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Las diagrammas suivsnts illustrant la mOthoda. 1 2 3 1 2 3 4 5 6 'ij^rt. ■: jjL ■ppapH MICROCOPY RESOLUTION TEST CHART (ANSI ond ISO TEST CHART No 2) 1.0 I.I 1.25 Hi ISO 12.8 ■ 3.6 2.5 2.2 1.8 1.6 ^ APPLIED IN/MGE Inc ^^ '653 tost Ma.n Slresl "^SS Rochester, New York 14609 USA •-^ (716) 482 - C300 - Phone ^1= (716) 288 - 5989 - Fo, A IMIAC TIC AL TREATISE THE STKKL SQUARE ASU Its Api)li('ati()n to Everyday Use ItKISG A.V K.XIIAISTIVK (()I.I,K( HON OF fiTKKI, HHIVKK I'KOIILKMS AND SOI.I TIONS, " Ol.D AND NEW, " WITH VANV OKIOINAI. AND fSKKri. ADDITIOXf*, KOK.MINO A COMPLKTK KNCYCI.OI'KDIA OK STEKI- SlilAKE KNOWLEIKiK, TOOETIIKK WITH A IIRIKK msTOKY OK THE HQIARE, AND nESCKIPTrON OF TAIII.ES, KEYS AND OTHEK AIDS AND ATTACHMENTS REVISED EDITION IS TWO VOLUMES HV FRED T. HODGSON Member o( Canadian Association of Archltt^'ts, Killtor of "Xatloiiul Bulliler," Author of "MoiliTiit'ariieiitry. " "('nmmoiiSciisi lluiiil- railinK" and other prai'tUal works on Hiillilinit, ctr. VOLUME II CniCAGO FREDERICK J. DRAKE & CO., PUBLISHERS l!K«t LaiXlUil J9«Oo. ^d.t.»,A«r CoMVmuHT liH.'t HV V .(■•iiRK K J. Drake * I'HK'AIIO. COPVBIHHT IMUU HY «'t». Fm(i>kruk .r Dkake St Cii. C'HII'AHO. 8 r»:«iM«C'- " ■ - ( -'>» PREFACE In preparing t' s second volume of practical work for steel ' jare users, it was itr'^Tative I should, in some measure, give ex? '■■.'. ■'% and rules that were somewhat similar to number that appeared in the firsi volume; though on close comparison it will be found that the simi- larity is apparent only, as different solutions of similar problems are rendered. Perhaps it is unnecessary to mention this, as the expert workman will recognize the differ- ence at once; but, in order to make explanation beforehand to the many thousands of readers, who, while ot being pcrts now in the use of the square, intend to 'jcome so as quickly as possible, I thou 'hi it better to mention the mat- ter in *] 's prefa. * The , . rchaser of this volume will find many things in it that are original, and many more that have been culled from the best work of experts and which have appeared in some one or other of the great number of trade journals that have been published in this country, in Eng- 3 Km'^i^: E^-^- :^m!m ^'T^rf^i^me'sm'^m^ t» ^ 4 PREFACE land and Australia, during the last twenty-five years. As before stated, there are a number of things original in the volume that have never before been published, some of my own and some that have been furnished by experts. I have made it my business to write to every per- son—at home and abroad— that I could hear of or read of, who had made a study of the steel square, asking for anything they might have that was new and useful on the subject, and tell- ing them I was preparing a new and exhaustive work on "The Steel Square." While all did not respond, I may state that over 75 per cent of those written to did; and while the great major- ity had nothing to offer, a groat many sent me "cuttings" from journals, on the subject, with diagrams and suggestions. Many, of course, of the problems sent me I could not make use of for obvious reasons, while several gentlemen sent me— for publication— a number of valuable prob- lems and solutions which I have embodied in the work and all the writers, without exception, wished me "God speed" in the work I had undertaken. It is now in order for me to publicly thank those who have so kindly aided me in getting together so much valuable material for the '§ M PREFACE 5 workman as will be found between the covers of these two volumes, and I am sure my readers feel as I do in the matter. Among those who might be named as having aided me materially, I may mention Mr. Woods, Mr. Reissman, Mr. Stoddard, and Mr. Penrose of Trafford Park, England, and Mr. Joseph Wil- cox of Sidney, Australia, to whom I tender special thanks. In conclusion, I may add, it is the intention of the publishers, should any new thing arise in connection with the steel square and its applica tions, to have the same embodied in the present volumes, or issue a supplementary volume if necessary. If by these volumes I have been instrumental in aiding and assisting the operative workman to earn a little more wages than formerly he received, or have helped him to better his condi- tion in any way, it will be a gratification and pleasure to me, and will, in some measure, repay me for all the effort I have expended in bringing together the matter contained in them. Fred T. Hudg.son. Collingwood, Ont., 1903. ft PREFACE TO SECOND EDITION. VOLUME II. In this volume, as in the previous one, there are mtroduced some new features which I feel con- vmced will prove very acceptable to the practical student and well repay him for any extra trouble in readmg or extra cost in procuring. A second preface to this volume, further than a simple an- Vol. 11 of The Steel Square and Its Uses,"' would seem an unnecessary labor, as the long preface in the volume of the first edition covers^ the enti,^" ground and. as a rule, the operative workman does not take much stock in a long-winded preamble. It ,s meat, not preliminaries, the active work- man looks for, and m this olume he will find plenty of the former and as few as possible of the latter 1 am convmced in my own mind, that the two volumes of th,s edition oflfer to the reader nearlv double the amount of good, honest and useful mate- ral, than can be found in any other similar work. nrnhl.n ^' ^'''' '"'^'^ ^"""^^ Solutions of useless problems as every example ofifered is a tried one and one hkely to confront the workman at anv hou; sale of this work may be attril,uted. While it is hardly fair to expect for this edition, the same grat- ifymg results that followed the first. I feel svitis- fied the publishers, as well as nvself. will ha^^ i^o reason to regret the -issuing of his second enlarged and improved edition. ^ r u- , T . ^^^°- T. Hodgson. Lollingwood. July, 1909. 6 THE STEEL SQUARE ?— tSSm i»' — •/•*• ».••■ fc*** A^** — •••• ^MM _•••• «••" ■ _«•• _ aaMit _«■ ■■■■'■ ■^■* ■ bi: U^M' 'Q\.!'u'\.,miT^-\.!i7ii^7^^^^^^^ ANOTHER STEEL SQUARE Since I wrote up a description of the various squares in the first volume of this work, I have been favored by a correspondent with the following description of a square which is the invention of a man in Augusta, Maine. I have not seen the square, nor have I seen a good illustration of it, but my correspondent says of it, "It is an ingenious tool for framing and can be adjusted so that work may be rapidly and accurately laid out, and it is capable of adaptation to all classes of buildings where wood- en frames are used. I have used one for a number of years, and i j^T|Tj;r['ip|i;^T|r^ Tp^T | T^r|T|T|r|;r| T^r|r|;iVi ' |;i ' |T | T r i TTT m 'titi 8 PRACTICAL USES OP find u quite handy for all kinds of work where fmber ,s employed." The description »ent 3 as follows: The body of the square has two tongues arranged „po„ it a. right angles. Three bars are so secured to the tongues that by loosenmg set-screws at the ends they may be placed at any points desired, the bolts sliding in slo s cut m the tongues. One of these bar, has a slot cut ,n it the greater part of its length, on wh,ch two gauge blocks are secured by thumb- forth and adjusted m any position desired. To the body of the tool, on one side, is secured a gauge, also capable of adjustment by means of crews. A bar, which is secured to one of the ongues and to the slotted cross-bar, is used in fratnmg roof timbers and the like where diag- onals are to be cut. It will be impossible to explain all the ways .n which the tool may be used. In laying out sTde iToi , '""T °" '""""■ "^^ ^-^'^ °" -e of the "i,"'"' "•" ''"""' ^''^•^ °f 'he body of the square will come at the outer edge of the mortise and there secured. A bar is then adjusted so that the space between it and the sin 'l " . t "" "'■""' °' '"^ -"«-■ For single work this is all that is necessary for mark- THE STEEL SQUARE 9 ing the sides of the mortises and tenons. For double work the second mortise is made by the adjustmer*- of the other two bars, so that all change of the tool or liability to error is avoided. To mark the ends of mortises and tenons, the end gauge is adjusted at the right distance from the tongue lying parallel with it, the tool moved alor T on the inside of the gauge, and outside of the tongue the ma'king is complete. For framing roof timbers the L r for this purpose is adjusted at the desired angle, and by this the Lead and foot of rafters or braces can be marked, without changing the tool, the marks for the pitch of roof being put upon the slotted cross-bar and end tongue. The two small gauges are espe- cially designed for use in cutting gains for shelv- ing, and being adjusted at the proper places on the cross-piece and secured, proper measure- ment will be given. As I stated before, I have not seen this square and am obliged to take the foregoing as being correct, but it seems to me that if t.'ic square is as useful as outlined it ought to be better -mown, but, ha ;^ing inquired at several large h ware houses regarding it, I find it entirely unknown among dealers. If any of my readers know any- thing of this square, as to its usefulness, price, to PRACTICAL USES op where it may be obtained, and will acJvise the oub of .t .s made at greater len, h n. future editions COLOR OF SQUARES It is only a ,W years a^o since any changes ^quare that has been in my possession for fifty steel surface, bur to-day it is oxi.lize.l and cov- ered w,th a coating that protects it from rust and f cm the weather. This happens to all po Lhed with mo,st fingers. This condition, however legible, indeed, ,n many cases the figures and mes become almost invisible, a very gr'eat ob^ec whe;e the fi"' r"" '"•"" -^>'-™"s cases where the figures have been mistaken and tin-. tt:x::'°"^'"'°''^^°"'"— -■-■^ A polished square should never be rubbed with su^chYbr^ "' """^ '""' ^"''^•--- While such r^bUne may make the sides of the square look bright and "tidy " it i, «„„ ,„ • ■ ■ squa e. efface some of the figures and markings, and leave the surface more susceptible to rul THE STEEL SQUARE II A little neat oil applied once in a while and rubbed with a soft rag is the best way to treat a polished square. Since nickel-pL-ting came in use, many squares are so treated, and this has many advantages, as under ordinary' circumstances there will be no rusting and the square will always be bright and "tidy" looking. Under a bright sun It is often difficult to read the figures or find the lines required, and, if the tool is exposed to the sun's rays for a short time, it will become so uot as to preclude handling. A careful workman, how- ever, will not allow his square — or any of his tools, for that matter — to be exposed to the sun for any length of time, s a two-foot blade often expands as much as one-eighth of an inch in length, when made hot — a condition the good workman cannot allow. Squares electroplated with copper, in my opin- ion, are much better than either polished steel or nickel-plated ones. The copper color is not so severe on the eyes, and the shadows cast by the cuttings in the figures and markings bring up the figures almost in relief, so that they arc readily seen. Age gives a copper-plated square a fine antique color which is restful to the eye and a protection to the square. la PRACTICAL UvSES OF If the square is allowed to oxidize, and is then polished on the surface, we get a fme copper fin- ish on the tool with dark oxichze.l figures and markings. Best of all is the blued or what may be term.d the "gun metal" square. These squares are oxidized or blued by some process unknown to me, and the figures and lines are filh^d in with some kind of white enamel, that brings them out m great shape. A very handsome blued and enameled crenelated square was sent me for examination by The Peck, Stov/ and Wilcox Company, of Southington, Conn., which lies before me as I write; and it seems to me that this method of bluing, and white-enameling the figures, is one of the greatest improvements made on the square for many years. To keep the copper, nickel-plated or blued squares in nice order, they should be rubbed over once in a while with good machine oil applied with a soft rag. By this simple process a square may be kept new m appearance for a long period of years. These few hints on the qualities of squares and their care will, doubtless, prove of value to my readers. THE STEEL SQUARE «J SOME ODD PROBLEMS The workman often is confronted with very curious problems, and is as often put to his wits' end to solve them. Here is a problem and its solution that will prove interesting, though it is not likely :<> \»: met very often: Suppose a box twelve inches square to be set Fii.. I on the ridge of a roof, as shown in Fig. i. The roof is half pitch, or rises 12 inches to the foot. Required, the cuts for each side. The corner X is 7}< inches from the ridge; therefore it is 7j4 inches lower than H. The cut for side A 14 PRACTICAL USES OP Is therefore VA to 12. X is also VA inches lowtr than Y. The cut for H is 7^ to q%. The cuts for the other i,icle are found in the same way. PlO. 3 For proof see Fig. 2. The side of the box being 12 inches, therefore, when set on the roof with one corner touching the ridge, or in any other position, it will reach 12 inches hori- zontally over the roof. And with a run of 12 inches you have a rise, or rather a fall in this instance, corresponding to the pitch of your roof. With the run on one side of your square and the rise or fall on the other, you will get the tonect cut eveiy time. THE STEEL SQUARE If Two other problems are given here which I am sure will prove interestinjjj: To determine by a steel square the result of any number, for example, 6, multiplied by the sin 45 \ Take 6 on both blade and tongue, and mark the line ac^f)c=f>. Then <: AVy/.m-^-The cuts for the bottom end cf th<^ pipe; the blade of the bevel to b<; a|,plien 45°. Mark alonK' the tongue. The angle made by the tongue and the line is the proper bevel. If the bevel is applied to mark the end of the higher halt of the pipe, we have on! to reverse the direction of the stock of the bevel so as to make an obtuse angle. These two problems are very much alike. Their demonstration is a good study in solid geometry. Suppose we wi.h to cut an opening in a roof for a round pipe or a tile, so that the pipe or tile will stand vertical through the opening. The exact form of the opening may be obtained by the following method, which is taken from "Car- pentry and Building" of New York: set the pipe or tile to be used on the roof at the point where the hole is to be cut. as shown at Fig. 4; plumb THE STEEL SQUARE ,7 it with a level or plumb and place a square alongside of it as is shown in the sketch. Keep. Fig. ing the blade of the square in contact with the pipe, move it around ^he circumference of the pipe, touching the roof at points about one inch apart anc makinjr a mark on the roof at each point of contact. When the entire circumfer- ence of the pipe has been traveled, join the points marked on the roof, and the figure out- lined will be a perfect t^liipse and of exactly the size required. Nothing remains but to cut the z8 PRACTICAL USES OF I hole, and if this is done correctly the pipe will be found to fit exactly. This method is simple and perfectly accurate, and as it requires no calcula- tion, it can be done by any workman who can handle the tools and pipe. Often men working in wood-working factories find it necessary to increase or decrease the speed of a shaft a few revolutions. To get just the size of pulley necessary for this increase or reduction of speed requires close calculation when figures are used, but a square will do it off-hand and correctly every time. For instance, it has been ascertained that a pulley 20 inches in diameter on a certain shaft will give a speed of 13 revolutions per second. If it be desired to reduce the speed to ii>^ revolutions per second, the 20-inch pulley must be removed and one of different size substituted. To solve this problem with a steel square, take 20 inches on the tongue and 13 inches on the blade. Place both of the marks on the edge of a straight board and draw a line parallel with the blade. Move the square parallel with this line until the tongue shows iij^ inches instead of 13. Notice the reading on the blade and edge of the board. This reading will be size of pulley required in place of the 20-inch concern, THE STEEL SQUARE 19 necessary to give the desired speed oi \i]4 revo- lutions per second. The principles involved in the calculations here made have been amplified and put into a more convenient shape for every- day use in the slide rule. In this perfected form some mechanics are familiar with the tool, but all do not recognize it in the slide rule. The principle upon which all slide rules work is that of the square and the piece of board above mentioned. There are several more problems of this kind that might be described with interest to the reader. To find the number of cogs in a wheel, pitch of cogs and diameter of wheel given, set the bevel to the pitch on the tongue and 3.14 on the blade. Given the diameter of a wheel to pitch line, and number of cogs, to find pitch of cogs. A wheel 70 inches diameter has 146 cogs; what is their pitch? 146 inches being too great to set on the square, we take proportional parts, setting the bevel to V inches or 83^ inches on tongue, and 4« inches or \^% inches on blades. Tighten the screw of fence, and move the bevel to 3.14 on blade, and the number given on the tongue multiplied by 8 will be the required pitch. li. we wish to divide a circle into a given num- ao PRACTICAL USES OF ber of parts, we proceed as follows: Multiply the radius by the corresponding number in column A, as per table, and the product is the chord to lay off on the circumference. Given the side of a polygon, to find the radius of the circumscribing circle. This problem has previously been tabulated, but by multiplying the given side by the number corresponding to the polygon in column B, in most cases we obtain the answer more expeditiously. No. of sides A or parts A B 3 • . Triangle 1.732 •5773 4 • . . Square I.414 .7071 5 • . Pentagon 1. 175 .8006 6 . . Hexagon radius side 7 • . Heptagon .8677 1-152 8 . . Octagon •7653 1.3065 9 • . Nonagon .6840 1.4619 ID . . Decagon .6180 1.6180 II . . Undecagon •5634 1-7747 12 . Dodecagon •5176 1.9318 Given the iHamclcr of tx circle, to find the side of a squdjc of equal ana. S^X. a bevel to q}i on the tongue and 1 1 inches on the blade. Then move the bevel to the diameter of the circle on the blade and the tongue gives the answer. When the circumference is ^n'vtn instead of the diam- THE STEEL SQUARE ai eter, set the bevel to s^A inches on tongue and I9>^ inches on the blade. To find the number of square yards in a given area we must proceed as follows: These prob- lems require a bevel of 9 on tongue, and the length or width of the surface to be measured on the blade. The bevel is then moved to the remaining dimension of the area on the tongue, and the number on the blade indicates the square yards contained. If the diameter of a circle is given, we can determine the circumference as follows: Set a bevel to 7 on tongue and 22 on blade; the answer will be found on the latter. In every case when a reverse problem is presented the bevel will solve it unchanged; we merely look for the answer on the other blade of the square. For instance, if, after sobirg the above, we are required to determine the diameter from the circumference, we still use the same bevel. SOME DIFFICULT PROBLEMS AND THEIR SOLUTION I am indebted to Mr. Fred La-^y of San Francisco, Cal., for some of the following inter- esting problems and their solutions. They make excellent practice for the young student who has made up his mind to learn all possi- 32 PRACTICAL USES OF ble concerning the square and what may be done with it. The workman is often confronted with prob- lems in oblique framing that are difficult to solve unless he possesses knowledge of a high order of solid geometry. *See how Mr. Lascy handles the square first in splayed work, then in oblique bevels. To construct on a base of any number of salient corners, a solid in which every two adjoin'ng faces slope together toward the hori- zon form;: : a hip, and the same rise, of any lengths. To de- termine by the steel square on a line all the angles that can be required for such constructions, fi st draw to a proper scale the run A B, Fig. 5. the rise BD, and the slope AD. This triangle is supposed ^' ^ to stand perpendicu- lar to the plane of the paper. Construct the angle ABC=B=any angle whatever. If all the corners of the base arc alike in degrees, B=i8o THE STEEL SQUARE ,3 divided by the number of corners. If the corners are unlike in degrees, consider each corner by itself and make 6=90°, minus half the degree, of this corner. Draw AC and BC, which is always the seat line of_'the hip. ABC is always half the corner of the base. All lines drawn from the run to the seat line must be at right angles to the run. By the steel square. Place the slope on the blade. Place AC (run tan B) on the tongue. Mark along the blade for the face cut against the hip line of boards which have the direction of jack rafters. Mark along the tongue for the face cut against the hip line of horizontal boards; for the top cut of purlins; for ihe top cut of a miter box to miter the horizontal boards. By the steel square. Place the slope on the blade. Place EF (rise tan B) on the tongue. Mark along the tonguv. for the miter cut across the square edp of horizontal boards; f jr the down cut of jrlin; for the down cut of a miter-box to m. -r the horizontal boards, lying flat in the box. If the boards be not mitered, we require the butt cut across the square edge of boards. For this draw BG, at right angles to the slope; make BK=AG, and draw KI=run cosine tan B. m »4 PRACTICAL USES OF . i By the steel square. Place the rise on the blade. Place KI on the tongue, and mark along the tongue for the 'ntt cut across the square edge. Make BH=seat, then HD=hip line with its top and bottom bevels. For the diedral miter at right angles to the hip line, which is half the angle between any two adjoining sloping faces. By the steel square. Place the hip line on the blade. Place EF (rise tan B) on the tongue, and mark along the tongue for the diedral miter, To back a hip rafter by a gauge mark, make B« = half the thickness of the hip, and draw nm to tht seat; from the toe of the hip rafter make Hp=um, and from p draw the gauge mark parallel to the hip. For a four-sided hipped roof, we may need the side cut across the top square edge of the hip rafter against the ridge. For this make Bv=ha\i the seat, and draw v^e at right angles to the seat, to meet bu at right angles to the run. By the steel square. Place QUV (seat cotan B) on the tongue; place the hip line on the blade and mark along the blade for the side cut, short- ening the hip according to the half-thickness of the ridge, thus: It B« = half-thickness of the ridge, draw a distance Bw parallel to the rise; where this line cuts the hip shows the shorten- THE STEEL SyUAKE 2S -if. ing. For a square corner, 6=45", tan F{= co- tan B=i. In this case tangents of B may be drawn or not. Again, let the raking molding of a pediment = AD, which is to miter against, as horizontal molding that forms with the run of the pediment any angles whatever. 8-90° minus half of this angle. The raking molding must be placed in the miter-box with that part of the molding which is vertical when in position against the side of the box. Lay the steel square on top of the box. Place AG (run cosine) on the blade; place AC (run tan B) on the tongue, and mark along the tongue for the top cut of the miter-box. Again, place the run on the blade; place the rise on the tongue and mark along the tonj,ue for the down cut of the miter box. The miter of the horizontal molding equals one-half the corner on the ground plan. If the diagram ADB equals half the gable end of a rectangular building, to miter the raking plan- ceer against the horizontal planceer, which slopes as the roof and runs along the eaves at right angles to the run of the gable, both planceers being in the same plane. By the steel square. xMace the slope AD on a« PRACTICAL USES OF the blade; place the run AB on the tongue; mark along the blade for the raking miter; ma k along the tongue for the horizontal miter. The horizontal planceer requires the wider board. In laying off angles for splayed work, lines as long as possible and as few as possible are the essentials of accuracy and comprehension of the subject. Another problem with its solutions follows: Given a hopper or hipped roof which stands on a base whose corners have any angle whatever, and whose sloping sides have the same run and the same rise of any length. Required the die- dral angle between any two adjoining sloping sides without using the hip line; or, what is the same thing, to find the backing of the hip raft- er without using the hip line. First lay off AB=run, Fig. 6, BD= rise, A D = slope line. Draw ACat right angles to the run, and lay off angle ABC=9o°, minus half the angle of the cor- ner of the base. Angle 6 = half angle of corner of THE STEEL SQUARE '7 base. Make BE = rise, and draw EF at right an- gles to BC, the seat of the hip line. On the edge of a board lay off t GF, Fig. 7,=slope AD. By the steel square on the lineT ^* * *^ GF, place EF on pio. 7 the tongue and apply the tongue as shown on the diagram, moving the blade until the blade cuts the point G. Mark along the tongue. EFG= one-hatf the required diedral angle=diedral miter. By calculation: Cos diedral miter=rise cos half-corner angle C. The demonstration may be studied in any book on solid trigonom- etry which treats on the right triangular pyra- mid. It is too long to be given here. Without using the steel square, draw a semicircle on FG, and taking FE in the compasses, mark the chord line EF. This gives the diedral miter EFG to back the hip rafter. The following problems in oblique bevels will prove both useful and instructive as well as interesting to the studious young workman: Given the run AC, the rise CB, Fig. 8, and the slope length BA of a stick of rectangular timber, which butts obliquely at the upper end against a vertical plane of indefinite length, and whose 38 PRACTICAL USES OF i seat line=CE. Angle between this seat and the given run = W, which may be any angle what- ever. Required the side bevel on the top face of the stick at the point B to fit this end of the -.tick against the vertical plane. First draw AE at right angles to the run, make AD^«1opeand draw DE. The required side bevel =:EDA. Demonstration: Let the triangle EDA re- volve on AE as on a ^k hinge; when the vertex D reaches tue required height of the rise, the triangle EDA will stand directly over the triangle EGA, the line ED will coincide with the vertical plane, the line DA will coincide with the top center line of the stick. By calculation, tan side Level = - ° ^^" ^ , ,,r - slope when \V ^ 45 , tan VV ^ unity. By the steel square on a line: Place AE (run tan W) on the blade, place the slope line on the tongue, mark along the tongue for the side bevel. Fig. 8 THE STEEL SQUARE »9 more When W is an < ccnvenient to ob v,. .»*^ i I following manner: From / "he center point I of the run, draw fg at right angles to the run. :: then AD=twice/^. If the stick should at its lower end butt obliquely against a vertical plane. the side bevel is obtained in precisely the same way as given. By this method we may obtain the side bevel at the lower end of certain jack rafters, making planers and rafting moldings. The other bevels at the top and bottom ot the stick do not require any remarks. Again, given the y: run KV, Fig. 9, the rise KA and the slope A V of a stick of rectangular tim- ■ berwhichatitslow- ered end V butts obliquely against a vertical plane whose ground line is in the direc fon of VC The corner K\C = . n.ay be of any size, and is supposed to stand perpendicu- lar to the plane of the paper. Required for the lower end of the stick the miter cut across the top face; also the down cut, so that this end of the stick may fit against the vertical plane 30 PRACTICAL USES OF ): whose ground line is VC. First, from the pc'nt K, and at right angles to the nin, draw I\C to intersect tht ground line of the vertical plane. From the point A, and at right angles to the slope, draw AB = KC; also draw BB; then AVB is the miter cut across the top face. Demon- stration: The triangle VKA being perpendicular to the plane of the paper, let the triangle VAB revolve on VA as on the hinge, until the line AB comes into the horizontal position over KC; then AVB is the required sloping triangle of which KVC is the plan. By the steel square on a line: Place KC on the blade and the slope VA on the tongue. Mark along the tongue for the top face miter. The end down cut may be marked by a bevel set to i.he angle IIAV. The stock of the bevel is applied along the bottom face of the stick with the blade of the bevel pointing upward along the side face. The most common applications are the rafters which butt obliquely against valleys, hips and ridges. If the line VA represents a raking planceer, which at the lower end of a gable miters around a square corner and against a horizontal planceer that slopes in accord with VA, then the angle // is always 45°, and CK and AB will each equal the run VK. The miter for the end of the hori- THE STEEL SQUARE 3« zontal planceer will be the angle ABV. The horizontal planceer is supposed to have its inside edge beveled to fit against a vertical plane; a square mark down the beveled edge is the down cut through the thickness of the mitered end of the wider horizontal planceer. VA and VK show the relation between the width of the two planceers. VA is a raking molding, mitering at point V around a square corner and against a horizontal molding. The lower (.nd of the ra- king molding, mitering at point Y around a square corner and against a horizontal molding. The lower end of the raking molding should be cut in a miter-box, with that part of the molding which is nailed against the gable placed against the side box. The foregoing are the proper cuts for the miter-box. The line KC, etc., may be drawn anywhere along the line KV. Here is an excellent graphical method of find- ing the areas of different figures. It is taken from "The American Machinist," and is worthy, I think, of a place in this work, because of its compactness and simplicity. When the area and diameter of any circle is known, by this method the area due to any other diam.eter, or a diameter due to any other area, may be determined. Suppose we take the diam- PRACTICAL USES OF eter 2 wJth the area 3.1416 as the known quantity from which to calculate all others. Any other diameter and area may be chosen, but this one IS the easiest to remember. Draw the indefinite straijrht line AB, Fi^r. 10, and. with a diam- eter equal to 3.1416. draw the semi -cir- cumference ADC. With a radius equal to 2, and with A as a center, cut the semi-circumference ADC in D. Throu^rh D erect the perpendicular HH. then wdl the distance Ai{ he a constant for every diameter and area. Let it be required to find by this dia^rram a diameter that has an area equal to 5. Lay off AF^^.s and draw the semi-circum, ference AGF, then will the distance AG be the required diameter. If the (hameter is ^men and the area required, take A as a center, and, with the given diameter as a radius, cut the line HF in G. Bisect the line AG at right angles with the hne. cutting AH in K, then will K be the center of a circle passing through A and G, and Its diameter. AF. will be the required area. This diagram is susceptible of a great variety of applications. The diameters on the line THE STEEL SQUARE 33 AB may be areas, capacities, weights, tensile strengths, or. in fact, anything that is made up ' ^ir-a anj , constant quah'ty. The distances ro-.n A .0 th . hne H E are always diameters, sides ^-^y ^qu.re or some constant component of area The distance AE is different for each kind of proposn.on. but is constant for every proposition of the same kmd. If we say that a bar of i-inch round ,ron has a safe tensile strength of ; 000 lbs, then we lay off AF-7 dr;,w rh 7^ ,. ^ ^^ ~/, draw the arc on th s diameter and lay off AG = ,. The position o G g,ves the location of the line HE. from which he ,ens,le .trength for any diam,-ter may be fo.,nd, A diameter AD will have a .eLile strength AC. icnsiie The advantage of thi. diameter lies in the fact that ,t i» impo^ible to remember all area. we,g ts, strength, etc., while it is comparati::,y , ""'• '-"°""' 't -natters no, whether the It is a well-known geometrical fact that the angle wuhin a semi-circumference is a i'h! hi! in 'T" "'-'"' -^y -■=« ='dvan ag?o ^h.s tn makmg of core-bo.es, in the manner t-i. ' r 34 PRACTICAL USES OP shown in Fig. it. If the core-box has been cut out accurately, then the square will touch at three places— th< two edges and a point in the curve. If it touches at only two places, one being on the curve, then it is not cut out deep enough; if it touches Fig. II only at the two edges, then it is cut too deep by the amount of clearance between the corner and the curve. By giving the square an oscillation, to make the corner sweep the entire surface of the curve, the accu- racy of the curve at that point may be ascer- tained, and by trying at several points the truth of the whole box may be determined. The square may thus be made to take the place of a templet in making of core-boxes, with the advantage that the square will fit any size, while a templet only fits one. There is an opportunity here for some one to get up a core-box plane that will produce any semi-circumference accu- rately and quickly. The only care that would have to be exercised by the workman would be the placing of two parallel metal strips as shown THE STEEL SQUARE 35 at a and b. These strips are, of course, not necessary when simply usin^ the square to test the accuracy of the work, but would be necessary in the use of a core-box plane made on this principle. If we wish to find the diagonal of a square or parallelogram, all we have to do is to set the blade of a bevel to 8^ inches on the tongue and 12^ inches on the blade. Then screw the bevel fast; and supposing the side of the square in question is ii inches, move blade to the ii-inch mark on the tongue, keeping blade against the square, when blade will touch 15 A inches on the blade, which is the required diagonal. There is no special reason for using 8->{ and 12^^; other nur''^ rs may be employed provided the propor- tic 70 to 90 exists between them. In the pro .v-in just solved, as in all that follow, the bevel being oncfe set to solve a particular ques- tion will solve all the others of the same kind, till the bevel is altered. To find the circumference of an ellipse or oval, we jiroceed as follows: Set 55^ inches on the tongue and 8^4 inches on the blade. Then set the bevel to the sum of the longest and shortest diameters of the ellipse on tongue, and the blade gives the answer. •»^ • ^f r «r 36 PRACTICAL USES OF If it is desired to find the side of tlie greatest square which ma> be inscribed within a circle we can accomplish it with the aid of the square as follows: The diameter of a circle being given, set the bevel to 8^ inches on the tongue and 12 inches on the blade. The answer will be found on the tongue. To inscribe three small circles within a large circle of given diameter, set to 6>^ inches on tongue and 14 inches on blade. Move the bevel to the given diameter on the blade and the required diameter appears on the tongue. Four equal circles require a bevel of 2.91 and 14. Tfl inscribe polygons zuithin circles. — In the fol- lowing table, set the bevel to the pair of numbers under the polygon to be inscribed: No. i f sides 345 6 7 8 9 10 11 12 Radius s6 70 74 ''''f, 60 98 22 89 80 8s o.j -' ' '^ equal to ^ -' .) bide . . 97 99 87 radius 52 75 15 55 45 44 If we require the radius of a circle which will circumscribe an octagon 8 inches on a side, we refer to column ?, take 98 parts on the blade and 75 on tongue, and tighten the bevel. As the side of a hexagon equals the radius of its circle, the side of an octagon must be less than the radius; hence we shift to 8 inches that end of the bevel iiniKrVc^f/. THE STEEL SQUARE 37 blade which gives the lesser number, in this case on t le tongue of the square, as the 75 parts to which the bevel was set are less than the 98. The required radius is then indicated on the blade. The following is another table, to be used for the same calculations: N*'"*^ No. of side Gauge points Triangle 3 1044 Square 4 8.49 Pentagon 5 7.06 Hexagon 6 6.00 Heptagon 7 5.24 Octagon 8 4.59 Nonagon 9 4.05 Decagon 10 3.71 Undecagon n -^^^g Dodecagon 12 3.11 In a circle 12 inches in diameter, the largest pentagon which may be inscribed is 5.24 inches on a side. Hence for pentagons the bevel is set at 12 inches and 5.24 inches. The number oppo- site each polygon gives its side when inscribed in a 12-inch circle. The first table is usually most convenient. IF/ifji the side of a polygon is given, to find its apothcw or perpcndirular.—'^^x. the bevel to the pairs of numbers in the table below. Thus, for a r 38 PRACTICAL USES OF heptapron. set 23 on tongue and 25 on blade, and the answer will appear on the latter. Sid es 3 4 5 8 9 ID II _^ 12 Apothem 9 i 20 13 25 40 40 20 29 28 Sides . 31 2 29 15 23 i,2> 29 13 17 15 The board measure.— A foot in board measure is I inch thick and i foot square. Set the bevel to 12 inches on the blade and the length of board in feet on the tongue. Then move the bevel to the width of board in inches, on the blade, and the area in square feet appears on the tongue. Whenever the 12 inches is set, whether on tongue or blade, there also must be set the width of board in inches. To lay off degrees with the steel square, con- sult a table of tangents, and from this table take the tangent of the angle required, using the first three figures from the left and calling them so many 64ths on an inch. Reduce them to inches, and then, with this quantity on one side of the square and 15^ inches on the other side, we v/ill have the figures for laying off the angle. Tables of natural tangents are usually calculated to the radius unit, and are therefore decimal fractions. This method is simply to multiply each by 1,000, thereby obtaining whole numbers. For exam- THE STEEL SQUARE 35 pie. let it be required to lay off an angle of 10° the natural tangent of which is 0.176327. Multi- plying this by rooo makes 176.327. Discarding the de-imal we have 176, and calhng the figures 64ths 01 an inch, we have \\\ or 23^ inches. The radius I treated in like manner makes ^F or ^S% inches. Now, taking 2V, inches on 'the tongue and 155^ inches on the blade of the square the blade gives the angle of 10°. and consequent-' ly the tongue gives 90° less 10°, or 80". There are other methods of laying off degrees with the square, several of which I have described and will describe hereafter. PROrORTIONAL REDUCTION OF MOLDINGS OR OTHER WORK There are many methods of doing 'lis work by lines, ordinates. and the pantagraph, but I do not know of many by the steel square. The fol- lowing, which may be new to many readers, has been in use for a long time: First draw the mold- ing bracket or other work, as shown at Fig 12 in a square as at A. B, E, F. Square out from' A and F lines meeting at B. Draw BE, and from E measure off the required projection of the reduced bracket, thus obtaining the point D bquare down from D to the line BE, thus loca- :iLxiiyMi^. 40 PRACTICAL USES OP tt i mm ting the point C. Then the line ED will be the width of the reduced bracket and DC its hei,;ht. Fig. 12 Now, at convenient points, their location and number being determined by the nature of the profile, as 2, 3, 4, 5, etc., square lines to the back edge, and also to the upper end of the brackets, all as shown in the sketch. Take the width ED of the reduced bracket on the blade 'of the square, and placing it, as shown in the engra- jcift-i^-.J »» >'tW"«. THE STEEL SQUARE 41 ving, against the corner E, carrying the tongue of square up until its edge strikes the outer corner A of the original bracket, draw a line along the blade, all as indicated by KE. Square down from points in this line to the points in the upper Hie of bracket, i, 3, 4, 5, etc., already obtained. Take the length of the line DC on the square, which is the length of the bracket after reduc- tion, and place it, as shown by the shaded square in the sketch, at E. Carry the square up until the blade strikes the corner at F. Mark along the blade of the square, thus producing the line EH, which is the back of the diminished bracket. From this line square out the line EG indefi- nitely; also square out the lines 16, 15, 14, 13, etc., from the points in the back edge of the original bracket, extending them indefinitely across the space the reduced bracket is to occupy. Tal a straight-edge and place it against the line KE and mark the points that have already been obtained in it. Then transfer these distances on to the line EG. If preferred, this may be done by the compasses, setting one leg at E and describing arcs from the several points in KE, striking the line EG. From the points thus located in EG lines are then to be carried at right angles to it, being produced until they meet . 1 iw^'«».:«%v "^^w 4» PRACTICAL USES OP he lines drawn from corresponding numbers on he inner edge of the original bracket. Then a line traced through these intersections will pre duce the profile of the bracket diminished. The number of f\xed points in the profile of the orig- inal bracket necessary' to be used will vary in different cases. Where the lines are long and regular less will be r^.quired than where they are short and irregular. To increase the size of a g.ven bracket the process here described is to be reversed. The same general rule may be also applied m drawing the profile of raking mold- jnp I thmk it will be seen that I have not here laid down an arbitrary rule. The principle on which It is founded is in laying down a line the ength of the required bracket, and dividing that line m the same proportion as the original bracket. // zs required to get the length of a hoop for a wooden tank, by the steel sguare.-^o accomplish th.s.pioceed as follows: First produce a circle to any desired scale, say i inch to the foot, and this on 24 feet would be 24 inches. Then place the heel exactly at the center as indicated in F«g. 13. and scribe closely to the square, cutting the circumference of the circle, as indicated by BC. I hen draw the chord intersecting the '4 THE STEEL SQUARE ^3 points n.C already referred to. The next step is to divide the segment equally, which is done by the line DE. Now three times the di- ameter plus the distance DE will give the required measure men . or circumference. To perform this by figures alone, take the diameter of the tank and multiply it by Fig. 13 31416. If the diameter of the tank is 24 Teet. for -amp e the equation is as follows: .4x3.1416 /5.09.S4, or multiply the diameter by 22 and divide by;. Thus 24x22-^7 = 75^ Fig. 14 The prober angle for ordinary door and win- . BEfi ! J 44 PRACTICAL USES OP dow s.lls is about , inch drop to the foot. A motho- .lifr.T,.ncc between 10 !i and 9, or I' Thi. im • , "•""'-" nuse nf ■ .*- '° •'»"""•''<■ hypothe- a ,h ■'" '""■""•■""*■ ^'«'"-«l"l triangle, ^'-l ■he ,>, one si,!,-. I, has already been :::;::*=" ""- '° '■-" •■- -'- s-i.-, or ;: .|.- ;^-'""™« square: what is the diam. ;>l.ndn,:al one with ,h,. same area of , u ::,^'^'- "— - -n <"n«„e and we 'iiitaiii 22, th<- answer .,n blade Tl,e length and an«l,:s of a brace of irregular "■n. ".• any r„n, ,nay be fonn.l rea.lily by at K, ^nsr th<; following ml,, usin.r ,1, " ^ - ^•''"".n;,,,ad:To^tie"'Takrr -™nle, the ease o, a brace of whi,h ht'un : .^" '""^t -'' •"-= l-i«lu 37 inches. Me., in .» ross the space for the length of the brae ""h ^ square will not do; accortlinRly then re • "; '"o lengths by four, which'^i'; :,";;; arms of the sqnare and measuring, across ',' --l-.-J l---ng.h will be obtained. To do this take a p.ece of board, join, one edtre an, r iraufrp rmri- f i ^^'^t- and run a feduye mark from the erJtrp fKo i • . eage the desired width. 46 PRACTICAL USES OP j i i II Then, placing the square so that the figures fall on the gauge mark, apply it four times, scribin.r along the blade and tongue respectively for the two ends. This gives the net length of the brace and the proper cuts for the joints. This may not be the best rule which can be employed for the purpose, but it is short and simple. Any ordi- nary carpenter can work it, and it is undoubtedly correct. At this point I show a few quick rules to give a square stick an octagonal shape. The rules given on the side of the square as shown in my description of squares in the first volume, while being perfectly correct so far as they go, do not work so well where fractions of an inch are involved, so the following methods of finding the points for the gauge lines are shown at Pig jc which shows the square as laid obliquely across the face of the stick so that just 6 inches will be on the stick. At iH inches from each corner draw gauge Imes. which will be the correct corners for the octagon. If the timb.-r is over six inches square lay twelve inches of the square upon the ''IG. 15 .JJ^^f^^^ *1L-^-, -J^ W^- THE STEEL SQUARE ^y face and gauge at 3^ inch -s from the corner If over 12 inches wide, layover me whole 24 inch^ of the square and prick off 7 inches from each end. and these points will be the gauge points. Indeed, .t is best to use the whole length of the Fig. 16 sqtiare in laying off work of this kind, and prick- •n^^ off 7 inches from each end of the square nn -tter what may be the width of the tX P^g. 16 shows this method quite ^-'early. The ;;, '.-a;^ j,.;;v. t..:v-; /.-?x » ■: V-' vr;.-' \v^.'.' \'<;-' >\:-'' v,-;,-' 'ov;;-' '%,» Hif» ■.> l^(>if'-« Rin ,-\ -> ,.A ,^ ,-\ ,-•» ^a ^ ' ■ ,^>'. ' J • ' •-' , '^^-r^ , ■■ »! ■ j'^ ii p f. . i i J r . II ■ J li' g A \> ^» V O -J u O >^ >kC«/V^ B<t,nihi l£r/»TV^(^ijt Fig. 19 ROOF FRAMING In th*^, cut I have illustrated a % pitch hip roof, 16x24 feet, rafters 16 inches apart. MAIN RAFTER One-third pitch rises 8 inches to the foot, and as 8 feet is half the width of the building, the run must be 8 feet. Therefore put the square on 12 and 8 and eight times gives the length and bevels (as illustrated). Notice how it is squared up at heel, and amount allowed for ridge. 1 THE STEEL SQUARE 51 CENTER nil' RAFTER As the diagonal of a foot is 17 inches, take 17 inches on blade in place of 12, and we have hip rafter (as illustrated). Now these methods are not new or original, as they have probably been used for ages, yet it is surprising how few carpenters know them. JACK RAFTERS My method for jacks is an original idea to me, yet it may have been used before 1 was born. I simply lay the square on the same as for com- mon rafter. If you wish them 16 inches apart, move th^ square up to 16 inches; if 18 inches apart, move up to 18, etc. The side cut is the length and run. Cut on length. If you wish to bevel top of hip, take length and rise. Cut on rise. OBSERVE ALL THE ILLUSTRATIONS Now remember the same method applies to all pitches. Run the same; simply change the rise to whatever rise the roof is to the foot. This applies to cornice as well as rafters. Do not be satisfied with this knowledge, but study the use of the square and go further, as there is no 'imit to what can be accomplished with it. r 5» PRACTICAL USES OF i PRACTICAL USE OF SQUARE AND RULE Study and fully understand the eight illustra- tions in this one little cut, and you will find, by thought and application, as the occasion requires, you have learned a great deal, as you will readily learn more. If you have a board 7 inches wide and wish to divide it into four equal parts, turn the rule until Fig. 20 it strikes eight inches, and mark at each 2 inches, as in No. i. Fig. 20. I use that almost daily not only in ripping up hoards but in drawing, etc If you happen to wish to square a board and THE STEEL SQUARE 53 do not have a square, take a rule and apply as shown in No. 2, Fig. 20. IN LAYING OFF RAFTERS Some may not like to place the square on once for every foot of run, as I illustrated in another cut. Also, if it is to go to a given height one may not wish to stop to figure the exact rise to the foot, figuring out the fraction, etc. Take a roof to be 7 feet 3 inches high and run 8 feet 5 inches. Put your rule on 71? and SA, and you will have VV or 1 1 feet 2 inches length of rafter, as seen at No. 3, Fig. 20. If you wish to hip the same roof, as it is 8 feet 5 inc ^is to the deck, the run of hip must be the diagonal of 8 feet 5 inches, which is 11 feet 11 inches, as illustration 4. The run being 11, 11, and the rise 7, 3, place the rule on them, and we have 14 feet, as shown in No. 3, Fig. 20. If you are buying lumber at $13 per M, and you wish to know what 7000 feet costs, place the square on 10 and 13, bring it down to 7 on the tongue, and we will find we have qtV on blade, or $9,10, which is the correct answer, as shown in No. 6, Fig. 20. If you wish to strike a circle and have nothing but a rule, apply as shown at No. 7. n i-: 11 PRACTICAL USES OP One noon a large crowd of workmen was asked by the foreman how to cut a third-pitch rafter so it would lay on half-pitch roof. It seemed to me to lay off half-pitch and then from that half-pitch line lay off % would cut it. I tried it, and we were all surprised to find it O.K., as shown at No. 8, Fig. 20. All of these problems, as formulated by Mr. Stoddard, are valuable in themselves because of their sim- plicity and because of their paving the way to many other things. Indeed, as I have often stated, there appears to be no limit to the use of the square; and I am sure there are hundreds of workmen scattered over the country that have found out things that can be done with this tool, of which we never hear, and I would like to impress on the minus of the readers of these volumes the fact that if they have any new "kink" they have worked out with the square, they will be doing a public good by sending a description of same to the publishers of this work, so that it may be published in future edi- tions, and thus saved to the trade. TO OBTAIN THE LENGTH OF A HOOP FOR A BARREL OR TANK BY THE STEEL SQUARE There are a number of ways by which the length of a tank or barrel hoop may be obtained, fe THE STEEL SQUARE 55 some of them being much easier than the one I am about to describe, as using a traveler, for instance, after the tub is standing, or stretching a tape-line arour.J the tub, and other ways; but where these methods cannot be applied — whiclr is very often — then the following method may be employed with profit: The diameter being known, the circumference may be obtained by the ordinary rule of multiplying the diameter by 3.1416, which will give the circumference nearly, then take a pair of dividers and strike a circle to a scale of say >^ or K inch to the foot; then pl ace the outside corner of a steel square to the center of the circle, as at A, Fig. 21. Referring now to the sketch, scribe along the outside of the square from B to A, and from A to C, then draw 56 PRACTICAL USES OF I a line from B to C, intersecting at the points where the lines previously drawn cut the circum- ference of the circle. Now obtain the center on the line BC, as at D. Take the square and place it with one edge at the center of the circle, cut- ting the line BC at D, and draw the line DE. Multiply the diameter of the circle by 3 and add the distance from D to E. For example, sup- pose the tank is 24 feet in diameter; the circum- ference would be 75 feet 6 inches; thus 3x24=72 plus the distance from D to E, which is 3 feet 6 inches, making 75 feet 6 inches. The sketch so clearly shows the method that further expla- nation would appear to be unnecessary. TO MEASURE INACCESSIBLE DISTANCES BY THE A'D OF THE SQUARE A number of wTiters on the s', -1 square have written on this subject and ha\ rot the matter down fine; but the best of the a. .icles I have met with are those of Lucius Gould of Newark, N. J., and A. W. Woods of Lincoln, Neb., the latter, in my opinion, being the better of the two, and it is from the latter that the following is largely taken, as it is placed before the readers in a sim- p unostentatious manner. Every mechanic knows that a triangle whose THE STEEL SQUARE 57 sides measure 6, 8 and 10 forms a true right angle and is the method commonly used in squar- ing foundations. But how many ever stopped to think what other figures on the square will give the same result? By referring to trigonometry we find only three places, using 12 inches on the tongue as a basis and measuring to the inches on the blade that do not end in fractions of i inch on the hypothenuse side. They are as follows: 12 to 5 equals 13, 12 to 9 equals 15, and 12 to 16 equals 20. Now, as we usually use a lo-foot pole to square up a foundation, we find that all of the above contain lengths greater than our pole, so we must take their proportions. The first contains numbers not divisible without fractions, consequently we will pass on to the next. We find that three is the only number that will equally divide all the numbers with quotients, as follows: 4, 3, and 5, but these are too srncul to obtain t h (' best results. fig. 22 5« r.M i I PRACTICAL USES OF Now let us examine 12. 16. and 20. They are «'vpr> numbers, and are divisible by 2 and 4. Fig 22 ^f we take one-half their dimensions, we have *:, 8. and 10. "he . beinjr convenient length^ and easily rcmrmoered. custom has settle 1 on tliese figures, i h: . are other places that 6, «. an « lo can be ''snd t ) ad. an .;^e. Simpi..;e /or .omc reason we want to know the C)stan^ on one arm of the square to 6, Fig. 30, on the other same as for a THE STEEL SQUARE 63 common rafter. This distance will mark the side, using the line formed by the arm of the square with the 6. Fig. 30 Cross over on the other side and cut, as shown. Fig. 31 AB, Fig. 31, represents the ridge or edge of the 4x4; line to the left of B the valley or side of the 4x4. It will be noticed that the ridge A is one-half the width of the width B. Of course this proposition holds good, and practical experience has so proven it. Any dis- tance may be used instead of those given, by applying the instructions as shown. lii ! 1" 1 3f t^t^ ^-«ST 64 PRACTICAL USES OP These instructions apply more particularly to roots when the pitches are equal, but there are many cases where the pitches are not the same on each side of the roof, and to meet this in- equality the following diagrams and explanations Fig. 3^ are given: Let us examine Fig. 32, here we have the valley AB and the run of the common rafters AC and AD, of unequal lengths. To obtain the cuts for the top of the valley rafter draw AE and I M THE STEEL SQUARE 05 AF at right angles to AB, extending them to the ridge line. Now AI^ and the length of the rafter on the square gives ihe cut ABE, and AF with the rafter gives the cut FBA, marking on the side representing the rafter. The lengths of these auxiliary lines may be obtained by laying the square on the roof plan and noting their lengths by the scale of the drawing, or, better yet, by a little simple proportion, as exemplified in Fig. 33, of the sketches— AC : AD :: AB : AE. Fig. 33 That is, take the runs of the common rafters, as AC and AD, on the square, place them on the edge of a board and mark along AC; then slide the square on the line AC until A'C equals CD; then A'E equals AE of Fig. 32. For the valley jacks use the length of rafter over AD with run 66 PRACTICAL USES OF 1 I AC for the angle DAB and the reverse of the opposite side. It is seldom if ever that a draw- ing is necessary. On the subject of the steel square as used in laying out roofs. I cull the following from an English source, which, while containing nothing new to those who have made a study of the steel square and its applications, yet is interesting as it offers another side light, as it were, to the sub- ject, and contains some things that may prove mstructive. Regarding the lengths and cuts of hip rafters on a pitch of 45°, the principle is the same in all pitches. Take the run and rise of the common rafter on both tongue and blade, and measure across, and the length of the common rafter is ascertained. Take the run on both tongue an^ blade, and measure across, take the distance obtained on the blade and the rise (12 inches) on the tongue, and measure across again, and the length and bevels of the hip rafter are found. Ihe above is to i-inch scale. Again, by taking 12 on both tongue and blade and measuring across, the actual length of rafter for I -foot run is found. Take 17 on the blade and 12 on the tongue, and measure across, and the length of hip rafter for i-foot run of common THE STEEL SQUARE 67 rafters is found; that is— if, say, the half-width of the building is 15 feet, take for the length of the hip the length of i-foot run 15 times, which is always 17 on the blade, and the rise for i foot on the tongue. If the instructions given are fol- lowed anyone will be able to get all the cuts, lengths and bevels for a roof of any pitch what- ever. Fig. 34 Supplementing the foregoing it maybe said that the line diagram, Fig. 34, shows the plan of one end of a hipped roof, the elevation of a pair of common rafters, and the development of the iour quarters of the hip. These will be sufficient to show clearly how the steel square is applied. ' 1 ■nv-mm^- av^vLa^a r .-rm -rai 68 PRACTICAL USES OP As both sides of the hip are alike, I have on the left side of the hip developed only one side. The process is as follows: First drop the point of the common rafter A to A', and draw a line from it to corner B. If this diagram is made on card- board to a scale of % inch to i foot, and the tri- angle formed cut through with a penknife from A' to B, and from A' to C, leaving from C to B as a hinge, also cutting through the lines from C to A, and from A to E, and folding this up on the line C to E as a hinge, raising the other tri- angle up and letting it rest on the first, one side of the hip will be represented in the position it would occupy when fixed; the points A and A' would stand plumb over the point D. Now apply the steel square, and note its position. Lay the tongue on the line C to B, which equals the run, and the blade on the line C to A, which is the length of the common rafter, while from B to A' is the length of the hip rafter. Marking alongside the blade, it will be seen, must always give the bevel for jack rafters. The numbered lines represent the jack rafters. On tne right side of the hip both carters are developed, because the run of one is :> feet, and t'.at of the other 14 feet. Now note the differ- ence in the application of the square. Take the THE STEEL SQUARE 69 run of the common rafter on the end, on the tongue, that is, from E to F (not from E to D), and the length of the common rafter on the right on the blade, and mark by the blade. This gives the top cut or bevel for the jack rafters on the right. Now take the run of the common rafter on the right side — that is, from D to E on the tongue, and the length of the common rafter on the end (which is the same length as on the left side) on the blade, and mark by the blade. This gives the top cut of the jack rafters for the end. If the triangles are cut and placed in posi- tions as suggested for the other side, the correct- ness of the measurement will be demonstrated. The length and bevels of the hip on the right side are obtained by taking the run of the end (16 feet) on the blade, and the run of the right side (14 feet) on the tongue, and measuring across, then taking the length thus obtained (which is the run of the hip rafter) on the blade and rhe rise (10 feet 8 inches) on the tongue, and measuring across, which giv^s the length, and likewise the bevels, of the hip rafter. G indi- cates the run of hip, and H the length of hip. The following is a useful application of the steel square: On the left of diagram, 8 and 12 inches on the square, cut the common rafters; if 70 PRACTICAL USES OF I i it rises 8 inches in i foot it will rise lo feet 8 inches on i6 feet. Using J<-inch to i foot scale, place the square on the line (Fig. 35) at 3 and 2, representing the 12 feet 8 feet. Now slide the square up, to bring 4 on the line, as shown by dotted lines, which gives 10 feet 8 inches rise in 16 feet, which may be stated thus,— 12 : 8 :: 16 ft. : ID ft. 8 in. It will be seen that the steel square, as a mechanical device, will solve problems both in square root and simple proportion. Perhaps the following examples on the subject which were submitted by correspondents to "Car- pentry and Building" may prove useful to my readers, as they contain several good ideas which are worth considering. I have made some slight THE STEEL SgUARE »i changes in the text in order to make it suitable to these pages, but this does not affect the sub- ject-matter in the slightest: In the lay-out shown at Fig. 36, we have a 3x6 valley rafter and find Fig. 36 the center of the face side at the point A, where it intersects the two ridges, as shown in Fig. ^-j, the center line being AB. Now measure one- half the thickness of the rafter— that is, ij^ inches from A, which gives the point C. Squar- ing across gives the points D and E. Connect- ing the points A, D, and E gives the angle cut i'i m P';1 - 'i 7i • V. practilal rsFs op where the ridges meet at I), as shown in sketch It must be evident that if C is raised to stand directly over B we get the angle f - slope of the valley rafter, also ti.c angles of bevels. In the Fjg. 37 diagram shown at Fig. 37. the square is set to show the method of getting the proper cuts and lengths. AB shows the horizontal line or seat of valley, while AC shows the run or length of val- ley rafter. CB on diagram, Fig. 36, shows the rise of the rafter. If the distance CB be used on the tongue of the square, and AB. the seat, set off on the blade, these will give the plumb and bevel cuts of the rafter; while the bevels shown in Fig. ^j give the cuts for sides of hip or valley. k: VALLEY RAFTER Fig. 3S A plan of the valley rafter as laid off for cut- ting bevels, is shown at Fig. 58. THE >TEEL sQUARE 73 Fi^' 39 shows position of rafter where ridges meet. Fig. 40 shows an elevation of the val- ley rafter in its proper position on the wall plates. VVc will sap- pose that the rise is 9 inches t' the foot run, as sli v.^ w n . in which case the ly- pio. 39 -%*^^ J ■ m Fia. 40 74 'it PRACTICAL USES OF pothenuse or line of rafter is ,5 inches to -foot run. F.g. 4, shows tl,e method of obtaining tTe PlO. 41 bevel. Take 15 on the blade and r^ on the tongue of the square, place it on the rafter as shown and the tongue will give the desired bevel marking along the blade. In order to frame a rafter against two ridge boards running at right angles, draw a line in the center of the rafter and reverse the square. This rule works on all pitches. Again, suppose the half-width of a roof having a pitch of 45° is 10 feet, and that an adjoining roof IS one-third pitch, then it will take 15 feet of It to make an equal rise. By the conditions of the problem we then have a rectangle loxi. feet by which to get the length of the valley rafter sought. A line drawn diagonally through this rectangle will give the run of the valley THE STEEL SQUARE 75 rafter. Lay off at right angles each way from the diagonal a distance equal to the rise, Fig. 42, Fig. 42 and connect it as shown in the diagram. This will give the length of the valley rafter. Let fall on each side of the diagonal a perpendicular equal to the half-width of the rafter terminating, at the sides of the figure. From B and C thus established let fall perpendiculars BA and CD »} i t| 1 "M A If " M 7« PRACTICAL USES OF from the line of the valley rafter. Then AB will be the backing or distance above the edge of valley to set the jacks for the 45° pitch, and CD will be the height above the edge of valley to set the jacks for the third pitch. The line on the plate will be ob- tained as shown in Fig. 43, using the square with lOon the tongue and 15 on the blade. From W. H. Croker, of Oril- lia, Ont., who is }■ 44 i J: i THE STEEL SQUARE 77 an excellent authority, I get the following on the same subject: Suppose the plan of che plates is AB, Fig. 44, in any given building, and the cor- responding rafters A'B'. Where the top lines of rafters intersect, marked i on the elevation, drop ' a plumb line 14 to intersect 4-6, made at an angle of 45°, and passing through the internal angle of the plates. At any point eaveward draw 2-3 horizontal, and from the point of intersection 3 drop the plumb line 3-6, and from where it inter- sects the line 4-6 draw 6-5 parallel to 2-3. Make 6-5 equal to 2-3. Then a line drawn as shown by 5-4 will be the plan of the center of the valley rafter. One-half of the thickness of the rafter laid off on each side of 4.5 will determine the relative position of the valley rafter to the plates In order that the student of this work may be armed with the proper theory underlying the formation of hip roofs, I submit the following which IS taken from Peter Xlcholson, whose methods for finding "working lines" in timber framing have never been excelled. Let abed, Fig. 45, U, the plan of a roof, wy the width or beam, ix the height of the roof, zox and i.'y the length of the common rafters- to find the length of x\^v. hip rafters from the data here given proceed as follows: W ^^■- .■ ii mmmm. I 78 f r t PRACTICAL USES OP Bisect each end of the plan a6, cd, in the points y and 4. and draw the plan ^r of the ridge line. li.. THE STEEL SQUARE ^^ Bisect the angles at a, b, c, dot the roof, by the lines as, 6s, cm, dm, meeting the plan of the ndge line in . and m and the lines as, 6s, cm dm are the plans of the hip rafters. From the pomt where the plans of the hip rafters meet the ndge Ime, draw a perpendicular to each of the h.p rafters, and set the height ix of the roof upon each perpendicular; and the hypothenuse of each nght-angled triangle will be the length of each hip rafter. Thus find th^ hip rafter over dm: Draw mz perpendicular to ./...; make ^uz equal to ix, and join mz; then mz is the length of the pnncipal rafter over dm. The hip rafter may be found very conveniently 'n the following manner: Produce iw to /• make tt equal to md, and draw tx, which will be equal to dz; and thus the remaining three will be found To find the backing of the hips, draw ^S at right angles to md; from the point /as a center find the radius of a circle which will touch the hne^.; make .^ equal to that radius, and jom^/and ,c; then the angle .^y is the angle of the back of the rafter. This method of find- ;ng the backing of hip rafters is said to be the -nvention of a Mr. Pope. In this illustration and 'description almost every possible shape of hip roof plan is involved, and from it hips and i ."fi •.a . 1 J >i. w : rm \t: 80 % I I PRACTICAL USES OP jacks may be determined with their lengths, bevels, and inclinations, without much trouble! It will be noticed that the steel square is not employed in this description, or in the illustr^x- tion. This is due to the fact that Mr. Nichol- son's works were published long before the American steel square came into general use. T: '1 illustration is the most complete of its kind known, and this is partly my excuse for its reproduction in this work. II 1 ti UNEVEN PITCHES Irregular, uneven, or unequal pitches, are simply different pitches in the same roof. When they are the same on all sides and the building is square, the hip. or valleys run in from the corners at an angle of 45 degrees, regardless of the rise of the roof; but should one side be steeper than the ad- H jommg side, or the «H gables be of different "^ pitch from the main roof, then the hips or valleys depart from the 45" angle. FlQ. 46 THE STEEL SQUARE gi Fig. 46 shows a roof plan with the one-third pitch on the main part, with a half-pitch gable. The seat and down cuts of the jack and com- mon rafters remains the same as in the even- pitch roof, except the top cut of the jack. I will not take up space to explain this cut at length, but will give that obtained by the square as follows: Take to scale the length of the left common rafter, on the blade and the run of the right common rafter on the tongue. CuToruerrJApr Fio. 47 Blade gives the cut of the left jack-z./s o feet 8 inches, run 1 1 feet, length of valley rafter 13 feet, Fig. 58. L '''''■■■' '^J^' ' ' f-fe- Fig. 59 As the rise of front gable is 6 feet 8 inclies and run 4 feet 6 inches, length of gable rafter 8 feet, As the length of common rafter on main roof '\ i>. :',j!AOTsai-cifr-«f^'" S?^ !^?^tJ^ i I MI I ' L . 90 PRACTICAL USES OF is 12 feet and run of gable 4 feet 6 inches, place the square on length and run cut on length, and Fig. 60 it gives sic- cut of main jack to fit valley. Fig. 60. Fig. 61 As the gable rafter is 8 feet and run of main roof 10 feet, length and run cut on length gives side cut of gable jack. Fig. 61. Most workmen who have followed all the examples given in this work are aware that the rise of the valley or hip taken on the square will give the seat and plumb cuts, but to cut the seat so that the top edge, backed or unbacked, will coincide with the plane of the common rafter is a problem that many are not so sure of; but go ahead and make the cut, trusting to luck, and if it doesn't come right, block up or cut down as the case may be, and the matter is dismissed for the time, only to reoccur on the next job. THE STEEL SQUARE In order to enable the workmen to get positive results, the following illustrations and text are submitted; they have appeared befor? in a different shape, but, as I stated in the outset, it is my intention to publish in thf" work every- thing that in my judgment, will oe of service to the reader and that is in anyway connected with the use of the steel square. The illustration shown at Fig. 62 exhibits the position of a hip or valley rafter when the roof is of equal pitch. A, being at the corner of plate for either a hip or valley. If the former, it sides will intersect the edge of the plate at B and B, or at C and C, for the latter. The distance from A to B and B, or C and C, is always equal to the diagonal of a square with sides equal to one-half the thickness of the rafter. If the rafter be 2 inches, then the distance will be lA inches. BC (along the side of the rafter) 's equal to the thic' of the rafter, and this measurement taken square out from the plate at BC, and by transferring the center as a, will give the different positions of the seat cuts with that of the common rafter. Now passing up to the common rafter, DE is the depth desired from the plate to the top edge of the rafter. 1 4 ?t PRACTICAL USES OF li M i 1 1 ii Fig. 62 fftmWyp^ f i % i fe f i t I 94 PRACTICAL USES OF with the hip swung parallel to the common rafter, AC and Ali being their respective lengths. Pig. 63 A method of laying out a hip roof and making a cardboard model for same was published in "Carpenter," some time ago by Mr. Henry D. Cook, of Philadelphia, Pa., and which is repro- mm^ T^i ^^mwmm<^' ;:$k?*iKi?»vi;^ h'>t'^*^Ml^ THE STEEL SQUARE 95 cluced here, as I think it worthy of a place in the present work. The most simpk fornj of hip roofs is tnat where the ground phm of the building makes right angles. In the ordinary hip roof but little constructive skill is required, the onlj points requiring particular attention are in finding the proper lengths and side cuts of jack rafters, and those can be made quite simple. To do this, sup- pose we get a piece of cardboard and commence laying down the ground plan of a building, which we will represent by letters A, B, C, 1 ). Next lay out the elevation of one pair of rafters B, E, C. shown at A, Fig. 64. Next lay down the seat of the hip at an angle of 45 ; on each side set off half thickness of hip which draw parallel with cent, line AF; from the seat line AF square out from F to G; make GF in A, Fig. 64, equal HE in the same diagram, and square out lines IJ and KL, and join AG, which gives the back line of the hip rafter; next layoff the seqts of the jack rafters on line AD, and make MN equal the given rafter BE or CE, and join I), N, on each s^de of which set off half thick- ness of the hip; next square over the seat lines of the jacks on line iD, and let them cut the seat of hip as represented on the plan; then with ■ m |.!l'' 'i IT a W^^--'!im^W^WM'JP: ^ f.g. 67 inches is^the length of the common rafter to the I -foot run, and the 8 inches rep- resent the rise. i' n 1,1 1 1 J For the edge bevel 01 purlin, lay the square flat agamst the edge of purlin with 12 inches on the tongue and 14H inches on the blade, as at Fig. '^9. and mark along the side of the tongue. This f^'ves the bevel or cut for the edge of the purlin. i .1 I I04 PRACTICAL USES OP The rafter patterns must be cut half tl j thickness of ridge shorter, and half th*: thickness of the hip rafter allovi-ed off the jack rafters. A few remarks regarding the backing of hi[) rafters and the getting of the pioper lengths of jack rafters, or 'cripples" as they are called in some sections of the country, and 1 have done with ordinary roof framing for the present. I have shown in several instances how the lengths of jack rafters and their bevels may be obtained, but I have not specially shown how these results are ol,taincd, so will devote some little space now to this purpose. Let us suppose AB and BC, Fig. 70, to be hips, and AD and CD val- leys laid out from any particular plan, then the jacks cutting in between valley and hip may be laid out as shown at FE; the bevels shov/ing the angles of the cuts, the plumb cuts being the same as for common rafters. The bevel at E shows the side cut against the hip, and the bevel at F the side cut against the valley rafter. Fic. 70 THE STEEL SQUARE 105 Another way to determine the length of jack rafters is given as follows: On the ste 1 square, take i? inches on the blade and the rise of the roof, 12-foot run, on the tongue, and measure the distance across. This length in inches, multiplied by the number of f<;et the jack ratters are to be on centers, will give the required length in inches. For example, if the roof rises II inches per foot run, measure the distance from II on the tongue to 12 on the blade of the square, which is 16^ incl.es. Now, supposing the jack rafters to be 16 inches, or i}i feet on centers, we have i6>'x 1^ = 21^ inches, which is the difference in the lengths t : the jack rafters. The lengths may also be found by first getting the length of the common rafter in inches for a 12-inch run, and multiply this by the distance in inches the jack rafters are to be from center to center, and divide the result by 12. This gives the difference in the length of jack rafters in inches. For example, if the rise is 12 inches and the run is 12 inches, the run of the rafter is nearly 17 inches. Now, 17 multiplied by 28 and divided by 12 gives 39-3 inches. This is the differ .ICC in the length of the jack rafters for a one-half pitch roof where the jacks are 28 inches io6 ii I PRACTICAL USES OF from centers. This rule will work on any pitch of roof. ^^ Mr. Hicks pives the following' rule, in his "Euilder's Guide," for obtaininjr the len^jths of jacks, which is somewhat similar to that already shown: Take the run of common rafter on the blade, .2 inches, and the len^nh lJ^,\ inches, on the tongue, and lay a straight edge across, as L "'""'' Fig. 71 shown in Fig. 71. Space the jacks on the blade of the square, which represents the run of com- mon rafter, and measure perpendicularly from the tongue to the straight edge on the line of each jack for their length. For cutting jacks for curved roofs, while not exactly within the scope of the steel square, yet the bevels may be laid off by that instrument as the reader will no doubt discover; so I give here Mr. Hicks' method of determining th T lengths. THE STEEL SQUARE 107 The curvature of these rafters will, of course, be: governed by the position th<:y ocf upy with rela- tion to the hips. The method offered is not new by any means, but is presented in a manner easily to be understood by the ordinary work- man. Let us suppose AI), rig. 72, to be the run of the common rafter, DE the rise.and AE the length and work line. To find the length of jack set off the run of jack AB and square up the rise BC to the work line of the com- mon rafter; then AC is the length of jack on the work line. This method* IS very simple, yet, as it is a new and novel way of finding the length of jack rafters, it will be well to point out a common mis- take which the inexperienced might chance to make. Bear in mind that A E is the length of com- mon rafter. BC is not the length of jack, as some might suppose, but the rise of jack; AC is the length of jack. The down bevel is the same as that of the common rafter. To find the bevel across the back, set off from D the length of Fig. 72 ' sl ll i if 1 if JjyuJ io8 PRACTICAL USES OF common rafter to F, and connect F with A, which shows the work iine of the hip. Now continu.- the hne BC to the work line of the hip, and the bevel at G will be the bevel across the top of jack. BG is also the length of jack, and wi!l be found to be the same as AC. When the bevel of the jacks is known all that is necessary is to square up the rise of each jack from the base line of common rafter AD to the work line AE, and take the length from A to the point where the rise of each jack joins the work line of common rafter, as shown. In connection with hip rafters for curved roofs, it may be well at this point to depart from the course pursued so far in the making of this book and give the ordinary lines fcr laying out such work without using the square for the purpose. We will suppose the lines A and B, Fig. 73, to represent the common rafter for a curvilinear roof, let B represent the co.nmon rafter, and C the valley rafter. In plan and profile, respect- ively, the curves of common rafters being given, first determine the seat, or base line, of valley rafter, which in roofs of this kind is curved. To illustrate, the common rafters were cut and put up on both sides of valley strips, tacked on one THE STEEL SgUARE 109 sulc parallel with eav<; or ridge, and the same number of strips on the: other side, tacked in the same manner and at the : ame verti' al hcij^ht; it Fir,. 73 is evident that their intersections would repre- sent the line of valley. Therefore, the curves of common rafters being drawn, divide, for exam- ple, B into any convenient number of parts, and through the points thus determined draw hori- zontal lines. The lengths of these lines are i ii ! 11 i « j S .1 5 .i.ij 4, -■? m *! no PKAlTICAL USES OP det(.'rmin<'(l l)y when- they cut th ift (' curve of com- mon rafter, and art; s«,t off on correspond rafter in plan. Then draw 1 ponding ines parallel with eaves or ridye, as shown in the sketch, and where these lines intersect is the base line C of valley rafter. The same points of intersection project on corresponding lines of profile, giving the line of valley rafter. To obtain the lateral curve of valley rafter, take distance on outline of valley rafter, as made' by .he horizontal lines already described, and draw lines, as shown, between X and Z at the right of the sketch. Draw a vertical line cutting all these lines, and set off on each corresponding line the same lateral distance as between the straight and curved base line in the plan. To find the top bevel of valley rafters, let the thick- ness be as shown at D in plan. Project the length of bevel by half the thickness on curved line in profile. Project the length thus found on corresponding line on upper face, as shown by XY at right of the sketch. From the two points thus determined draw a line, which will be the true bevel. In connection with curved roofs, the following is offered as being a good method of getting the side bevels and lengths for jacks in a hipped '^'^'i^iSf^^i^zmrmf. -■■■i. ■ hk' a ( r / F e .y THE STEEL SQUARE ,„ roof: Let CB be the top line of one of the com- mon rafters. In the diagram, Fijr. 74, CHU \^ supposed to stand upright on rise HC In shape all the jacks must he some parf;of the length of th<; com- mon rafter meas- ured from point C. On one common rafter lying on a F";. 74 flat surface with marked run and rise must be laid off all the jacks showing vertical cut also long and short top edges opposit.- to each other. On run make CH equal to F£G. long seat line of this jack. At right angles to run draw H K. the verti- cal cut. Then CK is the long top etlge of this jack. I- or the opposite short top edge :. aw a line parallel to the vertical cut at a distance back equal to IG taken from seat of jack. On top face of jack mark the side bevel from end of long edge to end of short ^.\^^. It ,s evident that when jack CK stands upright over its seat its bevelled top nd will fit against the hip face which stands over BC, because long top edge of uy-k stands over long seat line, and short top edne of jack >Un(is over short seat line; only if ABC is an angleof45MoesFG equal thickness of jack. For -I 112 PRACTICAL USES OP 1 1 I : - I ; :■ I each jack the side bevel will be different, but can be obtained in this manner. Before leaving the subject of hip, valley, and jack rafters with regard to their lengths and bevels, I think it will be in the interest of my readers to reproduce a system of lines first K ' A w [ — r \\. *2^y ^ %e ^^'^ M ^ X \ \ \ ^ X "'1 - M \ \ '\ /-\ i^ ' \ f J /\ V \ \ rt ^ \ \- /. A/ -7^ . /. /I M / F f \ N Fir,. 75 invented by Peter Nicholson, and simplified by Mr. Smith and published in his "Architect," a THE STEEL SQUARE "3 book that at one time had a deserved popularity. While the book is now seldom spoken of, this system of lines has been made use of by nearly all the late writers on constructive carpentry, with greater or lesser elaboration. On Fig. 75, the plan 1,1,1,1,1 represents the outside plate; 2,2, the ridge line; 3,3,3.3-3.3. the jack rafters of hip and valley; 4,4, the side bevel of jacks and the length of jack from corner of plate and ridge to side of hip and valley; 5, bevel at head of hip and valley; 6, bevel at foot of hip and valley rafter; 7 is a common rafter; 8, the bevel at head of common rafter, is the down bevel for all jacks on hips and valleys; 9,9 is the length of hip and valley rafter; 10 is the method of get- ting the bevel of back of hip. Draw a line at right angles with base line of hip, then set one foot of the dividers where this line crosses the base line, and the other where it crosses the hip- rafter line, and set the same distance on the base line, and draw lines from that point to the plate each way, which gives the bevel for hip, and, turned the other way up, it gives the hollow for the back of the valley. Line from a to d h the length of hip and valley dropped down to get the length of jacks. Lengths and bevels of all hips and valleys the same in same roof of same pitch. • -i 114 PRACTICAL USES OF ^ ^i Fig. 76 is a plan for framing a valley in a roof where one side is much steeper than the other, Fig. 76 as, for instance, one side rises, say, to feet in 8 feet, i-i is the wall line; 2-2 is the ridge line; 3 is the valley rafter; 4 is the bevel at the foot; 5 is the bevel at the head; 6 is the bevel of the jacks on the lowest pitch, also the length of same; 7 is the bevel of and length of jacks on the steep side; g is common rafter on the lower pitch; 10 is the down bevel on jacks of each side; 1 1 is the height of roof; 12 the base line of val- ley. The rafters will not matcl. on the valley as THE STEEL SQUARE "5 on an equal-pitch roof, as in Fig. 75. It will be seen that it will take seven jacks on the steep side, while it requires only four on the other side, but the bevels will all fit. BACKING HIP RAFTERS A writer on Building Construction has said: "In America there are more hips 'backed' in books and papers than there are in houses." Unfortunately this is too true. Most workmen ne\er think of backing a hip; they put in the timber just as it comes from the lumber yard, with the exception of cutting the bevels for point and heel of rafters. This is all wrong. All hips should he backed, in order to get a good strong and nearly perfect roof. When the hip is thin, being no more than two inches thi'^k, it is not so bad, yet it ought to be backed; bui when the hip is three or more inches thick, then under no circumstances should backing be omitted. In the present volume, as well as in the former one, I have shown some rules for getting baciv- ing for hips, but in order to have the principle well understood I present a few more examples showing how the angles may be obtained. The method shown at Fig. 'j'j is a quick one, and a correct one if the measurements are i ■ ' rl -p I i WM ate Ji i I ll ii6 PRACTICAL USES OP exactly taken. The diagram explains itself and requires no description. When the horizontal Fig. 77 or bottom cut for a hip rafter has been obtained, take one-half the thickness of the rafter and measure back from the toe or point toward the heel. This will give the point on the side of the rafter i-c gauge to. Then a line on the center of the top of the rafter in connection with a line gauged on the side, will give the bevel or back- ing. Another example of backing and I have done. Let us suppose AB and BC, Fig. 78, to represent the plates of the building, and BD the hip rafter. THE STEEL SQUARE "7 BE being the seat of the rafter. Take any point of the hip, as i. Draw a line at right angles to Fig. 78 this, producing it until it cuts seat BE, as shown in the point 2. From the po! t 2 thus estab- lished draw a line perpei licular to the seat, pro- ducing it until it cut ; line of plate AB. Transfer the distance 1 along the line repre- senting the seat of the rafter, thus establishing the point 4. Draw 3 and 4; then at 4 ••11 be given the bevel for use in backing the rafter. Fig. 79 shows the application of the bevel to the timber which will give the gauge points to work from. '■ li ■-. •-•t • t I H ii8 PRACTICAL USES OF These examples, with the ones on the same •bject illustrated and described in previous pages, shoulw prove quite am- ple and are varied enough to meet the requirements of most workmen no matter what may be their constitu- tional peculiarities. Fig. 79 FRAMING OCTAGONAL ROOFS, DOMES, BAYS AND OTHER OCTAGONAL WORK We now enter another phase of the carpen- ter's art, and one in which the steel square plays, or can be made to play, an important part. I have discussed the "octagon" pretty fairly in the first volume of this work, but very much more than I have said, or can say for that matter, may be said on the construction of octagonal work; in order, however, to make this work as com- plete as possible I have thought it necessary to present to the reader the following illustrations and descriptions, knowing from experience they will be useful. I have shown how the miters in polygons may be obtained by aid of the square and by other methods, and, as a sort of introduction to this chapter, I offer the following which I know will THE STEEL SQUARE 119 \ be acceptable to many of the readers of this book who have been fortunate enough to get a fair public school education: There are three kinds of angles: the right, the obtuse, and the acute. A right angle is an angle formed by two lines perpendicular to each other. An obtuse angle is greater than a right angle; an acute angle is less than a right angle. All angles of the octa- gon are obtuse. A right angle is equal to 90°. The angle ABC, _r - >. Fig. 80, which is one of the angles of the octagon, is 45' greater than a right angle, and is equal to 90°+45°= osi'" 135'. The octagon miter is an acute angle, and is found by bisecting 135°, Fig. 80 which is 'i''=67~ 30', which is shown at ABD. I will now show the proportionate length of each line in the octagon, Pig. 81; the diameter being one, the number on each line indicates its exact length in fractional parts of one. To lay out the miter or angle, place the square as shown at ABC; take 12 inches on the blade of the } m u^ , i t ■ 1 lao PRACTICAL USES OF square and 4^ on the tongue; tongue gives cut. Any other number will do as well, providing the "l proportion of 3827 and 1585 exists be- tween them. This is a simple and correct meth- od of finding the miter of an octa- gon, and will be found useful in solving many prob- lems that confront the workman from time to time. Pi'-. 81 BAV WINDOWS Often workmen are put to their "wit's end" when "laying out" an octagon bay window, owing to the surrounding conditions. The following is submitted, which shows how the faces or sides of the wmdow or other w^ork may be laid out with ease: First lay off a straight line DA. l-jg. ^2, to the length desired for one side of the win- dow, as indicated from A to B. Then from B to C make the length A of AB. The length CD IS to be the same as AB. Now, with the foot of THE STEEL SQUARE lai the compasses in D, and with radius DC, strike an arc as shown. Then, with the same radius • E -^ r. Fig. 32 from A as center, strike the second arc indicated. With the dividers set to the same distances and with C as center, strike an arc, cutting the arc ^iruck from A, thus establishing the point F; then, in the same way, using B as center, strike an arc cutting the opposite arc, establishing the point E. Draw the lines DE, EF, and FA, the result will be three equal lengths and three equal angles. To find the center of the octagon, draw lines through the points FB and EC until they intersect in the point G; then G will be the cen- ter as required. The lines FB and EC will be the seats of hips, if any are desired. To lay off i"' if il; .* laa PRACTICAL USES OP .T^'t 1 ii I i an octagon end of building, as is often done, divide the width of the building into 29 parts, and take 12 parts for each of the extreme spaces and 5 parts for the mean space, and proceed as above. If we wish to make the front side wider than the other side— for example, 2 feet wider— deduct 2 feet from the width of the building; divide the remaining space into 29 parts, take 12 parts each for the extremes and 5 parts plus 2 feet cut off for the mean space, and proceed as above, save that in crossing the arc at E wt must set the compasses 5 parts from C. or at I, all as shown in Fig. 83. And in crossing at F we set 5 parts from B, or in the point H, a*- shown. Then we have the front side 2 feet longe- than the others, and the angles the same. Three sides of any figure composed of more than four sides can be produced in the same general manner. How- ever, the ratio between tho mean part and the extremes will be different. Thus, in a figure of seven sides the mean part will be one-fou' h of the extremes. Whatever the mean part is, the Fig. 83 :i ) ' l-i^i - THE STEEL S(^UARE i»3 .ides will be equal and the angles at E and F will be the same. Fig. 84 OCTAGON TOWKKS .\Nn SPIRES Now iiat we know how to lay out the base of an octagon and how to lay off a part of the fijrure for a bay windc w or other similar work, it will be in order to see how the framing is done for an \ m 194 PRACTICAL USES OP octatronal tower, spire, or other simihr structure. Suppose we have a tower to erect which is partly over another roof, as shown at Fig. 84, where the intersections occur. It will be seen that the tower intersects the hip roof, as A,B,C,D,E,F,G. Fig. 85 Before the intersections shown by Figs. 84 and 85, and the timbers shown by Figs. 87, 88. and 89 lit THE STEEL SQUARE »«1 *l i-' i\ ! : t! I ' I ' I Fig. 86 mLM hu ^ m^:^*^> '^^g^Jil^^^, Fig. 87 can be properly set out, it will be necessary to ii_ mmmi^M^^3t^^'^^ii^J(fi--^.^^M- .5N&^..:^-m'x:'::^mri >< THE STEEL SQUARE ,,7 obtain the Intersections of the boarded surfaces {^n^ometri-ar-y. The method of doing this is shown ,>y Fig. r.r,, .n.d is as follows: Set out the half oc ayon A^iS B, which is the line of board- FiG. 88 ing. The other half, A3456B, it will be noticed, IS a little less, this being the line of rafters. To avoid confusing the diagrams with a number of lines, several of them have been omitted; it would of course form a smaller parallel octagon to those shown. Next set out line CD. which is the line of feet of rafters, and P:F, which is the line of face of the fascia board of the main roof, also the line of the main hips, as shown at GH. hi ij !i1 138 PRACTICAL USES OP At right angles to i-8 draw OP, and at right angles to this line set up OR, making it equal to the height. Join PR, which is the true inclina- tion of the sides of the lower roof. At any point along EF draw xy at right angles to it, and set up the pitch of the main roof as shown at xs. Fig. 8g Now take any point T on this pitch line, and project down at right angles to xy, meeting it as shown. From O mark off OV equal to the height TU. From V project across to W, .^^ff^fJIMi^3V»mm*L«_. THE STEEL SQUARE 139 parallel to 1-8, which will meet TU in a. Join Ea, which will give the intersection of the sur- face 0-1-8, and the main roof. For the next inter- section, from where Wa cuts 0-8 in ^, draw a line parallel to 7-8. Now produce TU, which meets the last line in/. Then from d draw through/ to meet 0-7 in ^. This gives the intersection of the main roof with the triangular portion, 0-7-8. The side B7 should be continued so as to meet EF in //. Join % and produce to G. Then ^G is half the intersection of the surface 7-0-6. Work- men having a knowledge of geometry will see that the principle of working this has been based on a problem in horizontal projection, the spe- cific p— blem being. "Given the horizontal traces and :linations of planes, find their intersec- tions.' If it is desired to obtain the developments of the several surfaces, they can be obtained in the following manner: Draw PO at right angles to 1-8, and OR at right angles to OP. Measure in OR the height. Join PR, which gives the incli- nation and true length of the center line of the full surfaces. Bisect line 2-3, and at right angles to it draw AK, making AK the same length as PR. Join 2K and 3K, which gives the true shape of each of the full surfaces. This development lki*i^9mWiWA% m ,V'.. 133 PRACTICAL USES OF can be used to shr vv the correct shape of the surfaces which intersect the roof. From i project up to meet tl.o line PR in //; make AZ equal to Fd'; then through Z draw line ,i^7.M parallel to A3; next make 3L equal to E8. Then 2LMK is the true shape of the side IFJO. From ^ draw a line parallel to 7-8, meeting 8-0 in K; then from the point K draw a line parallel to 1-8, and continue it to meet PR, and meeting />. Now measure off on the line AE a distance AX equal to P/. Through X draw a parallel to 2-3, meeting 3K in point X. Join N^/; then Ny/- is the true shape of the surface _ir/)o. From G draw GwHS parallel to B78P. Trom s project up to meet PR in r. Make A;/ equal to rP. Join X^^ and nv; then X//<'K is the true development of the surface JG^O. The method of obtaining the bevels of the several parts may now be described, the meatr-- of obtaining the backing of the hips br^ing fir'-t shown. At right angles to 0-4 set 0-9, and make it equal to the hef'^ht; join 4-9, which gives the true rake of the hips. At 11 is shown the bevel for the vertical cut of the top, and at 12 that for the foot. As will be seen, one edge of this bevel is adjacent to* the pitch line; the other, being horizontal, is drawn parallel to 0-4. For the *ll^'- :^vm^j!^^9m'tj^ THE STEEL SQUARE 131 backin{( of the hips, join 3-5, and from where this line meets 0-4 in point 10, draw an arc tangent to the pitch hne 4-9. From where the arc meets 4-10 in point 11, join to 5 and 3 as shown; then D is the bevel required. The bevel for where the hips meet each other is shown at 13. Reference to Fig. 8} will show where this bevel will be required, and also that the upper part of the mast or central post is octagonal. This allows the upper cuts of the hips to be made square through their thickness, and therefore no bevel is required. The develop- ment of the intersection shown at 3MXKe'^2 gives us the bevels for the feet of the hips and rafters :A, B, C, D, E, F, and G, Fig. 87. The bevels 14, 15, 16, and 17, Fig. 86, are the feet of the hips A, B, E, and F, respectively, Fig. 87. These bevels are for application after the hips have been backed. The bevels to apply to the backs of the jacks at C and D, Fig. 87, are shown at 18, Fig. 86. Whilst the bevel for the foot of G, Fig. 87, is shown at 19, Fig. 86, it will be noticed that the valley rafters shown by i, 2, 3, Fig. 87, have their upper edges in the same plane as the main roof; therefore, it will be necessary to obtain a bevel for the prej^aration of these edges. The geo- Jl M ^?&l^.V^.m*^- :W. '3* PRACTICAL USES OP metrical construction for this is as follows: From any point in the plan of the valley, as H, Fig. 87, draw the horizontal lines HK and HL at right angles to 2-3; then at any point on HL draw X Y at fight angles to it, and cutting the line HK in K. From where HK cuts the pitch of the roof, as shown at M, draw MX at right angles to HK. Then from L drop a perpendicular to MX as shown. Xext project Li* at right angles to XY, and make it equal in length to XO. Xow join KP; then with L as a center, draw an arc tan- gent to KP, meeting XY in R. Join RH, which gives the bevel required as shown at 25. The bevels for the jack are shown at 20 and 21, Fig. 86, whilst the bevel at 13 is for application to the tops of the jack rafter, or these bevels may be obtained by the steel square as shown in previous examples. The methods for obtaining the bevel for the jack rafters for the main roof may be obtained by the square, as shown by 22. 23, and 24, Fig. 86, respectively. The hip of the main roof, as will be noticed, requires supporting at the lower end. This is done by placing a dragon beam across the octagonal space, as shown at S and S, Figs. 87 and 88. Then the end of the hip should be notched, as shown at Fig. go. The ceiling joists in the octagonal space ..^H^f . 'w : '--' THE STEEL SQUARE . ,33 are fastened into the wall as shown on plan, Figs. 87 and 88. Of course, as is usual, -he ceil- Fio. 90 ing joists should be on the same plane as the ceiling joists in the main building; the ends of four of these, U, V, W, and Z, cannot be carried to any wall, therefore a trimmer is provided of stouter scantling to carry these ends, as shown in the plan and section, Figs. 87 and 88. The boarding is clearly shown in Figs. 84 and 85, and therefore does not require further description. There are other little points which are fully shown in the illustrations, but it has not been thought necessary to enlarge upon them here. m .4%^ ' %JM :! 1 i • 1 '34 PRACTICAL USES OF At Figs. 88 and 89 the man- ner of construction is shown, including projeciion of raft- ers over eaves. It will be noticed there is a center post to which the hips or corner rafters are nailed. This post is not absolutely necessary, but when it can be used it is a great help to rapid construction, and cer- tainly makes the work stronger. The pitch of a tower roof may be obtained along with all the bevel lines by a proper use of the square, as shown in P'ig.gi, which 'llus- trates some unusual pitches. It is evident that if the run of one foot is 12 inches the run of two feet must be double that or 24 inches. Therefore the rise must be that proportion of 24 inches. The first inch in rise is sV, CZH] the second ,'2, and the third t*ftt-i -SAtm ■i/ Pitch -2PiTeii -li Pitch iSj-I Pitch Fig. 91 lis ^j^..\^. THE STEEL S(JUARE ,35 14, the fourth /t,, etc. The twenty-fourth inch rise being equal, the span is therefore i pitch. As the rise continues above this point, it is simply a repetition of the above with a i prefixed, thus: Th- tvventy.fifth inch rise being a pitch', etc.; but we are now beyond the limits of the full scale as applied to the square, so we must reduce the scale. By letting the vertical line ai A represent the blade we will have reduced th(- scale one-half. The pitches would center at 6 on the tongue instead of 12. as in the full scale. We must now use the half inches above 12 on the blade for each inch in rise till we reach the twenty-fourth inch which will be equal to 2 pitches or 48-inch rise to the foot. For steeper pitches it is necessary to again chan - the scale. • If we let the blade rest at B the patches will center at 3 on the tongue (ma- king the scale % size), and by letting the K inches abo/e 12 on the blade represent the full inches in rise will give the cuts, etc.. from the forty- eighth inch rise to the ninety-sixth inch rise to the foot, or 4 pitches. One of the methods of laying off the lines for an octagonal roof having curved rafters is given at Fig. 92, where a method of obtaining the ! i - 1 'f lli f i' r 136 PRACTICAL USES (JF .:urves Is srlvcn. The plan is shown by the octag- onal figure, and we will suppose it to be 20 feet -8 4 Fic in d iameter and the rise of roof 25 feet. ABis THE STEEL SQUARE ,.„ the run of t^c common rafter and AC its rise. Divide AB into as many parts as may be neces- sary and square up from each of these points parallel to AC, and cutting the curved line CB, which in this case is struck with a sweep of 38 feet. Now divide the run of the rafter AD into the same number ' f equal parts as the run of the common rafter A'.i. P:rect perpendiculars from each of these points at rifrht angles to AD and set off from A to E the same distance as that from A to C on the common ratter. Next set off II, 22, 33, etc., on the hip to correspond with II, 22, 1% etc., on the common rafter, and con- nect these points. The result is the shape of the hip rafter. As for the jack rafters, their lengths depend on the number necessary. If one is sufficient, its length would be one-half that of the common rafter taken on the working line, and hy making the cut for the upper end through this point, as shown at 4 where we have the proper length of the jack. The plumb and ver- tical cuts of the jacks are the same as those on the common rafter. The figures on the steel square vvhich ^ive the cuts for an octagon are 7 inches and 17 inches; the 7-inch side giving the cuts. The figures on the square which will give the cuts for common rafter in this case are 5 I I : Tl" ^ »38 PRACTICAL USES OP inches on the tongu.. and ,2^ inches on the blade: the latter giving the upper cut and the tongue the lower cut. The figures giving the cuts for the hip rafter in this case are 5 and 5^ •nches on the tongue and ,2>^ inches on the blade; the latter giving the plumb cut. This will he readily understood for the reason that the run of a hip rafter on an octagon is one- twelfth greater than the run of the common ra ter. One method of obtaining cuts of the jac k ratters is shown on the lines Ati and liC It will be seen that it will work in any case, no matter what may be the pitch of the roof or the shape of the rafter. Obtain the plumb cut of the upper end of the rafter EC. which is the same as that of the common rafter. Then square across from A to H on the upper edge. Now. as 7 inches and ,7 inches on the square will give the cuts of the jacks if they are to have no rise at all, the same will work when they have a rise I ake seventeen-seventiis of the thickness of the stuff whjch is being worked and set it off square from the line BC to the outer edge, as CD. Then a hne fn.m A to C is the bevel c,f the jacks. All these cuts and lengths may be obtained by using the square, as has been shown in previou'i examples. THE STEEL SoUARE «39 Another method of obtaining the curves for hip and jack rafters is shown at Figs. 93 and 94. The lengths and bevels will, of ( ourse, be the same as though they were to be straight, and lay them out in that way tA/T" on boards wide enough i make the curve. It is ^est to ha v e the m planned and jointed on the back, then strike the curve of the common rafter. It may be struck with a trammel from one renter, as in ihe sketch, or of any shape that may i suit the fancy or condi-? tions of the case. Next.i divide the length of the' rafter on the jointed edge of the pattern into any number of equal parts and draw the lines, as i, 2. 3, etc., in the sketch, Fig. 94, on the same bevel as the plumb cut. Then proceed in the same ' "^- 93 i.Si. I40 « I i f| PRACTICAL USES OF manner wfth the board for the hip rafter, being careful to divide it into the same number of i "j^mik THE STEEL SQ^JARE ,4, equal parts, and draw the lines parallel with •ts plumb cuts. It will be found convenient to number the lines on the patterns the same as shown m the sketch; then, with the dividers or ruj^e, lay off i' on the hip equal to i on the jack. 2 2, 3 -3, etc. Then spring a light line or edg- ing and draw through the points thus obtained If the work is done correctly the two sides of the roof will meet exactly on that line. It will be readily seen that it makes no difference whether the hip IS to be set on a square, hexagon, octa- gon, or at the angle of any other regular figure providing run and length are first properly set off. Another method, which may be termed a geometrical method." is shown at Fig. 95 and to those who have any knowledge of geometry further explanation will be unnecessary. It is given here merely as a comparison and may perhaps, be found useful to a few readers. The plan and elevation shown at Fig 96 is almost self-explanatory. It simply shows the lengths and bevels of hips and jacks. The cuts >n both h.ps and jacks are the same as would be tor common rafters, except thut instead of a square cut across the back of the rafter it must he at a diagonal to fit against the side of the hip as shown by the dotted lines at A and B w !, 8 M* PRACTICAL USES OP :11 ! i i I I These lines are always vertical and the same dis- tance apart regardless of the pitch given. A diag- onal line from A to B across the back of the jack determines the angle. Fig. 97 illustrates this point. If there was not pitch at all then 5 and 12 would give the cut. These figures also give the start- ing points of lines A and B, which, since the rafters are of the same thickness, will remain at right angles the same distance apart. Thus, if the rafter be 2 inchesthick.thelinesAand • B will be 45i inches apart. The jack cut may also be found as follows: Take 5 on the tongue and the length of the common rafter for one foot run on the blade, the blade giving the cut. . '!*£ , 'T I «* ivKT '* "klV', / KJy vA "HE STEEL SQUARE •*i Ij ^ a»^»f■^^v Fn:. 97 I'i^. 98 is a modified diagram of Fig. 96. Ot:TAGO\ DORMERS I have thought fit to present to my readers a few .llustrafons showing how an octagonal dorn, „ ,„ ^^^ ^^^^.^^ ^^ ^ ^^.^^^^ roof by architects of note. But little more s necessary than the plans, sections, and eleva- ■ons showing the mode of construction to enable .he^rcader to understand the whole arrange- IV 99 shows, in a conventional w.j, how the "be s are arranged and fr,m«l in order to ■n^l^e the dormer bay strong and effective. The "M ■ n wjtri.^?sa^z < i f" !!«F r n »*4 PRACTICAL USES JP sidelights in a dormer of this kind are gener- ally fixtures, while the front sashes may be hinged and open in two leaves, or they may be so constructed as to hang with cords and weights. Fig. ICO shows a front elevation of the frame- work with sashes in place; and fall-back ga- ble of roof, rafters, and all studding are seen in position. Fig. loi shows a plan of the whole construction, in- cluding top of brick wall, plate, and gutter. This figure requires no further explanation. Figs. I02 and 103 show side elevations of por- tions of roof and dor-^ mer. Fig. 98 1HE STEEL SQUARE »45 Pig. 99 [:|lf| i^l i , ! • ^ t! r-Ki .■>«rrT'*B '.'<«3R1^ ^' tkr - " ■ ««».-«fc:>ifi- Tt: vOl 146 : i 1 I i PRACTICAL USES OP At Fi^r. ,04 a ground plan of the dormer proper is shown giving shape of corner and Fic. 100 angle posts. A little study of these examples will enable the reader to understand the prin- ciple of construction without further explana- tions. The steel square may be used to get every cut and bevel in this roof, also bevels for SWCT &«*&;« Xi.'SBKKr "¥8 THE STEEL SQUARE 147 the window sills, as shown in Fig. 105. and which IS explained as follows: Take the height of Fio. loi rise A to A' and set up from B to C each way Intersect the two lines at D.one at .n arc struck troni O as a center, and the other from E. The line from O to D will be the bevel for face of :j|j m 148 PRACTICAL USES OP if ! ' Fta. :o3 °A^. THE STEJiL SQUARE ,4, sill. To obtain the down cuts, drop a line from the point of overhang of sill to the line of inter- Pio. 104 section of the angle of the bay window. Set of! the thickness of sill parallel to face of window. * t«iht Fig. 105 Square up from where the point of overhang cuts the Ime of intersection of angle to the thickness of sill and draw a line from H to O. which will be the bevel or down cut. This, of course, gives he bevels for all sides of the sills. At Fig. 106 i show an octagon tower in place, or rather the !l •H m: -i i I': ir ri 11 li'^n ISO PRACTICAL USES OP method of framing same on a balloon building over a veranda. Sizes of timbers are shown. The example may be of use. Fig. 107 shows a plain octagonal tower finished. This is in connec- tion with a two- storied frame house that is sided outside. A por- tion of veranda and front entrance i s shown. The whole is made as plain and econom- f al as possible. .Vhile these last two figures have no direct connec- tion with the steel square, it is thought they may a'xfl' o xe Fig. 106 e useful as showing the work when finished. ^^,-yMksM^ ^nkJ^Si- u-^lt^J THE STEEL SQUAKE «S« f -i i jiiii i»w»iii»wiiiMiiii»»»i»iiiiiiihiiuiiiii(.wti»ywiiiiia'yt'g<*y'i!*''' Fiu, 107 if < ?! ..£1 j^ '.'. I, i I I i • a! ^ • i >i« PRACTICAL USES OF liOPPERS AND HOPPER BEVELS •he question of "hopper cuts' is one that • CMS o puzzle nearly all youi - workm- n-and n.any -Id n-s also. Inde-xi. I have known many ixctllenr workmen who coi Id cut every lin'.Lx ■ or a complicated hij) roof, on the ►-oun', I, HJ whose work was beyond suspicion, who couid not lay out the lines for the miter cuts of a hopper in a proper manner. In this . hapter I will endeavor to ^dve to the reader, in as simple manner as I know how, a number of the best methods employed by expert workmen for findinjr the proper lines for cuttin^^ hopper bevels, both by geometrical methods and by the use of the steel square, and to this end I have gathered up a number of methods, diagrams. and explanations .rom various sour es — many of which have heen published before. The shapes of hop- pers shown at Figs. 108, log, and no nre trian- gular, square, and hf v. agonal respectively, while Fig. ,1, shows a cover Fig. 108 THE STEEL SQUARE 153 or box lid, which requires a little special treatment. Fir, 109 I will treat all these figures in u geometrical ay, so that the reader may know the "reason why" It is nee -*- sary for C( - in cuts to ha e certain lin^s ihaf are at va- riance vith cer n lines in Fio. no apparently- simi- lar cables. The j^ e o m » • t r i c a 1 P r o b I e m s i n- volved in these cases consist (1) in ti di 4 Fii,, III dihedral angle be- liji ■4 154 PRACTICAL USES OF ■ ;! tween two planes; (2) in bisecting this angle, which gives the bevel; (3) in developing or obtaining the true shape of a plane surface. A geometrical problem of a very ordinary type, involving the first VlG. .H two of this character, is shown by Fig. 11 2, where two planes are indicated by their horizontal and vertical traces. The solution is as follows: First find the plane and elevation of the intersection odAE. Next obtain the true length of the inter- section by rebating it into the horizontal plane as shown by ^B. Through any point c on a draw de at right angles to ab. Then from point c, draw cf at right angles to «B. Then with c as center and / as radius cut ab in g. Join eg^ and dg; then dge is the dihedral angle between the two planes. Then, given two pieces of material of equal thick- .P*C«^^i^T?iff THE STEEL SQUARE ^S5 ness meeting, and it is desired to miter-joint them, the angle for this would obviously be half the dihedral angle, as A. Let this geometrical reasoning be applied to the case illustrated by Fig. 113, which, as will be seen, is a direct application of the problem just described, ad indicating the plane and a'6 the elevation of the line of intersection of the two adjacent plane surfaces. Draw 6" at right angles to ad, and make it equal to M; then joining a to /'" gives the line of intersection rebated into a horizontal plane. Then proceeding with the construction as explained at Fig. 112 (it will be Fig. 113 seen that Fig. 1 13 is similarly lettered), the angle }] iS6 PRACTICAL USES OP I between the two surfaces of the lid is obtained. Half this angle gives the bevel required for the mitered joint as there shown. In the cases considered the working has been Fig. 114 in front of the vertical plane and on the hori- zontal plane; that is, in the first dihedral angle of the co-ordinate planes. This has been the most convenient method, because the surfaces slope up and away from the observer; whereas in the cases shown by Figs. 114 to 116 the sur- faces slope upward and toward him, and there- i. THE STEEL SQUARE »57 fore, for a person who has the necessary knowl- edge of geometry, the simplest method will be to work under the horizontal plane, but in front of the vertical, that is in the fourth dihedral angle of the co-ordinate planes. Taking the case of Fig. 114, ab is the plan of the intersection of the two surfaces. At right angles to this, set up the line ^B, making it equal to BC. Join aB; then this line is the true length of the intersec- tion constructed upward into the horizontal plane. Then through c draw de at right angles to ab — it should be noticed here that half the line de coincides with B, the intersecting surfaces being equally inclined. Then from b draw bf 2X right angles to ^ B. Next, with f as a center and cj as radius, describe the arc cutting ab in g. Then joining dg and eg gives the bevel required, as shown at B. By imagining the object at Fig. 1 14 turned upside down, exactly the same kind of working as that just described would apply; but perhaps the problem will be simplified by imagining that the work is in the first dihedral angle of the co-ordinate planes. To many who understand geometry ihis method ot working this problem will commend itself as being sim- plest to imagine, although giving the same results. In Fig. 115 precisely the same reference I 'i i 1 • i| 158 PRACTICAL USES OF i ■■I • I . -i letters have been adopted, and it will be seen that the same problems and principles are involved. Fui. 115 The geometrical formula for finding the true shape of development of the sides is: Given the plane and inclination of a plane surface, deter- mine its true shape. A prcblem of this class is shown at Fig. 117. Let adc(/e be the plan of the given figure, its side ad being the horizontal plane; through aS draw a horizontal line, at right angles to which draw .vy as shown; next THE STEEL SQUARE '59 set out the angle of inclination of the figure as shown hy the line VT, which may be considerea Fio. ii6 as the vertical line of a singly oblique plane. From the plan set up projectors to this plane, then rebate back the figure into the horizontal plane by drawing the arc projections to xy, and from these, projecting at right angles to xy, and from cd and e parallel to xy, gives the points CU ^ 4l t \\ . ^ SB i6o PRACTICAL USES OF and E. Joining a to E, E to D, and D to C,and Kih gives the true shape required. Fig. tiy If the working of this problem has been care- fully followed and mastered, its direct application to cases shown at Figs. 113, 114, 115, and ii6 will be readily understood. The foregoing is meant primarily for geome- tricians, and secondarily for those who wish to know the "reasons why" of hopper bevels. Having shown this much, simply to satisfy the "learned" in "theoretical" carpentry, I will now proceed to show how all the lines and cuts may THE STKIiL SQUARE i6i Fig iiS. be obtained for this work by aid of the square alone and in a speedier and simpler manner. Let us suppose Via. 1 18 to represent a sec- tion through a hopper; then take a board 12 inches wide, joint one edge and draw a line of the side elevation according to the refine- ments. From this proceed to make a draft with the square i inch to a foot. Take 12 inches on the blade, hold 1 2 at A, and by it find how many inches rise the given inclination is to a toot. Draw a line by the tongue, as shown by BV, Fig. 119. At B draw square with AC, and again square on line AB through C, as shown by CL. From VB erect a perpendicular also to C, cutting VB in the point E. Fig. 119 By these several opera- tions we have a complete draft by which to solve the problem. AB is the given slant, and has 9 f I 1 ! i i \ 1 i6a PRACTICAL USES OP inches rise to a foot. AC is 12 inches, and CB is 9 inches. The length of AB, as indicated by the figures on the square, is 15 inches. Fifteen inches, therefore, is the width of the board required to cut the hopper. Use the foot draft of the hopper as follows: Fig. lao Take on the blade AB, and on the tongue CB, and apply the square to the board as shown in Fig. 120. Mark on the tongue, which will give the down-cut bevel. For a butt joint take EC on the tongue and CA on the blade and apply it as sh -^wn in Fig. 121. Mark on the square edge of th« Doard EC. The above principles will give the manner of backing a hip rafter. Suppose that AC, Fig. 119, were the seat of the hip rafter, and AB the li I*. THE STEEL SQUARE 163 length of the hip, by taking CL on the tongue and CB on the blade, marking on CL and setting Fig. 121 the bevel by that line, the exact backing of the hip will be obtained. The handle of the bevel is to be square across the hip rafter, as shown at CL in Fig. 119. A very good way, and one I have found to work out correct, and which I take from "Carpentry and Puilding," for obtain- ing the butt joints for hoppers, is as follows: Draw a line CD, Fig. 122, at the same an- gle with the straig h edge of the board ^ AB as the sides of fig. 122 i . i • *i i<4 PRACTICAL USES OP the hopper are to stand. Cut this line at any point, E, with a line, FG, drawn at right angles with AB. Divide CF into five equal parts, and through the fourth point thus established draw a line as shown from H, cutting E. Then HE will be the bevel of the butt joint- This rule will work no matter what the flare of the side or width of boards used, providing always the hopper is square-cornered. Fi ■ I . I > I ■ I ■ I . I ■ I , I , I ■ i>SM ■ I , I ■ I . I . I ■ I ■ I ■ I . I , I , Fig. 123 A handy rule is given as follows: Let Fig. 123 represent the slant of the hopper as shown by the line running from 12 to 12 on the square. Figs. 124, 125, and 126 show the applications. Fig. 123 represents the square with a line drawn from 12 to 12; this line shows the flare of the Tte. THE STEEL SQUARE 165 hopper. The distance from 12 to 12 on the diag- onal line is 17 inches nearly, as I have shown Fig. 124 previously. The angle of the miter on the hori Fig. 125 zontal is 45° or a true miter. The base line ' ^.i•^ ; I: "n ■^^ ^Mtfttl li ll i66 PRACTICAL USES OF ii i i from the corner of the square to 12 is t 2 inches, the pitch line is 5 inches longer tha- the ^)ase line (the difference b**»^wcen 17 and i 2 inches). Add 5 inches, the excess of the pitch from the base line, and we have 12 and 17 inches of bevel and miter. The application of this is shown in Figs. 124 and 125. For a butt-joint the sides slant two ways on each angle. The angle on the hori- zontal is a right angle. The excess of the pitch from the base line on the one side is 5 inches, and on two sides 10 inches, making an angle or Fig. T26 joint 12 and 10, as shown in Fig. 126. The lat- ter figure shows the application of the rule named; all the stuff being square-edged. This method when thoroughly understood is very sim- ple and effective. Another method, somewhat similar to the one THE STEEL SgUARE .67 already shown, is given herewih. To obtain the bevels of a hopper either by the square or lines, the rise to the foot of the sides being given, first ascertain th' hypothenuse from 12, taken on the blade -f th'- sqi:ar( , to the rise to the foot, taken on the tongue. 1 )ivi(le the square of 12 by the rise to the loot. Apply the squan to a straight -edge, taking' the hypothenuse on the blade and ^ on the tonj;u( . This will denote the surfuir lx.v. . .">' tin. apply the square to the straight-edge taking the hypothenuse on the blade and the rise to the foot on the tongue. This will denote the bevel of the miter-joint, so called. For a butt-joint, rake on the blade the quotient arising from the division of the square of 12 by the rise to the foot, and on the tong take the hypouienuse. Then the blade wi.i denote the bevel required. Another methxl on the same diagram is .as follows: Apply the square to a straight-edge, taking 12 on the blade, the rise to the foot on the tongue, and mark by the blade to obtain AD of Fig. 127, which represents the inclination of the sides. At random make BD perpendicular to .ABC. which represents the straight-edge, taking AD on the blade and AB on the tongue. This will give the surface bevel. Again, apply I , m i i , ii i| ' il II ; k f ; llii ii Its PRACTICAL USES OF the square, taking CD on the blade and BC on the tongue, which will denote the bevel for the miter-joints. For butt-joints apply AB taken on the blade and CI) taken on the tongue. The blade will denote the bevel required. The same results may be obtained by geornetrical construc- tion, as follows: Referring to Fig. 127, make DF equal to AB, and draw AF; make DG equal to CD and draw FG; make CE perpendicular to CD and equal to BC and draw DH. The designated acute angles at F will be the angle of the surface bevel. The designated obtuse angle at F will be the bevel for the butt-joint. The angle at F is the bevel for the miter-joint. The foregoing may be applied to roofs of one pitch over rectangular bases. Fig. 128 repre- sents a section of cornice. That which relates to the surface bevel is applicable to the surface bevel of the boarding, the outward bevel of pur- i i i i THE STEEL SQUARE 169 Pig. 138 lins which come in contact with each other or with hip rafters, the surface bevel of the planceer in a cornice similar to _ , the diagram, and the V^^^^^^^^^^^ edge bevel of a fascia. f^~^ That concerning miter- joints is applicable to the edge bevel of the boarding, the inward or down bevel of purlins, and the surface bevel of the fascia. Applying the square to a straight- edge, according to the directions given in the first and second methods above presented for obtaining the surface bevel, and marking by the blade, we obtain the edge bevel of jack rafters. In Fig. 127, FAD denotes the edge bevel of jack rafters. In order to properly get the crown molding for a cornice similar to the drawing, lay off on the edge of a miter-box the surface bevel of the board, and on the sides the edge bevel. Hopper cuts to a large extent are similar to the cuts required for fitting boards in or over ;i valley or hip roof; consequently the figures on the square that give the cuts for the roof boards must give the cut for a hopper of ti:e same pitch. si /] Y 1 1 1] 1 . ;.( 17© PRACTICAL USES OF ■ I; I -.i i Fig. 129 shows a hopper in different views, as follows: Beginning at the top is the top view of 171 Fin. 129 I the hopper. As far as this part is concerned all hoppers look alike, as there is nothing in this to distinguish the pitch. .\ext is the sectional or THE STEEL SQUARE »7i side view. In this is shown the thickness of the boards and the flare or pitch, which in this is the three-quarter pitch. Following (Fig. 129) is shown the four sides in the collapsible or ready to be p It together, followed with the top view of the edge of the board. Of course it is not necessary to lay out all of this diagram, or any of it for that matter; it is done by way of illustration. See the application of the square, which, in this case, is 12 and 215^. But why use these numbers? Because the flare given is the three-quarter pitch, or 12 and 18 on the square, and the hypothenuse of these num- bers is 2154, the tongue giving the side bevel. When working full scale it is always 12 on the tongue. For the miter bevel, the top edges should be first beveled so as to be level when in position. The miter would be at an angle of 45°, and any of the equal numbers on the square gives this cut; but if the edges are to be left square with the sides, as shown, the above will not work. To accomplish this, however, a very simple way is shown in diagram at A, or in Fig. 130 as follows: Lay off ^he base and the desired pitch, and on the latter measure the thickness of the Gj ^H ■"SSL-W^Effw i I!" 17a PRACTICAL USES OF board as at AB. From B draw a plumb line to base. BC is the width apart; the side bevels should be along the edge of the board. In case of large hoppers to be built to suit some particu- lar place, or regard- less of pitch, it is bet- ter to use the one- FiG. 130 inch scale as illustrated in Fig. 131. The figures "jYi and ijj^ give the side cut, while the section gives the mi- ter. This diagram is all that is necessary to find the cuts for any size hopper, and were it not fi)r the miter even this is unneces- sary. Another method is here offered which is taken from the old F"'*"- mi "American Builder," and which many of the older workmen will recognize, as it was at one ■ n -u -ti -M -1* '74 -17 -V -It ■ M -II -« f "?■ -s i^/ -» €"•' -T ;:< -t * IK -S 1 ' -1-; .... K V'. ■ 1 > A « « * 1 T 1 t 1 1 : 1 1 1 1 :r\ hoNVV 7i THE STEEL SQUARE 173 time in great favor. The lines E, D, H, A, ond i m h: >74 PRACTICAL USES UF C, G, B, Fig. 132, show the outside edges of the steel square, or squares; OO is the edge of the board. Half the width of the width of the top less half the discharge hole on the blade AB. The depth on the tongue BC gives the diagonal AC, the width of the side. Makt- AD on the blade equal AC, and DE on the tongue equal AB, and E is the bevel for face of stuff. Make GB equal FB; connect GA, and G is the miter cut for edge of stuff. Square a line from 00, cutting AB. To save extra lines use FB. Make AH (on AD) equal AI-. HK square from AD, and FM equal HK; connect MB, and M is the bevel for straight cut (to nail on, as we nail a box together, square instead of miter), the long point inside. If we make the line cutting at H square from AE instead of AD, as XH and FS equal NH, connect SB, and S is the trying bevel for straight cut. Bisect BSO; draw the miter line and it gives P the trying bevel for miter cut; the lines are extended and bevel P placed below the line for want of room above And yet another method: Suppose ABCD, Fig- i33> to represent the elevation of a box, the sides of which have a slope of 45°. Take C as a center and CB as radius, and describe the semi- circle under the line AB, as shown by Hi:j. THE STEEL SQUARE ,„ From C let fall a perpendicular through the line AB, intersecting the semicircle in the point E. Through the point E thus established draw a horizontal line or a line parallel to AB, *; making it in length equal to AB. From F'« 133 the point F thus established draw FC. Then the bevel at F will be the bevel to be used for the sides of the box or hopper. To find the bevel of miters at the corners with K as center and radius equal to the distance from K to the line CB, de- scribe an arc cutting the line KC, thus establish- ing the point L. From L draw the line LJ, cut- Fio. 134 ting the line AB at the point J, which represents the intersection of the arc first drawn with the AB. Then the bevel at L is the bevel of the line I if I t :l i>! 176 PRACTICAL USES OF miters at the corners. Fig. 134 shows the sides of the box laid out flat. Mr. Woods recently published in the "The National Builder," a paper on bevels and hop- per cuts generally, which, owing to its excel- lency, I have thought worthy of a place in this work. The author starts out by saying: "If there was no given pitch, then the sides would be vertical. Now with 12 on the tongue as cen- ter, draw an arc of same radius from the heel to a point directly above and square over the blade; Fig. 135 I'^if-. 1-56 12 and i2willgi\« the miter. Fig. 135. Simple enough, but do you know that this simple ruk- applies when there is a pitch given? At thr point where the am intersects the pitrh taken on the blade will give the miter, the blade giving the c"t See Fig. 136. 77 THE STEEL SQUARE "For the side bevel across the board, transfer the length of the pitch to the blade, the tongue giving the cut. These figures also give the side cut of the jacks for a roof of same pitch. The blade in this case giving the cut. "Now wc will give another method of finding the miter. In all roofs and hoppers there is an unseen pitch which we will call co-pitch. Assum- ing that the edges of our boards are square the ^4ia* Fig. 137 co-pitch v,-ould stand at an angle of qo' with the given pitch." See Fig. 137 The rule given in Fig. 136 for the side oevel will apply to the miter, but instead of using the length of the given pitch substitute that of the co-pitch, and by referring to Fig. 137 we find this ■ It ' I nil •i ii n i 178 PRACTICAL USES OP to be IS'I inches on the blade. No\ " for proof, see Fig. 138. Twclvt? and gyi first method, and 12 Fig. 138 and 15K second method. The blade giving the cut in the former, and the tongue in th( latter. 15;^ Miter '.'6 2l| Butt Miter r'O. ly, THE STEEL SQUARE «79 In Fig. 139 is shown all that is contained in the other figures and then some more. In this we run up against another pitch. It is an extenua- tion of the co-pitch to a point on a level with the starting point of the given pitch. This we will call complement pitch. The length of this pitch transferred to the blade will give the butt-joint or miter; the tongue giving the cut. In Fig. 140 is shown a dia- gram for a hopper of one- half pitch. In this all of the pitches are of equal length. < Therefore 12 and 163 1 will give all of the cuts; the tongue giving the cuts in each case. Figs. 139 and 142 are simply fillers and self-explanatory. While editor of the " Builder and Woodworker" of New York, I had considerable communication with the late Robert Riddell whose works in the 6o's, 7o*s, and 8o's were very popular, and on one occasion, in 1879. he sent to me the following diagrams and explanation of a method for obtaining the bevels and cuts for nearly all kinds of flared work or inclined framing. This prob- lem and solution, said Mr. Riddell, is offered Fig. 140 -' ''H MICROCOPY RESOLUTION TEST CHART (ANSI and ISO TEST CHART No. 2) 1.0 I.I 1.25 1^ ill 2.8 IIIIM 1^ !■■ 112 2 If ti^ '^ 1 -^ lll^-° »- ^ KbL^l- 1 ''^ 1.4 1.6 ^ /APPLIED IIVMGE Inc =; 1653 East Mam Street r.= "Rochester. Ne» Vork 14609 USA JS (716) 482 - 0300 - Phone 1^ (716) 288 - 5989 - Fo« ^ I l' ^ 1 11 ' ' :: i 111 i8o PRACTICAL USES OF for the first time for publication, as a reliable and simple method for obtaining bevels and cuts Fig. 141 for nearly all kinds of inclined framing, and for the finding of cuts for splay and flare work. FiQ. 143 The diagrams show how the cuts for hopper work may be obtained for angles other than m THE STEEL SQUARE i8i right ones, also for getting angles for corner posts having a double inclination. Fig. 143 Let us suppose the plans shown at Figs. 143 and 144 to be right angle figures, having sides which incline or flare equally to any desired angle. A corner post is also used which will incline same as the ends and sides. The junction at the angles may be either mitered or a butt, as either iU i8a PRACTICAL USES OF i ; Style of joint may be obtained when desired. To describe the problem, begin by drawing two R Fig. 144 oarallel lines, AB an ' DC, any reasonable dis- tance apart. Assume AC as inclination or flare of sides. From N square down a line making mBm\i\s^Ki-xi. '^^'^am'Z'-'-wr^ >j<, VjFr* THE STEEL SQUARE 183 NA and NR equal. From A square down a line cutting D; join RD, and in the angles thus formed find bevel 2 for cut on face of sides To find the bevel for miter on edge of stuff, take N as a center and describe an arc, touching the line AB and terminating at J. From B draw a line through J indefinitely. This gives bevel 3 for the miter. To find the corner post, proceed as follows: Make NC equal XD; join RC, and extend AN to cut RC at P, from which square up a line cut- ting at B. From N draw through B, thus form- ing both angles of the corner post, and giving bevel 4, which answers either for a butt-joint or the shoulder cuts on cross-rails of framing. Nothing can be more simple or more accurate than this method, and it may be easily tested by first drawing the "spread out," as shown on the upper part of the diagram on cardboard, and cutting through in the lines marked X, X, X, then fold on the lines marked O, O, O. Bring the points S, S together, and the mode of construction will readily be understood. The flare may be any angle, the results will be the same. Another problem and solution, and I am done. The problem I am now about to present is one l-.jr wsiaiics^A-^w.'m^^: i84 PRACTICAL USES OF that, when thoroughly grasped by the workman, will enable him to lay out the lines for every cone 'vable cut required in tapered framing, paneling, or splayed work of any kind when angles are thrown out of square. To Peter Nicholson is due the credit of this method of solution, but, as in many other things, ihe master hand of the late Robert Riddell improved and simplified it and clothed it in language easily understood by the American workman. Let us suppose the line AD, at Fig. 145, to be \ Fig. 145 a given base line on which a slanting side of hopper or box rises at any angle to the base line, as CB, and the total height of the work is represented by the line BE. By this diagrara it will be seen that the horizontal lines or bevels of the slanting sides are inaicated by the bevel Z. Having got this diagram, which, of course, is not drawn to scale, well in hand, the ground plan IJ THE STEEL SQUARE ..^s of the hopper may be laid down in such a shape as desired, with the sides, of course, having the slant as given in Fig. 145. Take T-*, 3S, Fig. 146, as a part of the plan, then set d! the width of sides equal to CB, as shown in Fig. 145. These are shown to intersect at PL above; then draw lines from PL through 2-3 until they intersect at C, as the dotted lines show. Take C as a cenrer, and with the radius A describe the semicircle AA, and with the same radius transferred to ( Fig. 145, describe the arc AB, as shown. Again with the same radius, set off arcs AB, AB on Fig. 146, cutting the semi- circle at B, as shown. Now draw through B, on the right, parallel with S3, cutting at J and F; square over FH and JK, anu join HC; this gives bevel X as the cut for face of sides which come together at the angles shown at 3. The miters on the edge iff are parallel with the dotted line L3, This is the acute corner of the hopper, and as the edges are worked off to the bevel Z, as shown in Fig. 145, the miter must be correct. Having mastered the details of the acute corner, the square corner at S will be next in order. The first step is to join KV, which gives the bevel Y for the cut on the fac( of sides on the ends which form the square corners. The m VM ^p I m ; ii i * m i86 PRACTICAL USES OF Pig. 146 Si- rmwv'mmma^rs^Md^^sr3SN?Mis^mx^*'^r?!F.,wmsmAm. * :(4'*itvJ!»-.^ II » THE STEEL SQUARE 187 method of obtaining these lines is the same as that explained for obtaining them for the acute angled corner as shown by the dotted lines. As the angles S T are both square, being right and left, the same operation answers both; that is, the bevel Y does for both corners. Coming to the obtuse angle P2, we draw a line BE on the left parallel with A2, cutting at E, as shown by dotted line. Square over at E, cutting TA2 at N; join NC, which will give the bevel VV, which is the an^le of cut for face of sides. The miters on edges are found by drawing a line parallel with P2. In this problem, like the orevious one, every line necessary to the cuttin. of a hopper, after the plan as' shown by the boundary lines 2, 3, T, S, is complete and exhaustive; but it must be understood that in actual work the spreading out of the sides, as here exhibited, will not be neces- sary, as the angles will find themselves when the work is put together. When the plan of the base— which is the small end of the hopper in this case — is given, and the slant or iuclination of the sides known, the rest may easily be obtained. In order to become thoroughly con- versant with the problem, I would advise the reader— as has often been before advised by HI If '''^tm^ txvt' Sd ■r"i«"i''3£!-?:- L-Tff.-xdfsi^Ei^s^! TTifl w?«iP«3?;-T'':i*»iaap^ri THE STEEL SQUARE 189 ana anpfles for those posts me set forth at con- siderable length in the text and drawings, and what I desire to show is that the corner pcit of a hopper is exactly, in miniature, similar to the corner post of a frame building having a double inclination; or, in other words, a tapered build- ing, with the exception that a hopper has its smallest area as a base, while a tapered building, a pyramid, has its largest area as a base. Now it must be evident that the lines giving the proper angles and bevels for the corner post of a hopper must of necessity give the proper lines for the corner post for a pyramidal building, such as a railway tank frame, or any si /.ilar structure. True, the position of the post is inverted, as in the hopper its top falls outward, while in the timber structure the top inc' u ^ inward; but this makes no difference it the theory, all the operator has to bear in mind is that the hopper in this case is reversed. Once the proper shape of the corner post has be^n obtained, all other bevels can readily be found, as the side cuts for joists and braces can be taken from them. A study of these two figures in this direction will lead the student up to a correct knowledge of tapered framing. I leave the subject of splays and bevels here. ! ij M ^'ZPfsmvA ■ -\^ - wjFwmr^^^ea^^^arr-mmm^^t^simKFivEmiMaeEBiaBcmj^is^iFr i !; ■mi i 1 ;; 'li i| II mm ^ -' fef 190 PRACTICAL USES OP but I may add that I have not, by any means, shown all the ways and means of fmdinjr solu- tions of problems presented, but I am persuaded the examples I have set forth are the best suited for the workmen of this country because of tlicir simplicity and of the manner in which they have been laid down. I think every possible kind of hopper and splay has been touched upon, and, if not, I am sure the rules given will enable any workman who has followed me closely to deal with the difificulty successfully. At Fig. 147, I show one of the usual methods of finding the circumference of aciicle; this is c Fig. 147 done by taking the radius AB on the compass, and with E as a center intersect the curve in U; then with the same radius, and C as a center. draw the arc from A, making it and DC equ .1; mi'miiS'G^m'fm |?TJ THE STEEL SQUARE to. draw from C, througl the point, cutting? in 2. which jjives 2K as one-fourth of the circumfer- ence of a :ircle havinjr AB as a radius. The same rest can be obtained almost instantly by using a . i-square having an ,'le of 60 and with it draw a line from C I " ii will cut the diameter at 2. It is generally supposed that this method is a correct one, or at least sufficiently so for prac- tical purposes. It is not so, however, and when used sometimes leads to error and trouble. A much more accurate method is shown at Fig. 148, which is obtained as follows: Take A as center and with B as a radius, describe the arc, and from A with an angle of 45°, draw a lin . cutting at E; w the same angle draw BJ; from J draw sq lare \^.in AB cutting the arc in 2; join it with E. .hen four times 2E wil! be found to be i ' -a! to tie circumference of a circle having the laaius AB. Suppose the diameter 2B of the circle shown at Fig. 149, to be 16 feet in diameter; make a similar figure or. a scale of one-quarter of an inch to the foot, then the exact circumference of this may be found by drawing AE with the angle 45°; then with the same angle draw BR, and fn a R draw quare with AB cutting the circle in D; :i m il' - i ] I i ► I] i>pan9v^ inches or 22 quarters, which is equal to the number of parts required. Draw line 4C; make 4R equal % inch; from R square over a line cutting in K. This gives RK, which will divide line No. 2 into 22 parts as required. Length of each I's". No. 3 on the right. This line measures, by a ■ ii ,1 Ui (it* I 196 PRACTICAL USES OF quarter-inch scale, 26 feet 6 inches. It is ""'■■'' ' ' ' Fig. 151 THE STEEL SQUARE 197 required to be divided into 17 parts. To do this, .aake AD on tongue of square (-qual AD, No. 3 on the ri^ht, and A3 on the blade of square to measure 4^ inches or 17 quarters; that being the number of parts required. Draw Hne 3D; make 3P equal % inch; from P square over a line cutting in J; then PJ divides line No. 3 into 17 par Length of each i'7". No. 4 on the right. This line measures, by a quarter-inch scale, 23 feet, and it is required to be diy'ded into 11 parts. This is done by ma- king AE on tongue of square equal AE, No. 4 on the right. Let A2 on blade of square measure 1 1 quarter inches, that being the number of parts required; draw line 2E; make 2F equal 5< inch; from F square over a line cutting in H; t;2n FH divides lines No. 4 into 11 parts. Length of each 2 Ii't". In dividing a space of any great extent, the quarter-inch scale will be found most convenient. To give a practical illustration: Suppose a line lo 1 e 96 feet long, and it is requi-ed t- divided into 48 parts. Begin in a systemat.^, way by squaring over a line on the surface of a board, \nd from its edge mark one foot on the line; one foot being equal to 48 inches. Next, meas- ure two feet from the line along the edge of I'm ^ if r :'t\ 1 :i : Sj i--*l|| 198 PRACTICAL USES OF board, two feet beiig equal to 96 quarter inches. Now proceed to find one part that wiil divide 96 feet into 48 parts. The answer is given by the method just explained. A little thought will enable the «tudent to use any scale that is divisible by the divisions and subdivisions laid down on the steel square, a matter 'ihat will enable him to divide lines of almost any length. When we know the side of a square we* cm readily find its diagonal by the steel square as follows: Suppose AB, Fig. 152, be the side of a square which measures, by a quarter-inch scale, 6 feet 10 inches. To find its diagonal, draw the line BC to the angle 45°; take A as center and strike an arc touching line BC, cutting in V. Join V, C. This gives bevel W; let it be applied to the square, Fig. 153, and at t! e distance AB, which is equal to the side of given square, lay a straight-edge against the blade of bevei and the line made by it cuts in mark C on the square, giving AC for the diagonal, which measures g feet 7 inches. This agrees exactly with line PR or BC, Fig. 152. Now let it be required to give the diagonal of a square, the sides of which are equal to A2, and measure 10 feet 6 inches; find its diagonal at [.-,4.7..- .AM<^.\ •A*it,^i..r,-^U. ^SSi «P1 THE STEEL SQUARE 199 Fig- 153 by making A2 e-ual the side of square; let bevel W and the straigtit-edge be applied as before; then the line from 2 cuts mark J on the square, giving AJ for the diagonal, which maas- r. ^1 • i i aoo PRACTICAL USES OP 11 res 14 feet 8 inches; tliis agrees with diagonal 2-3. I^'ig- 152. Fig. 153 Similar results may be obtained without any drawing by merely finding the numbers on the blade and ton^^ue of a squa : that will agree cr ' -1- -th^-ai'"" 4~ Y '^: --^.4-tt:' ^J.-tc^ THE STEEL SQUARE SOI equal the angle in bevel W, then let the bevel as now set be applied to the square, and we find that the blade of bevel agrees with mark 4"^" on tongue of square and 6"^" on the blade; so that the diagonal of any square being required, it is easily obtain d by setting a bevel in the manner staged. The answer will be correct by the angle in the bevel being accurate. Fig. 154 shows a construction which might be called an attempt at squaring the circle. Its ',. I flOii PRACTICAL USES OP solution has been thought an impossibility, and with all due regard to the opinion of others on this point, we think it quite possible to solve this diflficult problem by a new and simple method of construction, which is here given. The diameter of the circle measures 12 feet. From center B draw line BD at right angles with the diameter; draw from B and D, with the angle 45°, intersecting in F; through F draw 3E parallel with BD; draw through E parallel with the diameter, and from center B draw parallel with FB cutting line from E in L; and from L draw parallel with BD; also from E draw through center B, cutting in 2; draw through 2 parallel with the diameter cutting line from L in H, and we have now three sides of a square; the fourth, being made equal to one of these, completes a square, the area of which will be found equal to that of the circle; its diameter beinp 12 feet, and one side of the square 10 feet 5 inches. To remove all doubt as to the correctness of this solution, let us prove it in another way, by a right angle, or the steel square, shown at Fig. 155. Here make the distance AC equal the diameter of the circle; lay a straight-edge across the square, keeping its edge on point C. Take bevel K in the angle HAC and apply it THE STEEL SgUARE 303 to point C; bring the straight-edge against the blade of bevel, and we find a line cutting the Fig. 155 right angle at point H, giving the distance HA, which exactly equals one side of the square HL, Fig. 154, thus giving the same result by two different methods. q n I r PRACTICAL USLo OF The utility of the stetl square is now evident, by it the measurement of any surface may be instantly given, and by the same means we can find the capacity of anythinpr round or square. All that is necessary is to set the bevel to a certain number of parts or inches on the blade and tongue of the square. To explain this point, take bevel K as now set, and apply its stock against the blade of square; move the bevel until the blade cuts some members on the square that will cor- respond with the angle of the bevel ; its blade agrees with marks s"'A" and :,%", or 6"^" and 5"^"; either of these numbers will answer. The bevel being set in the manner stated, will not require any alteration, let the diameter of the circle be what it may. In a previous diagram I explained how this problem might be worked out by a different method. Both are correct, and the reader may adopt either one or the other. I will now endeavor to show how the square may be used in getting certain dimensions with- out much effort. Suppose we wish to find the superficial contents of a board or other material that is not more than one inch thick, we proceed as follows: Let Fig. 156 represent the square. The blade and tongue may be divided into any THE STEEL SgUARE ,05 number of parts by scale. V - may call a ,',, or an >i or a >i, etc., a foot, just as we wish to u -R >l (> i i* It 'f' ti V i * .. ". : ^-il I ■ I I "' 1 I I I I I I I I I I fa JJI •^?^ M 1*11 Fig. 156 ,■; mi 3o6 PRACTICAL USES OF I: J' meet the condition. Let the point 12 or 6. where the lines most converge, be a fixed point. Now assume a board to be 26 feet long and 6 inches wide. We know that the surface measurement of it is exactly 12 feet. The same result is given by the right angle. For example, draw the line 12-26, and parallel to it draw from the 6-inch mark. The line cuts in 13 feet, which is the answer. Find the measurement of a board 21 feet long and 19 inches wide. Draw the line 12-21, and parallel to it draw from the 19-inch mark. The line cuts in 33^^ feet, which is the answer. Give the measurement of a board 18 feet long and io>^ inches wide. Draw the line 12-18, and parallel to it draw the loK-inch mark. The line cuts in ist'V feet, which is the answer. Find the measurement of a board 18 feet long and 9 inches wide. The line 12-18 being already given, draw parallel to it from the 9-inch mark; and this line cuts in 13 feet 6 inches, which is the answer. What is the surface measurement of a board 29 feet long and 4 inches wide? Draw the line 12-29, and parallel to it draw from the 4-inch mark. The line cuts in 10 feet, which is the answer. THE STEEL SQUARE 907 In the foregoing it will be seen that the scale used is less than one inch; but whatever the scale, it must be made to represent an inch in the end; thus, if we use }^ inch, then we must multiply the result by 2 to make it into inches, and if we use a quarter-inch scale, then multiply by 4, and so on, in order to make the result into feet and inches. To reduce surface measure to square yards by aid of the square, we proceed as follows: There are 9 square feet in i square yard, as shown in Fig- 157. so we make 9 a constant number in this problem. Let us suppose any of the regular di- visions of the square i yard in length; be it a ^-inch, i-inch, or any other division. To desig- nate the sides, call the perpen- dicular biadc of square, and lower line tongue of square, and A the internal angle. Let 9B, Fig. 158, be the fixed point, and from it draw the perpendicular, which divide into any number of parts, each to equal those on the right angle. It is now required to give the number of square yards in a floor 22 feet long and 15 wide. Draw the line 9-22-H, and parallel to it draw from mark 155 cutting in H. This gives 14?^, to which add Fig. 15 7 -ill ■■ M I! iii !■ f; J 11 3o8 PRACTICAL USES OF 22, making 365^ square yards of floor, which is the answer. How many square yards of carpet will cover a floor 18 feet long and 13 wide? Draw the line 9-18-L, and parallel to it draw from mark 13-D cutting in F. This gives 8, to which add 18, making 26 square yards of carpet, which is the answer. To find the number of square yards in a floor which is 15 feet long and 11 wide. Draw the line 9-15-K, and parallel to it draw from mark ii-C cutting N. This gives 3K, to which add 15, making 18^ square yards in the floor; this is the answer. Now give the number of square yards on the surface ot a counter which is 10 feet long and 75^. wide. , Draw the line 9-10-R, and parallel to it, draw from mark 7K-P cutting in J. This gives 8}i square yards on the surface of counter, umch is the answer. It is now evident that the steel square may be made to give many other useful and practical ideas, besides those which have been shown. THE SQUARE IN IIANDRAILING For over 25 years I have felt certain that some genius will arise and show the world how THE STEEL SQUARE ,09 all kinds of handrailing may be "laid out" by the use of the steel square alone. I have wrestled with the subject often and long, but it has so far eluded me, though in a hazy way I have been able to get a glimpse of the relation between the square, the rise and tread, and an oblique cut cylinder. I know that a relation exists, and that relation and its perfect rendering will be discov- ered some of these days by a steel square expert; and a fortune awaits the man who makes the discovery and gives it to the public. This may seem heretical to the old time hand- railer who has waded through the mazy paths as marked down by the old master-hands of the science, and they may well be forgiven if they turn up their scientific noses at what I have said in this matter, and sneeringly call it so much "bosh." If thirty years ago any person had pre- dicted that the steel square could be made to accomplish what it now can in good hands, the prophet would have been s..t down as a foolish fellow, and his predictions "all bosh." Yet we see what has been done, and knowing what I do regarding the capabilities of the square, I do not hesitate for a moment regarding my reputa- tion as a prophet when predicting that all circu- lar and elliptical handrails will be laid out miL Ij i no PRACTICAL USES OF altogether by aid of the steel square and a piece of string before the end of the first quarter of FiQ. 158 ^^ ir THE STEEL SQUARE 211 the twentieth century. With this certainty before me, I urge all young men, and old ones too, to sometimes try and find the method I refer to. That it exists in the unseen land I am as confident as I am of penning these lines, and to the man who makes the discovery fame and wealth will be the reward. As an item in this direction, I give the follow- ing, which is by Mr. Penrose of England, and which was sent me for publication in this work. I had reached this point myself some years ago, but described it a little differently; but think on the whole, Mr. Penrose's way of putting it is perhaps better than mine, so give it as it came to me. In getting out face molds it has generally been considered necessary first to unfold the tangents ind get the heights, and by construc- tion get bevels. This methoc' is somewhat clifferen ough results are the same, but are produced more expeditiously, — a steel square, a pencil, and a pair of compasses being used. Take, for illustration, a side wreath mitered into the newel cap. The distance the newel stands out of line with the straight rail is usually gov- erned by the width of the hall, but where there is plenty of room it is a matter of taste. The fi 1 '\<- i I * > 1 (I lii I* k ■:t1l 1 S Sii - li « ' aia PRACTICAL USES OP distance the easement runs back is also a matter of cl jke. The method will apply no matter where the newel is placed, or whether the ease- ment is less or more than the one step of tlie example illustrated. What is meant by one step is, that the tangent of the straight rail continues < \v.--- Fig. 159 to the point 2, Fig. 159. The tanjrent 2-1 is level. To produce the face mold, lay the steel square in the position in- dicated by the lines i, 2, 3, 4, not the figure on the square at the points num- bered, and transfer them Fig. 160 to a piece of thin stuff, Fig. 160. Line 3-4 in Fig. 160 is indefinite. Now take the length of the long edge of the pitch board in the compasses, and with point 2, Pig. iTx). as a center, cut the line 3-4 in 4 and draw :-4. THE STEEL SQUARE »t$ Now 1-2 is the level, and 2-4 is the pitch tangent on the face mold. To get the bevels and width of the face mold at both ends, take the distance 34 on the blade of the square, and the height of a riser on the tongue of the squa'-e, apply to the edge of a board and mark by the tongue; this gives the bevel for the lower end of the wreath. Mark the width of the rail on the bevel; this gives the width of the mold at the lower end. Next take the distance 4-x on the blade of the square, and the distance shown on the pitch board by the line squared from its top edge to the corner, on the tongue of the square; apply to the edge of a board and mark by the tongue; this gives the bevel for the op end of the wreath. Mark the width of the rail on the bevel, and this gives the width of the mold at the top end. An allowance of 6 inches is made at the lop end to joint to the straight rail, and 2 inches at the bottom end to form the miter into the newel cap. The springing line is taken from the pitch board. Fig- 159. in which are shown the bevels and the pitch board will help to make clear the method used. The bevel at the back of the pitch board is for the bottom end of the wreath. The triangle has for its base the line 3-4, and for !v! \l^ I <- ill 314 PRACTICAL USES OP its height one riser. The hypothenuse is the length of 3-4, Fig. i6o, and Fig. i6o stands ovor Fig. 159. level on the line 1-2-3, and inclined from it in this cast at an angle of nearly 45°. The top end bevel is shown below the pitch board. The angle has for its base the distance 4-x, and for its height not one riser but the length of a line, from the corner of the pitch board squared from its top edge. This bevel will be understood better by placing the pitch board on the line 2-4 and applying the small tri- angle to it with its base on the line 4-x, and its point even with the top edge of the pitch board. It will then be at right angles to the top edge of the pitch board. In practice, a parallel mold is generally used, and the wreath piece is cut out; both thickness of plank and width of molding being equal to the diameter of a circle that will contain a sec- tion of finished rail. This is a good beginning, and if this much c^n be accomplished by the square, why not more on the same lines? In the eariier part of this work I have shown how the square may be employed in laying out strings for stairs, step ladders and similar work, and if a method or system for setting out hand- m.' THE STEEL SQUARE ai$ rails for circular and elliptical stairs by the square can be evolved, then nearly the whole science of carpentry and joinery may be devel- oped and explained by aid of that wonderful instrument, the American steel square. h nn , ^i^m ^E- I "6 PKACTICAL UiJES OF TABLES In the following tables are all the cuts necessary for obtaining the proper bevels to cut common rafters, hips, jacks, valleys and purlins, either by degrees or by the use of the steel square, for six different pitches, namely, quarter-pitch, one-third pitch, three-quarter pitch, half-pitch. one and a quarter-pitch, and one and a half-pitch. The figures to be employed on the iquare to get the proper bevels, are in the last column, right hand side of the tables. TABLE I QUARTER-PITCH ROOF «7 Degrees, or 6.inch Rise to T2-lnch Ron Descbiption. Common Rafter. Hips Jacks Valleys Purlins VerMcai . Purlins to Plane of Roof I- i ! 1 12x6 12x4i 12X6 12X4^ nm 27 18} 7 18J 60 71J 60 71i 46 65) Nat III 27 18} 42 18} 90 184 ^ 3 U £13 a 12X6 12X4J 12X10} 12x4r Square 13X13 a 1! n Sq. of 12x6 " :2x4i " i.^xe* " 12X4J " 12X'2 " 12Xdi THE STEEL byUARE «I7 TABLE a ONE-THIRD PITCH ROOr 34 Degrees, or 8-inch Rise to la-lnch Rijn Dehckiption. Common Rafter Hips Jacks Valleys Purlins Vertical Purlins to Plane of Roof 12X8 12x5j 13- 10 12X5J Square 12X14J S<|. of 12x8 " 12X55 " 12X8 " 12x5i " 12X12 " 12x«i TABLE 3 THREE-QUARTER I'lTCH ROOF 37 Degrees, or 9-inch Rise to 12-inch Run Description. Conmion Rafter. Hips Jacks Valleys Purlins Vertical. Purlins to Plane of Roof 12X9 12X6J 12X9 12x6i i- - ^ 37 P ot 071 53 62} 53 62| 45 58\ X2Z 37 27} 39 27} 90 130i 12X9 12X6} 12X9^ 12x6} Square 12X141 Sq. of 12X9 " 12X6} " 12x9 " 12X6} '• 12X12 " 12X7J tiS PRACTICAL USES OF TABLE 4 ONE-HALF PITCH ROOF 45 Degreei, or 1 3-inch Rise to isMtiuh Run DnCRIFTIliN. Common Rafter . Hipe Jacks Valleys Purlins Vertical . Purlins to Plane of Roof 3, S5 Si >Si 12X12 12X8^ 12X12 22x8i Art in Mi 55 45 34 i 45 55 45 64 8q. of 12x12 " 12X8i 12X 13 12X8J 1»X 12 12+8i TABLE 5 ONE AND ONE-THIRD PITCH ROOF S4 Degrees, or i6-inch Rise to la-inch Run DCSCRIPTION. Common Rafter. Hips Jacks Valleys Purlins Vertical . , Purlins to Plane of Roof J3 i 12X16 12X11? 12X16 12X111 a 54 44 54 44 ♦J t-« H t->u DC 86 44 36 44 45 51 O w f'' I P *j y: N 3 S" N 3 » rsi 54 44 32 44 90 132t 12X16 12xllf 12x7i 12Xll| Square 12Xl9i Sq. of 12xl») " 12X113 " 12xlt'' '• 12X11? " 12X1J " 12X»3 THE STEEL SyUAKE 319 TABLE 6 ONE AND ONK-IIAl.K I'lTCH ROOF 57 Degrees, or 18 inch Rise to 13-inch Run Dehcbiptjun. Common Itafter . HipM Jacks Vrtlleys Purlins Vertical . Purlins to Pliine of Roof fl |-Si £5 t^ 12X18 i2xr.J» 1','X1M 12X121 •a t 33 44 iJ3 44 45 50 m i2^ 57 46 30 46 ttO 120 III 12X18 12X12« 12x»i 12x12* Square 12x21 £-■3 >33 Sq. of 12X18 " 12X12| " 12X18 " 12X121 " 12X12 " 12x10 I have given the pitches of those roofs which are more generally used than any other, though the same rules which obtained the above figures could be continued indefinitely. It has been thought, however, that the foregoing examples were quite sufficient for all ordinary purposes. In these tables it will be seen that the bevels for purlin cuts have been given, both when the purlin is square or plumb with the horizon, and v\hen it sits with one of its sides against the raft- ers, or inclined with the roof. The square is made to produce all these bevels. ii iao PRACTICAL USES OF i A m m wM i : wi- ^^H' I t -^ ii Hi ' •■,\ f^H^Bj FOR ESTIMATING CONTENTS OF RAFTERS In the earlier pages of this work, I promised to publish a table wherein the contents of rafters might be estimated without being obliged to take actual measurements of the timbers, and to this end the annexed table. No. 7, is presented. In the lengths given, there is no provision made for projections over eaves, or for ridge poles. The measurements are from the face of the plate to the point of ridge. The length of any rafter is given for roofs having a pitch of one-quarter to one-half, and a span of from 8 feet to 50 feet. Xo provision is made for frac- tions of feet in width of building. The length of the rafter being obtained, and its sectional area being known, the contents may readily be found: Thus, suppose width of build- ing to be 34 feet, rafters to be 2x6 inches, sec- tional area; pitch 9-inch rise; then we have length of rafter by table, is 21 feet 3 inches, and as each foot in length of a rafter 2x6 inches con- tarns one foot board measure, we have 21 feet 3 mches as the amount of material in each rafter, board measure. So with all the other dimensions in the table. The lengths of rafters are given. Determine the sectional areas of rafters, and the contents may easily be found: THE STEEL SQUARE 921 w .c c c a (^ .2 O "o (3 M CIS H ^ ^ -g W ►^ -^ *.- a< •asm jOO'^H 00^ 00^ 00 TJ< OO'^ 00T*< 00 .i: —I Ti -< ^ ^ >-« o< Ti c< o I'* ?»_w rerjccM 00 ^ • »" •asm j5D0D0i>-iWTr«f»05O?I'^l0f»XO««C01C»Xi3 ir 1 -1 — < T~ »— — 1^ •?»?>/?) T< C^ ?< W CC CO OS CO cc ■^ 5«50C>»3503 = t=i.-;,-.OOt-^t-(M» OT OS ^= lO *; —-<>--< — .-I —.?> o< gi Tj ?) o> rr oa co 55 to r? So'^foosascoasox? J10l>XO-<05-^!C(-3SOCOOSl~tOXS;0(?Je0 1.'5® ±! "-" —I — —'—'—''?>'?) 0> ?< C> l?> C) ?t 05 OS CO CO •3STH U3UI-81 iJt^OlCCOSX) — !C— •'J"3SC»t- = OOC0 * ^® ^5« ^Ult-SDOS — SJ-*lIS5Cx3ST-.o>tW»XOS»-i(rJC01ffl i: "^ ^ -" -^ — — — 0< IM i?> O) O) ?! Ol CO 03 CO 05 •ssia •3sia qouioi i:iAcsi-i®oe<;c.-icci» "^' X * IC Ci -" O O Ol W th tiU5«e»(»OM05->t»r-05O— COtlCI^XCJ— iNOS ±t i-*^^— ^■^ ^^ '— — *o^ '?^c^ o> o> O 7> 0» CO 05 CO ; -*?>—»:^^r^ -S-JC— i.l.,x — i;o^®Or-ii.'sxscot-.-iN«co-Htcxo'j'i-»-cj« ^rt^t-wo^os-fo^xaso^cs-^^cr-xa — e} -^i; *^ ^^ '^ — ^^ ^* "^ -^ C> ?> ?? Ci gf o OJ CJ 05 05 CO » 3S ^ o to I- X » — ^ CO i-i 50 r- 00 o -^ ?> CO i." « r- X o t— .^^ — ' — — ~ — -^ -^ ^ gJ ?' gt Tl ?) CO CO — ;ji-x— f r?«t — »■— M ■^sjji — * ® '^ =='- =5 ooj Lo t- o 3 01 1.0 »- o d Mt- t- o 5 UOUT-H I '^ *"* *" »:'*«cj>xosoo}os-towxs5C— o»TCLO-jr-x© " r'.;^-'^ " — ' ^ — '-' o> o> o o» o< o.( g» o) CO •3stM O t- » O ioj ^- -J X O i-COlCJ'Ol^ ^■*l.0»X05O— OJCO'OtCI-XSiSOlCO-tlCSOt^X -^^ '-'"'-' — — — — '-"- o« oj e< c( oj o} g» (» •■>sra JounH e is H a ij o H H i: o t- X o -- = oj CO 1.0 '-r (- 3s « — o» -qTiff i- x o >- j'*i-o:c(-x©- .^^ "—'—'—' — — ' — — -• 01 0> 0» CT 0< C« 0< P< X © oj -r ffi X o o r- >-« .-1 « —I 01 sS^^^^SSig I I XR/W*MnBf rn. I i\ if PRACTICAL TREATISE ON THE STEEL SQUARE. VOL. II. The student will be expected to read carefully these papers before doing any work. His name and address will recjuire to be given on each page. He will be expected to write up the (jucstions in a neat and intelligent manner, using his own style and language, representing the answers in such a manner as will be intelligible — make all drawings as clear as possible, and wherever they can be lone render them in India ink. Let each answer be original, do not copy either from the instruc- tion paper nor from any other source. The paper used may be of any kind, provided that it is clean and durable. Do not attempt an answer until you have thoroughly grasped the subject. QUESTIONS. 1. Give description of "steel square" from Augusta, Maine. 2. Give some examples of how this square may be used. 3. Give description of color and coating of sev- eral steel squares, and what may be termed "the best." 222 i^^ If THE STEEL SCJUARE 223 4. Give description and sketch how to find by use of square, the cuts for each side of a roof hav- ing- a box 12 inches square set on the ridge of roof. 5. Show by sketch and descril)e how to deter- mine by a steel square tlie result of any number —for example, 6 multiplied by the sine 45°. 6. When the rise and seat of the corner posts are given of any rectangular framework wiiich slopes alike on all four sides, show how to find the cuts for the ends of tlie corner posts, and the blade of the bevel to be applied to the two faces adjoining to the ridge line. 7. When the run and rise of a common rtjof are given, and we wish to place upon it a perpen- dicular square pipe to stand upon the roof dia- mond-shaped, show how by the steel hf|uare to find the cuts for the bottom end of tlie pipe, and the blade of the bevel to be applied to the two faces adjoining the lowermost vertical edge. 8. Suppose we wish to cut an opening in a roof for a round pipe or a tile, so that the pipe or tile will stand vertical through the opening, show by sketch how the exact form of the opening may be obtained. 9. Describe how by use of a steel square the size of a pulley that would be required to replace one in use, if speed is reduced or increased. 10. When the pitch of cogs and diameter of a wheel are given, show how by use of steel square to find the number of cogs in wheel. ?24 PRACTICAL 'JSES OF l» 'l ' 2. i J 11. When the diameter of a circle is given, sl.^vv how to find the side of a square of equal area, by use of the steel square. 12. Show how by use of steel square to find till' number of square yards in a given area. 13. Show how by use of steel square, to deter- mine the circumference of a circle when the diam- eter is given. 14. When the area and diameter of any circle is known, show how the area due to any other diameter, or a diameter due to any other area, may be determined. lo. Gi'-e description and sketch showing the advantage of the steel square, in determining the proper depth of "core-boxes." U). Show how to find the diagonal of a sciuare or parallelogram by use of tl;e S(iuare. 1/. Show how to find the circuipference of an ellipse or oval, by use of the scpiare. 18. Show hcnv to 'ind the side of the greatest square which may be inscribed within a circle, by use of the square. 10. Show how to inscribe three small circles within a large circle of given diameter, set to 6yj inches on tongue and 14 inches on blade. 20. Show how to get the length of a hoop for a wooden t;ink by the steel square. 21. Show "arithmetically" how to find the length of hoop for same tank. 22. The proper angle for ordinary door and window sills is about one inch drop to the foot, ii THE STEEL SQUARE 225 show method of finding this incHnation by use of the steel square. 2i. The arms of a straight horizontal lever are 8 and 12. A weight of 9 lbs. is suspended from the shorter arm, what weight will balance it on the longer arm, show by use of steel square. 24. Show how much power is required to sup- port a weight of 4 lbs on an incline of 5 in 30, by use of the steel square. 25. A body is weighed in a false balance and in one scale appears to be 9 oz. and in the other 12 oz. What is its true weight? Show this first- ly by arithmetic and secondly by use of the steel square. 26. A spout is 20 inches square, what is the diameter 01 a cylindrical one with the same area of L.oss-section? Show how to obtain by use of the steel square. 27. Show how by use of the steel square the length and angles of a brace of irregular run or any run, may be obtained. 28. Show by descriptio-. and sketcli how to give a square stick an octagonal shape where frac- tions of an inch are involved. 29. Show how to divide a board 7 inches wide into ^our equal parts, by use of "the rule." 30. Show how to square a board by use of "the rule." 31. Show how to strike a circle and have noth- ing but a rule. 226 PRACTICAL USES OF -i I ■l\ .i : ? . 228 PRACTICAL L'SES OF 51. Give description aiul sketch sho\vin.4 feet wide, by use of the steel scjuare. 11 ^' \ H J t f •• 3 1 W- III" i! ■ i * - m -^ -• ■ 4 ■Hftl ^2 Ir *'- :MFJ^^^'r '»<... INDEX TO VOLUME II Author's Preface 3 Aii'ithrr Steel Square 7 All Kni,'lish Mithod of L'sitifj the Square loo An Octagon Towir 123 Anjjit's and Cuts in Octatjon Work 126 A Mithod of Laying Out f,"ur\<-d Rafters 136 Al! Mitt rs for Hopjxrs 163 Another Method of Obtaining Miters 168 A Method of Hopixr Lines 184 A rian of Lines for All Kinds of Hoppers 186 Acute Miters for Hoppers l86 B Bevels of Slopes 28 He\ tds of Window and Door Sills 43 liay Windows, Octagcn 48 Hacking of Hi|)s 115 Bevels for Backing 1 18 Bay Windows 12c Bevels for Hoppers 1 59 Butt Cuts for Hoppers 1 5g Bevels for Odd-shaped Hoppers 162 Bevels for All Kinds of Hoppers 186 Color of Squares 10 Care of Squares Generally 11 i i (I '■A •■ II if .'t i :\ ~ ' i fi' t M ' r ii INDEX TO VOLUME II Crenelated Squares '^^ Cutting for a Round Pipe through 'incline.' ".'.'"" H Center Hip Rafter * * ^ Concerning Roof Framing 1! Cutting Double Bevels .. . f- Croker's Method [[[] Cut of Right Jack l^ Cut of Left Jack Z^ Cuts for Jacks ^^ Cut of Main Jacks ^ Cut of Gible Jacks ^ Combination Diagram, by Mr', 'woods.' .' ?? Cardboard Diagram ^^ Curved Rafters ... ^^ Cove Rafters ............'. '°^ Curved Octagon Rafters !?2 Curved Rafters by Lines [ Co-Pitches and Other Pitches' 'for *Hoppers' .' I77 Corners for Hoppers ' * i/ Cuts for Six Different Pitches 216 Contents of Rafters for Different Lengths. ." .' '.'..'. ] 221 D Description of Complicated Squares ... Diedral Angles ^ Diagram for H ip Jack Rafters ......."' iy Double Curved Raftcrs-Ogcc " ' ' j , [ Diagram for Backing Hips - Describing an Octagon ^ Describing an Octagon Bay Window Wz Diagram of Octagon Tower Dormer Window ^^ Dormer Window Front ^"^j 140 •■."■■■? "^SKtiirw-^z tMM.9*iHiM%ifF INDEX TO VOLUME II m Page Dormer Window Plan . . . , 147 Diagram for Hoppers ice Diagram of Hopper Cuts 160 Dividing a Circle into Equal Parts 193 Different Pitches and Their Cuts 216 E Elevation of Octagon Timber Tower 127 Elevation— Another Vf w 128 Elevation of Dormer Window 147 Elevation of Hoppers 152 Elevation and Plan of Hoppers 170 F Fitting a Box Diagonally over a Ridge 14 For Raking Mouldings and Cornices 25 For Working Core Boxes 34 P>aming Octagon Roofs ng Framing for Dormer no Framing Octagon Bays 120 Framing Octagon Tower 126 Framed and Notch cl Timbers 133 Finished Sketch of Octagon Tower 153 Flares of Hoppers i^g Finding the Circumference of a Circle 190 G Graphic Method of Finding Areas of Circles 31 General Items by Stoddard 84 General Pitches of Six Kinds 216 H How to Obtain Length of Huups for Tanks 42 How to Bevel Window or Door Jambs 43 b I ^ li - 'If i 11 ■ I iv INDEX TO VOLUME II P&ffC Hips and Valleys, by Stoddard s6 Henry Cook's Method of Laying Out Roofs 96 Hick's Method of Roofing 106 Hoppers and Hopper Bevels 152 Hopper, Three-sided je^ Hexagonal Hoppers ,5 . Hopper Cuts by the Steel Square ] . 161 Handrailing 2i» Hips for Six Pitches 216 I In Laying Off Rafters 53 In Laying Out Jack Rafter Bevels .... ,,,, 68 Irregular Pitches [ -g J Jack Rafters , j Joints in Hopper Work ' . 162 Joints For Hopper by Steel Square 178 Joints in Splayed Work, by Riddell igi Joints for All Kinds of Hoppers ] . . 186 Jacks for Six Pitches 216 L Laying Out Sizes for Pulleys jg Laying Out Cogs in Toothed Gear ig Lines for Oblique Framing 22 Laying Off Hip-Backing .' ] 53 Length and Bevel of Jack Rafters " 71 Laying Off Valleys 75 Lengths of Jack Rafters— Another Method 104 Length of Cripples by the Square 106 Lines for Curved Rafters ,,n Ifi! INDEX TO VOLUME II v Page Lines for Hoppers 154 Lumber Measurement by the Square 210 M Multiplication by Aid of Square 15 | Main Rafters 50 Measuring Inaccessible Distances 56 Making Trestles 60 Method of Laying Out Timber Octagon Tower. . . 130 Miter Cuts for Hoppers 158 Method of Getting Joints for Hoppers, by Mr. W' ods 172 Miters for Obtuse and Acute Angle Hoppers 186 Measurements by the Square 207 N Nicholson's Method of Hip Roofing jg O Obtuse and Acute Angles 30 Octagon Bay Windows 48 Ogee Rafters 107 Octagon Framing 120 Octagon Bays 121 Obtuse and Acute Cuts for Hoppers i86 P Polygons of All Kinds 20 Proportional Reduction of Mouldings 39 Proportioninjj Spouts 45 Practical Use of Square and Rule 52 Pope's Method of Roofing 78 :. K ^ I N , n vi INDEX TO VOLUME II Pitches of Roofs ^q Positions of Hips and Valleys qi Plan of Octagon Roofs 124 Pitches and Scales for Towers 134 Perspective View of Framed Dormer 145 Plumb Cuts for Rafters 13S Plan -of Timber Work for Dormer Window 147 Plan of Octagon Tower 142 Pitches of Octagon Tower 143 Plan of Dormer Base 140 Plans of Hoppers 156 Planceer Cuts for Cornice im Plan and Elevation of Hopper 170 Pitch Line for Hoppers j -_, Problems in Handrailing by the Square 212 Pitches of Various Kinds 216 Purlin Cuts for Six Pitches 217 Q Quick Methods of Laying Out Octagon Sticks. . . 46 Quick Methods of Obtaining Hopper Cuts 163 Queries in Hopper Building lyg R Rules for Inclined Framing 23 Roof Framing co Run and Rise of Rafters 6; Run and Rise of Jack Rafters -q Run of Hips j^- Run of Val leys gj^ Rafter Patterns 104 Riddel I's Methods for Hopper Work 182 INDEX TO VOLUME II ▼« Page Remarks on Handrailing 214 Ratter Tables, Pitches, etc 217 S Some Odd Problems 13 Speeding Pulleys 18 Some Good Things 49 Side Cuts for Valley Rafters 72 Stoddard's Method of Roofing 81 Spacing Off a Rafter 85 Short Jacks 88 Sections of Hips and Valley Rafters 94 Smith's Improved Method of Roofing 112 Scale Elevations of Pitches 134 Steep Pitches, and How to Work Them 135 Side Elevation of Dormer 148 Skeleton Frame of Tower 150 Square Hoppers 153 Some Hopper Lines 175 Some Remarks on Hopper Work 180 Stair Railing 213 T To Find Area of Given Circle 20 To Find Number of Yards in Given Area 21 To Inscribe Polygons within Circles 36 To Find the Apothems of Polygons 38 To Obtain Length of Hoop for Barrel 54 To Measure across a River 58 To Measure the Height of a Standing Tree 59 Timber Framing in Octagon Tower 126 Triangular Hoppers 1 52 The New Hopper Lines 177 i 1 ^.1: viii INDEX TO VOLUME II T'le Square as a Calculating Machine 199 Tables for Rafters 216 To Find Cubical Contents of Rafters 221 U Unequal Pitches 76 Uneven Pitches 80 Uneven Valleys 114 V Valley Rafter Bevels 64 Valley Rafter and Cripple Cuts 73 Valley for Uneven Pitched Roof 1 14 Valleys for Six Pitches 216 VV Wood's Method for Hips and Valleys 83 Work on Cornices i6g Wood's Method of Working Hoppers 176 Review Questions 222 HOUSE PLAN SUPPLEMENT PERSPECTIVE VIEWS AND FLOOR PLANS of Fifty Low and Medium Priced Houses FULL AND COMPLBTB WORKING PLANS AND SPECIFICATIONS OF ANV OF THESE HUUSES WILL BE MAILED AT THB LOW PRICES NAMED, ON THE SAME DAY THE ORDER IS RECEIVED. Other Plans WH ILLUSTRATE IN ALL BOOKS UNDER THE AUTHORSHIP OF FRED T. HODGSON FROM 2% TO 50 PLANS, NONE OF WHICH ARE DUPLICATES OF THOSB ILLUSTRATED HEREIN. FOR FURTHER INFORMATION, ADDRESS THE PUBLISHERS. SEND ALL ORDERS FOR PLANS TO FREDERICK J. DRAKE & COMPANY ARCHITFCTURAL DEPARTMENT CHICAGO. ILL. i Fifty House Designs E-l= ff i>l: WITHOUT EXTRA COST to our readers we have added to this and each of Fred T. Hodgson's books published by us the perspective view and floor plans of fifty low and medium priced houses, none of which are duplicates, such as are being built by 90 per cent of the home builders of to-day. We have given the sizes of the houses, the cos* of the plans and the estimated cost of the buildings based on favorable conditions and exclusive of plumbing and heating. The extremely low prices at whirh we will sell these complete working plans and specifi- cations make it possible for everyone to have a set to be used, not only as a guide when build- ing, but also as a convenience in getting bids on the various kinds of work. They can be made the basis of contract between the con- tractor and the home builder. They will save mistakes which cost money, and they will pre- vent disputes which are never settled satisfac- torily to both parties. They will save money for the contractor, because then it will not be necessary for the workman to lose time waiting for instructions. We are able to furnish these complete plans at these prices because we sell sc many and they are now used in every known country of the world where frame houses are built. The regular price of these plans, when ordered in the usual manner, is from $50.00 to $75.00 per set, while our charge is but $5.00, at the same time furnishing them to you more complete and better bound. ALL OF OUR PLANS are accurately drawn one-quarter inch scale to the foot. We use only the best quality heavy Gallia Blue Print Paper No. loooX, taking every precaution to have all the blue prints of even color and every line and figure perfect and distinct. We furnish for a complete set of plans : ::-% ^ FRONT ELEVATION REAR ELEVATION LEFT ELEVATION RIGHT ELEVATION ALL FLOOR PLANS CELLAR AND FOUNDATION PLANS ALL NECESSARY INTERIOR DETAILS Specifications cc isist of several pages of typewritten matter, giving full instructions for carrying out the work. We guarantee all plans and specifications to be full, complete and accurate in every par- ticular. Every plan being designed and drawn by a licensed architect. Our equipment is so complete that we can mail to you the same day the order is received, a complete set of plans and specifications of any house illustrated herein. Our large sales of these plans demonstrates to us the wisdom of making these very low prices. •■«■ ADDRFSS ALL ORDERS TO FREDERICK J. DRAKE & CO. Arihitectural Vefiirtment CHICAGO. ILL. u Ml mn « ^^ak - » t i Ucz.f ' i '! |Bj 'J. |;-SH R '1^ 1 1 A s J3 o '2 ¥: © 1 v^. .• CO ^Jkt'' • <-* ''^/' c fl-' /^ M «?'•■• •3 i^''' c ^n .bf ^s. "55 o § ^Z' Q » -2 ^ c i is ;c i£ » eS to S M S I g •r ^* § o is iiil ^ • -■= ii ns. a N^ U r^ •M (A •M ** *m M •1 le ■ "" ^^ "* 0^ ^^ >«' 7f 1 «4 PQ ic c s 2 '■J LS, 8| .5 -S (Q to N ^ - o o c c8 C C 1^ 8. o k. a a s u ® ^ - .2 o. >. 'ifl 1^ t* 8 £5 a e« «• 8 -2 r: 5 - <" -B " J o 1 I I -;•■ I I ' s I i ■^'P m mM 1 s u 'Ja s r^ 2 CO c: o d V P ^ 3 J?. o CO ,i| ^^^^^^w7 TTTsa •I , j^ I^^B i S H CO « M S. js a a o .a "B £.2 01 ^ a ■« *' .2 S 3 o M .2 « .5 »■: S a 5 " Si o. H 8-H 95 ijy n 1 ' i H II J-.' " i ' CO ~ .6 t t/3 = ffi t 3 j3 a S .-" 3 u a (0 -5 00 J ■- s i5 2 c o I. 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S 5 ^ -2 £ S 1^ kH StCOrtO FLOOR PtflN-Jofl, f ^:^jL'i'(PS\.^e.4i:^.rjmk^.x^iii.i'iemh k^t^ ^^V e o 2 «■> e O o n "a In -1 % (so u . ~^m!tf.% :x ^dpi "SR-^*.* ' ■ —. ^fii '■■■»■ ' ■■■ .ifc. »— _ .iJ ; . ~ ~*-y «» r*i 5- - ' -t ' — ■- « ■■ ■ ji^- ■^ '" '? . .» e o o X V en C rt u O o 1 ^ flooi plan **• c o i« •• a S c -p 1 § d founda ns with ( sS^. Si Width, Length cellar an on. pecificatlo v.= n St leva tten M « r s con side typew srint and lete rr~ o> Sg E CqA tS f^ ^ai.,^ -^^'^ s a. ^ zn rrr'JS/ Ms P 0.(0 8 o O .•trt4,7««, be O 55 -5 "S. M •a ^ a - Ri rt 03 O L j m If - i BrP' H[fii' ' ■fl''''' V ' i \ Ll. ._._.. ' a. s b] .^£^S i#. *^ 2 -^ b4 r UJ N 10 xf 2 c 14 o c e !2— r ° C C o ^ « - c c 5 - o - -2 iS « 2 p.c5 o.„5^ o c o «* q a 3 3 ^ C C * iv? o t: "D o (0 00 "i- ™ K O c c TJ-OTJ — CSC (D n (1 ttf o I* J MICROCOPY RESOLUTION TEST CHART (ANSI and ISO TEST CHART No. 2) 1.0 I.I 1.25 ■ 50 i^ IM Li 1^ 1^ IP III 1.8 J /APPLIED IIVMGE Inc ^^- t653 tost Main Street S*— Rochester. Ne« ''<>"' '4639 uSA .^B (7t6) 482 - 0300 - Ptiore aas (716) 288 - 5989 - Fa. te^ll|iiwMt^^t_ i^^^m: ::^T<.7M'^^ms^^^^"'rm- •o c Q CO c O o « s iqs ro j= j= T3 M »J c p c (« o o « c . ■^§ o >~ C O c •J rt « "5. -c c rt o «> ■^ •£ •a -r c f 3 M ™ o V a, o n 4) p 03 O •a :i V o 5 CO C o o o. - W CS •tJ ** rt 3 !a^ rt « a. ^ ^' i: o O (4 5 « O 01 C4 (« o c ■^ *> n Zi 'n C <= > o V o S. OQ U T?3gBl!3 -,Ji.'; HI liv V ^ 2 c = o rt .2 V CD O : i )l .*?«a'^- ■jtfi o U en U N 55 ■£ « S o ^ ^ a. M ti "~ CO '*■ o • J- o jz £ > u 1^ B O n c . ft n •5.S •3.-g g 2 c * 3 m ° c *" o -a s c rt rt o (4 O ~ u V a U n o u 8 a c « — o CQ O II -.jiwS».^'#KI»IB™F«'?*^T5^??aS3r?:3aPi^ ^11 1 mm I "fMt -a o o V 4) 4) a 2 o o c P ° U. O ^■^-j^rja I ft ^''^^Ml^i% Mt^i^f^ :^'^T^'m^^^ u 9 W) u N - r- 2 c o I Ob c 5 "3. 5 gi § '^ c " o *^ O Q. .. _ E « — cow w s OJ 2 O tl! »q ir ■niaiiimKiXi h V^-^^'j;^.' 1^ h ff^ \ i ■Ffis;::':^«!!3i^^^KiJi^r •*j^'%.' S ^^If-'. ■^!^'^a^i V a.r- LI s^?""^'^' n^-. kj mtii Hijii \Mm ^^^^^^^^Tnoi mm i.^ '.t'^';rw.^9^^- s o o Ctf B o C/l C o o UJ "Q ^ T3 > .J ol i J= ■g « '? 5 « c Tl " o S "^ (4 c« n O u nl Si s K- o .. ~ c M I« .tJ c' "• > 8-S « o c u cu ^s 4J *^ n '' in F « t: o " w , o U j: o nl > 1 > i 1 i 1 :. . i; ' j if i U C o Full and complete plans and specifi- cations of this house will be fur- nished for $6.00. Cost of this house is from $3,650 io $3,700. ■»«!»I*Ymi-, .". C C0 -c o o tu o g ? -J •i c o > JO "S t> •o w •a c ft c o c "5. o o 3 s ^ o 's.-g g S ■^:§ C !* 3 W £ 1= "- O T3 — C <« CD O >- S cfl y :=: u (u o. u n o u c > n i> c »> i.' I 03 O T III lYIMllTf r' TTli lilHIll AJi .3 o B O u i-tOmmahMi a o 4-1 CO o U c o o J! 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J2 u a re u E i ~ -S ^ ^ — r u -c S C •— oj E Ul ■? s c «< « .t; 1^ t^ > >. C j/ - "H. 2 '° "" «j »i C -U U ^ bt •— Ui -. S.2 Ji "O ^ c I' >- = H c ^ •- 01 >. 'i •a >- w C fl « S? = I- H ra " x: u — ^ ^ (A ai r -i t^ ^ ■? U- X =- •« ™ S m u c !?=T1 " 1 c 1 X ^ « •- C i, *-• ""' — S u ;? 'o f ■■■'- :l i di J =1£ c ^HR T fi o 19 & ^ u — a is w. o C - 2 c-C-: be CO ^ o S f (« c U, U o o (0 < r< o o ^ j: ■£ \2 c CJ a. tr. *-* •c .-« CO u k. t/l '— re »-«i r/ >. «' ~ c ^ i< c U is o ■u 4-* re 13 rs Q c c* x 5^ 3 c fB _J ^. im (/J X j_, m o X n o >4 If) c '■J u. u c v; H — c :/) T rK c 4>J ti. c c u p: -. « 1/3 i; K c o o Si iT o o S f so $ ■= -2 -• .a J3 i« bii S <» w, «> ti< u 21 g o — — -g ja .S 3 nj c 5£ Z ^ o - c c -c U 5 C.S a2 J* o c. c 9 w T^ M o' Z Z o •>« CO Ui Q U. O to z < cu o o .J a. •J lb u • ^ k. X li^ Ij ^ •«> n; ^1 w ^ -• O ^ ^ > ^ •^ t/) 7B c "u :^ -J n / *• -pp ^ .-1 a, as O O -< b H It > u 'w o a o O in U C 7; 19 "2 § i .. . Q- be 5 c c — K c n: ^ r: tf] M tfj — L. tj ;: re r: u ^ o " »- in ^ o -. ^ * ?^ a. s o It 8 3 •9 - ^ 5 i " w '-E 'S^ o c £ 8 ^- r- •2 i CS c fii U 25 o o 2 Z o mm (/) (d Q U. O CA Z < § X 1- s V /. . - ^ -ti tt - r, - ■/. £ « C T- c ii >. = -J v; S ' - c 2 5= = ? ~ c c — ^ 5!:^ ? - — r; f. '■J. VjWp^:^nt?pm' 'f\ ■«.■• t: -l w C u « 2 o en « • >-' c y IC fii a. . ij .ti Cfl ' . - a! , t- - u / i ; - .2 w _• ji: j: *•-•*-* X ^ - . = ■" y. - c c K -B ^ ,y s w - <-i « o t' :^ u . c ^o I 1- ■0 ^ S CO 7] *^ c o - o " 1.-^ u* c s / -^ I 1 >- ■- 1 - ■« / .— r ? •/! • .M S :j Cv i/i c -^ C £/) y .- -c *^ c ^ (4 — 4-1 — ' t/J 3 O fc u 27 sr '.m ^.:m. r "T-l Mi i= I ^ 2(5 ..*_i;«i. o o as y. S 3 •c .2 CO ■a s — c *"^ (/) — 3 ra to t£ Co o — 18 ^ E - C T (J .- C >~ t^ ^•mim B o c 2 a >-i to U] Q b O < cu o o ir. o « s u '* 01 e .2 > u II II 5 " »_ o c o o k- w O C j2 "C o •c c 3 C •" ij; 41 £r'o = r: 4; CO !/. ,; e >- ■!»:. VA_ urn '* *V1 Remember We can mail out the same day we receive the ordei any complete set of working plans and specifications we illustrate in this book. Remember also That, if you are going to build, complete working •^lans and specifications always Save Money for both the owner and contractor. They prevent mistakes and disputes. They save time and money- They tell you what you will get a.id what yon ire to do. m -mii Estimated Cost It is impossible for any one to estimate the cost of a building and have the figures hold good in all sections of the country. We do not claim to be able to do it. The estimated cost of the houses we illustrate is based on the most favorable conditions in all respects and does not include Plumbing and Heating, Possibly these houses ^annot be built hy you at tht prices we name because we have used minimum materia? and labor prices as our basis. The home builder should consult the Lumber Dealer, the Hardware Dealer, and the Reliable Con- tractors of his town. Their knowledge of conditions in your particular locality makes them, and them only, capable of making you a correct estimate of the cost .-..f % : "m^.J>Mm4» /.% TO CORRECT MEASUREMENTS of areas and cubic contents in all matters relating to buildings of any kind. Illustrated with numerous diagrams, sketches and examples showing how various and intricate measure- ments should be tak«n :: :: :: :: :: :: :: :: :: By Fred T. Hodgson, Architect, and W. M. Brown, C.E. and Quantity Surveyor m /jiHIS is a real prartir.il bonk, ^*' phi'wing h'W all kinds of odd, ( iiMikfcd and ditticiili mea^^- ui t'liif nts in a y Ix; takc-n to ^e^ult• cnritTt rc-^nlts. Tliis wnik in no wav rontiirts with any wmk on t'-^tunalint; as it dtjfs nut nwv piire-i. nt'ithfr does il attempt ta deal with MUestinns of lal.itr or e^tiuiato lii'vv ninrh tlir I'xrciuion of rcr- lainwuKs will co-t. It sjniply dt-aU with tliM <|Mes,iions nf areas and riibic r* intents of anv 1,'iven wt'ik and shows liow liifir ait-as .uid rmittMils may i*-;idily be oliiained and fin- ni-hes for ttie reji'ilar estimator tluj data npon whi^h he can b.i'-t: liis prices. In fact, the U'lT k is a ureat aitl and .issist- ant to the re^nilar estimator ami of inestimable vahie to the g'jueral builder and contractor. ■* 'S'l 12mo, cloth, 300 pages, fully illustrated, price - $1.50 Sold by Booksellrrs ffenerally or sent postpaid to any address upon receipt of price by the Publishers FREDERICK J. DRAKE & CO. PUBLISHERS CHICAGO. U.S.A. * y\