at the lower end is not preferred by the author; in a case of this kind, when the inclination of pitches is less steep then an easing may be worked on the shank of wreath-piece and carried into the twist part with good results. At Fig. 9, the parallelogram FHWZ is shown projected from the tangent FHm elevation. Some prefer in this way to find the angle of tangents on the oitting jilane, then transfer them to the material for pattern by using a bevel. 140 Plate SO. PLATE 30. Plate 30. [Scale M^^=l foot]. Exhibits how to cotistruct the face-mould ivhen. the risers are placed at any point in the cylinder without reference to what effect they viay luwe on the wreath part of rail as is sometimes the case, ivith ordered rails, and also in the capping of iron balustrades. Pigs. 1 and 2. Shoio how to Und the length of tangents and treatment of the wreath over a cylinder 12" in diameter start- ing from the level to a rake, rvhen the face of No. 1 rise is 1" outside the spring of cylinder, ihe balusters are2''^2", bracket %" thick. Then the radius for the center line of rail will equal 6%'\ The rise is 8" by d" tread. The size of rail is 4}4" by 2%" . The rail is to be a "■right-hand rail." Fig. 1. Shows the plan. From the center O draw the semi-circle ACE io a radius of 6%'''' for the center line of rail. Enclose the semicircle "with the rectilineal parallelogram ABDE, draw OC, forming, the two square parallelograms OABC and OCDE. Prolong tangents BA and DEto the right for the direction of straight string. The dotted line shows the face of outer string. The face of No. 1 rise is one {1") inch outside the spring of cjiinder. Fig. 2. Shows the elevation of tangents, as developed from plan, they are folded. Let XX indicate the edge of drawing board, make AB equal AB on plan, Fig. 1, (G%"}, draw AE and BD perpendicular to XX, elevate No. 1 and 2 rise and tread, keeping the face of No. 1 rise 1" from the spring line AE. At B set up 4^^ to the under- side of rail plus the half thickness of rail riys^''+4'''=5^<^^) equals 5%^''to the center of rail as BC; draw CF parallel to XX. Through the center of baluster 00 draw the underside of rail, parallel with 00 draw the inclination for the center line of rail to intersect AE at h. Let it be observed that if the center line of rail Gh, were prolonged to intersect the perpendicular BD, it would fall below the point C, and as the point C is a fixed point, we will have to assume an inclination for the wreath-piece different from the reg- ular pitch and make what is termed a "cripple joint" at tlie shank end of mould. Then draw CJ for the assumed pitch, cutting AE at K. Tlie point J" is not arl)itrary, the stair-builder can use his judgment as to what inclination or length to make the shank JK. At C is sliown a section of rail. C 2 shows the increased witltli of mould for half the rail at the center joint. In the angle at K is shown the bevel for the center joint. At the shank joint of mould the square will be applied because there is no spring in tliis case. The joint at J is made at right angles to the true jiitch, and the bevel shown at J" gives the cut for tlie cripple joint as shown. Tlie shaded part shows the curve of wreath-piece flowing into the straight rail. In the hands of a skillful stair-builder this treat- ment of the wreath-piece will give a beautiful twist as the curve is carried more into tiie shank. The dotted lines drawn parallel with CJ show the thickness (5") of plank required, less thickness would answer by taking off the corners as shown at section N, Fig. 5 , for they will come off in moulding the wreath-piece. Plate 30. 141 The face-mould for this wreath-piece is shown at Fig. 5. Figs. 3 and 4. Repeats Figs. 1 and 2, only the former is shown for a wreath-piece landing, or from the rake to a level. The face of No. 1(5 rise is one inch (f) outside the spring of cyl- inder. The tangents AB, BC, CD and D.E7 coincide with those at plan, Fig. 1. Fig. 4. Shoivs the elevation of tcmgents, they hcing folded. Let XX indicate the edge of draught board, make AB equal AB, J-jg. 3. Draw AE and BD perpendicular to XX. Elevate No. 15 and 16 treads and risers, also floor line, keeping the face of No. 16 rise 1^' from the spring line AE. Draw the floor line OS. Through the center of baluster 00 draw the inclination for the underside of rail, parallel with 00 draw the center line of rail intersecting AE at h. At the floor line /S set up 4'^ to the under- side of rail and IM^^ more or o%^^ to the center C, then Cis a fixed point. Draw Cr parallel to XX, cutting AE at W. Now let it be observed that if the true inclination Gh were prolonged to intersect BD, it would raise above the fixed point C, which would be too high, so we will make a cripple joint at the shank same as at Fig. 2. Now in drawing this, to save time draw the assumed pitch to agree with that at Fig. 2, and make one face-mould answer for both wreath-pieces. Then make CF equal FK, Fig. 2. draw J^jff parallel to XX, draw CK prolonged to intersect GH at J, make the joint at J at right angles to GJ. The bevel in the angle at J gives the down cut for tlie cripple joint, the bevel in the angle at Cis applied at the center joint. Parallel with BD draw the half width of rail, cutting CK at 2. At C is shown a square section of rail and the dotted lines cutting the angles of block pattern are drawn parallel to JC and give the thickness of plank (4K^^'i required to form the twist. Prolong if2^ to equal the chord AC, Fig. o, as KU. Fig. 5. Shows the face-mould for both starting and land- ing. At any convenient place on the paper draw a line to equal CKJ, Fig. 4, as BAJ. At right angles to JB draw AO and BC each equal to AB, Fig. 3. Join CO for the rectilineal parallelogram OABC on the cutting plane. Proof. The diagonal OB aiul chord AC must each equal the distance UW, Fig. 4. If so, the parallelogram is correct. As all the angles are right angles, the face-mould will be a quarter of an ellipse with the joints on the axis lines. Make C 2 and C 3 each equal C2. Fig. 4, make A 4 and A 5 each equal half width of rail i'~}4), draw 4 6 and 5 7 parallel to AJ", make joint at J perpendicular to A J. The dotted line at J" indicates the overwood required for making the cripple joint. Pivot the trammel at the point Oand sweep the quarter ellipse in the usual way. The section N shows the bevel found in the angle at C, Fig. 4, applied from the face of crook. At section D the try square is shown applied. The shaded part shows the thick • ness of plank and the width at joints to saw out the crooks. The center point at section D should equal from the face of crook the distance 4 J, Fig. 4, and at section N the distance 3 C. Fig. 4, from the upper side of crook. Make J 8 equal Jh, Fig. 4, then J 8 will be the amount to deduct for shank when cutting the straight rail. In the two elevations, Fijrs. 2 and 1, the shader) parts show thf> t\YO wreath pieces, the one starting, and the other for a landinir; it •will be observed if they were bolted togetlicr at the center joint C. they would answer for a platform twist. The quarter turn connectini: the wreath-piece, is taken from plank the same thickness as for the straight rail, there being no spring bevel required at the shank of quarter turn. 142 Plate 30. Figs. 6 to 10. Shows a different treatmeyit of a landing and stai'ting twist. Fig. 6. Shows the plan, starting from the level to a rake similar to Fiq. 1. Ouly in this case the face of rise is placed on line with the diameter or spring line of cylinder, the rise is S'^'X^''^ tread, the tangents AB, BC, CD and DE, coincide with Fig. 1. Fig. 7. Shows the elevation; the tangents arc folded. Allow BA prolonged to indicate a base line; make BA equal tangent AB on plan, Fig. 6; draw AG and JBJ' perpendic- ular to BA. Now elevate No. 1 and 3 rise and tread, keeping the face of No. 1 rise on the spring line; from the base or floor line BA, set up V^ to the under side of rail plus the lialf thick- ness \l%^^\ of rail, \l%''+'^''=h%''] equals b%'^ to L. Draw LK for the center of level rail, ttms establishing the point L; through the center of balusters 00, draw the inclination for the under side of rail; parallel with 00, draw the center line of rail, intersecting AG at J". From L, and parallel to 00, draw Lm. (In tills case, Lm happens to come on line witli the upper side of rail.) The block pattern shows the section of rail at L. Make shank joint at pleasure, say at 2, and perpendicular to 00. The dotted lines show the thickness of plank required to foi'Tii llie twist, the thickness of rail bciiis? tr;iuKf'd oil' from the lower side, ajid all the over wood is to he removed from the top side of crook at the shank; 1/ 4 is the distance from the lower side of crook to center the block pattern ati. The bevel for tlie center joint is shown in the angh! atrn; as the plank in this case is canted but one way, only one bevel will be required; the try square will apply at the shank joint; the face-mould is shown at Fig. 10. Fig. 8. Shows the 2^toH from the rake to a level, sliailar to Fig. 3. Only the face of No. 16 rise is on line with the spring of cyl- inder. The tangents AB. BC, CD and DE, and also the tread and rise agree with Fig. 6. Fig. 9. Shoivs the elevation of tangents from plan. Fig. s, they arc folded. Let BA prolonged, indicate a base line or line of floor; per- pendicular to BA, draw BF and AG; elevate No. 15 and 10 rise and tread, keeping the face of No. 10 rise ou the spring line AG. Tlirough the center of balusters 00 draw the inclination for the under bide of rail, parallel with 00 draw the center ot rail intersecting AG at J; also draw the upjier side of rail /S :», to intersect the perpendicular from B at F. From the Hooi- line set up i^^ to the underside of rail plus the half thickness (I;?h^^) of rail (4^^+l>^8^^=-53^'/) ciuals .^,?h^^ U> L. Draw I,if ))r.>h.nged and parallel to BA, draw LM ))aralhl to SF; draw MN at right angles to AG; parallel to JBi^ draw the half width ot rail (^'i'^), cutting /S^ at X. The block pattern is shown centered at L, the dotted line shows the thickness (»f plank reiiuired for the wreath piece. [-1%^^.] Make shank joint at ])leasure, say ii 2. Let L U equal the chord AC on plan. Fig. 8. In the angle at L is shown the bevel for the center joint. Fig. 10. Shows the facc-mouhl. Make BA and AJ vi\WA\ FZ and Z 3 Fig. 9. From A and B and i>eri)endicular to jB draw BC and AO each equal t(» OC, Fig. 8, for the rectilineal parallelogram OABC on the cutting plane or plane of plank. Plate 30. 143 Proof. The diagonal BO and chord AC must be equal and of the same length as NU, Fig. 9. Make joint at J at right angles to tangent BJ. Let C 2 and C 3 each equal Fx, Fig. 9. Also make A 4 and A 5 each equal the half width of rail (2)^'^). Draw 4 6 and 5 7 parallel to AJ. Now pivot the trammel in aud trace the curves for the face-mould as has been explained, riie joint at C is ou liue with the major axis OC in this case because the plank is not sprung and the angles of the parallelo- gram on plan aud face-mould are all right angles. The section at N show's the tangent BC carried across the joint square to the face of plank and the dotted line run on from the face of crook equal to the distance L 4, Fig. 9. The bevel found in the augle atL, Fig. 9. is applied through the intersection. At section D the full thickness of rail is shown gauged from the face or upper side of crook and the surplus wood at the shauk is removed from the under side of crook. At elevation Fig. 7, tlie surplus wood at the shank is shown gauged from the under side of crook, and most of the surplus wood is removed from the top. It will l)o o1xsorv(>d that if the wrf ath-picce for Fior. 8 were bolted to tiiat of t'^ig. tit, it ■would answer for a platform twist, the face of risers hoins on line with the spring of cylinder. KJ is the correct height of wreath-piece from center to center of rail. Make J 9 on face-mould equal J 2 in elevation Fig. 9, for the amount to be deducted tor shank when jointing the straiglit rail. The quarter turns connecting the wreath-pieces are level as described for Fig. 5. Figs. 11, 12 and 13. Shows a third inctlwd how to obtain the face-mould when the risers are misplaced. Fig. 11. Shows the plan and a,qrecs icith Fi(j. 8; the face of No. 10 or landiiKj, rise is on line with the spring of cylinder. With (yyi^^ as a radius and O for a center draw tlie semicircle ACE. The doited line indicates the face of cylinder. Draw the diameter AOE, draw the direction of straight rail from A aud J5 at right angles to AE, Fig. 12. Shows the elevation of a step and rise and incli- nation of rail from ivhich to determine the length of tangents on the plan. Tlie rise is S^^ hy d" tread. Ijct XX iiulicate tlie edge of draught board, elevate No. 1.5 and 16 treatl and rise ami the lloor line at the lauding. From A draw the spring line AE indefinite. Througlx the center of balusters 00 draw the under side of rail, parallel with OO draw tlie center line of rail RJ and prolonged indefinite. From the lloor line set up 4^'' to the under side of rail and the half thick- ness of rail l-'a^^niore, or 5^tj^^tothe center of rail at B through B, draw a line parallel to the floor line, cutting the inclination for the center of rail at Cand prolonged indefinite. From Cdraw CD parallel to AB. Now return to ])lan, Fig. 11, draw tangent AD indefinite. Make AD equal AD, Fig. 12. Make UJ' equal AD and to tan- gent the bcmiciiele at F. Draw J^O for the joint on plan, then the wreaih-pieee will cover so much of the plan that lays between A and the joint at F. From F to E will be level, the solid lines bhows the pattern for the same, it being a little over a ((uarter of a circl". raialiel with tangents DA and DF draw FH and AH, From i^and at right angles to AD prolonged draw FG, draw the diagonal OD. Now return to the elevation Fig. 12, draw JM parallel to XX and prolonged to the left. Make ilfiV equal the chord A J', Fig. 144 Plate 30. 11. Make CK equal DG, Fig 11, make CQ equal DH, Fig. 11: also make CL equal DO on plau. Fig. 11; draw MQ prolonged, cutting the perpendicular from L at U. Fjom K and at right angles to RC draw KP. Bevels. The dotted line 2 4 is drawn parallel to and at any distance from the edge of l)oard XX\ draw 2 5 perpendicular to XJS. and equal to GF, Fig. 11. Make 2 3 equal KP, also make 2 4 equal the height MC, draw 5 3 and 5 4 to edge of board and the angles at 3 and 4 give the bevels required. Parallel with 2 4 draw the half width of rail, cutting the hypotheuuse of bevels at 7 and 8. Fig. 13. Shows the face-would. Make DA e(iual JC, Fig. 12; with A for a center, and the distance NC, Fig. 12. as a radius, draw arc at P; again, with D as a center, and DP, Fig. 11, for a radius, draw arc intersecting at F; join DP; parallel witli DA and DP, draw PH and AH for the parallelogram HADP on the cutiiug plane, that will agree when in position, with the parallelogram HADP on plan. Fig. U. Proof. The diagonal DH must equal the distance MQ, Fig. 12. Prolong DA (V to J for length of shank; make joint at J" at right angles to JD; make joint at F perpendicular to DF; pro- long joint at P indefinite; draw the diagonal DH io intersect the joint line from F at O; draw OA prolonged for the trapezium OADF on the cutting plane, that will coincide with the trape- zium OADP on plan. Fig. 11; proof: the diagonal DO must equal the distance MU, Fig. 12, if so, the quodrilateral is correct. Draw DC perpendicular to OP; make OC equal the radius OA, Fig, 11: then O is the center to pivot the trammel, and OP is the semi-major axis, and OA the semi-minor axis. Make J^ 2 and J?'3 each equal 4 8, Fig. 12; make J 6 and J 1 each equal 3 7, Fig. 12: draw 6 4 and 6 5 parallel with AJ; make C8 and Ctf each equal the half width of rail \^2,W]. Now pivot the trammel at O, with the arms at right angles to the minor axis OC then set from pencil to minor pin the distance O 8, and from pencil to major pin the distance O 2; now trace the curve from 2 to 4 for the concave side of mould; proceed in like manner to trace the convex and center line of mould. At section B the bevel shown in the angle at 3, Fig. 12, is applied from the face of crook; at section iVthe bevel shown in the angle at 4, is applied from the face of crook. Observe the bevels do not cross the tangents in their application, because the minor axis is not within the parallelogram, as shown. Fig. 14. Shows how to ylacc the risers in the cylinder m as to hai'e the rail the T)ropcr hei/jht at the landing, and make the torcath-piece ivithout nviy spring or Joint bevel at the shank; thus canting the plnnk only one way. This gives the most simple method to construct the face- mould; the rise is S^\ and tread 9'^ the elevation is first drawn, then the plan. ' liCt XX indicate the edge of drawing board; draw BP at right angles to XX; any where on the line BP, place the rise of pitchboard, and draw the inclination CJ" indefinite for the center of rail. Parallel with CJ" draw the underside of rail; at C, draw a section of rail,frora the under side of rail section, measure down half a rise [4''''] to the floor line V; draw the floor line parallel to XX, cutting the under side of rail at O, then O is the center of short baluster, and the baluaters are 2'^X2'^ then the face of No. Plate 30. 145 IG rise will be V^ totho left of O, and No. 15 rise will be ^" to the left of that again. Now cletermiue the radius for the cylinder say 6*", then the radius for the center line of rail will equal 6%''''. Make JBA equal ^",i"\ perpendicular to XX, draw AE, cuttinK JC at H; draw HM prolonged, and parallel to XX; draw CK parallel to XX, and of indefinite length for the direction of level rail. Fig. 15. Shows the plan. Draw the two right angles A OC and CO^ indefinite; make OC e(iual AB, Fig. 14; with OC [6%^^] as a radius, draw the semi-circle ACS; draw the tangents AB, BC, CD and DE. Prolong tangent BA and DE to the left for the direction of straight rail; place the face of No. 15 and 16 rise to agree with J isers in elevation, relative to the spring of cylinder; the face of No. 15 rise is located 7}^^^ from the diameter, then the face of No. 16 rise will be in the cylinder IJa^''. Keturn to Fig. 14, make HG ecpial the chord AC, Fig. 1.5: parallel with BF, draw the half width of rail [2 3^^']. cutting ffC at 3; tlie dotted line shows the thickness of plank [4'^] required for the wreath-piece. Fig. 16. Shoivs the face-mould, and is drawn in the same manner as at Fig. 5 and 10. Observe at sections N and D the block pattern is applied at the center of plank in this case; the bevel for section N is found in the angle at C, Fig. 14; there being no spring in this case, the try square is applied at the shank joint section D. Figs. 17 and 18. Shows the same principal as at Fig. 14 and 15 applied to a platform twist, the manner to find the location of risers in the cylinder, and the construction of face-mould repeats, and in this case is the same as shown at Fig. 16, the let- tering being the same. Let it lie noticed tlio face of No. 15 rise is T'/j" from the spring- of cyliiidor and the face of No. IS rise- is 0'^" from the spring line, the difF«>rence being eaual tohalf tlie thickness of balustei-. This method allows in this case two more balusters (4) to he placed in tlie cylinder on the platform. Also observe at section N, Fi.s,^ 1(5, that if the wrey.th-piece were turned upside down it would suit for the wreath- piece starting off the platform. This melliod of placing the risers in the cylinder makes the con- struction of face-mould and formation of the wreath an easy matter. One fault is in a "level to rake," and "rake to a level" twist, the joining of the wreath-piece with the level auarter turn at the center join t in small cylinders does not please the eye so well as wheu both the raking and level wreath pieces are sprung at the shanks, then the "helix " or twist line, in passing from V to E, Fig. 15, gradually lowers into the straight part of rail resulting in a graceful curve pleasing to the eye. Of course by using a spring bevel more work is required, and in cheap work the easiest method is mostly resorted to. The objective feature referred to above may bo remedied by cutting thelevel crook from stuff halt an incli thicker than the rail, and raising the (wist strip so as to c:u-ry the curve gradually from the rake to the level, and taking tlie surplus wood off at the top at the center jouit, and off from the undeiside at shank for the landing twist, and for thelevel part of twist starting the block jjattern is applied so as to take the surplus wood from the underside at the center joint, and froTn the upper side at the shank as shown at Figs. 22 and 23. This would increase the length of balusters V^" at each turn, but may be counter- acted by raising- or lowering the block pattern at the shank of the raking wreath piece, as shown at section JD, Fig. 10. As the center joint is plumb this treatment of the above wreath-pieces makes the center joint a splice joint and thereby detracts from its appearance to some e.xtcut. 1^6 Plate 30. Pig. 19. Shows the plan of a 12" cylinder for a platform stairs having two different pitcJies; the rise to the platform is 8"X^" tread, and off the platform the rise is 6"y(J2" tread; this of lea happens i7i a stairs goiny to the attic, where thefliglit off ihc platform has to he constructed to give head room at the landing under the rafters. The method here shown is to construct the face-mould so that one pattern and one sit of bevels will answer for both wreath- in c CCS.* AB, BC, CD and DE are the tangents for the center of rail on plan; the face of No. S rise is ^", and the face of No. 11 rise is lli" from the joint or spring of cylinder; No. 9 and 10 rise is iu the cylinder. Pig. 20. Shows tlie elevation of tangents, they being folded. Let XX represent the edge of drawing board; make AB equal AB, Fig. 19; perpendicular to XX, draw AE and BD; elevate Nos. 8, 9, 10 and 11 rise and treada, keeping the face of No. 8 rise b", and No, 11 rise 7}4" from the spring line AEj as shown. Through the center of balusters 00,00, draw the inclination of the under side of rail for both inclinations; parallel with 00, OO, draw the center of rail, intersecting at F; bisect the two pitches at G, and draw GF indetinite, cutting AE and BD at C and H; perpendicular to GF, and through fi"and C. draw JHP and LCM, cutting the center of rail FN at i-*and Q. Perpen- dicular to LM. draw QR: join CR and produced. At right angles to JP, draw PTe(iual to the chord AC, Fitr, 19; from C. and at right angles to FN. draw C 2; again, from P, and perpen-- dicular to CR produced, draw PZ; parallel with ML, draw the half width of rail, cutting CR and QP at 4 and 5 for the increased width of mould on the radial lines. Bevels. The dotted line Z7 Vindicates a gauge line run on parallel to edge of board XX; draw UW jx-rpendicular to XX, and equal to HC, Fig. 20; make Z70 equal C2, and Ul equal JP 3; draw W6 and W7 prolonged to edge of board. Then in the angle TJQ'W, is found the bevel for the shank joint, and the angle Z77W gives the bevel for the center joint. Pig. 21. Shows the face mould. At any convenient place on the pattern pai)erdraw SJ" indef- inite, make BA and AJeach eciual QP and PN, Fig. 30. Then with A as a center and TH, Fig. 20, for a radius, draw arc at C. Again with B as a cent( r and RC, Fig. 20, as a radius, draw arc intersecting at C, join BC. I'arallel with the tangents AS and BC draw CO and AO prolonged at A and C for the parallelo- gram OACB on the catling plane. ISIake AD equal QF, Fig- 20. .Toin DO and piolonged for the direction of the minor axis, then at right angles to DO and through O draw the direction of the major axis EF. Make O 2 e.i^O equals 15.5K inches, which reduced to feet (155K''^-^12=:12' 11>2^0 equals 12^ ll>a'^ for the horizontal run of the treads to tlie platform. Then the width of platform to the face of No. 16 ri.se will equal IS'' ^" minus 12^ \\)4," (18' ^"—W ny/^=5^ oy/') or 5' OK''' as sliowu on plan. Again, the thickness of rise equals V^ and allow 13^'' for uneven walls, the platform may be framed 4'' IC (o' oy^^ — 1''^4- iy^^=2}4'^)=i' lO'O for the neat width. The flight off the platforni has 5 risers and 4 full treads at 10''' each, then there should be 6" from the last rise to the face of joist at the landiug, so that the bearers may have good nailing against the joist. The width of platform from the face of No. l(j rise to wall equals 5' 03<". the distance then from the wall to face of joist at the landing A, (4X10"+5' 0>.i"=8' 10>i'O equals 8'' ioy, as shown on plan. Headway. For the headway count down from the landing, s;iy 16 risers at 7'-' each minus the width of joist 10''' in the secoiul story, thickness of floor 1" and 1" for the plastering (1 OX'' ^^= 113)— 10"+1"J-1":^100"-T-12"=8' 4") equals 8' 4" plumb over and includin,' No. 6 rise. In this case No. 6 rise should be deducted from b' 4", leaving 7' 9" for the head-room. So we will locate the face of trimmer plumb over the center of No. 5 step. This will give ample head-room. Thoi from the back of platform to the face of trimmer marked Cfatthe headway will equal 5' 03^" from the back of platform to the face of No. 16 rise plus 10 treads at 10" each .5' OK'^ + (10X10"=100")^12 =8' 4"+.5"+5' 03^"=13' 9M") equals 13' 93^".) And the length of quarter pace landing marked AJSon i)lau will equal 13' 934" minus 8' 103^" U3' 93-^"— 8' 10>^"=4' 11"), or 4' 11" for the length of trnnming at the landing, and the width 08 qiuirter puce will e lual the width of hall (10' 3") minus the half width of hall (."/ 13<") plus the radius of silver 12" and the thickness of fascia (132^0 «r outer string (10' 3"— [.5' 13'.<"^- vy'-\-i)4'^=-P,^ ^A' 0") equals 4' 0" for the width of landingfrom thorough wall to the face of header marked C, on plan. Fig. 1. This completes the trimming for the well of staircase. Tho joist at the landing should be well bridged and spiked to prevent sagging; if the walls are brick, the platform should be framed and l)uilt in when the vi'alls are going iq), by leaving a space over each joist, the i)latform may be levciled u]) when put- ting up the stairs. At the terminus of the lail, a block should bo built ill the wall ?/ 8" from the floor to its center, for to fasten the rosette and rail; if stud partitions, see that the block is set far enough back for the plastering to key. At h, h, are .shov/n the rough brackets underneath the stairs cut and nailed to 3''V4" bearers, liaving the grain i)erpendicular to the treads. At H is .shown the panel work under the lirst flight, having an arched opening into a coat closet or wash-stand; tho bize and variety of panels may bo arranged to suit the taste of architect or stair-builder. PrATR SI, 149 Eig. 3. [Scale X''=\ foot]. Shows the plan of cylinder 24^^ in diameter. The risers Nos. 15, 16, 17 and 18 are spaced off on the stretch- out AD, and are drawn into the cylinder at L and N to suit the lx)sition of balusters in the same manner as explained at Plate 15, which should be well studied. From D, on the stretchout, to F, equals half a tread (r/'), and is tlie face of No. 10 rise; and fronx F to K, and K to G, each equal to a full tread (IC), and are the face of Xo. 14 and 15 risers, thus establishing the face of No. 14 rise 6^^ from the spring line AC, as shown. Draw No. 17, IS and 19 rise opposite to No. Ifl, 15 and 14 rise, and locate the short balusters on No. 14 and V.i treads; then space off the intervening balusters equally, and curve the face of risers No. 16 and 17 to suit the balusters as shown. The outer solid line ABC shows the face of cylinder; the outer dotted line, indicates the center line of rail 3*i'^ from the cylinder line; the inner dotted line shows the thickness of braclcet {^i'^), and the inner solid line 1, 2, 3, 4, shows the projection o/' nosing, (IK^O- Fig. 4. Shown the semi-rirclc ABC, divided off into an odd number of staves (11); the chord or tlie segment for No. 1 stave is extended to the edge DE, of drawing board; for a convenient means to adjust the bevel F, a greater or less number of staves may be used, as the stair-builder may desire; this ^vi^lth of staves answers very well for the 'treads, and is intended to be made from stuff IK^^ thick. At Fig. 1, Plate 33, is shown how to obtain the length of staves. Figs. 5 and 6. [Scale 1}4^'^V\. Shows the convex and concave risers. The end of risers are slit, forming a veneer CC, about ^/^ thick, steamed, bent, glued and clamped to the cores A and B, for the proper curve; "cauls" are clamped down over the veneer to press them close and even to the core. Fig. 7. Shoivs plan of the turnout at the neivel, divided into three staves, and spliced to the outer string; then screwed from the back. The face of No. 4 rise is 5^^ from the spring of cylinder, the radius is 2^ Wi'^ and should be drawn on heavy paper, as shown at Fig. 9, Plate 32. The bevel for jointing the staves is shown at A. The cut-out for No. 14 rise is ^%" from the joint of cylinder; at No. 19 rise the cut-out of outer string is Q\i'^ from the joint of cylinder; the width of string at the internal angle of thread and rise is 6^''. The jointing of the staves, glueing and splicing to the strings has been described under a 1^' cylinder. Also preparing the steps and stepping up is the same as has been explained for a 7^^ and 13^^ cylinder, at Plates 23 and 35. Measuring for jointing the rail at the bench. The first length [11' 6''^], for jointing the rail, is shown taken from spring of turnout to the spring of cylinder; the second length [3'' 8''J is taken from the spring of cylinder to the face of last rise landing; the third length [4' ^y/'] is taken from the face of last rise landing to the spring of quarter cylinder; and the fourth and last length [5' IK^^] is taken from the spring of quarter cyl- inder to wall. All these lengths, and also the dimensions of pitch-board [7''X10^^]i should be entered in the order hook.' is^ Pr.ATE 32. PLATE 32. Plate 32. [Scale X^^=V]. ExMUts the construction of face-moulds for a stair-ease, Plate 31. Fig. 1. Shows the plan of the center line of rail, and the tangents for the same. Draw the diameter line AE, and the radius OC, indefiuito, and at right angles to each other; then witli 12yi^^ for a radius, and O as a center, draw the semi-circle ACE, for the center of rail; the dotted line shows the face of cylinder or outer string line. Enclose the semi-circle with the rectilineal parallelogram to tangent the semi-circle at the points A, C and E, and we have the two square parallelograms OABC and OCDE. on plan. Prolong tangents BA and DE, to the right for the direction of straight rail. The position of risers in the semi-circle are the same as at Fig. 3. Plate 31; the face of No. 14 and 19 rise is iV^ from the spring of cylinder. The balusters No. 1, 2, 3, &c,, are spaced olf on the center line of rail and drawn normal to the curve at their centers. Fig. 2. Shows the elevation of tangents; they are nn folded. Let XX indicate the edge of drawing board, Make AJB* BC, CD and DE, each equal AB, BC, CD ainl DE on j.lan, Fig. 1. Perpendicular to XX, draw AF, BG, CH, DJ and i7ir indefinite. Now elevate Nos. 1.", 14, 15, Ki, 17, 18. 19 and 20 risers and treads, keeping the face of No. 14 and 19 rise 6^^ from the spring lines AF and EK. Tlnoiigh the center of baluster SOO, on the riglit, and 00, on the left, draw the inclination for the underside of rail; parallel with 00, draw the center of rail, cutting JE7ifand JD at L and M, on the left. And on the right, cutting A^ and JBG^, at JV and P; connect PM, cutting CHat Q. At right angles to CH, draw QR and LS, cutting DJ at T and U; then ML is tlie lieight of the wreath-piece for the (puirter circle, fiom C to E, on plan, Fig. 1; and the height for the quarter circle from A to C is the same as shown at NV. Prolong LM to intersect RQ at W; also prolong QM to intersect EK at Y. Make LZ equal to the chord AC, Fig. 1; at M set off the half width of rail on each side of DJ; parallel with DJihuw the half width of rail, cutting the tangent MQ at 3, and tlie tangent LM at 3. From T, and at right angles to LW, draw r4; from U, and perpendicular to YQ, draw U 5. Bevels. Let 6 6 indicate the edge of drawing board, and the dotted line indicate a gauge line parallel with the edge of board. Make 7 8 perpendicular to 6 6, and equal to the radius |12M^^J OC, Fig. 1. Make 7 9 equal to T'4, and 7 10 equal J75; draw 8 9 and 8 10 prolonged to 6 6; the angle at 9 gives the joint bevel for the sliauk, and the angle at 10 gives the bevel for center joint. Fig. 3. Shouis the face-mould. At any convenient place on the pattern paper draw JD in- definite; make DE equal LM, Fig. 2; let EJ equal P/'' more or less, for length of shank. With E for a center, and the distance jRZ, Fig. 2, as a radius, draw arc at C; again, with JD for a cen- Pt-ate S2. 151 ter, and the tangent MQ, Fig. 2, as a radius, draw arc inter- secting at C; connect DC parallel with tangents DE and DC; draw CO and EO, establishing the center O for the trammel, and the parallelogram OEDCon the cutting plane that will agree when in position with the parallelogram OABC on plan Fig. 1. Proof. If the diagonal OD equals the distance ZY, Fig. 2, the angle at D must be correct. I'rolong OE and OC. Make EF equal MW, Fig, 3; draw OF produced, through O, and at right angles to OF, draw AB, for the direction of the major axis; make 2 equal OC, Fig, 1, (12M'^). Let 2 3 and 2 4 each equal the half width of rail (1%'^); make C5 and C(% each equal M 3, Fig. 2. Also E 7 and i7 8, each equal M 2, Fig. 2. Parallel with i^J", draw 7 9, and 8 10; make joints at J" and C perpendicular to the tangents DJ and DC, Now pivot the trammel at O, with the amis cen- tered on the axis line AB; then set from pencil to minor pin the distance 2, (12%^^), place the pencil at C, and slide the major pin until both pins drop into the grooves; then fasten the major pin, and trace the elliptic curve through the points C2E, for the center line of rail; proceed in the same manner to trace the concave and convex curves of face-mould. At section M and N, the bevels are shown applied through the center of plank; the shaded part shows the amount of over wootl to be removed, and also the thickness of plank required for the wreath-piece. Pig 4 Repeats Fig. 3, the lettering being the same, and indicates tlie fiice-inould, Fig. 3, turned over, and the tangents lined off on liie oijposite side; the figure is introduced liere to show the learner liow thu bevels are applied, so as to mate thuiu for a platform twist. (Jl)serve at N. Fig. 3, the bevel is applied so as to pitch the shank EJ, down; and at section N, Fig. 4, tlie bevel is shown applied from the face of i)lank, and the i-everse way, so as to pitch the shank EJ, up; and al the sliank, section M sliows the bevel apijliod so as to piicli the joint at C, down; and at Fig. 3, section M shows the bevel applied .so as to pitch the joint at O, up, thus mating the two crooks. Fig. 3 forms tlie wreath-piece landing on tlie platfoi'is; and Fig. 4, the M'roalh-piece olf the platform. The learner, bj' trying the two crooks together, will see if the bevels are applied correctly. Pig. 5. Shows the elevation of the two irrcatJirjyicces ; the iangoits being folded. ]"ig. 2 sliows the tangents unfolded; it will be obvious to the learner, after la little study, that for a platform twist, all the lines lUM^essary to obtain the face-mould may be found in tliis way, with- out the trou1)le of unfolding all the tangents. See Fig. 2 Plate 29; also, Figs. 9 and 11, Plate 33; unless it be required to find the lengths of odd baluster.s; then the development of tangents together with the treads and risers becomes necessary. How to obtain the length of odd balusters. Re- turn to Fig. 1; prolong tangents AB and i7Z) to the left; make BF and DG, each equal WT, Fig. 2. Connect FC and GC, lor the directors; from the center of b'llusters 2, 3, 4, draw lines parallel to CF, intersecting the tangeniS AB and BC, at 9, 10 and 11. Also from the center of balusters 6, 7 and 8, draw lines parallel to the director GC, intersecting the tangents at the points 12, 13 and 14, as shown. Now we have the position of balusters on the tangents on plan, relative to their position on the tangents, on the cutting plane. Keturn to elevation. Fig. 2, and transfer the location of each baluster from the tangents on plan. Fig. 1, to corresponding posi- tion in elevation, Fig. 2, as shown at 9, 10, 11. 12, &c. Setoff the under side of rail parallel to the line of tangents; draw the conlcr of baluster perpendicular to the treads, to intersect the under side of rail, as 9 2, 10 3, 11 4, 12 5, &c. \n2 I'l.ATE Sa. Now as the under side of rail is drawn through the center of the regular short balusters at 00, and the length of a regular short baluster is 2' V from the top of step to the under side of rail measured at its center; then the baluster marked No. 9, will be li'^ longer than a n-gular short one; anil the baluster marked No. 10, will equal 4>ft''^ longer; and No. 11 baluster will equal 13^^'' longer; No. 12 will be zy/^ longer; No. 13 will equal y^ shorter; No. 1-4 will equal \W longer; and No. 1.5 baluster will be K'^ shorter than a regular short baluster, ?J 0%^', from the top of No. 18 step to the under side of rail at the center of bal- uster, and a regular long baluster will be half a rise or SJ^'''' longer than a regular short one. The length of short baluster is taken for the standard because the under side of rail is drawn through the center of balusters 00 and 00: and the length of balusters at those points is naught, as shown in elevation.* Figs. 6, 7 and 8. Shmo the jAan, elevation and face- mould /(^r Vie Uirnont. Pig. 6. Shows the plan. Let D indicate the center of newel, set off 13^' to the face of outer string, and %^^ more to the center of rail at L. Draw LA indefinite. If the post be a pedestal newel, set the same to show the two sides to the best advantage; draw DB at right angles to the side of newel; make the end of wreath-piece at C to enter the newel }4^^' Make BA equal BC, locating the point A in this case 5'' from the face of No. 4 rise. From A and C, draw lines perpendicular to AB and BC, converging at 0; then with O for a center, and OA as a radius, draw the center line of rail from A to C; then AB and BC will be tangent to the curve at the points A and C, The dotted line at A shows the face of outer string, and makes the radius %''-' less for the curve of cylinder or 2^ IM^^- Locate the short bal- uster on No. 4 step, and space off the intermediate balusters to- wards the newel. Now curve the risers to suit the spacing of balusters and position of newel; parallel with the tangents BA and BC, draw A J* and CP for the parallelogram PCBA on plan; from C, and perpendicular to AB prolonged, draw CL. Pig. 7. Shows the elevation of tangents, they are folded; also the treads and risers. Let XX indicate the edge of drawing board. Make AB equal AB on plan Fig. 6, perpendicular to XX, draw AR and BD indefinite; now elevate Nos. 1, 2, 3 and 4 risers and treads, being careful to keep the face of No. 4 rise 5^^ from tlie spring AR. From the top of No. 1 step set up 6^'' to the underside of rail andSK^'' additional, plus the height of a rise {lo}4'^) to the center of rail at F, making 153^'''' from the floor line to the center of rail. Then the height of newel from the floor line to the center of rail will equal \b}i'' plus the length of a short baluster (9/ V), or (1' 3K''+2' l'^=3^ ^H^^) equal to 3^ 4K''^ •' *The standard lengths for balusters, as furnished to the market, are cut 2' 4" to 2' 0" for short ones, iiad 2' 8" to 2' 10" for loii^ Ijsilusters, ill proportion of one short baluster to three \on% ones; tlie standard leiig-tli for solid newels is 4' 0", or cut in scantlinaj len^iths of 8', ICK, 12' and 16'. and in squai-es of 5s, 6s, 7s, 8s and 10s; for angle newel.s, the lengths run from 5' ti" to 6' 0" long. Plank for stair-builders' use Is sawed iVa", 2%", :tH", 35g", iVs", i%", SJg" and OJi" in thickness, f loni forest timber, straight in the grain, and free from shakes, knots and sap wood. Plaie 32. loo Draw FE parallel to XX, cutting BD at W; through th<« center of balusters 00 draw the under side of rail; parallel to 00 draw the incliuatiou for the center of rail, cutting the spring and angle lines at R and K, and also cutting i^^ at J"; draw JCJE7*. Make EQ equal the diagonal BP, Fig. 6; make EM equal the chord ACon plan Fig. fi; let i?iV equal twice if W"; make WP equal BL, Fig. 6, from P, and at right angles to GJ prolongi d draw PH; make EL equal the diagonal BO, Fig. 6; draw NQ prolonged to intersect the perpendicular from L at Z. As the radius on plan in this case is 2'' 2^^ for the center line of rail, it will require a large trammel, and also more room to draw the curves of mould; therefore we will find points in the elliptic curve, from the chord line, through which to trace the curves of face-mould. Jteturn to plan Fig. 6. From the center O draw the half width of rail equally on each side of the center line (l.?i^O- Make BJ equal "WJ, Fig. 7; connect cTCforthe directing ordinate; draw (he chord AC; bisect AC at R; bisect JRCand RA at Tand S. From the points A, S, R, Tand Cdrawordinates parallel to JC, cutting the inside, center and outside of rail on plan at the points 2, :; and 4; fnnn A, and at right angles to CJ prolonged, draw AK; draw CM indefinite and at right angles to JC, cutting the curve of rail on plan at 5 and 6. Bevels. Return to elevation Fig. 7; let YY indicate a l)a?o line; draw 2 2 parallel to YY; draw 3 14 perpendicular to XX, and equal to AK, P'ig. 6. Make 3 5 equal to CL. Fig. 6: make 3 6 equal HP; also make 3 7 equal the height ER. Draw 14 7 and 5 G prolonged. The angle at 7 gives the bevel for the joint connecting the newel, and the angle at 6 gives the bevel for the shank joint. Parallel to 2 2, draw the half width of rail {i%'^), cutting the hypothenuse 5 G at 10; also make 2 9 and 2 15 each equal (75 and CG, Fig. G. From and 15, and parallel to' 2 2, draw 9 12 and 15 16. Fig. 8. Shou's the face-mould. Draw GJ indefinite; make RK equal RK, Fig. 7; with R for a center and the distance RM, Fitr. 7, as a radius, draw arc at C; again, with K for a center, and KE, Fig. 7, as a radius, draw arc intersecting at C; draw KC prolomred. Parallel to KR and KC draw CO and RO tor the parallelogram OCKR on the cut- ting plane. Proof. The diagonal OK must equal the distance QN, Fig. 7; if .so, the angle of tangents at K is correct. Make KJ equal KJ. Fig. 7; join JC f4^^^; then elevate Nos, 15, 14, 18 and 19 risers and treads, making Xo. It) rise 6 }<'''' from the spring line AD, and Xo. 14 rise 5%^^ from the spring line LE. The joints to connect the straight strings are made at Xo. 14 and 19 rise; the veneer is made the same thickness throughout, and the staves 1, 2, 3, &c., are shown extended beyond the veneer to allow for trimming off, and to have room for a screw to bring the stave and veneer down close on the drum. At Gf and F, the stave is shown wiiler, and should be glued to the veneer before bending over the drnm; keep the joint of stave to the springing lines AD and LE; after the two staves G and F, have been glued to the veneer, and the glueing is thoroughly dry, dampen the veneer with hot water; then clamp one end to the drum, being careful to keep the spring line on the veneer over the spring line on the drum. Then bend the veneer over the drum gently, and at the same time screwing down a stave at intervals temporarily, to bring the veneer down close to the drum; if the other spring line agrees with the spring on the drum, then clamp fumly to the drum, and leave stand until the veneer is dry, then glue and rub the staves to place, using a screw at each end to bring all down firm. This makes clean, nice work; the tread and rise is cut out, after being lifted from the drum. When laving off the lines on the veneer use a lead pencil; if a knife be used there will be danger of causing the veneer to kink. Fig. 4. Shnu-t; another method of doing the same thing by benOinij the i^cnecr AA over a drum B; then kertinrj and steam- ing the hacJi jmrt CC, and hending it over the veneer. In this case the back part CC must be of an even thickness throughout, say IK^', and the kerfs should be extended beyond the springing lines as shown, and to within Jg'^ of the back, if possible; the kerfs must be cut perpendicular to the treads, and made from straight-grained stuff"; the kerfed part should be well softened by steaming, then bent over the drum and let stand until dry, being careful to have the springing lines on the kerfed part parallel with the spring lines on the drum. After it has set, lift from the drum, tack on two braces, then soften the veneer and bend over the drum, being careful to keep the plumb lines on the veneer to the spring lines on the drum; leave the veneer in this way until dry; then glue and clamp the kerfed part down firmly to the veneer. Tfie distance apaH to m^ake the kerfs so as to close is shown next. Fig. 5. fScale H^^ equals 1^] Take a strip, the thickness being equal to the thickness of plank intended to be kerfed, say IJ^''; run a gauge line on to whatevH'r is allowed to remain solid, say }4^^, shown by the dotted line AB; then make a kerf OC with the saw that is to be used for the kerfing. Xow draw the arc of a circle DE to whatever radius required, in this case 12 \"^; then through the center O draw a straight line DO, and place the strip to the line, keeping the kerf to the center at O: now fasten with two nails at 2 and 3; then move the other end i'' until tlie kerf will close, and mark the edge of strip cutting the arc at H, then DH will be the distance apart to space off the kerf. The saw should liave guides clamped to the sides, so the kerfs will be all the same depth, for good work. 15C Pl-ATK 33. Pig. 6. Shows another incthod for constrnoA'infj the cyl- inder. Tlie piece intended to bend over the drum BB to form the cylin- der should he straight in the grain, and be made of an even thiclcness throughout. The dadoes are cut perpendicular to the treads, and all to the same depth. This is very speedily and neatly done on the saw table witli a dado head. The piece to be dadoed can be arrauged across the table to the proper inclination in a frame, and in a few minutes the woik is done. One or two grooves .should l)e cut lieyond the spring lines CC, as shown. Tliis is done to prevent too sudden de- part ure from the straight to the circular part, and thus avoid per- haps fracture. The string should be laid out with a pencil .so as not to aliraid tlie surface, us that may cause fracture or kinks. Tlie lay- ing out sliould be done before bending, but not cut out until the work is lifted from the drum. After the dadoing is done the work require to be well softened by by steam or hot water; a steam box is very handy and economical for work of this kind. By turning on tlie live .steam the work is soon softened, and the moisture is speedily evaporated from the worlc. After tho worlc is sufficiently softened clamp one end to the drum, keeping the spring lines opposile, and gently bend over tlie drum, adding a stay every few inches to avoid kinks, and clamp tho other end firmly to tlie drum, being sure the spring line on the work comes opposite or parallel to the spring line on the drum. Now .size the dadoes willitliin glue, and leave the work in tliis shape to dry out. Then fit hard wood Iveys into the grooves, glue and drive them in lightly, not too liard, as the woric may raise off the drum and cause kinks. After the glue is hard dress off the keys to the curve; then glue and nail on tlio back of cylinder two lines of haxd wood strips J4," thick and 2" wide; one line %" from the lower edge, and tlie otlier near the internal angle of the tread and rise. The strips laminated in two thicknesses would be be.st, as shown at 2 and 3. Tho distance apart to make the dadoes and the thickness of veneer, or solid part, is explained for Plate 28, Fig. 6. This treatment of circular strings for any circle over 10" radius makes good and substantial worlv. AH circular wall strings should be constructed in this way. c Fig. 8. Shows tlie lower end of cylinder increa.'^od in thickness to form grounds for plastering and offer solid nailing for a heavy plaster moulding D; S shows the lagging glued on to the back of cylinder, and the monlding D, is shown one-half full size; J' shows the veneer; A, the stave, and C, the plaster- ing. This heavy moulding will have to be constructed same as the wreath-piece of a hand rail, and tit to the cylinder in the shop and ptit up after the plastering is done. At Fig. 7, the tangents AB and BC are shown for tho plaster moulding D, the radius for the center of which Is IK'''' more than the radius of cylinder, or 13}4^\ Fig. 9. Shoivs the eUvation of ttmnents; they are folded. Let XX indicate the edge of drawing board. Make AB equal OA, Fig. 7, [1.3^'^]; perpendicular to XX, draw AC and BD indefinite; anywhere on XX, place the ]>itcliboard, and draw the inclination indefinite, cutting AC at JS, and BB at F; return to Fig. 3, draw CJ perpendicular to the treads, and to intersect the inclination GF at J; from J, and perpendicular to EL prolonged, draw JK; then KL is the height tliat the wreath- piece will raise for tlie quarter circle from A to C, Fig. 7. Now make EC ecpial the height KL, Fig. 3; perpendicular to AC, draw CD prolonged, intersecting EF prolonged at G; from D, and perpendicular to GE, draw DH; parallel with CD, di-aw FJ; draw DJ indefinite; From E, and at right angles to DJ, draw EK\ make B 3 equal half width of the plaster moulding [33^^^]; parallel with BD, draw the half width of moulding, cutting the tangents FE and DJ, at 3 and 4; Parallel with XX, draw EL indefinite; make EL equal the chord AC, Fig. 7; make JM equal JC. J'l.ATK 3:>. 157 Bevels. From the odi^^e of board XX, rnn the gango line ab; perpendicular to XX, draw Z)/ equal to tlie radius OA, Fi^'. "i", iX'^y/'), Make Z) A equal DH, and bg equal ^iT; draw fh and /g- prolonged, to XX for convenience when setting the bevels. Fig. 10. Shoiostlic face-mould. At any convenient place on the pattern paper, lay down the steel square, and draw the right angle JHC indelinito. Make HF and HE, each equal HF and HE, Fig. 9; with F for a center, and DJ, Fig. 9, as a radius, draw arc intersecting HC at C; Connect FC; parallel with FC and FE, draw EO and CO, establishing the center 0, and the parallelogram OEFC, on the cutting plane. Proof. The diagonal OF must equal the distance LM, Fig. 9, if so, the angle of tangents at F is correct. Make EN equal FG, Fig. 9; draw NO prolonged; through O, and at right angles to ON, draw PQ for the position of tram- mel; make joints at J" and C, at right angles to the tangents JF and FC. Make O 2 equal the radius OA, Fig. 7 [V.',}4J']- Let 2 .^ and 2 4, each equal the half width of moulding (2,V^); make C 5 and C G, each equal F 3, Fig. 9, and E ~ and E 8, each equal D 4, Fig. 9; parallel with JF, diaw 8 10 and 7 9 -^or the shank. Now pivot the trammel at O, and trace the elliptic curves of face-mould as has been described for the preceding Plates. At R and iS, the bevels are shown applied to cross the tangents; the block pattern shows the twist of rail and the thickness of plank required; the shaded parts show the surplus wood to be removed. Pigs. 11 and 12. Show another method hoiv to conMruct the wraith part of viouldlng hy making both tajigcnts eqxiol, and using a cripple joint at the shank. Fig. 11. Shoivs the elevation of tanoc?its ; they are folded. Let XX indicate a base line; make AB equal DC, Fig. 7; draw AC and BD perpendicular to XX; draw the inclination of the center line of rail as EZi make EC equal the height LK^ Fig. 3; draw CD and EL parallel to XX; EL cuts BD at N; bisect ND at F; connect CF; draw FE prolonged; make EL equal the chord AC, Fig. 7. From the center D, draw arc to tangent FC as DH; join HC for the bevel for both joints; par- allel to BD, draw the half width of rail to intersect FE at 3. On EZ, set off C/'' to >S for shank: make joint at S perpendicular to EZ; prolong FE to allow over wood for cripple joint, say to J; through J, draw 4 5 perpendicular to FJ tor a temporary joint on shank. Fig. 12. Shoivs the face-mould. Make £^C equal CL, Fig. 11; with Cand Etor centers, and .E'JF', Fig, n. as a radius, draw arcs intersecting at J* and 0; draw FE, FCEO and CO for the parallelogram OEFC on the plane of plank. Proof, The diagonal OF must equal the diagonal OB on plan Fig, 7; if so, the angle of tangents at -Pmust be correct; prolong FE to J, equal to EJ, Fig. 11. Prolong the radial lines OC and OE indefinite; make Joints at Cand J" perpendicular to the tangents FC and FJ; make C 5 and C 6, also E 7 and E S each equal F 3, Fig. 11. As both tangents in elevation are tlie same length in this case, the diagonal Oi^ becomes the direction of minor axis. Make O 2 equal OA, Fig. 7, for the semi-minor axis, Let 2 3 and 2 4 each equal the half width of rail (2^^). Now Ifts ' Plate 34. pivot the trammel at 0, with tlie arms at riglit angles to 2; then set from pencil to minor pin on tlie rod to equal O 4, and place llie pencil in the point at 6, then slide the major pin until both pins drop into the grooves, then fasten the major pin and trace the curve for tlie convex side of mould through the points 6, 4, 3; proceed in like manner to trace tlie curves for tlie concave and center lines of face-mould ; draw 7 9 and 8 10 parallel to EJ. Sections R and /S sliow the bevels applied. Observe the block pattern is first applied the same distance down from the face of crook at botli joints; then work off the sides of shank to the joint bevel; afterwards apply the cripple bevel shown in the angle at 4, and cut the cripple joint. See first that all lines on tlie temporary joint are squared over from the joint on to the upper and lower and also the vertical sides of shank, so that when the temporary joint is cut away the lines required on tlie cripple joint may be easily carried over the joint again. Now slide the block pattern down on the cripple joint the distance JS, Fig. 11; then lay off the direction of straight mould- ing at right angles to the cripple joint, and ease the angle to please the eye, and fit the lower end of string. PLATE 34. Plate 34. ErhiMts a tivo-story stnir-cnse with n half pace winding* in the fimt flight, (tnd lc qjutrtcr pace ^vindi■)UJ Jlifjht in the second stm-y. Figs. 1 to 4, are drawn to a }i^^ scale [scale H^^^^l foot]; and Figs. 5 and C> arc dnnvn to a %'' scale [scale %^^=1 foot]. The height of first story from top to top of joist is 11' 4''^; the hall is C/ 0^' wide in the rough; the joist is 9'^ deep in tlie second story; tlie back door under stairs is 6' 6^' high; and there is IC 8^' at tlie starting, and at the landing 5^ 2>" to tlie door jamb. The height of second story from top to top of joist is 10' 0''; depth of joist in the third tlf)or equals 9''; the soffit of stairs will clear the window in the second story as they land with a (piarter pace winding, and tlie window is set plumb over tlie opening lielow on the one side; the horizontal space between the walls for the stairs equals If 4''. The newel is l^'^/l", to be set in the center of hall; size of rail 4''X2K^^; balusters 2''X3"; height of base exclusive of tlie moulding is 8'' for the first story, and 7'' in the second story. fig. 1. Shows the plan of the first flight. The lieight of stoi-y is 11' 0", which rednced to inches equals 136", divided by 17 risers [136'''---17=8"] equals 8" for each vise, if we allow 25" for a constant, the relative proportion of *Thc plan of the a'oove stair-case is not intended as a model for the young man to copy from in practice, for wiudeis iu small and contracted halls are objectionable, and sliould be avoided at all times if possible. The grand circular or elliptical stair-case can he made an object of beauty, that cannot be excelled in stair-casing, In that ciise having more room, the winders can be increased in width at the outer string, and thus give to the rnil easy flowing curves, and a gentle ascent to the grade of steps; but as the stair-builder has to adapt the stair-case to the space allotted, hence he will have to exercise his judgment as to what will suit the space, and give the best I'esulls; the above plan Is given as an extreme, and is intended 1o sliow liow such stairs may be treated, the principle being the same for all. Plate 3-i. 159 tread to rise will equal 9^^ pr/^— (8'^+8''=16^')=9^^], then the pitch ot: stairs for the lirst flight will equal 8'^X9'^, if there be roum to allow a tread equal to 9i". The well hole is in the center of hall, and the cylinder is S'' diameter, the winders should be laid off on heavy paper. First lay oit' the line of rough walls-, parallel with the walls, draw the wall strings, allowing ";i" for plaster on brick walls, and 14," on stud partitions, then %" for the thickness of base, making \%" lor the stud partition wall, and \%" for the brick walls, that niust ))e allowed from the rough walls to face of the wall strings; the thickness of wall strings is usually \%", this thickness allows sufficient to work the grounds on the solid for the plaster- ing. Draw the cylinder in the center 3'' Q" from the walls; draw the outer string; set ott' in tins case from the string line \\" for the regular tread line around the cylinder. Right here the work- man may have to make several trials in spacing off the winders on the walking line, either shifting the walking line further out, or bringing it nearer to the well hole, so as to locate the first rise to admit an easing and the finish around door at the starting; the walking line should not be over 18'^ from the string line, and may be located at any point between, the nearer to the string line the better, as then the winders at the narrow ends will be increased in width. Now with the dividers set to %" begin at the center (No. 13 rise) of semi-circle, and space off towards the starting 11 threads on tlie walking line, which leaves about 18'''' from No. 1 rise to the face of door-jamb, to allow for the "trim" and easement on the wall string. Also, step off towards the landing five (5) full treads, and we have 9'^ from the face of No. 17 rise to tlie door- jamb; this is too scant if it could be avoided, but less has to do iu many cases. In all cases for a half-pace Vi^iuding stairs at the center (No. 13 rise) there should be a rise, or the center of a tread. In locating the risers at the outer string first space off the balusters around the cylinder, beginning at the center of cylinder (Xo. 12 rise in this case), and spacing them equal to half a tread or less; in this case they are less, so the balusters will appear evenly spaced. Nos. 8, 9, 14 and \h threads may be graduated to help the easing on the lower edge of string, while the balusters, iii- slead of being over the rise, may sit over on the treads to show them evenly spaced, as shown at step No. 17, Fig. 16, Plate 15. In stairs cramped for room this has to be done. When the points for the risers are spaced off on the cylinder and front string to connect the "llyers" or straight treads, then draw the face of risers to suit the balusters and through the points just found on the walking line to intersect the wall string. It will be observed the rise Nos. 11 and 13 do not radiate to the center of cylinder, and Nos. 9, 10, 14 and 15 cut the front string very obli(iue. This should be avoided whenever possible to do so by allowing llie risers to radiate from the center of cylinder and allowing but one graduating sl«*p outside the cylinder. This case has been intro- duced to give the learner an idea how to manage a flight when the steps have to be crowded. In small buildings, for want of space, the stairs are often too much crowded. Observe, that the temporary step patterns for Nos. 8, 9, 10 and 11 winders will answer for Nos. 1.5, 14, 13 and 12 winders by reversing the patterns. For tlie trimming of the well in the second story, allow %" from the back of No. 17 rise to the joist, or ^/ ^" from the wall, as shown. The width to trim the well ia the 160 Plate 34. secoud story at the starting of the second flight will equal the half width of hall (S' 0'''') plus the radius of cylinder (4^^), and the thictness of fascia (1^0 [^^ 0^^-t-4^^4-l^J=3'' 5^^J equals 3' 5^' as shown on plan. The trimmer at the starting of secoud flight should be at the back of cylinder, and at right angles to the wall, if that will allow enough head room. In this case there will be ample head room, the face of trimmer being over the middle of first tread, thus allowing the ceiling to carry around level to back of cylinder starting in the second flight, and at right angles to the level fascia. The soffit of the second flight will then curve down at right angles to the face string, instead of following the twisting line of treads and risers. The face of trimmer is sliown F/ 1}-^''^ from the face of joist at the lauding, or 8' i^i'^ from the wall; the lauding v/ill theu be trimmed out to 2'' T^'' by 3^ Tri" ^' in the rough. The dotted lines show the manner of jiutting in the Ijearers; dry 3'^ by 4^' scantling should be selected for this purpose, neatly cut, fit aud well nailed, aud rough brackets, with the grain per- pendicular to the treads. The brackets are nailed to the bearers, aud through the back of rise into the tread. Fig. 2. Shoics the elevation of threads and risers. AB shows the story rod; the height of story 11' 4^^ is divided into 17 risers; the first length (6' 8'^) to measure for the straight rail is shown taken from the spring of turnout to the spring of cyl- inder. No. 2 length (2'' V^ is taken from the spring of cylinder to plumb with the face of last rise. No. 17, aud the length of level (3'' 9'^) is taken from the face of No. 17 rise to the spring of cyl^ iuder starting of the second flight. Fig. 3. SJioxvs plan of the flight landiwj in the tliird story. The height equals 10' C, being reduced to inches equals 120''', aud divided by 15, the number of risers for a trial, equals i" for each rise, and the relative width of thread will equal 9". The well will be trimmed to the width of that in the second story 3' 5" wide, and the space between the walls is 11' 4"; allowing the space for the second flight to equal 11' 4" by 3' 5" in the rough. Allow 3' 0" at each of the quarter pace winders to the center of cylinder, phis 43'o" for j^raduating the treads at the spring of cylinders (3' 0"+3' 0"+4ii"-f4>^"=6' 9"), which equals 6' 9" to be deducted from the whole space, 11' 4" (11' 4" —6' 9"z=4' 7") equals 4' 7", to be divided by the width of a tread, (9") gives us six straight treads; that will place a rise in the cen- ter of well. In this case. No. 8 rise comes in the center. We will make the walking line 14" from the face of front string ou trial. Now, with the dividers set to the width of a tread (9"), begin at No. 8 rise and step off the treads either way ou the walking line; the face of Nos. 1 and 15 rise comes on line witli the face of level front string. This will answer very well; from No. .5 and No. 11 riser, space off tlie balusters in the cylinder to equal the spacing of tlie balusters on the straight steps (4K'''') or a little less in small cylinders. Tlie spaces in this case are less. Nos. 1 and 15 rise should be curved to suit the circular nosing without too much peak at the point of miter, and also help the easing at the joining of the cylinder with the level string. It will bo observed that Nos. 4 aud 11 winders are the same, only reversed. The front of the former is back of the latter, and one set of patterns will srnswer for the winders of both quarter paces in tliis case, thus saving time aud material when cutting tliem from the plank, aud also adds to the general appearance of the stairs. Platj: ?a. 161 After the measures arc taken for stairs of this kind, the first thing the stair-builder sliould do is to make a plan to a scale of say K*" to equal one foot. By so doing fewer mistakes will occur, and much time be saved. For the head-room. Count down from tlie landing 12 risers at 8^^ each equals 96^^ (H|/^=8' 0^^) equals 8' 0'^, minus the joist, flooring and plastering, say 11^^ will equal 7^ \" from the top of No. 3 step to the line of ceiling; this will answer. Fig. 4. Shoivs the dcvatioyis of treads, risers and strings. The story rod AB is divided into 15 risers. SSS, &c., show the springing of the cylinders. The measure for cutting the straight rail is shown taken from spring to spring of cylinder. No. 4 length is shown 7' 0M^^ and No. 5 length (6' 7^^) is taken from the spring of cylinder landing to spring of quarter cylinder, and from quarter cylinder to wall 3^ 0''^ measured on plan. The raking lengths are taken at tlie bench, and the level lengths after the stairs are stepped up, and all entered in the order book. Fig. 5. Exhibits plan of No. 1 cylinder, [Scale %'^=V\ Showing the angle that the steps and risers make with the outer string and cylinder. The nosing line is drawn parallel with the risers, and intersects with the nosing line, which is drawn par- allel with the face string; the face of risers is drawn to intersect the bracket line, determining two points through which to draw' the miters of nosings. It will be observed at No. 10 step the point of miter is very acute. To avoid so long and pointed miter it will be better to round it off, and connect the nosing and scotia of step with the return nosing, making a joint at right angles to the face string, as shown at A. The nosings of Nos. 11 and 12 steps are rounded off to connect the return nosings. Fig. 6. [Scale ^'"=1 foot], Shows the cylinder connecting the straight string; the shaded part shows the staves, the angle that the risers make with the face string, and the cylinder, for the cut-out of front string, is shown. The joint of cylinder is y/^ from the cut-out of No. 10 rise; Nos. 9, 8 and 7 treads are 4^^, 4K^^ and 9'^ respectively: Nos. 11, 12 and 13 rise come in the cylinder, and the staves can be divided off, glued up and trimmed to suit the direction of risers; the joint of cylinder is 1)2 ^^ from the cut-out of No. 14 rise, and Nos. 14, 15 and 16 treads are 4^^, ^y and 9^^ respectively. Fig. 7. [%''' Scale]. Shows the well hole, startmg second flight; tlirough the intersection of the steps and risers, with the return nosings aud brackets, draw the miter lines for the ends of steps; No. 1 and 2 step have the nosings rounded to avoid too long a miter, and allow the baluster to enter the mortise easily. The joint of cylinder is %'^ from the face of No. 4 rise. Fig. 8. \}i'^ Scale]. Shows the width of staves, they are made to suit the spacing of the risers; the spring line of cylinder is l}4'^ from the cut-out of No. 4 rise; and the cut-out of outer string for No. 4 and 5 treads are Z}4^^ and 9^'' respectivelj', * Fig. 9. [^4^^ Scale]. Shows the angle of steps and risers intersecting the bracket line and return nosing; from those two points draw the miters for the ends of Avinding steps, for the return nosings and brackets. The landing and starting rise, Fig. 7, is made concave and convex to help the width of cylinder connect- ing the level fascia or string, and also the spacing of the balusters; the face of No. 12 rise is shown !}£" from the joiut of cylinder. 163 Pi.AtR S5. Pig. 10. [%^^ Scale]. Shows the staves lined off to suit the spacing of risers, tlie cut-out of face string is shown for Nos. 10 and 11 tread to be 0'^ and o>.<^^, and the joint of cylinder is shown %^^ from the cut-out of No. 13 riser. Th.e plan of winders should be drawn fall size on nianilla paper, and rolled up until needed; the intersection of risers with the wall strings, give the width of winders, as shown on plan, Fig. 1* No. 8 tread is 15^' from face to face of rise, and No. 9 is Sf^; No. 10 is 8}i^^ from face of rise to wall string in the augle; and from the angle to face of No. 11 rise is 173<^^; No. 11 and 12 treads are 16K^^ each, and No. l:> tread is 173^''^ from face of rise to face of wall string in tlie angle; then S}4^^ from the face of string in the angle to face of No. 14 rise; No. 14 tread is 31^^ and No. 15 tread is 15''^ from face to face of rise; No. 16 tread is the regular width 9^\ When laying out the wall string, a very good way is to note downi these widths on a piece of paper, and thus avoid referring to the drawing. The face strings are shown to be 1'''' thick, and are halved on to the cylinders at their joinings, and should bo glued and screwed from the back for good work. PLATE 35. Plate 35. [Scale %''^=l foot.] Shoivs the front and ivall stri7igs, and the development of staves in the cylinder for the first jHilhL of winders, Plate 34. Fig. 1. Shmrs the front string, at the starting and landing, and broken at the middle, connecting which is shown the devel- opment of staves, as taken from the chords shown at Fig. G, Plate 34, and is intended to show how the length of staves may be ob- tained. Tlie stave for No. 10 tread is 3^^ wide on the chord; Nos. 11 and 12 is 3K^^ and No. 13 is shown 2^^ See Fig 6, Plate 34. The face string is spaced off with the dividers for the regular treads, as has been explained at former plates. No. 8 tread is 43^^^; No. 9 is 4^^, and the spring of cylinder AB l^H^' from the cut out of No. 10 rise; No. 16 is a regular tread; No. 15 tread is ^}i^^, and No. 14, ?>%'\ and the joint or spring of cylinder is 13'2^^ from the cut-out of No. 14 rise. The steps are IH^^ thick. It will be noticed at the landing F the string is notched li^^ to come level with the joist; make the width of level string at that point 1)^^ for joist, and %^' for plastering (9^^-f X'^=9X^0 equals *.)%^\ The width of inclining string is 6^^ from the internal angle of stop and rise to the lower edge of plaster. Opposite No. 15 and 16 rise the string is increased in width, and at No. 8 rise the width is diminished some to allow the casings a graceful curve. The easings connecting the staves are imperfect, and is done to get the length of staves in the rough. After the staves are glued up, the cylinder formed and connected to the strings, then the perfect easings are made agreeable to the eye and taste. A triangular piece, H, is glued on the lower end to relieve the acute angle formed by the inclination of the string with the floor line with an easing. The easing is short to allow the base board along the panel work to join flush and square with the string. JK shows the joint or spring of turnout 6''^ from the cut-out of No. 3 rise. The dotted line Jtf shows the thickness of step (IK^'') Put off the lower end of string, making No. 1 rise 6%^^ high, so that Plate 3S. 163' when the step is in place the rise will be equal in height to the rest. The first length for jointing the rail is is 6'' 8''''; the second is 2' V^; and the third is 3^ 9^^ on the level. Fig. 2. Shows the ivall string for the first 10 risers. EF shows a joint; to make this string in one piece would require the plank to be 22'^ wide. So wide a plank is not always at hand and it is better to make the string in two pieces. Let the piece containing the winders, take in so many regular treads as the width of plank will admit. In this case we have two and a half regular treads in addition to the winders, so we will make the joint EF at the center of No. 5 tread. To hud the bevel C, take the average of the treads by the rise; thus add together the width of each tread at the wall string, and divide by the number of risers to be included in the string piece, 1 8K^^-l-31^^+15'''+9''+9''^+43'.i ^^=65^^] [65^^-f-5=15'^], equals 15'''' as the average tread. Now with the steel square take 15^'' on the blade, and the rise %'' on the tongue, and apply from the edge FH, of plank. The tongue will give the bevel or plumb cut shown at C for each rise. Now space off the risers from the plan, Fig. 1, Plate 34, using the bevel to give the direction. Then set the dividers to the height of a rise [8^^], and mark each rise, and draw the treads at right angles to the rise, using the steel square. The regular treads at the lower end of string, are stepped off on the gauge line, as has been described. The joint EF, may either be lapped or grooved and tongued in the joint, and a strip nailed on the face of string to hold the joint firm. It may be removed when the string is in place. At the upper end \%^^ is added on to receive the wall string, Fig. 3, and a groove at AB, is shown to admit the tongue AB, Fig. 3. Pig. 3. Shows the ivall string No. 2. The two center treads. No. 11 and 13, are alike IGK'^XS^' rise and the bevel E, may be set to that pitch for the direction of each rise. The width of treads on the string 17^^^ IGH^^, H')H'', 17K^^ are taken from the plan. Fig. 1, Plate 34. At "the upper end, \%'^ is added on to receive string No. 4. The shaded part CD, shows a groove )4'' deep to receive the tongue DC, Fig. 4. The dotted line at the lower end shows the face of No, 1 string, and K''^ is added on to allow for tongue. Fig, 4. Shows No. 3 ivall string. The bevel for the direction of risers may be found by taking the average of the treads, the same as for No. 1 string. The width of treads on plan equals 8}^^^, 21'''', 1.5'^, 9''^, plus 11^^ for easment at the landing; equals 643^'^, which being divided by 4 risers, gives 163^ ''^ for the average. Then apply the steel square to the edge of plank, with 163^^' on the blade and 8^^ on the tongue, the tongue will give the bevel A for the direction of risers. Now line otf the treads at right angles to the risers; No. 14 rise is 8^^^ from the face of No. 2 string, shown by the dotted line; DC shows the limit of the tongue. The treads and risers are now lined otf on the wall strings; the next will be to ease off the angles and connect them in the corners with curves that will be graceful and pleasing to the eye. From the external angle of tread and rise set off 4^^ as shown by the arcs, as JZ, Fig. 2. Now it will be obvious, that at the lower 164. Pl^TE 36. end of No. ?> wall string, the easing to connect No. 2 string must raise np sufficiently liigh to clear No. 14 rise with the 4'''' face above the treads at the angle; T'''' is allowed at the lauding for height of base, and on trial we will set up ll''' at the lower end and trace the curve AXD for the upper edge of string, using a pliable strip. At D care must be taken to have the curve to tan- gent a line at right angles to the plumb cut DC. From the top of tread, at the upper end of No. 1 wall string, set up on trial G}i^^, and trace the curve for the upper edge of wall string to tangent the straight part at (r, also keeping the strip near to the ares, as consistent with a curve agreeable to the eye. Now at the lower end of No. 2 string set up 6}4^^ to agree with the easing at the upper end of No. 1 wall string; also set up 11''^ at the upper end of No. 2 wall string to agree with the lower end of No. 3 wall string, and ease oft' the angles at F and G. As the treads iu No. 2 wall string are near alike, the upper edge may be straight. It will be noticed that opposite Nos. 15 and 16 rise. No. 3 string, the curve does not tangent the arcs, passing above the former and under the latter. This cannot be avoided always. The stair-builder must make his curves agreeable to the eye and free from abruptness as /le possibly can make them, for in that consists the pride of his profession. Tiie upper and lower edge of wall string may next be trimmed off, and the steps and risers housed in. The nosings and scotias for the straight steps are marked on the string from the step and rise after being glued up, as has been explained for Plate 22, When marlcing the nosings and scotias on the strings for housings, for the winders, cut a piece of straight nosing to the oblique angle that the winder makes with the string for a pattern, and trace the contour . of nosing and seotia for the housing, as pre- viously Qxplaiued. PLATE 36. Plate 38. [Scale %''-=.l foot]. Exhihits the front and wall fstrimis for the flight Fuj. 3, Plate 34. Also the development of staves in the cxjUnder to obtain their lengths in the rough. Fig. 1. Shows the lower end of face string connecting the cylinder. AB shows the joint of cylinder 13^^^ from the cut-out of No. 4 rise. No. 4 tread is o}4''^ wide on the face string, and No. 5 tread equals y^''. The dotted line AC, shows a gauge line run on 6^^ from the lower edge as a guide for the internal angles of the regular treads and risers. The width of each stave is taken from the ciiord, Fig. 8, Plate 34. DS shows the joint of cylinder connecting the level string, which is 9X^'' wide to cover tlie rougli joist and receive the plaster. it will be noticed that No. 1 rise is 7M'''' Wgh» instead of 8''', the regular height of rise. This occurs from the step being 13^'" thick, and the tloor 1^^. By reducing the first rise }4^^, the first stave will then come close up under the flooring. This does not reduce the whole height any, for it will be seen at the landing, Fig, 2, a K'''' 1^5 added on, so that the level fascia will come chuck up under the iloor. and at the same time tkp easing will drop Ji^^ below the joist to receive the plastering. pLATr. 06. 16j5 The casing on the string and at the lower end in tlie cylinder is temporary, and intended to sliow the length to cut the staves in tlie rough. After they are glued up and the cylinder spliced to the strings, the easements can be better studied, and the defects remedied to please the eye. Fig. 2. Shoics the clevelophient of staves for the cylinder landiruj in the Hard story; the widths are taken from the chords Fiij- iO, Plate 31. AB shows the joint of cylinder connecting the inclining string, and %'^ from the cut-out of No. 13 rise. DC shows the joint connecting the level string. A triangular piece at E, will be required to glue on the lower edge of level string to help tlio casing at the joining of the cylinder. The stave at No. 12 tread is 2M''^ wiil^; No. 13 and 14 are 8^'' wide, and No. 15 is 3'^ wide. The dotted lines show the length of staves in the rough. Fig. 3. Shows the wall stHng No. 1 at the starting. No. 1 tread is 15^^ wide, and No. 2 is 23'^ wide on tlie w'all string, as taken from plan, Fig. 3, Plate 34. The part dotted Une CD shows the face of No. 3 wall string connecting; in the angle a X'^ is added to enter the wall string No. 2. Fig. 4. Slwws ivall string No. 2 As there are w inders at both ends, it had better be made in two parts. The bevels at X and O, for the direction of risers, may be found in the same manner, as explained for Fig. 3, Plate 35. Nos. 3 and 13 rise come in the corners in this case. AB and CD show grooves into which the tongue on No. 1 and No. 3 wall strings are fitted; EF shows the splice joint. Another and convenient way to line off the strings for veneers is to malce an elevation of each string to a scale drawing, as shown at Nos. 1, 2, 3, wall string, then draw a line GH. so as to divide the breadth 01 string at HG and EL about equal at it.s widest part; then set the bevel X, or O, to agree with the riser.-: and the line GH, or EL. The width of treads is taken from plan, Fig. 3, Plate 31, on a slip of paper for convenience. The bc\el at B, Fig, 5, may be found in the same way. Care must be taken to line off all the treads at right angles to the risers. By drawing the elevation of treads and risers, also the curves on the upper edge of string, to a scale, the young man will gain experi- ence in tlie use of the drawing instruments. At tlie lower end of No. I wall string 7^^ it allowed for the iioight of base; also the same is allowed at the landing, exclusive of the moulding. A rule for the width of base, ir.c.uding the moulding, rela- tive to the height of story, is one inch I'ci tvery foot in heiglit of story from floor to ceiling. A suitable easing is found in tliis case at the lower end of No. 3 string, by allowing 33<^^^ for height of string above No. 13 step at the angle. At the upper end of No. 3 string, from tlic top of No. 13 tread to the upper edge of wall .string will equal C8^'-f 3K^'=11K^'') HX'^ This also allows an easing agreeable to the eye. And at the lower end of No. 2 wall string a suitable easing is formed, leaving the siring 63-.^'''' above No. 3 tread. Then the height at the upper end of No. 1 string, above No. 3 tread, will e(|nal H^^ for rise, plus OK^^ to connect tlic lower end of No. 3 string, equals 14^'^ wliich allows a favorable easing at the ui)per end of No. 1 string, as shown. After the curves on the wall strings are made, mark all the casings on a board intended for th<; mouldings, and thus save time in working them out when needed. 166 Plate 37. PLATE 37. Plato 37. [Scale K^'=l foot]. Exliiljits how to find the lenrjth of tatujents and radius for the turnout at the newel past; also the face-mould for the same. Tlie facc-mouMs for the half pace winding in tlie first Jiiijht, and the tico quarter pace wind- i7i;/s', startinij and landing, in the upper flight, for the stair-case, Plate 34. Figs. 1 and 2. Show how to locate the radius of turnout relative to the height of newel post and inclination of rail. The newel post is 7'''X'i'^^- Through the center at O, draw the face of Ko. 1 rise perpendicular to one side of newel; extend the side of post by a line to A, as CA, for the center of rail shov n Ijy the dotted line. Parallel with No. 1 rise, draw No. 2 and 3 ri-< . Now draw a base line XX, Fig. 3, and ele\ate No. 1, 3 and o rise. Through the center of baluster O, O, draw the under side of rail, and parallel with 00, draw the center of rail AB indefinite. From the top of No. 1 step set up 6'^ to the under side of rail, and IK^' additional to the center of rail, (G'^+lii'^^=7K''\ equals 731^^ from the top of No. i step to tlie center of rail at F: from JP, and parallel to XX, draw a line 1o intersect the inclina- tion of rail at S; from B, let fall the perpendicular, intersecting the center line of rail, Fig. 1, at C; draw CO, intersecting the cap at D. Make CE equal CD for the length of tangents on plan. Perpendicular to the tangents, and from the points E aiul D, draw the radial lines, converging at F: with FE as a radlu-^, draw the curve for the center line of rail from E to D. Now draw the string, bracket and nosing lines, and miteis con- necting the return nosings, thus completing the plan of turnout for getting out the steps. The face of No. 3 rise is shown C' from the spring of turnout; the face of string is the solid Hue on plan. Fig. 3. Shows the j>leriiendicular to jE7C prolonged. Draw the diagonal CF, cutting (he chord line of rail at 3; draw the chord E 5; 5 3 shuws the amount {9/') of straight wood required to point of the mitca" from the verge of cap. Fig. 4. Shouts the elevation of tan, on plan, Fig. o. Perpendicular to XX, draw AD, BE and CF indefinite. Now elevate No.s. 1, 3 and 3 risers, making the face of No. 3 rise 6^'' from the spring line AD; through the center of balusters O, 0, draw the under side of rail, > Parallel to the under side of rail, draw the inclination for the center of rail indefinite, cutting AD and BE, at D and J; through J, and i>arallel with XX, draw KJJj, intersecting AD, at K, and CF, at N: then KD is the height of the wreath-piece. JD is the increased length of tangent EC, on plan; JN i^ a level tangent, aud remains the same length as C 5, on plan. Make Plate 37. 167 Kb. equal CF, Fig. 3. Now make K 4 equal the chord JEo, Fig. 3. Also make J 5 equal CH, ou plan. Fig. 3. From 5, and per- pendicular to DJ" produced, draw 5 6. Bevels. Parallel with XX, draw the gauge line PQ; per- pendicular to XX, draw QR equal to H 5, Fig. 3. Make QS equal 5 6; also make ^P equal the height KD; draw RS and RP produced to XX. lor convenience when adjusting the bevel. Parallel with XX, draw the half width of rail (2^0. cutting PR and SR, at 8 and 7. Now the bevel in the angle at P is for the joint at miter, and the bevel in the angle at iS is for the shank joint, and X 8 is the increased width of half the face-mould at the miter, and X 7 shows the increased width of half the mould at the shank. Make DG equal 6^^ for shank connecting the straight rail. Fig. 5. Shows the race-mould. With the steel square, draw the right angle ff 6 5 indefinite. Make 6 J and 6 D, equal 6 J and 6 D, Fig. 4. With J for a cen- ter, and C 5, Fig. 3. as a radius, draw arc cutting 6 5 at 5. Draw J h prolonged 2^' to point of miter. Parallel v»ith DJ and J 5, draw 5 O and DO, for the parallelogram ODJ5, that will agree, when in position, with the parallelogram FEC5, on plan, Fig, 3. Proof. The chord D 5, must equal the distance D 4, Fig. 4, and the diagonal OJ, must equal the distance Dh, Fig. 4, if so, the angle of tansents at J, must be correct. Prolong JD 6'^ to G, for shank. Make joint at G perpen- dicular to GJ\ from 5, and at right angles to J 5, draw a line indefinite to the left, also prolong the diagonal JO, to intersect the line from 5; and from the intersection (not shown), draw a line through D, for the point of contact. Make .5 3 and 5 3 eacii equal 8 X, Fig. 4. Make G 4 and G 10 equal 7 X. Fig. 4. Draw 4 8 and 10 9 piiralicl to GD; add on 2'^ to point of miter parallel to 2 3. Now trace the concave side of mould from 9 to 3, and for the convex side from 8 to 2, using a flexible strip, being careful to tangent the straight lines at the points 2, 3 and 8. 9. If it be desirable to find another point in each curve as 1 and 7, on the diagonal, then proceed as directed * for Fig. 4, Plate 24. Pig. 6. Shows the plan of an S^^ cylinder, a2"' more, and Nos. 8 and l(j *Mr. Sccor in his work on ILnnd-rallinfr has fully described this method for finding the points on the diazonal from tlio plan. Also Mr. .James U. Monckton, In his 1S78 edition on slalr-huilding, has de- scribed the points correctl.v o>i tlic diasonal from the plan. Others bav-o shown a center point on the diagonal, but not correct in all cases. At sections L and JV the dotted line Indicates the center of plank ; the tangent line is carried across the section .square to the face of crook, and intersects the dotted line; then tbrouRh Die intersection the bevel is shown appliod from the face of crook. The block pat- tern Is applied at right angles to tlie line made from bevel; the shaded p.-vrt indicates the thickness of plank required. Observe the bevels do not cross the tangents in their application. 168 Plate 37. rise are each i}4^^ still further out from the spring line AF of cylinder to the commencement of the regular treads (9^^). Next draw the tangents AB. BC. CD and DF, forming the two square parallelograms OCBA and OCDF on plan Fig. 6. Fig. 7. Shows the elevation of tangents and the treads and risers, ivhich are unfolded from plan Fig. 6. In a case of this kind, it will be more economical to find the average pitch of the tangents; then the drawing may be made on a board not over 24^^ wide and 9' 0^^ long. Thus, add together the width of treads on the center line of rail, and aroun.d on the tangeiits, including Xos. 7 and 16 treads. First, from the face of No. 7 rise to joint ot cylinder (9^^+5^^+4'^HM^'=183i^O> equals 18K'''' plus 4M^^ for the tangent AB; the opposite side will equal the same, 'Zo}i^\ Then the two tangents JSCand CD will equal 9K^^ more. The sum total (•2aii^^+23K^^+9K^'=50^0 wiil equal 56'^, which being divided by 10 risers, equals 5%^^ nearly lor the average tread. The rise is 8''''; therefore the average pitch will be 8^^ by ^%'\ Now let XX indicate the edge of drawing board; then apply the steel square to the edge of board, with 8'''' on the tongue and b]4^^ on the blade; the tongue will give the pitch FL for all the vertical lines. Now set a bevel from the edge of ])oard to Hie line FL, as shown, and draw the perpendiculars DK, CJ. BH a.iv\ AG, par- allel ioF L and equal to AB, BC, CD and DF, {iK^') Fig. 6. Next elevate the treads and risers, measured from the center line of rail on plan Fig. G, keeping No. 10 rise }4^^ from the spring of cylinder AG, and No. 14 rise J^''^from the spring FL; No. 9 tread measures 4^^, and No. S tread is 5^'; No. 7 tread equals 9^'' wide; the measurement on the opposite side is the same on tlie ceuter line of rail. Through the center of balusters O, draw the inclination for the underside of rail; parallel witli O, O and O, draw the ceuter of rail to intersect AG at M, and FL at N. The next is to give the inclination of tangents over the winders. This requires tlie best judgment of the stair-builder. By referring to Fig. 1, I'late 34, observe the direction of Nos- 9. 10 and 11 risers, they fall back froui the rail, while at Nos, 13, 14 and 15 rise, they come out towards the rail. Or, in other words, when going up the stairs at No. 10 step, the person will bo beyond the rail, and consequently the rail will feel high, and when coming down the stairs at No. 14 rise, the person will be forward of the rail, and the rail if set at the regular height would be too low. To obviate this, tlie rail must be ''lifted" over the winders at Nos. 13, 14 and 15 treads, and kept down to about the regular height or less on Nos. 8. 9 and 10 winders. If by lilting the rail, it .should come a little high, that will be a good fault, but if the stair-builder should miss it, and the rail be low, then the owner would lind fault, as then there would be danger when going down the stairs. The worst diflicuUy the stair-builder will have, and where his best judgment will be required in lifting the rail, will be over Nos. 13, 14 and 15 treads. At Nos. 8, 9, 10 and 11 treads, there will be no danger, as the direction of risers fall back of the rail, and the rail will naturally apjjcar high even when kept a little below tliu height of a regular short baluster. If th«! risers in the cylinder were to radiate from the center O, then tlie difficulty of regulating the lieight of tangents in the cylinder would be obviated, as the person would then be normal to the curve in the direction of any of the treads, but it is only where there is ample room that the winders can be so placed. Plate 37. 169 At Fig. 1, Plate 34, the diameter line is extended to the walking line, and happens to cut the walking line at Nos. 9 and 15 rise. Now let it be observed that the diameter line being extended cuts across Nos. 10 and 14 rise, thus indicating that the point on the walking line at No. 9 rise, is one rise lower than at the joint of cylii.der. At the opposite side the joint of cylinder will be two risers lower than at the point of intersection with the walking line, consequently we will have to lift the tangent at the angle D, on plan, say nearly two risers. Then set up from No. 13 tread to It \^", and from any point, say Q, on the line JPiV, draw the inclination lor the upper laugcnt, cutting the perpendiculars FL, and DK, at JL and I, and intersecting CJ at S. From I, draw the incli- i:alion of tangents to touch the arc at T, and cut the perpendic- ulars JC, HB and GA, at Z7, H and W, and prolonged to inlerstct MB at y. Perpendicular to FL draw LJ and UF, cutling DK at 2 and o. At right angles to AG, draw WC, cutting BH at 4; tlicn CU will be the height of the lower wri-ath-picce, and FL will be the height of the upper wreath- piece. ( Prolong HI to intersect iJ^at V; from 3, and at right angles to VU, draw 3 5; from ;3, and at right angles to LS, draw 2 19; from 4, and at right angles to HW, draw 4 6. Make C 22 and U 18 eacli equal the chord AC, on plan. Fig. 6. Parallel witli DK. draw the half width of rail (2^0. intersecting IL and UI, at 20 and 21. At W, allow 4^^ and at L, allov/ 2'^ for shank. Ease the angles at y and Q, to please the eye. Opposite the face of Nos. 8 and 10 rise, draw the face of rise to intereect the center of rail at 00 and 00, for points to raeasui'e from when jointing the straight rail. These points should be marked on the patterns, also the distance M, 00 (12}-r;^^), and N, 00 {V2y/'), to be con- venient when jointing the straight rail. Bevels. aa indicate the edge of drawing board, and the doited line indicates a gauge lin(! parallel with aa/ ix'ipendicular to aa, draw bf, ecjual to OC, on plan, Fig. 0. Make bh equal 2 It); also make bm equal 5 3; let bn equal 4 G; draw fm, fn and fli prolonged to the edge of board. Now in the angle at h is found the bevel for the shank joint 24, and in the angle at m is found the bevel for the joint at Z7on tiie upper wreath-piece. As the tangents UH and HW are of the same length, only one bevel will be required at the lower wreatli-piece. The angle at 72 gives the bevel for both joints of the lower wreath-piece. Fig. 8. Shoivs the face mould for the loivcr wreath-jnccc. Draw BAFio equal HWT. Fig. 7; with A for a center and U 22, Fig. 7, as a radius, draw arc at C; then with B for a cen- ter and BA as a radius, draw arc intersecting at C, join BC parallel witli AB r.nd CB; draw CO and AO prolonged. Priinf. 03 must equal the diagonal OD, Fig. G; if so, the angle of tangents at B must be correct. Make A 2, A S and C 4, C5 each equal J 21, Fig. 7: make OD equal the radius li%^^) of the center line of rail; let D G and D 7 each equal the half width of rail (2^0- Make joints at C and .F' perpcuidicular to tangents SJ* and BC; parallel with BF draw 2 8 and 3 9 lor the shank; now pivot the trannnel at O at right angles to BO, and set from pencil to minor pin the distance OD lor the center or fdVinij line of rail; then place the pencil in C, and drop the pins in the grooves, g 170 Plate 37. and fasten the major pin; then trace the curve CD A; repeat the operation lor the concave and convex side of mould. The bevels are shovtii applied through the center of plank at sections L and N; the )>evel at L is applied so as to elevate the joint at C, and at iVthe bevel is applied so as to pitch the joint at i^dowu. Fig. 0. Shows the face-mould for the upper irrcdth-piecc. The tangents being of unequal length, there will be two bevels required, as the shortest tangent connects with the easing at the upper end, the steepest bevel will apply to that joint.* Draw CB equal to tangent JZ7, Fig, 7. With C as a center and J 18, Fig, 7, for a radius, draw arc at A; with .B as a center an rise from plan. Fig. 10. From the face of No. 6 rise to B equals 1S}4^''; from B to D equals {)}.<^^, and from D to E e(iuals 4K^^ total (18};r^^^-93-2^^'^-4%^^=33'0 equals o-J^'', which being divided by G risers, equals 5X''' for tlie average tread. Then with the square, take 532^^ on the blade and the rise (8^^) on the tongue, and apply to the edge of l)oard XX; the tongue will give the line AJ required for the average jiitcli iioni the edge of drawing board. Parallel witli AJ, draw BK, CH, DG and E, equal to AB, BC CD and DE, on plan, Fitj;. 10. (t5:Ceach). Now draw the floor line Z/i, at right an- gles to CH. ,, * It is well to note this, as some use a parallel f ace-moulcl. Plate 37. 171 Make the position of No. 1 rise from D, equal tliat on plan, Fig. 10, {V)\ then elevate the risers and treads from the tan- gents on plan, making the risers parallel and the treads at right angles to the pcrpeEdicular lines, keeping No. 4 rise %^^ from the spring line AJ; No. 4 and No. 5 treads equal 4'^ and *y^ each. Now through the center of baluster mm, draw the incli- nation for the under side of rail. Parallel with m.m., draw the center of rail MN; it will be observed that the regular pitch intersects the spring of cylinder at N. On Ihe plan. Fig. 3, Plate 34, extend the diameter Uiie to cut the walking line at A; let it be observed tliat the joint of cylinder comes in No. 3 tread at C, and the diameter line cuts across No. 4 rise; and will rest at about the center of No. 4 tread, thus indicating the point A will he one rise and a half above the point C, on the inclination ot rail. Then that is about the height above No. 3 tread, that tlie in- clination of the upper tangent should cross the perpendicular AJ, in the elevation say at P; now allow fnan iVto Q sufllcient to make an easing on the straight rail to connect the shank of wreatli-piece. Then from Q, through P. draw tangent cutting BK at O. and prolonged to intersect CiJ at i?. From the floor line at L set up lialf a rise to the under side of rail taken from the first flight (4^^ in this case), and \W^ more to the center of rail, making SM''^ from the floor line LL to the center of rail at S. I'arallel to LL dvavv SC, cutting DG at T: connect TO, cutiing CH at U. From U and P, and at right angles to CH, draw UA and PiiT. cutting BK ai V and 9. then CU will be the height fin- the lower vireath-piece, and AP w ill be the height for the upper wreath-] )iece. Prolong Z70 to intersect PH at Y; make U7 and CS each e(iual to the chord AC, on plan Fig. 10. From 9, and perpendic- ular to Pi? produced, draw 9 10; from Vand at right angles to UY, draw V 12; at O draw the half widtli of rail (2^^) parallel with BK, cutting PR at 13 and OU at 14. Bevels. Parallel with LL draw the dotted line WIS: per- ])i'ndicular to LL draw WZ, equal to the radius OC, Fig. 10; nr.ike W 16 equal V 12, an^d Wl7 equal to 9 10; niake W 18 equal tlie height CU; draw Z IS, Z 17 and Z IG prolonged to LL, for the bevels required. Fig. 12. Shines the facc-mmi.ld at the starting. Draw the rectilineal parallelogram OEDC. making EO and DC, a!.so OC and ED enr-h equal UT and TC, Fig. 11. Proof. The chord EC and the diagonal OD must each equal Z78, Fig. 11. If so, the parallelogram is correct. Make E 2 and iJ7 3 each equal O 14, Fig. 11. Let C4 and C .') each ecpial the half width of rail (2^^). Pivot the Iranimel in O, with the arms centered on the semi major axis OE, and trace the concave and convex sides of face-mould, as has hvx n described; add on 2^'' or more straight wood at ^parallel to OE, to help the easing at the joint connecting the straight rail. The minor axis OC forms the joint at C. The section at JV" shows Ihe bevel found in the angle at IS. Fig. 11, ajiplied so as to pilch tlie joint at Cup, and at section iWthe square is shown applied from the face of crook, allowing the tangent ED to be horizontal. Fig. 13. Shows the face-mould for the upper torcath- plccc, connrotinr; the casement, Fiij. 11. Make CB equal tangent OU, Fig. 11, With C as a center and 7 H, Fig. 11, for a radius, draw arc at A; with J3 as a 173 Plate 37. center, and tangent OP, Fig" 11, for a radius, draw arc intersect- ing at A; join BA and produced; draw AO and CO parallel with BC and BA prolonged, for the parallelogram OABC on the cutting iilane. Proof. The diagonal BO must equal the distance i? 7, Fig. 11. If so, the angle of tangents at B is correct. Make AE equal P 19, Fig. 11; make joints at E and C at right angles to tangents BE and BC; make A ii and A o each equal O 14, Fig. 11; also make C 4 and C5 each eiiual O 13, Fig. 11. Let CF equal OY, Fig. 11; join OF, and pro- longed, for the direction of m.inor axis; make O J" equal to radius OC, on plan Fig. 10. Make J 6 and J 7 each equal the half width of rail (2^0; draw the shank 2 8 parallel with BA. Pivot the trammel at O with the arms at right angles to the somi-minor axis OJ, and set from pencil to minor pin the distance OJf for the ceiiter line of rail; then place the pencil in A, and slide the major pin until they both drop into the grooves, then fasten the major pin and trace the center or fnlUn(j line; pioceed in the same manner to trace the concave and convex bides of face-mould. At D and P, the sections show the bevels applied through the center of plank; the bevel found in the angle at 17, is applied at JD, and the bevel found in tlie angle at 16, Fig. 11, is applied at P, alwa3-s from the face or upper side of crook, which must be true, or out of wind. Fig. 14. Shows plan of the center line of rail for the quarter ijace winding landlncj in tlw third story. The diameter of cylinder is 8^^, and the radius for the center line of rail is ■i^i^''. Make AE equal OV,^^; v^itli a radius of 4^^^^ and O for a center, draw tlie srmi-circle ACE, for the center of rail. Draw the rectilineal paral!eloo;vam ABDE, tangent to the curve at the points A, C and E; prolong BA and DE for the direction of straight rail. In practice for a working drawing, the direction of winders and flyers showing how oblique they cut the string line, are supposed to be laid out on paper full size from the scale drawing. Fig. 3, Plate 34. The plan shows No. 11 tread measiires 4^^ on the center line, and No. 12 rise Ji^^ from joint or spring of cylinder. Fig. 15. Shows the devchypmcnt of tangcuts and tlie elevation of treads and risers measured frcmi the tangents and straight part on plan, Fig. 11. For convenience, first find the average tread from the face of No. 10 rise, around the tangents to face of No. 1.') rise; thus from No. 10 rise to spring of cylinder ('J'^+4'''^ Ji^^=13^X^^) e(iuals 133i^^, pl^^i AB, (4M^0' I'l"^ ±'''"»i -^ *<> f'i<-'« of No. l.'i rise (SK'O. equals (13M^^-f-4%^'+83.<'^=-27^0 l'<>i" tlie total 27^^ which behig divided by 5 risers (27^^H-.5=53id^^) equals 5>'s^^ nearly for the average tread. Let XX indicate the edge of board; then with b^'^ on the blade, and b" on the tongue, draw the direction of the perp-^i- dicular EF from the tongue. Parallel with EF, draw DG, CH, SJ'and AK indelinife. Now elevate Nos. 10, 11, 12, 13, 14 and 1.5 risers and treads, keeping No. 12 rise %^' from the spring line AK, as shown. Draw the floor line at the lauding; from the floor line set up 4^'' Plate 37. 1T3 to under side of rail, aud IH^^ more, or BH^^ to the eentor nf rail at L, At right angles to EF, draw LN, cutting EF at F, and DG at N. Through the center of balusters O, O, draw the under side of rail; parallel with 00; draw the ceuter of rail FM, inter- secting AK at M; draw arc equal to the half thickness of rail at the center of baluster No. 12 tread, then from the point N, draw a line tangent to the arc, intersecting the inclination MP at Q, and cutting the perpendiculars CH, BJ and AK, at R, y and iS. From S, and at right angles to AK, draw /SC, cutting JB at T; from B, and perpendicular to CH, draw RTJ, cutting GB at Y; then TJF is the height for the upper wreath-piece, and RC is the height for the lower wreath-piece; and NR, Rj and jS show the increased length of tangents in elevation; make joint at W at right angles to QS; ease off the angle at Q to please the eye, and to tangent the joint at W; make JJ 2 and C 3, each equal the chord CB, on plan. Fig. 14. Parallel with DG, draw the half width of rail, cutting the tangent NR at 4: from T, and at right angles to JR/S, draw TS. Bevels. As the tangent FN is level, there will be but one bevel for the landing wreath-piece, and as both tangents Rj and yS, for the lower wreath-piece, have the same inclination, there will be but one bevel required for both joints. Let aa indicate the edge of board, and the dotted line be a gauge line parallel to aa; draw 6 7 perpendicular to aa, and equal to OC, on plan, Fig. 14; make 6 8 equal T 5, aud 6 9 equal the height TJF; draw 7 8 and 7 9 produced, to edge of board aa for the bevels iu the angles at 8 and 9. Fig. 16. Shoivs the fncc-mould for the quadrant ACO, Fiq, 14. Make WAB equal WSY, Fig. 15. With A as a center, and R, 3. Fig. 15, for a radius, draw arc at C, again with B as a center and AB as a radius, draw arc intersecting at C; with the same radius and C, also A, as centers, draw arcs intersecting at 0; draw OA and OC indefinite, and we have the parallelo- gram OABC on the cutting plane, that will agree with the par- allelogram OABC, on Plan Fig. 14, whtn in position. Proof. The diagonal OB must ecjual the diagonal OB, on plan Fig. 14. Draw BO indefinite for the direction of minor axis. Make OD e.iuai OC, on plan Fig. 14; make A 2, AS, C4 and C 5 each equal N i. Fig. 15. Let D 6 and D 7 eaoh equal the half width of rail (2"). Make joints at C and W at right angles to the tangents BA and BC. Parallel with B Wdraw 2 8 for the shank. Kow pivot the trammel at O, with the arms at right angles to the semi-minor axis OD, Set from pencil to minor pin the distance OD; then rest the pencil in C, drop the pins in the grooves. Now fasten the major pin, aud trace the center curve of face-mould. Proceed in the same manner to trace the concave and convex sides of moidd. The sections G and H show the application of the joint bevel, which is taken from the angle at 8, Fig. 15. Fig. 17. Shov:s the facc-mouhl for the quadrant OEC, Fifj. 14. Diaw tlie rectilineal parallelogram OCDE, making OC and BE, also OE and CD, to equal FN and NR, Fig. 15, respect- ively. Proof. The diagonals OD and EC must equal F2, Fig. 15. Make C 2 and C 3 each equal the half width of rail (2^0 ; let 174 Plate 37. E 4: and J575 each equal Na, F.ig. 15. Now pivot the trammel at O, with the anus resting on the semi-major axis OS. Then set I'rom pencil to minor pin, equal to OC, and trom pencil to major pin to equal OE. Xow trace tlie center line on face- mould from ^to C; proceed in lilve manner to trace the concave side of face- mould tlirougli tlie points 4, 2, and convex side throngli the points 5, 3. Parallel with 4 5 add 2^^ of straight wood EF. to help the easing in wreath-piece, to connect the straiglit rail LF, Fig. 15. The bevel at section M is taken from the angle at 9, Fig. 15, aiul applied from the face and through the center of plank, so as to pitch the joint at C down. At section iVthe block pattern is applied parallel to the face of crook, and at the center of plank. Fig. 18. Shoivs the caacmcnt required at the Inndinfj of the Jirst flhjht. Through the center of balusters G, O draw the underside of rail parallel with O, 0; draw the center of rail AB indefinite. From the floor line set up 4'^ to underside of rail, and 1}4^^ more to C for center ofjrail (4"— IK^^^^oK^O. equals 5^^^. Parallel with floor line draw CB, intersecting AJ3 at B. Now ease off the angle ABC, to please the eye; plumb over the face of No. 17 rise; make a point D, showing the face of landing rise No. 17 on the easement pattern. Now marlc on the pattern the distance (14^^^) from D to join C, for the amount to allow for easement when joiuting the straight rail. Jointing the rail. No. l length. -The leno-th from th -pring of turnout to spring of cylinder equals G'' S^^ Fig. 2, Plate :'4; the length of shank on face-mould, Fig. 5, equals 6'^ and the distance, M, 00, Fig, 7, equals I2y/^, and from 00 to joint at E equals V Z", or 2'' ^\i" from M to joint at E, as the allowance for ramp. Then the dimension to cut No. 1 length (G'' 8'''' — (2'' "No. 2 length. From joint of cylinder to face of No. 17 rise landing equals ?J V, Fig. 2, Plate 34. The easement connecting the wreath-piece at the upper end measures from iVto 00 12)^''^; then 2M/' minus 12K^^ C^^ 1^^— 1^ 0K'^=1^ 0}4''), equals 1^ OK'''', as the distance between tiie two points 00, Fig. 7, and tlie point D, Fig. 18. This will allow QH'^ from D to joint at A, Fig. 18, and &li'^ from 00 to joint near P, Fig. 7. as shown. JVo. 5 length. The length from face of No. 17 rise to spring of cylinder starting second flight equals 3'' 0'^ Now the hori- zontal distance from D to C, Fig. IS, equals 14^^^ for the level length on easement landing of the first flight, and we have 2," of straight wood, EB, added on to the lower end of wreath-piece, Fig. 12 (143^ ^^+2^^:=!^ 4K''0. which equals V ^Y," to be de- ducted from the wJiole length (3'9'^— 1' 4K''^^2^ 4K^0. equals 2' 4K^^, the length to cut No. 3. iVo. 4 Length. The length from spring of cylinder starting to spring of cylinder landing in the second flight equals 7^ OM^', see Fig. 4. Plate 34. The length of easement. Fie. 11, measures from t.pring of cylinder at iVto Q, G'^, and from Q to joint at M, 9K'^; (G'^+9K^^=15X'0 equals 15K^' to allow for the easement; and for the ramp. Fig. 15, allow G'^ from M to the point plumb over the face of No. 11 rise, and lihi^^ more to the lower joint of ramp at P, which being added (14K'^--6'^=1' SK''^). equals V 8X^^ plus the l ength of easing al)ove'(15K'0 [1.8>2+1' 3K''=3' * Note.— The face of second lise is mostly taken as a fixed point from which to measure for the first lensrth in straight flights; in this case we have taken the measure from the spring of turnout. Plate 37. 175 8''], equals 3' 0'^ to be deducted from 7^ 0%'^ (7^ 0%^'— 3^ 0'' =4' OM^O. equals 4^ 0%-", the leugth to cut JNo. 4. JV'o. 5 Icnrjth. Is the second level, and is marked 6^ 7^' from spring of cj'Jinder landing in the third story to the spring of quarter cylinder. Now we have allowed 2^^ of straight wood FE, on face-mould, Fig. 17; then deduct 2'^ from the whole length, 6^ 7^^, equals 6^ 5'^ to cut No. 5 length. In case there be straight wood on the quarter turn, deduct the amount of straight wood in addition to the 2^^ allowed on the wreath-piece. The short piece to wall equals 3^ 0^^ in length. After the joints are made, bolted and dressed off, the rail is ready to hang. IIwGiNG THE Eail. In large stair shops it is customary for one set of men to step up, and another to hanj; rails and complete the job, while neither get them out. So a few words in addition to what has been said for a platform stair-case, will not be out of place here. When proceeding to hang the rail over winders, as in this case, a very good plan is to elevate the rail on stanchions notched out to receive the rail. Let the rail be raised or lowered to suit the height of the regular balusters, and plumb carefully on the convex side to points made at intervals on tlie steps and around the cylinder. The distance in to mark the points from the face of bracket line equals half the thickness of balusters (1") plus the half width of rail ('2"), equals 3" for the convex side of rail. Now when the rail is plumb to the points around the cylinder and the straight part, and also the height to suit the baluster is correct, then plumb through the center of cap for the center of newel on the floor, and take the height from floor to under side of cap for the height of newel. The newel may now be cut, set, and the rail placed in position, using stanchions for supports, set near the joints and out of the way for glueing and driving up the nuts. Next if the straight rail have any bends, they may now be straightened by l:)racing; plumb and bore for the balusters, dress off the under side of rail and put the balusters in place, using a thick glue for this purpose. After the balusters have been set and glued at intervals, remove the stancliious and complete glueing in the balusters. Some prefer to hang the rail at once on a few balusters; the stanchions make the rail more solid, and may save a broken twist, and is better for straightening the crooked part of straight rail, whicli often has to be done. If the stairs are got out and stepped up carefully, the newel and balusters may be cut at the bench. It will be noticed at Fig. 4, that the under side of rail is drawn through the center of baluster, and its length is naught at that point, while the under .side of rail is raised up 6" above the first step to allow for easing at the newel; then whatever height from top of step to under side of rail, we make the short baluster, the newel will be 6" longer from the top of first step to the under side of rail, or from the floor to under side of rail will be the height of arise [8"] more, or 14", in this case; for instance, the height of a short baluster from the top of the .step to under side of rail at the center is 2' I'J". and the newel is fi" higher than a short bal- uster, plus the height of first rise 8", will Ofjual for the whole height of newel from the floor to under side of cap 3' Sli", (2' in" i-C"-rS"= 3'3;i"). For pin top balusters allow lY," to enter the rail; the height of couare is half a rise for the short ones, and the length of turning bein" all the same, will make the squares for the base of long bal- uster equal the height of a rise; sometimes the balusters are square at the top and also at the base; the squares at the base and top for both balustei's being the same height, then the turning for the shafts will be the ditference of half a rise; .sometimes the squares at the base are one height and the shafts, also one length in turnino-. and the squares at the top of different lengths. To find the length of odd balusters under the wreath part of rail, study Fi"^s. 1 and 2, Plate 32; to prevent a confusion of lines, this is omitted liere, however, for common work, if the balusters have their shafts ail turned the same length, the difference will be in the square at the base, and will not be objectionable unless the propor-. tion is too much out of the way. ^r, ^ . ,. .,. Boeing the Rail. This is best done on the .stairs after the rail is hung, the plumbing and boring for the balusters is a short job for a flight of stairs; for Ijoring around the winders, use a ratchet or angle brace. 176 Plate 38. PLATE 38, Plate 38, Pigs. 1 and 2. [Scale }4^^=1^.\ Exhibits plan elevation of a doithle qiiaHcr-jxice winding stair-case, having a (inart£r pace and four risers to the landing in the second story; the quarter pace nt the starting has a circidar ^vall string, and tke circidar corner is ornamrrded with a nitch. The heiglit of story is V.y C from top to top of joist; the joist are 10'^ wide in the second story, and they are to be stripped across for lath and plastering witli V by 3^^ strips, malving IV^ for tlu; depth of joist. The width of hall is 8^ 2'^; the height of door under the stairs is 7^ C'^; the length of well to receive the stairs 13' 9'^; tlie rail is 4^' by 2K''. Balusters are to be 2'^ by 2^\ Fig. 1. Shows the plan, having 20 risers and starting with a quarter i>ace winding around a cylinder of 8'^ radius, and land- ing witli a ([uarter pace winding on a cyHnder of 6'^ radius to a level (piarter pace at No. 16 rise; thence with four flyers to the latuliug in tlie second story. The height of story from top to top of joist equals 12' 6^'', which reduced to inches equals 150'^ being divided by 20 risers (1.50''-^20=7K''), equals 7}4^^, as the heigiit of eacli rise, and if we use a constant of 25'^ in tiiis case, the relative width of tread to rise will be IC, or 7)4'^ by IC for the pitch. See rule for width of step in proportion to height of rise, page 87. The cylinders are laid otf so as to have the well hole in the center of well; tlien Nos. 2 and 16 rise will be about the same U'ugth as at the flyers or straight steps, ?/ 7" . The walking line is drawn \h" from the face and parallel with the outer string. No. 9 tread is placed at the center of well in this case, and the balance of treads is spaced ofl: on the walk- ing line either way equal to a regular tread (IC^). Nos. 1 and 2 treads are slightly curved and increased in width on the walking line. The length of Nos. 2 and 16 treads should never be less than those at Nos. 9 or 20, but if anything, they should be longer ?/' or A^" . If shorter the stair-case Avill have a contracted appear- ance. An incli or two at those points makes a great difference either way. Draw Nos. 7 to 11 and also No. 17 rise at right angles to the outer string; then locale the short balusters on Nos. 11 and 17 treads, and space off for the intermediate balusters on the center line of rail. The face of risers may now be drawn to suit the balusters, and through the points on the walking line to intersect the wall string. It will be observed at Fig. 4, that the baluster on No. 12 step is located near the center of tread; this is done to allow the bal- usters to appear near the same from centers; the tread is gradu- ated to aid the easing on tiie lower edge of string.. The face of wall string projects IK^'' from the line of studs, allowing %'' for lath and plastering, and \" for the thickness of base; No. 11 winding tread is 12'''' wide at the face of wall string No. 12 ami vi treads are 19);^'' and 26M''' each. No. 14 is C to the face of string in the corr.or, and 243^'''' from the corner to face of No. 1.5 rise. No. 15 wimiing tread is 243^j''' wide; No. 7 rise is ^" from the spring of circular wall string, and No. 2 rise is 6'^ from tlu> spring at the starting. Nos. 1, 2, 3, 4 and 5 treads arc each ISM^'' wide, and No. 6 is 1&}4^^ wide at the wall string. This quarter pace winding for a working drawing should be drawn full size on heavy paper. AA shows the bearers S'^X^'^ Plate 38. IW seautliug, to which rough brackets are nailed for supports; they are placed iu position so as to transmit the weight from the outer string to tlie walls. SB shows the bridging on the quarter pace opposite the bearers, the same should be done at the landing. Fig. 2. SJiftn's the clevntion. The height AB, [12^ 6^'] being divided into 20 risers at 7>^'^ each, the joist is shov, n 10'^ wide and f^ stripping for lathing, making 11^' as the depth to allow for joist. The height from floor to top of platform is 10'' 0'^. and the joist, plastering and flooring of platform, equals 12^^ and the carriage, lath and plastering, plumb under and including No. 15 rise, will equal 16^' mere, (12^^+16^^=-:^ 4'^) equals 2^ 4''^ Now tii(! door under the platform is 7' 0^^ phis 2^ 4^^ [7^ 0'^+ 2^ 4^^=9' 4^^J e(iuals 9' 4^', this deducted from the height of platform [10' 0"—^' i'^=0^ S^^] equals 8^' for the finish over the door plumb undcT No. 15 rise. Often a full finish over the doors and windows under the platforms ca-niot bo had. In this the stair-builder has to arrange as bL'st he can. but whenever practicable, let the finish be com- plete, as that improves the work. Headway. Count down from the landing 14 risers at 73^'' each (U >J>^''=Yf'''=S' 9'0, equals 8' 9'^ from the top of lloor to a]i(l including No. 7 rise. The width of joist, stripping; flooring, lath and plastering eciuals 13'^ Then S' 9^'' minus 1' 1^' equals 7' 8'^ as the height from No. 6 tread, plumb over No. 7 rise to the line of plastering. We will locate tlie face of trimmer plumb over the center of No. 6 tread, at the walking liiie. This will allow ample head room. This locates the trimmer in the second floor 9' 3''' from the wall for the length of well. For the quarter pace at the lauding iu the second story, the face of joist is located 6''' from the face of No. 20 rise, and 7' 2^' from the wall; the half width of hall is i^ \'^, and the radius of cylinder equals 6", and the thickness of level fascia is 1'^. Then tiie width to trim the huidiug [V 1^'— (6''-f l'')=3' 6^']. will equal 3' 6^' by •?/ V\ as shown on plau Fig. 1. SS shows the spring of cylinder. The lengths to measure for jointing the rail are shown taken from spring to spring of cyli)ider (6' 9'^) for No. 1 length; No. 2 length is taken from spring of cylinder to face of No. 20 rise (3' l}i'')\ No. 3 length is taken from the face of landing rise to tlio spring of quarter cylhider (2' 0^'), and No. 4 length is taken fnmi spring of quarter cylinder to wall on plan (4^1''); all in- clining lengths must be taken parallel to the true pitch, and all level lengths parallel to the floors. Fig 3. Shows the plan of cylinder at the newel draivn to a H^^ iicalc. The balusters are shown spaced off on the center line of rail, and the risers are drawn iu to suit; the radius OA equals 8'^ for the line of cylinder. The face of No. 6 rise is 2%'''' from the spring of cylinder. The cut-out of No. 6 tread is 5Ja''; No. 2 step should be increased in width, so as to give the proper .space between Nos. 2 and 3 balusters and at the newel post. Fig. 4. Shoivs the halutstcrs spaced off around the 12'^ crjlLnder, and face of risers drawn to suit the spacing of halustci's. The nosing line of steps intersects with the return nosings and gives the miters for the return nosings. The miter on No. 14 step is very acute, and may be relieved by rounding off at the 178 Plate 39. poiut, as shown in Fig. 5, Plate 34. The face of No. 13 rise is Z}4J' from the spring of cylinder. The width of Nos. 11 and 12 tread on the face string is 8>2^'' and 5^^ each. Fig. 5. Shows how to obtain the curve of grounds around the head of a nitch. The solid line AB sliows the line of plaster, and the dotted line CD shows the line of studs, the radius being 4^ 2^^; iJJ'shovvs the diameter of nitch. 'With Hnud J for centers, draw focus at Q from Q; through H and J draw the radial lines indehnite; parallel with HJ draw a line tangent to the semi-circle at K, in- tersecting the radial lines at L and M; tlien LM is the stretchout of the semi-circle HKJ; now divide the diameter HJ into any number of spaces, as 1, 2, 3, 4, &c.; then draw 1 X, 2 X, 3 X, &c., perpendicular to HJ, and intersecting the semi-ch-cle HKJ at 5, 5, &c. From Q and through 5, 5, &c.. draw lines to intersect the stretchout LM at (5, 6, &c. From L 6. 0, and iWdraw lines perpendicular to LM indeiinite. Draw PR indefinite and par- allel to LM, cutting the perpendicular at 7, 7, 7, &c., to indi- cate the edge of nitch. Now transfer 4 X, 3 X, Ac, to corresponding points 7 4, 7 3, &c., as shown, and throngli the points P, 1, 2, 3, 4, &c., trace the curve for the head of nicch. Then licrf between P and R, and bend to the required circle. PLATE 39. Plato 39. [Scale %^'=l foot]. Shows the front and ivall siriwj for the stair-case, Fvjs. 1 ami 2, Plate 38. Fig. 1. Shoivs part of the outer strinrj and the panel loork underneath, conncctlwj the cylinder at the starthifi. AB is the spring of cylinder, CD, EF ninl GH bhow the length of staves in the rough, and are taken from the cylinder. Fig. 3, Plate 38, in the same manner as explained for Figs..O and 0, Plate 34. Tiie cut-out for No. rise is 3''^ from the spring of cylinder, and the cut-out for No tread is 5 34'^'. The width of outer string below the internal angle of tread and rise is 7^\ The lower edge of string is eased olf in this case down to the lloor, and the i>anel work is made to suit.* The first length [6' 9^'] for rail is taken parallel v.ith th(! internal angle of the regular treads from joint to joint of cylin- ders. No. 1 rise is shown reduced to OJ^^^, to allow for the step. Fig. 2. Shows the \^]^]^cr end of outer string, Fig. 1, con- necting the 12''^ cylinder at AB, and also the lower end of face strings Fig. 3, connecting the cylinder at CD. Tiie regular ^t^aight treads 7, 8, 9 and 10, are si)accd off wiih the dividers, and the pilchboard is applied to the gauge line NP. . No. 11 tread is 8K^^ and No. 12 is 5^^ on the face string. The joint of cylinder is 3'a^'' from the cut-mit of No. 13 rise: LM, KJ, HG and EF show the lenutli of staves in i\w rougli; the angles R and S are eased otf to please the eye; tlic length iV W for the straight part of rail is shown taken from the spring of cylinder AB. The two cylinders may be tongned and glued to this string at the bench, and set up in the building. Plati: p,, in elevation. Then if lines be drawn from the center of each baluster in the same manner to intersect the under side of rail as shown for No. ?> baluster on No. 4 step, which intersects the tangent at L, and again intersects No. 4 step at Y, and the underside of rail at U; then No. 3 baluster is equal to UY, longer than a regular short baluster. The other balusters may be elevated in the same way. Fig. 6. Shows plan of the center line of rtdl for the 12" eiilinder, Fig. 4, Plate 38. The radius OA for the center line of rail equals fi%'"., AB, BC, CD and JDE show the tangents enclosing the semi circle ACE; draw the direction of the straight string AF and ECr. Now show the direction of the risers cutting the tangents. The face of No. 18 rise is 3%^'' from the spring of cylinder measured on the face of outer string, if measured on the center line of rail, it equals "/\ When proceeding to draw the elevation of treads and risers, take the width of treads from the center line of rail on the straight part, and in the cylinder follow the line of tangents. Fig. 7. SIioivs the clcvatmi of tmvjents; they hc\noints Cand .£7 always, if not, the work must be gone over again and corrected. If the elliptic curve should cross the tangents at the points C and E. or should it fall short, the learner will he sure the mould is not correct. The sections S and J" show the bevels applied through the center of mould. The bevel shown at 9. Fig. 7, is applierl at both joints. The distance LE shows the amount of straight wood (6''' to allow' when cutting the straight rail. Fig. 10. Shows the easement pattern landing in the second story. The under side of inclining rail is made to pass through the center of balusters O, O, on No. 19 and 20 tread, and the under side of level rail is lifted above the floor half a rise (o^'i'^); the face of No. ;20 rise is show-n on pattern at 2. The amount of straight wood is 3^^ as shown for the inclining part, and for the level from the face of No. 20 rise to the joint on easement, equals V o" to allow when cutting and jointing the rail. For con- venience, these measurements .should be marked on the pattern. Jointing the rail. The easement pattern over Nos. G and 7 treads. Fig. 2, shows \\}" from the spring of cylinder to the fixed point H, marked on the pattern. And tlie ramp pattern over Nos. 11 and 12 treads, Fig. 7, shows \' ^" from the spring of cylinder to the fixed point I, also mark on the pattern. These points shoidd be transferred to the crook, so as to be seen on the under side of rail, when bolted together for proving the exact lengths before leaving the shop. Tlie length from spring to spring of cylinder equals 6^ ^" for A^o. 1 length, (see Fig. 2. Plate 38); now deduct the two meusure- nienls almve (lO'^+l^ 9^^=:2^ 7^^) from the whole Icngtli (i')^ 9'^— 9/ V^\f 2") equals 4' 1" between the two fixed points H, Fig. 2, and I, Fig. 7. Platk 41. 185 For No. 2 Icngtii. TIic shank at Fig. 9, is 6'''' long, and the easement pattern at Fig. 10, shows 3'^ from the joint to the fixed point 2, and the whole length from spring of cylinder to face of landing rise, equals 3^ IK^'; from which deduct the above (3'' IK'^— (C^'+3^0=2MX^0 equals 2^ 4K^^ the length to cut No. 3. For No. 3 length. We have 1' ?>" to the joint on easement pattern to be deducted from the whole length '7/ Q" fiom face of No. 20 rise to spring of quarter cylinder, which (2^ 0^''— 1^ 3''''= 0^ 9'^) equals 9^^ for the short length, the shank of quarter turn may be increased, aad one joint made to answer; from quarter turn to wall is 4'' V. PLATE 41. Plate 41. [Scale M'^=l foot]. Exhibits two different face-moulds for a quarter j:)ac.ervinding, starting froni a newel, and having five winders; the regular tread is 10''''X~''^ rise. Fig. 1. Shows a quarter circle starting from a 6''''X6'''' neivel, with five winders. The radius OA for the face of outer string equals \2'\ the distance AB is 15''' from the face of string for the walking line. The regular tread 10^'', is spaced off on the walking line; No. 1 tread is swelled and increased to 11 1-^^'' on the regular tread line; the balusters should be spaced off equal, and the risers drawn from tiie l)alusters through the spacing on the walking line; in this case it will be ol)served that No. 3, 4 and 5 risers radiate to the center O, and No. 2 rise nearly so; now if the radius OA be prolonged to cut the walking line at B, it will be seen in this case that the line BO remains on No. G step from the intersection at the spring of cylinder to the intersection of the walking line, hence a person walking up or down the stairs, the hand will be normal to the curve of rail. Wherever the stair builder can arrange his winders in this way, more satisfaction will be the result. The center of newel is 5'^, and the point of mitre is 4''^ from the spring line OC, Fig. 2. On the face of wall string. No. 1 ticad is 15'^; No. 2, 18 J4''; No. 3 is 27^' to corner: No. 4, 27'^; No. 5, 19'^, and No. 6 is the regular tread IC wide; these measurements may be marked from the drawing made to the full size, on to the scale drawing, or a slip of paper for convenience, when lining off the wall string. At No. 3 winder, a temporary pattern made of thin lath is shown tacked and braced; this pattern, by turning over, will answer for No. 4 winder; patterns may be made for the other winders and all strujig up until needed, and the drawing, if made on paper, may be rolled up for future use. The lining out of the wail and front strings is the same as previously described. Fig, 2. Shows the plan of tangents AB and BC, inclos- ing tJie center line of rail ADC. The radius OA, of the quarter circle ADC, equals V2%^^; the center of newel is located C from tiie quarter circle, and the point of miter is 4'^ from the sjiring line OC; the face of No. 7 rise is 8}^^^ from the spring line OA; the dotted line ^^indi- cates the face of cylinder, the radius of which is 12^''. 186 Plate 41. Fig. 3. Shoios the cicvelopment nf ianoenis from plan Firj. 2, the dertilion nf treads and riftcrs, and aho the tanfjcnts for face-mould. Let XX iiulicato the edge of drav.'ing board; make AB and BC equal AB and BC, Fig. 2, perpeiidieular to XX: draw AD, i5i7aud CF, indefinite. At A, set up the licight of G risers, and elevate the treads and risers from their intersection with the tan- gents on plan, Fig. 2; mal?e the face of No. 7 rise 8^'^ from the spring line AD. Tln-ougli the center of balusters O, O, draw tlie underside of rail parallel with 00, ch-awtlie center of rail, cutting AD at G; mark tlie point H, on the center of rail, plumb over No. 7 rise, for a fixed point when cutting the straight rail. Now 4^^ to the right, and jiarallel with CF, draw tlie point of miter JK; from the top of first step set up r/^ to the underside of rail, and I'^i more, or QH^^ to the center of rail at Z. We are now ready to elevate the tangents; if a radial line be drawn to the center of No. 2 step. Fig. 1. on the walking line, as OC, it will be seen. The line is all on No. 2 step, and indicates that tlie direction of steps is very near normal to the curve of cylinder or rail, and the position at Cwill tlierefore admit of keeping the rail down to the regular height of a short baluster. Tlien we will in this case give the lower tangent the inclination of No. 2 tread and rise, as sliown at SS: draw the underside of rail parallel with SS: draw the tangent imlefiuite for the center of rail, cutting CF at L and BE at N, and at the same time intersecting a level line from Z, say at n, so as to allow a small easing into tlie cap. From N, draw the upper tangent to intersect GH, at any point beyond G, sufficient to form an easing connecting the wreath with the straight rail, we will say at M, and cutting AD at P, prolong FN to intersect CF at Q; parallel with XX draw LR, cutting BEKi T; parallel willi XX draw GU. cutting LN. ]>ro- diiced at V. From 27, and perpendicular to VL, draw J7W; from T, and perpendicular to PQ, draw TY; parallel with BE, draw the lialf width of rail, cutting LN and iVPatUandlO. Make R 12 equal the chord ACou plan, Fig. 2; make P 1'.) equal twice NU, prolong the tangent NL to intersect JK at in. Bevels. The dotted line 14 1.5, indicates a gauge line par- allel with XX; at right angles to XX, draw 1.5 16, ecpial to tlic radius OC, Fig. 2; make 15 17 equal UW, and 15 14 equal TY, join 14 and 1(>, and 17 16; the angles at 14 and 17; give the bev- els required; make P IS equal 2^^ for shank on face-mould. Fig. 4. Shoivs the face-mould. At any convenient place on the pattern paper draw JBClo, equal to NL lo. Fig. ;]; with C as a center and P 12. Fig. H, for a radius, draw arc at A; again, with B as a center, and tlu^ tangent PN, Fig. :;. for a radius, draw arc intersecting at A; draw BA prolonged, parallel with AB and BC, draw CO, and AO pro- duced, for the parallelogram OABC, on the cutting plane, or plane of plank. Proof. The diagonal OB must equal the distance 12 19, Fig. 3; if so, the angle of tang(>nts at B must be correct. Make CD equal NV, Fig. 3, draw DO, ]>roduced, for the direction of the minor axis; make OF equal the radius OC, Fig. 2; make F2 and F3 each equal tlie half width of rail [2'^^ make A 4 and A 5 each e(iual N'J, Fig. 3 ; make CO and C7 each equal iVlO, Fig. 3; draw 8 and 7 '.) parallel with B 13; make joint at 13, perpen- dicular to tangent B 13, and the joint at H, make at right angles to tangent AB ; draw 4 10 parallel to HB. Now pivot the tram- Plate 41. ISt incl at 0, and set from pencil to minor pin the distance OF, for the center of rail on the face-mould; then place the pencil in tlie point at A, and drop the pins in the si'ooves; then fasten the major pin and trace the cnrve for the center of rail on th(^ face- mould shown by the dotted line AFC; now draw the concave and convex sides of mould in the same way. The bevel shown in the angle at IT, Fig. 3, is applied at sec- tion L, and the bevel shown in the angle at 14, Fig. 3, is shown applied at section N. For the joint at L, connecting the newel cap, make the joint and apply the bevel in the usual way. After the bevels have been applied, and the wreath piece dressed off to the plumb on the concave and convex sides, then draw the line 2 2 through the center of crook as shown at section L, and apply the bevel .shown in the angle at 13, Fig. 3, from the joint and through the points 2, 2 giving the plumb cut JK, Fig. 3, from which lay off the center of rail Zn, Fig. 3, the distance to 13 Z above the point 13. A short piece may be glued on the upper side of crook to increase the tliickness of plank for the easing, if required, as shown at 00. The section at L shows the block pattern lifted up from 13 to Z, at the center. Now shape the easing at right angles to the joint JK, as shown. To obtain the length of odd balasters, make CH on plan, Fig. 2, equal VU in elevation. Fig. 3; join OH for the director on plan. From the center of Nos. 3, 4 and 5 baluster, draw lines to intersect CB and BA at 9, B, and 10; now transfer the point marked 10 on No. 5 tread, also the poiut 9 on No. 3 tread, and the intersection at B on No. 4 tread on plan, Fig. 2 to corre- sponding treads in elevation. Fig. 3, as shown. The bxiluster on No. 4 tread happens to come on the angle line BE; now elevate the balusters parallel to the perpendicular, to intersect the underside of rail as shown. As the short baluster at O is naught, then the baluster on No. 3 step will be equal to }4^^ longer, and on No. 4 step \H'^, and on No. 5 step Ifi^^ longer than a regular short baluster. Fig. 8. Shoivs hoiu the easing at the upper end of ivrrath- plcce may be irnrkcd on the shanli of same, and tlius avoid the short cashaj HM, Fig. 3. , PA shows tise tangent NG, Fig. 3, prolonged; also GB for the inclination GH, Fig. 3. Through 5, draw the joint 3 4 per- pendicular to the straight rail BG; setoff on each side of PA, the half thickness of plank required for the wreath-piece, shown at seclion N, Fig. 4; make a temporary joint at A, at right angles to AP, as 2 3; to this joint apply the. bevel in the nsual waj% shown at N, Fig. 4, and work off the wrtath-piece to the plumb on tlie concave and convex sides; then apply the bevel .shown at 2, from the temporary joint, for the cripple joint 3 4. Now drop the block p.ittern on the joint just made the distance A 5, below the center of jtlank for tin; center of rail as shown; then ease off into the wreath part, from a tangent at right angles to the joint 4 3, thus saving tlie short straight easing. Fig. 5. Shov-s another treatment of the plan, Fig. 2; the radius [12%'''] for center of rail, the position of treads and ris- ers being the same. AB, BC, are the tangents; the face of No. 7 rise is SK'"' from the spring of cylinder. 188 Plate 41. Fig. 6. Shows the elevation of treads and risers, al&o the tancjents. The upper tangent is allowed to have the same inclination in this case as the straight rail; hence the wreath-piece will contain its own natural easing all in the wreath part of rail, at the upper end. In this case a joint is made at the spring line at the lower end, and a small easing is added to connect the newel cap, as shown. Let XX indicate the edge of drawing board, make AB and SCeach equal AS, BC, on plan, Fig, 5. Perpendicular to XX draw AD, BE and CF indefinite. Now elevate the treads and risers from the tangents on plan. Fig. 5, making No. 7 rise ^}4'' from the spring line AD. Through the center of balusters O, O, draw tiie underside of rail. Parallel with 00, draw the center of rail, cutting AD at P, and BE at N, and prolonged to intersect CF at Q; from the top of No, 1 step set up 5^^ to the underside of rail, and lU^'' more, or 6)^^'' to the center of rail at Z; draw Zn parallel to XX. From the center of baluster on No, 2 tread, draw the arc for center ot inclining rail. From N, draw a line to tangent the arc and intersect the horizontal line from Z at n, and also cutting CJP at L; make joint at L perpendicular to DN, and draw easing to suit. From P draw PU parallel with XX, cutting LN pro- duced at V. From L draw LR parallel with XX, cutting BE at T; from T and at right angles to PQ, draw Tjr; from U, and at right angles to LY, draw WU; parallel with BE, draw the half width of rail (:y^) cutting the tangent PiV at 10, an.d the tangent NL at 9; make R 12 equal the chord AC, on plan, Fig. 5; at P allow G'-' for straight wood on the shank of face-mould; make P 19 equal LQ. Bevels. Let 14 15 indicate a gauge line parallel with XX; perpendicular to XX, draw 15 16 equal to the radius OC, Fig. 5. Make 15 18 ecpial Ty, Fig. 0, and 15 17 equal WU, Fig- 6; draw 16 18 and 16 17 prolonged to edge of board. Fig. 7. Shoivs the fucc-moidd lined ojf from the chord line, by the 7(se of ordinates transferred from the plan direct. Make BC equal NL, Fig. 6; with C as a center, and P 12, Fig. 6, for a radius,' draw arc at A; again with S as a center, and PN for a radius, draw arc intersecting at A: join BA and prolonged to H, allowing 6'^ for shank. Parallel with AB and BC, draw CO and AO, for the parallelogram OABC, on the cutting plane. Proof. The diagonal OB must equal the distance 12 19 Fig. 6, if so, the angle of tangents at B must be correct. Make joints at H'and C perpendicular to the tangents AB and BC; make CD equal NV Fig. C; join OD and produced for the points in face-mould, through which to trace the concave and convex curve using a pliable strip. Draw the inside and outside curve of rail on plan, Fig. 5; make Cff equal UV Fig. 6, connect OH for the director; bisect the clioni AC at D; bisect AD and DC at E and F; now parallel with OH, and from the points A, E, D, Pand C, draw ordinates to cut the concave and convex curves of rail ou plan, as E. 2, 3, 4. &c. Now return to Fig. 7, bisect the chord AC at D; biscci DA and DC at E and F; parallel to the director OD, draw ordinates indefinite from the points A, E, D, F and C, then transfer the points on the ordinates ou plan, Fig. 5, to corresponding ordinates Plate 42. 189 ou face-mould, using the chords for base lines; make OJ equal the radius OC on plan, Fig. 5; make J" 4 and J^ each equal the half ^^idtl^of rail (;y^); make C2 and C3 each equal iVlO, Fis:. 6, also make A G and A 7 eat-li CMjual iV9, Fig. 6; parallel with IrlB, draw 6 8 and 7 9 for the width of face-mould at the shank. The curves may now be drawn through the points just found, using a pliable strip. At sections P and J2 the bevels are shown applied to cross the tangents. Bahisters. If required to find the length of odd baluster in the quarter circle on the center line of rail, then from the center of each baluster ou phm. Fig. 5, draw lines parallel witli the directing ordinate OH, to intersect the taugeuts as at XXX, Fig. 5. Now transfer their intersection with the treads ou tangents on plan, to corresponding treads in elevation. It will be noticed that the balusters ou No. 4 winder does cross No. 4 rise and ou to No. 3 wiuder on plan, and is set over on line wilh No. 4 tread in eleva- tion. Fig. 6, as the proper place for its length. Now parallel with tli(^ risers, draw the balusters to intersect the underside of rail at //, »fec., for their respective lengths, as explained for Fig. 3. It will be observed that after tlic d i reel or Oi^fou plan, is located, the position of risers under llie tanirtuts in elevation, relative to the cutting plane may bo establLslicd; for each ordinate on face-mould is horizontal and parallel to corresponding ordinates on plan, and all parallel to the directing ordinates. It will be oI)served at Fig. 0, tlie tangent PN, is in line with the center of straight rail, thus avoiding the easing shown at Fig. 3. This has caused the rail to raise at iV-, and as a result, the balusters are longer, and ihe rail will feel liigh when standing on No. 4 step; while at Fig. 3, the rail will Jiave an agreeable lieight; but it is a good fault that the rail on winders be high better than to feel low. On quarter and half pace winding stairs the rail should never be constructed so that the wreath-piece would line with tlie nosings, unless the risers ail radiate to the center, and at the same time, be for a large cylinder, or in a circular stair-case, then the wreath may be constructed so that when placed on the stairs, the underside of rail will touch every nosing. When as in this case the tangent PN, Fig. 0, is the continuation of tlie straight rail; there shordd not be more than one graduating step outside the cylinder, or the space of a regular trciid, may start from the spring of cylinder at the upper end of v.inders, while at the lower end, they should be graduated; one fault to tlus method is, without the use of graduated steps at the upper end, Die easing on the lower edge of front string will be difticult to malce s;itisfactory, as the string has to be increased more than usual at t he angle in the forma- tiou of the casing couuecting the regular treads with the winders. PLATE 42. Plata 42. Exhibits jilan (Fig. 1) of a half pace divided into ticu quarter paces Inj pldcmij two risers across the half pace. This is done in many cases to gain room and avoid winders. The well hole is 10''^ diameter, and the pitch is 8^^ wide by 9^^ tread. No. 14 tread is placed in the center of half pace, and the balusters are to be spaced olf on the center line of rail equally, and then the risers are curved to t-uit the balusters and the miters at the return nosings. Also at Fig. 2 is shown a quarter pace winding; the radius of quarter cylinder is 13^'. In this case we have placed No. 16 tread in the center, and spaced the olluns either way on the walking line. Nos. 13 and 20 are the fust square risers outside the wind- ing treads. Locate a short baluster ou Nos. 13 and 30 tread; 190 Plate 42. then space the iuterveuing balusters equally, and draw the risers through the points on the walking line to suit the balusters. The scale for Figs, 1 and 2 is }/i'' equal 1 foot. Fig. 3. [Scale M^^^l^-] Shotvs ijlan of the W cylindefi; Fl'j. 1. The dotted line indicates the string, and solid line the center line of rail, AB. BC, CD and JD^ show the tangents enclosing the center line of rail. The balusters are spaced off around the cylinder, and the concave and convex risers are curved into the cylinder to suit the balusters. The face of Nos. 12 and 17 rise are 2>4'''^ from the spring of cylinder, the radius of cylinder being 5'^ and balusters 2^'X2^''. Then the radius for the center line of rail will equal 5%''^ or the diameter for the center line of rail will equal IIK^^ Fig. 4. Shows the elevation ^f taiujents, treads and risers, unfolded from the tangents on 'ijlan Fig. 3, Let XX indicate the edge of drawing board; the points A, B, C, D and J57show perpendiculars draAvn from XX, and indi- cating the spring A and angle B, center C and angle D and spring E, on plan Fig. .3. Now elevate the risers and treads from the tangents on plan Fig. C, being careful to keep the face of No. 12 and 17 rise 2^i'^ from the spring lines A and E, as shown on plan Fig. 3. Throu>:h the center of balusters O, and O, O draw the underside of rail; parallel with 00, on the right, draw the center of rail, cutting the spring line at E, and to intersect the angle and center lines at B and F, and also intersecting the spring line on the left at G. Now D becomes a fixed point. At the external angle of No, 12 rise draw arc equal to the half thickness of rail (I i^'^), and from JD draw a line to tangent the arc and intersect the center of rail at H, and cutting the center, angle and spring lines at C B and A; from A, and square to the spring line C, draw AJ. cutting the angle line B at K: then JC will be the height fm- the lower wreath-piece; from C, and square to the spring line E. draw CL, cutting the angle line D at M; then LE is the height for the upper wreath-piece. Square to LE, draw EN, cutting the angle line D at P; prolong the taiiiient CD to intersect iV^ at Q; from B, and per- pendicular to CQ, draw PR; from M, draw MS stiuare to EF\ make CT and JU equal the chord AC, Fig. 3: draw KV at right angles to AB; parallel with the angle line D, draw the lialf width of rail, cutting the tangent DE at 2, and the tangent CD at 3; ease the angle on the left at H to please tlie eye; make joint on easement so as to allow 3'''' of straight wood on the shank of face-mould. Bevels. Let 4 5 indicate a gauge line parallel with XX; square to XX, draw 4 6, and equal to the radius OC, Fig. 3; make 4 7 equal PR, make 4 8 equal KV, also make 4 5 to equal MS; draw 6 5, 6 8 and 6 7 to the edge of board. As the two tangents AB and BC for the lower wreath-piece are both the same length, and inclination, one bevel only will be required for both joints, the angle at 8 gives that bevel. The tangents CD and DE for the upper wreath-piece are of different inclinations and will require two bevels; the angle at 5 gives the bevel for shank, and the bevel iu tUe angle at 7 gives the bevel for ceuter joiut. Plate 42. 191 Fig. 5. Exhibits the face-mould for the lower ivreath-piccc. Make AB equal the tangent AB, Fig. 4: with A as a center and CU, Fig. 4 for a radius, draw arc at C: again with B as a center, and BA for a radius, draw arc intersecting at C, join BC. rarallel with BA and BC, draw CO and AO for the parallelo- gram on the cutting plane, that will coincide with the parallelo- gram O ABC on plan. Fig. 3. when in position. Proof. The diagonal BO must equal BO on plan. Fig. 3. Draw the diagonal BO for the direction of minor axis; prolong the radial lines OA and OC; make A '2, A 3. also C 4, eacli equal jD 3, Fig. 4; make OD equal the radius OC, Fig. 3; make D 6. D 7, each equal the half width of rail [2^^J; make from A to F equal 3'' for shank; make joint at F square to the tangent. Now draw the right angle at O. and trace the concave and convex sides of mould by using the rod as described at Fig. 9, Plate 13. The sections at Jif and J" show the bevels applied to cioss the tangents, and through the center of plank. The shaded part shows the width at the joints to saw out the crooks. At the crossing of minor axis, %^^ wider than the mould on the concave side, is all the over-wood that will be required in this case; that amount will allow the twist of rail; at this point the bevels blend, and the section is at right angles to the face of crook. Fig. 6. Exhibits the face-mouJa for tlie upper wreath-piece. Make CD equal tangent CD, Fig. 4, with C, as a center, and NT, Fig. 4, for a radius; draw arc at E; again, with D as a center and DE, Fig, 4, for a radius, draw arc intersecting at E, draw DE. prolonged 6'^ to H; parallel with DE and DC, draw CO and EO, for the i^arallelograni on the cutting plane. Proof. The diagonal DO must equal TF in elevation. Fig. 4; make CA equal DQ, Fig, 4. Draw OA indefinite for the direction of minor axis. Make OB equal the radius OC, Fig. 3. Prolong the radial lines OC and OE: make E 2 and E 3 each equal D 3, Fig. 4; make B .5 and B 4 each equal the half width of rail (2^0; make C6 and C7 each equal D 2, Fig. 4. Make joints at Cand iiZ" square to the tangents CD and DE. Draw 3 9 and 2 S parallel with DH. Draw the right angle to the minor axis at O, and trace the concave, convex and center curves of face-mould, using a rod to make points in the curve, then use a pliable strip in tracing the curves, -j The section at P shows the bevel at the angle 7, Fig. 4, applied through the center of plank; the section at R shows the bevel found in the angle at 5, Fig, 4, applied through the center of plank, for the shank end of wreath-piece. Fig. 7. Exhibits the plan for the quarter pace icinding. [Scale %"=l foot]. The dotted line indicates the face of outer string, and the solid line the centre line of rail. The radius for the center line of rail equals VZ%''\ the tangents AS and SC inclose the quarter circle and are at right angles to each other; the face of No 14 and 19 rise is 9Ai from the spring of cylinder; Nos. 13 and 19 treads are 634^'' on the center line of rail; Nos, 13 and 20 treads are the regular width, 9''', Fig. 8. Exhibits the development of tangents AB and BC, from the plan. Fig. 7, and also the treads and risers. Let JCX indicate the edge of drawing board; at right angles to XX draw the spring and angle lines A, B, C; elevate the treads and risers from the tangents on plan, keeping the face of ]9'2 Plate 43. Nos. li and 19 rise, 3)^'''', from the spring lines; Nos. 13 and 19 treads are 6H^^ wide, and Nos. 13 and 30 are 9^^ wide. Note, at Fig. 3, the radial lines intersect the regular tread line by crossing No. 19 rise, and also crossing No, 14 rise, tlius indicating tliat the rail should be kept np at No. 19 rise, while at No. 14 the rail should be kept dowu."^" Through the centre of balusters 0, O and O. drav/ the under- side of rail parallel with the underside of rail; draw the center of rail, intersecting the spring line at D on the left and at .Eon the right; now we will allow the direction of upper tangent to cross the spring line at C, level with No. 19 tread, making a small easing to connect the wreath-piece. Take a point say at G, and draw GC prolonged, cutting the angle line at JB, and intersecting the spring line at J". At the center of balusi-er on No. 14 tread draw arc e and E there are llli'^\ mark this on the pattern, as shown. Bevels. The dotted line 4 5 indicates a gauge line parallel witli XX; make 4 S on the perpendicular line equal the radius OC Fig. 7: make 4 6 equal MP; let 4 5 equal KQ\ draw S 6 and '^ Irorn the lace of cylinder; tlie tread i.s iO-'^ by 7^^ rise, No. 12 ri.M! is placed on tlie center of half pace, and the rejrulai' tread (10^^) is spaced oil from that, riglit and left on tlie walking line. Tiie balusters are then si)aced olf equally, and llie risers drawn thnnvuh tiie points on tlie walkint;- line to the wall siring and tlie outer string to suit the spacing of balusters; the balusters are ■Z^^ by 2'', and rail o'^by i'\ double moulded. Fig. 2. [Scale K'^=l foot.] Exhihils the phtn for the center line of rail; Hie radiiifi OA for the semi-circle ABC equals lO^ii^''. The dotted, line slioics the face of outer striiuj. The face of Ncs. 8 and 16 rise are 514^^ from the sjiring of cylinder on the outer string, the width of No. 7 and 10 treads, is .'■/^ and No. (i and 17, are regular treads 10'^ wide on tlie horse; the balusters are ',i^^X,-y^. *Mr. Simon DeGraff claims the first to advance the system of making three pieces in the wreath around a semi-circle. 10 194 Plate 41 Divide the semi-circle ABC into three equal parts * as AD, DE and EC; draw the joints at D aiul^, radiating to the center O, draw the tangents AF, FD, DG, GE, EH and HC, each perpendicular to the radial lines OA, OD, OE and OC; they must equal each other in this rase to b :^ correct. Parallel with tanL'cnts AF and ED, draw DJ and AJ for the parallelogram AFDJ, on plan; tlse semi-cirde being divided into three parts at ^and D, then the tangent A J' will equal half the diagonal OF, and the diagonal Ji^will equal the length of tangent AF: prolong the tangent AF. From D, and at right angles to AF prolonged, draw DK. Fig. 3. Shows the development of tangents from the plan. Fig. 2; the treads and. riscvfs are elerated from the tangents on plan. Let XX indicate the edge of drawing board; now find the average pitch from tangents on plan, as explained at Plate ill; then draw the perpendiculars AB, CI), EF, GH, JK, SM and NP, to the average pitch, from the edge of board XX, all parallel to each other and the same distance apart, and eijual to AF, FD, DG, GE, EH and HC, on plan, Fig. 2. Now elevate the treads and risers from the tangents on plan, being careful to keep Nos. 5 and 16 rise 3>^^^ from the spring lines AB and NP. Nos. 7 and 10 treads are each 5'^ and Nos. (5 and 17 treads are W^ each; the risers and treads are sho\^ n by the dotted Ihies, as taken from the tangents on plan Fig. 2. Through the center of balusters 0,0 and O, Otlraw the undertudr of rail parallel with the underside of rail; draw the center line of straight rail, cutting the spring line AB at Q for the lower end. and at the ujiper end prolong the center line to intersect the per- pendicular JKa% R, and cutting Z,iVf at -S, and NP At N. At the center of baluster on No. 8 tread draw arc eipial to the half depth of rail (3''^); from ^Sdiaw a line to tangent the arc and in- tersect the center of rail at T, and al?o cutting the perpendiculars at A, C, E, G and J. Make joints at E and J, as they are ]ioints of eontactf witli the center line of rail, as .shown at JDand JE7, on plan Fig 'Z. At right angles to AB, draw AF, culling the perpendicular CD a( D, then FE will l)e the height for the lower Avreath-i)iecc; perpendicular to EF d\i\\^ EK, cutting GH i\t H, then K.T vviil be the height for the middle wreath-piece; at right angles lo JK draw JP, cutting the perpendicular from L at M, then PN is the height for the upper wreath-piece. Paiallel with JP diaw NL, cutting JS prolonged at u: nuike LV, MW .\ud D 'I each equal i^ir on plan, Fiif. ti; from V, and perpendicular to tangent JS produced, draw Y .\, and from W, and at right angles to NR draw W 4; from y, and at right angles to AC, draw j^O; make FZ and P 9 each equal the chord AD on plan, Fig. 2. "The radius of any circle will divide that circle into 6 enunl parts, or the sciid-circlu into :jc(iual parls, l)CC!uise the radius is tins chord of (SO doEcreos of the cir(;lc. Whcu dividiiis a circle into .sc;i- iiionts, it will I)o found the l)etter uay to make the cliord of each .se"nier)tei|\i:il the radius of the circle, Ihen llic lenj^tli of each tan- •lent AF, VB, on plan, will equal the dia;_-onal FJ of tlieii' parallelo- 2r:inis, and will aLso e(iual one-half thediay;onal OF. of the trapezium i)AVD, both on plan and face-mould, as shown at Figs. 4 and 5, and also at Figs. 5, 6 and 7, Plate 44. +At these points of contact the direction of the tangents cannot be cban<;ed. The points C, O and S are the angles, and answer to the atislcs F, G, Jf, on plan Fiii. 2. At these points the directioji of tan- ■rents may be changed. The student in stair-building will do well to remeuibc'r this. Plate 43. 195 Bevels. Let 5 5 indicate the edge of a board, and the dotted line indicates a gauge line; perpendicular to 5 5 draw 20 21 equal to KD on plan, Fig. 2: make 20 22 equal V3, and 20 23 equal jr2. and 20 24 equal W 4; draw 21 22, 21 23 and 21 24 prolonged to the edge of board 5 5 for the bevels required; parallel with 5 5 draw the half width of rail, cutting the hypoth- enuse of triangles at a, b and d. Fig. 4. Shoios the face-mould for the lower wreiith-{yiece connecting the ramp. Make AB equal .AC in elevation. Fig. 3; with A as a center, and EZ, Fig. 3 for a radius, draw arc at C; again with JB as a center, and BA for a radius, draw arc intersecting at C join BC; parallel with BA and BC draw CD and AD for the parallelogramASCZ) on the cutting plane. Proof. The diagonal BD must equal the diagonal FJ on plan. Fig. 2. if so, the parallelogram is correct. Make BO equal FO, on plan, P'ig. 2, and OF equal the radius AO on plan. Fig. 2. Make AH equal 3" for the shank connecting the ramp; make joints at If and C perpendicular to tangents AB and BC. Draw the radial lines AO and CO for the points of tangencji make F 2 and F 3 each equal the half width of rail {IH/^); make H 4, Ho. also C6, C 7. each equal 5 b, Fig. 3; parallel with HB draw 4 8 and 5 9. Now pivot the trammel at O, and trace the concave and convex sides of face-mould. If preferred, bisect the chord AC, and draw the ordinates as has been explained. By making a joint at A. shown by the dotted line, this face-mould will answer for the middle wreath-piece, because the length of tan- gents and heights is equal in both cases. The sections at L and N show the application of the bevel. Only one bevel is required for both joints, and is shown at the angle 23, Fig. 3. The same bevel applies for the middle wreath- piece. Observe the bevels cross the tangents. Fig. 5. Shoirs the faee-mould for the uiqicr irrcath-picre. In this case, the tangents being unequal, two bevels will be re(iuired, and also observe from the elevation. Fig. 3. they will not cross the tangents, as the long tangent JS produced cuts the horizontal line LN between the perpendicular LM Aud the point V. The learner will tind this explained at Fig. 9, Piate 14. Make AB eijual to JS, Fig. 3. With A as a center and 9 N, Fig. 3, tor a radius, draw arc at C; then with B as a center, and tangent SN, Fig. 3 for a radius, draw arc intersecting at C: draw reproduced to F, equal to (>" for the length of straislit wood on shank. Parallel with AB and JBCdraw CO and AO for the parallelogram on the cuttintr plane. Proof. The diagonal BO must equal the distance MR. Fig. 3. Prolong tangent AB, to D, equal to Su, Fig. 3; join DC lor the director; the curve of face mould may now be drawn by ordinates or with the trammel, if with the trammel, prolong the diagonal BO e()ual to itself, to H, through H, and parallel to the director DC, draw HJ" equal to the radius OA, Fig. 2. 10>'4 ", then JH is the semi-minor axis; draw the radial line HC pro- longed, which gives the point of tangency on mould; make joints at A and F perpendicular to tangents BA and BF; make F2 and F3 each equjil 5 d. Fig. 3; draw 2 4 and 3 b parallel with the tangent BC; make A 6, A 7 each equal 5 a. Fig. 3; make J 8 and J" 9 'iach equal the half width of rail (IK^O • Now pivot the trammel at H, with the arms perpendicular to JH, and set 196 Plate 43. from pencil to minor pin the distance HJ for the center lino on face-mould, then place the pencil in C and the minor pin in the groove, at the same time sliding the major pin on the stem until ))oth pins drop into the grooves, then fasten the major pin and trace the center line; proceed in liiie manner to trace the con- cave and convex sides of mould. Ramp. Mark tlie face of Xo. 7 rise on the rarnp ah T; then TQ sliows the amount (10^^) to allow when jointing the rail; at the upper end the sliank of face-mould shows 6^' to be deducted. The sections at M and iVshow the application of the bevels. The bevel shown at 22, Fig. 3, is applied at M. and the bevel shown at 24, Fig. 3, is applied at N. Observe they do not cross the tangents. The shaded part shows the required widtli to saw out the crook from the plank, also the thickness of i)lank required to contain the twist of wreath-piece. If ordinates be preferred for tracing the cni-ve of face-mould instead of a trammel, then prolong tangent EH, on i)lan Fig. 2. to Z», equal to Z»Z7, Fig. 3. Join LC for the director on plan; DC is the director on mould Fig. .5. Now bisect the chords, draw ordinates and transfer points from plan to face-mould, as described in former plates. Fig. 6. [Scale k'^^^l foot,]. Exhibits the ])l(m of a quarter pace windlmj at the laiidlmj. The cylinder being struck from two different radii, 12''^ and 6'\ as shown for the cjdinder line; tlie tread line on the winders is \?/' from the face of cylindcM-; tlie balusters are 2^'' by S^', and the rail is Z^^ by 4'", double moulded; the rise is Q'', and the regular tread 10'^ Fig. 7. Shows the plan of the center line of rail ACS. struck from Die centers M and N, their radius hciiaj VZ%'^ and 0%'^, respectively. [Scale %^^=1 foot.] Draw the tanirents AB, BC. CD and DE at riglit angles to the radial lines. Tlie tangents for the upper wreath-piece loiui an acute angle at D. arid the two lower tangents form an obtuse angle on plan at B-, parallel with AB and .BC draw CO and AO; draw the chord CE. and the diaguiuil BO produced to M. Bisect CE at F: draw the diagonal FD. The face of Xo. 14 rise is l]^/' from tlie spring line of cylinder; No. 1^ tread is S>.^^; No. 12, the regular tn-ad, W^. From C draw CH per- pendicular to AB i)rolonged; again from C draw CJ perpenilic- ular to tangent DE. Fig. 8. Slioics the tniKjcnts unfolded from plan, and the cleviUion of the treads and risers from the tawjents on plan; also the increased leiKjth of the tangents in elevation, for the facc-moulil. Let XX indicate the edge of drawint; I)o;uil. then with the bevel set to tlie average piteli. draw A3, CD. EF. GH and JK, to correspond to tangents AB, BC, CD and DE. on ])lan, Fig, 7. Now elevate the treads and risers, keeping tJie face of No. 14 rise l^i'^ from the spring line AB, as shown on plan. No. 13 tread is S}-^^' wide, and No. 12 tread is the regular width HV^. Tlirough tli*^ center of balusters O, 0, tlruw the underside of rail. I'arallel witli llie underside of rail, draw the center of rail, to cut the spring line AB at L. Mark the fac-, of No. 13 rise on tlie center line of rail' at M, then ML equals 12M'''' to allow when jointing the straight rail to the ramp. At the lauding set up half a ri&e (4^'') to the under- Plate 43. 19% side of rail, plus the half thickness of rail {2^^), equals C/^ to the center of rail from the floor line. Then draw FN parallel to the floor line. Now J* is a fixed point. At the external angle of No. 14 rise, draw arc equal to the half thick- ness of rail, and from the fixed point JP, draw the inclination of tangents, to tangent the arc and intersect the center of inclining rail. This happens to be at Q. The inclination of tangents cuts the perpendiculars at A, C, ^andP, and the level tangent at P and N. From A, and perpendicular to AB, draw AF, cutting the angle line C at D. Then FE is the height for the lower wreath-piece. From E, and perpendicular to EF, draw EH, then HP is the height for the upper wreath-piece. Make HR equal the chord EC, on plan. F\s. 7. Let HS equal the diagonal DF, on plan, Fig. 7. Make DU equal BH, on plan, and PV equal DJ, on plan, Fig. 7. From V, and perpendicular to tan- gent EP prolonged, draw VW, and from U, and perpendicular to tangent AC, draw Ujr. Bisect PH at Z. Make FT equal the chord AC, ou plan, Fig. 7. Bevels. Let ?> 3 indicate the edge of board, and 4 4a gauge line, make 4 5 equal HC on plan, Fig. 7, and 4 6 equal JC on plan, Fig. 7. Make 4 7 equal Uy, Fig. 8, and 4 8 equal VW, Fig. 8, and 4 9 equal the height HP, Fig. 8; draw 6 4,68 and 5 7 prolonged to 8 3. Then the angle at 7 gives the bevel for both joints on the lower wreath-piece, as both tangents have the same inclination. The angle at 8 gives the bevel for the lower joint on the upper wreath-p'ece, and the angle at 4 gives the bevel for the upper joint. Parallel with 4 4 draw the half width of rail, cutting the hypotheuuse of triangles at a, b and d. Fig. 9. Shows the face-mould for the lower tvrenth-jriece. Make tangent A3 equal AC, Fig. 8: with A as a center, and TE, Fig. 8 for a radius, draw arc at C; again with B as a center, and BA for a radius, draw arc intersecting at C, join BC; par- allel with AB and BC, draw CO and AO for the parallelogram ABCO on cutting plane. rroiif. The diagonal BO must equal the diagonal BO on plan, Fig. 7. Prolong BO to e(iual JBilf on plan. Fig. 7; through A and Cdraw the radial lines MC a.ni\ MA to give the points of tangency; prolong BA .y to D for straight sliaidt, as shown at ramp in elevation. Fig. 8; make joints at D and C square to the tangents; make 2)2 and D'S, also C4 and C5 each equal 7 a. Fig. 8; make MF equal the radius MC, Fig. 7. Let FG and J* 7 each equal the half width of rail \IM^^\- parallel with DB draw a S and 3 9. Nov/ pivot the trammel at M, with the arms at right angles to MF, then set from pencil to minor pin the dis- tance MF, place the pencil in the point A and drop the pins into the grooves, and trace the center line of rail through the points AFC; repeat the operation for the concave and convex side of face-niould. At sections P and N, one bevel is shown applied to both joints taken from the angle at 7, Fig. 8. Pig. 10. Shoivs the face-mould for the landing wrcath- irlcce. ^ Make tangent AB equal EP, Fig. 8. With A as a center, and RP, Fig. 8, for a railius, draw arc at C; again with B as a center, and PN for a radius, draw arc intersecting at C; draw BC prolonged to D, 'i," for straight wood connecting the level rail. Draw the chord line CA perpendicular to tangent CB; i98 Plate 44. draw C£7 indefinite. Bisect tlie eliord AC at F; draw tlie diag- onal BF prolonged, ir.teisecting CE at H; draw HJ parallel with JSC, and equal to EN, {G%^^), Fig, 7. Proof. The diagonal BF must equal the distance SZ, Fig. 8. Make J 2 and J 3 each equal the half width of rail (IK''''). Make joints at D and A square to the tangents AB and BC; make 0*4 and C 5 equal 9 d, Fig. 8, and A 6, A 7, each equal to 8 b. Fig. 8. Now pivot the trammel in H with the arms resting in the major axis CH; then for the center line of rail, set from pencil to minor pin the distance HJ, and from the pencil to major pin the distance HC, now trace the curve through the points C, J" and A for the center line of rail on mould, repeat the operation for the concave and convex curves of mould; 2'^ of straight wood is added on at D to help the easing at the joint connecting the straight rail. At sections L and M the bevels are shown applied to cross the tangents; the bevels shown in the angles 8 and 9, Fig. 8 are applied at sections ilf and L, respectively. If a trammel be not at hand, then bisect the chords and use ordinates as previously described, BC will be the director on face-mould, and DE will be the director on plan, Fig. 7. For the face-mould, Fig. 9, the diagonal will be the director, both on the mould and on plan, because both tangents have the same inclination. At N, Fig. 3, observe the tangent NS is drawn with the same inclination as the straight rail, thus lifting tlie falling line of rail at the upper end of winders; if kept down at tliis point, and a short easing made to connect the wreath rail, the wreath would be improved and be still high enough; it is thouglit well to show the langents in elevation in this way, so as to develop a face-mould having the minor axis outside the mould and near the .ioint, and thus exhibit an extreme in hand-railing, in a practical way; for in practice the stair-builder has various shapes of moulds to make, and how to construct and prove their correctness is the aim here sought. PLATE 44. Plate 44. Exhibits the construction of face-moulds for a flkjht of ijcometrlcal stairs circular on plan and starting from a newel. Fig. 1. [Scale H^'—'^ foot.] Shows the plan. The regular tread equals 10^''' by 6%'''' rise; they are spaced off on the regular tread line 18 inches from the outer string, the corners remaining square to give more room, and by using a wall rail will suit the ascent and descent of old people better than if the wall string were concentric with tlie outer string. For effect, in grand circular stairways the wall string should lie circular to correspond with the front string, and the wall space ornamented with niches for statuary. The rail is 4^^ wide by h'^ deep, to be double moulded. Pig. 2. [Scale %^''==1 foot.] Sliows phtn of tangents. The radius for the cylinder equals 1.5^^, and the balusters are 2^^ by 2''^, and by allowing }i^^ for the projection of bracket, the radius for the center line of rail will equal 15%^''. At the newel the cylinder is contracted, which increases the length of first step Pr.ATK 44. 199 and make the stairs more inviting to ascend. Tlie direction of risers and tlireads are given on plan; tlie widtli of rail (4'''') is laid ott" on plan, so that ordinates may be used in drawing the face- moulds. The plan of rail is divided into live wreath-pieces; the chord of the segments for Nos. 3, 3, 4 and 5 wreatli-pieces equals the radius OB: tlie joints shown at A, B, C and D all radiate to the center 0/ the joint at E shows 2^^ of straight wood added to connect the straight rail on the level. Draw the tangent FH perpendicular to radius F^, and tan- gents AH, AJ, BJ, BK, CK. CL, DL, DM aud EM, all per- pendicular to the joints A, B, C, D and E; parallel with the tangents iZlFand HA, draw FN and AN; parallel with tangents AJ and BJ, draw BP and AP, forming the parallelogram AJBP on plan. The parallelograms for Nos. .3, 4 aud 5 are the same, because the chords BC, CD, DE, are of one length. From F No. 1 and perpendicular to tangent AH, draw FG\ from A No. 3, and at right angles to BJ prolonged, draw AS; from C No. 4, and pei-pendicular to tangent DL prolonged, draw CT; from D No. 5, and perpendicular to tangent EM prolong- ed, draw DU. The treads, risers, joints of rail and tangents on plan being located, we are ready to draw the increased length of tangents in elevation. Pig. 3. Exhibits the elevation of tojiigents, treads and risers, as developed from tangents on plan Fig. 2, for the in- creased length of tangents for the face-moulds. To economize room, first find the average pitch on the line of tangents; then elevate the treads and risers from the tangents on plan. The stretchout of tangents from F io M equals, we will say, 943^^'' which being divided by 14 (the number of risers) equals say 6% ^'' for the average tread; then set a bevel to the average pitch, Q%^' by Q%^' rise, and draw the perpendiculars F, H, A, J, B, K, C, L, D. Jf and E. Now elevate Nos. 1 to 15 treads and risers from the tangents on plan Fig. 2; then determine which tangents will be level. Tangent FH connects the newel cap, and should be level, or nearly so, k) make the curve connect- ing the cap graceful as possible, we will give the tangent HF'No. 1 a slight inclination, and by forcing the curve at the miter joint, a more graceful curve will be obtained than if the tangent HF in elevation was to remain level. Again, tangent ME, at the lauding, should be level, for a full easing at that point is re- quired. From the top of No. 1 step set up to F, the height to under- side of rail, 5'^ plus the half depth of rail [2^^^], equals 7K''; at the landing set up to E, i'' to the underside of rail, plus the half depth of rail f3K'^] equals ^H" to the center of rail at E: draw EM at right angles to the spring line E; now M and H are fixed points; from the external angle of Nos. 4 and 12 rise, draw arcs equal to the half depth of rail \2yi"\ and draw the inclina- tion of tangents, cutting the perpendicular at H, A, J, B, K, C, and L; now L and M are fixed points, draw ML prolonged to cut the perpendicular C at N; draw FH prolonged, cutting the perpendicular A at D; from F draw the horizontal line FO, cutting the perpendicular H at P, and also cutting the tangent AH" prolonged at y; draw AQ at right angles to AO, cutting the perpendicular J" at JZ; draw BS at right angles to BQ, cutting the perpendicular K at T; draw CU square to CS, cutting the perpendicular L at v; from D, and at right angles to DU, draw DiV, cutting the perpendiculars M aud L 9.1 X aud Z, and Riso cutting the tangent CL prolonged at J, 200 Plate 44. ~~' Now OA is the hckiht for No. 1 wreath-piece; 13 Q ami CS are the hcMjlds for Nos, 3 and 3 wreath-piece, they beiiij? botli the- same. DJJ is the heujht for No. 4 wreatli-piece, and XJkZ" tlic hcUiht for No. 5 wreatli-piece. Malce Oa equal tlie cliord FA on plan; prolong (JA to the right and make Ab equal the diagonal NH on plan; malte Qd equal the chord BA No. 3 on plan; make Ui and Wiif equal the chords DC and DE Nos. 4 and 5 on plan. Fig. 3; make Pni equal GH No. 1 on plan. Fig. 2; let Rn equal JS No. 3 on plan; make Vt and Z-j equal iT No. 4 on plan; let Mg equal UM No. .5 on plan, Fig. 3. From the points rti, ii, t, c and g, draw Jiie, i27i, ti, CP and ^/S square to the tangents AH, J A, DL, CL and DM produced. Bevels.* Return to plan, Fig. 3, make HV No. 1 equal Py, Fig. li, draw VF indefinite for the director; from A, and perpendicular to FV, draw AW; make Wa equal OA, Fig. 3; join Aa, and in the angle at a tlie bevel is found for the joint F, at the miter cap; for the joint at A on No. 1 wreath-piece, makt? GrJC equal me, Fig. 3; join XF, and in the angle X is found the bevel as shown. For Nos. 3 and 3 wreath-piece, make Sy equal nh, Fig. 3; join yA for the bevel required for the two wreath-iiieces on each joint; for No. 4 wreath-piece, make Tb equal CP, Fig. 3, and Td equal tl, Fig, 3; join be and dc, and in the angle b is found the bevel for the joint at C, and the bevel shown at d applies at joint marked D; for No. 5 wreath-piece, make Uni eiiual gS. Fig. 3; make Un equal the height XM, Fig, 3; connect mD and i2l> and the angle at m gives the bevel for the joint at D, and the angle at n gives the bevel for the upper joint at E. Now draw the half width of rail [3^^] parallel with the tan- gents to cut the hypothenuso of triangles forming the bevels; for No. 1 wreath-piece tlie dotted line cuts at 3 and 3, and for Nos. 3 and 3 wreath-piece the dotted line cuts at 4; for No. 4 wreath- piece the dotted line cuts at .^) and 6, and*it No. .5 wreath-piece the dotted line cuts at 7 and 8. Pace-moulds, Fhj. 4 shows the faee-monhl for No. 1 torailh-inece, fttortlng from the newel post. Make AH equal AH, Fig. 3; with A as a center, and Aa, Fig. 3, for a radius, draw arc at F; with iJas a center, and HF. Fig. 3, for a radius, draw arc intersecting at F; draw HF pro- longed, parallel with HA and HF, draw FN ami AN, for the parallelogram NAHF, on the cutting plane. Proof. The diagonal HN must equal the distance Db, Fig. 3; if so, the parallelogram is correct. Make Hy equal Hy, Fig. 3. Connect yF for the director on face-mould. Draw F 3 indefinite, and at right angles to Fy. Make joint at A at right angles to AH. Make F2 and F 3, each equal a 3, Fig. 3. Make A 4 and A .5, each equal X 3, Fig, 3. Draw the chord AF. Bisect the chord AF at 3; bisect BA and BFai D and C; from the points A, D, B and C, draw ordinatcs parallel with the director jy,- now on plan, at No. 1, bisect AF at B; bisect BA and BF at D and C. From the points A, D, B, C and F, draw ordinates parallel with the director FV, to cut the concave, convex and center line '■> *In practice, a sood plan is to take the bevels off the drawing on to a piece of 1)oard, and liaii,;,' up until needed, tlien the drawmg board can be laid away. To take an angle from the drawing with a bevel, first lay down the steel square to one of the lines, then adjust the bevel from the square to the other liue. Plate 44. 20-1 of rail on plan; then transfer the points on the ordinates from the plan Fig. 3 to corresponding ordinates on face-mould Fig. 4, and trace the curves through the points, using a flexible strip. At F add on 1%" for miter on the line of tangent HF, but make tlu; .splice joint at right angles to tlie director Fy, instead of the tangent HF, as is done for a butt joint; by drawing the joint at right angles to the director Fy, the joint will be plumb when the wreath-piece is elevated into position, and at the same time will be square to the face of plank, because the joint is parallel to the major axis. Observe in this case the joint made from the director cuts tlie rail oblique at F, on plan Fig. 3, and not normal to tiie curve, as it would if drawn at right angles to the tangent HF, Fig. 2; hence the block pattern must be increased in width equal to the oblique cut at this joint. The sections M and JVshow the be^'els api»lied through the center of plank; they cross the tangents in their application. The block pattern is applied square to the lines made from tlie b«vels; the shaded parts show the thickness of plank and the amount of overwood to be taken oft". Fig. 5. Shows tlie face-mould for Nos. 2 and 3 wreath- piece. Make BJ equal the tangent BJ, Fig. 3; with B for a center and Bd, Fig. 3, as a radius, draw arc at A; then with J" for a center and JB as a radius, draw arc intersecting at A, join AJ; parallel with JB and JA draw AP and BP. Then PBJA will be the parallelogram on the cutting plane. Proof. The diagonal JP nuist e(iual the diagonal JP on plan Fig. 2. Make joints at A and B perpendicular to tlie tan- gents BJ and AJ; as both tangents are of eiiual length, the diagonal JIP becomes the director on the face-mould, and also on plan Fig. a. Draw the chord AB on mould and also on plan; bisect them at CF and D. Draw the ordinates parallel with the directors J'J'at Fig. 5, and also on phm Fig. 2; then transfer the distances from the plan Fig. 2, to the face-mould. Fig. 5, using chords for base line-; make A 2, A 3, and B i, B 5. each equal yi, Fig. 2. Now trace the curve through the points for the con- cave and convex sides of mould. ' If a, trammel or rod is desired in i)i'ofereneo to tiic ordinates, (.hen extend the diagonal JP iudelinite; make PO equul to the di;i;j;onal J/' in lliis ease. From the points 2, 3, 4, 5, draw lines indefinite, and at riiilit a.nirles to the joiuts; from O, di'aw the. radial lines OI> and <>A, indefinite, which gives the points of cotitaet, as G, 7, 8, 9 on the j-adi-'.l lines; at rij^'iit angles to OJ draw UK, to indieiite tlie tranunel and uss the rod, malving points, througli which draw tiie elliptic curves, using a pliable strip. The sections at M find N sliow the bevel at y, No. 2, Fig. 2, applied through the center of plank; observe they cross the tangents. Fig. 6. Exhibits the facc-nwidd for No. 4 ivr calh-piccc on plan. Make CLI equal CLI, at No. 4, in elevation, Fig. 3. Witii Cas a center, and Z>/, Fig. 3, for a radius, draw arc at D; with Zf as a center, and tangent LD, Fig. 3, for a radius, drav/ arc iidersccting at D, Connect LD; jiarallel with DL ami CL, draw CQ luid DQ for the parallelogram QDLC, on tiie cutting plane, or plane of plank. < " Proof. The diagonal LQ must equal the distance VN, Fig. 3. Make joints at D and C square to the tangents DL and CL, 203 Plate 44. join D/for the director; make B 8 and D 9 each equal D 5 No. 4 on plan, Fis- 2; malie C4, C5 each equal h G No. 4 on plan; now draw the chord line DC. Fig. 6, also on plan, Fig. 2; make LI, Fig. 2 equal Zi, Fig. 3, draw ID for the director on plan. Bisect the chords as previously explained, and draw tlie ordinates parallel with the directing ordinate DI to cut the inside, center and out- side of rail on plan, Fig. 2; now transfer the points on the ordinates on plan at No. 4, Fig. 2, to corresponding ordinates, Fig. 6, using the chords as base lines, then trace the elliptic curves through the points shown. If a trammel or rod be i)reterred, then prolong the diagonal LQ indefinite, make Qo equal LiQ; from O, draw the radial lines OC and OD indefinite; at right angles to the joints, draw 3 2 and 9 3, also 4 6 and 5 7 for the points of contact; draw OA parallel to 1?J and equal to the radius OS on plan, for the semi minor axis; draw the direction of nuijnr axis through O and at right angles to OA; make AF aiul AB each equal the half width of rail [2^^]; now with a rod, find points in the elliptic curve, through which trace the curves using a pliable strip. The sections T and S show the bevels applied through the center of plank, so as to cross the tangents; the shaded parts indi- cate the thickness of plank and alt-o the width at the joints to saw out the crooks from the plank; at the minor axis, 14" or less is all that will be required over the true width of rail, to allow for the twist of rail. Fig. 7. Exhibits the face^iouhl fur Xo. 5 wrcath-imcc on plan. Make tangent i)ikr equal DM, Fig. 3. With jD as a ceuter, and Eh, Fig. 3 for a radius, draw arc at E; with Jlf as a center, and EM, Fig. 3 for a radius, draw arc intersecting at E, draw tangent ME, which becomes the director, make joints at M aud E perpendicular to the tangents; at E, 2" of straight wood is added on to help the easing connecting the straiglit rail on tlie level; parallel witli tangents MD and ME, draw SR and DR for the parallelogram RDME o\\ the eutting plane. Proof. The diagonal MR must equal the distance EX, No. 5, Fig. 3 ; make D 2 and D 3, each equal m 8. at No. 5, Fig. 2. Make E 4 and E 5, each equal n 7, at No. .5 Fisr. 2. Draw chord DE; bisect the same at A, also bisect AD and AE at C and B; now ]>arallel with the direc- tor, draw ordinates iudefiinite. Bisect the chord DE, on plan, Fig. 2, in like manner, and draw ordinates parallel with the director ME, to cut the concave, center and convex sides of rail; now transfer the points on the ordinates at No. 5, Fig. 2, to corresponding ordinates on face-mould. Fig. 7, using the chord lines DE as a base; now trace the curves through the points, using a pliable strip. The sections Wand y, show the bevels applied through the center of plank; the shaded parts show the thickness of ])lauk, and also the width at the joints to saw out the crooks from the plank; observe the bevels do not cross the tangents in this case. If a trannnel be preferred to the onlinaies, then prolong the diagonal MR, and also tlie joint line 5 E 4, to inter.sect at O, then OE wiil be the scnii-major axis; the senii-iuinor axis will be at right angles and ecjual to the radius EO, Fig. 3, {lo^i"). Draw the radial line OD prolonged; at right angles to the joint, draw 2 7 and 3 6 for the points of taugency. Proof. RO must equal MR in this case. The ordinates, however, wiil be found more convenient, as less space will be required in the construction of the moulds. t- Plate 45. 203 Balusters. If the length of odd balusters be required under the tangent in elevation, Fig. 8; the directors on plan. Fig, 2, for each wreath piece will give their location on the tangents, and may be transferred to their position in elevation as previ- ously described. Self-Supporting Stairs. Self-supporting stairs circular on plan may be constructed in several ways. The main object is to have them secure at the land- ing and at the starting with a substantial system of carriage under- neath the steps. The string for tlie concave side sliould he made from lYz" plank, and dadoed on the bade; then bent over a drum and keyed with hard wood li:eys set in glue. The string for the convex side should be dadoetl and keyed in the same manner, and after being set in place a thin veneer may be bent over the same to cover the keys and joints. The steps and risers may be constructed in the usual way, all laid off from the working plan, which is drawn full size on the floor or large drawing board, ^\hen the stairs are to be set up in tlie building they should be lined off on the door, and the strings and risers set plumb over their lines on the floor. The string for the concave side should l)e set first and fastened to the joist above and floor below, and supported temporarily between by setting up several scantling and screwing them to the face of string. The.se scantling should be well braced, holding the string rigid and plumb over the lines on the floor. Now the string for the convex side may be set up true to the lines on the floor, and supported temporarily in the same manner, as the concave string. Fhice the supports at the face of string, and brace them well to hold the string firm and true to the lines on floor. The steps and risers, being tongued blocked and all well glued, may now be set in place. Then take strips ^2" thick and 3" or i" wide, glue and nail them with the .3" way in a vertical position to the back of the concave string, extending them down into the joist at the start- ing, and also up into the joist at the landing. Keep the strips up to line with the under edge of risers; then the sotHt will be regular. At intervals cut in brackets underneath each step, having the grain perpendicular to the tread. These brackets are nailed to the laminated soffit, and triangular blocks maj' be glued and nailed to the step, and also the bracket. After tlie soffit is covered in this way then strip for plastering, or a better way is to panel the soffit with Lincrusta Walton. Iron bolts should be placed across the stairs par- allel with and at the bade of every otlier rise; at the starting a spandrel should fill the triangular space to four feet high, which will give additional strength. At the landing two by half-inch flat iron L2"X'/2"j shiuldbelet into the joist and soffit 03 stairs where the soffit of stairs ease olT to to »he level. These irons should be well bolted tx) thesam*. . PLATE 45. Plate 45. E.rMblts the conatructlon of the face-mould for a st(tir-casc eUlptical on plan, and staHing from a scroll. Fig. 1. [Scale ?X^'=1 foot ] Shows the plan. The direction of risers in an elliptical stairs should be normal to the curve, or near to that as possible, so the rail may be con- structed more readily to an even height for the hand, and at the same time obtain easy flowing and graceful curves; and if the front string bo open, having return nosings and brackets, long miters on the return nosings will be avoided; or should tlio outer string be close, the nosings and string lines will show more even, and a more satisfactory job will be the result, and the angles of strings and risers will be less liable to collect the dust. Observe No. 16 tread is increased in width to receive two balusters, and thus allowing the rail tocase oft' gracefully at the landing. The con- struction of such stairs requires the skill of a practical stair-builder. In the construction of the wall string, dado the back of string parallel with risers. Bend over a drum and insert dry, hard wood keys, well glued. A joint may lie made at the center, and the drum m.'ide high enough to take in No. 10 rise. Then both the upper and lower pieces may'be bent over the same drum l)y turning the notched 204 Plate 45. edges toward each other, and thus save room and material. The outer striiis may be made in the same way; if for a close string, while on the drum lag out to the required thickness to admit the veneer on the convex side. Then house in the treads and risers; cross- tongue and bolt the joints of string if made in two pieces; light, thin mouldings may be easily bent and nailed to place; or if large and lieavy mouldings for panels; they should be worl<;ed from the solid plank for flrst-class worlc. If large mouldings are used as string mouldings, tlien work each member out sep;irately and nail tliem to place. A little ingenuity displayed in this line will avoid the kerling of mouldings, wliich should never be done, particularly when the finish is in hard wood. In splicing winders or steps plough and xise slip tongues cut across the grain obliquely, and of very dry iiiaterial; beads may be steamed and bent to plaoe; if large, they will require to be nailed every few inches. Tlie steaming of light v.ood discolors the g'rain, and should be avoided for liard wood finish, unless of walnut, which being a darlv wood, the steaming is not objcctionalile. For intenmdiate rails between the string and hand rail, which is intended to support balusters or panels of spindle woj'li; these intermediate rails nuiy be laminated in several thiclcnesses of strips by clamping and gluing three or four pieces at a time; over the drum the strips sliould be tiiin enough to bend easily, the clamps may be constructed on the drum, and wedges used every few inches; paper should be laid over tlie drum before gluing, to prevent the work adhering to the drum; string mouldings may be built up in the same manner, and worlT, cutting perpendicu- lar L at U; then TE'is the height for No. 4 wreath-piece. From E, and scjuare to perpendicular F, draw E, NN; then NN, F is the hcUjht for No. 5 wreath-piece. From F, aud square to per- pendicular G. draw FW, cutting perpendicular N at X; tlu'u WG will be the hckiht for No. (5 wreath-piece. It will be observed that the tangents EM and MF for No. 5 wreath-piece are the same length, and also height as for No. 3 wreath-piece, and need not be described further, as the face- mould and bevels for No. 3 will apply to No. 5 wreath-piece; for by turning No. 3 wreath-piece upside down it will answer for No. ri. Prolong the tangent CJ" to intersect BQ at a; extend tan- gent JB J" to intersect PC at b: prolong tangent DK to intersect CS at d: prolong tangent CK to intersect DR at e. Make of equal the chord AB. Fig. 2; let Pg-and Ck'^o. 2 equal the chord BC, Fig. 3; make J^ii equal the chord CD, Fig. 2; make 2Vcqual the chord DE, Fig. 2; let Win equal the chord FG, Fig. 2, and Wl equal the diagonal NT, Fig, 2; make Dn equal the diagonal MSov KQ, Fig. 2. Make SP equal KIT, on plan. Fig. 2, Let kr equal KV, Fig. 2. Make uq equal Z/W on plan, Fig. 2. Let Ns equal Njr, on plan. Fig. 2. From Q, No. 2 and square to CJ, draw Qt: from XX, and square to tangent BJ prolonged, draw XX Z; from p, and square to tangent DK prolonged, draw pv; from r, and square to tangent CK prolonged, draw i', 00; from q, and square to tangent DL, draw qw; from S, and square to tangent P2V prolonged, draw Sy- Parallel with JQ, draw the half width of rail (2^0 > cutting the tangents SJ"and CJ" prolonged, at IS and 20. Bevels. Now return to plan, Fig. 2, and draw the bevels aud the increased width of face-moulds at the joints. Make Ja 206 Plate 45. No. 2 equal Qt, Fig. 3. Join aB; make Jb No. 2 equal XX Z, Fig. 3. Join bC, which gives the bevels for No. 2 wreath-piece. No. 1 wreath-piece has one tangent raking, and one level, hence there will be but one bevel required, and that is found in the angle at B, Fig. 3. No. 3 has both tangents iucliuiug, and of different inclinations, hence two bevels will be required. Make Nd No. 3, equal pv. Fig. 3. Join dC: make Ve No. 8, equal r, 00, Fig. 3. Join eD, and we have the bevels in the angles at d and e for No. 3 wreath-piece. No. 4 wreath-piece has both tangents Inclining the same, therefore only one bevel will be required for both joints. Make W/No. 4, equal QfW, Fig. 3. Join fE, the angle at i gives the bevel for No. 4 wreath-piece at both joints. No. 5 wreath- piece is the same as No. 3. No. 6 wreath-piece has one tangent inclining and the other horizontal, and as they form an obtuee angle, two bevels will bp required. Make xg^o. 6 equal the height WG, Fig. 3; join gf; make yh equal ys. Fig. 3; join hG for the l)evels in the angles at g and h for No. 6 wreath-piece. For the increased ■width of moulds at the joints. Draw parallel with tlie tangents the half width of rail to inter- sect hypothenuse of bevels; as for No. 3 wreath-piece at j aiul 1; for No. 4 wreath-piece at m; for No. 6 wreath-piece the Hue cuts at n and p. Fig. 4. Shows the face-mould for No. 1 wreathr^iece; one tangent U incUning and the other horizontal in elevation, and as the tangents on plan form a light angle, only one bevel is re- quired, and also the face-mould, may he drau-n icith tin; trammel. Make BH equal BH, Fig. 3; make HA scjuare to HB, and equal to the radius OB on plan Fig. 2; draw AO and BO parallel with HB and HA for the rectilineal parallelogram OAHB on the cutting plane. Proof. The chord AB and diagonal HO must equal Bf in elevation. Fig. 3. Make A 2 and A 3 each equal J" IS, Fig. 3; make B 4 and B 5 each eciual the half width of rail (2''); now pivot the trammel at O, with the arms on the major axis A O, and trace the concave, convex and falling lines of face-mould. The section at M shows the bevel found in the angle at B, Fig- 3 applied through the center of plank; the section at iV shows the try square is applied, and the block pattern shows the thickness and width to saw out the crook. Fig. 5. Exhibits the face-nundd fur Xo. 2 wreath-piece; as the plan is the segment of a true curve, this face-^nould may also be draicn with a trammel. Make CJa equal CJa,, Fig. 3. With C as a center, and Cg, Fig. 3 for a radius, draw arc at B; again, with J" as a center, and JB, Fig. 3 for a radius, draw arc inter^ecting at B, connect BJ; parallel with JB and JC, draw CP and BB for the parallelo- gram JPBJC on cutting plane. Proof. The diagonal JB must equal the distance bK, Fig. 3. Connect Ba for a director; parallel with Ba draw the direc- tion of minor axis PJ^ indefinite; make Pi^ equal the radius BC, Fig. 2; make PG and F7 each equal the half width of rail [2" |. Let C2 and C3 each ecjual JlS, Fig. 3, and B 4, B H each equal J 20, Fig. 3; pivot the trammel at P, w ith the arms at right angles to the minor axis PF: then for the center line on face- mould, set from pencil to minor pin the distance BF, now place Plate 45. 20? the pencil in the point at J5, and the minor pin in the major groove, at the same time slide the major pin to drop into the minor groove, fasten the major pin and trace tlie center line on mould. Proceed in the same manner to trace the concave and convex sides of mould. Make joints at JB and Cat right angles to the tangents SJ" and CJ. The section at A shows the bevel taken from the angle at h. Fig. 2, and the section at D shows the bevel taken from the angle at a, Fig. 3. They are applied through the center of plank, and the shaded parts show the over wood to be removed. Fig. 6. SJioivs the face-mould for No. 3 wreathrpiecc, which is termed a '■'■ utrcath-picce with an intermediate easing," or accommodation ivreath; not because both tnnijetits are in- clining, but that one tangent has a greater inclination tlian the other. Make DKd equal DKd. Fig. .3; with D as a center and Dh, Fig. 3, for a radius, draw arc at C; again, with JK" as a center and .ZTC, Fig. 3, as a radius, draw arc intersectim; at C; connect ICC; parallel with CK and DK draw DQ and CQ for the parallelo- gram QCKD on the cutting plane. Proof. The diagonal KQ must equal the distance en. Fig. 3. Join Cd for the director; draw the chord CD; bisect CD at 2; bisect 2 Cand 2 Z) at 3 and 4; parallel with the director Cd draw ordinates indefinite from tlie points 3, 2, 4, and D; now on plan Fig. 2 bisect the chord CD, in like manner at the points 3, 2, 4 and D; make K XX equal Sd, Fig. 3. Join C XX for the director on plan; parallel witli the director C XX draw ordinates from the points 3, 2, 4 and D to cut tlie concave, center and con- vex side.s of rail on plan at 5, 6 and 7. Now transfer the points 2, 5, 6 and 7 from the ordinates on plan Fig. 2 to corresponding ordinates on face-nu.uld. Fig. 0; make joints at Cand D. square to the tangents CK and DK; let C8 aiul C9 each equal dJ, Fig. 2. Let D 10 and Z> 11 etiual le. Fig. 2. Through the points trace the concave, center and convex sides of face-monld. The section A sliows the bevel found at e. Fig. 2, and at .section B, the bevel found in the angle at d, Fig. 2. The bevels are applied from tlie upper or face side of crook, and through the center of plank; observe they cross the tangents. Fig. 7. Exhihlla the face-mouUl for No. 4 wreath piece; the tangents both have the same iyiclitintion. hence only one bevel ivill be required for both joints, and the wreath-piece is termed a v^reuth-piece without an easing. Make DL equal DL, in elevation. Fig. 3. With Z) as a cen- ter, and the distance EJ, for a radius, draw arc at E; again, witli Zf as a center, and LD for a radius, draw arc intersecting at E; join LE; parallel wiih LE and LD, draw DR and ER, for the parallelogram RDLE, on the cutting plane. Proof. The diagonal LR must etiual tlie diagonal LR, on plan. Fig. 2. In all cases when the two tangents have the same inciination and length, the diagonal becomes the director. Make tlie joints at E and D at right angles to the tangeut-s LE and LD. Draw the chord DE; also ttie diagonal RL bisects tlie chord at .5. Bisect 5 E and 5 JD at 6 and 7. Parallel with the director LR, draw ordinates indefinite from the points Z), 7, 5, 6 and E; draw the diagonal RL, on plan, Fig, 2, bisects the chord DE at .5. Bisect 5 E and 5 jD at 6 and 7. Parallel with the director LR, draw ordinates to cut the concave, center and con- vex ides of rail, then transfer the points on the ordinates, Fi^. 3, •:i08 Plate 45. to correspoudiug ordiuates ou the face-uiould, Fig. 7, as 5, 8, 9, 10. For the increased width of mould at the joints, make E 3 and E 3 also D 4 and D 5, each equal fm. Fig. 3. The concave, center and convex curves of face-mould may now be drawn through the points indicated, using a flexible strip. The bevels for both joints as shown at sections H and J, are found in the angle at /, Fig. 3, the bevel is applied so as to cross the tangents; the shaded parts show the width at the joints to sav»' out the crooks. At the narrow part of mould %'' of overwood may be allowed on the concave side, and less on the convex side. At that part of the mould the bevels bleud, and the side of rail is at right angles to the face of plank. Fig. 8. Exhibits the face-moxiJd for No. 6 urcath-plccc;* one tangent GN, is horizontal, and the other NF, is inclining; tfie ivreath- piece is termed "a wreath-piece %vith a full easing.^' Make FN equal FN, in elevation. Fig. 3. With JP as a center, and the distance Gni, Fig. S, for a radius, draw arc at G; again, with iVas a center, and GN, Fig. 3, for a radius, draw arc intersecting at G; join NG; let it be observed NG I)ecomes the director for the ordinates. Parallel with NG and NF, draw FT and GT, and we have the parallelogram TGNF, on the cutting plane. Proof. The diagonal NT must iqual the distance Gl in elevation, Fig. 3; if so, the angle of tangents is correct. Draw the chord GF; bise'ct GF at 18; bisect F 18 and G 18 at 30 and 19. From the points 19, 18, 9.0 and F, draw ordinates indefinite, and parallel to the director GN. In like manner on plan, Fig. 3, bisect the chord in IS, 19. ;20 and draw the ordinates parallel with the director GN to cut the concave, center and convex curves of rail on plan, as shown at 18, 31, 33 and 33. Now transfer points from plan to corresponding lines on face- mould, Fig. 7, For the increased width of mould at the joints, make F 3 and F 3 each equal hp ou plan, Fig. 3; also make G 4 and G 5 each equal gn on plan, Fig. 2, then trace through the points for the concave, center and convex curves of mould, using a pliable strip. Add on two or more inches at the joint G, to help the easing in the wreath-piece to connect the straight rail. The bevel shown at section L is tsiken from the angle at g, P'ig. 8, and the bevel shown at section F is taken from the angle at h. Fig. 3. Observe they do not cross the tangents, as there is no point in the mould that is equal to the true width of rail [4"]. The bevels are applied from the upper side of crook, the shaded parts show the width at the joints to saw out the crook. Balusters. If it be required to find the length and location of balusters under the wreath rail, then make J" 34 No. 3 ou plan, equal Qa, at No. 3 in elevation, connect 34 B for the director of No. 3 wreath- piece on plan. Fig. 3. Also make E 2b, No. 5 wreath-piece ou plan equal Sd in elevation, Fig. 3. Connect S 25 for the director of No. 5 wreath-piece on plan. The small circles indicate the location of balusters on plan. At Nos. 1, 3, 3, 4, .5 and 6 wreath-piece on plan draw from the center of each baluster parallel with their directors, lines to intersect the tangents; the black dots show their intersection on the tangents. i^ Now transfer the position of balusters on the tangents to their respective treads in elevation, being careful to space them * Note.— To facilitate reference from plan to cloviition, and also face-moulds, numbers are used, which correspond to each other. Pr.ATE 45. 9^: between the perpendicxilars, as thej- are shown on the tangents on plan. The small circles at each tread show their location in elevation. From the center of each baluster draw perpendiculars to intersect the ruider side of rail as shown. The height of scroll from the top of No. 1 step to tlie imder side of rail we will say is 2'' Z", and the elevation Fig. 3 shows No. 1 baluster to be 1" from top of No. 1 step to the under side of rail. Then this one inch (\") has to be deducted from the odd balusters, as shown in elevation; thus, the second baluster measures 2 1^: '"' f rom the top of step to underside of rail; then (2M^^— l''+"i' 3'^=2MJ^'0; No. 2 baluster equals 2' A:}i" from top of step to under side of rail. The third baluster on No. 1 step equals 5 J^^^; then 5 K''' plus V Z" and minus \" equals 2' 'IM" for No. 3 baluster, from the top of No. 1 step to uuder side of rail when in position. The balusters on No, 4, 7, 8, 13, treads, and the first baluster on the landing No. 17, are the same length as the first baluster; those on No. 3, 0, 9, 12 and 14 treads are "^A" longer than the first; the balusters on No. 10 and 11 treads are %'' longer; on No. 5 tread a M" longer; on No. 2 tread 2>.<^^ longer; on No. 15 tread V longer; on No. 16 tread 4^'' longer, and the second bal- uster at the landing No. 17 equals IM^^ longer than the first baluster; the balusters on the level will be Z" longer than the first. The balusters under a wreath having an easing, may vary a trifle, but this system will be correct enough for all practical purposes. If desired, the exact location of each haluster may bo found on the face-mould and transferred to the wreath-piece in the rouKli. When the mould is shifted on the tangents to its exact position on the croolc, tlien tlie center of eacli baluster may lie pricked through, and also the direction of eacli ordinate, then bored for the balusters to a templa,te made to the pitch. To do this, extend the center of balusters to intersect the tan- gents as shown in elevation. Fig. .3, for No. 4 wreath-piece: now transfer from tlie tangents in elevation to face-mould. Fig. 7, as shown by the dots; then parallel witli the director, draw lines to intersert the center or falling line of mould, tlius establishing- tlieir location. To find the pitch to cut the template for the wreath-piece. Fig, 6, make E, PP, No. 5 in elevation. Fig. 3, equal the shortest distance from D to tlie director C, XX No. 3 on plan. Fig. 2; draw RR, PP, Fig. 3 for the anales at RR or PP, which gives the cut for the template PP, E, RR, to guide the bit when boring. For No. 7 mould the template is made to tlie pitch TEJ No. 4 in elevation, Fig. 3. For Fig. 8 mould the template is made to the pitch XgF No. 6 on plan. Fig. 3, and held at right angles to the ordinates, as ab, Fig. 7, or 6 a. Fig. 6, r As tlie face-mould is drawn the upper or back of rail is shown. The opposite side will be the underside or lircnat of rail. The boring of a rail for balusters is easily done, iind the ijcst results will be obtained by doing the boring on the stairs after the rail is hung and in Its proper place. If the balusters be square at the top, then each baluster is plumbed to place and filled in between. The lower end of scroll, containing the eye, may be worked from a tliick piece of plank by marking out the plank to the pattern No. 1 on plan Fig. 2, and sawing out the same neat and square from the face of plank, and making the joint to the bevel shown by the dotted lines at No. 1 in elevation, I<'ig. 3; thus avoiding the face-mould No. 1, Fig. 4, and also the joint at A, on plan Fig. 2. In tliis case the twist line must lie carried around so as to give a graceful curve. The re- ciprocal scroll shown at Fig. 5, Plate G, may be applied here with pleasing ett'ect. 210 Pi.ATK 46. PLATE 46. Plate 46. Exhibits 14 profiles of hand-rails one-half full size. Figs. 3 and 4 show the "pew back" pattern; Fig. 8 suits well for an altar or office rail; Fig, 9 for capping; Fig. 11 for cap- ping on iron; Fig. 13 suits well for a wall rail. For square top balusters, the underside or "l)reast" of rail should be rebated out, as at A and S, Figs. 3 and 6; then filled in between. Fig. 5 shows the straight rail made in two pieces. These rails are classed as '-double rails," with the exception of Nos. 9 and 11. Straight Rail,. If the material is dry, the straight rail may be made from plank as the cheapest way ; 11 very large double rails, then use inch stuff glued to a pine core, as shown at x'ig. 3, or in several thicknesses, as shown at Figs. 2 and 7. "When ripping out the straight rail select plank that have squared edges and thoroughly dry. Tlien they will not spring much, if any. If the plank be springey, then rip up the center and joint off straight. If the rail be bowed up or down, it will be easy to straighten on the stairs when putting in the balusters, but if crooked sideways, it is more difficult to straighten, but it can be done by bracing the rail a little over straight the other way, and allowing tbe braces to remain over night after the balusters have all been well glued to place; then when the braces are removed the rail will come about straight. To do this requires one to have some experience in hanging rails. By thoroughly wetting the concave side of the crooked part, then forc- ing it out of straight the other way and leaving until dry, the rail will be about straight. Short kinks in the straight rail will some- times occur in this way: the material being cross-grained, or a knot curl on the side of rail, will cause short kinks, particularly if the stuff be not thoroughly dry. When rails are worked out for some time before they are required these defects should be avoided much as possible, for a rail may be worked out nice and straight, then placed on the rack until v.'anted, and upon examination will be found crooked. This is a difficulty the stair-builder labors under, and hence the necessity that the material for straight rail should be made from the best lumber, straight in grain and thoroughly dry. Black walnut requires one year for every inch in thickness to dry, and then it must be well "stuck" on strips up from the ground, where the air can circulate freely around the pile. Owing to being a dark wood It does not dry out so soon as any of the light-colored woods, and if near the ground will never dry. It is the best native wood we have to-day for hand-rails, balusters and newels; the sap part is objec- tionable, and should not be used in first-class work, but in second- class work may be stained* to imitate the heart wood. The sap in ash wood is like the heart wood, and tlierefore may be used. The sap part of oak should be avoided, as it is mostly damaged by the worm, and in cherry the sap .should be cut away, as it is very inferior to the heai't wood. The Wreath Part. For double rails the wreath-piece should be made in one piece from well sea.soned lumber, as it is very tedious work to make the Avreath part of double rails in two or more pieces when fitting them together. It may )>e difficult to find plank thick enough for large double rails. In that case the thickness may be in- creased by gluing the required amount to the underside of crook. This, however, sliould never be done, if possible, a.s the glued joints work out to a feather edge, and, if exposed to dampness, may ruin the wreath; plank thick enough for the wreath rail should in all cases be used for flr.st-class work. PLATE 47. Plate 47. Shf>ws the construction of an open newel stair' case having a qxiailmmjular well'-hole and close front strimj, pnishcd irith string mouidings and rosettes. ^- To color the sap of hlack wahmt. Take one-half gallon of water, one-half pound of dry burnt umber, one-quarter pound of rose pink, one-quarter pound of Vandyke brown; mix thoroughly, apply with a sponge, allow to dry and rub off; when oiled will be uniform with Plate 47. 211 Pig. 1. [Scale H^^=l footj. Shows the plan having 18 risers. The main newel is S'^yiS'^, and the minor newels are This style of a stair-case belongs to the dog-leg type, having newels in the angles, and the rails to butt the newels, either straight or with the old style "ramp and knee" or "goose neck ; sometimes the minor newels are allowed to extend up and finish with turned finals, at other times the rail is mitered around the newel, forming a cap with turned top, and the lower end of newel ex- tends down below the ceiling, far enough to receive the lower edge of string and finish with a drop. Eig. 2. [Scale %''=\ foot]. Shoxos the plan of the minor newel at Nos. 8 and 9 rise. The face of risers is located at the center of newels; A and B show the front string housed into the posts half an inch; the dotted lines show the steps housed into the posts and string M^'. Fig. 3. SJwws the elevation of four sides A, B, C, D, of post, Fig. 2, and also the strings connecting the same. * The treads, risers and strings are housed into the newels, and the string mouldings are allowed to butt against the posts. The post is mortised to receive the rail, which should be tenoned and bolted at the joint. -' If hard wood finish, the strings may be veneered on a pine core. H shows the core, and K the veneer ^'^ thick, which is glued and well clamped to the pine core. J shows the shoe grooved out on the underside to straddle the string, and on the upperside to receive the balusters. L shows the filler cut in between the balusters. N shows the plaster, and P a moulded strip to cover the joint of plaster. R shows the carriage. S, the joist at the platform and lauding. The posts and strings may be chamfered, paneled and moulded in a variety of ways to suit the style and taste of the architect and stair-builder. The post at Nos. 12 and 13 rise is lined off similar to Figs. 2 and 3. Fig. 4. Shows the plan of post at the landing No. 18 rise Tlie face of rise being at the center of post. Fig. 5. Sfioias the elevation of post and the raking and level strings connecting the same. C, D, E, F, shows the sides of post lined off to correspond •with plan, Fig. 4. The rail T extends to the post straight. V shows the floor. V, the rough flooring, X shows the stripping underneath the joist for lathing. N indicates the plastering. The rail is set V %" from the floor to the top on the level, and on the rake 2' ^" from the top of step to the upper side of rail, plumb over the face of rise. c- Fig. 6. Shows a method of tnakiiuj and gluing up box or b%iilt posts. C and E are the two narrow, and D and Pare the two wide pieces; J" and J" show the piue core. The sides C, D, E and F should be made from two inch stuff, and of perfectly dry material. A jointer, to joint and take the sides out of wind, and a pony planer to size them to an even thickness, are the best tools for preparing the material for box or pedestal posts. The pine core J J is made to the proper width and glued to the wide sides D and F, first, then let dry, afterwards glue the internal angles and the edges of the narrow pieces, put a clamp at each end and one at the center, so as to draw up the naiTOW pieces close to the pine core; then put all the clamps on the wide pieces D and F that will be thought necessary to make the joints close. 31^ Tlate 4S. Carpentry and Joinery, Carpentry. Is properlj' divided into tliree branches, con- stnictivc, descripthic and mcchaniad. Constructive Carpentry. Is divided into '-carpentry and joinery." tlie distinguishing featnre being less tools required tor the carpent(U' proper, who takes charge and erects the naked carcase of the building. Then afterwards the joiner prepares the doors, sash, shutters, stairs and other finish for and com- pletes the structure. In our country the diit'ereiiee, however, is very artificial, as most carpenters have all the tools reqiured in the art of joinery, with the exception of the stairs. In large cities that branch of joinery is made a separate business. Constructive Carpentry shows the practice of shaping and jointing the different pieces of wood in the erection and com- pletion of a building, according to the design and intention of the architect. Descriptive Carpentry. Shows the lines or methods for forming various kinds of work on plan, elevation and detail, acccmling to and by the rules of geometry. Mechanical Carpentry. Shows how to arrange the dif- ferent timl)ers of a building relative to their strength, and the strain to which they may lie subjected. This brancli of carpen- try is becoming inore studied by the young carpenter who asjiires to the upper rung in the ladder of that noble and honored occu- pation. The best and most practical books for the young American to study in this branch of carpentry is the "American House Carpent(!r," and also "Transverse. Strains," by Mr. H. G, Hatfield. In this branch of cariientry we will give a fev\' rules to calcidate the strength of timber, aiul recomnienil the student to the dilTerent works herein rel'evred to lor a more extensive elucidation vii the subject. PLATE 48. Plate 48. ExhihUs the construction of the ''Hip,^' "Faf- Inf (tnd ''JdcJi" Rafters for a roof having the angles on q)l(in, aJL right angles. "Fig. 1. [Scale }4^^=1 foot]. Shows two right angles of a lmildi)tg to he hipcd. t KG, GA and AL indicate the outside line of walls; ZZ, the inside of walls. Bisect GA at N; make GK and AL, each equal AN; draw KL parallel to GA: at right angles to GA, draw a line from N to intersect KL at H; draw GH and AH for the seat of the center line of hip. Now space off for the seat of jack rafters. B, B, &c. Plate 4S. 211 Pigi 2. Shoios the elevation of "common rafters.'^ GA is the width of building (12'' C''), and agrees with GA. Fig. 1. JCX shows the walls, yy, the ceiling joist resting ou the "wall plates" W; bb indicates the "raising plate" flush with the walls on the outside. Bisect GA nth; draw hD perpendicular to AG; make tlie height from top of raising plate at J to D, to equal one-third the span GA {i^ 0'^), which is termed a "third pitch." Draw DG and DA to the top edge of raising plate bb, for the length of common rafter from the "toe" to the center of "ridge plate" at F; allow the half thickness of ridge plate less, when cutting the rafter. Now return to Fig. 1. Draw HE at right angles toHG, and equal to hD, Fig. 2. Draw EG from the top edge of raising plate b to the center of ridge plate at F, for the length of hip. When cutting the hip to the proper length, deduct the half thick- ness of ridge plate, which is equal to Ed on the inclination of hip as shown, carried up from the intersection of the seat of hip with the scat of ridge plate at H. To find the length of jack rafters. Make HJ, Fig. i equal DA, Fig. 2, the length of common rafter from the toe to the center of ridge plate. From J, draw a line parallel to LA, inter- secting GA prolonged; at S, draw SH \or the center line of hip, set off the half thickness of hip parallel to SH, Proof. SH must equal the length of hip GE; CH is the same length; now extend the jacks from L to J shown by the dotted lines. Then JS, 2 3, , and the purlin be parallel with the walls, then the down cut on purlin will be a right angle to the upper edge, and the side cut against the hip will be equal to the angle that the seat of purlin makes with the seat of hip on plan. In this case the bevel shown in the angle at U, Fig. 1, gives the side cut. "■ *This method of flndins the b:if:KiiiK for the hip rafter is ascribed to William Pope l)y Godfrey Eii'lsards, in his translation of the First Book of Andre-w Pulladio, 187'o.— Stuart's Dictionary of Arch, 214 Plate 48. - At B is shown another position for the purlin, the sides being oblique to plan, or perpeudicular to the inclination of the common rafter. The method to find the cuts for a purlin in this position is shown at Figs. 3, 4 and 5. [Scale >2''^= 1 foot.] Fig. 3. Shows a section of purlin ^" by 9". Apply the bevel shown at 2), Fig. 2, from the upper side of purlin, and draw the plumb line AC across the section; at right angles to AC draw .£72 and D .3. Fig. 4. Shoivs how to find the doivn cut against the hip. Let AB indicate the "arris" of purlin shown at E, Fig. 3; parallel with AB draw the depth of purlin at C(9'^); set a gauge to equal D 3, Fig. 3. The dotted line DE indicates tlie gauge line drawn parallel to AB\ now set a bevel to the angle that the purlin makes with the hip on ))lan, which in this case, the hip being square, the bevel will be an angle of 4.5 degrees, and is shown in the angle at U, Fig. 1. Then apply the bevel found in the angle at U, Fig. 1, from the arris AB as FG, intersecting DE at 2; from 2, and at right angles to AB, draw 3 3; join F'S for the bevel in the angle at AF o, required for the down cut. Fig. 5. Sliows how to find the side cut of inirlin against the hip rafter. Let AB iudicate tlie arris E, Fig. 3. Make BC e(3ual t\w breadth AE, Fig. 3 [5^^J. Let BJE7 equal E 2, Fig. 3. Draw tlie gauge line BE parallel to AB; from the arris AB, apply the bevel found in the angle at U, Fig. 1, cutting the gauge line DE at 3. At right angles to AB, draw 3 4. Join Hi for the bevel required in the angle AH4. This method of finding the bevels for the cut of purlin hutting the hip, is practical, as they may be lined otf on the tim- ber, when the material is on the "trestles." * Fig. 6. Exhibits the rear and part of a main biiUdiiig on ))lan. AE, FC and BD indicate the line of walls. Tin; interna' angle at J* must have a valley to form a junction with the rear wing; also the plane of that part of roof between the hip and valley will intersect with the plane of roof on the opposite side forming a short hip. The width AB, of main building is 32' 0", and the wing CD, 20^ 0''. How to lind the length and cuts of hip and valley rafters, also the length and cut of jack rafters between the hip and valley rafter. Bisect -E7G at H. draw a line from JTindelinite and i)aralli'l io AE; imi\<{^ HL ^'<[Ui\\ HE, connect JE7Z/ for the seat of long hip on •'♦■nter line; join LG tor the seat of short hii> on the center line; s and valley rafters. The down cut for all jack and common rafters are alike, and the bevel is shown at O, Fig, 7. The bevel for the foot of all common and jack rafters that rest on the raising plate is shown at X, Fig. 7. The bevel at Z will give the cut of hip against the ridge plate at W, but must be applied from the inner side 4 4 after tiie hip is backed. The valley rafter will need no backing, the edge being left square in this case. If exposed aud the sheathing to form a finish, then the top edge of rafter must be worked into a V shape, the bevel being the same as for the backing of the hip. As the valley rafter forms a right angle with the hip at Q on the plan, then the cut against the hip rafter WFwill be a square rut, if applied before any backing is done to the valley rafter; 7, 7, shows the raisiug plate; the "foot" of rafter is notched out, form- ing a "heel" to butt the raising plate and ueutralize the tlirust of ratter. 216 Tlate 49. Fig. 9. Slioivs a ffraphicnl method to determine the stiffest beam that can be cut frorii round timber. ii't AB indicate the diameter of a round losj; bisect AS at C; with A and B as centers, draw arcs from Cto interstct the circumference of loj^ at E and D\ join AE, EB. BD and AD, forming the rectiliutal parallelogram AEBD lor the size of beam. PLATE 49, Plate 49. Shoivs an cc-fiy latthod lyw to find the lewjth of /(//> ((nd Jar); rafters, and cuts for the same, when the angles of bu'ddbiij are obtuse or acute. Fig. 1. [Scale M"=l foot]. Shows the plan. AB, jBCand CD sliow tlie outside line of walls: ZZ sliovvs the inside line of waiis; vy, &c., indicate the line of raising plate, wliich lias to be well nailed to the joist to resist the thrust of rafters; XX show the "lookout joist" for cornice. Bisect AD at G, draw GH iiarallel to AB for the center of ridge plate; bisect the angle ABC, also the angle BCD, at E and F; draw BE, also CF prolonged, to intersect GH at K; then ifB and iiCC are the seat lines for the center of hips. On each side of GH, set off the half thickness of ridge plate, also set olf the half tliickness of hips on eacli side of the seat lines BK and CK. Now space off 2, U, V, W, &c., for the seat of com- mon and jack rafters, and draw US, Q 4, R 5, M 6, &c., at riglit angles to the line of walls, to intersect the seat of hips at P., 3, 4, 4, 5, 5 and 6, 6; in this case the rafters occupy position directly over the joist. Fig. 2. E.vliihiis the Icwjih of common and jack raff cm, ■in elevation, the Icnrjth of each jacli is aiid 4 7 all perpendicular to the center line KL; join G .">, G G and G 7 for the bevels. The bevel in the angle at 5 is the side cut. and is the same as shown in the angle at 6, Fig. 4. Tiie bevel in the angle at G gives the butt cut, aud the bevel in the angle at 6 gives the miter cut if required. Fig. 8. Shoi'-s the bevels when the angle on jylan is a right angle. Let GH and HE indicate the sides of box, and HB the miter on plan corresponding to HB on plan. Fig. 4. Ohseive the dotted lines that are drawn from the points A and B, Fig. 5; parallel to the center line KL, intersect the miter BH on plan at 2 and 3; from the points 2 aud 3, draw lines at right angles to the center line KL to intersect the outside dotted lines at 5, 7 ami 6; draw JEf 5, H6 and iJ7 for the bevels. The bevel in the angle at 5 gives the side cut, aud is the same as shown iu the angle at 7, Fig. 4. The bevel iu the angle at H gives the butt cut, and the bevel in the angle at 6 gives the miter cut if required. " The student, by a diligent study of these eight figures, may find the cuts for any splay. •330 Plate 50, f'igs. 9 and 10. Show how to find the veneer for a cir- cular door-head having splayed jambs; the splay being carried around the head agreeable to the jambs. Fig. 9. Shows the plan. AB and CD show the splay of jambs. Prolong AS and CD to converge at O: with O for a center, and OD and OC as radius, draw the arcs DE and iJi^' indefinite. rig. 10. Shows the elevation of arch. BD is the lesser diameter, and corresponds to BD, on plan, Fig. y. From the center O, draw the semi- circle BCD; with the dividers, divide the semi-circle into 11 equal parts, as 1, 3, 3, &c. Now return to plan, Fig. 1, and with the dividers, step off on the arc DE, the same number of equal spaces, as 1, 2, 3, &c., to H: draw the radial line OH to J for the length of veneer DEHJFC required. This veneer is made from thin slutf and bent over a semi- conical drum made to the lines AOC, then tapered staves glued on the back, and cleaned off afterwards, thin strips may be glued on the back to stiffen. The radial lines from O, will give the taper for the staves, as shown at EF\ lor painted work, the tsplayed head may be glued up with staves. For the joint bevel of staves, draw the two staves ab. dc. Fig. 10. Draw the radial line ob: return to Fig. 9, make Kb equal Ob, Fig. 10; draw ba parallel to OC; from K, and at right angles to ba, draw Kc; return to plan, from O, and per- pendicular to Ob, draw a line indetinite; prolong the back of stave be, to intersect the perpendicular from O at Z; make Of equal Kc, Fig. 9. Join fZ, and the angle at /, gives the joint bevel for the staves. Pigs. 11 and 12. Shows the above jyrinciplc applied to the con8tructlo)i of a sphtyed "Pue Bark'' circular o)i plan. Fig. 11. Shows the plan of circnhir pue. [Scale Js'^= 1 foot]. O is the center, and OA shows the radius for the intersection of the back with the seat. At Fig. 13, J>^ shows the splay of back. Prolong the incli- nation of the splay ED to intersect the perpendiculars from O. and from the intersection (not shown), sweep the curves jDi^ and EH indefinite. Now divide the line ACE. Fig. 11, into any number of equal parts, as 1, 2, 3. 00 B:=—j^=-^~- ^g, ^- =12,500 lbs. It is also proven from experiment that if the same beam be firmly fixed at one end ;md weiehted at the free end, as shown at Fig. 9, the breaking weight would equal one-fourth the first ex. ample, or ""* =^1562.5 lbs., and the formula would read 4 „ bd^C 2^'X10'^X10'^X500 ^=-4'L =- 4X1*3- -'^'''■' ''''• And for a uniformly di.stributed load, as .shown at Fig, 10 would be doubled, or equal to 3125 lbs. It is al.so a fixed rule that if the beam be firmly fixed at both ends and weighted at the center, as shown at Fig. 11, then the Mechanical Carpentry 336 breaking weight would be one and a half times the first example, or 9375 lbs., and the formula would read : JB= j^ = ^^7 =9,375 llw., and the uniformly distributed load, as shown at Fig. 12, would be doubled, or equal to 18,750 lljs. If the beam be firmly fixed at one end and supported at the other, as shown at Fig. 13, then the breaking weight would equal 1}X times the first example, or 7812 lbs., and the fonuula would read: 1 H bdPC 13^X2''X10''X10''X500 lbs. Br= j^ = jgT =-7,813 Ih->. and the uniformly distributed load, as shown at Fig. 14, would be double, or equal to 15,634 lbs. The foregoing rules as set forth, are for the breaking strength of timber, and are established from the average of several tests made on each kind of wood; the bars selected for the tests are supposed to be perfect, or nearly so, hence a factor of safety is reouired to avoid accident. Mr. G. Hatfield puts the factor of safety for the above formula at 3 and 4; The New York building laws also provide a factor of 3 for safety. Valuable data has recently been added to the transverse strength of timber by Professor Lanza and his students at the Massachusetts Institute of Technology. They experimented on full sized timbers, and made the discovery, that the usual factor 3 for safely is too low for the j*?neral run of timber as supplied from the mills and yards. The knots proved to be the weakest part of the timber. A factor of 6 will nearly harmonize Mr. Hatfield's formula with Prof. Lanza's experiments. Then if the breaking weight in the first example be divided by 6, the result will be the safe or working load for the timber. The formula would then read : equals 1,041 lbs. for the safe load concentrated at the center W, Fig. 7. And the unifonnly distributed load would be doubled, or equal to 2082 lbs,; the formula would read : S=^-J^^ S equals safe load. The safe load for the same beam firmly fixed at one end and leaded at the free end, as shown at Fig. 9, the formula would read; ^ bd^C 2^^X 10'' X10 ^"X500 ,,,,, „ ^-476ir — Ixexre" "^''" ^'''•= and the unifoimly distributed load, Fig. 10, would equal twice that, or 520 lbs. for a sale load. And the safe load for the beam Fig. 11, firmly fixed at both ends and load at the center, would be formxUated thus : « iHbd^-C ii/X3"Xlo"Xlo"X500 , ,^, . „ 8^—Q-^-- ^n^^, -==1..62 o lbs.; 226 Mechanical Cakpextry. and for the uniforrrflj' distributed load, Fig. 12, the safe load would be twice 1562 lbs, or 3125 lbs.; aud tlie formula would be ; The safe load for the beam, Fig. 13, firmly fixed at one end aud supported at the other, is fonnulated thus : IH bd^C iKX3'^Xio''Xio^'X500 ^=~-6~ir~= 6Xl6^ ^ =1'203 lbs. equals 1,303 lbs. And the uniformly distributed load, Fig. 14, would be twice 1,302, or equal to 2,601 lbs. for the safe load. Formulated thus: 2,iHbd^'C 2X13^X2^" Xio^^ XiQ^^XSOO „ ^„, ,^ S^ >^l, = 6X16'>^ ^^'^^^ ^^^- The best form for rectangular beams is when the proportion of the breadth Is to the depth, (or nearly so), as 5 is to 7; as for ex- ample 3"X4" scantUng, 4"X6". e"X8", 8"X10", 9"X12", 10"X14" timbers. How to find the reaction at the walls, when a load is concen- trated at any point other than at the center of the beam. Of course, if a beam be loaded at the center, as at W, Fig. 7, one-half the load is transmitted to each of the supports A and JB, and the reaction at each of the supports isequal to one-half the load at W; but if the weight be changed towards either end, then the reaction at the walls will be in the inverse proportion to its distance from each end. Fig. 7, Plate 49. Shows a beam 16' 0'^ between supports. Load, say 100 lbs. at C, 4' 0'^ from the support at A; required the reaction at the supports A and JB. * Rule. Multiply the weight at C (100 lbs.) by the distajice to the near support (4'' 0''-'), and divide tbe product by the length AB (16'' 0"), for the reaction at B; then subtract tiie reaction at 100V4'' O" B from the total load, for the reaction at A. Thus — , ., :=r 16' 0" 25 lbs. that are transmitted to the support B; then 100 lbs. minus 25 lbs. (100—25=75 lbs.) equals 75 lbs. carried to the near sup- port A. Again, suppose the beam, Fig. 15, to be loaded at three dif- ferent points C, D aud E. AB equals 16' 0"; AC equals 4^0"; Ai> equals 7' 0"; AjE7 equals 10' 0", from the near support A. The weights C, D aud E, equal 150, 200 and 900 lbs. respectively, or 1,250 lbs. for the total load, exclusive of the beam. Proceed as before, and find the reaction at the remote support B for each weight, separately; then add the results togetlier and subtract that sum from the total load, for the reaction at the near support A. 4' O'^XfSO Thus A C= . ft 77)7/ — "^37.5 lbs. for the reaction at B. n' 0'''X200 A.P— — ■, ^/ n// — —87.5 lbs. for the reaction at B. lo' 0" 10' 0"'X900 A.ff= — .g/ Q// — =562.5 lbs. for the reaction at B. * A cipher is here added to denote inches. If inches should occur in any of tliese calculations, then reduce the inches to the decimal of a foot, to facilitate calculation. For instance, should the measurement be 4' 9", then tlie item would read, four feet and seventy-five hundredths [4.75']. MECHAXTfAT, CARPENTRY. 23T Then 37.54-87.?>-|-562.5 equals 687.5 lbs. of the weight car- ried to the supviort B: hence the amount carried to the support A equals the weights C, D, E, 150^200+900=1250 lbs., minus the total reaction at the support B. Thus, 1250—687,5=563,5 for the total reaction at support A. In the above calculation the weight of beam has not been con sidered. One-half the weiglit of beam should be added to each total reaction. This problem is useful to the mechanic when framing around the well of stair-ways and stcylijihts where one trimmer raay have to support several concentrated weights. Strains. Rule — Tlie strain or bending moment at any point in a beam is equal to ti.e reaction at the support multiplied by its distance from the pouit selected. The weight at the center of beam, Fig. 11, equals say 200 lbs. Then, of course, the reaction at the supports A and B equals 100 lbs. pins the half weight of beam or trimmer; then let be required to find the strain at the point D. the reaction at A equals 100 lbs. Thus, AB equals 8^0^^; then 8^^X100=800 lbs. for the strain at the point D, for a single load. Find the strain at the point C, 3^ 0'^ from the support A; AC equals 3'' 0^^; then 3.0 :<100 ]bs.=300 lbs. as the strain at the point Cfrora the single load at D. Find the strain at the point E; BE equal 5' 0'^\ then b' d'^ XlOO lbs.=:500 lbs. as the strain at point E from the single load. Let it be observed at Fig, 11 the weight is at the center of beam, and the strain is found at any other point from the reaction at the supports. At Fig. 15 three weights intercept each other in their passage to the supports. The weight at C is 4'' 0^^ from the support A; the weight at D is 7'' Q'^ from A. and the weight JE7 is lO'' O^'' from the support A, and the reaction at the sup- port A equals 562.5 lbs. To ascertain the strains at the points C, D and E, Fig. 15, commence at support A, Strain at C equals reaction at AX A 0=562.5X4' 0^^=2250 lbs. Strain at D equals reaction at AX-AjD-Wat CxCD= 562.5x7' 0^'— 150x3=3487.5 lbs. Strain at E equals reaction at AxAE—W at C CAKP^:^rTRY. stirup irons for heavy work. The joint bolts may be omitted in the tail joist. Posts and Struts. Are subject to compressive strains. The rule mostly followert is Mr. Tredgolds, mortified by Mr. Shalor Smith, C. E., of Baltimore. The formula reads for square or rectangular timber posts thus : C <«04)] equals 70,237 pounds as safe load. Observe that if the post sup- ports a beam of white pine, the upper end of post must have a bolster or pillow of some hard wood, or else the post would sink into the pine girder, as shown at column 5, table 7, where tiie shearing force across the grain per square inch for white pine is 800 pounds, and the area of cross section of post equals 80 square inches. Tlien 80X800=64,000 pounds, allowed to crush the girder ^^ of an inch; hence the post should be capped with an oak bolster'^' projecting a foot on each side. The effect of the post on the oak bolster from the load of 70,237 pounds would be immaterial, as seen from column six, table 7, wheie 1060 pounds is allowed per sipiare inch for a sensible impression. Hence, 1060X80=84,800 pounds as the amount to load the oak bolster to leave a sensible impression. Posts should be carefully set, being plumb and well bedded, if allowed to incline; then in addition to the load they carrj', a strain is set up in the fibers that is not provided for in the dimen- sions; hence accidents may occur. Metal or wrought iron columns set over each other, their flanges should all be trued up, being centered in a lathe. It is not safe to bed the flanges in sheet lead, the lead is liable to work an injury, especially under a heavy and concentrated weight. For wood posts supporting heavy loads, the posts should have metal caps and shoes in the form of sockets, to receive the ends of posts; the caps may be recessed to receive the end of the next post, and also project beyond the post as a support for the girder, or a metal cap may be fit down over the end of post, having two sides continued up, forming a shoe to receive the ends of girder, and the metal sides forming supports for the shoe to receive the next post al)Ove. For warehouses and factories, the above construction will give the best satisfaction. The end of wooden posts should never b(! set directly on a wooden girder, as the wood will shrink, often more on one side than the other, thus cause the post to deflect from * Oast iron preferred. Mechanical. Cakpentry. 23y a perpendicular, and result in cracks in the walls, floors out of level, and fi:ial ruin of tlie building. If the posts or girders be made from green or part dry lumber, they should be allowed to stand until dry before paintine. as the paint will close up the pores, and thus prevent the moisture from reaching the surface; dry rot will then commence at the center of timber, weaken, and perhaps cause accident. By boring a hole in the center of the posts to admit air, will prevent dry rot; the caps and shoes should be cast with a lioHovv space, so as to admit a free circulation of air through the post. If posts be made from green timber, the strength of same v/iil be reduced one-half:' therefore the constant for crushing, in No. -Z Column, Table 7, should not be taken at more than one-half. Pig. 15, Plat9 50. Exhibits /wio to find tlic strain from a load on a brace or strut; the same rule applies to a pair of raf- t&'s; and from the strain, hoio to calculate the dimensions of timber required to s^ipport the load with safety. Let AC and AS pass through the center or neutral axis of the strut and tie, and SC equal the perpendicular height, connect the two at the center of timber, forming the triangle of forces ABC. Now the length of strut from center to center of timbers is 10^ 0'^. set out 6' O" from the perpendicular height, which is 8' 0", a)id loaded with 40 tons at its apex; there are two struts meet- ing at the same point; therefore each strut has to support 20 tons each; required the strain in each strut. Rule. As the perpendicular height {8'' 0") is to the iveight {20 tons), so Is the length of strut ( W 0^^) to the strain in the strut. Example: 8' 0": 20:: 10: 35 equals 25 tons, for the strain in the strut AC. Kequired the strain in the tie AB. Rule. As the perpendicular height {8' 0") is to the load (20 tons), so is the horizontal, tie (6'' u") to the strain in the tie. Example; 8' 0^^:25 :: 6' 0" :15 equals 15 tons, for the strain n the horizontal tie AB. Then 25 tons tiu^ strain in the stnxt AC being reduced to pounds (2240X25=56,000 lbs.), equals 5(5,000 pounds, and 15 tons the strain in the tie AB reduced to pounds (2240X15=33,600 lbs.) equals 33,600 poimds. Required the dimensions of a spruce strut to safely support the compressive sti'ain of 56,000 pounds. Nov/ from column tv>'o. Table 7, we find the crushing force per square inch for spruce pine eqiuils 0862, and if we allow a factor of 6 for safety, or 1000 pounds per square inch, then 5^^0q'>|>=56 square inches in section, or a timl)er 7^^ by 8^'' Vv'ill give ample strength, provided the stiffness is sufficient in pro- portion to the length, for when tlie length of a column or strut exceeds 15 times its least diameter there is danger froin bending. Now to ascertain if the above 7X8 timber is stiff enough for flexiire, proceed by the formula of Shalor Smith's for the strength of columns or posts, as previously shown. Thus ; 6862X56 Safe load= — =21,778 lbs. e[i+(iHgl!^X.o«4)] equals 21,778 pounds, this being too light ; try an 8'^XIO'^ yellow pine timber, by the above formula, wliich will sustain a safe pres- sure of 63,663 pounds ; this allows ample for stiffness of strut AC, W lOTlg. 230 Mechanicai. Carpentry. Next required the dimensiou of the tie AB to safely resist the tensile strain of 33,600. From table 7, column 3, we find the tensile strain for a yellow pine to be 11,400 pounds to a square inch, and if we use a factor of 6 for safety, (1^400=1900), or 1,900 pounds per square inch, then the area of tie (33.600-h1, 900=17^), will equal 17 square inches, or a timber of yellow pine 2X^^X,"i^'' will be ample to neu- tralize the strain on the tie AB, ALGEBRAIC SIGNS AND SYMBOLS. -\- Plus — Sign of addition. — Minus— Sign of subtraction. X Times — Sign of multiplication, -f- Divided by — Sign of division. : Is to — Sign of ratio. : : So is — Sign of equality of ratio 4 : 8 : : 1 fi : 32. = Equals — Sign of equality. □ Signifies square inches. [3 Signifies cubic inches. ■\/ Radical sign of square root, f/ Eadical sign of cube root. 1 Represents length. b Represents breadth. d Represents depth. h Represents height. -^"i"^ Indicates the length is to be added to the l)readth and d divided by the depth. -^^ Indicates the length to be multiplie<.l by the breadth and d divided by the depth. * " Indicates the breadth to be subtracted from tlie length d and divided by the depth. i-Z)3 Indicates the square of the length is to be multiplied by the cube of the breadth. 1 1 Indicates the square root of the lengtli is divided by the f b cul)e root of the breadth. l~7bd^ ~ \ -^ ) +l=Z. Indicates the breadth is multiplied by the square of the depth and divided by the height plus unity, and the square root of this sum, multiplied by 6, and the pro- duct equals Z. [ ] Bracket — Indicates that all the figures within are to be taken together as one. — — Bar — Indicates that the figures over which it is placed are to be taken together. Mkchaxicai, Cakpentby. 331 12"= ] ' ■SQ"=^ 'S ' = 72"= 6 ' = 198"== 1(5.5' = r,920"= 660 '=■ TABLE 1. Long Mkasure. ya. 1 a •• = I fath 5.5 •• = 2.75 " JO " =110 " 63,360"==5,280 '=1,760 " =880 " =320 1 French iueter=39. 37 inches. 1 " ceutimeter=.about ?« of an inch. Ipch. or pole 40 •• =lfnr. =8 " =>lml. TABLE 2. Measure of Surface. (Superficial.) 144 square inches = 1 square foot. 9 square feet SO^square yards 40 square rods 4 square roods 640 square acres 1 square mile 1 square yard, 1 square rod — 1 square rood =-- 1 square acre 272 >4 square feet. 10,890 square feet. 43,560 1 square acre = 43,560 square feet. 1 square mile =27,878,400 square feet. 1 square section, land measure. 1 square==100 sq. ft. architect's aud builder's measure TABLE 3. CfBic Measure. 1738 cubic iriches== 1 cubic foot. 27 cubic feet 128 cubic feet 34.75 cubic feet I cubic foot 1 cubic foot 1 cubic foot ■= 1 cubic yard. == 1 cord. ■=■ 1 perch of etone. =2200 cylindrical inches. =3300 spherical inches. =6600 couical inches. TABLE 4. Averdupois Weight. — U. S. Standard. 16 drachms =1 ounce. 16 ounces =1 pound. 28 pounds =1 cwt.— L12 pounds. 4 quarters or 113 lbs.=l hundred weight. 20 hundred weight =1 ton=2340 pounds. 14 pounds =1 stone. 100 pounds =1 quintal, American Commercial ton=3000 pounds. 2?2 Mechaxicai. Carpe^ttky. TABLE 5. Dry Measi'RE. 2 piuts=l quart=67.2 cubic inches. 4 quarts=l gallou=268.8025 cubic inches. 2 gallons ==1 peck=537.605 cubic inches. ■i peck?=l bushel=2150.43 cubic inches. [Winchester.] TABLE 6. Liquid Mkasuke. 4 gllls=:l piut='.28.875 cubic inches. •2 pints=l quait.^57.75 cubic inches. 4 tiuaits=l gallon=2ol cubic inches. <2 a o * r~< o o ■It "HI ii u i, r^ ," rs =5 C co.-.i 7 ^ '?a P^ I"": (■ as «-> «5 ^ . 4) « cH i ? 2 « *: 55 . 2 I a sd g§ ri^^ifT-^ ht5.r i: 'is ■~ "s ■* oc ^ t>. o ;2 OC ■* O C^ 5<> -H -i a ■7.£~ Ms C 3S o C^ W 35 5.t«a2 .a -^ O 3 s M •o ;c --I ^t -^ — r; 2 -r o o o -» ■* o e: -T u~_:r S cr. o c>o — "— ' M '-c'a's-'if: IS -^ o"oo Mechanical Carpentby. TABLE 8. Crushing Strength of Building Material. Cru.shing Weinht Materiai.. per sq. inch in pounds percu. foot in pounds. Fox Island Granite, 14,875 Gilmore 164 Maine. Quincy 17.750 " \m Massachusetts. Hurricane Island " 14,425 ** 166 Bay of Fundy 11,813 " 162 Joliet, 111. Limestone 12,775 ** 1,58 Illinois. Com. Italian Marble 11.250 168 Berea Sandstone, 8,300 133 Ohio. Brown 9,8;50 " 140 Little Falls, N.Y. Amherst " 6,650 138 Ohio. Cleveland " 6,800 140 " Massillon 8,750 131 " Medina 17,2.50 ( 375 150 New York Brick (Common), \ to 1 S.O-IO Thurston 113 Brick-work, ordin'y, 300 to 500 112 Brick-work, good in cement, 450 to 1000 112 Press Brick, i2,ono Kidder Mortar, common, 120 to 240 Haswell 98 " year after setting, 440 to 580 Rondelet Good Cement, pure... 7,500 81 Concrete, 600 125 TABLE 9. Average weight per cubic foot for materials used in the con- struction and loading of buildings. Woods. POtTNDS. Apple tree 49 Ash 52 Beech 43 Birch (American) 40 Butternut ovhite walnut) 23 Cedar— red 40 Cedar— Canadian 56 Cherry 44 Chestnut 41 Cypress 27 Dogwood 47 Ebony 79 Elm 36 Fir (Norway spruce) 33 Blue gum 52 Hemlock 26 Hickory 58 POUNDS. Larch 33 Lignum vitee 83 Locust 46 Mahogany (Honduras) 40 Mahogany (Saa Domingo) 50 Maple 49 Oak, white 50 Oak, live 59 Poplar, or white wood 24 Pine, Georgia 48 Pine, white 28 Rosewood 45 Red wood 23 Spruce 30 Walnut, black 33 Sycamore 37 Metals. POUNDS. Brass cast 525 Copper cast 5.55 Cast iron 450 Gold (standard) 1108 Lead 712 POUNDS. Silver (standard) 644 Steel 490 Tin cast 459 Wrought iron 485 Zinc cast 450 '.:U MECHAXTf'AT. nAlIPKXxnV. Stones. POUNDS. Alabaster 173 Asphalt 150 Asphaltum. 87 Bricks, pressed IM Bricks conirnon hard i'l'y Bricks, soft KtO Brick work, press. 140 Brick work, ordinary 113 CeiiienI, Portland Si Cement, Rosendale 5(5 Coal, anthracite, solid 93 Coat, anthracite, hroken loose CA Coal, bituminous, solid S4 Ooal, broken loose 49 Earth loose 7t) Earth rMunied 95 Earth, with gravel 126 Glass, coimnon window 157 Gla.ss, plate 172 POUNDS. Gianite 170 Glass, flint 192 Gypsum 143 Lime, quick .53 Lime, stone 1B9 Lime, stone, broken loose 96 Marble 170 Masonry.granite or limestone 165 Masonry of ruble. 140 M asonry of sandstone.dressed 144 Mortar, hardened 103 Porphyry. 180 Quartz 165 Rotten stone 124 Sand, coai'se 112 Sand, moist 130 Slate, American 175 Tile^ „.. „ „. 115 Misc ellan eous. POUND.S Ashes, wood. . 58 Bark, Peruvian 49 Butter 59 Coke 27 Camphor 62 Charcoal 26 Cotton, baled 20 Pat 58 Gunpowder 57 Gneiss, common 168 Hay, baled 17 Ice 58 Ivory 114 Plaster of paris 73 Petroleum 55 Platinum 1342 lied lead. 559 Rosin 69 Salt, coarse 45 Salt, wet 140 iPcnrNDs- J?nOW, jTIsTfalleb 5 to 12 Snow, moistened by rain. .15 to 50 Sulphur 125 Saltpeter 131 Tar 63 Water, rain at 60° F 62'2 Water, sea (U Wax, bees 60 Whale bone 81 Mercury at32°F 849 Mud, wet fluid 120 Mud, dry compact 110 Talc 156 Soap _ _ _ 56 Sugar 100 Honev 90 Milk.'. 64 Fire brick 137 Clay. 125 TABLE 10. Weight per Lineal Foot of Square and Round Iron. ■Fry.] .Thickness Squai-e Bound Thickness Square Round I or Bar in in or Bar in in Diameter. Pounds. Pounds. Diameter. Pounds Pounds. h 0.0132 0.0104 H 3.5790 2.03.50 4 0.0.526 0414 1 3.3680 3.6450 A 0.1184 0.0930 IH 5.3630 4.1330 1 0.3105 0.1653 IK 7.578 5.953 i"s 0.3390 0.2583 IM 10.310 8.101 a 0,4736 0.3730 2 13.470 10..580 .7_ 0.6446 0.5063 Wa 17.050 13.S90 . i 0.8420 0.6613 3K 31.050 16.530 I 1.8160 1.0330 2% 35 470 30.010 3 4 1 .8950 1.4880 3 30.310 23.810 -^ To find the weight of a scjuare bar H'^ thlck.and 1)4'^ wide and "i'' long from the above table, take the decimal .3105. opposite %, and multiply by 6. or take one-sixth of tlie sum opposite 1)4 (7.578) for the weight of a foot lineal; thus, 7.578^6—1.263 pounds. Mechanicat, Carpe??tuy. fSS TABLE 11. To Find the Weight of Castings from their Patterns. Multiply weight of white pine pattern by 16 for cast iron. " " " " " '• 18 " bi'ass. " " " " " " 19 " copper. " " 25 " lead. TABLE 12. The Weight of Various Metai>s Per Superficiat, Foot. [Fry.] Thicivness in trac- tions of an Inch. g M O tr. a o •6 Hi 6 a lbs. lbs. lbs. lbs. lbs. lbs. lbs. lbs. _)L 2.344 2.526 2.. 552 2.734 2.891 3.708 2.344 k 4.687 5.0.'52 5.104 5.469 5.781 7.417 4.687 A 7.031 7.578 7.656 8.203 8.672 11.125 7.031 \ 9.37.'5 10-104 10.208 10.938 11.. 563 14.833 9.375 ^6 11.719 13.630 13.760 13.673 14.453 18.540 11.719 I 14.063 15.156 15.312 16.406 17.344 23.250 14.063 /e- 16.406 17.683 17.865 19.141 20.234 25.9.58 16.406 1 18.750 20.208 -20.417 21.875 23.125 29.667 18.7.50 ^ 28125 30-312 30.6-25 33.813 34.688 44.500 28.1-35 1 37.. 500 40.417 40.833 43.750 46.2.50 59.333 37.500 TABLE 13. Weight per Superficial Foot on Roofs from Various Causes. — [Thurston.] LBS. Weight of >§' slating on 1 inch sheathing 6.75 of {>g slating on 1 inch sheathing 9.00 of 1^ slating on 1 inch sheathing 11.25 if slate-felt be used, add 25 or laid in mortar, add 3.00 of snow on roof '30.00 from wind pressure — [Trautwine] 40.00 for roofs not over seventy-five foot span. W^eight, if covered with corrugated iron, unboarded 28.00 " if plastered below the rafters 38.00 if corrugated iron on sheathing boards 31.00 if plastered below the rafters 41.00 if slated on laths 33.00 if slated on 1^ inch boards 35.00 " if plastered below the rafters 46.00 " if shingled on lath 30.00 " if plastered below the tie beam ... 40.00 " if roof be from 75 to 100 foot span add to each 4.00 5J36 Miscellaneous. Excavators' Memoranda. Exca/atlng is computed by the cubic yard of 27 cubic feet; aud paving by the superficial yard of 9 square feet. To ascertain the numl>er of cubic yards, multiply the length by the breadth, aud that sum by the deptli, ail in feet; then divide by 27. If to find the number of bushels, divide by 2,150.42; or to find the number of gallous- divide by 231. If the plot to be excavated is uneven, aud apt to lead to dis- putes after being excavated; then survey the portion to be dug out, with a Comstock level, by taking levels at regular intervals over the surface, and from the average calculate the amount of earth to be removed, before commencing the work. To take tha levels, locate the instrument at the highest point, say four feet above the earth surface. Now the level of founda- tion below the earth surface at the high point is 10''; then take levels at intervals over the plot to be excavated from the level of instrument to the earth surface, and deduct the perpendicular heights from the 10''. Now suppose levels at regular intervals for 25 points have been taken, and the remainders summed up and divided by 25 will give the average depth of excavation or earth to be removed, TABLE 14. Clean dry Sand aud Oravcl in excavation, will retain a ver- tical face of one foot for a short time without caving in = to 1 foot. Moist Sand and ordinary Surface Mould = 1 to 3 feet. Loamy Soil, well drained, = 5 to 10 " Clay, = 9 to 13 " Compact Ch-avel Soil, for a short time.. =10 to 15 " Hurst. A cubic yard of earth, when dug up, will occupy from 1 ^^ to 1)4 cubic yards. Paving Brick should measure 9''''X4K'"X1^''''. and weigh from 4 to 4K pounds each. One yard of paving requires 32 paving brick, laid flat, and 83 on edge. 14 cubic feet of Chalk weighs 1 Ton, 18 " " Clay " 1 " 31 '' •• Earth '• 1 " 19 •• •' Gravel " 1 '• 22 " •• Sand •• 1 " MlSCELI-AXEOXTO. TABLE 15. SST? Number of Cubic Feet to be Eemoved, axd the Number of Bkick Eequiked fok Wells from 3' 0'' to 12' 0'' nf Diameter. O d) 1 i fSfo a a o« be 6^S .2g -^2 62.2 03 ® a.= So h ^5f a.a o JC *" ^ "Z z » ;2: 3' 0^' ,••2 11.0446 66 7/6// 1 63.6174 347 3^6^' >3 14.1862 77 8^0^' 70.8823 368 4/0/' J^ 17.7205 88 8' 6^' 78.54 390 4' 6'' K 21.6475 99 9'0^' 86,5903 413 5'0'' H 25.9673 110 10' 0'' 103.8691 457 5' 6'' K 30.6796 121 10' 6^' 113.0976 479 6'0'' K 35.7847 132 11' 0" 123.7187 500 6' 6'' 1 50.2656 302 11' 6" 132.7326 523 -/()// 1 56.7451 3-24 12' 0" 143.1391 546 Stonemasons' Memoranda. Cellar walls should be constructed of stoue, carried up above the ^ound. Brick should not be used for outside cellar walls, as the dampness will rise by capillary attraction injurhig the joist and render the rooms unhealthy; this may be obviated by providing a damp course of slate and pitch under the first floor of joist. FoGtiirgs. The New York City Building Laws require them 13" wider than the thickness of wall next above them, and IS" projection aroiuid all foundations for piers, colurans, posts or pil- lars, and 18" in thickness, and all laid in cement mortar. Stoue foundations shall be at least 8" thicker than the wall next above, to a depth of 16 feet below the curb level, and shall be increased 4" in thickness for every additional five feet the wall is deeper. All fouudation walls, either of brick or stone, are re- quired to be built in cement mortar, for ali high buildings. For ordinary dwelling houses of two or three stories, the basement walls are usually of common ruble 18" thick, built in commoa mortar, and having a footing course of heavy stone projecting 6" on each side whenever practicable. For high and massive buildings in the city of Chicago, owing to the yielding nature of the soil, the entire excavation is some- times covered over with rail road iron, bedded in concrete; then the foundation walls are started otf the concrete. Where the soil is compressible, piling is resorted to as a base for foundation walls; wooden piles should not be used iu dry- soils, but in water they have proved to be durable; they are usually 20 to 35 feet loog, owing to the depth required for a solid base. 238 MiSCELLAXEOrS. Load on piles. When a wooden pile refuses to sink over )4 inch under a blow from a ram, weighing 2,500 lbs., falling 30 feet, it is considered home. Brevet Major John Sanders, U. S. Engineer, gives a formula for a safe load, thus : =Safe load. 8 R equals the weight of ram in pounds. h equals the height of fall in inches. d equals the distance the pile sinks from the last blow in inches. Example. Required the safe load of a pile, the ram weigh- ing 1,200 pounds, dropping 6 feet and driving the pile 3 inches. 1,200X72-^8 ^ =3,600 pounds as the safe load; if the pile refused to sink over }i'' under the above conditions, then the safe load would be 43,200 pounds. For the crushing strength of stones and mortars, the student is referred to Tables 7 and 8. Ruble masonry is valued by the perch, containing 24% "ubic feet. To lay a perch of ruble masonry, it requires about 3 bushels of sand and 1 bushel of lime; for good ruble work, the stone on the corners and jambs should be lapped, and through stones every five superfical feet in all straight walls, and no stones built in on edge, but on their natural bed. The proportion for mortar is 3 bushels of sand to 1 of lime. Concrete is made by mixing gravel, broken stone, brick or slag, with 1 part of cement, 6 parts of clean sautl and other solid components, and \}4 parts of water; and thorouglUy mixed and carefully rammed in place; and is measured by the actual contents. MeuHunivj Stonework. Girt around the building, adding twice the thickness of walls at the internal angles, including all openings that are not over 4 feet wide, in walls under two feet thick; also including all openings five feet wide, when the walk-; rae two feet to 2' 6" in thickness. When the openings are over 5 feet wide, add one thickness of wall to the width of each jamb only. Footings measured same as main wall. All arches to be girt and one-half. All safety arches to be considered extra. All abutments that project out from the wall six inches and under to have 12 inches added on each end. All abutments abo^e 6 inches, aud not over 18 inches, to be girt measure only. All piers that are built by themselves, and are 3 feet square and under, to be girt and one-half by one foot thick; aud all pier- that are three feet square and upward, to be ftieasured solid, with 9 feet added for each foot in height for corners. Circular walls to l>e girt and one-half. Recesses and slots to be measured solid. No reductions made for cut stone trimmings and lintels. Quarry measure in all cases to be solid. Cut Stone Setting. Measure vault covers, flagging and ashler by the superficial foot. Coping aud belt courses by the lineal foot; all other cut stone by the cubic foot. — Hand Book of the Pittsbinyh Builders' Exchawje. ': ■ For the number of brick per superficial foot in a wall, add seven brick for everv half brick the wall is thick. MlftCELLAXEOtTS. 239 BncKiayers' Memoranda. TABLE 16. Nv-ifBEn OF Hai,k Bkicks in the Thickne.«5s of Wall per St'pekficial foot of avall surface. 1 3 21 4 j 5 6 28 35 42 7 fi No. of brick per super- ficial feet 7 14 49 56 Example : Required the number of brick in a wall 40 feet long, 25 feet high and 4 half brick in thickness. Under 4 in the above table is 28 brick to the superficial foot. Then 40X-25X-38=28,000; equals 28,000 brick. The size and quality of brick vary in different localities. The standard size in Western Pennsylvania is 8>.2^4J^X2^ thick, and weight 5 to 6 pounds each, and for a good quality they should not absorb over one-twentieth their weight of water when dry. A bricklayer's hod measures 9^'X9'^X16" long, and holds 16 brick; the Y angle should be 60 degrees. Ladder rungs should not be over 9^'' from centers; 1000 brick closely stacked occupy about .56 cubic feet; 1000 old brick cleaned and loosely stacked occupy about 72 cubic feet; 1000 brick require about '6^i bushels of lime and 9 bushels sharp sand. A load of mortar measures 1 cubic yard, or 27 cubic feet; a double load measures twice the above quantity. Strength of Brick Work. Mr. Trautwine, Civil Eugi- neer. states ordinary brick work cracks wilh 20 to 30 tons pres- Bure per square foot, or 311 to 466 pounds per square iucli, and for good brick work in cement, 770 to 1088 pounds per square inch. The New York City building laws require that brick cellar walls shall be four inches thicker than the walls next above, and the in- crease below the curb level is the same as for stone foundations, which see : For buildings not over 55 feet high, external walls. 13''' thick; exceeding 55 feet and not over 80 feet, \&'^ to top of jirst story, thence if not over 40 feet to be 12" thick; more than 80 feet then 4^' to be added for every 15 feet the building is higher tlian 80 feet. If brick be used for foundation walls, they must be laid in cement for all large buildings. ' Mr. F. E. Kidder, March :30ll), 1882. at the U. S, Arsenal in Watertown, Mass., made some experimental tests on brick piers, for the purpose of comparing different kinds of cements with common mortar and sand. The common mortar was taken from a building in course of erection near by. Seven ditferent tests are shown in Table 17. The Portland cement used in the build- ing of these piers is known as Brooks, Shoobridge & Co.'s cement. 240 MlSCKI^LAKEOrS. •>108J0 %e.i]g^ •sajnuua n]lS8^ JO a rat X •Si^'Bp 92 'sqinotn f :aSV •spunod u{ 4q§ie^v^ S 8 8 8 o «s •asjnoo JO q^Suai a ■//2lX,/8 Jaid JO 9zts oj paonamraoo .i9ia oj n? aqi qoiDAV jopun 5 ' ^ ui 'bs .led a.mssyj({ | 3 S •jaid JO tri S S t- « 9 -^ CO Ha- I •3 ;3 6 6 ^ il9 ■c : s-i c : rt r* • ;-.^ ■ c ;_ -» ^ ta 3 O C -c ^ 0) cj ■"> 3S- 8grt ■^ i a: +i c3 *^ K ^ c c « c ® ■ : s: CS r; «* ^ o o 55 ^ TABLE 18. Table Showing the Thickness of Walls for Dwellings AND Stores, as Set Forth in the Building Laws of THE City of Chicago. Basement and two stories Basement and three stories Division walls, basement and two stories Moi'e than three stories Division walls, basement and three stories Division walls, basement and four stories When fli'st story, or basement and first story, are for shop or stores, then: Two stories and liasement Three stories and basement. Four stories and basement Three-story building, division wall Four-storj' building, division wall 12" 15" 12'/ 16" 12" 16" 12" 16" 2(V' 12" 16" 8" 12" 8" 16" 12" 12" 12" 16" 16" 12" 16" 12" 8" li" 12" 12" 12" 12" 12" 1?" Miscp:lla>'eous. g^l Plasterers' Memoranda. TABLE 19. MATEKIAL BEQUIEED for 100 YAKDS OF PLASTEIJING, THltEE COATS COMMON. Lime, 12 buslieis. Sand, 36 " llair, 2 " 8 d Nails (fiae) for joist spaced 2' C' from centers, 6>^ pounds. 3d " " " " V A^' " " 8K " 3d " " " " 1^0^^ " " 10>^ " Number of lath required, 1,600. Latlis are cut 4^ 0'^ by \%^^ by %^^ standard. One bushel of hair equals 8 pounds standard. One yard of plastering, three coat work, requires nearly one- half peck of uuslacked lime, and nearly a peck and a half of sand, and 16 lath. One load of mortar or sand meaures one cubic yard, and will fill 21 hods. A plasterer's hod measures IS*" by 15'' by 33'' long, the V is 60 degrees, and contains a bushel of mortar. Plain plastering is valued by the yard, of 9 square feet; when the material is furnished by the plasterer, deduct the half of all openings over 9 square feet, and add all openings that are 9 square feet and under; deduct all openings measuring 100 square feet and over. When materials are not furnished by the plasterer, no deduc- tions for openings are made. All cupboards, closets, pantries, and all circular work, are measured double full height to the ceiling; Softils of staircases are measured one and a half times. All arrises, quirks and cham- fered corners are measured by the lineal foot, yiucco cornice is measured by the square foot. Oirt around all members, and add one foot lineal for every miter for the length, by (lie girt; any cor- nice or moulding less than 12" girt, shall be measured as 12" girt. Ellipsis are measured three times. For eurichmeuts, charge by the piece. 12 243 Miscellaneous. CD "D C CD (D CO C CD a s_ CO O o 2 n sitcrooooooooooooo cooocsooooo iiTHC>}r;;scs'Mi-':x— t^s-ort — =: ?J>sxrt-tt~or5;ocs r^ 1- T-1 !>! C! ■?! ?I ra ?; — '»■* ^ O in !<; -^ -^ -^ "O 3 ii^iSr^oacs5TC* O X -* O X -J- 9. i?- M oo O ;= O •-= O •-= O -w O 5S O tC O ^ O -JS o ■-= o » o o t-5-->C:-*XO'*XC-j'*-ixcoc-?-*53xo M 5jo--ooooo!=c:oooocc;o©ooooooc.oo o^-H^^-^xo^^3xog,jj;gxoji^g.50^5^ CI § IhOmX-*OX-*OX-*ff!XT)»-*jjooxo-j.';>.i--st-X!3-.ooi^w2^>2;2;gi^5oc» 0-. -"?■* ts 3: -^ rs o o: o f? o a; -.3 :« o a^ -^ CO o c» --D CO o 31 «5 M & O o o rt iM ro m ■* m o -2 f- X 3-. 3! o —■ ^ 72 2 ;2; 'i: '^ 3 ^ QO 5.i-<»itoxT)-*ox->*C'Co->*ox-*ox-fOx-tox'* >> o o o .^ N c! CO -1"+ m- -o I- X X C-. o o -H c! 32 2 ;2; 3 S O otS:iol-IN05T)(tHfflr-iCCCr50>OOt.~Clffi-*THrOr-(XCOO>Q "« if" o 5 v^\rf . fe O O O O — H — i C^ CJ C 1 f O ?D r|l "»St »n Lt »r^ to Ol^ I- i* X X 35 3s n> 5hCIM-*XO-1'0COi*XO-*0CO-^C0O'*XO-*XO-*I00 ooooo— !l-lI-^l^l(M<^Jcofom-*■*T^OL';mooo£-^^l•- M JT^ W M O 3; O CO O 05 O JC !0 OS O M to C-. O rO O' 31 O C;? O 33 ■ o & = o o o o -4 -i -i -H « 7J r^ c> M ro .r< CO -f -t. -* ^ .- .= o o e>! i>ooooooo^rt^.H-i^Mc-?':>c>c-»c}ro coco coco » - >'''oO'-l(MC0-*>nt0t>.X33O-^Ol-l(MC0-*Ot0t~XS5Oi-! S^OOOOOOOOOOOOO-H-H— !—•-<-< — —1-Hr-l—I.H •saq 'mi OUl ^^^,_^N„^...t^l,CC=.0;HC,cg;5;5t=^X20jHO^g5 Miscellaneous. 243 oooooooooooo© t» i;: X -H -1. t^ o M o a 2j in 3c GO O ?0 to C: •?? in »-- O M '-^ 31 (M O {- i- l- 1- =C 00 X ~. 01 0-. Oi O li>a. -*ox■*ox-!^ox-* — ^ to 3; r^ -♦♦ t^ o e-i m X o ^tt 'js S -^ ; tc -J I^ t^ t- X X X X sa J (i> -3 o -^ o w o -i o -^ o -^ o i o! in t- o ^■> 1.- 1>- o N 1-c t- o ■ ^■■^•JSZO t~ t- I- i- X X X X C. ii-*xo-*xO'*xo-^xo taxOMmr-oiM-^i— 3;— •-* m 1.'; tc 05 o as t- 1- 1— 1- 1- x x i>(M-^!OOOOOSo »o in 'JD :c to tc tc 1-- 1* ^ o X '.O •* M o o X to Tti C3 o 2; 2 *~ 2 ~ ?S '-~ ^ ^ '3 ?-J ^ ■-? m m to to to to &»-*OX-l -^ m to X 05^-*tOXO©ClT}rointot-xt33or3 e • C'j M CO ro C-: r^ :o r-. ~ ~ -I- ■i? 000; m to t 00000 ©000 • X a: o ^H ci ro *ti in to c! ■r^C't ?* c^t c-irororocr?;^ro» ©OXt ©o— 'n '-^C-^OOXtO-^C^© !^3-*v^in'Oi-x::io 03i to: or X3;< >©o;to:c©C5to?t© > — - ^^ c-i ro ^ -h in to t- o to t- - 'XTti©X-*©0G'*O ■t ^-^ ^-t T-H ^-t ^^ c'? o? c-^ '^ j -M o> 'M & t- t-l CR ■♦ -^ to —I X CO ; OtOOtO©tO©tDOtOOtO© c!c^ V 1 V V ^ ^ ^ 30 ^ ;§t ^^ Tfl ao 00 TH ■* ao ® OJ 35 Tt< ^ s^ s^ v V ^ ^ ^ 00 CO CO CO JO crj 00 00 ■* ^ ~W\ -3 = J '^ 3 ~ >-• 2 ^ " ^ in C! a C^ «s rs v' a en &l-n©coxTHto-^•.^Ci»:!t,o &! © © -J -^ s^j s>> ^1 CO ?? -ti t« 1 n & -* X © ^ X CiC X X CJ! Oi 3^ o-*x ©oo ri -f ^ © & cn to o: © CO — N— -^ CO X X X to to to to i - 1 - t - ; - v & c3 tj rl Ti a <1) ^ ri > ■a -o a r. eS io a> :; H CJ M ■* in to I- X 3: o T-i © C! CI Cl 51 T! 01 W CI CJ CJ CJ CJ CO ■>»i m to t- CO OS © :H CJ CO -* '1 ■-" cjocjcjdcicocecococococo Q- ;3 & ^ i© bet:: o o '%^ 5) ti > c3 ?- ^ rt 'r. ^^ 2:v ^ a> ri c/> "-; U OJ ^ ^ a, .-> "•^ "•^ «='Z! I— 1 k=H ::: a p- -0 :=- & C8 o 5 ^ rt 3 r-i ^ 3 a ^ a fS CO V ;5t ■3305 ^ o. -^.^ ol 2 <= ^ c« ei c-J ;S a; -' < Oh 5 244 Miscellaneous. TABu^E 21. Number of Shingles and Nails tek 100 Squaue Feet. Laid to the Nurnhcr per 100 weather. sq uiire feet. 4// 1,000 41^// 890 5" 800 5H^' 727 (S" 667 Kuiiibei- of 5 d nails. 5 4>^ 4 3^ 3K TABLE 22. The standard width of shingles is 4^^. 1000 feet of sheathing on rafters spaced 2' d" from centers requires 29 lbs. lOd nails. 1000 feet of sheathing on rafters spaced I'C'^ from centers requires 40 lbs. lOd nails. 1000 feet of sheathing on rafters spaced 1^4^' from centers requires 45 lbs. lOd nails. 1000 feet of sheathing on rafters spaced \' ^" from centers requires GO lbs. lOd nails. 1000 feet llooring joist spaced \' ^" from centers requires 40 lbs. 8d nails. 1000 feet ^" weather boarding studding spaced 1' ^" from centers requires 3.")>2 lbs. 8d nails. 1000 feet lath on studding spaced V \" from centers requires 5 % lbs. od tine. For flooring and weather boarding allow one-fifth for waste and matching. MiSCELLAKEOtrS. ^. TABLE 23. Approximate number of cut axd wire xails to the pound. Leugtli in inches Cut nails Wire finisli- ing Wire fence nails Wire slating Wire roofing Wire com- mon 2(1 1 1,558 411 411 1,200 3d fine. 1% 760 3d IH 480 980 329 251 720 4d iVi 300 760 209 165 432 5d 1% 2G0 500 142 142 142 300 6d 2 160 350 124 103 252 7d 2H 128 275 92 ISO 8d 2X 92 190 82 132 9d 3M 72 173 62 105 lOd 3 60 137 50 87 12d ^H 44 98 38 66 Itid 3K 32 81 30 51 sod 4 24 71 23 35 30d iH 18 27 40d 5 14 21 .50d 5K 12 15 God 6 10 12 6d ^^'If'' nails. 2 80 8d " 2}i 50 lOd " 3 39 IM " 314 24 Speeding Pulleys. Tlie diameter in inches and number of revolutions of the driver helng given, required the number of revolutions of tl\c driven. Rule. Multiply the diameter of the driver in inches by its number of revolutions, and divide by the diameter of the driven in inclies, for its number of revolutions. The dkancter in inches and number of revolutions of the f7ru'C/i helncj given, required the diameter of the driver for a given nuraber of revolutions. Rule. Multiply the diamever of the driven in inches by its number of revolutions, and divide by tlie given number of revo- lutions, for the diameter of the driven required in inches. The diameter in Indies and number of revolutions of tlie driven being given, required the diameter of tlie drivendn inches. Rule. Multiply the diameter of the driven in inches by its number of revolutions, and divide by the number of revolutions of driver for its diameter iu inches. Pulleys. Are of two Icinds, "straight face " for shifting belt, and "crown face " for non-shifting. Belting. To give durability to fast running belts, grease them well with castor oil. '246 MiSCEttAiTEOtTS. When endless belts can be used splice them about ^", glue and clamp, let stand over niglit, then peg with wooden pegs set in glne. This is better by far than leather lacing. The best lacing for belts in last speed is annealed brass wire of No. 55 gauge for narrow belts. Weight of Grindstone. Rule. Square the diameter in inches and multiply by the thickness in inches, and that sum again by the decimal 0.06363. for the number of pounds. To find tJie pressure per sqiiare inch of water in a Umh 10' square and W deep. A cubic foot of water at a temperature of 60 degi-ees F. weighs 62 K pounds, or 8.3 pounds to the gallon; at 213 degrees weight is 59.80, or 8.3 pounds per gallon. The pressure per square inch of a column of water one foot high equals 0.434 pounds; hence a column of water 12 inches square and 10 feet high at a temperature of 60 degrees will exert a pressure on the bottom of tank equal (10X62^=625 lbs.) to 625 poinids, or equal (625-^-144=4.34) to 4.34 pounds per square inch nearly; then 120^^X130^^X4.34=62.496, equals 6,i,496 pounds as the pressure on the bottom of the tank. The pressure on the sides of tank diminishes as the height decreases; at the top the pressure is nothing, while at the bottom the pressure per square inch is equal to the height (10') in feet multiplied by 4.34; hence the average pi-essure on one side of the tank will equal one-half 4..34 (3.17) multiplied by its area in inches; thus, 120X130X2.17=31248; equals 31,248 pounds on one side of the tank. Ohie. The cohesion of solid glue, Mr. Bevaii found to be 4,000 pounds per square inch; his experiments on pieces of wood glned together, req^uired a force per square inch, of 350 to 715 pounds to separate them. Good glue is very hard and tough, and of a brown color; is transparent if held to the light, the fracture is ragged and oblique to the edge, is almost tasteless, and no bad smell, swells when soaked in cold water, requires to be boiled before it will thoroughly dissolve, requires about t«n times its w-eight of water; it also forms a stiff jelly when cold, which is a fair test for good glue. Poor glue is dark and cloudy, breaks easy, having a straight or conchoidal fracture and glass edge, is easily dissolved in cold water; if exposed to dampness, will emit a bad smell in a short time. To prepare tlie glue. Place the amount of glue to be used, in a bag of some strong material, then pound with a mallet; place the glue in tiie pot, cover with clean cold water, let stand over night; then place the pot in the kettle filled with water, let boil, and stir well, remove the scum, and when the glue will run from the brush, smooth and free, having no lumps, is ready for use. Apply hot, the hotter the glue is the better will be the joint. In cold weather, warm the joints to be glued, but not too hot to burn tlie glue; after applying the glue, rub the joint well, to lessen the film of glue in the joint, and thus form a grain and suction in the joint: the work may now be clamped and let stand until dry, which for heavy work will require from two to three days, for light work less time will suffice; if the glue is a little thick, only apply it to one side of the joint. A spoonful of whitening to a pot of glue is said to improve the strength of same; a little alcohol will keep the glue sweet; fresh glue is always the best. When re- melted its strength decreases, and if burnt it is worthless. Clean boiling water should be used to thin down if too thick; the pot should be thoroughly cleansed when making fresh glue, as the old MlSCKI.T. ANEOUS . 247 will taint and ruin the new. When not in use, place the pot in a cool place apart from the kettle; if a cover be placed over the pot, the moisture will keep the glue from crisping on the sides. 2'uiiiers' Cement. To one pound of melted rosin, add a quar- ter pound of pitch; whilo boilins:, add brick dust until considered thick enough, roll into sticks same as grafting wax. When turn- ing rosettes or other light work, by heating the above cement the work may be attached to the face-plate, and removed with a light tap from the hammer; in winter add a little tallow. Glossary of Technical Terms — AND — GENERAL INDEX. Aaron's Rod. — An enrichment consisting of a straight rod from ■which almond leaves are repre- sented sprouting on each side; the term has been applied incor- rectly to a rod around which a serpent is coiled. — [Audsley. Abacas. — The upper number in the capital of a column, on which the architrave in classic and the springers in Gothic architecture immediately rest; in the Tuscan, Doric and Ionic orders it is rec- tangular; in the Corinthian and Composite orders the abacus is curved outward at the angles termed the horns; the curve is ornamented at the center with a rosette, termed the rose of the abacus. Abntment. — A construction of stone, brick or other material which receives the thrust of an arch, vault or strut. Acroferia. — A small pedestal placed on the apex or angles of a pedement for the support of a statue or other ornament. Acute Angle.— Page 21. Albarnom.- Sap -wood. Alcove.— A recess in a room for a bed, sometimes curtained oiT and Jiid from view during the day. AJJar-ISail.— The railing of stone, marble, metal or wood in front of the communion table, and in front of which communicants kneel while receiving the sacra- ment; the height from the top of kneeling step is about 2' 2". Algebraic Symbols.— 230. Asiijle-Bracket— A bracket placed in an interior or exterior anple, and not at right angles with the wall; to fuid the length and curve. 222, 223. Ajuglc-tie.— The timber that gives support to the clragon-hcam in a roof; the diagonal piece cut in the anjjlcsof a square frame to redvxco the same to an octagon. AMglc-Beaitoiie.«i.— Sometimes termed throiiijli .^tonex, are such as extend through the thickness of wall; in good ruble work they should be built in every S'i or 3 feet super- ficial of wall surface, and in very thick walls every course should be heart hounde.d. Boss.— An enrichment at the inter- section of groins, or cross vaulted ceilings. Bossagc.— Stones that are left pro- jecting from a wall to be orna- mented in the future. Boudoir.— French, a trrm used to designate a room especially ap- Eropriated to the mistress of the ouse, as her sitting room. (Gwilt). Boxing;.- Window frames are such when boxed out at the sides to re- ceive sash weights ; for inside shutters in first-class houses the jambs are boxed, either splayed or square to the window frame, to re- ceive the inside shutters ; the shut- ters are then termed ho.v shxittero. Brace or Strut. — An inclined piece of timber forming a triangle to give strength to a building or any part of the same. It is also termed a strut. To find the strain from a given load, 229; from a given strain to find the dimension of timber, 229. Bracket.— How to diminish the size, 27, 105. Breadth.— The greatest width of a body at right angles to its length. Bressunin»er. — The meaning is restri(^ted to a beam or lintel across the opening of a shop front, to support the superincumbent load. Brick-L.ayers.— Memoranda, 239. Brick-Work. — 'When brick are laid in a wall at right angles to the face of wall, it is termed a header. When laid parallel to the face of wall it is termed a stretcher. When every alternate brick is laid as stretcher and header, and to break joints with the next course above and also under, it is termed Flem- Uh bond. Brid{fingr.— Strips IWXS", cut and fixed from the lower edge of one joist to the upper edge of the next, crossing each other, is termed her- riny-hone, tru.'is or ci-o.-^s-hridijing. The joist is usually bridged in this way every five feet of their length ; the strength gained by bi'idging in this way is three times over those that are not bridged. (See Hat- field's Transverse Stairs). Builder. — One who contracts for the erection and completion of a building in all its parts. IV Glossary a.nd Index. Building— An edifice constructed for use or cou v^enience, as a house, a church, a shop, &c. Btittr«!l- umns when projecting three-quar- ters or less from a wall, and (»i,sm- lated columns when they stand clear of the walls. <'oinmon Hafter,— The rafter to which the sheathing is nailed; it is supported by the purlin, which in turn are supported by the prin- cipal rafters, in a trussed roof. Common Rafter.— 315. Concentric. — Circles, 24, and el- lipses, 33. Concrete.— Proportions of cement and stone, 238. Concave.— 34. and Convex, 24. Cone.— Eule to find the area, 29. Conic Sections.— 29, 30. Conge.- A co^e. same as the apo- poge and the Cavetto. Console.— An ornamental bracket placed on the pilaster of a frontis- piece to support the cornice; also to ornament the keystone of an arch; termed, also, Ancone and Trvsses. Conservatory.— A building for the propagation and preserva- tion of rare plants and flowers. A superior greenhouse, it should be in a very dry situation, and the walls should be at least three bricks thick. [Stuart. Coping^. — A course of stone or wood on top of a wall for a pro- tection. When the coping is one thickness it is termed par-allel coping; when thicker in the mid- dle and the top slanting two ways it is termed saddle-baeh coping; when thinner on one edge than the other, it is termed feather- edge coping. Corbel.— A term denoting a pro- jecting stone or piere of timber which supports a weight or strain. A row of corbels connected with small arches or otherwise is termed a corbel table. Cornice. — The upper division of the entablature directly over the frieze; when the frieze is omitted the cornice is termed an archi- trave cornice. Corona.— The outer fascia of a cornice to which the crown moulding is nailed; the corona is allowed to drop below the plan- cier, and thus form a drip. Corridor.— A passage or gallery, from which an entrance may be had to various apartments; some- times running around a quad- rangle. Coved-Ceiling.— The walls of an apartment made to join the cell- ing with a curve instead of a right angle. Crockets,— Ornaments of foliage, or animals used to decorate the angles of spires, pinnacles or ga- bles. Crown nionldingr> — The upper moulding of a cornice, the cyma recta being mostly used for that purpose, as it forms a good water drip. Crossetts.- The projection or ears on arch stones to allow one arch stone to bang on the adjacent stone; the breaks in architraves around openingr, as doors, win- dows, &c. Curtail-Step.— The first step in a stairway when the end is finished, in the form of a scroll; also termed the scroll step; how to draw, 33; manner of constructing. 35. Cupper Gauge.— Construction of, 99. Cupola. — A dome or spherical vault crowning an edifice. Cutting Plane.— A plane cutting a solid into two parts in any direc tlon. In hand railing, the plane on which is developed the face- mould for the wreath-piece, 38. 43, 48, 51, 54, 47. C.vlindcr,— In geometry, a solid flsfure whose base is a circle and whose curved superflces is every- where at an equal distance from the axis or line supposed to pass through the middle. The term throughout this book is applied to the wreath or circular part of the outer string. To find the location of rises in a jjlatform cylinder, so the wreath-piece will raise the proper height without springing the plank, 144; to determine the position of risers in a platform cylinder, so the inclination of rail will have the common pitch on the center lino of rail, 70; to establish the position of rise in a quarter cylinder, so the wreath-piece will raise the proper heighten the level and not spring the plank. 144; to locate the position of rise In a quarter cylinder, so the wreath- v> Glossary and Ixdex. piece may be constmeted in one piece when there are llyers above au"l below the cylinder. 71, T;J; to flud the position of risers in a quarter cylinder having flyers iihove and below, so that two wreath-pieces may form the twist, and that they m;iy be constructed from the least thickness of plank, 73; to find the position of risers in cylinders, so the wreath part of outer siring may have the same inclination as the straight part, 70. 09, 100; to determine the location of risers in a cylinder at turnout, 84, 119, 123. 120; number of staves in a platform cylinder, 149; ve- neering a cylinder over a drum, I'A, 155; thickness of vencrr, rule for, 125; dadoing and bending over a drum, 127; distance apart to make the grooves. 127; glueing up staves, 102, 10:3; that is, veneered, 127; how to place the risers in a semi-circular cylinder starting and landing, so that one set of face- moulds will answer for both wreaths, 73. 75; how to place the risers when less or more than a quarter circle, 80; bow to place the risers in platform cylinders. 69; how to place the risers in a f)latform cylinder on the center ineof rail. 70; joints of cylinder. 121; to find the length of staves, 122, 123, 124, 165. Cyina-reeta.— An Ogee moulding with the concave portion above. When the convex part is above, it is termed a cyma-reversa moulding or talon. DimIo.— The middle division of the pedestal of a column, termed tlie die. The term is also applied to the wainscoting around a room, vestibule or hall. (See pedestal.) A plane also termed cuttinfj- thrush ; used to cut grooves in a board across the grain, termed dadoint) or (jaininu- 150. Damp-Course.— A course of slate or asphalt laid iu cement on out- side walls about 18" above the earth surface to prevent the dampness rising in tlie wall. Dead Sliore,— An upright piece of timber built in a wall to sup- Dort a superincumbent weight until the brick which is to carry the load has set or become hard. Dentils.— Small cubes resembling teeth arranged in a course to or- nament a cornice or plain sur- face. Their width is one-half the length, and the space between eacli dentil is one-half their width, and the pro.ioction is equal to their face width. Details. — Drawings made to a large scale or full size, furnished by the architect to tlie builder, and termed workinu drawimj^. Diag'Oiial. — 32. Diameter. — 24. Dimension.- The length, breadth or thickness of a body. Directing Ordinate. — A line th.at governs the direction of an ordinate. 43, 44. Discliargring Arch. — An arch built in a wall over a lintel, or above another arch to discharge the weight to the piers. Also termed a relieving arch. Dog-lLeifg-ed Ntairs. — Such as are solid between the upper flights; or those which have no well hole; the center of rail and balusters of the dilferent flights being in one plane. — [Nichohion. 86-93. Dome. — A circular, elliptical or polygonal covering of a part or whole of a building. The dome of the Pantheon at Kome is spherical in form, and 1423^ feet in diameter. Door.— The entrance into a house or an apartment. Door-Frame — The wooden jambs and head enclosing a door; also including the sill if made of wood. Dormer — A window in the roof of a house having the frame in a vertical position. When the win- dow lies in the plane of the roof it is termed a skylight. Dormitory. — A large sleeping apartment containing many beds. Dove-Tail.- A tenon in the .shape of a dove's tail which is made to flt into a mortise shaped like a trapezoid, forming a joint mucli used by cabinetmakers in mak- ing drawers, boxes, &c. 104, 105; pattern, 98. Dowel.— A pin used in a joint fo give strength and prevent the joint from shifting, termed ttowe!- iny the joint. 98. Dragon Bean»— A sliort beam forming a seat for the foot of a hip rafter, and is held in ijlacc by the mujU tie. Draught- Board. — And draught- ing instruments, 3. Drawing-Knife.— 37. DrawiiiifS. — In carpentry the de- scription of a building made on paper to a scale, describing the dilferent parts thereof, both in- terior and exterior, by plans, ele- vations, cross-sections and details. They are considered the prop- erty of the architect when the building is completed. DrawJnar-Roona. — The room to which the company retire after a meal. Draw-Bore and Pin.— In fram- ing, the hole through the tenon is bored closer to the shoulder by Vi" than the hole in the header is from the cheek; this allows the oak pin to draw the shoulders up close. When the tenon extends through and the pin driven on the outside of trimmer it is term- ed througlihore and pin. Dressinp-s- In abrick front, stone is often used for lintels, sills, arches, quoins, cornices, termi- nals, string courses and otlier facings, which is termed stone dre-tsings. The front may be of stone and brick or terracotta used for dressings; if wood be used it would be termed wooden dressings. Glossary and Index. Dwarf Walls.— Low walls of less height than the story of a build- ing; sometimes the joist of a ground floor rest upon dwarf walls; and the enclosures of courts arc frequently formed by them, with a railing of iron on their top. — [NichnUon. Die or Dye.— The plain shaft of a pedestal. Eav«.— The lower edge of a roof which overhangs the wall to carry the drip beyond its outer surface. The eave course in shingling should be in three thicknesses tapered. EaHiiiff.— To draw. 161, 163, 1G.5; pattern to make for cutting straight rail, 111, 138; starting from a newel, 114. Cchiiius.— An ovolo, a member in the capital of a column under tlie abacus, which is carved into the egij-and-ilart niouldiny ; also term- ed einj-iutd-tonijue or egg-andan- chor moulding. £lli|>sis.— To draw with a tram- mel, 31; with a string, 3:2; straight edge, 31; described on the cutting plane, 31: rule to lind the area, 31; and also the circumference, 31. £ilipiic. — Winding stairs and manner of constructing the same, 203, 201. Embattled Biiildin;;. — Resem- bling a castle having merlons and emhni^urets on the top of walls, forming a parapet above tiie roof, as in castelated architecture in mediajval times. Einbo^sted Work. — In Gothic architecture a kind of sculptured work; the figure is raised and chiseled into foliage heads and animals, serving as a .sfop at the intersection of ribs in groined ceilings, and also weather mould- ings around doors and windows. Any raised figure relieving a plain surface may be termed em- bossed work. Knibrasure. — Also termed Crc- ncUrs; ilie spare between the mer- lons, for the battery iu ancient miliiiiry architecture. Eende4-a;;4>n. — A polygon of eleven sifh's. Eiilablat lire.— The whole of the parts of an order, above the aba- cus; in the Greek, Roman and Italian architecture it is divided into three divisions, the architrave, frieze and cornice. Entasis.- la the Greek classic or- ders, a slight swell in the shaft of a coluniu or baluster Kqiiilalerjil.— triangle. 23. Es«'a|>e.- The cove at the bottom and top of the shaft of a column, where the shaft e.'^cajics from the l:>asc, and also the capital; also termed the apoplnjuc. Kx.lornal AMjflp,"— 22 Excavalorx.— Memoranda, 236. Eye.— Eye of a scroll is the circle from which the spiral line com- mences; the center of a volute, 35, 36 Facade. — The main front of a building. Face-Moald.— The profile of a cyl- indric section as it is developed on the cutting plane; the pattern is used to line off the crook for the wreath-piece of a hand-rail: to draw for a turnout slightly inclining at the newel, 152, 153; for a turnout level at the newel, 109, 110; for level to a rake, 73, 77, 135; for rake to a level, 73, 80, 133; for platform twist, 111; without springing the plank, 114; when the risers are misplaced, 110, 113; when the pitch off the platform is different to the pitch to the platform, to draw one mould to answer for both wreath-pieces, 116; for a quarter turn starting and landing, 110, 111; for over winders in a cylinder greater than a quarter circle, starting from a newel, 180, 181; to connect the straight rail at)ove and below the quarter pace, 80, 82; for winders in a senu-circle having' two wrealh-pieces, 107, 168, 18:3, 184; for winders in a semi-circle hav- ing three wreath- pieces, 193; for winders in a quarter circle to connect the straight rail above and below the quarter circle and work the ramp in the wreath Eiece, 138; for a quarter circle aving windei's starting from a newel, 185; to draw over winders to contain the easing connecting the straight rail at the upper or lower end, 187, 138; for a circular well hole on plan, 198: elliptical, 203, 209, 64, 69; wreath-piece with a full easing for a quarter circle on plan, 130, -37; for a wreath-piece with an intermediate easing for a quarter circle on plan, 41; for a wreath-piece with no easing over a quarter circle on plan, 46: for a wreath-piece with a full easing over a less than a quarter circle on plan, 49; for a wreath- piece with an intermediate eas ing over a less than a quarter circle on plan, 53; for a wreath- piece with no easing over a less than a quarter circle on plan, 56; for a wreath-piece with an inter- mediate casing over a greater than a quarter circle on plan, 62; for a wreath-piece with an inter- mediate easing over an elliptic curve on plan, (55: for easing pat- terns andwreath-piece for a wall rail on the wide end of winders in box stairs, 95; for 6" to 20" plat- form cylinders, 1.34, 135; for a wreath over winders in a cylin- der struck from different radii, 196; to draw so that one set of moulds and bevels will answer for a wreath starting, and also landing, the plan being a semi- circle, 75; to draw so that one set of moulds and bevels will answer for a wreath-piece starting or landing over a plan le.s.s or greater than a quarter circle, 76, 80; to draw for the wreath part of a plaster moulding underneath the wreathed part of outer string. 1.57; application to the plank, for sawing out the crooks, 51, 57: VIU Glossary and Index. sliding the face-mould on the crook 39, 44, 48, 51,55, 58,112, 113 long- est tangent to be taken first when drawing the face-mould, 50; cor- rect points on radial lines for the trace of mould, 50; over winders in a semi-circle starting from tlie level, 170; over winders in a semi-circle landing on the level, 172; instructions in locating the Inclining tangents in elevation over winders for the falling line of rail, 189; points on the joints not the correct points througli which the curve will pass, 54, 56; the block pattern does not give the correct contour of the squared section of wreatli-piece when the .joint is made from an inclining tangent in the curved part of wreath-piece, 40, 49; when drawing the face-mould by this method take the longest tangent first, 50. Factor for Safety.— 225. FalliusT lilne.— An imaginary line passing through the center of a wreathed hand-rail, as may be con- ceived in the center of a round rail; in the tangent system of hand railing the falling line an- swers to the center elliptic curve on a face-mould, and may be so termed the falling line of the wreath rail, IT3. Fnllin;; Mould— A parallel mould made equal to the depth of rail, and from pasteboard or other flex- ible material, used to trace the twist lines of a wreath-piece by bending the same around the vei'- tical sides of a cjiiudric section af- ter being worked off to the pitch bevel, then tracing the upper and lower sides of rail; the old masters used two, one lor the inside and the other for the outside, 41. FaHCia.— The level casing of the joist around the well hole of a staircase is termed the kvcJ fascia. In a cornice the baud under the planceer is termed the hack fascia and the band under the crown moulding which drops down below the planceer is termed the front faticia. In the classic orders, the back fascia is termed the frieze; and the front fascia is termed the corona. Festoon.— Dr.ipery, garlands knot- ted at intervals, much used in the classic style of architecture in dec- orating the frieze of the Ionic and Corinthian orders. The garlands are heaviest at the centre and ta- per towards the points of 8usi)en- sion. Fillet. — A narrow flat band; it marks the division of the flutes in the Ionic and Corinthian columns, but is missing iu the Doric colunui, where the flutes sire allowed to in- tersect, forming one arris instead of two, as iu the former. Flnlal. — The bunch of foliage which terminates pinnacles, cuno- pies, pediments, &c., in Gothic architecture.— L/'fir/fcr. FInisli.— A term in joinery for the iutcrior finishing of a house such as doors, architra%^es, mouldings, base, &c. FIre-Place.— The place in a room for the Are. The stone under the Are is termed the hearthstone. The peri)endicular sides are termed the jambs. The grate contains the Are, and the ornamental iron work arou nd the fire place is termed the aratc-front. Fish-PIato.— A piece of wood or iron bolted on to the sides of a scarfed tie-beam to prevent the joint from pulling apart. Flanjfe.— A projectinir rib or rim for strength, as a guide, or for at- tachment to another object.— Kniiiiit's American Mechanical Dic- tionary. Flasliing'.- In a roof, pieces of tin or lead, shingled or slated in and turned up along the walls and chimneys to prevent the rain and snow from beating in. In good work cap ffashina is built in tlie walls and turned down over the flashings, 89- Flexible. — That which has the property of bending; contrary to stiftnoss and brashness. Flig^ht of Stairs. — A series of steps from one landing to another. A one-story staircase may be com- posed of one, two or more flights, and arc designated as the lower, upper, or middle flight (as the case maybe) of the staircase. If the flight runs from story to story it is termed one fliylit of stairs. If there is one half pace, it is termed two fligJils of stairs. If the staircase is divided into two quarter paces, it is termed a staircase of three Uiuhts. Floors.— Load likely to come upon, 139. FSii«ili .foints.— In masonry, when the stones chip off at their hori- zontal joints, because the stones are allowed to pinch at their outer edge, being dished out in their beds. In joinery, two surfaces are said to l)e flush when both form a junction even and parallel, having the same plane as the frame work of a door when ready for the pain- ter. Fl H tcs —Vertical coves on the shaft of columns. The Doric column has twenty flutes arouud the shaft ; the Ionic, Corinthian and Com- posite have twenty-lour; the Tus- can has none. Flycr«.— The straight steps iu a stair-case are termed Jlyers, in con- tradistinction to winders. Focus.— 31. Fol4liiii;-noor. — The term do- notes such doors that are hung in pairs and fold at the center from opposite jambs; the rebate on one side is made to fold on the other, and a Itead worked on the arris to hide the joint; inside and outside shutters fold on each other in the same manner. There may fje two, three or more do fyining^s.— The wood work to which the door is hung, and the inside finish is nailed. Jamb Stones.— The stones form- ing the openings in walls for doors and windows; every other stone should cross and finish through the entire thickness of wall, and never built on their edge. Jerken-IIeatl.- A truncated ga- ble. Jib Door.— A secret door that stands flush with the wall, without any projecting mouldings, having the base and surface carried across for concealment. Jogrg-le.- A projection in a joint to prevent its sliding; the joints of a straight arch are sometimes cut with a joggle for that purpose. Joinery.— The art or practice of preparing and fixing woodwork for the inside and outside finish- ings of houses, 212. Joint.— The abutting of two pieces forming either a dose or open joint. A clean close joint is the pride of a joiQcr. Joist.— The timber to which the flooi'ing boards, lath or stripping for lath are nailed. Formula to calculate their transverse streng-th 224; stiffness to prevent the crack- ing of plaster, 129. Kerf.— The cut made by a saw; sys- tem of bending by kerfing, 155. Key.— Pieces of wood cut and fit and glued into grooves when bend- ing a cii'cular stair string over a drum to hold the string to the re- quired curve, are termed keyri. Foldinri lii^ys are used to fasten the newel post to the floor; also to draw the veneer close to the block in a curtail step; also used in scarfing to draw the joint close. Keystone.— The highest and mid- dle stone in an arch. Sometimes the keystone is allowed to project from the face of arch and is mould- ed and carved. Knee.— That part of a hand-rail connecting the newel at a land- ing with the ramp is termed a rcunp and knee, or goose-neck. All straight casings of a hand-rail that are concave on the back are term- ed ramps; and those that are con- vex are termed knees. King-Post.— Or king-bolt, thecen- ter post, or bolt in a roof truss; when used the truss is termed a king-post or king-bolt truss. Isabel. —In Gothic architecture, the drip or hood mouldinij of an arch when it is returned square.— i>7it- art. KoJ>l».v.— An enclosed space from which entrance is made into one or more apartments. I^o^sif* —An open space recessed in a wall from which to obtain a view of the surroundings. I.nflrer, or I-osiver- Boards.— Boards set inclining in an aperture and spaced so as t() admii air and exclude rain; used in bell towers, stables, factories, to carrj- oS the smoke and allow a free circulation of air, I^ozftsige.— A quadrilateral, 23. I^of t.— A room in the I'oof of a build- ing; a gallery or small chamber raised within a larger apartment, or in a church, as a music loft, a singing loft, a rood loft, &c.— Par- 1;cr. Mahoffany. — A wood sometimes used for doors, sash and hand rails of stairs; the best quality comes from the mountains of .Jamaica, that which comes from Honduras being soft and often spongy; it should be air dried and not forced. The genuine San Domingo mahog- any is easily identified by the pores being filled with a white substance resembling flour. The Mexican mahogany is better than the Hon- duras. M«*an.— In mathematics, the aver- age of the sum of all the quanti- ties. Measiire.—That by which extent or dimension is ascertained; table of board, 242. 243. ]tle<-haiiicHl Carpentry. — 212. >!i«!ia»-«'al Arcliiloclnro. — The architecture of Eugland, France, Germany and Italy during the middle ages, including the Nor- man and early Gothic styles, from A. D. 4fX) to 1500. ullcdallion.— An oval, circular or sometimes square tablet, on which are embossed figures, busts and the like. Moinher. — Different parts of a moulding, or the separate parts of a cornice, entablature or column, as the base, fillet, torus, conge, or capital. 2H[«>tev — The French unit of length, 231. Mezzanine Story.— Two stories within the height of one story, mostly divided at the platform of a stairway ; it is also termed the en- tresol story. Mixtilinear.— Angle, 21. 5Io«li31ioii.— A cornice ornament similar to a bracket, having a greater projection than height. Monkey. — Used as a weight in driving piles ; fi-om its weight and force to ascertain the stabil- ity of a di-iven pile, 23S. Mortar.— A mixture of lime and sand used to cement stone and brick in a wall and plaster in a building; is composed of two-thirds clean river sand and one-third well burnt lime. Hydraulic mortar is made from cement, and used in the construction of piers or walls under water, as it soon hardens by the action of the water. Mortise- -- A rectangular hollow made in a timber with auger and chisel, into which is ^:t a tenon made on the end of another piece of timber. The sides of the mor- tise are termed the checks. JloiiUiings. — The ornamental forms applied to the projecting and receding members of an order. The contour of the Grecian mould- ings have a curve of a conic sec- tion; the Roman form of the moulding was arcs of the true cir- cle. In the five orders thei-e are but eight regular mouldings, the (ivolo, the Tcdou. the Cyma, the Cavetto, the Torus, the Astraf/nl, the Scotia and the FiUct. The pro- jection of a moulding is about eijual to its height as a rule. Mnilion - Franse. — A window frame divided into two or more openings by vertical posts. (See window.) Mmitin.- The short pieces in a door that are cut between the rails are termed muntins. Net Measure Is such when no allowance is made for waste of material. Neutral Axis.— That plane in a beam in which the tensile and compressive forces terminate, and therefore nothing. 227. 22!i, 131. Ne-tvel Post.— In close winding stairs the shaft around which the steps and risers wind at their nar- row ends. In open slaii's the post at the starting is termed the vc.wcl pout; if posts be placed at the angles insteacl of cylinders, they are termed annlc ncvils; box or built, 211; solid, 152; standard lengths, 152. Xirhe.— A recess in a wall for the reception of a statue, vase or other ornament; they were a decorative feature and much used in the mid- dle ages. 178. Glossary axb I^dex. xiu Wouason.— 22. BTornial liine. — In geometry a line drawn perpendicular to any line is normal to that line. To find the normal line anywhere in the elliptic curve, 31. No«iing' of Steps.— The projecting moulding on the edge of a step, 92, Oblonff.— A rectangle of unequal dimensions. Obtnse Anjfie-Triaiigle.— 21. Octagon. -To draw, 27. Open Strinsf.— An outer string, finished with return nosings and brackets, the balusters being dove- tailed into the steps. Orders.— There are five ordei-s. the Tuscan, Doric, Ionic, Corinthian and Composite. Each have their ornaments; and the understand- ing and application of which, con- stitutes the foundation of all ex- cellence in the art of architecture. -IGwUt. Out to Ont.— Any dimension taken to its utmost bounds. Outer Strinsr.— The front or face string bounding the well hole of the staircase, if trimmed witli bracket and return nosings, is termed an open tct-.-ndc. The float- rule is of various lengths; wnon very long two men are required to handle it; with the tloat-rule or iitraiiilit cdieceof tim- ber in a truss roof resting on the principal rafters and snpiiortiug the common rafters and their load To find the cuts against the hip rafter, 211. 213. QnadruKglc. . . Any figure with four angles and four sides. 22. ei) narrow channel at its greatest projection to ob- scure the joint. Kadial f^inex. Lines projec*^ing in different directions from a com- mon center; also termed convery- hm lines. Kadi us, 24. Rafter, ...The inclining timber in a roof on which the sheathing is nailed: to find the length of com- mon 213, 21.5, 216; to find the cuts, 213, 217, Rake. An inclination or slope, as the pitch of a stair case or roof, or anything that inclines to the horizon. Ramp... A concave bend rising up. (See knee) to locate the points on pattern for jointing the rail, lOB. Raising'- Plate. . The board se- cured to the ends of the ceiling joist to receive the heel of rafters. Rail. ..In joinery framing, the hor- izontal pieces in doors, sash, shut- ters or wainscoating. Rebate... A rectilineal channel worked on the edge of a piece of stuff. Rectangle. ..A figure whose an- gles are all right angles ; 22. Rectilinear. A figure whose boundaries are right lines. Relieving Arcli. ..An arch built over a stone or wood lintel, or over a flat arch, to carrj' the load to the pier. Retaining Wall... A heavy strong wall constructed with a batter to prevent a bank of earth from sliding down. Rhomboid. A quadrilateral; 22. Rhombus.. .22. Reveal. A rectilineal rebate or recess at the sides of an opening for a door or window between the frame and the face of wall. Ri'b ..A curved timber, forming an arch, to which the lath are nailed; 221. Ri«lge. The comb or highest part of a roof, where a pair of rafters butt the riiige }jlate. Right Angle. An angle contain- ing 90 degrees; 22, 21. Right iJne. A line absolutely straight; 21. Rise. .One of the divisions Info which the height of a story is di- vided for a stair-case: 'JO. Riser. The vertical board between two steps. Sand. Proportion to lime in mor- tar, 241. Sash. The frame holding the glass in :i window Scalfolfl.. An assemblage of scant- ling, planks or boards erected to support workmen in the construc- tion of a building. Scalene Triangle. 23. Scarfing. The splicing of two pieces of timber together to in- crease their length. Scotia. — A cove inoulding, the contour of which is a (juarter cir- (de or a quarter ellipsis. Scribing.— The fitting up of any piece of woi'l; to an uneven sur- face with the dividers. Scroll.— A name given to spiral ornaments, as the termination sometimes given to a hand-rail at its starting, and al.so the vo- lutes of the I(>inf and t'orinthian capitals. When a hand-rail starts with a wreatlK-d .scroll the first step is termed the scroll or curtail GLOSbAiiv a:si) 1^.UEX. XV step. When a hand-rail starts from a newel with a straight easement worlved into a scroll on top of the newel, it is termed a ccrtical scroll. Hov/ to draw, '65; reciprocal scroll to draw, 35: how the eye may be formed to I he best advantage, 309. Seasoned 'I'iinber. — Such as rendered sufficiently dry to be used for the rough framing tim- ber. Sectionof a Bnildin^.— A ver- tical plane or planes through any portion of a structure represent- ing the walls, doors, windows, flues and finish on that particular plane. Segiiieiit— 24. Se«ni-€ircle. 24. Self-Knpportiiigr Stairs. ... 203; mode of construction, 203. Shalt.. ..The body of a column be- tween the base and the capital. Sbakey I^tiiiiber. ..That in which the growths or annual rings are disunited by the action of winds and frost. Sbaiik of a M'reHtta-Piece... The straight part. Sliearingf.... Strength of timber per square inch, Table 7. Suingles... .Number to the square, Shoe. ...The inclined piece at the foot of a conductor to turn the water from the building-; if long and cast of metal it is termed a boot; also an iron socket at the foot of a rafter or strut to re- ceive the thrust. Shiifters. .An appliance to shut out the light from a window. Sin^^Ie or IJonblw Worked Uoor.... Means that the door is either moulded on one or both sides. Skew- Back. ... In curved or straight arches, the abutment that slopes to receive the end of an arch. Soffit.. The ceiling or underside of a stair-case or archway, cor- nice or entablature. S|»aiidril....Tlie triangular panel- ing under a flight of stairs is termed a upaiidrU. S|»lay. ./rhe term is applied to whatever hasone side oblique to the otlier, as the v/alls or linings around doors or windows are sometimes splayed to admit light. Bevels for splayed woi-l<, 218; to lind the veneer for a splayed cir- cular door-head, and also of a ]]ew-back, 220; to find the veneer for a circular door-liead in a cir- cular wall, the jambs being par- allel, 220, 221. Spring Bevel. The bevel ap- plied to the shank of a wreath- piece. Tlie t)ovel applied to t!ie center .ioint is termed tlie iiitcli- hcvft, and l)oth are sometimes lei-med the joinf-hcPKlf. S<|ii»re. To construct, 2f>. Stairs. A series of steps leading frOTii one level to another. The horizontal distance from the face of first rise to the face of the rise landing is termed the run. Stair-Case. ...The structure in a building by which communica- tion is made with the ditterent stories, and includes all the flights in a well for that stair- way. When of one flight, land- ing on the ne.xt floor, is termed a level landiiKj stair-case; when di- vided into two flights, running parallel to each other, is termed a half pace-stair-ca^e ; if winders fill tlie half pace, it is then term- ed a Italf pace ivinding stair-case; if winders fill a quarter pace, it is termed a quarter pace ivindiug stair-case; if winding in two quar- ter paces, it is termed a double quarterpaceiviitdiv()stai7--casc;il all the steps arc winding and circu- lar on plan, it is termed a cireu- lar (leomctrical stair case; if the plan be an ellipsis, it is termed an elliptical stair-cnse; when stairs are supported only at the start- ing and landing, the sides being free, they are termed self-support- iiifj stairs. To construct box, 5)9, 01,92; to line off the string for bo.x, 90; putting up box stairs, 90: preparing open stairs, 100, 101, 107, 108; setting up open stairs, lOfi. ]07,92:to lineofl winding stairs 1.58, 159, 160, 170; building self-support- ing stairs, 203; building elliptical stairs. 203; directions lor lining oir strings, 163, 1(54, 165, 178, 179: rule to proportion the step to rise (Blondel s), 87, 99 ; to construct for a six-foot hall, 99; to con- struct for a, two-story and seven- foot hall, 116; flight outside steps, 87; to construct step-ladder, 89; to construct box stairs, 90. Staves of a t-ylinder. To ob- tain their lengths, 102, 165; man- ner of preparing theiii by ma- chinery, 102; number to a semi- circle, 101. Stay... Same as brace. Step. A step of a stair-case in- cludes from the face of rise to the verge of nosing, 90; preparing tiie steps, mitering, dovetailing and gluing, 128, 104, Kkj; graduating the flyers and winders, 73; pat- tern for circular end steps, 104, 108; gluing up Step and rise, 105; if crooked, 105; nosing the step, 105; mitering step for nosing, 108; grooving and tongueing, lOS; thickness to length of step, 80; oak for outside steps, 89; making step-ladder, 89; making cellar steps, 88; cutting out steps, 127. Stile.. .In joinery the vertical pieces that bound a piece of framing and are made ridged to the horizontal pieces by tenon, mortise and glue. Stirrup iron.. ..To calculate the strength of, 131; their value in framing, 227. Stereotoniy. The science or art of cutting solids into certain ligures or sections, 43. Stonemasons.. Riemoianda, 237, 238; measuring stonework, 238. XVI GLOS.SAKY AM) iMtEX. Story....Comprehends the distance from one floor to that of another. 90. Story-Rod.. ..In stair-building is used to measure off the height of story from top of ^ioist in one story to the top of joist in the next preparatory to stepping off the number of risers required; 90. Strain.. .A force exerted tending to disarrange or destroy tlie structure or cohesion of any of its parts. To calculate the shear- ing strain across the grain, 228; to calculate the tensile strain, 2-30 231; to calculate the trans- verse strain, 224, 225; to calculate the compressive strain, 22;); to calculate the stiffness of a post or strut for stiffness, 229; to And the strain at any point, 227; to find the strain in a strut or brace, 229; to find the size of timber for a given strain, 227. Stretchout....Of a semi-circle, to draw, 28. String.... In joinery that which supports the steps at lioth ends; that at the wall side is termed the ivall string; and that at the open side or well hole is termed the front, or outer stritig; the curved part is termed the cylin- der; tlie part that is level and lining the rough joist is termed the level string. Strut. ...See brace- Stiilf. . A general term for wood used by joiners. Stylo. ..The different classes of arcliitPcture. SHb*Plinth. .A second and lower plinth placed under the principal one in columns and pedestals.— INeirUtnds. Ta!>l«s.. Table 1, long measure, 2:U; 2, superficial measure, 2:jl; 3, cuVjic measure, 231; 4, avordu- poise measure, 231 ; 7, resistance to compression, shearing, tensile and transverse strain, 2:j2; 8, crushing sti-cngth of building material, Vii; 9, average weight per cubic foot for materials used in the construction and loading of buildings, 233, 231; 10, weight per lineal foot of .scjuave and round iron, 234; II. to find the weiglit of castings from tlieir pat- terns, 23.'»; 12, the weiglit of vari- ous metals per superficial foot, 235; 13, weight per superficial foot on roofs from various caiis<;s, 235; 14, vertical reposeof different .soils for a short time, 23!); 15, num- ber of cubic fiiet to bo removed, and nuiiiber of briclc re(|uired for wells from 3 to 12 feet in di- ameter, 237; 17, shows seven tests of the strength of brick piers built in cement and com'mon mortar, WO; 58, shows the thick- ness of walls for dwellings and stores as .set forth in tlie building laws of the city of Chicago, 240; 19, materials required for 100 yards of plastering-, three coats, 241; 20, number of shingles and rails per 100 s(|uare feet, 244; 21, showing the number of nails re- quired for fixing 1,000 feet super- ficial and lineal, 244; 22, approxi- mate number of cut and wire nails to the pound, 244; 23, board measure, 242, 243. Tail- Joist.. ..Short joist tailed into the header, as around fire-places, stairways, &c. Tall-Trimmer.. ..A trimmer next to the wall into which other joist are framed to pass a flue. Tangent.. ..31, 24. TanK.. ..To calculate the pressure per square inch of water on the sides and bottom, 245. Tenon.. ..A projecting rectangular prism formed on the end of a tim- ber to be inserted into a cavity of the same form. Tensile. -A strain tending to pull apart, the strain being given to find size of timber required for safety, 230. Thrust.. ..The force exerted by an arch or pair of rafters. Tie-Ream.— The beam that con- nects the foot of two principal rafters, and neutralizes the thrust of same from injuring the wall and preserving their equil- librium. Tie-Roil. .. An iron rod perform- ing the ofBce of a tie-beam. Trammel... A device for drawing the ellipsis; to construct, 33: sub- stitute for, tjO. Transom... A window above a door separated from the door by a transom rail, or continuation of the imijo.st moulding. Transver.se.. -Across, at right an- gles to the length; formula for transverse strains, 224. Tra»e«inm.. .23. Trapezoid... 23. Tread... A t read of a stair includes the horizontal distance from face of one rise to the face of next; or the horizontal cut of the outer string, 90; rule to proportion the tread to suit the rise, 87. Triangle... 22. Trimmer... A double joist into wliich a header is framed to car- ry the tail joist, as around the lire-place, stairways and sky- lights Trn.s!* Roof. . .Is composed of tie- lieam, inincipal rafters, king or queen bolts and struts, so ar- ranged as to carry tin; purlins, common rafters, sheathing and other weights to the v/alls which support tliem. Turners' Cement. 24G. Twist lAne of a Ilitnd Rail. The ln;lix, or screw line forin.iiig the lAvist of a wi-ealh-piecc, of a hand rail, found by lioixling a pliable strip around the wreath- piece to agree with the minor axis and the axisof bloi-lc jjat tern a.s shown at p. 113. When there are two or more wreath pieces connecting each other, also if any vamps or casings, bolt them all together, after being worked off to the plumb; then apply th( flexible strip from end to end foi the twist lines, both on the con Glossary and Index. xvii cave and convex sides, the arris of block pattern at the joints, and minor axis will he a guide for the flexible strip. Valley- rafter... The rafter at the internal angle of a roof, as opposed to the hip, which forms an external angle; to obtain the length 215; find the length and cuts 214; find the cuts of Valley and jack rafters 21.!). Vault.. .A passage or room cover- ed with an arched ceiling of brick or stone. Veneer... Method of veneering a cylinder; preparing the Veneer 154, 1.55, 156; to prepare for winders 72; its thickness 125. Vertix...The point from which two or more lines radiate. Vertical-plane.. .One that is per- pendicular to the horizon. VoInte...A spiral scroll in the Ionic, and Composite Capitals. Wall... A structure enclosing a building and supporting its parts. 'Wall-string... How to line off for flyers, iiO; for winders, 94. Walnut. ... For hand-rails and finish, 210. Well.. .The vertical space in which the stair-case is built; trimming the well, 99, 100, 116, 117, 118. Well-bole... In a stair-case, the vertical clear space between the line of nosings, or the greatest projection of any mouldings around the cylinder. Winders... The steps in a stair- case that are wide at one end, and narrow at the other, used to turn a corner instead of a level pace; patterns as a handy means to mark out, 94, 185. Window... An opening in a wall to admit light, and ventilate the room. Wire.. .For sewing a belt, 245. W^reathed-Rail. . . The twisted part of a hand-rail, as required around the cylinder in a con- tinued hand-rail stair-case; a piece of work having two curves, or double curvature is termed wreathed. To find the twist line, 11-3; gluing up or building up crooks for the wreath part. 210; best worked from thick stulT, 210; Wreath-piece.. . Part of a wreath. Wreatta-strine'*..The cylinder in a stair-case having a double curvature as when cut and moulded to suit the inclination of stairs. Yard. . . A measure of three feet. Zac. . A tool for cutting slate. KN Plate 1. I' l.^ 12 A 18 18.0 h G E E F 15 3 N 16J?" 6 L__JO Qeometry Plate 1 6 V 9 13 A >B / /; 18 / /"S im" / 12 r Plate 2. BZQ / \ \ \ / \ \ / \ \ I \. V' A-- " B Geometry Plate 3. fi^ A » 10 Q ** ' B Problems ( 4 \ 1 ^- - ' ' .1 [ t=0=3 Plate 3. 9 ! 5 c >" « II 10 « ^v e 1 2 3 « ' e W«5432»8 Plate 4. !: \ 'o;: \ ^'L n ^'\- 9 \ r^ ■\.\-- ,-^ ~. --'" > /^\^, -i \-.- V 1 ^ Plate 5. / 2 Plate 5. V^ Ellipsis ' ,,/-- f ■ - K 1 \.^ ' ^-^^ Plate 6. B1234 567 Plate 6. Plate 7, Plate 7. 15' Radial Line /^ Plate 8. Plate 9 Plate 9 I I ^'iTioiii" "'BadiarLine'""o Plate 10. v. Plate 10. Plate 11. ) Tangent ^M \ :4 i -^ ^ 1 OB '^ pi^"**is;^' Plate 11. r^ Plate 12. Plate 12. V la ■3 // y ^ / / / / / •a H f/ I / / 2 ] EJevalioQ A f/ ~~-jh C e Plate 13. Bnt e.e ce Mou B Tangent ^--^^^^t m 1 ''49^ f-j^~^4: Face Mould Plate 13. ' TnQgeat _^ - -c/ r i--^ Sj i- ««»-C JII.VIU1U . 1^4 Face Mould I Plate 14. Plate 14. E H D !•> / \ / < / _N 11 F '/ >^ a bo A ^/. /-Elevation B X M ) L ; ^,-^, , 8 Plate 15. Scale 3^^ I' rr ilO Plan :\r-- 16' J r Plan ; Plate 15. Plate 16. Plate 17. Scale 3^=r Plan « 6 > 14 Plate 18. Plate 18. Plate 19. Scale 1-4=^1 Plate 19. V / 6 ■ Plan r -4 64-9 1 1 r .4 10 0' ■ 9 16 8 16 S 6 : 4 ^' : 3 L IB J 1 la p |4 Is la r? LL3 -■ 10 11 12 13 14 16 Plate 20. 3 --if - - -?"• ^M. \ DC 9 - ^ff(i}mj;,"fiuu""""'^ ^ Plate 20. X Scale 3^=t' ^^^-^ --"■"''^ ~:rr==— — T^- r," '.i- ^^^- Plate 21. 1^ Plate 21. Plate 22. ^ Plate 22. Plate 23 Plate 23. ■ftt (»A 10 : :-::::^^ ^ \ IB Plate 24. Plate 25. Scale 1-4—1 7 -a* I h >-4 8.4" D •1 ^e1^ 15 14' 18' 12 « 9" 8 i 7 1 ' 8. a P ^1 i'B B ;; a A.' :;-4U'; :aa 13 13 . C*. . . . .1.1- B~nsjB^ Hi' 4'. I " *)K- -^.z Plan Plate 25. Scale i:«=l I Jof --'A Plan ; € r Plate 26. Plate 27. Plate 27. Plate 28. jutv^^» ^^ Scale a^=i'' Plate 29. Scale 3-4=t' 70h" , 6- 'A ^2 Floods Llne^ 4:, g glevatjoD 15 Plate 30. Scale 3-4=- 1 ' Plate 30. ^ ::^ s. - S'H^ . "fe^ r\^ ^ ^-^^ 4 » x\ F «.\ |. 17 J Eleratlon / . //. % '^ 7 ■* ^v r>. ^' 'r3K* 5r -^ 8H 4-.-.l0"---4 6 1 Pit Plate 40, Plate 41. s.y. Plate 42. Bcale 1-4=1 Plate 42. 1.'^ Bc«la 1-4=1 r*— r Plate 43. Face Mould Scale 3^'4=1' Plate 43. Plate 44. m Scale a^=l' 10^ 9\ 10-3 O N ^i^ ^K "^ 15« No-^ / -"^v. No-1 .0-- Pian -^•^ 1 W c/ 10-5 .■-(^ I t m u €^ \ 1 ,_^ i^ M ^- ■ o V Plate 44. r-;Jv;:;/-\----- jpian l^^X. ,.;j\- l.s ^ ^ 13" 14 ' !&■' Plate 46. V Plate 45. Plate 46. QQsas r^ Plate 47. 9'- 6' Plate 47. C Plate 48. Plate 48. TAB in feet. Length of for 1" mon raft mals of For the In coll foo And m Cutoff in the euqals And th Equals Multipl Equals Multipl Equals TABLE OF LENGTHS AND CUTS FOB COMMON AND HIP RAFTERS FOR A HIP SQUARE ON PLAN; ALSO, CUT OF JACK RAFTERS AGAINST HIP. THE LENGTH OF RAFTERS, FOB A FOOT RUN, IS GIVEN IN FEET . DECIMALS OF A FOOT, P.TCH.. 1 Vl 1 A i J 1 * 1 5 i 1 1 1 J i i 1 1 1 1 GOTHICt a 1 a g 1 1 w 1 g a 1 sa 1 f § $ » 1 a § 1 S a 1 » § 1 s c 1 - c 1 - « 1 » » 1 » § 1 f a 1 w a 1 » i 1 1 a 1 » £ 1 » Out of common rafter, 12» X 2" 12" X 2.4" 12"X2.066" 12" X 3" 12" X 3.428" 12" X 4" 12" X 4.8" 12" X «" 12" X 8" 12" X 12" 12" X 10" 12" X 18" 12" X 20.7846'' Out of hip rafter. 17" X 2" 17" X 2.4" I7"X2.86a" 17" X 3" 17" X 3.428" 17" X 4" 17" X 4.8" 17" X 6" 17" X 8" V' X 12" 17"X16" 17"X18" 17" X 20.7846" Out of jack rafter t a),'ainst the hip. f 12" X 12A" ]2"X12/8" 12"X12-A" 12" X 12li" 12"X12H" 12"X12H" 12" X i2ir 12" X 13H" 12"X14JI" 12" X loH" 12"X20" 12"X2li!" 12"X24" Length of common] ratter, for 12'' run, } in feet. ) 1.138' 1.0198' 1.0344' 1.0308' 1.010' 1.0541' 1.0770' 1.1181' 1.2019' 1.4142' 1.6667' 1.8028' 1.9999' Length of hip rafter 1 tor 12" run of com- ■ nion ratter, in feet. 1.42M' 1.4307' 1.4340' 1.4380' 1.4452' 1.4654' 1.4721' 1.5023' 15«57' 1.7341' 1.9454' 2.0832' 2.8376' Length of common rafter for 1" run. in • decimals of a foot. o.osiy O.OS49!f 0.08.54' 0.0859' O.OSOff 0.08784' 0.0898' 0.09317' O.lOOlff 0.117a.¥ 0.1389' 0.1502' 0.1066' Length of hip ratterl fof 1" run of com- V mon rafter, in deci- 1 mals of a foot. J O.llSlf 0.U92' O.ll'.B' 0.1197' 0.1201' 0.1313' 0.1227' 0.12.-12' 0.1305' 0.i415' 0.II121' o.nw 0.1805' EXAMFIiE. — Required the cuts and Length of a rafter K pitch h(ViAng a run of 14' 7"; also the cuts and length of htpfor the same. For 1 I line for common rafter, take for o foot in length the constant And multiply the siime by the length of rafter in feet, [14'] And the decimal .2906, which multiply by 12' E(iuals for the length of rafter S r 2lade gives the cut, also cut of hip rafter against the ridge plate, provided it is applied on the plane of backing and from the side in which the jacks run parallel to the ridge plate — [if applied from the opposite side then the tongue gives the cutj. The blade also gives the miter cut for the gable mouldings and planceer in a raking cornice; the tongue gives the miter cut for the purlin against the hip. sheathing, gutter stop, and level planceer to suit the pitch of rafters ; 4 and 5 show the foot and plum cuts : their several lengths are described at Figs. 1 2, and 5. For the side cut of p^irlin. — Take KP on the blade and MN on the tongue. The tongue gives the cut, also the cut for edge of sheathing, gutter stop, and level planceer to suit the pitch of rafters. Bevel for the backing of hip rafter. — Take DR on the blade and DS on the tongue. The tongue gives the bevel for the one side, and together with the bevel for the opposite side will give the angle for the saddle on hip if required. The points a^b, c, d, e, Fig. 5, show the lengths of jack rafters for the side GDJ; the half thickness of hip must be de- ducted. Remember in deducting the half thickness of hip, or ridge plate when required, it must be measured square to the plane of the cut and not square to the plumb cut. The lengths Ga, Gb, &c , as here drawn, are to the center of hip, and the hips to the center of ridge plate. Appendix No. 1. ?h 1 ^ . 1 ^ \ ^ R is 1 i 1 1 1 1 1 '\ >1 y A r<^^ -1 _ Run 1 H \ \ ^T" Rise 1 8 7 / / \ \ \ - 1/ c e / ^ \\ ^ / ^ 5\ \ j / V /a / b \ s. / / c D y / W / B P Fig. 1. — {Scale, one-eufhth inch equals one foot). Shoivs one side of a roof haiying a % 'pitch and a square hip and valley. How to find the length of valley and cripples hetioeen the valley and hip. A, S, C, D, E, F, shows the plan of roof ; EB shows hip, DH shows seat of valley ; ab. cd, e/ shows the seat of cripples ; CA is a gable and AJ3"the ridge. Draw JBJ'for the rise ; make .5.2" equal EF; join JJTfor the length of common rafter. From H, and perpendicular to AB, draw a line to intersect the common rafter at M, then MH is the rise of valley rafter. Make HN equal HD ; join MN for the length and cuts of valley rafter. For the length of cripple ab. — From the points a and b erect perpendiculars to inteisect the common rafter JK at 2 and 3 ; the valley and hip being both parallel, then the distance 2 3 is the length for all the cripples between the hip and valley rafters. For the length of cripples between the ridge and valley ra/ter.— Make AP equal the rise HM; join PC for the length and cuts of the gable rafter. From the points a, c, e, erect perpendiculars to CA, cutting PC at 4, .5 and 6 ; then P4, Po and Pa show the length of cripples e7, C8 and a9 on plan. Fig. 2. {Scale three-eighths inch equals one foot). Sliows the template for marking the heel and side cuts of jack rafters. A shows the side and B the top. The cuts for the cripples at the valley rafters are the same as for a hip rafter, only the plumb cut is the reverse way, the long point being on the under side, ^ Plate 49. Plate 49. I Plate 50. Plate 51, flPPEflDlX Ho. 2. PLATE 52. Plate 52. [Scale >s=i J Square Hip and Octag"on Roof Framing'. Fig'. 8. Shows a practical and simple metJiod lioio to devclope the length and cuts of the common hip and jack rafters on the pitching plane, for a square hit) and for any pitch. Let ABCD be the plan for two square hips, and AF, FB he the seal of hips; draw the seat of jack rafters to intersect the seat line of hip, as FF, et, di'f and JO give the down and foot cuts for the hip Fr/; for the cut of jacks against the hip, jirolong dr to equal its length Ar, as (If/, join Aff for the bevel at ry. For the cut of hip against the ridge plate at _//; take any point on the seat FT* of hip, say at the intersection of the jack from I at II, draw tirj at right angles to FH; from ii and parallel to the base line Ali, draw a line to intersect the hip as i(S,' make i((i equal 'f'i, join Lfi for the bevel as shown. This completes the drawing: for a pair of rafters, two hips and the length of () jack rafters and the cuts for the same for a |^ piteh roof; the bevel as shown for the cut of hip against the ridge plate is applied before backing. For a }^ pitch the principle is the same; let F{r be j^ the span AB, join ^Ifr and (xBy the intersection with the seat of jacks gives the length of jack on the common rafter Afr as at z and s/ Atf is the length of the jack to stand over its seat et, and so of all the rest respectively. For the length and cuts of hip, join (rJy for the cut of hip against the ridge plate, from /.■ draw h-2, make W.T equal fJ2. join .)L for the bevel at ,5.' for the cut of jack against the hip prolong the seat of jack ef to equal its increased length As as eh, join Ah for the bevel as shown. I^or a J/t, pitch this system is the same. Let FII equal % the span AB, join AJl and Blifor a pair of common rafters and their cuts; join HJ iox the length and cuts of hip rafter; for the length of the jacks prolong their seats to insert the common rafter at x and y: then Al/ is the length of jack to stand over its seat et, and so of all the rest; for the cut of jack ?gainst the hip, prolong the seat Bit to equal its increased length Ap. as bf, join Af for the bevel required at/'. For the miter of hip against the ridge plate, prolong /« to ir, draw *r -i^ parallel to ^IB,- make /^ 7 equal J-t:, join 7 Li for the bevel as shown. This is the most simple method to line off a sipmre hip. less linps ard room is required by this system: if the hip be backed less lines may be used, the bevels sb.own at/, r/and h, for the jacks will also give the miter cut for their respective hips ngaiubt the ridge-plate if applied on the plane of backing. Pl.^te 49 shows the most practical method to obtain the difl'erent bevels, also the backing. The young man should first study how to draw the lines on the board for a hip and valley ronf. then from the drawing thus made to a %" scale, proceed to frame the mof. For a square liip it is very easv to a])i)lv the sfpiare for all the princijial cuts, but for a hip or valley that" accommodates two pitches it is more difticult. a drawing then is necessary, and if the student (irst learns to draw otf the roof, he afterwards will be better qnali- Med to tind his way out when framing one more complicated, even v.\X\i the steel square. Fig' 9, [Scale y% =i ] Shams the end of a building finished ivith acta zona I corners. The octagon is carried up to the comb of main roof instead of a gable; one side of the octagon to admit windows is shown increased in widlh, the cornice is to project the same at the eaves and to be rak- ing; unless the workman has had some experience in the framing of such a roof he may be taken unawares. Let AB, BC, CD, DE and EF, Fig. 9, indicate the outside measurements of walls and dll, HJ^ JK, KL and iJXlhe projection for the ends of ratters to receive the cornice. Now from the points II, «/, K and L draw lines to the centre O for the seat lines of hips, also space off and draw the seat of jack rafters as may be desired. Let it be observed that the seat of hips does not intersect the angle of walls at the poiitts BCL and E, but cuts to one side, tliis cannot be avoided in a roef with two jiitches and a raking fornice, for the look-outs (or a cor- nice having the same projection all around and of different pitches, the points of hips and other ratters must be on the same level, therefore we must draw their seats from the utmost projection. If the cornice was a level one then the seat of hips may be drawn from the points BCD aud E. Now ^J^ equals the span of building and OG, OJM and Of, show the seat of common railers and they are equal from the centre O; but the seals OA", Otl are longer, therefore we have two pitches; Ihe planes 03IL, OQH and OJK have the same pitch; but the planes OHtJ and OKL have a flatter pitch; their length, pitch and location of the wall plates must be determined by drawing them in elevation. Fig. 10. S/i07i'S in elevation the pitch and length of the different rafters, also the location of the tvall plates, all on the pitching plane. Draw the base line XX ; from h erect the perpendicular ho equal to the rise of roof; make hu equal OA. Fig. 9; draw OCI prolonged to intersect the horizontal projection of rafter at C; from C and parallel to J^TJE^draw ctll indefinate; now the points of all the rafters must terminate at this line cm as shown. At ff is shown the wall plate; next set off the width of look-out required for the cor- nice and draw the back of rafter intersecting ho at V, then VO is the heighth above the plate on line with the wall at ci for all the rafters. Make hd equal Ot on plan Fig. 9; draw 0(l prolonged to intersect the line mc at draw SK, thus establishing the position of the wall plates for the flatter pitch relative to the wall plates for the steeper pitch; observe the plate at s drops down below that at ^''=3' 10^'] in elevation 3', io}i-, the rafter at/' on plan buts the end of ridge plate and scales [6' 3" plus i' 7}4" equals?', loj^^ ' ] in elevation 7, 10^' for the total length to the long points; the length of jacks for the ])lanes OrJK and 1,31 arc the same as for the plane OCH. The length of jack yh on plan scales [2, 7^ ''-|-i, 6^=4, 2] in elevation 4', 2 for its total length; the jacks at 11 and /« on plan miter between two hips, and scale [6', 7 -j-i , 6^=8, lyi''] to the long point 8, i^ in tleva- tion; the common rafter GO miters to the hip OH and scales [6 , 7>^ -|-i'> 7/^ =8 > 2' ] in elevation 8 2 . The two hips OL and O/i" miter to the ridge plate and scale [7, 3 '4" I » 9/^=9 > O /^] for their length 9,0^'; the hips OrT and OK on plan, miter against the ridge and adjacent hips as shown and scales [7', o"-(- 1', 9^ =8', 9^ ] for their length in elevation 8', gl^z '. Observe the lower end of hip is cut to the bevel at i and the length is measured square from the point at q,- make all measurements on the back of rafters by first lining oft for the projection of cornice to the line of wall, then the exact length from wall to the point of rafters; use a template shown at Fig. 7 to line off" all look-outs and cuts for the different rafters. Observe at t on plan the seat of hip intersects the line of wall to one side of the angle CV now in mitering the planceer the centre of hip will be the centre of miier, this will look bad, as the miter should run from angle e/ to angle C/ to accomplish this two methods may be adopted, either to cut the look-outs for the planes OCrH. OJIi. and OJlLi by dropping as shown by the dotted lines at C and q Fig lO and thus make the pitch for the planceer agree with the lesser pitch OSj or to make the pitch of planceer for the lesser pitch OS agree with the greater pitch 0(f by reducing the look outs at ,S' as shown by the dotted line, this reduction will only be for the three jacks on each side for the odd pitch, the hips and all other ralters will remain straight; the better way then will be to make the underside of look- outs for the lesser pilch agree with those of the greater pitch in this case; this will allow the raking planceer to miter from angle to angle. Bevels. The bevels at « and /'give the foot and down cut for all the common and jack rafters for the steep pitch; the bevels shown at Jt and *■, give the foot and plumb cuts for the lesser pitch; the bevels at fl and o give the foot and down cuts for all the hips. J^or the cut of hip at the lower end to agree loith the square ends of the Jacks. From b and perpendicular to the back of rafter oc draw hb, draw hi paralell to JSlAT, join bi for the bevel shown at // this bevel gives the cut on side of hips for the planes OGII, 03IL and OJK; for the opposite side of hips on the planes OH'f und OKL Fig. 9; from JiL and perpendicular to the back of rafter SO draw ivH; draw wy parallel to JCX, join yk for the bevel required. J^or the mitre cut of hips HO and LO against the ridge- plate. Apply the bevel shown at (i to the sole cut of hip as shown for Plate 49, this bevel will also give the cut on top of the hip at the lower end for one side, for the opposite side on the lesser pilch the bevel at */" applied in the same way will give the cut. For the cut of jacks against the hips. On plan Fig. 9 prolong lib to equal its increaseed length in elevation as Cic, join c to the side ot rafter at y; the bevel at c gives the miter cut lor all the jacks on the planes OGM, Ot/K and OJML, the bevel in the angle at // gives the cut for the sheathing and edge cutof corona, also the cut of the two hips HO and LO against the ridge plale, also for the back of hips at the lower end for the above planes if applied after backing. The same bevels may be found from the sole cut of com- mon rafter by applying the bevel at 3, also the miter cut for the hips Oltand OJa-i their upper ends by applying the bevels shown at 4 and i) Fig. 9 For the cut of jacks against the hips on the planes OH'T-And OKL; prolong the seat of jack (fit to equal lit in elevation, as (fl join ii'i the bevel at i gives the cut, the bevel at V gives the cut of sheathing also the edge cut of corona and the cut on the back of hips at the lower end if applied after backing. To find the bevel to miter the plnvceer. If the planceer was to miter so as to agree with the hips as drawn, then tlie bevels slifiwn at H and K would give the initer cut for their respective pitches, and the bevels shown at z and j would give the cut on the edge; also for the edge cut of sheathing. But the look-outs being all cut to agree with the regular pitch oa/ then from E Fig. 9, and perpendicular to KL draw JEx prolonged to equal cct Fig. lo, as at 7% join »'X, the bevel at L gives the miter cut fur llie p'anceer ; for ike cut on the edge of planceer. At Jig. Ji draw the right angle ,Hp)il, let sil be the inclination of the look-out on its under side, and f^i)t equal the pilch of the miter over the seat EL on plan Fig. 9; from p and perpendicular to Sll., draw pf from t and paralicl 10 j)}n draw f}%wjth P for a centre and 2iV for a radius draw arc to intersect sn at/', join/'p for the bevel required. J^07- ilie cut of corona or outer fascia on the eaves Gli, 311^ and tJK Fig- 9 Return to Fig, 10, wiih h for a centre and hi lor a radius draw arc to intersect Vd at J. join j7> for the bevel required. For the bevel on the ea.\esllrf and/i[L, Fig. 9; at Fig. lO take Ic for a centre and Icj/ for a radius drnw arc to intersect the back of rafter vn at z .'"or the bevel required. These two bevels also give the edge cut of sheathing on their respective planes, also in case purlines are used they give the side cut against the hip rafters. F.igS- 11 and 12. [Scale ^■^"==i'] Shows the plan and eleva- tion of a spire on an octagon base attd how to obtain the lengths and cuts of timbers. Fig 11. Shows one half the pian. Draw the rectangle ^2>^/i. for one half the plan, biscei Alt at I.; w:th (r lor a center and the diagonal (rL for a radius, draw the arc from L, to K; make HT'^GC, CrJJ, equal jf//, JLE and Lh\ for the seat of the hips; (i, a, shows twisted irons boiled to the girls and hips for binders at the several bays. Fig. 12. Shoius the length and cuts of the hips, girts and struts in elevation and development Let ^1 JJiuilicate a base line, draw X/lT perpendicular to ^S, make IjD and jLi!^ equal 7v/> and KE on plan Fig. 1 1 ; also make EA and EB, equal EA and EH on plan Fig. 1 1. Make EC, equal EC on plan. Now determine the height of spi^e as EI, join IB and I A, for the inclination of the side, join JC for the inclination of the hip; I/// shows tl'C girts. To find the development of one side of the spire ; make Ltf equal ^il, join JI) and JE, pavaliel to »//> and tlE, draw the face of backing; make Lzz, equal AtJiJ, for the spacing of the girts. Now draw the struts as shown, tins gives all the miters or face cuts for the struts and girts for the different bays; the beve' shown atj> gives all the miter cuts fn- the girts; the bevels at 5 and t#, give the miter cuts f>r the struts in the first bay; the bevels shown at jj» and d, gives the miter cuts for ihe struts of the second bay; those for the upper bay are shown in lite proper place. The bevels at it, t and V give the cu;s at the crossing o( the stmts. Now for the side cut of the girts and struts against the hips ; drav/ the dotted lines sq, pO and mn at the center of the struts and to intersect ihe sides rJD, EE, and also the perpendicular tJE, at the points a, t and r. Mzke Ah, Ac and ^1?> equal I^u, Et and Er; from It, e and i"=l '] Shows plan and elevation of an Ogee Octagon roof and ho'du to line off the same. Fig" 13. Shows the plan Let AH, SD, AC and CI> indicate the sides of a square; draw the diagonals BC and AD intersectmg at O; with B for a center and B O for a radius, draw ark to JV, make Dili" CL, CJ, AH, AG, Bf and BP equal i>.V; join MN, JL, HG a'nd/'"P for the sides of the octagon. I-)raw the seat lines HN,GM, J-jLand p./" for the hips; the seal of jacks may also be drawn square lo the walls, and sp.iced as desired. Draw Ji_0 perpendicular lo »JH, divide KO into any number of parts. 2S,K2,tiH^S 4, £■<•; from the points i, 2, :i, 4, t£c. draw lines at right angles lo KO, intersecting the seat line JIO at e. f, on plai plan draw a line perpendic ,IXB at r for the length ? 00 lO o r ^ t ; ( i I I or D n UC SOUTHERN REGIONAL LIBRARY FACILITY A 000 606 521 3 '■-f^mtm^^-m^ DCSB LIBRARY tM jt^ i^^iljtftf^'i'"' •—