,transcriber's ,notes3 ,underscores 80 before and after a word or phrase indicate italics in the original text4 ,equal signs 80 before and after a word or phrase indicate bold in the original text4 ,small capitals have been converted to ,,solid capitals4 ,illustrations have been moved so they do not break up paragraphs4 ,typographical errors have been silently corrected4 ,added ,,index to ,,toc4 ,,stair-,,building ,,and ,,the ,,steel ,,square ,a ,,manual ,,of ,,practical ,,instruction ,,in ,,the ,,art ,,of ,,stair-,,building ,,and ,,hand-,,railing1 ,,and ,,the ,,manifold ,,uses ,,of ,,the ,,steel ,,square ,,part ,i--,,stair-,,building ,by ,,fred ,t4 ,,hodgson ,,author ,,of 8,,modern ,,carpentry10 8,,architectural ,,drawing1 ,,self-,,taught10 ,,etc4 ,,member ,,of ,,ontario ,,association ,,of ,,architects ,,and ,,morris ,,williams ,,writer ,,and ,,expert ,,on ,,carpentry ,,and ,,building ,,part ,,ii--,,the ,,steel ,,square ,by ,,morris ,,williams ,,illustrated ,,chicago ,,american ,,technical ,,society #aiag ,,copyright1 #aiaj1 #aiaf1 ,,by ,,american ,,technical ,,society ,,copyrighted ,,in ,,great ,,britain ,,all ,,rights ,,reserved ,,introduction ,on entering a building1 almost the first thing that meets the eye is the staircase and unconsciously it is made to serve as an indicator of the quality of the architecture4 ,if the design is poor or the construction faulty1 this flaw immediately gives the visitor a bad impression of the whole building4 ,furthermore1 stair-building is a rather difficult subject and the principles involved are very little understood1 which is evidenced by the fact that the layouts as furnished by architects in their plans are often improperly done4 ,probably more mistakes occur in connection with the stairway of a building than with any other construction feature4 ,it is with the idea1 therefore1 of giving a complete though simple presentation of the construction methods as applied to standard design of staircases1 that this book has been prepared4 ,the article discusses straight and winding stairs1 stairs with well hole1 layouts for curved turns1 the proper proportions of rise and width of tread1 the design of hand railings and many other problems1 the solution of which will be found very useful4 ,coupled with this article is a most instructive section on the ,steel ,square1 containing many applications of this useful instrument to roof and other types of construction4 ,illustration3 ,,greyrocks1 ,,rockport1 ,,mass4 ,frank ,chouteau ,brown1 ,architect1 ,boston1 ,mass4 ,,contents ,,part ,i ,,stair-,,building ,,page ,stair construction #a ,definitions #b ,setting out stairs #h ,pitch-board #aj ,well-hole #ah ,laying out close-string stair #bb ,open-newel stairs #cb ,stairs with curved turns #cd ,geometrical stairways and hand-railings #dc ,wreaths #dc ,tangent system #dd ,bevels to square wreaths #fj ,how to put curves on face-mould #fh ,arrangement of risers #gd ,,part ,,ii ,,steel ,,square ,introductory #a ,specifications for steel square #a ,miter and length of side of polygon #d ,steel square in roof framing #g ,general problems #h ,heel cut of common rafter #ac ,hips #ac ,heel cut of hips and valleys #af ,,index #bh ,illustration3 ,,hall ,,and ,,partially ,,enclosed ,,staircase ,,in ,,long ,,hall1 ,,greyrocks1 ,,rockport1 ,,mass4 ,frank ,chouteau ,brown1 ,architect1 ,boston1 ,mass4 ,,part ,i ,,stair-,,building ,introductory4 ,in the following instructions in the art of ,stair-building1 it is the intention to adhere closely to the practical phases of the subject1 and to present only such matter as will directly aid the student in acquiring a practical mastery of the art4 ,stair-building1 though one of the most important subjects connected with the art of building1 is probably the subject least understood by designers and by workmen generally4 ,in but few of the plans that leave the offices of ,architects1 are the stairs properly laid down2 and many of the books that have been sent out for the purpose of giving instruction in the art of building1 have this common defect--that the body of the stairs is laid down imperfectly1 and therefore presents great difficulties in the construction of the rail4 ,the stairs are an important feature of a building4 ,on entering a house they are usually the first object to meet the eye and claim the attention4 ,if one sees an ugly staircase1 it will1 in a measure1 condemn the whole house1 for the first impression produced will seldom be totally eradicated by commendable features that may be noted elsewhere4 ,it is extremely important1 therefore1 that both designer and workman shall see that staircases are properly laid out4 ,stairways should be commodious to ascend--inviting people1 as it were1 to go up4 ,when winders are used1 they should extend past the spring line of the cylinder1 so as to give proper width at the narrow end 7see ,fig4 #gb7 and bring the rail there as nearly as possible to the same pitch or slant as the rail over the square steps4 ,when the hall is of sufficient width1 the stairway should not be less than four feet wide1 so that two people can conveniently pass each other thereon4 ,the height of riser and width of tread are governed by the staircase1 which is the space allowed for the stairway2 but1 as a general rule1 the tread should not be less than nine inches wide1 and the riser should not be over eight inches high4 ,seven-inch riser and eleven-inch tread will make an easy stepping stairway4 ,if you increase the width of the tread1 you must reduce the height of the riser4 ,the tread and riser together should not be over eighteen inches1 and not less than seventeen inches4 ,these dimensions1 however1 cannot always be adhered to1 as conditions will often compel a deviation from the rule2 for instance1 in large buildings1 such as hotels1 railway depots1 or other public buildings1 treads are often made #ah inches wide1 having risers of from #b inches to #e inches depth4 ,definitions4 ,before proceeding further with the subject1 it is essential that the student make himself familiar with a few of the terms used in stair-building4 ,illustration3 ,fig4 #a4 ,illustrating ,rise1 ,run1 and ,pitch4 ,the term rise and run is often used1 and indicates certain dimensions of the stairway4 ,fig4 #a will illustrate exactly what is meant2 the line ,a ,b shows the run1 or the length over the floor the stairs will occupy4 ,from ,b to ,c is the rise1 or the total height from top of lower floor to top of upper floor4,a ,the line ,d is the pitch or line of nosings1 showing the angle of inclination of the stairs4 ,on the three lines shown--the run1 the rise1 and the pitch--depends the whole system of stair-building4 ,the body or staircase is the room or space in which the stairway is contained4 ,this may be a space including the width and length of the stairway only1 in which case it is called a close stairway1 no rail or baluster being necessary4 ,or the stairway may be in a large apartment1 such as a passage or hall1 or even in a large room1 openings being left in the upper floors so as to allow road room for persons on the stairway1 and to furnish communication between the stairways and the different stories of the building4 ,in such cases we have what are known as open stairways1 from the fact that they are not closed on both sides1 the steps showing their ends at one side1 while on the other side they are generally placed against the wall4 ,sometimes stairways are left open on both sides1 a practice not uncommon in hotels1 public halls1 and steamships4 ,when such stairs are employed1 the openings in the upper floor should be well trimmed with joists or beams somewhat stronger than the ordinary joists used in the same floor1 as will be explained further on4 ,tread4 ,this is the horizontal1 upper surface of the step1 upon which the foot is placed4 ,in other words1 it is the piece of material that forms the step1 and is generally from #a to #c inches thick1 and made of a width and length to suit the position for which it is intended4 ,in small houses1 the treads are usually made of -inch stuff4 ,riser4 ,this is the vertical height of the step4 ,the riser is generally made of thinner stuff than the tread1 and1 as a rule1 is not so heavy4 ,its duty is to connect the treads together1 and to give the stairs strength and solidity4 ,rise and ,run4 ,this term1 as already explained1 is used to indicate the horizontal and vertical dimensions of the stairway1 the rise meaning the height from the top of the lower floor to the top of the second floor2 and the run meaning the horizontal distance from the face of the first riser to the face of the last or top riser1 or1 in other words1 the distance between the face of the first riser and the point where a plumb line from the face of the top riser would strike the floor4 ,it is1 in fact1 simply the distance that the treads would make if put side by side and measured together--without1 of course1 taking in the nosings4 ,suppose there are fifteen treads1 each being #aa inches wide2 this would make a run of #ae #aa #afe inches #ac feet #i inches4 ,sometimes this distance is called the going of the stair2 this1 however1 is an ,english term1 seldom used in ,america1 and when used1 refers as frequently to the length of the single tread as it does to the run of the stairway4 ,string-,board4 ,this is the board forming the side of the stairway1 connecting with1 and supporting the ends of the steps4 ,where the steps are housed1 or grooved into the board1 it is known by the term housed string2 and when it is cut through for the tread to rest upon1 and is mitered to the riser1 it is known by the term cut and mitered string4 ,the dimensions of the lumber generally used for the purpose in practical work1 are #i inches width and -inch thickness4 ,in the first-class stairways the thickness is usually #a inches1 for both front and wall strings4 ,illustration3 ,fig4 #b4 ,common ,method of ,joining ,risers and ,treads4 ,fig4 #b shows the manner in which most stair-builders put their risers and treads together4 ,t and ,t show the treads2 ,r and ,r1 the risers2 ,s and ,s1 the string2 ,o and ,o1 the cove mouldings under the nosings ,x and ,x4 ,b and ,b show the blocks that hold the treads and risers together2 these blocks should be from #d to #f inches long1 and made of very dry wood2 their section may be from #a to #b inches square4 ,on a tread #c feet long1 three of these blocks should be used at about equal distances apart1 putting the two outside ones about #f inches from the strings4 ,they are glued up tight into the angle4 ,first warm the blocks2 next coat two adjoining sides with good1 strong glue2 then put them in position1 and nail them firmly to both tread and riser4 ,it will be noticed that the riser has a lip on the upper edge1 which enters into a groove in the tread4 ,this lip is generally about -inch long1 and may be -inch or -inch in thickness4 ,care must be taken in getting out the risers1 that they shall not be made too narrow1 as allowance must be made for the lip4 ,illustration3 ,fig4 #c4 ,vertical ,section of ,stair ,steps4 ,illustration3 ,fig4 #d4 ,end ,section of ,riser4 ,illustration3 ,fig4 #e4 ,end ,section of ,tread4 ,if the riser is a little too wide1 this will do no harm1 as the over-width may hang down below the tread2 but it must be cut the exact width where it rests on the string4 ,the treads must be made the exact width required1 before they are grooved or have the nosing worked on the outer edge4 ,the lip or tongue on the riser should fit snugly in the groove1 and should bottom4 ,by following these last instructions and seeing that the blocks are well glued in1 a good solid job will be the result4 ,fig4 #c is a vertical section of stair steps in which the risers are shown tongued into the under side of the tread1 as in ,fig4 #b1 and also the tread tongued into the face of the riser4 ,this last method is in general use throughout the country4 ,the stair-builder1 when he has steps of this kind to construct1 needs to be very careful to secure the exact width for tread and riser1 including the tongue on each4 ,the usual method1 in getting the parts prepared1 is to make a pattern showing the end section of each4 ,the millman1 with these patterns to guide him1 will be able to run the material through the machine without any danger of leaving it either too wide or too narrow2 while1 if he is left to himself without patterns1 he is liable to make mistakes4 ,these patterns are illustrated in ,figs4 #d and #e respectively1 and1 as shown1 are merely end sections of riser and tread4 ,illustration3 ,fig4 #f4 ,side ,elevation of ,finished ,steps with ,return ,nosings and ,cove ,moulding4 ,illustration3 ,fig4 #g4 ,front ,elevation of ,finished ,steps4 ,fig4 #f is a side elevation of the steps as finished1 with return nosings and cove moulding complete4 ,a front elevation of the finished step is shown in ,fig4 #g1 the nosing and riser returning against the base of the newel post4 ,often the newel post projects past the riser1 in front2 and when such is the case1 the riser and nosing are cut square against the base of the newel4 ,illustration3 ,fig4 #h4 ,portion of a ,cut and ,mitered ,string1 ,showing ,method of ,constructing ,stairs4 ,fig4 #h shows a portion of a cut and mitered string1 which will give an excellent idea of the method of construction4 ,the letter ,o shows the nosing1 ,f the return nosing with a bracket terminating against it4 ,these brackets are about #e_/af-;inch thick1 and are planted 7nailed7 on the string2 the brackets miter with the ends of the risers2 the ends of the brackets which miter with the risers1 are to be the same height as the riser4 ,the lower ends of two balusters are shown at ,g ,g2 and the dovetails or mortises to receive these are shown at ,e ,e4 ,generally two balusters are placed on each tread1 as shown2 but there are sometimes instances in which three are used1 while in others only one baluster is made use of4 ,an end portion of a cut and mitered string is shown in ,fig4 #i1 with part of the string taken away1 showing the carriage--a rough piece of lumber to which the finished string is nailed or otherwise fastened4 ,at ,c is shown the return nosing1 and the manner in which the work is finished4 ,a rough bracket is sometimes nailed on the carriage1 as shown at ,d1 to support the tread4 ,the balusters are shown dovetailed into the ends of the treads1 and are either glued or nailed in place1 or both4 ,on the lower edge of string1 at ,b1 is a return bead or moulding4 ,it will be noticed that the rough carriage is cut in snugly against the floor joist4 ,illustration3 ,fig4 #i4 ,end ,portion of ,cut and ,mitered ,string1 with ,part ,removed to ,show ,carriage4 ,fig4 #aj is a plan of the portion of a stairway shown in ,fig4 #i4 ,here the position of the string1 bracket1 riser1 and tread can be seen4 ,at the lower step is shown how to miter the riser to the string2 and at the second step is shown how to miter it to the bracket4 ,fig4 #aa shows a quick method of marking the ends of the treads for the dovetails for balusters4 ,the templet ,a is made of some thin material1 preferably zinc or hardwood4 ,the dovetails are outlined as shown1 and the intervening portions of the material are cut away1 leaving the dovetail portions solid4 ,the templet is then nailed or screwed to a gauge-block ,e1 when the whole is ready for use4 ,the method of using is clearly indicated in the illustration4 ,strings4 ,there are two main kinds of stair strings--wall strings and cut strings4 ,these are divided1 again1 under other names1 as housed strings1 notched strings1 staved strings1 and rough strings4 ,wall strings are the supporters of the ends of the treads and risers that are against the wall2 these strings may be at both ends of the treads and risers1 or they may be at one end only4 ,they may be housed 7grooved7 or left solid4 ,when housed1 the treads and risers are keyed into them1 and glued and blocked4 ,when left solid1 they have a rough string or carriage spiked or screwed to them1 to lend additional support to the ends of risers and treads4 ,stairs made after this fashion are generally of a rough1 strong kind1 and are especially adapted for use in factories1 shops1 and warehouses1 where strength and rigidity are of more importance than mere external appearance4 ,open strings are outside strings or supports1 and are cut to the proper angles for receiving the ends of the treads and risers4 ,it is over a string of this sort that the rail and balusters range2 it is also on such a string that all nosings return2 hence1 in some localities1 an open string is known as a return string4 ,illustration3 ,fig4 #aj4 ,plan of ,portion of ,stair4 ,illustration3 ,fig4 #aa4 ,templet ,used to ,mark ,dovetail ,cuts for ,balusters4 ,housed strings are those that have grooves cut in them to receive the ends of treads and risers4 ,as a general thing1 wall strings are housed4 ,the housings are made from to inch deep1 and the lines at top of tread and face of riser are made to correspond with the lines of riser and tread when in position4 ,the back lines of the housings are so located that a taper wedge may be driven in so as to force the tread and riser close to the face shoulders1 thus making a tight joint4 ,rough strings are cut from undressed plank1 and are used for strengthening the stairs4 ,sometimes a combination of rough-cut strings is used for circular or geometrical stairs1 and1 when framed together1 forms the support or carriage of the stairs4 ,staved strings are built up strings1 and are composed of narrow pieces glued1 nailed1 or bolted together so as to form a portion of a cylinder4 ,these are sometimes used for circular stairs1 though in ordinary practice the circular part of a string is a part of the main string bent around a cylinder to give it the right curve4 ,notched strings are strings that carry only treads4 ,they are generally somewhat narrower than the treads1 and are housed across their entire width4 ,a sample of this kind of string is the side of a common step-ladder4 ,strings of this sort are used chiefly in cellars1 or for steps intended for similar purposes4 ,a ,,note4--,the measure for the rise of a stairway must always be taken from the top of one floor to the top of the next4 ,setting ,out ,stairs4 ,in setting out stairs1 the first thing to do is to ascertain the locations of the first and last risers1 with the height of the story wherein the stair is to be placed4 ,these points should be marked out1 and the distance between them divided off equally1 giving the number of steps or treads required4 ,suppose we have between these two points #ae feet1 or #ahj inches4 ,if we make our treads #aj inches wide1 we shall have #ah treads4 ,it must be remembered that the number of risers is always one more than the number of treads1 so that in the case before us there will be #ai risers4 ,the height of the story is next to be exactly determined1 being taken on a rod4 ,then1 assuming a height of riser suitable to the place1 we ascertain1 by division1 how often this height of riser is contained in the height of the story2 the quotient1 if there is no remainder1 will be the number of risers in the story4 ,should there be a remainder on the first division1 the operation is reversed1 the number of inches in the height being made the dividend1 and the before-found quotient1 the divisor4 ,the resulting quotient will indicate an amount to be added to the former assumed height of riser for a new trial height4 ,the remainder will now be less than in the former division2 and if necessary1 the operation of reduction by division is repeated1 until the height of the riser is obtained to the thirty-second part of an inch4 ,these heights are then set off on the story rod as exactly as possible4 ,the story rod is simply a dressed or planed pole1 cut to a length exactly corresponding to the height from the top of the lower floor to the top of the next floor4 ,let us suppose this height to be #aa feet #a inch1 or #acc inches4 ,now1 we have #ai risers to place in this space1 to enable us to get upstairs2 therefore1 if we divide #acc by #ai1 we get #g without any remainder4 ,seven inches will therefore be the width or height of the riser4 ,without figuring this out1 the workman may find the exact width of the riser by dividing his story rod1 by means of pointers1 into #ai equal parts1 any one part being the proper width4 ,it may be well1 at this point1 to remember that the first riser must always be narrower than the others1 because the thickness of the first tread must be taken off4 ,the width of treads may also be found without figuring1 by pointing off the run of the stairs into the required number of parts2 though1 where the student is qualified1 it is always better to obtain the width1 both of treads and of risers1 by the simple arithmetical rules4 ,having determined the width of treads and risers1 a pitch-board should be formed1 showing the angle of inclination4 ,this is done by cutting a piece of thin board or metal in the shape of a right-angled triangle1 with its base exactly equal to the run of the step1 and its perpendicular equal to the height of the riser4 ,it is a general maxim1 that the greater the breadth of a step or tread1 the less should be the height of the riser2 and1 conversely1 the less the breadth of a step1 the greater should be the height of the riser4 ,the proper relative dimensions of treads and risers may be illustrated graphically1 as in ,fig4 #ab4 ,illustration3 ,fig4 #ab4 ,graphic ,illustration of ,proportional ,dimensions of ,treads and ,risers4 ,in the right-angle triangle ,a ,b ,c1 make ,a ,b equal to #bd inches1 and ,b ,c equal to #aa inches--the standard proportion4 ,now1 to find the riser corresponding to a given width of tread1 from ,b1 set off on ,a ,b the width of the tread1 as ,b ,d2 and from ,d1 erect a perpendicular ,d ,e1 meeting the hypotenuse in ,e2 then ,d ,e is the height of the riser2 and if we join ,b and ,e1 the angle ,d ,b ,e is the angle of inclination1 showing the slope of the ascent4 ,in like manner1 where ,b ,f is the width of the tread1 ,f ,g is the riser1 and ,b ,g the slope of the stair4 ,a width of tread ,b ,h gives a riser of the height of ,h ,k2 and a width of tread equal to ,b ,l gives a riser equal to ,l ,m4 ,in the opinion of many builders1 however1 a better scheme of proportions for treads and risers is obtained by the following method3 ,set down two sets of numbers1 each in arithmetical progression--the first set showing widths of tread1 increasing by inches2 the other showing heights of riser1 decreasing by half-inches4 ,,treads1 ,,inches ,,risers1 ,,inches #e #i #f #h #g #h #h #g #i #g #aj #f #aa #f #ab #e #ac #e #ad #d #ae #d #af #c #ag #c #ah #b ,it will readily be seen that each pair of treads and risers thus obtained is suitably proportioned as to dimensions4 ,it is seldom1 however1 that the proportions of treads and risers are entirely a matter of choice4 ,the space allotted to the stairs usually determines this proportion2 but the above will be found a useful standard1 to which it is desirable to approximate4 ,in the better class of buildings1 the number of steps is considered in the plan1 which it is the business of the ,architect to arrange2 and in such cases1 the height of the story rod is simply divided into the number required4 ,pitch-,board4 ,it will now be in order to describe a pitch-board and the manner of using it2 no stairs can be properly built without the use of a pitch-board in some form or other4 ,properly speaking1 a pitch-board1 as already explained1 is a thin piece of material1 generally pine or sheet metal1 and is a right-angled triangle in outline4 ,one of its sides is made the exact height of the rise2 at right angles with this line of rise1 the exact width of the tread is measured off2 and the material is cut along the hypotenuse of the right-angled triangle thus formed4 ,the simplest method of making a pitch-board is by using a steel square1 which1 of course1 every carpenter in this country is supposed to possess4 ,by means of this invaluable tool1 also1 a stair string can be laid out1 the square being applied to the string as shown in ,fig4 #ac4 ,in the instance here illustrated1 the square shows #aj inches for the tread and #g inches for the rise4 ,illustration3 ,fig4 #ac4 ,steel ,square used as a ,pitch-,board in ,laying ,out ,stair ,string4 ,to cut a pitch-board1 after the tread and rise have been determined1 proceed as follows3 ,take a piece of thin1 clear material1 and lay the square on the face edge1 as shown in ,fig4 #ac4 ,mark out the pitch-board with a sharp knife2 then cut out with a fine saw1 and dress to the knife marks2 nail a piece on the largest edge of the pitch-board for a gauge or fence1 and it is ready for use4 ,fig4 #ad shows the pitch-board pure and simple2 it may be half an inch thick1 or1 if of hardwood1 may be from a quarter-inch to a half-inch thick4 ,fig4 #ae shows the pitch-board after the gauge or fence is nailed on4 ,this fence or gauge may be about #a inches wide and from to inch thick4 ,fig4 #af shows a sectional view of the pitch-board with a fence nailed on4 ,illustration3 ,fig4 #ad4 ,fig4 #ae4 ,fig4 #af4 ,showing ,how a ,pitch-,board is ,made4 ,fig4 #ae shows gauge fastened to long edge2 ,fig4 #af is a sectional elevation of completed board4 ,in ,fig4 #ag the manner of applying the pitch-board is shown4 ,r ,r ,r is the string1 and the line ,a shows the jointed or straight edge of the string4 ,the pitch-board ,p is shown in position1 the line #h represents the step or tread1 and the line #g shows the line of the riser4 ,these two lines are of course at right angles1 or1 as the carpenter would say2 they are square4 ,this string shows four complete cuts1 and part of a fifth cut for treads1 and five complete cuts for risers4 ,the bottom of the string at ,w is cut off at the line of the floor on which it is supposed to rest4 ,the line ,c is the line of the first riser4 ,this riser is cut lower than any of the other risers1 because1 as above explained1 the thickness of the first tread is always taken off it2 thus1 if the tread is #a inches thick1 the riser in this case would only require to be #f inches wide1 as #g-a #f4 ,the string must be cut so that the line at ,w will be only #f inches from the line at #h1 and these two lines must be parallel4 ,the first riser and tread having been satisfactorily dealt with1 the rest can easily be marked off by simply sliding the pitch-board along the line ,a until the outer end of the line #h on the pitch-board strikes the outer end of the line #g on the string1 when another tread and another riser are to be marked off4 ,the remaining risers and treads are marked off in the same manner4 ,illustration3 ,fig4 #ag4 ,showing ,method of ,using ,pitch-,board4 ,sometimes there may be a little difficulty at the top of the stairs1 in fitting the string to the trimmer or joists2 but1 as it is necessary first to become expert with the pitch-board1 the method of trimming the well or attaching the cylinder to the string will be left until other matters have been discussed4 ,fig4 #ah shows a portion of the stairs in position4 ,s and ,s show the strings1 which in this case are cut square2 that is1 the part of the string to which the riser is joined is cut square across1 and the butt or end wood of the riser is seen4 ,in this case1 also1 the end of the tread is cut square off1 and flush with the string and riser4 ,both strings in this instance are open strings4 ,usually1 in stairs of this kind1 the ends of the treads are rounded off similarly to the front of the tread1 and the ends project over the strings the same distance that the front edge projects over the riser4 ,if a moulding or cove is used under the nosing in front1 it should be carried round on the string to the back edge of the tread and cut off square1 for in this case the back edge of the tread will be square4 ,a riser is shown at ,r1 and it will be noticed that it runs down behind the tread on the back edge1 and is either nailed or screwed to the tread4 ,this is the ,american practice1 though in ,england the riser usually rests on the tread1 which extends clear back to string as shown at the top tread in the diagram4 ,it is much better1 however1 for general purposes1 that the riser go behind the tread1 as this tends to make the whole stairway much stronger4 ,housed strings are those which carry the treads and risers without their ends being seen4 ,in an open stair1 the wall string only is housed1 the other ends of the treads and risers resting on a cut string1 and the nosings and mouldings being returned as before described4 ,illustration3 ,fig4 #ah4 ,portion of ,stair in ,position4 ,illustration3 ,fig4 #ai4 ,showing ,method of ,housing ,treads and ,risers4 ,the manner of housing is shown in ,fig4 #ai1 in which the treads ,t ,t and the risers ,r ,r are shown in position1 secured in place respectively by means of wedges ,x ,x and ,f ,f1 which should be well covered with good glue before insertion in the groove4 ,the housings are generally made from to inch deep1 space for the wedge being cut to suit4 ,in some closed stairs in which there is a housed string between the newels1 the string is double-tenoned into the shanks of both newels1 as shown in ,fig4 #bj4 ,the string in this example is made #ab inches wide1 which is a very good width for a string of this kind2 but the thickness should never be less than #a inches4 ,the upper newel is made about #e feet #d inches long from drop to top of cap4 ,these strings are generally capped with a subrail of some kind1 on which the baluster1 if any1 is cut-mitered in4 ,generally a groove1 the width of the square of the balusters1 is worked on the top of the subrail1 and the balusters are worked out to fit into this groove2 then pieces of this material1 made the width of the groove and a little thicker than the groove is deep1 are cut so as to fit in snugly between the ends of the balusters resting in the groove4 ,this makes a solid job2 and the pieces between the balusters may be made of any shape on top1 either beveled1 rounded1 or moulded1 in which case much is added to the appearance of the stairs4 ,illustration3 ,fig4 #bj4 ,showing ,method of ,connecting ,housed ,string to ,newels4 ,illustration3 ,fig4 #ba4 ,method of ,connecting ,rail and ,string to ,bottom ,newel4 ,fig4 #ba exhibits the method of attaching the rail and string to the bottom newel4 ,the dotted lines indicate the form of the tenons cut to fit the mortises made in the newel to receive them4 ,fig4 #bb shows how the string fits against the newel at the top2 also the trimmer ,e1 to which the newel post is fastened4 ,the string in this case is tenoned into the upper newel post the same way as into the lower one4 ,the open string shown in ,fig4 #bc is a portion of a finished string1 showing nosings and cove returned and finishing against the face of the string4 ,along the lower edge of the string is shown a bead or moulding1 where the plaster is finished4 ,a portion of a stair of the better class is shown in ,fig4 #bd4 ,this is an open1 bracketed string1 with returned nosings and coves and scroll brackets4 ,these brackets are made about inch thick1 and may be in any desirable pattern4 ,the end next the riser should be mitered to suit2 this will require the riser to be inch longer than the face of the string4 ,the upper part of the bracket should run under the cove moulding2 and the tread should project over the string the full inch1 so as to cover the bracket and make the face even for the nosing and the cove moulding to fit snugly against the end of the tread and the face of the bracket4 ,great care must be taken about this point1 or endless trouble will follow4 ,in a bracketed stair of this kind1 care must be taken in placing the newel posts1 and provision must be made for the extra inch due to the bracket4 ,the newel post must be set out from the string inch1 and it will then align with the baluster4 ,illustration3 ,fig4 #bb4 ,connections of ,string and ,trimmer at ,upper ,newel ,post4 ,illustration3 ,fig4 #bc4 ,portion of ,finished ,string1 ,showing ,returned ,nosings and ,coves1 also ,bead ,moulding4 ,illustration3 ,fig4 #bd4 ,portion of ,open1 ,bracketed ,string ,stair1 with ,returned ,nosings and ,coves1 ,scroll ,brackets1 and ,bead ,moulding4 ,we have now described several methods of dealing with strings2 but there are still a few other points connected with these members1 both housed and open1 that it will be necessary to explain2 before the young workman can proceed to build a fair flight of stairs4 ,the connection of the wall string to the lower and upper floors1 and the manner of affixing the outer or cut string to the upper joist and to the newel1 are matters that must not be overlooked4 ,it is the intention to show how these things are accomplished1 and how the stairs are made strong by the addition of rough strings or bearing carriages4 ,illustration3 ,fig4 #be4 ,side ,elevation of ,part of ,stair with ,open1 ,cut and ,mitered ,string4 ,illustration3 ,fig4 #bf4 ,plan of ,part of ,stair ,shown in ,fig4 #be4 ,fig4 #be gives a side view of part of a stair of the better class1 with one open1 cut and mitered string4 ,in ,fig4 #bf1 a plan of this same stairway1 ,w ,s shows the wall string2 ,r ,s1 the rough string1 placed there to give the structure strength2 and ,o ,s1 the outer or cut and mitered string4 ,at ,a ,a the ends of the risers are shown1 and it will be noticed that they are mitered against a vertical or riser line of the string1 thus preventing the end of the riser from being seen4 ,the other end of the riser is in the housing in the wall string4 ,the outer end of the tread is also mitered at the nosing1 and a piece of material made or worked like the nosing is mitered against or returned at the end of the tread4 ,the end of this returned piece is again returned on itself back to the string1 as shown at ,n in ,fig4 #be4 ,the moulding1 which is -inch cove in this case1 is also returned on itself back to the string4 ,the mortises shown at ,b ,b ,b ,b 7,fig4 #bf71 are for the balusters4 ,it is always the proper thing to saw the ends of the treads ready for the balusters before the treads are attached to the string2 then1 when the time arrives to put up the rail1 the back ends of the mortises can be cut out1 when the treads will be ready to receive the balusters4 ,the mortises are dovetailed1 and1 of course1 the tenons on the balusters must be made to suit4 ,the treads are finished on the bench2 and the return nosings are fitted to them and tacked on1 so that they may be taken off to insert the balusters when the rail is being put in position4 ,fig4 #bg shows the manner in which a wall string is finished at the foot of the stairs4 ,s shows the string1 with moulding wrought on the upper edge4 ,this moulding may be a simple ogee1 or may consist of a number of members2 or it may be only a bead2 or1 again1 the edge of the string may be left quite plain2 this will be regulated in great measure by the style of finish in the hall or other part of the house in which the stairs are placed4 ,b shows a portion of a baseboard1 the top edge of which has the same finish as the top edge of the string4 ,b and ,a together show the junction of the string and base4 ,f ,f show blocks glued in the angles of the steps to make them firm and solid4 ,illustration3 ,fig4 #bg4 ,showing ,how ,wall ,string is ,finished at ,foot of ,stair4 ,illustration3 ,fig4 #bh4 ,showing ,how ,wall ,string is ,finished at ,top of ,stair4 ,fig4 #bh shows the manner in which the wall string ,s is finished at the top of the stairs4 ,it will be noticed that the moulding is worked round the ease-off at ,a to suit the width of the base at ,b4 ,the string is cut to fit the floor and to butt against the joist4 ,the plaster line under the stairs and on the ceiling1 is also shown4 ,illustration3 ,fig4 #bi4 ,showing ,how a ,cut or ,open ,string is ,finished at ,foot of ,stair4 ,fig4 #bi shows a cut or open string at the foot of a stairway1 and the manner of dealing with it at its junction with the newel post ,k4 ,the point of the string should be mortised into the newel #b inches1 #c inches1 or #d inches1 as shown by the dotted lines2 and the mortise in the newel should be cut near the center1 so that the center of the baluster will be directly opposite the central line of the newel post4 ,the proper way to manage this1 is to mark the central line of the baluster on the tread1 and then make this line correspond with the central line of the newel post4 ,by careful attention to this point1 much trouble will be avoided where a turned cap is used to receive the lower part of the rail4 ,the lower riser in a stair of this kind will be somewhat shorter than the ones above it1 as it must be cut to fit between the newel and the wall string4 ,a portion of the tread1 as well as of the riser1 will also butt against the newel1 as shown at ,w4 ,if there is no spandrel or wall under the open string1 it may run down to the floor as shown by the dotted line at ,o4 ,the piece ,o is glued to the string1 and the moulding is worked on the curve4 ,if there is a wall under the string ,s1 then the base ,b1 shown by the dotted lines1 will finish against the string1 and it should have a moulding on its upper edge1 the same as that on the lower edge of the string1 if any1 this moulding being mitered into the one on the string4 ,when there is a base1 the piece ,o is of course dispensed with4 ,the square of the newel should run down by the side of a joist as shown1 and should be firmly secured to the joist either by spiking or by some other suitable device4 ,if the joist runs the other way1 try to get the newel post against it1 if possible1 either by furring out the joist or by cutting a portion off the thickness of the newel4 ,the solidity of a stair and the firmness of the rail1 depend very much upon the rigidity of the newel post4 ,the above suggestions are applicable where great strength is required1 as in public buildings4 ,in ordinary work1 the usual method is to let the newel rest on the floor4 ,illustration3 ,fig4 #cj4 ,showing ,how a ,cut or ,open ,string is ,finished at ,top of ,stair4 ,fig4 #cj shows how the cut string is finished at the top of the stairs4 ,this illustration requires no explanation after the instructions already given4 ,thus far1 stairs having a newel only at the bottom have been dealt with4 ,there are1 however1 many modifications of straight and return stairs which have from two to four or six newels4 ,in such cases1 the methods of treating strings at their finishing points must necessarily be somewhat different from those described2 but the general principles1 as shown and explained1 will still hold good4 ,well-,hole4 ,before proceeding to describe and illustrate neweled stairs1 it will be proper to say something about the well-hole1 or the opening through the floors1 through which the traveler on the stairs ascends or descends from one floor to another4 ,fig4 #ca shows a well-hole1 and the manner of trimming it4 ,in this instance the stairs are placed against the wall2 but this is not necessary in all cases1 as the well-hole may be placed in any part of the building4 ,the arrangement of the trimming varies according as the joists are at right angles to1 or are parallel to1 the wall against which the stairs are built4 ,in the former case 7,fig4 #ca1 ,a7 the joists are cut short and tusk-tenoned into the heavy trimmer ,t ,t1 as shown in the cut4 ,this trimmer is again tusk-tenoned into two heavy joists ,t ,j and ,t ,j1 which form the ends of the well-hole4 ,these heavy joists are called trimming joists2 and1 as they have to carry a much heavier load than other joists on the same floor1 they are made much heavier4 ,sometimes two or three joists are placed together1 side by side1 being bolted or spiked together to give them the desired unity and strength4 ,in constructions requiring great strength1 the tail and header joists of a well-hole are suspended on iron brackets4 ,if the opening runs parallel with the joists 7,fig4 #ca1 ,b71 the timber forming the side of the well-hole should be left a little heavier than the other joists1 as it will have to carry short trimmers 7,t ,j and ,t ,j7 and the joists running into them4 ,the method here shown is more particularly adapted to brick buildings1 but there is no reason why the same system may not be applied to frame buildings4 ,usually in cheap1 frame buildings1 the trimmers ,t ,t are spiked against the ends of the joists1 and the ends of the trimmers are supported by being spiked to the trimming joists ,t ,j1 ,t ,j4 ,this is not very workmanlike or very secure1 and should not be done1 as it is not nearly so strong or durable as the old method of framing the joists and trimmers together4 ,fig4 #cb shows a stair with three newels and a platform4 ,in this example1 the first tread 7,no4 #a7 stands forward of the newel post two-thirds of its width4 ,this is not necessary in every case1 but it is sometimes done to suit conditions in the hallway4 ,the second newel is placed at the twelfth riser1 and supports the upper end of the first cut string and the lower end of the second cut string4 ,the platform 7#ab7 is supported by joists which are framed into the wall and are fastened against a trimmer running from the wall to the newel along the line #ab4 ,this is the case only when the second newel runs down to the floor4 ,illustration3 ,fig4 #ca4 ,showing ,ways of ,trimming ,well-,hole when ,joists ,run in ,different ,directions4 ,if the second newel does not run to the floor1 the framework supporting the platform will need to be built on studding4 ,the third newel stands at the top of the stairs1 and is fastened to the joists of the second floor1 or to the trimmer1 somewhat after the manner of fastening shown in ,fig4 #bi4 ,in this example1 the stairs have #af risers and #ae treads1 the platform or landing 7#ab7 making one tread4 ,the figure #af shows the floor in the second story4 ,this style of stair will require a well-hole in shape about as shown in the plan2 and where strength is required1 the newel at the top should run from floor to floor1 and act as a support to the joists and trimmers on which the second floor is laid4 ,perhaps the best way for a beginner to go about building a stairway of this type1 will be to lay out the work on the lower floor in the exact place where the stairs are to be erected1 making everything full size4 ,there will be no difficulty in doing this2 and if the positions of the first riser and the three newel posts are accurately defined1 the building of the stairs will be an easy matter4 ,plumb lines can be raised from the lines on the floor1 and the positions of the platform and each riser thus easily determined4 ,not only is it best to line out on the floor all stairs having more than one newel2 but in constructing any kind of stair it will perhaps be safest for a beginner to lay out in exact position on the floor the points over which the treads and risers will stand4 ,by adopting this rule1 and seeing that the strings1 risers1 and treads correspond exactly with the lines on the floor1 many cases of annoyance will be avoided4 ,many expert stair-builders1 in fact1 adopt this method in their practice1 laying out all stairs on the floor1 including even the carriage strings1 and they cut out all the material from the lines obtained on the floor4 ,by following this method1 one can see exactly the requirements in each particular case1 and can rectify any error without destroying valuable material4 ,illustration3 ,fig4 #cb4 ,stair with ,three ,newels and a ,platform4 ,laying ,out4 ,in order to afford the student a clear idea of what is meant by laying out on the floor1 an example of a simple close-string stair is given4 ,in ,fig4 #cc1 the letter ,f shows the floor line2 ,l is the landing or platform2 and ,w is the wall line4 ,the stair is to be #d feet wide over strings2 the landing1 #d feet wide2 the height from floor to landing1 #g feet2 and the run from start to finish of the stair1 #h feet #h inches4 ,the first thing to determine is the dimensions of the treads and risers4 ,the wider the tread1 the lower must be the riser1 as stated before4 ,no definite dimensions for treads and risers can be given1 as the steps have to be arranged to meet the various difficulties that may occur in the working out of the construction2 but a common rule is this3 ,make the width of the tread1 plus twice the rise1 equal to #bd inches4 ,this will give1 for an #h-;inch tread1 an #h-;inch rise2 for a #i-;inch tread1 a #g-;inch rise2 for a #aj-;inch tread1 a #g-;inch rise1 and so on4 ,having the height 7#g feet7 and the run of the flight 7#h feet #h-;inches71 take a rod about one inch square1 and mark on it the height from floor to landing 7#g feet71 and the length of the going or run of the flight 7#h feet #h inches74 ,consider now what are the dimensions which can be given to the treads and risers1 remembering that there will be one more riser than the number of treads4 ,mark off on the rod the landing1 forming the last tread4 ,if twelve risers are desired1 divide the height 7namely1 #g feet7 by #ab1 which gives #g inches as the rise of each step4 ,then divide the run 7namely1 #h feet #h inches7 by #aa1 and the width of the tread is found to be #i inches4 ,great care must be taken in making the pitch-board for marking off the treads and risers on the string4 ,the pitch-board may be made from dry hardwood about -inch thick4 ,one end and one side must be perfectly square to each other2 on the one1 the width of the tread is set off1 and on the other the height of the riser4 ,connect the two points thus obtained1 and saw the wood on this line4 ,the addition of a gauge-piece along the longest side of the triangular piece1 completes the pitch-board1 as was illustrated in ,fig4 #ae4 ,the length of the wall and outer string can be ascertained by means of the pitch-board4 ,one side and one edge of the wall string must be squared2 but the outer string must be trued all round4 ,on the strings1 mark the positions of the treads and risers by using the pitch-board as already explained 7,fig4 #ag74 ,strings are usually made #aa inches wide1 but may be made #ab inches wide if necessary for strength4 ,illustration3 ,fig4 #cc4 ,method of ,laying ,out a ,simple1 ,close-,string ,stair4 ,after the widths of risers and treads have been determined1 and the string is ready to lay out1 apply the pitch-board1 marking the first riser about #i inches from the end2 and number each step in succession4 ,the thickness of the treads and risers can be drawn by using thin strips of hardwood made the width of the housing required4 ,now allow for the wedges under the treads and behind the risers1 and thus find the exact width of the housing1 which should be about -inch deep2 the treads and risers will require to be made #a inches longer than shown in the plan1 to allow for the housings at both ends4 ,before putting the stair together1 be sure that it can be taken into the house and put in position without trouble4 ,if for any reason it cannot be put in after being put together1 then the parts must be assembled1 wedged1 and glued up at the spot4 ,it is essential in laying out a plan on the floor1 that the exact positions of the first and last risers be ascertained1 and the height of the story wherein the stair is to be placed4 ,then draw a plan of the hall or other room in which the stairs will be located1 including surrounding or adjoining parts of the room to the extent of ten or twelve feet from the place assigned for the foot of the stair4 ,all the doorways1 branching passages1 or windows which can possibly come in contact with the stair from its commencement to its expected termination or landing1 must be noted4 ,the sketch must necessarily include a portion of the entrance hall in one part1 and of the lobby or landing in another1 and on it must be laid out all the lines of the stair from the first to the last riser4 ,the height of the story must next be exactly determined and taken on the rod2 then1 assuming a height of risers suitable to the place1 a trial is made by division in the manner previously explained1 to ascertain how often this height is contained in the height of the story4 ,the quotient1 if there is no remainder1 will be the number of risers required4 ,should there be a remainder on the first division1 the operation is reversed1 the number of inches in the height being made the dividend and the before-found quotient the divisor2 and the operation of reduction by division is carried on till the height of the riser is obtained to the thirty-second part of an inch4 ,these heights are then set off as exactly as possible on the story rod1 as shown in ,fig4 #cc4 ,the next operation is to show the risers on the sketch4 ,this the workman will find no trouble in arranging1 and no arbitrary rule can be given4 ,a part of the foregoing may appear to be repetition2 but it is not1 for it must be remembered that scarcely any two flights of stairs are alike in run1 rise1 or pitch1 and any departure in any one dimension from these conditions leads to a new series of dimensions that must be dealt with independently4 ,the principle laid down1 however1 applies to all straight flights of stairs2 and the student who has followed closely and retained the pith of what has been said1 will1 if he has a fair knowledge of the use of tools1 be fairly equipped for laying out and constructing a plain1 straight stair with a straight rail4 ,plain stairs may have one platform1 or several2 and they may turn to the right or to the left1 or1 rising from a platform or landing1 may run in an opposite direction from their starting point4 ,when two flights are necessary for a story1 it is desirable that each flight should consist of the same number of steps2 but this1 of course1 will depend on the form of the staircase1 the situation and height of doors1 and other obstacles to be passed under or over1 as the case may be4 ,in ,fig4 #cb1 a stair is shown with a single platform or landing and three newels4 ,the first part of this stair corresponds1 in number of risers1 with the stair shown in ,fig4 #cc2 the second newel runs down to the floor1 and helps to sustain the landing4 ,this newel may simply be a #d by #d-;inch post1 or the whole space may be inclosed with the spandrel of the stair4 ,the second flight starts from the platform just as the first flight starts from the lower floor1 and both flights may be attached to the newels in the manner shown in ,fig4 #bi4 ,the bottom tread in ,fig4 #cb is rounded off against the square of the newel post2 but this cannot well be if the stairs start from the landing1 as the tread would project too far onto the platform4 ,sometimes1 in high-class stairs1 provision is made for the first tread to project well onto the landing4 ,if there are more platforms than one1 the principles of construction will be the same2 so that whenever the student grasps the full conditions governing the construction of a single-platform stair1 he will be prepared to lay out and construct the body of any stair having one or more landings4 ,the method of laying out1 making1 and setting up a hand-rail will be described later4 ,stairs formed with treads each of equal width at both ends1 are named straight flights2 but stairs having treads wider at one end than the other are known by various names1 as winding stairs1 dog-legged stairs1 circular stairs1 or elliptical stairs4 ,a tread with parallel sides1 having the same width at each end1 is called a flyer2 while one having one wide end and one narrow1 is called a winder4 ,these terms will often be made use of in what follows4 ,the elevation and plan of the stair shown in ,fig4 #cd may be called a dog-legged stair with three winders and six flyers4 ,the flyers1 however1 may be extended to any number4 ,the housed strings to receive the winders are shown4 ,these strings show exactly the manner of construction4 ,the shorter string1 in the corner from #a to #d1 which is shown in the plan to contain the housing of the first winder and half of the second1 is put up first1 the treads being leveled by aid of a spirit level2 and the longer upper string is put in place afterwards1 butting snugly against the lower string in the corner4 ,it is then fastened firmly to the wall4 ,the winders are cut snugly around the newel post1 and well nailed4 ,their risers will stand one above another on the post2 and the straight string above the winders will enter the post on a line with the top edge of the uppermost winder4 ,illustration3 ,fig4 #cd4 ,elevation and ,plan of ,dog-,legged ,stair with ,three ,winders and ,six ,flyers4 ,platform stairs are often constructed so that one flight will run in a direction opposite to that of the other flight1 as shown in ,fig4 #ce4 ,in cases of this kind1 the landing or platform requires to have a length more than double that of the treads1 in order that both flights may have the same width4 ,sometimes1 however1 and for various reasons1 the upper flight is made a little narrower than the lower2 but this expedient should be avoided whenever possible1 as its adoption unbalances the stairs4 ,in the example before us1 eleven treads1 not including the landing1 run in one direction2 while four treads1 including the landing1 run in the opposite direction2 or1 as workmen put it1 the stair 8returns on itself40 ,the elevation shown in ,fig4 #cf illustrates the manner in which the work is executed4 ,the various parts are shown as follows3 ,illustration3 ,fig4 #ce4 ,plan of ,platform ,stair ,returning on ,itself4 ,illustration3 ,fig4 #cg is a section of the top landing1 with baluster and rail4 ,illustration3 ,fig4 #ch is part of the long newel1 showing mortises for the strings4 ,illustration3 ,fig4 #cf4 ,elevation ,showing ,construction of ,platform ,stair of which ,plan is ,given in ,fig4 #ce4 ,illustration3 ,fig4 #cg4 ,section of ,top ,landing1 ,baluster1 and ,rail4 ,illustration3 ,fig4 #ci represents part of the bottom newel1 showing the string1 moulding on the outside1 and cap4 ,illustration3 ,fig4 #dj is a section of the top string enlarged4 ,fig4 #da is the newel at the bottom1 as cut out to receive bottom step4 ,it must be remembered that there is a cove under each tread4 ,this may be nailed in after the stairs are put together1 and it adds greatly to the appearance4 ,we may state that stairs should have carriage pieces fixed from floor to floor1 under the stairs1 to support them4 ,these may be notched under the steps2 or rough brackets may be nailed to the side of the carriage1 and carried under each riser and tread4 ,there is also a framed spandrel which helps materially to carry the weight1 makes a sound job1 and adds greatly to the appearance4 ,this spandrel may be made of #a-;inch material1 with panels and mouldings on the front side1 as shown in ,fig4 #cf4 ,the joint between the top and bottom rails of the spandrel at the angle1 should be made as shown in ,fig4 #db with a cross-tongue1 and glued and fastened with long screws4 ,fig4 #dc is simply one of the panels showing the miters on the moulding and the shape of the sections4 ,as there is a convenient space under the landing1 it is commonly used for a closet4 ,illustration3 ,fig4 #ch4 ,string ,mortises in ,long ,newel4 ,illustration3 ,fig4 #ci4 ,mortises in ,lower ,newel for ,string1 ,outside ,moulding1 and ,cap4 ,illustration3 ,fig4 #dj4 ,enlarged ,section of ,top ,string4 ,illustration3 ,fig4 #da4 ,newel ,cut to ,receive ,bottom ,step4 ,in setting out stairs1 not only the proportions of treads and risers must be considered1 but also the material available4 ,as this material runs1 as a rule1 in certain sizes1 it is best to work so as to conform to it as nearly as possible4 ,in ordinary stairs1 #aa by #a-;inch common stock is used for strings and treads1 and #g-;inch by -inch stock for risers2 in stairs of a better class1 wider and thicker material may be used4 ,the rails are set at various heights2 #b feet #h inches may be taken as an average height on the stairs1 and #c feet #a inch on landings1 with two balusters to each step4 ,in ,fig4 #cf1 all the newels and balusters are shown square2 but it is much better1 and is the more common practice1 to have them turned1 as this gives the stairs a much more artistic appearance4 ,the spandrel under the string of the stairway shows a style in which many stairs are finished in hallways and other similar places4 ,plaster is sometimes used instead of the panel work1 but is not nearly so good as woodwork4 ,the door under the landing may open into a closet1 or may lead to a cellarway1 or through to some other room4 ,in stairs with winders1 the width of a winder should1 if possible1 be nearly the width of the regular tread1 at a distance of #ad inches from the narrow end1 so that the length of the step in walking up or down the stairs may not be interrupted2 and for this reason and several others1 it is always best to have three winders only in each quarter-turn4 ,above all1 avoid a four-winder turn1 as this makes a breakneck stair1 which is more difficult to construct and inconvenient to use4 ,illustration3 ,fig4 #db4 ,showing ,method of ,joining ,spandrel ,rails1 with ,cross-,tongue ,glued and ,screwed4 ,illustration3 ,fig4 #dc4 ,panel in ,spandrel1 ,showing ,miters on ,moulding1 and ,shape of ,section4 ,bullnose ,tread4 ,no other stair1 perhaps1 looks so well at the starting point as one having a bullnose step4 ,in ,fig4 #dd are shown a plan and elevation of a flight of stairs having a bullnose tread4 ,the method of obtaining the lines and setting out the body of the stairs1 is the same as has already been explained for other stairs1 with the exception of the first two steps1 which are made with circular ends1 as shown in the plan4 ,these circular ends are worked out as hereafter described1 and are attached to the newel and string as shown4 ,the example shows an open1 cut string with brackets4 ,the spandrel under the string contains short panels1 and makes a very handsome finish4 ,the newels and balusters in this case are turned1 and the latter have cutwork panels between them4 ,illustration3 ,fig4 #dd4 ,elevation and ,plan of ,stair with ,bullnose ,tread4 ,illustration3 ,fig4 #de4 ,section through ,bullnose ,step4 ,bullnose steps are usually built up with a three-piece block1 as shown in ,fig4 #de1 which is a section through the step indicating the blocks1 tread1 and riser4 ,fig4 #df is a plan showing how the veneer of the riser is prepared before being bent into position4 ,the block ,a indicates a wedge which is glued and driven home after the veneer is put in place4 ,this tightens up the work and makes it sound and clear4 ,figs4 #dg and #dh show other methods of forming bullnose steps4 ,illustration3 ,fig4 #df4 ,plan ,showing ,preparation of ,veneer before ,bending into ,position4 ,fig4 #di is the side elevation of an open-string stair with bullnose steps at the bottom2 while ,fig4 #ej is a view showing the lower end of the string1 and the manner in which it is prepared for fixing to the blocks of the step4 ,fig4 #ea is a section through the string1 showing the bracket1 cove1 and projection of tread over same4 ,illustration3 ,fig4 #dg4 ,fig4 #dh4 ,methods of ,forming ,bullnose ,steps4 ,illustration3 ,fig4 #di4 ,side ,elevation of ,open-,string ,stair with ,bullnose ,steps4 ,figs4 #eb and #ec show respectively a plan and vertical section of the bottom part of the stair4 ,the blocks are shown at the ends of the steps 7,fig4 #ec71 with the veneered parts of the risers going round them2 also the position where the string is fixed to the blocks 7,fig4 #eb72 and the tenon of the newel is marked on the upper step4 ,the section 7,fig4 #ec7 shows the manner in which the blocks are built up and the newel tenoned into them4 ,the newel1 ,fig4 #di1 is rather an elaborate affair1 being carved at the base and on the body1 and having a carved rosette planted in a small1 sunken panel on three sides1 the rail butting against the fourth side4 ,open-,newel ,stairs4 ,before leaving the subject of straight and dog-legged stairs1 the student should be made familiar with at least one example of an open-newel stair4 ,as the same principles of construction govern all styles of open-newel stairs1 a single example will be sufficient4 ,the student must1 of course1 understand that he himself is the greatest factor in planning stairs of this type2 that the setting out and designing will generally devolve on him4 ,by exercising a little thought and foresight1 he can so arrange his plan that a minimum of both labor and material will be required4 ,illustration3 ,fig4 #ej4 ,lower ,end of ,string to ,connect with ,bullnose ,step4 ,illustration3 ,fig4 #ea4 ,section through ,string4 ,fig4 #ed shows a plan of an open-newel stair having two landings and closed strings1 shown in elevation in ,fig4 #ee4 ,the dotted lines show the carriage timbers and trimmers1 also the lines of risers2 while the treads are shown by full lines4 ,it will be noticed that the strings and trimmers at the first landing are framed into the shank of the second newel post1 which runs down to the floor2 while the top newel drops below the fascia1 and has a turned and carved drop4 ,this drop hangs below both the fascia and the string4 ,the lines of treads and risers are shown by dotted lines and crosshatched sections4 ,the position of the carriage timbers is shown both in the landings and in the runs of the stairs1 the projecting ends of these timbers being supposed to be resting on the wall4 ,a scale of the plan and elevation is attached to the plan4 ,in ,fig4 #ee1 a story rod is shown at the right1 with the number of risers spaced off thereon4 ,the design of the newels1 spandrel1 framing1 and paneling is shown4 ,illustration3 ,fig4 #eb4 ,plan of ,bottom ,part of ,bullnose ,stair4 ,illustration3 ,fig4 #ec4 ,vertical ,section through ,bottom ,part of ,bullnose ,stair4 ,only the central carriage timbers are shown in ,fig4 #ed2 but in a stair of this width1 there ought to be two other timbers1 not so heavy1 perhaps1 as the central one1 yet strong enough to be of service in lending additional strength to the stairway1 and also to help carry the laths and plaster or the paneling which may be necessary in completing the under side or soffit4 ,the strings being closed1 the butts of their balusters must rest on a subrail which caps the upper edge of the outer string4 ,illustration3 ,fig4 #ed4 ,plan of ,open-,newel ,stair1 with ,two ,landings and ,closed ,strings4 ,the first newel should pass through the lower floor1 and1 to insure solidity1 should be secured by bolts to a joist1 as shown in the elevation4 ,the rail is attached to the newels in the usual manner1 with handrail bolts or other suitable device4 ,the upper newel should be made fast to the joists as shown1 either by bolts or in some other efficient manner4 ,the intermediate newels are left square on the shank below the stairs1 and may be fastened in the floor below either by mortise and tenon or by making use of joint bolts4 ,illustration3 ,fig4 #ee4 ,elevation of ,open-,newel ,stair ,shown in ,plan in ,fig4 #ed4 ,everything about a stair should be made solid and sound2 and every joint should set firmly and closely2 or a shaky1 rickety1 squeaky stair will be the result1 which is an abomination4 ,stairs with ,curved ,turns4 ,sufficient examples of stairs having angles of greater or less degree at the turn or change of direction1 to enable the student to build any stair of this class1 have now been given4 ,there are1 however1 other types of stairs in common use1 whose turns are curved1 and in which newels are employed only at the foot1 and sometimes at the finish of the flight4 ,these curved turns may be any part of a circle1 according to the requirements of the case1 but turns of a quarter-circle or half-circle are the more common4 ,the string forming the curve is called a cylinder1 or part of a cylinder1 as the case may be4 ,the radius of this circle or cylinder may be any length1 according to the space assigned for the stair4 ,the opening around which the stair winds is called the well-hole4 ,illustration3 ,fig4 #ef4 ,stair ,serving for ,two ,flights1 with ,mid-,floor ,landing4 ,fig4 #ef shows a portion of a stairway having a well-hole with a #g-;inch radius4 ,this stair is rather peculiar1 as it shows a quarter-space landing1 and a quarter-space having three winders4 ,the reason for this is the fact that the landing is on a level with the floor of another room1 into which a door opens from the landing4 ,this is a problem very often met with in practical work1 where the main stair is often made to do the work of two flights because of one floor being so much lower than another4 ,a curved stair1 sometimes called a geometrical stair1 is shown in ,fig4 #eg1 containing seven winders in the cylinder or well-hole1 the first and last aligning with the diameter4 ,in ,fig4 #eh is shown another example of this kind of stair1 containing nine winders in the well-hole1 with a circular wall-string4 ,it is not often that stairs are built in this fashion1 as most stairs having a circular well-hole finish against the wall in a manner similar to that shown in ,fig4 #eg4 ,illustration3 ,fig4 #eg4 ,geometrical ,stair with ,seven ,winders4 ,sometimes1 however1 the workman will be confronted with a plan such as shown in ,fig4 #eh2 and he should know how to lay out the wall-string4 ,in the elevation1 ,fig4 #eh1 the string is shown to be straight1 similar to the string of a common straight flight4 ,this results from having an equal width in the winders along the wall-string1 and1 as we have of necessity an equal width in the risers1 the development of the string is merely a straight piece of board1 as in an ordinary straight flight4 ,in laying out the string1 all we have to do is to make a common pitch-board1 and1 with it as a templet1 mark the lines of the treads and risers on a straight piece of board1 as shown at #a1 #b1 #c1 #d1 etc4 ,if you can manage to bend the string without kerfing 7grooving71 it will be all the better2 if not1 the kerfs 7grooves7 must be parallel to the rise4 ,you can set out with a straight edge1 full size1 on a rough platform1 just as shown in the diagram2 and when the string is bent and set in place1 the risers and winders will have their correct positions4 ,to bend these strings or otherwise prepare them for fastening against the wall1 perhaps the easiest way is to saw the string with a fine saw1 across the face1 making parallel grooves4 ,this method of bending is called kerfing1 above referred to4 ,the kerfs or grooves must be cut parallel to the lines of the risers1 so as to be vertical when the string is in place4 ,this method1 however--handy though it may be--is not a good one1 inasmuch as the saw groove will show more or less in the finished work4 ,illustration3 ,fig4 #eh4 ,plan of ,circular ,stair and ,layout of ,wall ,string for ,same4 ,another method is to build up or stave the string4 ,there are several ways of doing this4 ,in one1 comparatively narrow pieces are cut to the required curve or to portions of it1 and are fastened together1 edge to edge1 with glue and screws1 until the necessary width is obtained 7see ,fig4 #ei74 ,the heading joints may be either butted or beveled1 the latter being stronger1 and should be cross-tongued4 ,fig4 #fj shows a method that may be followed when a wide string is required1 or a piece curved in the direction of its width is needed for any purpose4 ,the pieces are stepped over each other to suit the desired curve2 and though shown square-edged in the figure1 they are usually cut beveled1 as then1 by reversing them1 two may be cut out of a batten4 ,illustration3 ,fig4 #ei4 ,fig4 #fj4 ,methods of ,building ,up ,strings4 ,panels and quick sweeps for similar purposes are obtained in the manner shown in ,fig4 #fa1 by joining up narrow boards edge to edge at a suitable bevel to give the desired curve4 ,the internal curve is frequently worked approximately1 before gluing up4 ,the numerous joints incidental to these methods limit their uses to painted or unimportant work4 ,illustration3 ,fig4 #fa4 ,building ,up a ,curved ,panel or ,quick ,sweep4 ,in ,fig4 #fb is shown a wreath-piece or curved portion of the outside string rising around the cylinder at the half-space4 ,this is formed by reducing a short piece of string to a veneer between the springings2 bending it upon a cylinder made to fit the plan2 then1 when it is secured in position1 filling up the back of the veneer with staves glued across it2 and1 finally1 gluing a piece of canvas over the whole4 ,the appearance of the wreath-piece after it has been built up and removed from the cylinder is indicated in ,fig4 #fc4 ,the canvas back has been omitted to show the staving2 and the counter-wedge key used for connecting the wreath-piece with the string is shown4 ,the wreath-piece is1 at this stage1 ready for marking the outlines of the steps4 ,fig4 #fb also shows the drum or shape around which strings may be bent1 whether the strings are formed of veneers1 staved1 or kerfed4 ,another drum or shape is shown in ,fig4 #fd4 ,in this1 a portion of a cylinder is formed in the manner clearly indicated2 and the string1 being set out on a veneer board sufficiently thin to bend easily1 is laid down round the curve1 such a number of pieces of like thickness being then added as will make the required thickness of the string4 ,in working this method1 glue is introduced between the veneers1 which are then quickly strained down to the curved piece with hand screws4 ,a string of almost any length can be formed in this way1 by gluing a few feet at a time1 and when that dries1 removing the cylindrical curve and gluing down more1 until the whole is completed4 ,several other methods will suggest themselves to the workman1 of building up good1 solid1 circular strings4 ,illustration3 ,fig4 #fb4 ,wreath-,piece ,bent around ,cylinder4 ,illustration3 ,fig4 #fc4 ,completed ,wreath-,piece ,removed from ,cylinder4 ,illustration3 ,fig4 #fd4 ,another ,drum or ,shape for ,building ,curved ,strings4 ,illustration3 ,fig4 #fe4 ,laying ,out ,treads and ,risers around a ,drum4 ,one method of laying out the treads and risers around a cylinder or drum1 is shown in ,fig4 #fe4 ,the line ,d shows the curve of the rail4 ,the lines showing treads and risers may be marked off on the cylinder1 or they may be marked off after the veneer is bent around the drum or cylinder4 ,there are various methods of making inside cylinders or wells1 and of fastening same to strings4 ,one method is shown in ,fig4 #ff4 ,this gives a strong joint when properly made4 ,it will be noticed that the cylinder is notched out on the back2 the two blocks shown at the back of the offsets are wedges driven in to secure the cylinder in place1 and to drive it up tight to the strings4 ,fig4 #fg shows an #h-;inch well-hole with cylinder complete2 also the method of trimming and finishing same4 ,the cylinder1 too1 is shown in such a manner that its construction will be readily understood4 ,stairs having a cylindrical or circular opening always require a weight support underneath them4 ,this support1 which is generally made of rough lumber1 is called the carriage1 because it is supposed to carry any reasonable load that may be placed upon the stairway4 ,fig4 #fh shows the under side of a half-space stair having a carriage beneath it4 ,the timbers marked ,s are of rough stuff1 and may be #b-;inch by #f-;inch or of greater dimensions4 ,if they are cut to fit the risers and treads1 they will require to be at least #b-;inch by #h-;inch4 ,in preparing the rough carriage for the winders1 it will be best to let the back edge of the tread project beyond the back of the riser so that it forms a ledge as shown under ,c in ,fig4 #fi4 ,then fix the cross-carriage pieces under the winders1 with the back edge about flush with the backs of risers1 securing one end to the well with screws1 and the other to the wall string or the wall4 ,now cut short pieces1 marked ,o ,o 7,fig4 #fh71 and fix them tightly in between the cross-carriage and the back of the riser as at ,b ,b in the section1 ,fig4 #fi4 ,these carriages should be of #c-;inch by #b-;inch material4 ,now get a piece of wood1 #a-;inch by #c-;inch1 and cut pieces ,c ,c to fit tightly between the top back edge of the winders 7or the ledge7 and the pieces marked ,b ,b in section4 ,this method makes a very sound and strong job of the winders2 and if the stuff is roughly planed1 and blocks are glued on each side of the short cross-pieces ,o ,o ,o1 it is next to impossible for the winders ever to spring or squeak4 ,when the weight is carried in this manner1 the plasterer will have very little trouble in lathing so that a graceful soffit will be made under the stairs4 ,illustration3 ,fig4 #ff4 ,one ,method of ,making an ,inside ,well4 ,illustration3 ,fig4 #fg4 ,construction and ,trimming of #h-;,inch ,well-,hole4 ,the manner of placing the main stringers of the carriage ,s ,s1 is shown at ,a1 ,fig4 #fi4 ,fig4 #fh shows a complete half-space stair2 one-half of this1 finished as shown1 will answer well for a quarter-space stair4 ,illustration3 ,fig4 #fh4 ,under ,side of ,half-,space ,stair1 with ,carriages and ,cross-,carriages4 ,another method of forming a carriage for a stair is shown in ,fig4 #gj4 ,this is a peculiar but very handsome stair1 inasmuch as the first and the last four steps are parallel1 but the remainder balance or dance4 ,the treads are numbered in this illustration2 and the plan of the handrail is shown extending from the scroll at the bottom of the stairs to the landing on the second story4 ,the trimmer ,t at the top of the stairs is also shown2 and the rough strings or carriages1 ,r ,s1 ,r ,s1 ,r ,s1 are represented by dotted lines4 ,this plan represents a stair with a curtail step1 and a scroll handrail resting over the curve of the curtail step4 ,this type of stair is not now much in vogue in this country1 though it is adopted occasionally in some of the larger cities4 ,the use of heavy newel posts instead of curtail steps1 is the prevailing style at present4 ,in laying out geometrical stairs1 the steps are arranged on principles already described4 ,the well-hole in the center is first laid down and the steps arranged around it4 ,in circular stairs with an open well-hole1 the handrail being on the inner side1 the width of tread for the steps should be set off at about #ah inches from the handrail1 this giving an approximately uniform rate of progress for anyone ascending or descending the stairway4 ,in stairs with the rail on the outside1 as sometimes occurs1 it will be sufficient if the treads have the proper width at the middle point of their length4 ,where a flight of stairs will likely be subject to great stress and wear1 the carriages should be made much heavier than indicated in the foregoing figures2 and there may be cases when it will be necessary to use iron bolts in the sides of the rough strings in order to give them greater strength4 ,this necessity1 however1 will arise only in the case of stairs built in public buildings1 churches1 halls1 factories1 warehouses1 or other buildings of a similar kind4 ,sometimes1 even in house stairs it may be wise to strengthen the treads and risers by spiking pieces of board to the rough string1 ends up1 fitting them snugly against the under side of the tread and the back of the riser4 ,the method of doing this is shown in ,fig4 #ga1 in which the letter ,o shows the pieces nailed to the string4 ,illustration3 ,fig4 #fi4 ,method of ,reinforcing ,stair4 ,illustration3 ,fig4 #gj4 ,plan ,showing ,one ,method of ,constructing ,carriage and ,trimming ,winding ,stair4 ,types of ,stairs in ,common ,use4 ,in order to make the student familiar with types of stairs in general use at the present day1 plans of a few of those most likely to be met with will now be given4 ,fig4 #gb is a plan of a straight stair1 with an ordinary cylinder at the top provided for a return rail on the landing4 ,it also shows a stretch-out stringer at the starting4 ,fig4 #gc is a plan of a stair with a landing and return steps4 ,fig4 #gd is a plan of a stair with an acute angular landing and cylinder4 ,fig4 #ge illustrates the same kind of stair as ,fig4 #gd1 the angle1 however1 being obtuse4 ,fig4 #gf exhibits a stair having a half-turn with two risers on landings4 ,fig4 #gg is a plan of a quarter-space stair with four winders4 ,fig4 #gh shows a stair similar to ,fig4 #gg1 but with six winders4 ,illustration3 ,fig4 #ga4 ,reinforcing ,treads and ,risers by ,blocks ,nailed to ,string4 ,illustration3 ,fig4 #gb4 ,plan of ,straight ,stair with ,cylinder at ,top for ,return ,rail4 ,illustration3 ,fig4 #gc4 ,plan of ,stair with ,landing and ,return ,steps4 ,fig4 #gi shows a stair having five dancing winders4 ,fig4 #hj is a plan of a half-space stair having five dancing winders and a quarter-space landing4 ,fig4 #ha shows a half-space stair with dancing winders all around the cylinder4 ,fig4 #hb shows a geometrical stair having winders all around the cylinder4 ,fig4 #hc shows the plan and elevation of stairs which turn around a central post4 ,this kind of stair is frequently used in large stores and in clubhouses and other similar places1 and has a very graceful appearance4 ,it is not very difficult to build if properly planned4 ,the only form of stair not shown which the student may be called upon to build1 would very likely be one having an elliptical plan2 but1 as this form is so seldom used--being found1 in fact1 only in public buildings or great mansions--it rarely falls to the lot of the ordinary workman to be called upon to design or construct a stairway of this type4 ,illustration3 ,fig4 #gd4 ,plan of ,stair with ,acute-,angle ,landing and ,cylinder4 ,illustration3 ,fig4 #ge4 ,plan of ,stair with ,obtuse-,angle ,landing and ,cylinder4 ,illustration3 ,fig4 #gf4 ,half-,turn ,stair with ,two ,risers on ,landings4 ,illustration3 ,fig4 #gg4 ,quarter-,space ,stair with ,four ,winders4 ,illustration3 ,fig4 #gh4 ,quarter-,space ,stair with ,six ,winders4 ,illustration3 ,fig4 #gi4 ,stair with ,five ,dancing ,winders4 ,illustration3 ,fig4 #hj4 ,half-,space ,stair with ,five ,dancing ,winders and ,quarter-,space ,landing ,illustration3 ,fig4 #ha4 ,half-,space ,stair with ,dancing ,winders all around ,cylinder4 ,,geometrical ,,stairways ,,and ,,hand-,,railing ,the term geometrical is applied to stairways having any kind of curve for a plan4 ,the rails over the steps are made continuous from one story to another4 ,the resulting winding or twisting pieces are called wreaths4 ,illustration3 ,fig4 #hb4 ,geometrical ,stair with ,winders all ,around ,cylinder4 ,wreaths4 ,the construction of wreaths is based on a few geometrical problems--namely1 the projection of straight and curved lines into an oblique plane2 and the finding of the angle of inclination of the plane into which the lines and curves are projected4 ,this angle is called the bevel1 and by its use the wreath is made to twist4 ,illustration3 ,fig4 #hc4 ,plan and ,elevation of ,stairs ,turning around a ,central ,post4 ,in ,fig4 #hd is shown an obtuse-angle plan2 in ,fig4 #he1 an acute-angle plan2 and in ,fig4 #hf1 a semicircle enclosed within straight lines4 ,projection4 ,a knowledge of how to project the lines and curves in each of these plans into an oblique plane1 and to find the angle of inclination of the plane1 will enable the student to construct any and all kinds of wreaths4 ,the straight lines a1 b1 c1 d in the plan1 ,fig4 #hf1 are known as tangents2 and the curve1 the central line of the plan wreath4 ,the straight line across from n to n is the diameter2 and the perpendicular line from it to the lines c and b is the radius4 ,a tangent line may be defined as a line touching a curve without cutting it1 and is made use of in handrailing to square the joints of the wreaths4 ,tangent ,system4 ,the tangent system of handrailing takes its name from the use made of the tangents for this purpose4 ,in ,fig4 #hf1 it is shown that the joints connecting the central line of rail with the plan rails w of the straight flights1 are placed right at the springing2 that is1 they are in line with the diameter of the semicircle1 and square to the side tangents a and d4 ,the center joint of the crown tangents is shown to be square to tangents b and c4 ,when these lines are projected into an oblique plane1 the joints of the wreaths can be made to butt square by applying the bevel to them4 ,all handrail wreaths are assumed to rest on an oblique plane while ascending around a well-hole1 either in connecting two flights or in connecting one flight to a landing1 as the case may be4 ,in the simplest cases of construction1 the wreath rests on an inclined plane that inclines in one direction only1 to either side of the well-hole2 while in other cases it rests on a plane that inclines to two sides4 ,illustration3 ,fig4 #hd4 ,obtuse-,angle ,plan4 ,fig4 #hg illustrates what is meant by a plane inclining in one direction4 ,it will be noticed that the lower part of the figure is a reproduction of the quadrant enclosed by the tangents a and b in ,fig4 #hf4 ,the quadrant1 ,fig4 #hg1 represents a central line of a wreath that is to ascend from the joint on the plan tangent a the height of h above the tangent b4 ,illustration3 ,fig4 #he4 ,acute-,angle ,plan4 ,illustration3 ,fig4 #hf4 ,semicircular ,plan4 ,in ,fig4 #hh1 a view of ,fig4 #hg is given in which the tangents a and b are shown in plan1 and also the quadrant representing the plan central line of a wreath4 ,the curved line extending from a to h in this figure represents the development of the central line of the plan wreath1 and1 as shown1 it rests on an oblique plane inclining to one side only--namely1 to the side of the plan tangent a4 ,the joints are made square to the developed tangents a and m of the inclined plane2 it is for this purpose only that tangents are made use of in wreath construction4 ,they are shown in the figure to consist of two lines1 a and m1 which are two adjoining sides of a developed section 7in this case1 of a square prism71 the section being the assumed inclined plane whereon the wreath rests in its ascent from a to h4 ,the joint at h1 if made square to the tangent m1 will be a true1 square butt-joint2 so also will be the joint at a1 if made square to the tangent a4 ,in practical work it will be required to find the correct geometrical angle between the two developed tangents a and m2 and here1 again1 it may be observed that the finding of the correct angle between the two developed tangents is the essential purpose of every tangent system of handrailing4 ,illustration3 ,fig4 #hg4 ,illustrating ,plane ,inclined in ,one ,direction ,only4 ,illustration3 ,fig4 #hh4 ,plan ,line of ,rail ,projected into ,oblique ,plane ,inclined to ,one ,side ,only4 ,in ,fig4 #hi is shown the geometrical solution--the one necessary to find the angle between the tangents as required on the face-mould to square the joints of the wreath4 ,the figure is shown to be similar to ,fig4 #hg1 except that it has an additional portion marked 8,section40 ,this section is the true shape of the oblique plane whereon the wreath ascends1 a view of which is given in ,fig4 #hh4 ,it will be observed that one side of it is the developed tangent m2 another side1 the developed tangent a 7 a74 ,the angle between the two as here presented is the one required on the face-mould to square the joints4 ,in this example1 ,fig4 #hi1 owing to the plane being oblique in one direction only1 the shape of the section is found by merely drawing the tangent a at right angles to the tangent m1 making it equal in length to the level tangent a in the plan4 ,by drawing lines parallel to a and m respectively1 the form of the section will be found1 its outlines being the projections of the plan lines2 and the angle between the two tangents1 as already said1 is the angle required on the face-mould to square the joints of the wreath4 ,the solution here presented will enable the student to find the correct direction of the tangents as required on the face-mould to square joints1 in all cases of practical work where one tangent of a wreath is level and the other tangent is inclined1 a condition usually met with in level-landing stairways4 ,fig4 #ij exhibits a condition of tangents where the two are equally inclined4 ,the plan here also is taken from ,fig4 #hf4 ,the inclination of the tangents is made equal to the inclination of tangent b in ,fig4 #hf1 as shown at m in ,figs4 #hg1 #hh1 and #hi4 ,illustration3 ,fig4 #hi4 ,finding ,angle between ,tangents4 ,in ,fig4 #ia1 a view of ,fig4 #ij is given1 showing clearly the inclination of the tangents c and d over and above the plan tangents c and d4 ,the central line of the wreath is shown extending along the sectional plane1 over and above its plan lines1 from one joint to the other1 and1 at the joints1 made square to the inclined tangents c and d4 ,it is evident from the view here given1 that the condition necessary to square the joint at each end would be to find the true angle between the tangents c and d1 which would give the correct direction to each tangent4 ,in ,fig4 #ib is shown how to find this angle correctly as required on the face-mould to square the joints4 ,in this figure is shown the same plan as in ,figs4 #ij and #ia1 and the same inclination to the tangents as in ,fig4 #ij1 so that1 except for the portion marked 8,section10 it would be similar to ,fig4 #ij4 ,to find the correct angle for the tangents of the face-mould1 draw the line m from d1 square to the inclined line of the tangents c d2 revolve the bottom inclined tangent c to cut line m in n1 where the joint is shown fixed2 and from this point draw the line c to w4 ,the intersection of this line with the upper tangent d forms the correct angle as required on the face-mould4 ,by drawing the joints square to these two lines1 they will butt square with the rail that is to connect with them1 or to the joint of another wreath that may belong to the cylinder or well-hole4 ,illustration3 ,fig4 #ij4 ,two ,tangents ,equally ,inclined4 ,illustration3 ,fig4 #ia4 ,plan ,lines ,projected into ,oblique ,plane ,inclined to ,two ,sides4 ,illustration3 ,fig4 #ib4 ,finding ,angle between ,tangents4 ,fig4 #ic is another view of these tangents in position placed over and above the plan tangents of the well-hole4 ,it will be observed that this figure is made up of ,figs4 #hh and #ia combined4 ,fig4 #hh1 as here presented1 is shown to connect with a level-landing rail at a4 ,the joint having been made square to the level tangent1 a will butt square to a square end of the level rail4 ,the joint at h is shown to connect the two wreaths and is made square to the inclined tangent m of the lower wreath1 and also square to the inclined tangent c of the upper wreath2 the two tangents1 aligning1 guarantee a square butt-joint4 ,the upper joint is made square to the tangent d1 which is here shown to align with the rail of the connecting flight2 the joint will consequently butt square to the end of the rail of the flight above4 ,the view given in this diagram is that of a wreath starting from a level landing1 and winding around a well-hole1 connecting the landing with a flight of stairs leading to a second story4 ,it is presented to elucidate the use made of tangents to square the joints in wreath construction4 ,the wreath is shown to be in two sections1 one extending from the level-landing rail at a to a joint in the center of the well-hole at h1 this section having one level tangent a and one inclined tangent m2 the other section is shown to extend from h to n1 where it is butt-jointed to the rail of the flight above4 ,illustration3 ,fig4 #ic4 ,laying ,out ,line of ,wreath to ,start from ,level-,landing ,rail4 ,wind around ,well-,hole1 and ,connect at ,landing with ,flight to ,upper ,story4 ,this figure clearly shows that the joint at a of the bottom wreath--owing to the tangent a being level and therefore aligning with the level rail of the landing--will be a true butt-joint2 and that the joint at h1 which connects the two wreaths1 will also be a true butt-joint1 owing to it being made square to the tangent m of the bottom wreath and to the tangent c of the upper wreath1 both tangents having the same inclination2 also the joint at n will butt square to the rail of the flight above1 owing to it being made square to the tangent d1 which is shown to have the same inclination as the rail of the flight adjoining4 ,as previously stated1 the use made of tangents is to square the joints of the wreaths2 and in this diagram it is clearly shown that the way they can be made of use is by giving each tangent its true direction4 ,how to find the true direction1 or the angle between the tangents a and m shown in this diagram1 was demonstrated in ,fig4 #hi2 and how to find the direction of the tangents c and d was shown in ,fig4 #ib4 ,illustration3 ,fig4 #id4 ,tangents ,unfolded to ,find ,their ,inclination4 ,fig4 #id is presented to help further toward an understanding of the tangents4 ,in this diagram they are unfolded2 that is1 they are stretched out for the purpose of finding the inclination of each one over and above the plan tangents4 ,the side plan tangent a is shown stretched out to the floor line1 and its elevation a is a level line4 ,the side plan tangent d is also stretched out to the floor line1 as shown by the arc n m4 ,by this process the plan tangents are now in one straight line on the floor line1 as shown from w to m4 ,upon each one1 erect a perpendicular line as shown1 and from m measure to n1 the height the wreath is to ascend around the well-hole4 ,in practice1 the number of risers in the well-hole will determine this height4 ,illustration3 ,fig4 #ie4 ,well-,hole ,connecting ,two ,flights1 with ,two ,wreath-,pieces1 ,each ,containing ,portions of ,unequal ,pitch4 ,now1 from point n1 draw a few treads and risers as shown2 and along the nosing of the steps1 draw the pitch-line2 continue this line over the tangents d1 c1 and m1 down to where it connects with the bottom level tangent1 as shown4 ,this gives the pitch or inclination to the tangents over and above the well-hole4 ,the same line is shown in ,fig4 #ic1 folded around the well-hole1 from n1 where it connects with the flight at the upper end of the well-hole1 to a1 where it connects with the level-landing rail at the bottom of the well-hole4 ,it will be observed that the upper portion1 from joint n to joint h1 over the tangents c and d1 coincides with the pitch-line of the same tangents as presented in ,fig4 #ib1 where they are used to find the true angle between the tangents as it is required on the face-mould to square the joints of the wreath at h4 ,in ,fig4 #hi the same pitch is shown given to tangent m as in ,fig4 #id2 and in both figures the pitch is shown to be the same as that over and above the upper connecting tangents c and d1 which is a necessary condition where a joint1 as shown at h in ,figs4 #ic and #id1 is to connect two pieces of wreath as in this example4 ,in ,fig4 #id are shown the two face-moulds for the wreaths1 placed upon the pitch-line of the tangents over the well-hole4 ,the angles between the tangents of the face-moulds have been found in this figure by the same method as in ,figs4 #hi and #ib1 which1 if compared with the present figure1 will be found to correspond1 excepting only the curves of the face-moulds in ,fig4 #id4 ,the foregoing explanation of the tangents will give the student a fairly good idea of the use made of tangents in wreath construction4 ,the treatment1 however1 would not be complete if left off at this point1 as it shows how to handle tangents under only two conditions--namely1 first1 when one tangent inclines and the other is level1 as at a and m2 second1 when both tangents incline1 as shown at c and d4 ,illustration3 ,fig4 #if4 ,finding ,angle between ,tangents for ,bottom ,wreath of ,fig4 #ie4 ,illustration3 ,fig4 #ig4 ,finding ,angle between ,tangents for ,upper ,wreath of ,fig4 #ie4 ,in ,fig4 #ie is shown a well-hole connecting two flights1 where two portions of unequal pitch occur in both pieces of wreath4 ,the first piece over the tangents a and b is shown to extend from the square end of the straight rail of the bottom flight1 to the joint in the center of the well-hole1 the bottom tangent a in this wreath inclining more than the upper tangent b4 ,the other piece of wreath is shown to connect with the bottom one at the joint h in the center of the well-hole1 and to extend over tangents c and d to connect with the rail of the upper flight4 ,the relative inclination of the two tangents in this wreath1 is the reverse of that of the two tangents of the lower wreath4 ,in the lower piece1 the bottom tangent a1 as previously stated1 inclines considerably more than does the upper tangent b2 while in the upper piece1 the bottom tangent c inclines considerably less than the upper tangent d4 ,the question may arise3 ,what causes this8 ,is it for variation in the inclination of the tangents over the well-hole8 ,it is simply owing to the tangents being used in handrailing to square the joints4 ,the inclination of the bottom tangent a of the bottom wreath is clearly shown in the diagram to be determined by the inclination of the bottom flight4 ,the joint at a is made square to both the straight rail of the flight and to the bottom tangent of the wreath2 the rail and tangent1 therefore1 must be equally inclined1 otherwise the joint will not be a true butt-joint4 ,the same remarks apply to the joint at #e1 where the upper wreath is shown jointed to the straight rail of the upper flight4 ,in this case1 tangent d must be fixed to incline conformably to the inclination of the upper rail2 otherwise the joint at #e will not be a true butt-joint4 ,illustration3 ,fig4 #ih4 ,diagram of ,tangents and ,face-,mould for ,stair with ,well-,hole at ,upper ,landing4 ,the same principle is applied in determining the pitch or inclination over the crown tangents b and c4 ,owing to the necessity of jointing the two wreaths1 as shown at h1 these two tangents must have the same inclination1 and therefore must be fixed1 as shown from #b to #d1 over the crown of the well-hole4 ,the tangents as here presented are those of the elevation1 not of the face-mould4 ,tangent a is the elevation of the side plan tangent a2 tangents b and c are shown to be the elevations of the plan tangents b and c2 so1 also1 is the tangent d the elevation of the side plan tangent d4 ,illustration3 ,fig4 #ii4 ,drawing ,mould when ,one ,tangent ,is ,level and ,one ,inclined over ,right-,angled ,plan4 ,if this diagram were folded1 as ,fig4 #id was shown to be in ,fig4 #ic1 the tangents of the elevation--namely1 a1 b1 c1 d--would stand over and above the plan tangents a1 b1 c1 d of the well-hole4 ,in practical work1 this diagram must be drawn full size4 ,it gives the correct length to each tangent as required on the face-mould1 and furnishes also the data for the layout of the mould4 ,fig4 #if shows how to find the angle between the tangents of the face-mould for the bottom wreath1 which1 as shown in ,fig4 #ie1 is to span over the first plan quadrant a b4 ,the elevation tangents a and b1 as shown1 will be the tangents of the mould4 ,to find the angle between the tangents1 draw the line a h in ,fig4 #if2 and from a1 measure to #b the length of the bottom tangent a in ,fig4 #ie2 the length from #b to h1 ,fig4 #if1 will equal the length of the upper tangent b1 ,fig4 #ie4 ,from #b to #a1 measure a distance equal to #b-a in ,fig4 #ie1 the latter being found by dropping a perpendicular from w to meet the tangent b extended4 ,upon #a1 erect a perpendicular line2 and placing the dividers on #b1 extend to a2 turn over to the perpendicular at a2 connect this point with #b1 and the line will be the bottom tangent as required on the face-mould4 ,the upper tangent will be the line #b-;h1 and the angle between the two lines is shown at #b4 ,make the joint at h square to #b-;h1 and at a square to a-#b4 ,illustration3 ,fig4 #ajj4 ,plan of ,curved ,steps and ,stringer at ,bottom of ,stair4 ,the mould as it appears in ,fig4 #if is complete1 except the curve1 which is comparatively a small matter to put on1 as will be shown further on4 ,the main thing is to find the angle between the tangents1 which is shown at #b1 to give them the direction to square the joints4 ,in ,fig4 #ig is shown how to find the angle between the tangents c and d shown in ,fig4 #ie1 as required on the face-mould4 ,on the line h-#e1 make h-#d equal to the length of the bottom tangent of the wreath1 as shown at h-#d in ,fig4 #ie2 and #d-e equal to the length of the upper tangent d4 ,measure from #d the distance shown at #d-f in ,fig4 #ie1 and place it from #d to #f as shown in ,fig4 #ig2 upon #f erect a perpendicular line4 ,now place the dividers on #d2 extend to h2 turn over to cut the perpendicular in h2 connect this point with #d1 and the angle shown at #d will be the angle required to square the joints of the wreath as shown at h and #e1 where the joint at #e is shown drawn square to the line #d-e1 and the joint at h square to the line #d h4 ,fig4 #ih is a diagram of tangents and face-mould for a stairway having a well-hole at the top landing4 ,the tangents in this example will be two equally inclined tangents for the bottom wreath2 and for the top wreath1 one inclined and one level1 the latter aligning with the level rail of the landing4 ,illustration3 ,fig4 #aja4 ,finding ,angle between ,tangents for ,squaring ,joints of ,ramped ,wreath4 ,the face-mould1 as here presented1 will further help toward an understanding of the layout of face-moulds as shown in ,figs4 #if and #ig4 ,it will be observed that the pitch of the bottom rail is continued from a to b1 a condition caused by the necessity of jointing the wreath to the end of the straight rail at a1 the joint being made square to both the straight rail and the bottom tangent a4 ,from b a line is drawn to d1 which is a fixed point determined by the number of risers in the well-hole4 ,from point d1 the level tangent d #e is drawn in line with the level rail of the landing2 thus the pitch-line of the tangents over the well-hole is found1 and1 as was shown in the explanation of ,fig4 #ie1 the tangents as here presented will be those required on the face-mould to square the joints of the wreath4 ,in ,fig4 #ih the tangents of the face-mould for the bottom wreath are shown to be a and b4 ,to place tangent a in position on the face-mould1 it is revolved1 as shown by the arc1 to m1 cutting a line previously drawn from w square to the tangent b extended4 ,then1 by connecting m to b1 the bottom tangent is placed in position on the face-mould4 ,the joint at m is to be made square to it2 and the joint at c1 the other end of the mould1 is to be made square to the tangent b4 ,illustration3 ,fig4 #ajb4 ,bottom ,steps with ,obtuse-,angle ,plan4 ,illustration3 ,fig4 #ajc4 ,developing ,face-,mould1 ,obtuse-,angle ,plan4 ,the upper piece of wreath in this example is shown to have tangent c inclining1 the inclination being the same as that of the upper tangent b of the bottom wreath1 so that the joint at c1 when made square to both tangents1 will butt square when put together4 ,the tangent d is shown to be level1 so that the joint at #e1 when squared with it1 will butt square with the square end of the level-landing rail4 ,the level tangent is shown revolved to its position on the face-mould1 as from #e to #b4 ,in this last position1 it will be observed that its angle with the inclined tangent c is a right angle2 and it should be remembered that in every similar case where one tangent inclines and one is level over a square-angle plan tangent1 the angle between the two tangents will be a right angle on the face-mould4 ,a knowledge of this principle will enable the student to draw the mould for this wreath1 as shown in ,fig4 #ii1 by merely drawing two lines perpendicular to each other1 as d #e and d c1 equal respectively to the level tangent d #e and the inclined tangent c in ,fig4 #ih4 ,the joint at #e is to be made square to d #e2 and that at c1 to d c4 ,comparing this figure with the face-mould as shown for the upper wreath in ,fig4 #ih1 it will be observed that both are alike4 ,in practical work the stair-builder is often called upon to deal with cases in which the conditions of tangents differ from all the examples thus far given4 ,an instance of this sort is shown in ,fig4 #ajj1 in which the angles between the tangents on the plan are acute4 ,in all the preceding examples1 the tangents on the plan were at right angles2 that is1 they were square to one another4 ,illustration3 ,fig4 #ajd4 ,cutting ,wreath from ,plank4 ,illustration3 ,fig4 #aje4 ,wreath ,twisted1 ,ready to be ,moulded4 ,fig4 #ajj is a plan of a few curved steps placed at the bottom of a stairway with a curved stringer1 which is struck from a center o4 ,the plan tangents a and b are shown to form an acute angle with each other4 ,the rail above a plan of this design is usually ramped at the bottom end1 where it intersects the newel post1 and1 when so treated1 the bottom tangent a will have to be level4 ,illustration3 ,fig4 #ajf4 ,twisted ,wreath ,raised to ,position1 with ,sides ,plumb4 ,in ,fig4 #aja is shown how to find the angle between the tangents on the face-mould that gives them the correct direction for squaring the joints of the wreath when it is determined to have it ramped4 ,this figure must be drawn full size4 ,usually an ordinary drawing-board will answer the purpose4 ,upon the board1 reproduce the plan of the tangents and curve of the center line of rail as shown in ,fig4 #ajj4 ,measure the height of #e risers1 as shown in ,fig4 #aja1 from the floor line to #e2 and draw the pitch of the flight adjoining the wreath1 from #e to the floor line4 ,from the newel1 draw the dotted line to w1 square to the floor line2 from w1 draw the line w m1 square to the pitch-line b4 ,now take the length of the bottom level tangent on a trammel1 or on dividers if large enough1 and extend it from n to m1 cutting the line drawn previously from w1 at m4 ,connect m to n as shown by the line a4 ,the intersection of this line with b determines the angle between the two tangents a and b of the face-mould1 which gives them the correct direction as required on the face-mould for squaring the joints4 ,the joint at m is made square to tangent a2 and the joint at #e1 to tangent b4 ,illustration3 ,fig4 #ajg4 ,finding ,bevel1 ,bottom ,tangent ,inclined1 ,top ,one ,level4 ,illustration3 ,fig4 #ajh4 ,application of ,bevels in ,fitting ,wreath to ,rail4 ,in ,fig4 #ajb is presented an example of a few steps at the bottom of a stairway in which the tangents of the plan form an obtuse angle with each other4 ,the curve of the central line of the rail in this case will be less than a quadrant1 and1 as shown1 is struck from the center o1 the curve covering the three first steps from the newel to the springing4 ,in ,fig4 #ajc is shown how to develop the tangents of the face-mould4 ,reproduce the tangents and curve of the plan in full size4 ,fix point #c at a height equal to #c risers from the floor line2 at this point place the pitch-board of the flight to determine the pitch over the curve as shown from #c through b to the floor line4 ,from the newel1 draw a line to w1 square to the floor line2 and from w1 square to the pitch-line b1 draw the line w m2 connect m to n4 ,this last line is the development of the bottom plan tangent a2 and the line b is the development of the plan tangent b2 and the angle between the two lines a and b will give each line its true direction as required on the face-mould for squaring the joints of the wreath1 as shown at m to connect square with the newel1 and at #c to connect square to the rail of the connecting flight4 ,illustration3 ,fig4 #aji4 ,face-,mould and ,bevel for ,wreath1 ,bottom ,tangent ,level1 ,top ,one ,inclined4 ,the wreath in this example follows the nosing line of the steps without being ramped as it was in the examples shown in ,figs4 #ajj and #aja4 ,in those figures the bottom tangent a was level1 while in ,fig4 #ajc it inclines equal to the pitch of the upper tangent b and of the flight adjoining4 ,in other words1 the method shown in ,fig4 #aja is applied to a construction in which the wreath is ramped2 while in ,fig4 #ajc the method is applicable to a wreath following the nosing line all along the curve to the newel4 ,the stair-builder is supposed to know how to construct a wreath under both conditions1 as the conditions are usually determined by the ,architect4 ,illustration3 ,fig4 #aaj4 ,finding ,bevels for ,wreath with ,two ,equally ,inclined ,tangents4 ,the foregoing examples cover all conditions of tangents that are likely to turn up in practice1 and1 if clearly understood1 will enable the student to lay out the face-moulds for all kinds of curves4 ,bevels to ,square the ,wreaths4 ,the next process in the construction of a wreath that the handrailer will be called upon to perform1 is to find the bevels that will1 by being applied to each end of it1 give the correct angle to square or twist it when winding around the well-hole from one flight to another flight1 or from a flight to a landing1 as the case may be4 ,illustration3 ,fig4 #aaa4 ,application of ,bevels to ,wreath ,ascending on ,plane ,inclined ,equally in ,two ,directions4 ,illustration3 ,fig4 #aab4 ,finding ,bevel ,where ,upper ,tangent ,inclines ,more ,than ,lower ,one4 ,the wreath is first cut from the plank square to its surface as shown in ,fig4 #ajd4 ,after the application of the bevels1 it is twisted1 as shown in ,fig4 #aje1 ready to be moulded2 and when in position1 ascending from one end of the curve to the other end1 over the inclined plane of the section around the well-hole1 its sides will be plumb1 as shown in ,fig4 #ajf at b4 ,in this figure1 as also in ,fig4 #aje1 the wreath a lies in a horizontal position in which its sides appear to be out of plumb as much as the bevels are out of plumb4 ,in the upper part of the figure1 the wreath b is shown placed in its position upon the plane of the section1 where its sides are seen to be plumb4 ,it is evident1 as shown in the relative position of the wreath in this figure1 that1 if the bevel is the correct angle of the plane of the section whereon the wreath b rests in its ascent over the well-hole1 the wreath will in that case have its sides plumb all along when in position4 ,it is for this purpose that the bevels are needed4 ,illustration3 ,fig4 #aac4 ,finding ,bevel ,where ,upper ,tangent ,inclines ,less ,than ,lower ,one4 ,a method of finding the bevels for all wreaths 7which is considered rather difficult7 will now be explained3 ,first ,case4 ,in ,fig4 #ajg is shown a case where the bottom tangent of a wreath is inclining1 and the top one level1 similar to the top wreath shown in ,fig4 #ih4 ,it has already been noted that the plane of the section for this kind of wreath inclines to one side only2 therefore one bevel only will be required to square it1 which is shown at d1 ,fig4 #ajg4 ,a view of this plane is given in ,fig4 #ajh2 and the bevel d1 as there shown1 indicates the angle of the inclination1 which also is the bevel required to square the end d of the wreath4 ,the bevel is shown applied to the end of the landing rail in exactly the same manner in which it is to be applied to the end of the wreath4 ,the true bevel for this wreath is found at the upper angle of the pitch-board4 ,at the end a1 as already stated1 no bevel is required1 owing to the plane inclining in one direction only4 ,fig4 #aji shows a face-mould and bevel for a wreath with the bottom tangent level and the top tangent inclining1 such as the piece at the bottom connecting with the landing rail in ,fig4 #id4 ,illustration3 ,fig4 #aad4 ,finding ,bevel ,where ,tangents ,incline ,equally over ,obtuse-,angle ,plan4 ,illustration3 ,fig4 #aae4 ,same ,plan as in ,fig4 #aad1 but with ,bottom ,tangent ,level4 ,second ,case4 ,it may be required to find the bevels for a wreath having two equally inclined tangents4 ,an example of this kind also is shown in ,fig4 #id1 where both the tangents c and d of the upper wreath incline equally4 ,two bevels are required in this case1 because the plane of the section is inclined in two directions2 but1 owing to the inclinations being alike1 it follows that the two will be the same4 ,they are to be applied to both ends of the wreath1 and1 as shown in ,fig4 #aje1 in the same direction--namely1 toward the inside of the wreath for the bottom end1 and toward the outside for the upper end4 ,illustration3 ,fig4 #aaf4 ,finding ,bevels for ,wreath of ,fig4 #aae4 ,in ,fig4 #aaj the method of finding the bevels is shown4 ,a line is drawn from w to c1 square to the pitch of the tangents1 and turned over to the ground line at h1 which point is connected to a as shown4 ,the bevel is at h4 ,to show that equal tangents have equal bevels1 the line m is drawn1 having the same inclination as the bottom tangent c1 but in another direction4 ,place the dividers on o1 and turn to touch the lines d and m1 as shown by the semicircle4 ,the line from o to n is equal to the side plan tangent w a1 and both the bevels here shown are equal to the one already found4 ,they represent the angle of inclination of the plane whereon the wreath ascends1 a view of which is given in ,fig4 #aaa1 where the plane is shown to incline equally in two directions4 ,at both ends is shown a section of a rail2 and the bevels are applied to show how1 by means of them1 the wreath is squared or twisted when winding around the well-hole and ascending upon the plane of the section4 ,the view given in this figure will enable the student to understand the nature of the bevels found in ,fig4 #aaj for a wreath having two equally inclined tangents2 also for all other wreaths of equally inclined tangents1 in that every wreath in such case is assumed to rest upon an inclined plane in its ascent over the well-hole1 the bevel in every case being the angle of the inclined plane4 ,illustration3 ,fig4 #aag4 ,upper ,tangent ,inclined4 ,lower ,tangent ,level1 ,over ,acute-,angle ,plan4 ,third ,case4 ,in this example1 two unequal tangents are given1 the upper tangent inclining more than the bottom one4 ,the method shown in ,fig4 #aaj to find the bevels for a wreath with two equal tangents1 is applicable to all conditions of variation in the inclination of the tangents4 ,in ,fig4 #aab is shown a case where the upper tangent d inclines more than the bottom one c4 ,the method in all cases is to continue the line of the upper tangent d1 ,fig4 #aab1 to the ground line as shown at n2 from n1 draw a line to a1 which will be the horizontal trace of the plane4 ,now1 from o1 draw a line parallel to a n1 as shown from o to d1 upon d1 erect a perpendicular line to cut the tangent d1 as shown1 at m2 and draw the line m u o4 ,make u o equal to the length of the plan tangent as shown by the arc from o4 ,put one leg of the dividers on u2 extend to touch the upper tangent d1 and turn over to #a2 connect #a to o2 the bevel at #a is to be applied to tangent d4 ,again place the dividers on u2 extend to the line h1 and turn over to #b as shown2 connect #b to o1 and the bevel shown at #b will be the one to apply to the bottom tangent c4 ,it will be observed that the line h represents the bottom tangent4 ,it is the same length and has the same inclination4 ,an example of this kind of wreath was shown in ,fig4 #ie1 where the upper tangent d is shown to incline more than the bottom tangent c in the top piece extending from h to #e4 ,bevel #a1 found in ,fig4 #aab1 is the real bevel for the end #e2 and bevel #b1 for the end h of the wreath shown from h to #e in ,fig4 #ie4 ,illustration3 ,fig4 #aah4 ,finding ,bevels for ,wreath of ,plan1 ,fig4 #aag4 ,fourth ,case4 ,in ,fig4 #aac is shown how to find the bevels for a wreath when the upper tangent inclines less than the bottom tangent4 ,this example is the reverse of the preceding one2 it is the condition of tangents found in the bottom piece of wreath shown in ,fig4 #ie4 ,to find the bevel1 continue the upper tangent b to the ground line1 as shown at n2 connect n to a1 which will be the horizontal trace of the plane4 ,from o1 draw a line parallel to n a1 as shown from o to d2 upon d1 erect a perpendicular line to cut the continued portion of the upper tangent b in m2 from m1 draw the line m u o across as shown4 ,now place the dividers on u2 extend to touch the upper tangent1 and turn over to #a1 connect #a to o2 the bevel at #a will be the one to apply to the tangent b at h1 where the two wreaths are shown connected in ,fig4 #ie4 ,again place the dividers on u2 extend to touch the line c2 turn over to #b2 connect #b to o2 the bevel at #b is to be applied to the bottom tangent a at the joint where it is shown to connect with the rail of the flight4 ,fifth ,case4 ,in this case we have two equally inclined tangents over an obtuse-angle plan4 ,in ,fig4 #ajb is shown a plan of this kind2 and in ,fig4 #ajc1 the development of the face-mould4 ,in ,fig4 #aad is shown how to find the bevel4 ,from a1 draw a line to a1 square to the ground line4 ,place the dividers on a2 extend to touch the pitch of tangents1 and turn over as shown to m2 connect m to a4 ,the bevel at m will be the only one required for this wreath1 but it will have to be applied to both ends1 owing to the two tangents being inclined4 ,sixth ,case4 ,in this case we have one tangent inclining and one tangent level1 over an acute-angle plan4 ,illustration3 ,fig4 #aai4 ,laying ,out ,curves on ,face-,mould with ,pins and ,string4 ,in ,fig4 #aae is shown the same plan as in ,fig4 #aad2 but in this case the bottom tangent a is to be a level tangent4 ,probably this condition is the most commonly met with in wreath construction at the present time4 ,a small curve is considered to add to the appearance of the stair and rail2 and consequently it has become almost a 8fad0 to have a little curve or stretch-out at the bottom of the stairway1 and in most cases the rail is ramped to intersect the newel at right angles instead of at the pitch of the flight4 ,in such a case1 the bottom tangent a will have to be a level tangent1 as shown at a in ,fig4 #aae1 the pitch of the flight being over the plan tangent b only4 ,to find the bevels when tangent b inclines and tangent a is level1 make a c in ,fig4 #aaf equal to a c in ,fig4 #aae4 ,this line will be the base of the two bevels4 ,upon a1 erect the line a w m at right angles to a c2 make a w equal to o w in ,fig4 #aae2 connect w and c2 the bevel at w will be the one to apply to tangent b at n where the wreath is joined to the rail of the flight4 ,again1 make a m in ,fig4 #aaf equal the distance shown in ,fig4 #aae between w and m1 which is the full height over which tangent b is inclined2 connect m to c in ,fig4 #aaf1 and at m is the bevel to be applied to the level tangent a4 ,illustration3 ,fig4 #abj4 ,simple ,method of ,drawing ,curves on ,face-,mould4 ,illustration3 ,fig4 #aba4 ,tangents1 ,bevels1 ,mould-,curves1 etc41 from ,bottom ,wreath of ,fig4 #ie1 ,in which ,upper ,tangent ,inclines ,less than ,lower ,one4 ,seventh ,case4 ,in this case1 illustrated in ,fig4 #aag1 the upper tangent b is shown to incline1 and the bottom tangent a to be level1 over an acute-angle plan4 ,the plan here is the same as that in ,fig4 #ajj1 where a curve is shown to stretch out from the line of the straight stringer at the bottom of a flight to a newel1 and is large enough to contain five treads1 which are gracefully rounded to cut the curve of the central line of rail in #a1 #b1 #c1 #d4 ,this curve also may be used to connect a landing rail to a flight1 either at top or bottom1 when the plan is acute-angled1 as will be shown further on4 ,illustration3 ,fig4 #abb4 ,developed ,section of ,plane ,inclining ,unequally in ,two ,directions4 ,illustration3 ,fig4 #abc4 ,arranging ,risers around ,well-,hole on ,level ,landing ,stair1 with ,radius of ,central ,line of ,rail ,one-,half ,width of ,tread4 ,to find the bevels--for there will be two bevels necessary for this wreath1 owing to one tangent b being inclined and the other tangent a being level--make a c1 ,fig4 #aah1 equal to a c in ,fig4 #aag1 which is a line drawn square to the ground line from the newel and shown in all preceding figures to have been used for the base of a triangle containing the bevel4 ,make a w in ,fig4 #aah equal to w o in ,fig4 #aag1 which is a line drawn square to the inclined tangent b from w2 connect w and c in ,fig4 #aah4 ,the bevel shown at w will be the one to be applied to the joint #e on tangent b1 ,fig4 #aag4 ,again1 make a m in ,fig4 #aah equal to the distance shown in ,fig4 #aag between the line representing the level tangent and the line m #e1 which is the height that tangent b is shown to rise2 connect m to c in ,fig4 #aah2 the bevel shown at m is to be applied to the end that intersects with the newel as shown at m in ,fig4 #aag4 ,the wreath is shown developed in ,fig4 #aja for this case2 so that1 with ,fig4 #ajj for plan1 ,fig4 #aja for the development of the wreath1 and ,figs4 #aag and #aah for finding the bevels1 the method of handling any similar case in practical work can be found4 ,how to ,put the ,curves on the ,face-,mould4 ,it has been shown how to find the angle between the tangents of the face-mould1 and that the angle is for the purpose of squaring the joints at the ends of the wreath4 ,in ,fig4 #aai is shown how to lay out the curves by means of pins and a string--a very common practice among stair-builders4 ,in this example the face-mould has equal tangents as shown at c and d4 ,the angle between the two tangents is shown at m as it will be required on the face-mould4 ,in this figure a line is drawn from m parallel to the line drawn from h1 which is marked in the diagram as 8,directing ,ordinate of ,section40 ,the line drawn from m will contain the minor axes2 and a line drawn through the corner of the section at #c will contain the major axes of the ellipses that will constitute the curves of the mould4 ,illustration3 ,fig4 #abd4 ,arrangement of ,risers ,around ,well-,hole with ,radius ,larger ,than ,one-,half ,width of ,tread4 ,illustration3 ,fig4 #abe4 ,arrangement of ,risers around ,well-,hole1 with ,risers ,spaced ,full ,width of ,tread4 ,illustration3 ,fig4 #abf4 ,plan of ,stair ,shown in ,fig4 #abc4 ,illustration3 ,fig4 #abg4 ,plan of ,stair ,shown in ,fig4 #abd4 ,illustration3 ,fig4 #abh4 ,plan of ,stair ,shown in ,fig4 #abe4 ,illustration3 ,fig4 #abi4 ,drawing ,face-,mould for ,wreath from ,pitch-,board4 ,the major is to be drawn square to the minor1 as shown4 ,place1 from point #c1 the circle shown on the minor1 at the same distance as the circle in the plan is fixed from the point o4 ,the diameter of this circle indicates the width of the curve at this point ,the width at each end is determined by the bevels4 ,the distance a b1 as shown upon the long edge of the bevel1 is equal to the width of the mould1 and is the hypotenuse of a right-angled triangle whose base is the width of the rail4 ,by placing this dimension on each side of n1 as shown at b and b1 and on each side of h on the other end of the mould1 as shown also at b and b1 we obtain the points b #b b on the inside of the curve1 and the points b #a b on the outside4 ,it will now be necessary to find the elliptical curves that will contain these points2 and before this can be done1 the exact length of the minor and major axes respectively must be determined4 ,the length of the minor axis for the inside curve will be the distance shown from #c to #b2 and its length for the outside will be the distance shown from #c to #a4 ,to find the length of the major axis for the inside1 take the length of half the minor for the inside on the dividers3 place one leg on b1 extend to cut the major in z1 continue to the minor as shown at k4 ,the distance from b to k will be the length of the semi-major axis for the inside curve4 ,illustration3 ,fig4 #acj4 ,development of ,face-,mould for ,wreath ,connecting ,rail of ,flight with ,level-,landing ,rail4 ,to draw the curve1 the points or foci where the pins are to be fixed must be found on the major axis4 ,to find these points1 take the length of b k 7which is1 as previously found1 the exact length of the semi-major for the inside curve7 on the dividers2 fix one leg at #b1 and describe the arc ,y1 cutting the major where the pins are shown fixed1 at o and o4 ,now take a piece of string long enough to form a loop around the two and extending1 when tight1 to #b1 where the pencil is placed2 and1 keeping the string tight1 sweep the curve from b to b4 ,illustration3 ,fig4 #aca4 ,arranging ,risers in ,quarter-,turn between ,two ,flights4 ,the same method1 for finding the major and foci for the outside curve1 is shown in the diagram4 ,the line drawn from b on the outside of the joint at n1 to w1 is the semi-major for the outside curve2 and the points where the outside pins are shown on the major will be the foci4 ,illustration3 ,fig4 #acb4 ,arrangement of ,risers around ,quarter-,turn ,giving ,tangents ,equal ,pitch with ,connecting ,flight4 ,illustration3 ,fig4 #acc4 ,finding ,bevel for ,wreath of ,plan1 ,fig4 #acb4 ,to draw the curves of the mould according to this method1 which is a scientific one1 may seem a complicated problem2 but once it is understood1 it becomes very simple4 ,a simpler way to draw them1 however1 is shown in ,fig4 #abj4 ,the width on the minor and at each end will have to be determined by the method just explained in connection with ,fig4 #aai4 ,in ,fig4 #abj1 the points b at the ends1 and the points in which the circumference of the circle cuts the minor axis1 will be points contained in the curves1 as already explained4 ,now take a flexible lath2 bend it to touch b1 z1 and b for the inside curve1 and b1 w1 and b for the outside curve4 ,this method is handy where the curve is comparatively flat1 as in the example here shown2 but where the mould has a sharp curvature1 as in case of the one shown in ,fig4 #aja1 the method shown in ,fig4 #aai must be adhered to4 ,illustration3 ,fig4 #acd4 ,well-,hole with ,riser in ,center4 ,tangents of ,face-,mould1 and ,central ,line of ,rail1 ,developed4 ,with a clear knowledge of the above two methods1 the student will be able to put curves on any mould4 ,the mould shown in these two diagrams1 ,figs4 #aai and #abj1 is for the upper wreath1 extending from h to n in ,fig4 #id4 ,a practical handrailer would draw only what is shown in ,fig4 #abj4 ,he would take the lengths of tangents from ,fig4 #id1 and place them as shown at h m and m n4 ,by comparing ,fig4 #abj with the tangents of the upper wreath in ,fig4 #id1 it will be easy for the student to understand the remaining lines shown in ,fig4 #abj4 ,the bevels are shown applied to the mould in ,fig4 #aje1 to give it the twist4 ,in ,fig4 #ajf1 is shown how1 after the rail is twisted and placed in position over and above the quadrant c d in ,fig4 #id1 its sides will be plumb4 ,illustration3 ,fig4 #ace4 ,arrangement of ,risers in ,stair with ,obtuse-,angle ,plan4 ,illustration3 ,fig4 #acf4 ,arrangement of ,risers in ,obtuse-,angle ,plan1 ,giving ,equal ,pitch over ,tangents and ,flights4 ,face-,mould ,developed4 ,in ,fig4 #aba are shown the tangents taken from the bottom wreath in ,fig4 #ie4 ,it was shown how to develop the section and find the angle for the tangents in the face-mould1 in ,fig4 #aac4 ,the method shown in ,fig4 #aai for putting on the curves1 would be the most suitable4 ,illustration3 ,fig4 #acg4 ,arrangement of ,risers in ,flight with ,curve at ,landing4 ,fig4 #aba is presented more for the purposes of study than as a method of construction4 ,it contains all the lines made use of to find the developed section of a plane inclining unequally in two different directions1 as shown in ,fig4 #abb4 ,illustration3 ,fig4 #ach4 ,development of ,face-,moulds for ,plan1 ,fig4 #acg4 ,arrangement of ,risers in and around ,well-,hole4 ,an important matter in wreath construction is to have a knowledge of how to arrange the risers in and around a well-hole4 ,a great deal of labor and material is saved through it2 also a far better appearance to the finished rail may be secured4 ,in level-landing stairways1 the easiest example is the one shown in ,fig4 #abc1 in which the radius of the central line of rail is made equal to one-half the width of a tread4 ,in the diagram the radius is shown to be #e inches1 and the treads #aj inches4 ,the risers are placed in the springing1 as at a and a4 ,the elevation of the tangents by this arrangement will be1 as shown1 one level and one inclined1 for each piece of wreath4 ,when in this position1 there is no trouble in finding the angle of the tangent as required on the face-mould1 owing to that angle1 as in every such case1 being a right angle1 as shown at w2 also no special bevel will have to be found1 because the upper bevel of the pitch-board contains the angle required4 ,the same results are obtained in the example shown in ,fig4 #abd1 in which the radius of the well-hole is larger than half the width of a tread1 by placing the riser a at a distance from c equal to half the width of a tread1 instead of at the springing as in the preceding example4 ,in ,fig4 #abe is shown a case where the risers are placed at a distance from c equal to a full tread1 the effect in respect to the tangents of the face-mould and bevel being the same as in the two preceding examples4 ,in ,fig4 #abf is shown the plan of ,fig4 #abc2 in ,fig4 #abg1 the plan of ,fig4 #abd2 and in ,fig4 #abh1 the plan of ,fig4 #abe4 ,for the wreaths shown in all these figures1 there will be no necessity of springing the plank1 which is a term used in handrailing to denote the twisting of the wreath2 and no other bevel than the one at the upper end of the pitch-board will be required4 ,this type of wreath1 also1 is the one that is required at the top of a landing when the rail of the flight intersects with a level-landing rail4 ,in ,fig4 #abi is shown a very simple method of drawing the face-mould for this wreath from the pitch-board4 ,make a c equal to the radius of the plan central line of rail as shown at the curve in ,fig4 #acj4 ,from where line c c cuts the long side of the pitch-board1 the line c a is drawn at right angles to the long edge1 and is made equal to the length of the plan tangent a c1 ,fig4 #acj4 ,the curve is drawn by means of pins and string or a trammel4 ,in ,fig4 #aca is shown a quarter-turn between two flights4 ,the correct method of placing the risers in and around the curve1 is to put the last one in the first flight and the first one in the second flight one-half a step from the intersection of the crown tangents4 ,by this arrangement1 as shown in ,fig4 #acb1 the pitch-line of the tangents will equal the pitch of the connecting flight1 thus securing the second easiest condition of tangents for the face-mould--namely1 as shown1 two equal tangents4 ,for this wreath1 only one bevel will be needed1 and it is made up of the radius of the plan central line of the rail o c1 ,fig4 #aca1 for base1 and the line #a-b1 ,fig4 #acb1 for altitude1 as shown in ,fig4 #acc4 ,the bevel shown in this figure has been previously explained in ,figs4 #aje and #ajf4 ,it is to be applied to both ends of the wreath4 ,the example shown in ,fig4 #acd is of a well-hole having a riser in the center4 ,if the radius of the plan central line of rail is made equal to one-half a tread1 the pitch of tangents will be the same as of the flights adjoining1 thus securing two equal tangents for the two sections of wreath4 ,in this figure the tangents of the face-mould are developed1 and also the central line of the rail1 as shown over and above each quadrant and upon the pitch-line of tangents4 ,the same method may be employed in stairways having obtuse-angle and acute-angle plans1 as shown in ,fig4 #ace1 in which two flights are placed at an obtuse angle to each other4 ,if the risers shown at a and a are placed one-half a tread from c1 this will produce in the elevation a pitch-line over the tangents equal to that over the flights adjoining1 as shown in ,fig4 #acf1 in which also is shown the face-mould for the wreath that will span over the curve from one flight to another4 ,in ,fig4 #acg is shown a flight having the same curve at a landing4 ,the same arrangement is adhered to respecting the placing of the risers1 as shown at a and a4 ,in ,fig4 #ach is shown how to develop the face-moulds4 ,illustration3 ,,finished ,,roof ,,truss ,,in ,,first ,,presbyterian ,,church1 ,,syracuse1 ,n4 ,y4 ,tracy ,swartwout1 ,architects2 ,ballantyne ,evans1 ,associated4 ,reproduced by courtesy of 8,the ,architectural ,review40 ,,part ,,ii ,,the ,,steel ,,square ,introductory4 ,the ,standard ,steel ,square has a blade #bd inches long and #b inches wide1 and a tongue from #ad to #ah inches long and #a inches wide4 ,the blade is at right angles to the tongue4 ,the face of the square is shown in ,fig4 #a4 ,it is always stamped with the manufacturer's name and number4 ,the reverse is the back 7see ,fig4 #b74 ,the longer arm is the blade2 the shorter arm1 the tongue4 ,in the center of the tongue1 on the face side1 will be found two parallel lines divided into spaces 7see ,fig4 #a72 this is the octagon scale4 ,the spaces will be found numbered #aj1 #bj1 #cj1 #dj1 #ej1 #fj1 and #gj1 when the tongue is #ah inches long4 ,to draw an octagon of #h inches square1 draw an #h inch square and then draw a perpendicular and a horizontal line through its center4 ,to find the length of the octagon side1 place one point of a compass on any of the main divisions of the scale1 and the other point of the compass on the eighth subdivision2 then step this length off on each side of the center lines on the side of the square1 which will give the points from which to draw the octagon lines4 ,the diameter of the octagon must equal in inches the number of spaces taken from the square4 ,on the opposite side of the tongue1 in the center1 will be found the brace rule 7see ,fig4 #c74 ,the fractions denote the rise and run of the brace1 and the decimals the length4 ,for example1 a brace of #cf inches run and #cf inches rise1 has a length of #ej4ia inches2 a brace of #db inches run and #db inches rise1 has a length of #ei4dj inches2 etc4 ,illustration3 ,fig4 #a4 ,face ,side of ,tongue of ,steel ,square1 ,showing ,octagon ,scale4 ,illustration3 ,fig4 #b4 ,back of ,blade of ,steel ,square1 ,showing ,rafter ,table4 ,illustration3 ,fig4 #c4 ,back of ,tongue of ,steel ,square1 ,showing ,brace ,measure4 ,illustration3 ,fig4 #d4 ,back of ,blade of ,steel ,square1 ,showing ,essex ,board ,measure4 ,on the back of the blade 7,fig4 #d7 will be found the board measure1 where eight parallel lines running along the length of the blade are shown and divided at every inch by cross-lines4 ,under #ab1 on the outer edge of the blade1 will be found the various lengths of the boards1 as #h1 #i1 #aj1 #aa1 #ab1 etc4 ,for example1 take a board #ad feet long and #i inches wide4 ,to find the contents1 look under #ab1 and find #ad2 then follow this space along to the cross-line under #i1 the width of the board2 and here is found #aj feet #f inches1 denoting the contents of a board #ad feet long and #i inches wide4 ,to ,find the ,miter and ,length of ,side for any ,polygon1 with the ,steel ,square4 ,in ,fig4 #e is shown a pentagon figure4 ,the miters of the pentagon stand at #gb degrees with each other1 and are found by dividing #cfj by #e1 the number of sides in the pentagon4 ,but the angle when applied to the square to obtain the miter1 is only one-half of #gb1 or #cf degrees1 and intersects the blade at #h-bc_/cb1 as shown in ,fig4 #e4 ,illustration3 ,fig4 #e4 ,use of ,steel ,square to ,find ,miter and ,side of ,pentagon4 ,by squaring up from #f on the tongue1 intersecting the degree line at a1 the center a is determined either for the inscribed or the circumscribed diameter1 the radii being a b and a c1 respectively4 ,the length of the sides will be #h-bc_/cb inches to the foot4 ,illustration3 ,fig4 #f4 ,use of ,steel ,square to ,find ,miter and ,side of ,hexagon4 ,if the length of the inscribed diameter be #h feet1 then the sides would be #h #h-bc_/cb inches4 ,the figures to use for other polygons are as follows3 ,triangle #bj-be_/cb ,square #ab ,hexagon #g ,nonagon #d ,decagon #c ,in ,fig4 #f the same process is used in finding the miter and side of the hexagon polygon4 ,to find the degree line1 #cfj is divided by #f1 the number of sides1 as follows3 #cfj #f #fj2 and #fj #b #cj degrees4 ,now1 from #ab on tongue1 draw a line making an angle of #cj degrees with the tongue4 ,it will cut the blade in #g as shown2 and from #g to m1 the heel of the square1 will be the length of the side4 ,from #f on tongue1 erect a line to cut the degree line in c2 and with c as center1 describe a circle having the radius of c #g2 and around the circle1 complete the hexagon by taking the length #g m with the compass for each side1 as shown4 ,illustration3 ,fig4 #g4 ,use of ,steel ,square to ,find ,miter and ,side of ,octagon4 ,in ,fig4 #g the same process is shown applied to the octagon4 ,the degree line in all the polygons is found by dividing #cfj by the number of sides in the figure3 #cfj #h #de2 and #de #b #bb degrees4 ,this gives the degree line for the octagon4 ,complete the process as was described for the other polygons4 ,by using the following figures for the various polygons1 the miter lines may be found2 but in these figures no account is taken of the relative size of sides to the foot as in the figures preceding3 ,triangle #g in4 and #d in4 ,pentagon #aa #h ,hexagon #d #g ,heptagon #ab #f ,octagon #ag #g ,nonagon #bb #i ,decagon #i #c ,the miter is to be drawn along the line of the first column1 as shown for the triangle in ,fig4 #h1 and for the hexagon in ,fig4 #i4 ,in ,fig4 #aj is shown a diagram for finding degrees on the square4 ,for example1 if a pitch of #ce degrees is required1 use #h-ac_/cb on tongue and #ab on blade2 if #de degrees1 use #ab on tongue and #ab on blade2 etc4 ,illustration3 ,fig4 #h4 ,use of ,square to ,find ,miter of ,equilateral ,triangle4 ,in ,fig4 #aa is shown the relative length of run for a rafter and a hip1 the rafter being #ab inches and the hip #ag inches4 ,the reason1 as shown in this diagram1 why #ag is taken for the run of the hip1 instead of #ab as for the common rafter1 is that the seats of the common rafter and hip do not run parallel with each other1 but diverge in roofs of equal pitch at an angle of #de degrees2 therefore1 #ag inches taken on the run of the hip is equal to only #ab inches when taken on that of the common rafter1 as shown by the dotted line from heel to heel of the two squares in ,fig4 #aa4 ,illustration3 ,fig4 #i4 ,use of ,square to ,find ,miter of ,hexagon4 ,in ,fig4 #ab is shown how other figures on the square may be found for corners that deviate from the #de degrees4 ,it is shown that for a pentagon1 which makes a #cf-;degree angle with the plate1 the figure to be used on the square for run is #ad inches2 for a hexagon1 which makes a #cj-;degree angle with the plate1 the figure will be #ac inches2 and for an octagon1 which makes an angle of #bb degrees with the plate1 the figure to use on the square for run of hip to correspond to the run of the common rafters1 will be #ac inches4 ,it will be observed that the height in each case is #i inches4 ,illustration3 ,fig4 #aj4 ,diagram for ,finding ,pitches of ,various ,degrees by ,means of the ,steel ,square4 ,fig4 #ac illustrates a method of finding the relative height of a hip or valley per foot run to that of the common rafter4 ,the square is shown placed with #ab on blade and #i on tongue for the common rafter2 and shows that for the hip the rise is only #f-g_/af inches4 ,illustration3 ,fig4 #aa4 ,square ,applied to ,determine ,relative ,length of ,run for ,rafter and ,hip4 ,the ,steel ,square as ,applied in ,roof ,framing4 ,roof framing at present is as simple as it possibly can be1 so that any attempt at a new method would be superfluous4 ,there may1 however1 be a certain way of presenting the subject that will carry with it almost the weight assigned to a new theory1 making what is already simple still more simple4 ,the steel square is a mighty factor in roof framing1 and without doubt the greatest tool in practical potency that ever was invented for the carpenter4 ,with its use the lengths and bevels of every piece of timber that goes into the construction of the most intricate design of roof1 can easily be obtained1 and that with but very little knowledge of lines4 ,illustration3 ,fig4 #ab4 ,use of ,square to ,determine ,length of ,run for ,rafters on ,corners ,other than #de4 ,in roofs of equal pitch1 as illustrated in ,fig4 #ad1 the steel square is all that is required if one properly understands how to handle it4 ,what is meant by a pitch of a roof1 is the number of inches it rises to the foot of run4 ,illustration3 ,fig4 #ac4 ,method of ,finding ,relative ,height of ,hip or ,valley per ,foot of ,run to that of ,common ,rafter4 ,in ,fig4 #ae is shown the steel square with figures representing the various pitches to the foot of run4 ,for the -pitch roof1 the figures as shown1 from #ab on tongue to #ab on blade1 are those to be used on the steel square for the common rafter2 and for pitch1 the figures to be used on the square will be #ab and #i1 as shown4 ,illustration3 ,fig4 #ad4 ,diagram to ,illustrate ,use of ,steel ,square ,in ,laying ,out ,timbers of ,roofs of ,equal ,pitch4 ,to understand this figure1 it is necessary only to keep in mind that the pitch of a roof is reckoned from the span4 ,since the run in each pitch as shown is #ab inches1 the span is two times #ab inches1 which equals #bd inches2 hence1 #ab on blade to represent the foot run1 and #ab on tongue to represent the rise over the span1 will be the figures on the square for a -pitch roof4 ,for the pitch1 the figures are shown to be #ab on tongue and #i on blade1 #i being of the span1 #bd inches4 ,the same rule applies to all the pitches4 ,the pitch is shown to rise #d inches to the foot of run1 because #d inches is of the span1 #bd inches1 the pitch is shown to rise #h inches to the foot of run1 because #h inches is of the span1 #bd inches2 etc4 ,illustration3 ,fig4 #ae4 ,steel ,square ,giving ,various ,pitches to ,foot of ,run4 ,the roof referred to in ,figs4 #af and #ag is to rise #i inches to the foot of run2 it is therefore a -pitch roof4 ,for all the common rafters1 the figures to be used on the square will be #ab on blade to represent the run1 and #i on tongue to represent the rise to the foot of run2 and for all the hips and valleys1 #ag on blade to represent the run1 and #i on tongue to represent the rise of the roof to the foot of run4 ,why #ag represents the run for all the hips and valleys1 will be understood by examining ,fig4 #ai1 in which #ag is shown to be the diagonal of a foot square4 ,in equal-pitch roofs the corners are square1 and the plan of the hip or valley will always be a diagonal of a square corner as shown at #a1 #b1 #c1 and #e in ,fig4 #ad4 ,in ,fig4 #ah are shown pitch1 pitch and pitch over a square corner4 ,the figures to be used on the square for the hip1 will be #ag for run in each case4 ,for the pitch1 the figures to be used would be #ag inches run and #d inches rise1 to correspond with the #ab inches run and #d inches rise of the common rafter4 ,for the pitch1 the figures to be used for hip would be #ag inches run and #i inches rise1 to correspond with the #ab inches run and #i inches rise of the common rafter2 and for the pitch1 the figures to be used on the square will be #ag inches run and #ab inches rise1 to correspond with the #ab inches run and #ab inches rise of the common rafter4 ,illustration3 ,fig4 #af4 ,method of ,laying ,out ,common ,rafters of a -,pitch ,roof4 ,it will be observed from above1 that in all cases where the plan of the hip or valley is a diagonal of a square1 the figures to be used on the square for run will be #ag inches2 and for the rise1 whatever the roof rises to the foot of run4 ,it should also be remembered that this is the condition in all roofs of equal pitch1 where the angle of the hip or valley is a #de-;degree angle1 or1 in other words1 where we have the diagonal of a square4 ,it has been shown in ,fig4 #ab how other figures for other plan angles may be found2 and that in each case the figures for run vary according to the plan angle of the hip or valley1 while the figure for the height in each case is similar4 ,illustration3 ,fig4 #ag4 ,method of ,laying ,out ,hips and ,valleys of a -,pitch ,roof4 ,in ,fig4 #ad are shown a variety of runs for common rafters1 but all have the same pitch2 they rise #i inches to the foot of run4 ,the main roof is shown to have a span of #bg feet1 which makes the run of the common rafter #ac feet #f inches4 ,the run of the front wing is shown to be #aj feet #d inches2 and the run of the small gable at the left corner of the front1 is shown to be #h feet4 ,the diversity exhibited in the runs1 and especially the fractional part of a foot shown in two of them1 will afford an opportunity to treat of the main difficulties in laying out roof timbers in roofs of equal pitch4 ,let it be determined to have a rise of #i inches to the foot of run2 and in this connection it may be well to remember that the proportional rise to the foot run for roofs of equal pitch makes not the least difference in the method of treatment4 ,to lay out the common rafters for the main roof1 which has a run of #ac feet #f inches1 proceed as shown in ,fig4 #af4 ,take #ab on the blade and #i on the tongue1 and step #ac times along the rafter timber4 ,this will give the length of rafter for #ac feet of run4 ,in this example1 however1 there is another #f inches of run to cover4 ,for this additional length1 take #f inches on the blade 7it being a foot run7 for run1 and take of #i on the tongue 7which is #d inches71 and step one time4 ,this1 in addition to what has already been found by stepping #ac times with #ab and #i1 will give the full length of the rafter4 ,illustration3 ,fig4 #ah4 ,method of ,laying ,out ,hips and ,rafters for ,roofs or ,various ,pitches over ,square ,corner4 ,the square with #ab on blade and #i on tongue will give the heel and plumb cuts4 ,another method of finding the length of rafter for the #f inches is shown in ,fig4 #af1 where the square is shown applied to the rafter timber for the plumb cut4 ,square ,no4 #a is shown applied with #ab on blade and #i on tongue for the length of the #ac feet4 ,square from this cut1 measure #f inches1 the additional inches in the run2 and to this point move the square1 holding it on the side of the rafter timber with #ab on blade and #i on tongue1 as for a full foot run4 ,it will be observed that this method is easily adapted to find any fractional part of a foot in the length of rafters4 ,in the front gable1 ,fig4 #ad1 the fractional part of a foot is #d inches to be added to #aj feet of run2 therefore1 in that case1 the line shown measured to #f inches in ,fig4 #af would measure only #d inches for the front gable4 ,heel ,cut of ,common ,rafter4 ,in ,fig4 #af is also shown a method to lay out the heel cut of a common rafter4 ,the square is shown applied with #ab on blade and #i on tongue2 and from where the #ab on the square intersects the edge of the rafter timber1 a line is drawn square to the blade as shown by the dotted line from #ab to a4 ,then the thickness of the part of the rafter that is to project beyond the plate to hold the cornice1 is gauged to intersect the dotted line at a2 and from a1 the heel cut is drawn with the square having #ab on blade and #i on tongue1 marking along the blade for the cut4 ,the common rafter for the front wing1 which is shown to have a run of #aj feet #d inches1 is laid out precisely the same1 except that for this rafter the square with #ab on blade and #i on tongue will have to be stepped along the rafter timber only #aj times for the #aj feet of run2 and for the fractional part of a foot 7#d inches7 which is in the run1 either of the two methods already shown for the main rafter may be used4 ,the proportional figures to be used on the square for the #d inches will be #d on blade and #b on tongue2 and if the second method is used1 make the addition to the length of rafter for #aj feet1 by drawing a line #d inches square from the tongue of square ,no4 #a 7see ,fig4 #af71 instead of #f inches as there shown for the main rafter4 ,hips4 ,three of the hips are shown in ,fig4 #ad to extend from the plate to the ridge-pole2 they are marked in the figure as #a1 #b1 and #c respectively1 and are shown in plan to be diagonals of a square measuring #ac feet #f inches by #ac feet #f inches2 they make an angle1 therefore1 of #de degrees with the plate4 ,in ,fig4 #ah it has been shown that a hip standing at an angle of #de degrees with the plate will have a run of #ag inches for every foot run of the common rafter4 ,therefore1 to lay out the hips1 the figures on the square will be #ag for run and #i for rise2 and by stepping #ac times along the hip rafter timber1 the length of hip for #ac feet of run is obtained4 ,the length for the additional #f inches in the run may be found by squaring a distance of #h inches1 as shown in ,fig4 #ag1 from the tongue of the square1 and moving square ,no4 #a along the edge of the timber1 holding the blade on #ag and tongue on #i1 and marking the plumb cut where the dotted line is shown4 ,in ,fig4 #ah is shown how to find the relative run length of a portion of a hip to correspond to that of a fractional part of a foot in the length of the common rafter4 ,from #ab inches1 measure along the run of the common rafter #f inches1 and drop a line to cut the diagonal line in m4 ,from m to a1 along the diagonal line1 will be the relative run length of the part of hip to correspond with #f inches run of the common rafter1 and it measures #h inches4 ,illustration3 ,fig4 #ai4 ,diagram ,showing ,relative ,lengths of ,run for ,hips and ,common ,rafters ,in ,equal-,pitch ,roofs4 ,illustration3 ,fig4 #bj4 ,method of ,determining ,run of ,valley for ,additional ,run in ,common ,rafter4 ,the same results may be obtained by the following method of figuring3 ,as #ab 3 #ag 33 #f #f #ab 7#ajb -------- #h - #f #h ,in ,fig4 #ai is shown a #ab-;inch square1 the diagonal m being #ag inches4 ,by drawing lines from the base a b to cut the diagonal line1 the part of the hip to correspond to that of the common rafter will be indicated on the line #ag4 ,in this figure it is shown that a #f-;inch run on a b1 which represents the run of a foot of a common rafter1 will have a corresponding length of #h inches run on the line #ag1 which represents the plan line of the hip or valley in all equal-pitch roofs4 ,in the front gable1 ,fig4 #ad1 it is shown that the run of the common rafter is #aj feet #d inches4 ,to find the length of the common rafter1 take #ab on blade and #i on tongue1 and step #aj times along the rafter timber2 and for the fractional part of a foot 7#d inches71 proceed as was shown in ,fig4 #af for the rafter of the main roof2 but in this case measure out square to the tongue of square ,no4 #a1 #d inches instead of #f inches4 ,illustration3 ,fig4 #ba4 ,corner of ,square ,building1 ,showing ,plan ,lines of ,plates and ,hip4 ,fig4 #bb4 ,corner of ,square ,building1 ,showing ,plan ,lines of ,plates and ,valley4 ,the additional length for the fractional #d inches run can also be found by taking #d inches on blade and #c inches on tongue of square1 and stepping one time2 this1 in addition to the length obtained by stepping #aj times along the rafter timber with #ab on blade and #i on tongue1 will give the full length of the rafter for a run of #aj feet #d inches4 ,illustration3 ,fig4 #bc4 ,use of ,square to ,determine ,heel ,cut of ,valley4 ,in the intersection of this roof with the main roof1 there are shown to be two valleys of different lengths4 ,the long one extends from the plate at n 7,fig4 #ad7 to the ridge of the main roof at m2 it has therefore a run of #ac feet #f inches4 ,for the length1 proceed as for the hips1 by taking #ag on blade of the square and #i on tongue1 and stepping #ac times for the length of the #ac feet2 and for the fractional #f inches1 proceed precisely as shown in ,fig4 #ag for the hip1 by squaring out from the tongue of square ,no4 #a1 #h inches2 this1 in addition to the length obtained for the #ac feet1 will give the full length of the long valley n m4 ,the length of the short valley a c1 as shown1 extends over the run of #aj feet #d inches1 and butts against the side of the long valley at c4 ,by taking #ag on blade and #i on tongue1 and stepping along the rafter timber #aj times1 the length for the #aj feet is found2 and for the #d inches1 measure #e inches square from the tongue of square ,no4 #a1 in the manner shown in ,fig4 #ag1 where the #h inches is shown added for the #f inches additional run of the main roof for the hips4 ,illustration3 ,fig4 #bd4 ,steel ,square ,applied to ,finding ,bevel for ,fitting ,top of ,hip or ,valley to ,ridge4 ,the length #e is found as shown in ,fig4 #bj1 by measuring #d inches from a to m along the run of common rafter for one foot4 ,upon m erect a line to cut the seat of the valley at c2 from c to a will be the run of the valley to correspond with #d inches run of the common rafter1 and it will measure #e inches4 ,how to ,treat the ,heel ,cut of ,hips and ,valleys4 ,having found the lengths of the hips and valleys to correspond to the common rafters1 it will be necessary to find also the thickness of each above the plate to correspond to the thickness the common rafter will be above the plate4 ,in ,fig4 #ba is shown a corner of a square building1 showing the plates and the plan lines of a hip4 ,the length of the hip1 as already found1 will cover the span from the ridge to the corner #b2 but the sides of the hip intersect the plates at #c and #c respectively2 therefore the distance from #b to #a1 as shown in this diagram1 is measured backwards from a to #a in the manner shown in ,fig4 #ag2 then a plumb line is drawn through #a to m1 parallel to the plumb cut a-#ag4 ,from m to o on this line1 measure the same thickness as that of the common rafter2 and through o draw the heel cut to a as shown4 ,in like manner the thickness of the valley above the plate is found2 but as the valley as shown in the plan figure1 ,fig4 #bb1 projects beyond point #b before it intersects the outside of the plates1 the distance from #b to #a in the case of the valley will have to be measured outwards from #b1 as shown from #b to #a in ,fig4 #bc2 and at the point thus found the thickness of the valley is to be measured to correspond with that of the common rafter as shown at m n4 ,illustration3 ,fig4 #be4 ,steel ,square ,applied to ,jack ,rafter to ,find ,bevel for ,fitting against ,side of ,hip or ,valley4 ,in ,fig4 #bd is shown the steel square applied to a hip or valley timber to cut the bevel that will fit the top end against the ridge4 ,the figures on the square are #ag and #ai4 ,the #ag represents the length of the plan line of the hip or valley for a foot of run1 which1 as was shown in previous figures1 will always be #ag inches in roofs of equal pitch1 where the plan lines stand at #de degrees to the plates and square to each other4 ,the #ai taken on the blade represents the actual length of a hip or valley that will span over a run of #ag inches4 ,the bevel is marked along the blade4 ,the cut across the back of the short valley to fit it against the side of the long valley1 will be a square cut owing to the two plan lines being at right angles to each other4 ,in ,fig4 #be is shown the steel square applied to a jack rafter to cut the back bevel1 to fit it against the side of a hip or valley4 ,the figures on the square are #ab on tongue and #ae on blade1 the #ab representing a foot run of a common rafter1 and the #ae the length of a rafter that will span over a foot run2 marking along the blade will give the bevel4 ,illustration3 ,fig4 #bf4 ,finding ,length to ,shorten ,rafters for ,jacks per ,foot of ,run4 ,the rule in every case to find the back bevel for jacks in roofs of equal pitch1 is to take #ab on the tongue to represent the foot run1 and the length of the rafter for a foot of run on the blade1 marking along the blade in each case for the bevel4 ,in a -pitch roof1 which is the most common in all parts of the country1 the length of rafter for a foot of run will be #ag inches2 hence it will be well to remember that #ab on tongue and #ag on blade1 marking along the blade1 will give the bevel to fit a jack against a hip or a valley in a -pitch roof4 ,in a roof having a rise of #i inches to the foot of run1 such as the one under consideration1 the length of rafter for one foot of run will be #ae inches4 ,the square as shown in ,fig4 #be1 with #ab on tongue and #ae on blade1 will give the bevel by marking along the blade4 ,to find the length of a rafter for a foot of run for any other pitch1 place the two-foot rule diagonally from #ab on the blade of the square to the figure on tongue representing the rise of the roof to the foot of run2 the rule will give the length of the rafter that will span over one foot of run4 ,illustration3 ,fig4 #bg4 ,finding ,length of ,jack ,rafter in -,pitch ,roof4 ,illustration3 ,fig4 #bh4 ,finding ,length of ,jack ,rafter in -,pitch ,roof4 ,the length of rafter for a foot of run will also determine the difference in lengths of jacks4 ,for example1 if a roof rises #ab inches to one foot of run1 the rafter over this span has been found to be #ag inches2 this1 therefore1 is the number of inches each jack is shortened in one foot of run4 ,if the rise of the roof is #h inches to the foot of run1 the length of the rafter is found for one foot of run1 by placing the rule diagonally from #ab on tongue to #h on blade1 which gives #ad inches1 as shown in ,fig4 #bf4 ,this1 therefore1 will be the number of inches the jacks are to be shortened in a roof rising #h inches to the foot of run4 ,if the jacks are placed #bd inches from center to center1 then multiply #ad by #b #bi inches4 ,in ,fig4 #bg is shown how to find the length with the steel square4 ,the square is placed on the jack timber rafter with the figures that have been used to cut the common rafter4 ,in ,fig4 #bg1 #ab on blade and #ab on tongue were the figures used to cut the common rafter1 the roof being pitch1 rising #ab inches to the foot of run4 ,in the diagram it is shown how to find the length of a jack rafter if placed #af inches from center to center4 ,the method is to move the square as shown along the line of the blade until the blade measures #af inches2 the tongue then would be as shown from w to m1 and the length of the jack would be from #ab on blade to m on tongue1 on the edge of the jack rafter timber as shown4 ,this latter method becomes convenient when the space between jacks is less than #ah inches2 but if used when the space is more than #ah inches it will become necessary to use two squares2 otherwise the tongue as shown at m would not reach the edge of the timber4 ,illustration3 ,fig4 #bi4 ,method of ,determining ,length of ,jacks ,between ,hips and ,valleys2 also ,bevels for ,jacks1 ,hips1 and ,valleys4 ,in ,fig4 #bh the same method is shown for finding the length of a jack rafter for a roof rising #i inches to the foot of run1 with the jacks placed #ah inches center to center4 ,the square in this diagram is shown placed on the jack rafter timber with #ab on blade and #i on tongue2 then it is moved forward along the line of the blade to w4 ,the blade1 when in this latter position1 will measure #ah inches4 ,the tongue will meet the edge of the timber at m1 and the distance from m on tongue to #ab on blade will indicate the length of a jack1 or1 in other words1 will show the length each jack is shortened when placed #ah inches between centers in a roof having a pitch of #i inches to the foot of run4 ,illustration3 ,fig4 #cj4 ,method of ,finding ,bevels for ,all ,timbers in ,roofs of ,equal ,pitch4 ,illustration3 ,fig4 #ca4 ,method of ,finding ,bevel #e1 ,fig4 #cj1 for ,fitting ,hip or ,valley ,against ,ridge when not ,backed4 ,when jacks are placed between hips and valleys as shown at #a1 #b1 #c1 #d1 etc41 in ,fig4 #ad1 a better method of treatment is shown in ,fig4 #bi1 where the slope of the roof is projected into the horizontal plane4 ,the distance from the plate in this figure to the ridge m1 equals the length of the common rafter for the main roof4 ,on the plate a n n is made equal to a n n in ,fig4 #ad4 ,by drawing a figure like this to a scale of one inch to one foot1 the length of all the jacks can be measured and also the lengths of the hip and the two valleys4 ,it also gives the bevels for the jacks1 as well as the bevel to fit the hip and valley against the ridge2 but this last bevel must be applied to the hip and valley when backed4 ,it has been shown before1 that the figures to be used on the square for this bevel when the timber is left square on back as is the custom in construction1 are the length of a foot run of a hip or valley1 which is #ag1 on tongue1 and the length of a hip or valley that will span over #ag inches run1 on blade--the blade giving the bevel4 ,fig4 #cj contains all the bevels or cuts that have been treated upon so far1 and1 if correctly understood1 will enable any one to frame any roof of equal pitch4 ,in this figure it is shown that #ab inches run and #i inches rise will give bevels #a and #b1 which are the plumb and heel cuts of rafters of a roof rising #i inches to the foot of run4 ,by taking these figures1 therefore1 on the square1 #i inches on the tongue and #ab inches on the blade1 marking along the tongue will give the plumb cut1 and marking along the blade will give the heel cut4 ,illustration3 ,fig4 #cb4 ,method of ,finding ,back ,bevel #f1 ,fig4 #cj1 for ,jack ,rafters1 and ,bevel #g1 for ,roof-,board4 ,illustration3 ,fig4 #cc4 ,determining ,miter ,cut for ,roof-,board4 ,bevels #c and #d are the plumb and heel cuts for the hip1 and are shown to have the length of the seat of hip for one foot run1 which is #ag inches4 ,by taking #ag inches1 therefore1 on the blade1 and #i inches on the tongue1 marking along the tongue for the plumb cut1 and along the blade for the heel cut1 the plumb and heel cuts are found4 ,bevel #e1 which is to fit the hip or valley against the ridge when not backed1 is shown from o w1 the length of the hip for one foot of run1 which is #ai inches1 and from o s1 which always in roofs of equal pitch will be #ag inches and equal in length to the seat of a hip or valley for one foot of run4 ,these figures1 therefore1 taken on the square1 #ai on the blade1 and #ag on the tongue1 will give the bevel by marking along the blade as shown in ,fig4 #ca1 where the square is shown applied to the hip timber with #ai on blade and #ag on tongue1 the blade showing the cut4 ,bevels #f and #g in ,fig4 #cj are shown formed of the length of the rafter for one foot of run1 which is #ae inches1 and the run of the rafter1 which is #ab inches4 ,these figures are applied on the square1 as shown in ,fig4 #cb1 to a jack rafter timber2 taking #ae on the blade and #ab on the tongue1 marking along the blade will give the back bevel for the jack rafters1 and marking along the tongue will give the face cut of roof-boards to fit along the hip or valley4 ,illustration3 ,fig4 #cd4 ,laying ,out ,timbers of ,one-half ,gable of -,pitch ,roof4 ,it is shown in ,fig4 #cj1 also1 that by taking the length of rafter #ae inches on blade1 and rise of roof #i inches on tongue1 bevel #h will give the miter cut for the roof-boards4 ,in ,fig4 #cc the square is shown applied to a roof-board with #ae on blade1 which is the length of the rafter to one foot of run1 and with #i on tongue1 which is the rise of the roof to the foot run2 marking along the tongue will give the miter for the boards4 ,illustration3 ,fig4 #ce4 ,finding ,backing of ,hip in ,gable ,roof4 ,other uses may be made of these figures1 as shown in ,fig4 #cd1 which is one-half of a gable of a roof rising #i inches to the foot run4 ,the squares at the bottom and the top will give the plumb and heel cuts of the common rafter4 ,the same figures on the square applied to the studding1 marking along the tongue for the cut1 will give the bevel to fit the studding against the rafter2 and by marking along the blade we obtain the cut for the boards that run across the gable4 ,by taking #ai on blade1 which is the length of the hip for one foot of run1 and taking on the tongue the rise of the roof to the foot of run1 which is #i inches1 and applying these as shown in ,fig4 #ce1 we obtain the backing of the hip by marking along the tongue of the two squares1 as shown4 ,it will be observed from what has been said1 that in roofs of equal pitch the figure #ab on the blade1 and whatever number of inches the roof rises to the foot run on the tongue1 will give the plumb and heel cuts for the common rafter2 and that by taking #ag on the blade instead of #ab1 and taking on the tongue the figure representing the rise of the roof to the foot run1 the plumb and heel cuts are found for the hips and valleys4 ,by taking the length of the common rafter for one foot of run on blade1 and the run #ab on tongue1 marking along the blade will give the back bevel for the jack to fit the hip or valley1 and marking along the tongue will give the bevel to cut the roof-boards to fit the line of hip or valley upon the roof4 ,with this knowledge of what figures to use1 and why they are used1 it will be an easy matter for anyone to lay out all rafters for equal-pitch roofs4 ,illustration3 ,fig4 #cf4 ,laying ,out ,timbers of ,roof with ,two ,unequal ,pitches4 ,in ,fig4 #cf is shown a plan of a roof with two unequal pitches4 ,the main roof is shown to have a rise of #ab inches to the foot run4 ,the front wing is shown to have a run of #f feet and to rise #ab feet2 it has thus a pitch of #bd inches to the foot run4 ,therefore #ab on blade of the square and #ab on tongue will give the plumb and heel cuts for the main roof1 and by stepping #ab times along the rafter timber the length of the rafter is found4 ,the figures on the square to find the heel and plumb cuts for the rafter in the front wing1 will be #ab run and #bd rise1 and by stepping #f times 7the number of feet in the run of the rafter71 the length will be found over the run of #f feet1 and it will measure #ac feet #f inches4 ,if1 in place of stepping along the timber1 the diagonal of #ab and #bd is multiplied by #f1 the number of feet in the run1 the length may be found even to a greater exactitude4 ,many carpenters use this method of framing2 and to those who have confidence in their ability to figure correctly1 it is a saving of time1 and1 as before said1 will result in a more accurate measurement2 but the better and more scientific method of framing is to work to a scale of one inch1 as has already been explained4 ,illustration3 ,fig4 #cg4 ,finding ,length of ,rafter for ,front ,wing in ,roof ,shown in ,fig4 #cf4 ,according to that method1 the diagonal of a foot of run1 and the number of inches to the foot run the roof is rising1 measured to a scale1 will give the exact length4 ,for example1 the main roof in ,fig4 #cf is rising #ab inches to a foot of run4 ,the diagonal of #ab and #ab is #ag inches1 which1 considered as a scale of one inch to a foot1 will give #ag feet1 and this will be the exact length of the rafter for a roof rising #ab inches to the foot run and having a run of #ab feet4 ,illustration3 ,fig4 #ch4 ,laying ,out ,timbers of ,roof ,shown in ,fig4 #cf1 by ,projecting ,slope of ,roof into ,horizontal ,plane4 ,the length of the rafter for the front wing1 which has a run of #f feet and a rise of #ab feet1 may be obtained by placing the rule as shown in ,fig4 #cg from #f on blade to #ab on tongue1 which will give a length of #ac inches4 ,if the scale be considered as one inch to a foot1 this will equal #ac feet #f inches1 which will be the exact length of a common rafter rising #bd inches to the foot run and having a run of #f feet4 ,it will be observed that the plan lines of the valleys in this figure in respect to one another deviate from forming a right angle4 ,in equal-pitch roofs the plan lines are always at right angles to each other1 and therefore the diagonal of #ab and #ab1 which is #ag inches1 will be the relative foot run of valleys and hips in equal-pitch roofs4 ,in ,fig4 #cf is shown how to find the figures to use on the square for valleys and hips when deviating from the right angle4 ,a line is drawn at a distance of #ab inches from the plate and parallel to it1 cutting the valley in m as shown4 ,the part of the valley from m to the plate will measure #ac inches1 which is the figure that is to be used on the square to obtain the length and cuts of the valleys4 ,illustration3 ,fig4 #ci4 ,method of ,finding ,length and ,cuts of ,octagon ,hips ,intersecting a ,roof4 ,it will be observed that this equals the length of the common rafter as found by the square and rule in ,fig4 #cg4 ,in that figure is shown #ab on tongue and #f on blade4 ,the #ab here represents the rise1 and the #f the run of the front roof4 ,if the #ab be taken to represent the run of the main roof1 and the #f to represent the run of the front roof1 then1 the diagonal #ac will indicate the length of the seat of the valley for #ab feet of run1 and therefore for one foot it will be #ac inches4 ,now1 by taking #ac on the blade for run1 and #ab inches on the tongue for rise1 and stepping along the valley rafter timber #ab times1 the length of the valley will be found4 ,the blade will give the heel cut1 and the tongue the plumb cut4 ,in ,fig4 #ch is shown the slope of the roof projected into the horizontal plane4 ,by drawing a figure based on a scale of one inch to one foot1 all the timbers on the slope of the roof can be measured4 ,bevel #b1 shown in this figure1 is to fit the valleys against the ridge4 ,by drawing a line from w square to the seat of the valley to m1 making w #b equal in length to the length of the valley1 as shown1 and by connecting #b and m1 the bevel at #b is found1 which will fit the valleys against the ridge1 as shown at #c and #c in ,fig4 #cf4 ,illustration3 ,fig4 #dj4 ,showing ,how ,cornice ,affects ,valleys and ,plates in ,roof with ,unequal ,pitches4 ,in ,fig4 #ci1 is shown how to find the length and cuts of octagon hips intersecting a roof4 ,in ,fig4 #cf1 half the plan of the octagon is shown to be inside of the plate1 and the hips o1 z1 o intersect the slope of the roof4 ,in ,fig4 #ci1 the lines below x y are the plan lines2 and those above1 the elevation4 ,from z1 o1 o1 in the plan1 draw lines to x y1 as shown from o to m and from z to m2 from m and m1 draw the elevation lines to the apex o1 intersecting the line of the roof in d and c4 ,from d and c1 draw the lines d v and c a parallel to x y2 from c1 drop a line to intersect the plan line a o in c4 ,make a w equal in length to a o of the elevation1 and connect w c2 measure from w to n the full height of the octagon as shown from x y to the apex o2 and connect c n4 ,the length from w to c is that of the two hips shown at o o in ,fig4 #cf1 both being equal hips intersecting the roof at an equal distance from the plate4 ,the bevel at w is the top bevel1 and the bevel at c will fit the roof4 ,again1 drop a line from d to intersect the plan line a z in d4 ,make a #b equal to v o in the elevation1 and connect #b d4 ,measure from #b to b the full height of the tower as shown from x y to the apex o in the elevation1 and connect d b4 ,the length #b d represents the length of the hip z shown in ,fig4 #cf2 the bevel at #b is that of the top2 and the bevel at d1 the one that will fit the foot of the hip to the intersecting roof4 ,when a cornice of any considerable width runs around a roof of this kind1 it affects the plates and the angle of the valleys as shown in ,fig4 #dj4 ,in this figure are shown the same valleys as in ,fig4 #cf2 but1 owing to the width of the cornice1 the foot of each has been moved the distance a b along the plate of the main roof4 ,why this is done is shown in the drawing to be caused by the necessity for the valleys to intersect the corners c c of the cornice4 ,the plates are also affected as shown in ,fig4 #da1 where the plate of the narrow roof is shown to be much higher than the plate of the main roof4 ,the bevels shown at #c1 ,fig4 #dj1 are to fit the valleys against the ridge4 ,illustration3 ,fig4 #da4 ,showing ,relative ,position of ,plates in ,roof with ,two ,unequal ,pitches4 ,illustration3 ,fig4 #db4 ,method of ,finding ,bevels for ,purlins in ,equal-,pitch ,roofs4 ,in ,fig4 #db is shown a very simple method of finding the bevels for purlins in equal-pitch roofs4 ,draw the plan of the corner as shown1 and a line from m to o2 measure from o the length x y1 representing the common rafter1 to w2 from w draw a line to m2 the bevel shown at #b will fit the top face of the purlin4 ,again1 from o1 describe an arc to cut the seat of the valley1 and continue same around to ,s2 connect ,s m2 the bevel at #c will be the side bevel4 ,,index ,,part ,,page ,b ,bevel ,i1 #dd ,bevels1 application of in fitting wreaths to rail ,i1 #eh ,bevels to square wreaths ,i1 #fj ,bottom steps with obtuse-angle plan ,i1 #ef ,bullnose stair ,i1 #cc ,bullnose steps ,i1 #cb ,bullnose tread ,i1 #bi ,c ,carriage ,i1 #af ,carriage pieces ,i1 #bh ,circular stairs ,i1 #be ,close stairway ,i1 #b ,cove ,i1 #bh ,curved steps and stringer1 plan of ,i1 #ed ,cylinder ,i1 #ce ,d ,dog-legged stairs ,i1 #be ,f ,face-mould ,i1 #fh ,face-mould developing 7obtuse-angle plan7 ,i1 #ef ,flyer ,i1 #be ,foci ,i1 #gj ,g ,geometrical stair ,i1 #dd ,geometrical stairways and handrailing ,i1 #dc ,k ,kerfing ,i1 #cf ,n ,newel post ,i1 #ad ,o ,open-newel stairs ,i1 #cb ,p ,pitch-board ,i1 #aj ,plan lines ,i1 #dh ,platform ,i1 #bj ,platform stairs ,i1 #bf ,projection ,i1 #dd ,q ,quarter-space stair with six winders ,i1 #dc ,quick sweep ,i1 #cg ,r ,rail1 plan line of ,i1 #df ,rise and rim ,i1 #c ,riser ,i1 #c arrangement of ,i1 #gd ,rough brackets ,i1 #bh ,s ,springing the plank ,i1 #gd ,stair with five dancing winders ,i1 #dc ,stair-building ,i1 #a-ge definitions ,i1 #b introductory ,i1 #a laying out ,i1 #bb open-newel stairs ,i1 #cb platform ,i1 #bj stairs with curved turns ,i1 #cd strings ,i1 #g housed ,i1 #g notched ,i1 #h open ,i1 #g rough ,i1 #g staved ,i1 #h well hole ,i1 #ah ,stairs1 plans and elevation turning around central post ,i1 #dd ,stairs1 setting of ,i1 #h ,stairs1 types of in common use ,i1 #da ,steel square ,,ii1 #a-bg applied in roof framing ,,ii1 #g applied to jack rafter ,,ii1 #ag back ,,ii1 #a back of blade1 showing essex board measure ,,ii1 #c back of blade1 showing rafter table ,,ii1 #b back of tongue1 showing brace measure ,,ii1 #c bevels1 method of finding for all timbers in roofs of equal pitch ,,ii1 #bj blade ,,ii1 #a board measure ,,ii1 #a brace rule ,,ii1 #a common rafters1 method of laying out ,,ii1 #aa face ,,ii1 #a face side of tongue1 showing octagon scale ,,ii1 #b finding backing of hip in gable roof ,,ii1 #bb finding length of rafter for front wing in roof shown in ,fig4 #cf ,,ii1 #bd finding length to shorten rafters for jacks per foot of run ,,ii1 #ah giving various pitches to foot of run ,,ii1 #aj heel cut of common rafter ,,ii1 #ac hips ,,ii1 #ac hips and rafters for roofs of various pitches over square corner1 method of laying out ,,ii1 #ab hips and valleys1 how to treat the heel cut of ,,ii1 #af hips and valleys of a -pitch roof1 method of laying out ,,ii1 #aa jack rafter1 finding length of ,,ii1 #ah jacks between hips and valleys1 method of determining length ,,ii1 #ai laying out timbers of one-half gable of -pitch roof ,,ii1 #bb laying out timbers of roof with two unequal pitches ,,ii1 #bc method of finding bevels for purlins in equal-pitch roofs ,,ii1 #bg method of finding length and cuts of octagon hips intersecting a roof ,,ii1 #be octagon scale ,,ii1 #a rise and run ,,ii1 #a tongue ,,ii1 #a use of to find miter of equilateral triangle ,,ii1 #f use of to find miter of hexagon ,,ii1 #g use of to find miter and side of pentagon1 hexagon and octagon ,,ii1 #d1 #e ,story rod ,i1 #h ,straight flights ,i1 #be ,string-board ,i1 #c ,strings ,i1 #g housed ,i1 #g methods of building up ,i1 #cg notched ,i1 #h open ,i1 #g rough ,i1 #g staved ,i1 #h ,t ,tangent inclined-lower tangent level ,i1 #fc ,tangent system ,i1 #dd ,tangents and face-mould1 diagram of ,i1 #ec ,tangents unfolded ,i1 #ej ,tread ,i1 #c ,treads and risers1 housing of ,i1 #ac ,trimming joists ,i1 #ai ,twisted wreath raised to position ,i1 #eg ,w ,well-hole ,i1 #ah ,well-hole connecting two flights ,i1 #ea ,winder ,i1 #be ,winding stairs ,i1 #be ,wreath1 cutting from plank ,i1 #eg ,wreath with two equally inclined tangents1 finding bevels for ,i1 #ei ,wreath twisted1 ready to be moulded ,i1 #eg ,wreaths ,i1 #dd