FRANKLIN INSTITUTE LIBRARY PHILADELPHIA Class <^.S.Mr Book hl..J^..(^.. Accession.../^. out, all unbound periodicals, and such text book.^ as ought to be found in a library of reference except when rfequircMl by Committees of the Institute, or by members or holders of seooa^ class stock, who have obtained the sanction of the Committee. ThJ^eCond class shall include those books intended for circulation. Artici.e VI.— The Secretary shall have authority to loan to Members and to holders of second class stock, anv work belonging to the s-kcond (;la.ss, subject to the following regulations: Section A— No individvial shall be permitted to have more than two books out at one time, without a written permission, signed by at least two members of the Library Committe ; nor shall a book be kept out more than two wkeks ; but if no one has applied for it, the former bor- rower may renew the loan. Should any person have applied for it, the latter shall have the preference. Section 2.~K fink ok ten cents per week shall be exacted for the detention of a book beyond the limited time ; and if a book be not re- turned withm three months it shall be deemed lo.st, and the borrower shall, in addition to his fines, forfeit its value. Section 3.— f^honXd any book be returned injured, the borrower shall pay for the injury, or replace the book, as the Library Committee may direct ; and if one or more books, belonging to a set or sets, be lost, the borrower shall replace them or make full restitution. AiiTK-i.E VIL— Any person removing from the Hall, without permis- sion from the proper authorities, any book, newspaper or other property in charge of the Library Committee, shall be reported to the Committee, who may inflict any fine not exceeding twenty-five dollars. Aktjcle Viri.— No member or holder o'f second class stock, whose annual contribution for the' current year shall be unpaid or who is in arrears for fines, shall be entitled to the privileges of the Library or Keadmg Room. Article IX.— If any member or holder of second class stock, shall refuse or neglect to comply with the foregoing rules, it shall be the duty of the Secretary to report him to the Committee on the Library. Article X.— Any Member or holder of second class stock^ detected 111 mutilating the newspapers, pamphlets or books belonging to the Insti- tute shall be deprived of his right of membership, and the name of the' offender shall be made public. THE STEEL SQUARE. THE CARPENTERS' Steel Square, AND ITS USES. BEING A DESCEIPTION OF THE SQUAKE, AND ITS USES IN OBTAINING" THE LENGTHS AND BEVELS OF ALL KINDS OF RAFTERS, flIPS, GROINS, BRACES, BRACKETS, PUR- LINS, COLLAR-BEAMS, AND JACK-RAFTERS; ALSO, ITS APPLICATION IN OBTAINING THE BEVELS AND CUTS FOR HOPPERS, SPRING MOULDINGS, OCTAGONS, STAIRS, DIMINISHED STILES, ETC., ETC., ETC. ILLUSTRATED BY OVER FIFTY WOOD-CUTS. Editor'' Sf ''t^if '"'j^tflci^ 'hu'd ' PFJodworier}' '' ' NEW YOEK: THE INDUSTRIAL PUBLICATION COMPANY. 1880. COPTBIGHT 6ECCKED. Some time ago, the author of this little work contributed a series of papers on the Steel Squake and Its Uses, to the American Builder, and since their appearance, he has received hundreds of letters from as many persons residing in various parts of the United States, Canada, Australia and New Zealand, in which the writers requested him to publish the papers in book form. Partly in compliance with these requests, and partly at the solicitation of personal friends, together with a knowledge that a cheap but thorough work of the kind, would be of service to all persons who have occasion to use a steel square, he has consented, with the aid of the present enterprising pubUshers, to issue the work as now offered. It is only of late years that American workmen have begun fully to understand the capabilities of the steel square ; and even now, only a few of the best workmen have any idea of what can be accomplished with it when in skilful hands. It is not claimed that the rules und methods shown in this little work are either new or original ; they have been known to advanced workmen for many years past; but it is claimed that they have never before been brought together and put in so handy a shape as 1^ iv PREFACE, in the present book ; and it is furttier claimed tliat many of the rules herein illustrated and explained, have never appeared in print previous to the publication of the papers on the subject in the magazine referred to above. Should this little volume prove of service to the man who toils with axe, saw and plane, for his daily bread, and profitable to the publishers who risk their money on its publication, it will have fulfilled its mission, as designed by THE AUTHOE. New York, 1880. CONTENTS. PAKT I. Preliminary . - - - - The Square as a Constructive Tool Lines and Scaies Description of Scales . . . Board Rule - - Brace Eule - - - - - Octagon Scale - - - - ■ Fence for Square - - - - Application of Square - - - To lay out Eafters - - - - To Cut Collar Beams, Gables, Pitches, etc. Hip Rafters, Cripples, etc. Backing for Hips How to Cut a Mitre Box - Stairs and Strings . - - Miscellaneous Rules . - - Board Measure - - - - Proportion of Circles Centering Circles . . - How to Describe an Ellipse How to Describe a Parabola Bevels for Hoppers Bisecting Circles Cutting Spring Mouldings vi C O K T E N T S. PAET II. PAGE Theoretical Eafters - - - - - - _ 46 Bevels and Lengths for Hips, Jacks, and Purlins - - 48 Divisions of Widths - - - - . . .50 Bisection of Angles 50 Diminishing Stiles - - - - - ■. .51 PAET III. Octagons ; - - - - - . -52 To Find the Diagonal of a Square - - - - 5G Polygons 57 Circles - -- -- -- .58 To Lay out Anglos ■ - - - . . - 60 Bevels of Hoppers ------ 60 Widths of Sides and Ends of Hoppers - - - - 61 Corner Pieces for Hoppers ----- 62 Roofing 63 Lengths and Bevels of Hip-Rafters - - - - 64 Backing of Hips ------- 64 Irregular Hip Roofs ------ 65 Side-bevels of Jacks - - - - . . . 66 Trusses ------- .g? THE CARPENTERS' STEEL SQUARE, AND ITS USES. /f ^^'^ PART T. Preliminary. — There is nothing of more importance to a young man who is learning the business of house-joinery and carpentry, than that he should make himself thoroughly conversant with the capabilities of the tools he employs. It may be that, in some of the rules shown in this work, the result could be attained much readier with other aids than the square ; but the progressive mechanic will not rest satis- fied with one method of performing operations when others are within his reach. In the hand of the intelligent mechanic the square be- comes a simple calculating machine of the most wonderful capacity, and by it he solves problems of the kinds continu- ally arising in mechanical work, which by the ordinary methods are more difficult to perform. The great improvement which the arts and manufactures have attained within the last fifty years, renders it essential that every person engaged therein should use his utmost exertions to obtain a perfect knowledge of the trade he 73 8 THE STEEL SQUARE professes to follow. It is not enough, nowadays, for a per- son to have attained the character of a good workman ; that phrase implies that quantum of excellence, which con- sists in working correctly and neatly, under the directions of others. The workman of to-day, to excel, must under- stand the principles of his trade, and be able to apply them correctly in practice. Such an one has a decided advan- tage over his fellow-workman ; and if to his superior know- ledge he possesses a steady manner, and industrious habits, his efforts cannot fail of being rewarded. It is no sin not to know much, though it is a great one not to know all we can, and put it all to good use. Yet, how few mechanics there are who will know all they can ? Men apply for employment daily who claim to be finished mechanics, and profess to be conversant with all the ins and outs of their craft, and who are noways backward in demanding the highest wages going, who, when tested, are found wanting in knowledge of the simplest formulas of their trade. They may, perhaps, be able to perform a good job of work after it is laid out for tficm by a more compe- tent hand; they may have a partial knowledge of the uses and application of their tools ; but, generally, their know- ledge ends here. Yet some of these men have worked at this trade or that for a third of a century, and are to all appearances, satisfied with the little they learned when they were apprentices. True, mechanical knowledge was not always so easily obtained as at present, for nearly all works on the constructive arts were written by professional archi- tects, engineers, and designers, and however unexception- able in other respects, they were generally couched in such language, technical and mathematical, as to be perfectly AND ITS USES. 9 unintelligible to the majority of workmen ; and instead of acting as aids to the ordinary inquirer, they enveloped in mystery the simplest solutions of every-day problems, dis- couraging nine-tenths of workmen on the very threshold of inquiry, and causing them to abandon further efforts to master the intricacies of their respective trades. Of late years, a number of books have been published, in which the authors and compilers have made commend- able efforts to simplify matters pertaining to the arts of car- pentry and joinery, and the mechanic of to-day has not the difficulties of his predecessors to contend with. The workman of old could excuse his ignorance of the higher branches of his trade, by saying that he had no means of acquiring a knowledge of them. Books were beyond his reach, and trade secrets were guarded so jealously, that only a limited few were allowed to know them, and unless he was made of better stuff than the most of his fellow-workmen, he was forced to plod on in the same groove all his days. Not so with the mechanic of to-day ; if he is not well up in all the minutae of his trade, he has but himself to blame, for although there is no royal road to knowledge, there are hundreds of open ways to obtain it ; and the young mechanic who does not avail himself of one or other of these ways to enrich his mind, must lack energy, or be altogether indifferent about his trade, and may be put down as one who will never make a workman. I have thought that it would not be out of place t9 pre- face this work on the " Steel Square," with the foregoing remarks, in the hope that they may stimulate the young mechanic, and urge him forward to conquer what at best are only imaginary difficulties. A willing heart and a lO THE STEEL SQUARE clear head will most assuredly win honorable distinction in any trade, if they are only properly used. The "Square," as a constructive tool, must of necessity have found a place in the " kit " of the earliest builders. Evidences of its presence have been found in the ruins of pre-historic nations, and are abundant in the remains of ancient Petra, Ninevah, Babylon, Etruria, and India. South American ruins of great antiquity in Brazil, Peru, and other places, show that the unknown races that once in- habited the South American Continent, were famiHar with many of the uses of the square. Egypt, however, that cradle of all the arts, furnishes us with the most numerous, and, perhaps, the most ancient evidences of the use of the square ; paintings and inscriptions on the rock-cut tombs, the temples, and other works, showing its use and appli- cation, are plentiful. In one instance, a whole " kit " of tools was found in a tomb at Thebes, which consisted of mallets, hammers, bronze nails, small tools, drills, hatchets, adzes, squares, chisels, etc.; one bronze saw and one adze has the name of Thothmes III, of the i8th dynasty, stamped on their blades, showing that they were made nearly 3,500 years ago. The constructive and decorative arts at that time Avere in their zenith in Egypt, and must have taken at least 1,000 years to reach that stage. Consequently, the square must have been used by the workmen of that coun- try, at least, four thousand years ago. The British Museum contains many tools of pre-historic origin, and the square is not the least of them. Hercu- laneum and Pompeii contribute evidences of the importance of this useful tool. On some of the paintings recently dis- AND ITS USES. 11 covered in those cities, the different artisans can be seen at home in their own workshops, with their work-benches, saw-horses, tools, and surroundings, much about the same as we would find a small carpenter shop of to-day, where all the work is done by hand ; the only difference being a« change in the form of some of the tools, which, in some instances, had been better left as these old workmen de- signed them. Lines and Scales. — The lines and figures formed onffiF' squares of different make, sometimes vary, both as to their position on the square, and their mode of application, but a thorough understanding of the application of the scales and lines shown on any first-class tool, will enable the student to comprehend the use of the hnes and figures ex- hibited on other first-class squares. To insure good results, it is necessary to be careful in the selection of the tool. The blade of the square should be 24 inches long, and two inches wide, and the tongue from 14 to 18 inches long and inches wide. The tongue should be exactly at right angles with the blade, or in other words the " square " should be perfecdy square. To test this question, get aboard, about 12 or 14 inches wide, and four feet long, dress it on one side, true up one edge as near straight as it is possible to make it. Lay the board on the bench, with the dressed side up, and the trued edge towards you, then apply the square, with the blade to the left, and mark across the prepared board with a penknife blade, pressing close against the edge of the tongue ; this process done to your satisfaction, reverse the square, and move it until the tongue is close up to the knife mark ; if 12 THE STEEL SQUARE you find that the edge of the tongue and mark coincide, it is proof that the tool is correct enough for your purposes. The next thing to be considered is the use of the figures, Hnes, and scales, as exhibited on the square. It is sup- posed that the ordinary divisions and sub-divisions of the inch, into halves, quarters, eighths, and sixteenths are un- derstood by the student; and that he also understands how to use that part of the square that is sub-divided into twelfths of an inch. This being conceded, we now pro- ceed to describe the various rules as shown on good squares. Description. — On the Frontispiece we show both sides of one of the best squares in the market. The following instructions refer to this plate. The diagonal scale is on the tongue at the junction with blade, Fig. i, and is for taking off hundredths of an inch. The lengths of the lines between the diagonal and the perpendicular, are marked on the latter. Primary divisions are tenths, and the junction of the diagonal lines with the longitudinal parallel lines enables the operator to obtain divisions of one hundredth part of an inch; as, for example, if we wish to obtain twenty-four hundredths of an inch, we place the compasses on the " dots " on the fourth parallel line, which covers two primary divisions, and a fraction, or four tenths, of the third primary division, which added' together makes twenty-four hundredths of an inch. Again, if we wish to obtain five tenths and seven hundredths, we oper- ate on the seventh line, taking five primaries and the frac- tion of the sixth where the diagonal intersects the parallel AND ITS USES. 13 line, as shown by the " dots," on the compasses, and this gives us the distance required. The use of this scale is obvious, and needs no further explanation. Board Measure —Under the figure 12, Fig. 2, on the outer edge of the blade, the length of the boards, plank, or scantling to be measured is given, and the answer in feet and inches is found under the inches in width that the board, etc., measures. For example, take a board nine feet long and five inches wide; then under the figure 12, on the second line will be found the figure 9, which is the length of the board ; then run along this line to the figure directly under the five inches (the width of the board), and we find three feet nine inches, which is the correct answer in " board measure." If the stuff is two inches thick, the sum is doubled ; if three inches thick, it is trebled, etc., etc. If the stuff is longer than any figures shown on the square, it can be measured by dividing and doubling the result. This rule is calculated, as its name indicates, for board measure, or for surfaces i inch in thickness. It may be ad- vantageously used, however, upon timber by multiplying the result of the face measure of one side of a piece by its depth in inches. To illustrate, suppose it be required to measure a piece of timber 25 feet long, 10 x 14 inches in size. For the length we will take 12 and 13 feet. For the width we will take 10 inches, and multiply the result by 14. By the rule a board 12 feet long and 10 inches wide contains 10 feet, and one 13 feet long and 10 inches wide, ID feet 10 inches. Therefore, a board 25 feet long and 10 inches wide must contain 20 feet and 10 inches. In H THE STEEL SQUARE the timber above described, however, we have what is equivalent to 14 such boards, and therefore we multiply this result by 14, which gives 291 feet and 8 inches, the board measure. Fig. 3. The " board measure," as shown on the portion of the square. Fig. 3, gives the feet contained in each board ac- cording to its length and width. This style of figuring squares, for board measure, is going out of date, as it gives the answer only in feet. Brace Rule. — The " brace rule " is always placed on the tongue of the square, as shown in the central space at x. Fig. I. This rule is easily understood ; the figures on the left of the line represent the " run " or the length of two sides of a right angle, while the figures on the right represent the exact length of the third side of a right-angled triangle, in inches, tenths, and hundredths. Or, to explain it in another way, the equal numbers placed one above the other may be considered as representing the sides of a square, and the third number to the right the length of the diagonal of that square. Thus the exact length of a brace from point to point having a run of 33 inches on a post, and a run of the same on a girt, is 46-67 inches. The brace rule varies AND ITS USES. 15 somewhat in the matter of the runs expressed in different squares. Some squares give a few brace lengths of which the runs upon the post and beam are unequal. Octagonal Scale. — The " octagonal scale," as shown on the central division of the upper portion of blade, is on the opposite side of the square to the " brace rule, " and runs along the centre of the tongue as at s.s. Its use is as iol-; ;v lows : Suppose a stick of timber ten inches square. : Make a centre hne, which will be five inches from eac|i edge ; set a pair of compasses, putting one leg on any of, the^ main divisions shown on the square in this scale, and the ,.,othe]^-. ^ leg on the tenth subdivision. This division, pricked off" from the centre line on the timber on each side, will give the points for the gauge-lines. Gauge from the corners both ways, and the lines for making the timber octagonal in its section are obtained. Always take the same number of spaces on your compasses as the timber is inches square from the centre line. Thus, if a stick is twelve inches square, take twelve spaces on the compasses ; if only six inches square, take six spaces on the compasses, etc., etc. The rule always to be observed is as follows : Set off from each side of the centre line upon each face as many spaces by the octagon scale as the timber is inches square. For timbers larger in size than the number of divisions in the scale, the measurements by it may be doubled or trebled, as the case may be. I have now fully described all the lines, figures, and scales that are usually found on the better class of squares now in use ; but, I may as well here remark that there are squares in use of an inferior grade, that are somewhat dif- i6 THE STEEL SQUARE ferently figured. These tools, however, are such as can not be recommended for the purposes of the scientific carpenter or joiner. Fence. — A necessary appendage to the steel square in solving mechanical problems, is, what I call, for the want of a better name, an adjustable fence. This is made out of a piece of black walnut or cherry 2 inches wide, and 2 feet 10 inches long (being cut so that it will pack in a tool chest), and inches thick; run a gauge line down the centre of both edges; this done, run a saw kerf cutting down these gauge lines at least one foot from each end, leaving about ten inches of solid wood in the centre of fence. We now take our square and insert the blade in the saw kerf at one end of the fence, and the tongue in the kerf, at the other, the fence forming the third side of a right-angle triangle, the blade and the tongue of the square forming the other two sides. A fence may be made to do r ' i Fig. 4. pretty fair service, if the saw kerf is all cut from one end as shown at Fig. 4. The one first described, however, will be found the most serviceable. The next step will be to make some provision for holding the fence tight on the square ; this is best done by putting a No. 10 inch screw in each end of the fence, close up to the blade and tongue ; having done this, we are ready to proceed to busi- ness. AND ITS USES. 17 Application. — The fence being made as desired, in either of the methods mentioned, and adjusted to the square, work can be commenced forthwith. The first attempt will be to make a pattern for a brace, for a four-foot " run." Take a piece of stuff already pre- pared, six feet long, four inches wide and half-inch thick, gauge it three-eighths from jointed edge. Take the square as arranged at Fig. 5, and place it on the prepared stuff as shown at Fig. 6. Adjust the square so that the twelve-inch lines coincide exactly with the gauge line o, o, o, o. Hold the square firmly in the posi- tion now obtained, and slide the fence up the tongue and blade until it fits snugly against the jointed edge of the prepared stuff, screw the fence tight on the square, and be sure that the 12 inch marks on both the blade and the tongue are in exact position over the gauge-line. I repeat this caution ^ because the successful completion of the work depends on exactness at this stage. We are now ready to lay out the pattern. Slide the square to the extreme left, as shown on the dotted lines at x, mark with a knife on the outside edges of the square, cutting the gauge-line. Slide the square to the right until the 1 2 inch mark on the tongue stands over the knife mark on the gauge-line ; mark the right-hand side of the square cutting the gauge-line as before, repeat the process four times, marking the extreme ends to cut off, and we have the length of the brace and the bevels. Square over, with a try square, at each end from the gauge-line, and we have the toe of the brace. The lines, s, s, shown at the ends of the pattern, represent the tenons that are to be left on the braces. This pattern is now com- AND ITS USES. 19 plete ; to make it handy for use, however, nail a strip 2 inches wide on its edge, to answer for a fence as shown at k, and the pattern can then be used either side up. The cut at Fig. 7, shows the brace in position, on a re- duced scale. The principle on which the square works in the formation of a brace can easily be understood from this cut, as the dotted lines show the position the square was in when the pattern was laid out. It may be necessary to state that the " square," as now arranged, will lay out a brace pattern for any length, if thfe angle is right, and the rim equal. Should the brace be of great length, however, additional care must be taken in the adjustment of the square, for should there be any departure from truth, that departure will be repeated every time the square is moved, and where it would not affect a short run, it might seriously affect a long one. To lay out a pattern for a brace where the run on the beam is three feet, and the run down the post four, proceed as follows : Prepare a piece of stuff, same as the one operated on for four feet run; joint and gauge it. Lay the square on the left-hand side, keep the 12 inch mark on the tongue, over the gauge-line, place the 9 inch mark on the blade, on the gauge -line, so that the gauge-line forms the third side of a right-angle triangle, the other sides of which are nine and twelve inches respectively. Now proceed as on the former occasion, and as shown at Fig. 8, taking care to mark the bevels at the extreme ends. The dotted lines show the positions of the square, as the pattern is being laid out. Fig. 9 shows the brace in position, the dotted lines show 20 THE STEEL SQUARE where the square was placed on the pattern. It is well to thoroughly understand the method of obtaining the lengths and bevels of irregular braces. A little study, will soon enable any person to make all kinds of braces. If we want a brace with a two feet run, and a four feet run, it must be evident that, as two is the half of four, so on the square take 1 2 inches on the tongue, and 6 inches on the blade, apply four times, and we have the length, and the bevels of a brace for this run. ^ For a three by four feet " run, take 12 inches on the ^ tongue, and 9 inches on the blade, and apply four times, because, as 3 feet is ^ of four feet, so 9 inches is ^ of 12 inches. Rafters. — Fig. 10 shows a plan of a roof, having twenty-six feet of a span. The span of a roof is the distance over the wall plates measuring from A to A, as shown in Fig. 10. It is also the extent of an arch between its abutments. There are tv/o rafters shown in position on Fig. 10, The one on the left is at an inclination of quarter pitch, and AND ITS USES. marked B, and the one on the right, marked C, has an inclination of one-third pitch. These angles, or inclinations rather, are called quarter and third pitch, respec- tively, because the height from level of wall plates to ridge of roof is one-quarter Dr one- third the width of building, as the case may be. At Fig. 1 1 , the rafter B is shown drawn to a larger scale ; you will notice that this rafter is for quarter pitch, and for convenience, it is supposed to consist of a piece of stuff 2 inches by 6 inches by 1 7 feet. That portion of the rafter that projects over the wall of the building, and forms the eve, is three or more inches in width, just as we please. The length of the projecting piece in this case is one foot — it may be more or less to suit the eve, but the Hne must continue from end to end of the rafter, as shown on the plan, and we will call this line our working line. We are now ready to lay out this rafter, and will proceed as follows : We adjust the fence on the square the same as for braces, press the fence firmly against the top edge of rafter, and place the figure 12 inches on the left-hand side, and the figure 6 in on the right-hand side, directly over the working line, as shown on the plan. Be very exact about getting the figures on the line, for the quality of the 22 THE STEEL SQUARE work depends much on this; when you are satisfied that you are right, screw your fence tight to the square. Com- mence at No. I on the left, and mark off on the working hne; then shde your square to No. 2, repeat the marking and cont'nue the process until you have measured off thirteen spaces, the same as shown by the dotted lines in the drawing. The last Hne on the right-hand side will be the plumb cut of the rafter, and the exact length required. It will be noticed that the square has been applied to the timber thirteen times. The reason for this is, that the building is twenty-six feet wide, the half of which is thirteen feet, the distance that one rafter is expected to reach, so, if the building was thirty feet wide, we should be obliged to apply the square fifteen times instead of thirteen. We may take it for granted, then, that in all cases where this method is employed to obtain the lengths and bevels, or cuts of rafters, we must apply the square half as many times as there are feet in the w_idth of the building being covered. If the roof to be covered is one-third pitch, all to be done is to take 12 inches on one side of the square and 8 inches on the other, and operate as for quarter pitch. We shall frequently meet with roofs much more acute than the ones shown, but it will be easy to see how they can be managed. For instance, where the rafters are at right-angles to each other, apply the square the same as for braces of equal run, that is to say, keep the 1 2 mark on the blade, and the 12 mark on the tongue, on the working line. When a roof is more acute, or " steeper " than a right-angle, take a greater figure than twelve on one side of the square, and twelve on the other. AND ITS USES. 23 Whenever a drawing of a roof is to be foHowed, we can soon find out how to employ the square, by laying it on the drawing, as shown in Fig. 12. Of course, something depends on the scale to which the drawing is made. If any of the ordinary fractions of an inch are used, the intelli- gent worlynan will have no difficulty in discovering what figures to make use of to get the "cuts" and length de- sired. Sometimes there may be a fraction of a foot in this divis- ion; when such is the case, it can be dealt with as follows : S suppose there is a fraction of « a foot, say 8 inches, the half of which would be 4 inches, or yi of a foot; then, if the roof is quarter pitch, all to be done is to place the square, with the 4 inch mark on the blade, and the 2 inch mark on the tongue, on the centre line of the rafter, and the distance between these points is the extra length required, and the line down the tongue is the bevel at the point of the rafter. On Fig. 13, is shown an application of this method. All other pitches and frac- tions can be treated in this manner without overtaxing the ingenuity of the workman. Fig. 14. AND ITS USES. 25 Sufficient has been shown to enable the student, if he has mastered it, to find the lengths and bevels of any com- mon rafter ; therefore, for the present, we will leave saddle roofs, and try what can be done with the square in de- termining the lengths and bevels of " hips," valleys,«!and cripples. Fig. 15. Fig. 14 shows how to get bevels on the top end of vertical boarding, at the gable ends, suitable for the quarter pitch at Fig. 10. At Fig. 15, is shown a method for finding the bevel for horizontal boarding, collar ties, etc. Hip Rafters. — Fig. 16, is supposed to be the pitch of a roof furnished by an architect, with the square applied to the pitch. The end of the long blade must only just enter 26 THE STEEL SQUARE AND ITS USES. 27 the fence, as shown in the drawing, and the short end must be adjusted to the pitch of the roof, whatever it may be. Fig. 1 7 shows the square set to the pitch of the hip rafter. The squares as set give the plumb and level cuts. Fig. 18 is the rafter plan of a house 18 by 24 feet; the rafters are laid off on the level, and measure nine feet from centre of ridge to outside of wall; there* should be a rafter pattern with a plumb cut at one end, and the foot cut at the other, got out as previously shown. (Figs. 16, 17, 18, P.) When the rafter foot is marked, place the end of the long blade of the square to the wall line, as in drawing, and mark across the rafter at the outside of the short blade, and these marks on the rafter pitch will correspond with two feet on the level plan ; slide the square up the rafter and place the end of the long blade to the mark last made, and mark outside the short blade as before, repeat the application until nine feet are measured off, and then the length of the rafter is correct ; remember to mark off one-half the thickness of ridge-piece. The rafters are laid off on part of plan to show the appearance of the rafters in a roof of this kind, but for working purposes the rafters i, 2, 3, 4, 5, and 6, with one hip rafter, is all that is required. Hip-roof Framing.— We first lay off common rafter, which has been previously explained ; but deeming it ne- cessary to give a formula in figures to avoid making a plan, we take pitch. This pitch is }i the width of building, to point of rafter from wall plate or base. For an example, always use 8, which is ^ of 24, on tongues for altitude; 12, the width of 24, on blade for base. This cuts common rafter. Next is the hip-rafter. It must be understood that 28 THE STEEL SQUARE the diagonal of 12 and 12 is 17 in framing, and the hip is the diagonal of a square added to the rise of roof ; there- fore we take 8 on tongue and 17 on blade; run the same number of times as common rafter (rule to find distance of hip diagonal a^ + a^ + b^ = y^). To cut jack rafters, divide the numbers of openings for common rafter. Suppose we have 5 jacks, with six openings, our common rafter 12 feet long, each jack would be 2 feet shorter. First 10 feet, second 8 feet, third 6 feet, and so on. The top down cut the same as cut of common rafter ; foot also the same. To cut mitre to fit hip. Take half the width of building on tongue and length of common rafter on blade ; blade gives cut. Now find the diagonal of 8 and 12, which is 14*", call it 14 7-16, take 12 on tongue, 14 7-16 on blade ; blade gives cut. The hip-rafter must be beveled to suit jacks ; height of hip on tongue, length of hip on blade ; tongue gives bevel. Then we take 8 on tongue 18^ on blade ; tongue gives the bevel. Those figures will span all cuts in putting on cornice and sheathing. To cut bed moulds for gable to fit under cornice, take half width of building on tongue length of common rafter on blade; blade gives cut ; machine mouldings will not member, but this gives a sohd joint ; and to member properly it is neces- sary to make moulding by hand, the diagonal plumb cut differences. I find a great many mechanics puzzled to makes the cuts for a valley. To cut planceer, to run up valley, take heighth of rafter on tongue, length of rafter on blade ; tongue gives cut. The plumb cut takes the height of hip-rafter on tongue, length of hip-rafter on blade; tongue gives cut. These figures give the cuts for yi pitch only, regardless of width of building. AND ITS USES. 29 For a hopper the mitre is cut on the same principle. To make a butt joint, take the width of side on blade, and half the flare on tongue ; the latter gives the cut. You will observe that a hip-roof is the same as a hopper in- verted. The cuts for the edges of the pieces of a hexagonal hopper are found this way : Subtract the width of one piece at the bottom from the width of same at top, take remainder on tongue, depth of side on blade ; tongue gives the cut. The cut on the face of sides : Take 7-12 of the rise on tongues and the depth of side on blade ; tongue gives cut. The bevel of top and bottom : Take rise on blade, run on tongue ; tongue gives cut. Fig. 19 exhibits two methods of finding the "backing " of the angle on hip-rafter. The methods are as simple as any known. Take the length of the rafter on the blade, and the rise on the short blade or tongue, place the square on the line D E, .the plan of the hip, the angle is given to bevel the hip-Tafter, as shown at F. This method gives the angle, only for a right-angled plan, where the pitches are the same, and no other. The other method appHes to right, obtuse, and acute angles, where the pitches are the same. At the angle D will be seen the line from the points k l, at the intersec- tion of the sides of the angle rafter with the sides of the plan. With one point of the compass at D, describe the curve from the line as shown. Tangential to the curve draw the dotted line, cutting a, then draw a line parallel to A b, the pitch of the hip. The pitch or bevel, will be found at G, which is a section of the hip-rafter. This problem is taken from " Gould's Carpenters' AND ITS USES. 3 1 Guide," but has been in practice among workmen foF many years. Fig 20. Fig. 20 exhibits a method of finding the cuts in a mitre box, by placing the square on the line a b at equal dis- tances fironi the heel of the square, say ten inches. The bevel is shown to prove the truth of the lines by applying it to opposite sides of the square. Stairs. — In laying out stairs with the square, it is neces- sary to first determine the height from the top of the floor on which the stairs start from, to the floor on which they are to land ; also the " run " or the distance of their hori- zontal stretch. These lengths being obtained, the rest is easy. Fig. 2 1 shows a part of a stair string, with the " square " laid on, showing its application in cutting out a pitch-board. As the square is placed it shows lo inches for the tread and 7 inches for the rise. To cut a pitch-board, after the tread and rise have been 32 THE STEEL SQUAKE determined, proceed as follows : Take a piece of thin, clear stuff, and lay the square on the face edge, as shown in the figure, and mark out the pitch-board with a sharp knife ; Fig. 21. then cut out with a fine saw and dress to knife marks, nail a piece on the longest edge of the pitch-board for a fence, and it is ready for use. Fig. 2 2 is a rod, with the number and heighth of steps for a rough flight of stairs to lead down into a cellar or elsewhere. Fig. 23 is a step-ladder, sufficiently inclined to permit a person to pass up and down on it with convenience. To lay off the treads, level across the pitch of the ladder, set the short side of the square on the floor, at the foot of the string, after the string is cut, to fit the floor and trimmer joists. Fasten the fence on the square, as shown at Fig. 5. The height of the steps in this case is nine inches, so it will be seen that it is an easy matter to lay off the string, as the 34 THE STEEL SQUARE long side of the square hangs plumb, and nine inches up its length will be the distance from one step to the next one. Fig. 24 shows the square and fence in position on the string. The opening in the floor at the top of the string shows the ends of trimming joists, five feet apart. Fig. 25 shows how to divide a board into an even number of parts, each part being equal, when the same is an un- even number of inches, or parts of an inch in width. Lay the square as shown, with the ends of the square on the edges of the board, then the points of division will be found at 6, 12, and 18, for dividing the board in four equal parts; or at 4, 8, 12, 16, and 20, if it is desired to divide the board into six equal parts. Of course, the common two-foot rule will answer this purpose as well as the square, but it is not always convenient. Fig. 26 shows how a circle can be described by means of a " steel square " without having recourse to its centre. At the extremities of the diameter, a, o, fix two pins, as shown ; then by sliding the sides of the square in contact with the pins, and holding a pencil at the point x, a semi- circle will be struck. Reverse the square, repeat the pro- cess, and the circle is complete. Miscellaneous Rules. — ^The following rules have been tested over and over again by the writer, and found reliable in every instance. They have been known to advanced workmen for many years, but were never pubhshed, so far as the writer knows, until they appeared in the American Builder, some years ago : AND ITS USES. 35 Measurement. — Let us suppose that we have a pile of lumber to measure, the boards being of different widths, and say 1 6 feet long. We take our square and a bevel with a long blade and proceed as follows : First we set the bevel at 1 2 inches on the tongue of the square, because we want to find the contents of the board in feet, 12 inches being one foot ; now we set the other end of the bevel blade on the 16 inch mark on the blade of the square, because the boards are 16 feet long. Now, it must be quite evident to any one who would think for a moment, that a board 1 2 inches, or one foot wide, and 16 feet long, must contain 16 feet of lumber. Very well, then we have 16, the length, on the blade. Now, we have a board 1 1 inches wide, we just move our bevel from the 12 inch mark to the 11 inch mark, and look on the blade of the square for the true answer ; and so on with any width, so long as the stuff is 16 feet long. If the stuff is 2 inches thick, double the answer, if 3 inches thick, treble the answer, etc. Now, if we have stuff 14 feet long, we simply change the bevel blade from 16 inches on the square blade, to 14 inches, keeping the other end of the bevel on the 12 inch mark, 1 2 inches being the constant figure on that side of the square, and it will easily be seen that any length of stuff within the range of the square can be measured ac- curately by this method. If we want to find out how many yards of plastering or painting there are in a wall, it can be done by this method quite easily. Let us suppose a wall to be 12 feet high and 18 feet long, and we want to find out how many yards of plastering or painting there are in it, we set the bevel on the 9 inch mark on the tongue (we take 9 inches because 9 36 THE STEEL SQUARE square feet make one square yard,) we take i8 inches, one of the dimensions of the wall, on the blade of the square; then after screwing the bevel tight, we slide it from 9 inches to 12 inches, the latter number being the other dimension, and the answer will be found on the blade of the square. It must be understood that 9 inches must be a constant figure when you want the answer to be in yards, and in measuring for plastering it is as well to set the other end of the bevel on the figure that corresponds with the height of the ceiling, and then there will require no movement of the bevel further than to place it on the third dimension. This last rule is a very simple and very useful one ; of course " openings " will have to be allowed for, as this rule gives the whole measurement. If the diagonal of any parallelogram within the range of the square is required, it can be obtained as follows : Set the blade of the bevel on 8^ in. on the tongue of the square, and at 1 2^ in. on the blade ; securely fasten the bevel at this angle. Now, suppose the parallelogram or square to be 11 inches on the side, then move the bevel to the 1 1 inch mark on the tongue of the square, and the answer, 159-16, will be found on the blade. All problems of this nature can be solved with the square and bevel as the latter is now set. There is no particular reason for using 8^ and 12^, only that they are in exact proportion to 70 and 99. 4^ and 6 3-16 would do just as well, but would not admit as ready an adjustment of the bevel. To find the circumference of a circle with the square and bevel proceed as follows : Set the bevel to 7 on the tongue and 22 on the blade ; move the bevel to the given diameter on the tongue of the square, and the approximate answer AND ITS USES. 37 will be found on the blade. When the circumference is wanted the operation is simply reversed, that is, we put the bevel on the blade and look on the tongue of the square for the answer. If we want to find the side of the greatest square that can be inscribed in a given circle, when the diameter is given, we set the bevel to 8^ on the tongue and 12 on the blade. Then set the bevel of the diameter, on the blade, and the answer will be found on the tongue. The circumference of an ellipse or oval is found by set- ting 5^ inches on the tongue and 8^ inches on the blade ; then set the bevel to the sum of the longest and shortest diameters on the tongue, and the blade gives tlje answer. To find a square of equal area to a given circle, we set the bevel to 9 ^ inches 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 answer will be found on the tongue./ If the circumference of the circle is given, and we want to find a.^*^' square containing the same area, we set the bevel to 5^ inches on the tongue and 19^ inches on the blade. On Fig. 27 is shown a method to determine the pro- portions of any circular presses or other cylinderical bodies, by the use of the square. Suppose the small circle, n, to be five inches in diameter and the circle r is ten inches in diameter, and it is required to make another circle, z, to contain the same area as the two circles n and r. Meas- ure line a, on the square d, from five on the tongue to 10 on the blade, and the length of this line a from the two points named will be the diameter of the larger circle z. And again, if you want to run these circles into a fourth one, set the diameter of the third on the tongue of the square, 38 THE STEEL SQUARE and the diameter of z on the blade, and the diagonal will give the diameter of the fourth or largest circle, and the same rule may be carried out to infinite extent. The rule is reversed by taking the diameter of the greater circle and laying diagonally on the square, and letting the ends touch whatever points on the outside edge of the square. These points will give the diameter of two circles, which com- bined, will contain the same area as the larger circle. The same rule can also be applied to squares, cubes, triangles, rectangles, and all other regular figures, by taking* similar dimensions only; that is, if the largest side of one triangle is taken, the largest side of the other must also be taken, and the result will be the largest side of the required tri- angle, and so with the shortest side. In Fig. 28 we show how the centre of a circle may be de- termined without the use of compasses ; this is based on the principle that a circle can be drawn through any three points that are not actually in a straight line. Suppose we take A B c D for four given points, then draw a line from a T Fig. 27. AND ITS USES. 39 to D, and from b to c ; get the centre of these Hnes, and square from these centres as shown, and when the square crosses, the line, or where the lines intersect, as at x, there will be the centre of the circle. This is a very useful rule, and Fig. 28. by keeping it in mind the mechanic may very frequently save himself much trouble, as it often happens that it is ne- cessary to find the centre of the circle, when the compasses are not at hand. In Fig. 29 we show how the square can be used, in lieu of the trammel, for the production of ellipses. Here the square, e d f, is used to form the elliptical quadrant, A B, instead of the cross of the trammel \ h I k may be simply pins, which can be pressed against the sides of the square while the tracer is moved. In this case the adjust- ment is obtained by making the distance, h /, equal to the semi-axis minor, and the distance / k, equal to the semi-axis major. 40 THE STEEL SQUARE Fig. 29. Fig, 30 shows a method of describing a parabola by- means of a straight rule and a square, its double ordinate and abscissa being given. Let a c be the double ordinate, andD B the abscissa. Bisect d c in f; join b f, and draw F E perpendicular to b f, cutting the axis b d produced in E. From B set off b g equal to d e, and g will be the focus of the parabola. Make b l equal to b g, and lay the rule on straight-edge h k on l, and parallel to a c. Take a string, M F G, equal in length to l e ; attach one of its ends to a pin, or other fastening, at g, and its other end to the • end M, of the square m n o. If now the square be slid along the straight-edge, and the string be pressed against The two arms of a horizontal lever are respectively 9 inches and 13 inches in length from the suspending point; a weight of 10 lbs. is suspended from the shorter arm, and it is required to know what weight wUl be re- quired to suspend on the long arm to make it balance. Set a bevel on the blade of square at 13 inches and the other end of the bevel on the 9 inch mark on tongue of square, then sHde the bevel from 13 inches to 10 on the blade of square, and the answer will be found on the tongue of the square. It is easy to see how this rule can be reversed so that a weight required for the shorter arm can be found. Fig. 3 1 shows how to get the flare for a hopper 4 feet across the top and 1 6 inches perpendicular depth. Add to the depth one-third of the required size of the discharge 42 THE STEEL SQUARE HALFOFDISCmce^ 3"j HALF W/IDTH OF HOPPER TOP Fig. 31. hole (the draft represents a 6-mch hole), which makes 18 inches, which is represented on the tongue of the square. (The figures on the draft are 9 and 12, which produce the same bevel.) Then take one-half, 24 inches of the width across the top of the hopper, which is represented on the blade of the square. Than scribe along the blade as rep- resented by the dotted lines, which gives the required flare. (The one-third added to the depth is near enough for all practical purpose for the discharge.) Fig. 32. AND ITS USES. 43 Fig. 32 shows how to apply the square to the edge of a board in order to obtain the bevel to form the joint. Using the same figures as in Fig. 31, scribe across the edge of the board by the side of the tongue, as shown by dotted lines. The long point being the outside. 14- ^ / //X N Fig. 33. On Fig. 33 we show a quick method of finding the centre of a circle: Let n n, the corner of the square, touch the circumference,. and where the blade and tongue cross it will be divided equally ; then move the square to any other place and mark in the same way and straight edge across, and where the line crosses a, b, as at o, there will be the centre of the circle. 44 THE STEEL SQUARE I and 2, Fig. 34, are taken from Gould's Wood- Work- ing Guide. The portion marked a, exhibits a method of finding the lines for eight-squaring a piece of timber with the square, by placing the block on the piece, and making the points seven inches from the ends of the square, from which to draw the lines for the sides of the octagonal piece re- quired. At the heel of the square is shown a method of cut- ting a board to fit any angle ^ with the square and compass, j by placing the square in the " angle, and taking the distance from the heel of the square to the angle a, in the com- pass ; then lay the square on the piece to be fitted, with the distance taken, and from the point a, draw the line a B, which will give the angle to cut the piece required. At 2 is shown a method of constructing a polygonal figure of eight sides ; by placing the square on the line a b, with equal distances on the blade and tongue, as shown ; the curve lines show the method of transferring the dis- tances; the diagonal gives the intersection at the angles. AND ITS USES. 45 The following is a good method for obtaining the cuts for a horizontal and raking cornice ; it is correct and simple ; the gutter to be always cut a square mitre. The seat or run of the rafter on the blade r c, Fig. 35, the rise of the roof on the tongue a, c, mark against the tongue, gives the cut for the side of the box a, c. The diagonal Fig. 35. A, R, which is the length of the rafter on th^ bl^de a, d, the seat of the rafter on the tongue d s, mark kgaipst the blade gives the cut across the box a, d. d, a, c is the mitre cut to fit the gutter ; then if we square across the box from A, it gives f, a, c the cut for the gable peak. 4^ THE STEEL SQUARE PART II. The following useful applications of the square were kindly furnished for this work, by Mr. Croker; several of them are new and original : Consider the blade of the square as representing the spaji of a building, but without any reference to actual or scale measurement. Next, some particular portion of the blade is to be taken as the rise of the supposed if a third, fourth, or half pitch is required, then a third, fourth, or a half of the blade is conceived as the rise which with half the blade solves the pitch. From this it will be seen that half the blade is always taken as the base of the theoretical co?nmon rafter. Where we have to deal with irregular pitches— by which is meant those pitches which are not a quarter, sixth, third, half, etc., of the building- then the square is to be applied to the irregular pitch with the blade lying in the direction of the pitch and the centre of the blade at the intersection of pitch and base line of the common rafter, and the resulting distance on the tongue, where it intersects the base line, is the distance to be taken as the rise of the theoretical rafter. Let us now take a hip-roof over a square plan (for all the rules apply only to square planned building), and the prac- tical problems supposed to need solution are : Length of common rafters, the plumb and level cuts ; length of hip- rafter, its plumb and level cuts ; bevel of jacks and sheet- AND ITS USES. 47 ing boards against the hips ; " backing " of the hip-rafter, top and down bevel of a purlin mitering against the hip with its surface in line with the plane of the roof. If the student can readily and inteUigently solve these problems, he will be in a position to make extensions in the principles involved. Let the width of building under consideration be 24 feet wide, and of one third pitch. Fig. 36. Let I, 12, Fig. 36, be the base of the theoretical common rafter, eight inches rise, equal to one third of the blade, be- cause it is a third pitch; mark along the blade and extend the heel, making it and 12 equal to half the width of the actual building to a scale of inch to a foot; this is a much better scale to work by than an inch one, being larger and more legible, eighths being inches, sixteenths, ^ inches, etc., thus enabling very accurate measurements to be taken. By the way, it is a good plan to have the square stamped off" on the eighths side at every i y^, inches for feet, for more readily counting the scale ; then mark along the tongue at B, which gives b 1 2 the length of common rafter ; level cut on blade and plumb cut on tongue. Next take the rise of the theoretical common rafter on the tongue, and 1 7 inches 48 THE STEEL SQUARE on the blade, as the theoretical base of the hip-rafter; place the square as shown at Fig. 37 ; then multiply the ^^i? ^ ^ I ^n^- r- Fig. 37. actual base of common rafter 12, (Fig. 36.) by 1*414 = i6'968 feet, or 17 feet, practically, which set off on blade at a 1 7 ; mark on tongue at b, then B 1 7 is the length of hip- rafter. For the bevels of jacks and sheeting-boards against hips take the diagonal b 12 — theoretical rafter — Fig. 36, on the blade with half the blade — the theoretical base — and place the square as shown at Fig. 38, then mark along the blade for bevel of jacks, and along tongue for bevel of sheeting-boards. Fig. 38. AND ITS USES. 49 For the " backing " of hip, take the diagonal of the theoretical hip-rafter, 8, 17 (Fig 37), on the blade, and its rise — 8 inches — on the tongue, and place square as shown at Fig. 39 ; mark by the tongue which gives the bevel re- FiG. 39. quired. To get the upper bevel of a purlin lying in the plane of the roof, take the bevei at tongue (Fig. 38), for the down bevel take the blade distance 147 — 16 (Fig. 38) on the blade with the theoretical rise — 8 ; place the square as shown at Fig. 40 ; mark by the tongue which gives the bevel required. Fig. 40. Fig. 41 shows how any length or breadth within the extent of the blade of the square can be instantly divided into any equal parts. Let a and b represent the edges of a board, say 8^ inches, wide, to be divided into 5 equal parts; take any 5° THE STEEL SQUARE convenient 5 parts, say 1 5 inches, because 5X3 = 15, placing heel of square fair to edge b, and 1 5 to edge a ; mark off" at every 3 inches on blade, as shown, and draw lines through these points, which will divide the board as required. We will here show how the square can be used to solve problems in proportion; for instance, if 1500 feet of boards cost $10.75, what will 600 feet cost ? Take 15 on the blade B Fig. 41. and 1075 on the tongue, and place the square as shown at Fig. 41, then count from 15 towards b, and from this point draw parallel to tongue ; 6 a, this is the answer re- quired. Fig. 42. Fig. 43. AND ITS USES. 51 Figs. 42 and 43 show quite a novel and useful way of bisecting any angle. Let a i 2, a b be the given sides of an acute angle to be bisected. At any convenient point as c square c D from c 12. Now take c d on the tongue, and the sum of a d and a c on the blade of the square, place as shown in the Figure, then mark by the blade, which is the bisection required. If the angle is obtuse, as a b, a f, (Fig. 42), produce a convenient distance, as a c, square over c D, take c d on the tongue, and the sum of a d, a c, on the blade, place square as shown, and mark by the tongue for the required bisection. Fig. 44. Fig. 44 shows a handy way of finding the bevel of rails to diminish door stiles. Place the square fair with the known joint a b, mark by the tongue, then the resulting bevel at a is the same as that at b. 52 THE STEEL SQUARE PART III. The following rules have been gathered from various sources, chiefly, however, from jDapers recently published in the Scientific American Supplement, by John O. Connell, of St. Louis, and from papers contributed to the Builder a?id Wood- Worker, by Wm. E. Hill, of Terre Haute, Ind. Fig. 45. * Fig. 45 shows how an octagon can be produced by the aid of a steel square. Prick off the distance a o equal to half the distance of the square ; mark this distance on the blade of the square from b to o, place the square on the *Win. E. HUL AND ITS USES. 53 diagonal, as shown, and square over each way. Do the same at every angle, and the octagon is com- plete. To obtain the same figure with the compasses, pro- ceed as follows : Take half the diagonal on the com- passes, make a little over a quarter sweep from c, and at the insersection at D and c, then D and c form one side of an octagonal figure. Again: take a piece of timber twelve inches square, as at Fig. 46 ; take twelve inches on the blade and tongue from a to b, and a to c, mark at the point A, operate similarly on the op- posite edge, and the marked points will be guides for guage-lines for the angles forming an octagon. The remaining three sides of the timber can be treated in the same manner. These points can be found with a carpenter's rule as ioUows: Lay the 4 54 THE STEEL SQUARE rule on the timber, partly opened, as shown, in the cut, "prick off" at the figures 7 and 17 as at a and b, and these points will be the guides for the gauge-lines. I'he same points can be found by laying the square diagonally across the timber and " pricking " off 7 and 17. To make a moulder's flask octagonal proceed as follows : The flask to be four feet across. Multiply 4 X 5 (as an octagon is always as 5 to 12 nearly), which gives 20; di- vide by 12, which gives feet, cut mitre to suit this measurement, nail into corners of square box, and you have an octagon flask at once. Another method of constructing an octagon is shown at Fig. 47. Take the side 2& a b for a radius, describe an arc cutting the diagonal at d\ square over from d to e, and the point e will then be the gauge-guide for all the sides. AND ITS USES. 55 Another method (Fig. 48) is to draw a straight hne, c b, any length; then let a b and a c corresponding figures on the blade and tongue of the square, mark along either and measure the distance of required octagon ; move Fig. 48. the square and mark also. Now use the square the same as before, and the marks c b and b d are the points required. Fig. 49 shows the application of a long bevel to a square, by which some calculations can be made with greater ease and quickness than by the usual arithmetical process. The largest size of carpenter's bevel placed under the framing square will answer in nearly every case. One edge of each blade should be perfectly straight and the edge of L should be cut out in several places to see the blade e, when placed under the square. The two blades should be fastened together by a thumb-screw. There 56 THE STEEL SQUARE should be three holes in l, one near each end and one in the middle, and a notch filed by each hole, so that the blade e, may be shifted when necessary. Fig. 49 *To Find the Diagonal of a Square by this instrument, set the blade e 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 1 1 inches, move blade E to the II inch mark on the tongue, keeping blade l against the square, when blade e will touch 15 9-16 inches on the blade, which is the required diagonal. There is no special reason for using 8^ and i2}i; other numbers may be employed provided the proportion of 70 to 99 exists between them. In the problem just solved as in all that follow, the bevel being once set to solve a particular ques- *J. O. Connell. AND ITS USES. 57 tion will solve all the others of the same kind, till the bevel is altered. Polygons Inscribed in Circles.— In the following table, set the bevel to the pair of numbers under the polygon to be inscribed. No. of sides. 3 4 5 6 7 8 9 lo ii 12 Radius 56 70 74 Side 60 98 22 89 80 85 Side 97 99 87 equal to 52 75 15 55 45 44 radius. If we require the radiu§ of a circle which will circum- scribe an octagon 8 inches on a side, we refer to column 8, take 98 parts on the blade and 76 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 blade which gives the lesser number, in this case, on the 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. We will now explain the figures used in stepping round a circle forming inscribed polygons from three to twelve sides : Set bevel or fence to 1 2 on blade, and the number opposite each polygon on tongue ; move to diameter of circle ; answer of the side of polygon on tongue. Names. No. of Sides. Gauge Points. Triangle 3 lo'^o Square 4 8 '49 Pentagon 5 7'°S Hexagon 6 6"oo Heptagon 7 5'2i Octagon 8 4 '60 i^onagon 9 4" Decagon 10 371 Un decagon n 3 '39 Dodecagon 12 3'ii 58 THE STEEL SQUARE To divide a circle into a given number of parts, multiply the corresponding number in column one and the product is the chord to lay off on circumference. The side of a polygon is known, to find the radius of a circle that will circumscribe : Multiply the given side by the correspond- ing number opposite of polygon in column two. No. o_f ^ 7igle of Sides. Name of Polygon. Angle. Polygon. Column 1. Columns 3 Triangle 120 60 1-732 -5,7, 4 Square 90 90 r-^t^ I Pentagon 72 108 1-77^ ^ Hexagon 60 120 Radius. Side. 7 Heptagon 513.7 ,384-7 8677 i-.S2 I O'^t^g"" 45 135 7653 1-3071 9 Nonagon 40 ,40 440 1-4863 o Decagon 36 144 .e/go i-6i8i II Undecagon 328-11 1473-" -5634 1-77,4 Dodecagon 30 15^' .^5,^6 ,.775+ The side of a polygon is known, to find the length of perpendicular : Set bevel or fence to the tabulated numbers below. Example: The side of an octagon is 12, set bevel to 23 on tongue, 27 11-16 on blade. Blade gives the answer. No. of Sides. 34567 8 9 10 II 12 ^EZ%^f^'-- 9 '3° 13 273-4 277.10 503-4 281-2 313-4 26 Side of Polygon 31 i-s 2 351.4 15 26 23 37^* iS 1-2 301-2 14 To Inscribe three Equal Circles in a circle of given diame- ter. Set to 61^ on tongue and 14 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. The following also, is another use for the square and bevel combined. If a person is drawing a machine on a scale of 1 inch to the foot, he may simply lay a common rule under the AND ITS USES. 59 square, touching the 1 2 inch mark on the blade, and the ii^ inch mark on the tongue; he then possesses a con- trivance by which he may easily reduce from one scale to the other. For instance, if a piece of stick 2 3^ inches square is to go into the construction, the draughtsman finds the g}( inch mark on the blade, that is 2^ inches back from the 12 inch mark, and measures square out to the rule. This distance is the reduced section of the stick. A straight mark, drawn on a table or a drawing board, serves as well as a rule. Conveyors' shaft 5 inches in diameter, 12 feet long, pitch of flights 9 inches ; make a posteboard template ; multiply- ing the diameter by 3-1416 gives the base, and the 9 is the altitude. The paper would be 9 inches altitude, 15 71-100 base ; draw a line along shaft, place altitude or 9 inches along this line, scribe along the hypothenuse ; this gives the spiral course of flight. This principle also teaches how to cut round sticks of straight timber by marking along base of template, take square on each end the same as taking a stick out of wind, before striking lines. The cuts for the edges of the pieces of a hexagonal hop- per are found by subtracting the width of one piece at the bottom, viz.,. the width of same at top, and taking the re- mainder on the tongue, and depth of side on blade. The tongue gives the cut. For the cut on the face of the sides, take 7-12 of the rise on the tongue, and the depth of side on the blade. The tongue gives the cut. The bevel for the top and bottom edges is found by taking the rise on the blade, and the run on the tongue ; the latter gives the ;'*Jc'"fii\(i'l1iexu': :6'* a^ octagon;.! licpiic-- for: the face of 6o THE STEEL SQUARE the- board and also the edge, substract the rise from the width of side ; take the remainder on the tongue and width of side on blade ; the tongue gives the cut. The edge of the stuff is to be square when applying the bevel. The bevel for the top and bottom edges of the sides is found by taking the rise on the blade, and run on the tongue, the latter gives the cut. This makes the edges horizonatal. The edges are not to be beveled till the four sides are cut. To lay off Angles of 60° and 30°.— Mark any number of inches, say 14, on an indefinite line. Place the blade against one extremity of this distance, and the 7 inch mark of the tongue at the other. The tongue then forms an angle of 60° with the indefinite line, and the blade an angle of 30°. To Find the Bevels and Width of Sides and Ends of a Square Hopper. — Fig. 50. The large square represents the upper edges of the hopper and the small one the lower edges, or base. The width of the sides and ends is found in this way : Take the run a d on the tongue, and the per- pendicular height a d on the blade. It is thus found in the same manner as the length of a brace. To find the cut for a butt joint, take width of side on blade and half the length of the base on tongue ; the latter gives the cut. For a mitre joint take width of side on the blade and perpen- dicular height on tongue ; the latter gives the cut. For the cut across the sides of the boards, take the run . AND ITS USES. 6i a b on the tongue, and the width of side on blade ; the tongue gives the cut. The inside corners of the sides and ends are longer than the outside, so if a hopper is to be of d Fig. so. a certain size, the lengths of ends and sides are to be meas- ured on the inside edge of each piece, and the bevels struck across the edges to these marks. This is only in case of butt joints. Of course if the hopper is to be square, the thickness of the sides must be taken from the ends. If the top and bottom edges are to be horizontal, the bevel is thus found : Take the perpendicular height of hop- per on the blade and the run on the tongue, the latter gives both cuts. A hopper can be made by the above method by getting the outside dimensions at top and bottom, and the perpendicular height. In large hoppers pieces are put down along the corners 62 THE STEEL SQUARE to Strengthen them. The length, and the bevel to fit the corner are thus found : Suppose the top of hopper is 8 feet, and the bottom i8 inches square. Find the diagonals of each, subtract the one from the other, and half the re- mainder is the run for the comer piece. From the length of this run, /, and the rise, a b, we find the length of the corner piece. To find the bevel or backing, take on the blade the length of the corner piece and on the tongue the rise ; the latter gives the bevel. Another method is to draw the line, /, to represent the seat of the corner piece, set off square with this the line m, of the same length as the run, a b. Then draw n o, which is the length of the corner piece. To find the backing, draw a line, /, anywhere across /, at right angles therewith, and at its intersection with line, /, strike a circle tangent to ;z o. From the point of intersection of the circle with /, draw lines to the extremities of The angle made by these lines is the bevel or backing. Another method generally employed for finding the bevels of hoppers is to bevel the top and bottom edges of the sides and ends to the angle they are to stand at, then to lay a bevel set to a mitre, or angle of 45°, on the beveled edge, and that will lay off a mitre joint, while a try-square will lay off a butt joint. An angle of 45° will mitre only those boxes with sides which are vertical and square with each other. When the sides and ends of a rectangular box or hopper are of the same width, that is, when sides and ends slope at equal angles, the bevels, either butt or mitre, are found as for square hoppers. When a hopper has the sides and ends of different widths, that is, when sides and ends stand at different angles. AND ITS USES. both having the same rise, find the cuts for each from its respective rise, run and width. Roofing. — Fig. 51. A hip-roof with two corners out of square is given an example, the dimensions of which are : width 15 feet, rise of roof 5 feet, length 30 feet on the Fig. 51. shorter side, 33 feet on the longer. The timbers a d, c d, E G, E G, are the hip rafters ; j j the jack rafters. The seats of each hip rafter should form a square, so that each pair of jack rafters, j j, for instance, may be cut of equal length. Lengths and Bevels of Hip-Rafters.— We will first con- sider those on the square end of the roof. In order to find their length, it is first necessary to obtain their run, which is found as follows : Take half the width of building on both blade and tongue, whence is obtained the length of seat from G to E, at the intersection of the dotted lines. By similar use of the square, this length with the rise of roof, gives the length of the hip-rafter. The lengths of all the rafters €4 THE STEEL SQUARE should be measured along the middle, as the dotted lines show. This is the full length; half the thickness of the ridge-pole is to be taken off, measured square back from the bevel. The bevel of the upper end of a hip-rafter is called the down bevel. It is always square with the lower end bevel, hence these bevels are found by the parts taken on the square to find the lengths of the hip-rafters. Another method is to take 1 7 inches on the blade and the number of inches of rise to the foot, that is, the rise in inches di- vided by half the width of roof in feet — on the tongue. , The tongue gives the down bevel, the blade the lower end bevel. The reason for the foregoing is that when the hip-rafters are square with each other, the seat of the hip is the diagonal of a square whose side is half the width of building. The diagonal of a square with a 1 2 inch side is 1 7 inches nearly. So if the rise of roof in -i foot is 6 inches, the rise of hip- rafter will be that only in 17 inches. The directions here given assume that the hip-rafter abuts the ridge-pole at right angles, but as the ground plan of the roof shows that they meet at an acute angle, another bevel must be considered, called the side bevel of the hip-rafters. Were there no slope to the roof, the bevel where they meet the ridge pole would be an angle of 45°, as the hips would be square with each other. When a pitch or slope is given, the hips depart from the right angle, and therefore the side bevels are always less than 45°. Take the length of hip on the blade, and its run on the tongue ; the blade gives the cut. Backing of the hip-rafters. The backs of the hip-rafters must be beveled to lie even with the planes of the roof. This bevel must slope from the middle toward either side. AND ITS USES, 65 It is found by taking the length of hip on blade, and the rise of the roof on tongue. The latter gives the bevel. To find the lengths of the jack-rafters : Suppose there are to be four between the corner and the first common rafter; then there are five spaces, which, by dividing 7 foot 6 inches by 5, are i foot 2 inches from centre to centre of jacks. The rise of roof, also divided by 5, gives i foot rise for the shortest rafter. The run is i foot 6 inches ; as both rise and run are given, the length down and lower bevels are found therefrom. The next jack has double the rise, run and length of the first ; the following one three times, and the fourth four times. All the measurements are to proceed on or from the middle lines of the jacks. The side bevel of all the jack-rafters is obtained by taking the length of a common rafter on the blade and its run on the tongue ; the bevel on the blade gives the result. Fig. 52. Let us now consider the end of the building out of square. Fig. 52 illustrates the method of laying down the seats of the hips. To find the lengths of these hips, the lengths of the seats must be got by taking half the width of building on blade, and the distance from the end of the dotted line crossing the roof, to the corner on the tongue. The length 66 THE STEEL SQUARE of the seat so obtained taken on the square, with the rise of the roof, gives the length of the respective hip-rafter. The down and lower end bevels are found as in the pre- vious hip-rafters. To obtain each side bevel, add the dis- tance from the dotted line to the corner and the gain of the hip-rafter; take the sum on the blade, and half the width of building on the tongue ; the latter gives the cut. The lengths, etc., of the jack-rafters on the side, are de- termined as at the square end of the roof ; the side bevel being found by taking the length of a common rafter on the blade, and the distance from the dotted line to corner on the tongue. The latter showing the bevel. The lengths of jack-rafters on the end. Assuming there are to be four jacks between the corner and the centre in- cluded, half the length of the end of the roof must be di- vided by 5. One side of the roof being 3 feet longer than the other, we place 3 feet, on tongue, and 1 5 feet, the width of building, on the blade, and thus obtain the distance from corner to corner on the end of the roof. Half this divided by 5 gives the distance of the jacks apart. The distance from where the middle Hnes of the hips meet to the middle point of the end of the roof is also to be divided by 5, the quotient giving the run of the shortest rafter. The rise is the same as for the jacks on the square end. These rules give the full length of rafter, so that when hips come against a ridge-pole or jacks against a hip, half the thickness of pole or hip, squared back from their down bevels, must be taken off. Side bevels of these jacks are obtained by adding the distance from the dotted line to the corner to the gain of a common rafter in running that distance; take this on the AND ITS USES. 67 blade, and half the width of building on the tongue. The blade gives the bevel. Trusses. — Fig. 53. a is the straining beam, b the brace, T the tie beam. Generally the brace has about one-third the length of tie beam for a run. From the rise and run find the length and lower end bevel of the brace. After marking the lower end bevel on the stick, add to it just what is cut out of the tie beam. The bevel of the upper end of the brace where it butts against the straining beam is found in the following manner. Take the length of the brace, or a proportional part, and mark it on the edge of a board ; take half the rise of the brace on the tongue, lay it to one of these marks on the board, and move the blade till it touches the other mark on board. A line drawn along the tongue gives the bevel for both brace and strain- ing beam. The angle made between brace and strain- ing beam is thus bisected. Lay off the measurements from the outside of the timbers. Put a bolt where shown, with a washer under the head to fit the angle of straining beam and brace. A Fig. 53. 68 THE STEEL SQUARE It will be seen in the foregoing work, that the " Steel Square " is capable of solving quite a number of useful prob- lems, but it must not be supposed that all its operations have been shown in this work. The skilled workman, after a careful study of the operations herein set forth, will, when occasion requires, invent applications of the square that are not. now suspected ; and it is quite possible that future editions of this work may contain many of these new dis- coveries. F. T. H. CATALOGUE OP Books and Periodicals PUBLISHED AND FOR SALE BY THE INDUSTRIAL PUBLICATION COMPANY, 14: Dey Street, New York. ^"Any of these Books may be obtained from any Bookseller or Newsdealer, or will be sent Free by mail to any part of the United States or Canada ON RECEIPT OF PRICE. ' The Amateur's Handbook of Practical Information, For the Workshop and the Laboratory. Second Edition. Greatly Enlarged. Neatly Bound - - 15 cents. This is a handy little bool?, containing just the information needed by Amateurs in the Worlishop and Laboratory. Directions for making Alloys, Fusible Metals, Cements, Glues, etc.; and for Solder* ing, Brazing, Lacquering, Bronzing, Staining and Polishing Wood, Tempering Tools, Cutting and "Working Glass, Varnishing, Silvering, Gilding, Preparing Skins, etc., etc. The New Edition contains extended directions for preparing Polish- ing Powders, Freezing Mixtures, Colored Lights for tableaux. Solu- tions for rendering ladies' dresses incombustible, etc. There has also been added a very large number of new and valuable receipts. Rhymes of Science: Wise and Otherwise. By O. W. Holmes, Bret Hart, Ingoldsby, Prof. Forbes, Prof. J. W. McQ, Eankine, Hon. E. W. Eaymond, and others. With Illustrations. Cloth, Gilt Title. - 50 cents Section Cutting. A Practical Guide to the Preparation and Mounting ol Sections for the Microscope ; Special Prominence being given to the Subject of Animal Sections.' By Sylvester Marsh. Reprinted from the London edition. With Illustrations. 12mo., Cloth, Gilt Title, • 75 cents. This is undoubtedly the most thorough treatise extant upon sectioi.) cutting in all its details. The American edition has been greatly enlarged by valuable explanatory notes, and also by extended direc- tions, illustrated with engravings, for selecting and sharpening knives and razors. A Book for Beginners with the iVIioroscope. Being an abridgment of " Practical Hints on the Selection and Use of the Microscope." ByJohnPhin. Fully illus- trated, and neatiy and strongly bound in boards. 30 cts. This book was prepared for the use of those who, having no know- ledge of the use of the microscope, or, indeed, of any scientific appar- atus, desire simple and practical instruction in the best methods of managing the instrument and preparing objects. How to Use the Microscope. A Simple and Practical Book, intended for beginners. By John Phin, editor of "The American Journal of Microscopy," Second Edition. Greatly Enlarged, with 50 illustrations in the text and 4 full-page engravings printed on heavy tint paper. 12mo., Neatly bound in Cloth, Gilt Title. .... 75 cents. The IVIicroscope. By Andrew Eoss. Fully Illustrated. 12mo., Cloth, Gilt Title. 75 cents. This is the celebrated article contributed by Andrew Eoss to the "Penny Cyclopedia," and quoted so frequently by writers on the Microscope. Carpenter and Hogg, in the last editions of their works on the Microscope, and Brooke, in his treatise on Natural Philoso- phy, all refer to this article as the best source for full and clear information in regard to the principles upon which the moderr achromatic Microscope is constructed. It should be in the library of every person to whom the Microscope is more than a toy. It is written in simple language, free from abstruse technicalities. NEW DESiaisrs FOE Fret or Scroll Sawyers. MR F T HODGSON, whose admirable series of articles on the USE OF THE SCROLL SAW are now in course of publication in the Young Scientist, has pre- pared for us a series of SEVENTEEIV I>ESIOIVS, of which the following is a list : No. I.— This shows one side, back, and bottom, of a pen rack. It may be made of ebony, walnut, or other dark wood. u i j j No. 2.— Design for inlaying drawer fronts, table tops, box lids, and many other things. It is a sumach leaf pattern. j r ■ v j a^- No 3.— Design for a thermometer stand. It may be made of any hard wood or alabaster. The method of putting together is obvious. No 4 -This shows a design for a lamp screen. The open part may be covered with tinted silk, or other suitable material, with some appropriate device worked on with the needle, or, if preferred, ornaments may be painted on the silk, etc. with^he neemc^or,^ J containing visiting cards. Will look best made of white holly. No' 6 —A placque stand, it may be made of any kind of dark or medium wood. No 7'— A design for ornaments suitable for a window cornice. It should be made of black walnut, and overlaid on some light colored hardwood. , , „ No 8-A design for a jewel casket. This .will be very pretty made of white holly and lined with blue velvet. It also looks well made of ebony lined with crimson. No 0.— Frame. Will look well made of any dark wood. No. 10.— Frame. Intended to be made in pairs. Looks well made of white holly, with leaves and flowers painted on wide stile. , No II.— Horseshoe. Can be made of any kind of wood and used for a pen rack. When decorated with gold and colors, looks very handsome. , , , . No 12 -Design for a hinge strap. If made of black walnut, and planted on a white or oaken door, will look well. , i-ifT,j a No i3.-Design for a napkin ring. May be made of any kind of hard wood. No. 14.— Hinge strap for doors with narrow stiles. No. 15. — Centre ornament for panel. No. 16. — Corner ornament for panel. No. 17. — Key-hole escutcheon. These designs we have had photo-lithographed and printed on good paper, so that the outlines are sharp, and the opposite sides of each design symmetrical. Common designs are printed from coarse wooden blocks, and are rough and unequal, so that it is often impossible to make good work from them. The series embraces over forty different pieces, and designs of equal quality cannot be had for less than five, ten or fifteen cents each. We offer them for twenty-five cents for the set, which is an average price of only one cent and a halt each. Mailed to any address on receipt of price. INDUSTRIAL PUBLICATiON CO., 14 Dej Street, New York. THE AMERICAN Journal of Microscopy. PROSPECTTJS-SIXTH YEAR-1881 The object of the Journal of Microscopy is to diffuse a knowledge of the best methods of using the Microscope; of all valuable improvements in the instrument and Its accessories; of all new methods of microscopical investigation, and of the most recent results of microscopical research. The Journal does not address itself to those who have long pursued certain special lines of research, and whose wants can be supplied only by elaborate papers, which, from their thoroughness, are entitled to be called monographs rather than mere articles. It is intended rather to meet the wants of those who use the microscope for purposes of general study, medical work class instruction, and even amusement, and who desire, in addition to the information afforded by text-books, such a knowledge of what others are doing as can be derived only from a penodical. With this object in view, therefore, the publishers propose to make the Journal so simple, practical and trustworthy, that it will prove to the advantage of every one who uses the microscope at all to take it. ILLUSTRATIONS.— The Journal will be freely illustrated by engravings representing either objects of natural history or apparatus connected with the microscope. TRANSACTIONS OF SOCIETIES.— The American Journal of Micro- scopy is not the organ of any Society, but it gives the proceedings of all Societies whose officers send us a report. As the Journal is devoted ^vlwlly to Microscopy and is in good form and size for binding, no better medium can be had for preserving the scientific records of any Society. Matters of mere business routine we are frequently obliged to omit for want of room. ^ EXCHANGES.— An important feature of the Journal is the exchange column by means of which workers indifferent parts of the country are enabled, without expense except for postage, to exchange slides and materials with each other. ' Published Monthly at $1.00 a year. SPECimEIVS FREE. Four copies for three dollars. Those who wish to economize in the direction of penodicals, would do well to examine our clubbing list. Foreign Subscribers.— The Journal will be sent, postage paid, to any counti-y in the Postal Union for $1.24, or 5 shillings sterling per year. English postage stamps American currency or American postage stamps taken in payment. In return for a postal order or draft for £\ 5s., five copies of the Journal will be furnished and mailed to different addresses. Make all drafts and postal orders payable to John Phin. □SaoIs. 'XT'oluxneis. Vol. I is out of print. We are occasionally able to complete volumes which, when bound, we offer for $1.50. Of Vols. II, III and IV we have a few copies on hand. Price J 1. 50 each. The JotjRNAL OF Microscopy, from its very nature, is a visitor to the very best families, and its value as an advertising medium has therefore proved to be much above that of average periodicals. A few select advertisements will be inserted at the rate of 30 cents per line, nonpariel measure, of which twelve lines make an inch. Address AMERICAN JOURNAL OF MICROSCOPY, p. O. Box 2852, New Yorlk. Thirty-seventh Year of Republication! THE London Lancet for 1881. i monthly Joarnal of British and Foreign medicine, Physiology, Surgery, Chemistry, Criticism, Literature and IVews* EDITED BY m- Tlie American Edition, as piil)lislied l»y the In- dustrial Publication Co., contains EVEItYTmr¥G relating- to Medical Matters tliat appears in tlie original London e