THE CARPENTERS AND JOINERS' - INSTRUCTOR. LONDON : PRINTED BY THOMAS DAVISON, WHITEFRIARS. % Lig 1 WIN”WNWNlHlHWiHMNW}fIfHNINhIJHMNWiir!IWHWHNHINHIWU \WWINWNWW!WNJIlNI/IIIHHHHHHMHNHHllIII\IH‘NHWWNIWUWNWM”NWWWWHUMIWWW ”"L, ‘mnw he .I.WM‘"—'j}l::[}}|[||[||| C m, [ h. C 14g: f ”HI“ : Mee HM " "@ - “V‘Hw "1“ fl. f s ' | [ i a London Publghed by Thomas legg 73, Cheapstde. BR € Fafa » m 2, rkeerK Ail ry CARPENTERS AND JOINERS INSTRUCTOR In CEOME T RILCAL LIKE S8. THB STRENGTH OF MATERIALS, AND MECHANICAL PRINCIPLES OF FRAMED WORK. EMBELLISHED WITH NUMEROUS ENGRAVINGS. BY THOMAS MARTIN, evir enoisegEn, AUTHOR OF THE CIRCLE OF MECHANICAL ARTS. _ LONDON: PRINTED FOR THOMAS TEGG, 73, CHEAPSIDE; J. CUM MING, DUBLIN; AND R. GRIFFIN AND €0. GLASGow. 1826. \ A MC to Kut wed Xi 7Wbé€4 MS: va'f'v “(UN- DESIGN PREF AC E. Tur following Treatise on Carpentry and Joinery, originally written and compiled by Martin, forming an article of a practical Eneyclopedia, denominated the Circour or tur Arts, will be found to be a very useful book ; not only as it respects the Journey- man, but the Master will receive many use- ful hints, both with regard to the Strength of the Materials which he employs, and the Construction which depends on the Theory of Mechanical Carpentry. - This part of the work has considerable merit. - Mr. Martis has not only collected his information from the best authorities, as the names of Pro- {A * NAL AQ vi PREFACE. fessor Rosisox, Banks, &c. will sufficiently testify ; but he has himself added and collected many valuable experiments on the Strength of Materials. The geo- metrical principles of Construction were taken principally from the works of N1- cmorsow, whose labours in this department have been long known to the Public: the designs are principally from the Encyclo- pedias, some from and Nicnorsox. This Treatise, as now offered to the Public, to make an independent work, has been greatly enlarged from the same and other recent authorities. 'The Editor therefore indulges the pleasing hope, that his readers will not be disappointed in finding the in- formation which they require. DIRECTIONS TO THE BINDER: Plate I. s ak f . s* to face page 2 IL. A : A & 4 A & e > b f t s 8 IV. f . s . i {98 I1 v.: ; & R o s a 12 VJ. s 6 A * a ~- ~I6 VIL: § 3 b s "t 30 VIIL. vigg % & ; > 19 1x: § k i & A "320 Ne : f : s as 97, XT. : 4 4 . ® "y- 49 XIL. f : 2 s a sa 44 XIIL. : f f s ec #C! XIV: * s « & cs 02 XV. 4 o F A R 480 VL c A % & i 408 XV IL. 4 $ * : ¢ ~~ XVIIL % / * 4 << 1189 XIX. s f - 3 % i , *: ~A20G XX. f R $ ; a * 198 XXL. 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CARPENTRY is the art of cutting out, framing, and joining large pieces of wood, to be used in building. Joinery is also the art of working in wood, or of fitting various pieces of timber together, for the ornamenting of certain parts of edifices, and is called by the French, menwuiserie, " small work." Both these arts are subservient to architecture, being employed in raising, roofing, flooring, and ornamenting buildings of all kinds. The rules in carpentry are much the same as those of joinery ; the only difference is, that carpentry includes the larger and rougher kinds of work, and that part which is most material to the con- struction and stability of an edifice; while B 2 THE CARPENTERS' AND joinery comprehends the interior finishing, and ornamental wood work. Carpentry and Joinery may very properly be considered separately. Under the former head, we shall enumerate the most useful tools made use of by carpenters, and then divide the article into three distinct divisions ; the first of which will treat of the most advantageous modes by which timbers in general may be connected together ; the second will describe and illustrate the several parts of constructive carpentry, such as centres, roofs, domes, niches, &c.; and the third will comprise some investigations and ob- servations relative to the strength and stress of timber. ENUMERATION OF THE MOST USEFUL CARPENTERS TOOLS. References to Plate 1 of Carpentry. Figure 1 represents The Axe. 2 # .s. .t; Gross, indeed, would be the errors in 90 THE CARPENTERS' AND constructing them, if much danger were to be apprehended of their falling from a want of equilibrium. In like manner, a dome of car- pentry, whatever may be its shape or con- struction, can hardly fall, unless some of its parts should fly out at the bottom; and this can be easily prevented by fixing an iron hoop around the bottom, or connecting straps with the joining of the trusses and purlines. Archi- tectural beauty requires that a dome should spring almost perpendicularly from the wall; and if we admit this, it may be readily perceived that the thrust exerted to force out the walls is very small; only it is necessary to guard against the operation of this thrust, and that is, where the tangent inclines about 40 or 50 degrees to the horizon, at which place it will be proper to make a course of firm horizontal joinings. We are of opinion that domes of carpentry may be raised of great extent, as sufficient room appears to offer itself for improving this very interesting branch. The Halle du Bled, which was constructed by an ingenious carpenter named Molineaux, is 200 feet in diameter, although it is not more than a foot in thick- ness, and yet it appears to possess abundant strength. - Molineaux, being convinced by his JOINERS' INSTRUCTOR. 91 mechanical experience, that a very thin shell of timber might be constructed so as to be nearly in equilibrio, and that when hooped or firmly connected horizontally it would have all the requisite stiffhness, presented his ingenious project to the magistrates of Paris, who, having doubts of its practicability, submitted the plan to the consideration of the members of the Aca- demy of sciences. Such of these as were com- petent to the task, investigated the principles of the intended construction, and were immedi- ately struck with its propriety, expressing their astonishment that a thing which appeared to be so very obvious should have escaped the attention of preceding carpenters. The Aca- demy, accordingly, presented a very favour- able report of the plan, which was immediately carried into execution, and, being soon com- pleted, it now stands as a monument, a proof of the inventive genius of Molineaux, and is justly considered one of the most curious ex- hibitions in Paris. The construction of this dome is very simple ; the circular ribs of which it is composed consist of planks nine feet long, thirteen inches wide, and three inches thick, and each rib consists of three of these planks bolted together in such a manner that two joints are contrived to meet. _ A rib is begun, 02 THE CARPENTERS' AND for instance, with a plank of three feet long standing between one of six, and another of nine feet; and this is continued to the head of it. At various distances these ribs are con- nected, horizontally, by purlines and iron straps, which act as so many hoops to the whole con- struction. When the work arrived at such a height that the interval between the ribs formed two thirds of the original distance, every third rib was discontinued, and the space between was left open and glazed. When the work had been carried so much higher that the distance of the ribs formed one third of the original distance, every second rib (now consisting of two ribs very near each other) was in like manner discontinued, and the open space was glazed. At a small distance above this, a cir- cular ring of timber was framed into the ribs, by which means a wide opening was made in the middle; over which is a glazed canopy, with an opening between it and the dome, to allow the heated air to escape. Every beholder is struck with the grandeur and the simplicity of this construction, and every unprejudiced mind cannot fail to confess, even from the idea we have endeavoured to give, that it must form a beautiful and magnificent object. In the construction of some domes one great JOINERS' INSTRUCTOR. 98 difficulty is to be overcome, and that is, when they are unequally loaded, by having to support a heavy lantern, or cupola, in the middle. In such a case, if the dome were only a mere shell, it must be crushed in at the top, or the force of the wind operating on the cupola might remove it and its supporter out of their place. The dome of St. Paul's cathedral is a model of propriety in this particular method of con- struction, and much valuable information may be derived from considering the principles on which it was erected. ('These are shown in Fig. 3, of Plate 15, where amBcoBsa, represents the dome turned over with bricks, two feet in length, which were made on purpose. EFGG FEE exhibit a cone of bricks one foot six inches in thickness, and visible through the opening c c. - This cone, aided by the timber work of the dome, supports a cupola, con- structed of Portland stone, nearly forty-four feet in height, and twenty-one feet in dia- meter. The timber work z z, is ingeniously tied together with iron cramps, run with lead into the stones, xm, N, 0, P, and then bolted through the hammer beams, ux, 11, K x, and 1 1. From these principles it may be per- 094 THE CARPENTERS' AND ceived, that the timber-work derives consi- derable support from the brick cone. The construction of this dome affords a strong proof of the profundity of Sir Christo- pher Wren's mathematical judgment, and his unrivalled excellence in constructive carpentry. Fig. 4. represents a dome raised over the Register office at Edinburgh, by Messrs. James and Robert Adam. _ In this instance the con- struction of this dome appears agreeable to me- chanical principles, and consequently is de- serving of attention, particularly when it is con- sidered that the span is 50 feet clear, and the thickness only 44% feet. The principle of a Norman roof is ingenious and simple. The rafters all butt on joggle king posts, aA F, B 6, cH, &c. (as in Fig. 5), and then braces or ties are disposed in the intervals. The ties, 1 B, 4 D, are in a state of extension, while the king post, cn, is compressed by them. Towards the walls on each side, as for instance, between s and F, r and 1, they act as braces, and are themselves compressed. The ends of these posts were generally ornamented with knots of flowers and other devices, and even the whole texture of the truss was exhibited and dressed out. JOINERS' INSTRUCTOR. f 95 Short timbers may be employed in these kinds of constructions, and this very circum- stance imparts additional strength to the truss : for the reason, that the angle which the brace or tie makes with the rafter is more open. All struts, likewise, may be removed from the walls, which demands attention. If the pieces a F, BF, L F, were to be removed, then the remain- ing diagonal pieces will act as ties, and the pieces directed to the centre will act as struts. The application of this principle to a flat roof, or to a floor, will be productive of advan- tages similar to those which we have before stated. For instance, a floor, such as a bo, having the joist in two pieces a b, be, with a strut b d, and two ties, will require a much greater weight to break it, than if it had been a continued joist, like a c, of the same dimen- sions. Moreover a piece of timber operating as a tie, is much stronger than the same piece if applied as a strut; since in the latter situation it is exposed to bending, and when bent is much less able to withstand a very great strain. It must be acknowledged, however, that this advantage is balanced by the great inferiority, in point of strength, of the joints. The joint of a tie depends wholly on the pins; for which 96 THE CARPENTERS' AND reason, ties are never used in heavy works with- out strapping the joints with iron. In the roofs we are describing, the diagonal pieces of the middle part only, act entirely as ties, while those towards the sides act as struts or braces. In addition to what has been observed, this method of trussing a floor cannot often be intro- duced on account of the very great additional height required for the truss-work ; and more- over, if rooms are under a floor of this con- struction, some additional timbers will be re- quired in order to form the ceiling: and as the braces in this construction perform the office of ties, they are in a state of tension, and have a tendency to draw out the joints, as already ob- served; and as iron cannot be depended upon for any great length of time, such a truss will be much more secure by placing the angle up- wards in the same position as it is usual to con- struct a roof ; by this method the two braces will act by compression, and the joints will be forced together. - In short, the whole will stand without being secured with iron, and in order to make a level floor of it, nothing more is ne- cessary, excepting to extend a beam over the top, which may be supported from the fixed points of the truss. JOINERS' INSTRUCTOR. 97 The order of carrying on a building, the dimen- sions of timbers, their distances, and the places of their insertion. As different buildings begin in different manners, some with cellars, and some with living apartments, the order of preparing the timbers will vary accordingly ; we shall therefore pro- ceed to one particular instance, which will help to regulate any other order of proceeding with the work, f Lintelings, which are the timbers laid over the heads of apertures, very soon occur.. Their thickness ought never to be less than as many inches as the aperture has feet in its width. Thus if a door or window is five feet wide, the thickness of the lintels ought not to be less than 5 inches. Some authors recommend that lintels should be laid on short pieces of wood, vulgarly called templets, laid across the wall close to the side of the aperture, as long as they can be made so as not to appear in the front of the wall. Bond timbers are those which are inserted in walls, at certain heights or at certain places, for the fixing of various wooden finishings in joinery ; they are therefore made flush with the It 98 THE CARPENTERS' AND inside faces of brick walls; but their greater use is to prevent settlements in buildings. The old authors say, that bond timbers should be dovetailed at the angles; but this is not sufficient to prevent one side of a building from descending, while the other remains stationary : they should either be mortised and tenoned through and through, or halved and bolted together; and for greater security in the best work, they may even be strengthened with diagonal ties at the angle, which may also be boited through each piece. By the thickness of horizontal timbers which are laid in walls, we mean the breadth of the faces which are placed vertically in the work. Where bond timbers are carried all round apart- ments or rooms, or entirely round a building, the thickness of bond timbers will depend upon the mass of the work over them; but where they are inserted for finishings and only partially in- serted, their thickness must be the thickness of a brick. Some old authors think it would not be amiss to place bond timbers, at the distance of 6 feet, through the whole height of the building ; but in our opinion they ought to be used with great caution; for as the moisture ries out of the timber, these ligatures will JOINERS' INSTRUCTOR. 00 shrink and cause the walls to bulge, which wilt not only produce a very unpleasant effect to the eye, but will endanger the building by weak- ening the walls, and make them liable to fall. Therefore, in good work, bond timbers ought to be dispensed with, and if necessary, other means ought to be resorted to -which will be equally effective in point of strength; but as neither stone nor iron will answer the purpose of fixing, we will recommend plugging, built in with the brick work. We come now to the consideration of floors. A4 floor, in carpentry, is the timber work for supporting the boards upon which we walk. A row of timbers employed in floors is called joisting. When a floor consists only of one row of timbers, it is called a common joist floor. Framed floors are those where the ends of joists are supported by a large beam of timber, called a girder, which is mortised from each rertical side to receive the tenons which are cut on the ends of the joists.s When a framed floor consists of only one row of joists, the floor is said to be single framed. When the joists on each side of the girder support another row of timberstaralle/l to the girder, the flgoris called .a double floors »The srow. of timbers H 2» 100 THE CARPENTERS' AND which are fastened to the girder by mortise and tenons are called binding joists, and those timbers which are supported by the binding joists, are called bridging joists. To a double framed floor there is another row of small timbers, attached to the binding joists, for sup- porting the lath and plaster; and are either nailed to the underside of the binding joists, or fixed to them by means of mortise and tenon. In some single joisted floors every third or © fourth joist is made deeper than the intermediate joists, and the ceiling joists are fixed to the deep joists, the one crossing the other at right angles. This construction is adopted to the prevention of sound, which must suffer an in- termission by reason of the space between the timbers. As no timbers must enter a wall where there are fire-places or flues, the ends of the joists, instead of being supported by the wall at such places, must be supported by a piece of timber parallel thereto by mortise and tenons, and this piece of timber must be fixed by mortise and tenons at each end, to the nearest joists to such fire-place or flue ; each of these joists is called a trimming joist, and the piece of timber which supports the joists leading to the fire-place or flues is called a trimmer. As the trimming JOINERS' INSTRUCTOR. 101 joists have also to support the intermediate joists, they ought to be in thickness equal to the breadth of the common joists, increased by a sixth part of that breadth. In double floors, the under sides of the binding joists are frequently framed flush with the under side of the girder, and about three or four inches below the top, in order to receive the bridging joists. Some old authors direct that the bridging joists should be pinned down to the binding joists; but this is unnecessary, and besides, it weakens the binding joists; this practice is therefore inadmissible. It was formerly the practice to place the binding joists about three feet or three feet six inches distant from each other ; the mean distance of the present practice is about five feet. Single floors consisting of the same quantity of timber are much stronger than framed floors ; but a preference is sometimes given to framed floors in superior buildings on account that they are not so liable to fracture the ceilings, and because they conduct sound more imperfectly than a common joist floor, and hence it is that single floors can only be employed in inferior buildings. | Framed floors differ from double floors only in the binding joists being framed to girders. 102 THE! CARPENTERS' AND In single floors where the joists exceed 8 feet bearing, pieces of board ought to be inserted in the spaces between the joists in a vertical position, and nearly the whole depth of the joists, and in one continued line at right angles to the joisting. . The pieces of timber thus in- serted are called struts, and the floor is said to ~ be strutted ; the struts ought not to be driven in with great force, but their ends should be in closes contact with the vertical sides of the joists, and should be fixed thereto with a nail at each end. The strutting of a floor is of great use when . the joists are thin and deep, in preventing their buckling by pressure ; but for this purpose there is another method called keying, which consists in framing short pieces of timber between the joists; but as the mortises which receive the tenons weaken the joists, and as the keys cannot be in a straight line, and since this method adds considerably to the expense, this practice is not so eligible as that of strutting. Single joist flooring may be used to any ex- tent not exceeding sixteen feet ; but whea it is desirable to preserve the ceiling free from cracks, and to prevent the passage of sound, a framed floor is necessary. ; The ceiling joists in double floors are generally JOINERS' INSTRUCTOR. 108 put in after the building is up ; if therefore they are fixed by means of mortises in the sides of the binding joists, to receive tenons on their ends, the space between every other two mor. - tises must be grooved out alternately upon the opposite sides of the two adjacent binding joists ; by this means the ceiling joists may easily be put in their places by inserting the tenon in each ceiling joist in the mortises at one end, and sliding the tenon on the other end along the groove in the are of a circle, until the ceiling joist come at a right angle with the binding joist. The long mortises or grooves in the sides of the binding joists are called chace mortises or pulley mortises. & The ceiling joists may be 13 or 14 inches ° apart ; the thickness of the bridging joists and ceiling joists need not be greater than what is sufficient to resist splitting by the driving in of the nails in order to fix them. It has been found by experience, that two inches is a suf- ficient thickness for the purpose. In double framed floors the distance of bridging joists in the clear ought to be about 12 inches, and should never exceed 13. It is a good practice to plane the upper edges of the bridging joists straight, because when the board- ing is laid, the faces for walking upon will be 104 THE CARPENTERS' AND more regular than if the boards had been laid down upon the edges of the bridging joists when rough from the saw. The straighting of the edges of the binding joists will not only give greater facility to the making of a level floor, but will contribute greatly towards making sound work ; for when the tops of the joists on which the flooring boards are laid are uneven, it will be impossible to avoid furring up the joists, or, what is still worse, inserting chips in the hollows, which will give way in the nailing of the boards to the joists ; and thus, when finished, walking upon the floor will occasion that disagreeable creaking noise occasioned by the unsoundness of the work. - The general practice is to make binding joists half as thick again as common joists, so that if a common joist be two inches thick, the binding joists may be three inches thick. In fixing the ceiling joists to the binding joists, it is better to notch the edges of the ceiling joists and nail them to the undersides of the binding joists, than to mortise and chace them to the sides of the binding joists. GIRDERS. Girders should always be placed. upon walls which are solid underneath; but when it be- FOINERS' INSTRUCTOR, 105 eomes necessary to lay them over apertures, the lintels should be sufficiently strong to support them. We cannot recommend the practice of laying girders obliquely across a room, since it divides the binding joists so very unequally. All joists should be laid with a camber up- wards, so as to raise the middle of the floor about three quarters of an inch higher than the sides of the room; and a similar observation applies to ceiling joists, viz. that the under horizontal sides should rise with a concavity, so that the middle of the ceiling should be three quarters of an inch above the margins at the cornices or walls. The distance of girders from each other or from walls should never exceed ten feet. Girders should always be made of timber of the greatest dimensions that can be found, and particularly those which have long bearings. When the bearing exceeds twenty feet, it is difficult to procure timber of sufficient di- mensions. The only method is to allow a suffi- cient thickness between the surface of the boarding and the ceiling, since it is found by experiments that have been made, that a truss - girder is not even so strong as a solid beam of the same depth: the reason is obvious ; for 106 THE CARPENTERS) AND braces which have only a small inclination to the horizon throw the most enormous com- pression on their abutments, which consequently must give way, and the effect of trussing will be rendered useless ; but if a sufficient height were allowed for trussing, girders might be made ca- pable of supporting any weight whatever. Two feet or even three feet in the height of a building would be an ample allowance for framing girders of sufficient strength, and would not occasion any considerable expense to the structure; but would give solidity to the walls by having a greater distance between the aper- tures, and would therefore allow more room for the display of ornaments. But where the depth is limited, and the bearing considerable, girders ought to be made solid, of cast iron. In order to equalize the strength of solid girders, builders frequently cut them longi- tudinally along the middle, and turn the ends of the flitches contrary to what they were at first in the solid, and apply the sawn sides so as to face each other, and then bolt the two flitches together in a sufficient number of in- tervals; but it is evident that, since the holes made for the passing of the bolts will weaken the timber, very litile strength will be gained. JOINERS' INSTRUCTOR. 107 This process, however, affords the opportunity of examining the timber which, in large trees, is frequently found in a state of decay. - When this process of reversing and boiting is used, the two sawn sides of the timber should not be brought in contact, but should be separated by parallel pieces of wood, so as to allow a suffi- cient circulation of air to pass between the two flitches of the beam thus bolted. To prevent the sagging of short girders, it is usual to cut them camber: that is, to cut them with an angle in the midst of their lengths, so tliat theis middle shall rise above the level of their ends, as many half inches as the girder contains times ten feet. And, indeed, girders of the greatest length, although trussed, should be cut camber in the same manner. It may be proper here to notice; that the cambering of girders does not prevent them from sagging, though perhaps it may obviate their becoming concave on the upper side. With regard to trussing girders, the flitches should not be cut to a camber, but brought into this state in the act of trussing. JOISTS. The next order is joists, of which there are 108 THE CARPENTERS' AND five kinds, viz. common-joists, binding-joists, trimming-joists, bridging-joists, and ceiling- joists. In Plate 16, Fig. 1, is the plan of a double floor in two bays, divided from each other by a girder; the wall plates are represented by a, ayia, &c. The binding joists by b, b, b, b, &c. The girder by C, C; &c. And the bridging-joists by d, d, d, d, &e. The ceiling joists are represented by e, e, e, &c. Fig. 2 is a transverse section of the floor. And Fig. 3, a longitudinal section of the same. As the girders are always used in floors, they are here connected with the drawings of the floor. Fig. 4, a vertical longitudinal section through the middle of a girder, constructed with two trusses abutting upon each side of the middle bolt. Fig. 5, a vertical longitudinal section of a girder, constructed with two trussing pieces abutting upon two queen bolts, with a strutting piece between them. Fig. 6, Plan. JOINERS' INSTRUCTOR. 109 Fig. 7, a transverse section of each of these girders through the abutments. Fig. 8, the section of the middle bolt taken in a transverse direction. Fig. 9, section of the middle bolt taken in the longitudinal direction of the girder. The trussing pieces should be let into the sides of the girder about an inch and a half, and should only be tight at the ends, otherwise they will be useless, for they will bend with the gir- der, and consequently their length will shorten; but it is evident if they are left free, and if a pressure be laid upon the girder, the girder will become bent to a certain degree, and con- sequently shorter, in all its parts; but since the trussing pieces remain of the same length, they will endeavour to keep the girder from shortening. In screwing the girder together, the head of the bolts should be previously greased, so as to permit it to slide freely by the ends of the trusses. In screwing the girder together, the nut of the king or queen bolts ought to be turned, while another person strikes the head of the bolt with a mallet, which will cause it to start at every time it is struck, and thus the pressure against the nut will be strengthened. 110 'THE CARPENTERS AND Proceed again to turn the nut till it resist the force applied; repeat the process until the girder become sufficiently cambered : this may be about an inch in twenty feet. PARTITIONS. Partitions are frames of timber-work for di- viding the space contained by the exterior walls into rooms. Partitions are usually lathed and plastered, and sometimes the spaces between the timbers are filled up with brick-work. A partition ought to be so constructed as to be capable of supporting its own weight in whatever situation the door is placed, or whether there is a door in the middle or two doors near the ends. Partitions that rest upon a solid wall do not require trussing ; but when there is no support, except at the ends, or at two given fixed points, the braces ought to be so disposed as to discharge the weight of the whole mass upon these points ; and it is better to support a partition by the extreme walls it is connected with than upon any solid from the bottom : for in the settlement of the walls, the partitions will be cartied along with them; but if sup- ported from the ground by light materials, the JOINERS' INSTRUCTOR. 111 walls and partitions will descend unequally, and cause large fissures and cracks in the ceilings and in the plaster upon the walls and par- titions. _ f When a partition is supported at each end by walls of unequal heights, the wall which is the most ponderous will sink in a much greater degree than that which is the lighter ; therefore in this case whatever care may be taken with the framing of the partitions, it will not be possible to avoid the cracking and splitting of the plaster on the walls and ceilings ; such con- sequences should be guarded against in the design of the architect. ROOF. A roof is the cover of a building for pro- tecting its inhabitants from disagreeable changes of weather, and from the depredations of evil- disposed persons; but a roof in carpentry is the timber framing made to support the actual covering of tile, slate, lead, &c. As the roof may be made one of the prin- cipal ties of a building, it should not be made too heavy to burden the walls, nor too light to be incapable of keeping them together. The principal timbers of a roof are the wall 112 THE CARPENTERS) AND I plates, tie beams, principal rafters, common rafters, pole plates, purlins, king posts, queen posts, struts, straining beams, straining sills, ye: s The wall plates are those timbers on which the superstructure of the roof is raised, placed upon the wall heads. Hence, since the pressure of the roof is wholly upon the wall plates, the wall plates should be made of sufficient thick- ness and breadth to distribute the weight of the roof to the best advantage. Tie beams are pieces of timber in a row reaching across the interior, and extended over the wall plates, for keeping the building to- gether. The upper surfaces of all timbers which are level across their breadth or thickness, are called the backs of these timbers. Principal rafters are timbers placed in two rows, one row from each opposite wall, so that their backs may be in a plane inclined to the horizon at a given angle. King posts are vertical pieces of timber, fixed upon the middle of the tie beams for supporting the upper ends of the principal rafters, so that when the two inclined sides of the roof meet in a ridge line, each tie beam and each pair of prin- JOINERS' INSTRUCTOR. . "MS cipal rafters may form a triangular frame, di- vided by the king post into two equal and similar right angled triangular parts. Queen posts are vertical pieces of timber, distributed in pairs, two queen posts being fixed to each tie beam at an equal distance from the centre, for supporting the upper ends of the principal rafters. ‘ Straining beams are those timbers in a trun- cated roof extending between the heads of queen posts, for supporting the flat on the top. Strutts are pieces of timber, each extending from the bottom of each king post or queen post to the under surface of the principal rafter, in order to support that rafter. Purlins are pieces of timber parallel to the horizon, supported upon the backs of the prin- cipal rafters. Pole plates are horizontal timbers at the foot of the principal rafters, parallel to the purlins, and are supported upon the ends of the tie beams. Common rafters are rows of parallel timbers supported upon the purlins between their ex- tremities, and at the lower end by the pole plate. Ridge piece, a timber board, placed in the I 114 THE CARPENTERY' AND meeting of the two sides of a roof, for sup- porting the upper ends of the common rafters. So that in triangular roofs, the tie beams sup- port the principal rafters ; the principal rafters support the king post; the king post supports the middle of the tie beam and the strutts ; the strutts support the bearing of the principal rafters between their extremities ; the principal rafters support the purlins, and the pole plates are supported by the tie beams; and lastly, the common rafters are supported by the pole plates and the purlins. The first thing required in a large roof is, the trussing, next the raftering, then comes the boarding, which must be prepared for the plumber, with proper tilting fillets at the lower edge, in order to raise the first course of slates which begins at the eaves to a propet. in- clination, so that the succeeding courses as they rise may lap, at the same inclination, upon each other, as the planes of the slates forming the first course. The tilting fillets are pieces of a triangular section, about 4} inches broad and 4 of an inch thick, formed by cutting boards of these dimen- sions diagonally. Skylights constructed in a roof should have JOINERS' INSTRUCTOR. "A15 a proper gutter at the head and one on each ' sige. In order to prepare for the guttering, grooves should be cut in stone-work, when the building is constructed of that material, for flashings, and the vertical sides of the lead which lines the gutters. - These grooves are by working masons called raggles, particularly in the north. These ragglings are afterwards sometimes filled with oil putty ; but the best way of securing the lead is to fill them with melted lead, which may be stiffened with the mallet and blunt chissel. sorFITs. A soffit is defined by theoretical carpenters to be the covering of any surface with wood, arranged on a plane. In a straight wall which flues equally all round, it is requisite to describe a soffit with a circular head. On aB, or cp, being sides of the plan aBco, Fig. 1, Plate 17, describe a semicircle, and pro- duce ac, and Bp, to meet in £. Then with s, as a centre, and the distances Ea, and Ec, describe the ares ar, and co: next divide the circumference of the semicircle into any convenient number of equal parts (say ten), and laying off a similar number of divisions 63 116 THE CARPENTERS' AND either from a to r, and join r r, and the re- quired soffit will be completed. Next in a circular wall, which flues equally all round, it is required to describe a soffit with a circular head. Let acoB, as in Fig. 2, represent the given plan. Having described upon a s the semi- circle ans, lay out the soffit in the same manner as before directed. From the several points of division 1, 2, 3, (made use of to effect the pre- ceding,) draw the perpendicular ordinates, 11, QQ, 88, 44! " Join' tl, P2, r8, and r4;08s0 as to intersect the curve lines am, of the plan acpB, in the points a, 5, c, d; through these points draw lines parallel to aB, in order to intersect a¥, in #i, ", 0, J; next with the: centre 'r, and distances rm, r»; ro, and rp, describe ares so as to intersect respectively r1, r2, r3, and F4, in the points a, b, c, d, then by these points the half of one edge of the soffit will be found, from which the other half may be readily pricked. The reverse edge is found in the same manner. Let it now be required to find the develope- ment of a semicircular headed door or window in a curved wall. Let arco, Figs. 3 and 4, be the plan of the wall, as and pc, being the jambs necessary, JOINERS' INSTRUCTOR. T17 and draw ac perpendicular to aB, intersecting cp, produced to r, if necessary. On ar de. scribe a semicircle a 5 r. Divide the quadrant £5 into any number of equal parts (which let for instance be 5), and lay these parts from ® to c, marking the points 1, 2, 3, in rae. From the points 1, 2, 3, &c. in the semicircle, and from the points 1, 2, 3, &¢. in the straight line rc, draw lines parallel to AaB, or nc, and let the lines drawn from the points 1, 2, 8, &c. in the semicircular are meet the curve line ap of the concave face of the wall in the points a, b, c, d, &c. and the curve line sc of the convex face in ~theqpoints:a.>/, ¢, d, &e. and the straight line ac in the points 1, 2, 3, &c. Make the perpendicular distances 1 a, 26, 3¢, &c. from the points 1, 2, 3, &c. in so equal to the cor: responding lines 1¢, 265, 3¢, drawn through the points 1, 2, 3, &c. in the chord line as, and we shall have the edge n» a be, &c. c, of the soffit; the other edge cab c, &c.. will be found in the same manner; the straight line Eo, mak- ing the entire circumference of the semicircu- lar atc. Again, let it be required to stretch out a soffit, when a door or window having a semicitr- cular head cuts into a straight wall in an oblique direction. 118 THE CARPENTERS' AND In Fig. 5, let be the plan; and at the point B erect the perpendicular Br, so as to intersect pa, produced in ®, and on Bz describe a semicircle; let the circumference of this be divided into any number of equal parts (say ten), and let the ordinates be drawn from the several points of division across the plan arcp ; on EF lay off the several divisions 1, 2, 8, &c. of the semicircle; then, when the ordinates have been all drawn across, and traced off from the plan as the figures and letters direct, the required soffit will be completed. For more complete information upon this sub- ject, we refer to Nicholson's Carpenters' New Guide. GRoOoIN#. Groins are the angular curves made by the mutual intersections of semi-cylinders or arches, and may be considered as either regular, or ir- regular. A regular groin is properly so called when the intersecting arches, whether semi- circular, or semi-elliptical, are of similar di- ameters and heights ; an irregular groin, is pro- perly so called where one of the arches is semi- circular, and the other is semi-elliptical ; thus Fig. 1, Plate 18, exhibits a perspective re- presentation of a regular brick groin, the arches JOINERS' INSTRUCTOR. - - 119 of which are semicircular, of the same diameter, and intersect each other at right angles ; and, since any oblique section of a cylinder produces a regular elliptic curve, it is evident that the angular ribs of such groins will be semi-ellipses, having their transverse axis horizontal, and their semi-conjugate vertical. This also will be the case when the intersecting or side arches are elliptical, for it is easy to conceive that their ordinates will coincide with those of the body arch a, which from being a semicircle, conse- quently produces an elliptic arch. It may now be perceived that the intersecting arches B, B, are formed by the erection of what workmen call the jack ribs, a perspective view of which is exhibited at Fig. 2, where 2, 8, 4, show the manner of their being fixed on the body arch a, after it has been boarded over. To keep these jack ribs true, and in a right line at the top, good workmen place a transverse board upon the crown of the arch, (as shown in the diagram here referred to) fixing it suffi- ciently low to receive the thickness of the cover- ing, that the body and intersecting arches may be perfectly even when the whole is covered With boards, as in the [perspective Fig. 1, which represents the state of the groin when ready for turning the brick work over the arches. 120 THE CARPENTERS' AND Let it therefore be observed that the body arch A, must be entirely covered before the erection of the jack ribs, whose seat on the body arch, and their several heights, as shown at 2, 3, 4, Fig. 2, may be readily found by inspecting the figure. Fig. 3, presents the plan of an irregular brick groin, whose body arch a, forms a semicircle, - and whose intersecting or side arches B, B, are semi-ellipses. It is next required to find the mould for the jack ribs. Let a 6 c, Fig. 4, be the body arch, and a d e, the intersecting elliptic arch ; draw similar ordinates to both arches, as 1, 2, 3, &c. make them intersect each other, and produce them each way at pleasure. Make g A, equal to the circumference or girt of ab, and gk, equal to the girt of a 1 2 8 d. Divide g A, and g k, into four equal parts, because the qua- drantal ares of the semicircle and semi-ellipses were so divided ; then draw through each di- vision perpendicular lines, so as to cut at right angles the ordinates which were drawn out at pleasure ; and if curves be drawn through the points of their intersections 5, 6, 7, 8, 9, 10, they . will produce correct moulds, by which the mi- tering of each arch, or their respective coverings may be described, Suppose it were required to mark aline on the body arch at Fig. 3, con- JOINERS' INSTRUCTOR. 121 trived so as to touch the extremity of each jack rib at the base; in such case the mould 5, 6, 7, Fig. 4, must be taken, and if this be made of thin pliable wood, the end A, is fixed to the crown of the body arch at A, Fig. 3, when after se- curing. it to) that- point,. the. other end : 5; is pressed, and with a pencil the required curve is traced out. - After a similar manner the other mould, 8, 9, 10, may be applied to the 'in- tersections of the elliptic arch, and if we suppose it to be covered, as is shown at Fig. 2, and the arch to be drawn back and separated from the body, the mould 8, 9, 10, bent over the boards, will be found to coincide with their ends, pro- vided the arches are of the same dimensions, which is not the case, however, in this example, although in theory the principles are entirely the same. In fixing the ribs of the body arch at Fig. 1, c d are strong wooden posts, and i ¢ are the ends of beams extending the whole length of the groin, and supported by posts under each rib. - The girders of each rib lie between, which are omitted, however, at e, in order to give a clear view of the internal parts of the arch. These long beams also act on the principle of wedges, as may be seen at e ; so that when the brick-work is properly set, they are eased gra- 122 THE CARPENTERS' AND dually, and the wooden ribs, beams, and posts, are easily struck and cleared away. Let us next consider the plan of an ascending or descending groin, Fig. 5, and also of the side arches c, in order to find the intersection of the angles, and the moulds for describing the cur- vature of the intersecting arches, the general principle of which problem, is the same as that of the preceding. - To project the present figure, let one half of the body rib s, be divided into four equal parts, as at 1, 2, 3, and let the lines be drawn from these points, to meet the per- pendiculars 5, 6, and continued round in cir- cular ares from the centre 5, to meet 5, 7, the base line produced : from these points let lines be drawn parallel to the line of inclination 5, 8, to meet the semicircle c on both sides. The points on the first side being 1, 2, 3, &c. From the several points 1, 2, 3, &e. draw lines through the plan parallel to the base of the rib B, and draw lines from the corresponding points of the outer are of the rib B perpendicular to the base, to intersect the lines drawn from the rib c, and draw the curve lines forming the plan of the groims. From the centre of the circular are forming the rib c, draw a line perpendicular to the inclined line 5, 8, intersecting the are of the rib c in 5, and produced out toe. Maked e JOINERS' INSTRUCTOR. 123 equal to the developement of half the are of the right section of the cylindroid, of which the end is the rib s; marking the points a, 6, c, d, in the straight line 5 e above c, which corre. sponds with the points 4, 3, 2, 1, in the outer are of the rib s. - Through the points ¢, 6, ¢, 4, draw lines perpendicular to 5 e, to meet lines drawn from the points 1, 2, 8, &c. in the outer arc of the rib c, perpendicular to 5, 8, and curves being drawn through the points of meet- - ing, will be the edges of the moulds for setting the jack ribs by. - These two moulds are to be applied in the same manner as pointed out in the foregoing example. ‘ This example is taken from one originally given by Mr. Peter Nicholson. The figure delineated at Fig. 6, represents a groin, whose intersecting, or side ar ch is Gothic, and under pitch, that is, it has its per- pendicular height less than the height of the arch a, which forms a semi-circle. From s, the intersecting arch, project a line from 1, round to 1, at the body arch a, and from 1, let fall a per- pendicular to s, the centre of the side arch, at a, draw the cord line 3, 4, 5, and let it be di- vided into four equal parts ; draw lines also from 2, through each division, and produce them until they cut the arch, then from 1, draw lines through the points on the arch to the perpendi- 124 ~THE. CARPENTERS' AND cular 0 p ; take o p, and place it on the perpen- dicular at the extremity of the side arch B, so as to create the same divisions, then let the cords of the side arch be drawn, and divided into four equal parts, and proceed as before at A ; finally, draw lines from 1, to 0 p, so as to intersect the lines which issue from the centre 2, and the in- tersecting points will give the curve of the side rib p. When the plan line p s, of the angular rib is drawn, let s g, be drawn perpendicular to it, then take the height of the intersecting arch 1, 2, and place it to 9 ; then let the cord line p g, be drawn, and proceed with the rest as before, and the angular rib E, will be produced as re- quired. - The same operation may be performed by ordinates in the common way, as is repre- sented on the opposite side of the figure. This groin is intended for plaster, and conse- quently very great exactness is required in the formation of the angular ribs, and the utmost correctness is necessary also on the under side, to produce regularity and smoothness in the ceiling. Fig. 7, represents the plan of another plaster groin, whose body arch a, forms a semi-ellipsis, and whose intersecting ones B, B, are semi-cir- cular ; D, is the angular rib, described by or- dinates, as appears by the corresponding nu- merals, which method we have already described. JOINERS' INSTRUCTOR. 125 The plan exhibited at c, shows the jack and angular ribs, as shown in the preceding ex- amples. The next object of the reader's attention is the plan of a curved groin, which may be ap- plied, either to brick work or plaster, though in the present instance, it will be confined to the latter. Fig. 8 exhibits the plan of a cylindro-cylindric, or Welsh groin, the body or intersecting arches of which are composed of semicircles or of similar segments, whose intersections meet on a curve surface, and therefore their plan will be a curve, as in the foregoing example. When the intersecting arch s has been divided into any number of equal parts, let lines be drawn at pleasure, perpendicular to its base. From the points 1, 2, 3, 4, let lines be carried round to the body arch a, as at 1, 2, 3, 4; from whence draw lines perpendicular to the base line of a, and where these intersect at 1, 2, 3; 4, draw a curve line, which will give the plan of the an- gular rib of the arch. The rib itself at o may be found as usual, and to this the numerals afford a direction. - To find the mould necessary to describe the curvature of the intersecting arches when laid on the body arch a, take the girt of the angular rib », in the inside, at 1, 2, 3, 4, and draw a right line at rE ; then take the 126 THE CARPENTERS' AND ordinates from the cord line of the plan of the angular rib, and place them respectively at 1, 2, 3, 4, which will produce the required mould. The angular rib in this figure will be curved both ways, similar to the shape of the circular groin, which has been beforedescribed ; the same means of describing the former may be made use of, as were adopted with respect to the latter. Cylindric groins were, we believe, first intro- duced into carpentry by Mr. William Pain, showing only how the plan was to be found ; but Mr. Peter Nicholson afterwards showed the construction of the angle rib, and he has shown, from mathematical principles, that the plan of the groin lines are hyperbolas, of which the asymptotes pass at right angles through the centre of the severy. Angle brackets are those which are placed in the mitres of large coves or cornices, in order to support the lath which is to be covered with plaster worked in the form of a cove or into mouldings. Plate 19, Fig.1, shows the method of diminish- ing or enlarging a bracket, so as to retain the proportions of the given bracket. For instance, let sBcprr, be the edge of the given bracket which is to be lathed ; let ar, be the wall line, ard ar, the line which is to come in contact with the ceiling. JOINERS' INSTRUCTOR. 127 Draw the diagonals ac, ap, and ar, and let aB, be the projection of the given bracket. Draw the straight lines bc, ode de, 'and ef; pa- rallel respectively to b¢,c¢d, de, and ef, meeting the diagonals in the points c, d, e, then abede f, will be the diminished bracket required. In the same manner, if a bedef, be con- sidered as the given bracket, we shall find the enlarged bracket aBcorr. Though cornices are generally equal and similar on all sides of a room, yet as there may be cases where they are dissimilar, we shall here show how to describe such as occur of this kind. In Fig. 2, let asc be the external angle made by the faces of two walls of a room, and let hm, be the projection of a cove upon one side, and Ham the projection on the other side: on the base Am, place the bracket Anropg, and on the base mam, place the bracket unorag. Let the entire heights ua, A4, be equal to each other. Draw aq, perpendicular to as, and a4, perpendicular to Bo, meeting the extreme points Q, g. In one of these heights ma, take any number of points, r, s, Tt, and in the other height 2a, make Ar, As, At, respectively equal to ur, gs, gt. Draw rx, so, and TP, parallel to aq, meeting the curve in the points x, o, p. In like manner draw r», so, and tp, parallel to ag, 128 THE (CARPENTERS' Ann meeting the curve in the points », 0, p. - Draw the lines x», or, pr, and qc, parallel to as, and in the same manner draw lines 21, or, pr, and qc, parallel to sc. Through the points s, 0,5, r, 6, draw the curve BpEF6, which will be the - seat of the mitre of the two coves. ~ Fig. 8, is the plan of a return angle, bracket- ing at a right angle. Fig. 4, exhibits the method of findmo an angle bracket for a plaster cornice. Let Ear be a right angle: make az, and apr, each equal to the projection of the cornice, and join Er. Let be considered as the given bracket, and let ar be in the same straight line with ap.. Draw «6, 1c, xd, parallel to rp, meeting Ep, in the points byes: dyrandanter- secting Ar, in the points B, c, D. Draw p/, bg, ci, and ah, perpendicular to sp. Make af,; bg, ci, ch, dk and dl, respectively equal to AF, BC, ct, CH, DK, pr, and join fg, & A, Ai, ik, and Ir ; then Erp fgAikl will be the almle bracket re- quired. I Figs. 5, and 6, are cove brackets described in the same manner. Fig. 7, angle cove upon an obtuse angle. Fig. 8, an angle cove upon an acute angle. PLATE 20. To cover the surface of a dome with boards JOINERS' INSTRUCTOR. 120 for slating, so 'that the joints may be in horizonal planes. Let aro, No.1, be a voettical section .of the dome through its centre. - Conceive the dome to be divided into numerous zones by planes parallel to the base, so that the portions of the spherical surfaces of these zones may be all equal ; and since the number of zones are supposed to be very great, the spheric surfaces will be very narrow, and consequently will not differ much from the surfaces of the frustums of cones ; therefore upon this supposition the two edges of each board will be in concentric cireles.=.. Therefore from the centre p, draw oB, per- pendicular to the base ac, divide the qua- drant Bc into as many equal parts as the number of frustums to be covered, and let the. points . of - division -be e, f; 'g, k.; !, &e. - To form the edges of any board, let, for example, the place: -of «the. board be at a/. Produce g/, and the axis pB, to meet each other ing2 ;- from 2, as a centre, .with the di- stances 22, 2A, as radii, describe the ares gr, and A5, and so gArr will represent the board at this place, and this board will cover a zone of the dome passing through the points & and h. In the same manner every remaining board K 130 THE CARPENTERS' AND may be formed; but when the boards to be formed approach near to the bottom, it be- comes very inconvenient to find centres for their description. To supply another method to remedy this inconvenience is therefore de- sirable. For this purpose bisect A7 in p, iX in g, kl in r, &c. Join pa, ga, ra, &c. inter- secting the axis Bp in the points, v, x, z, &c. and parallel to ac draw pu, gw, ry, &c. also meeting the axis DB, in the points u, w, ap Then at any convenient place, No. 2, draw pp, and from u, near the middle of pp, make up, up, each equal to up No. 1. Draw uv, No. 2, perpendicular to pp, and make uv, No. 2, equal to uv, No.1; then by means of an angle moved upon pins in the points », p, describe the are pup. Through the points p, p, draw ox and mi, parts of each radius. Make vx and v1, as also P1, and pa, likewise pa and px, each equal to half the breadth of a board, and describe the con- centric ares cHt1 and KLM in the same manner. In No. 3, the extent is supposed to be too great for the length of a board. Draw qq, in which take any point, w, near the middle, and make wg, wg, each equal to wg, No. 1. In No.3, draw w e perpendicular to q¢, and make JOINERS' INSTRUCTOR. 181 wae equal to wa, No. 1. In No. 8, jbin qa. Bisect the angle twg@, by the straight line gs, and bisect g @ in r, and draw rs perpendicular to ga. Then through the three points g, 8, a, describe an are of a circle with the angle. Then make go, qP, XN, xq as before, and describe the board norgq. No. 4, shows how a shorter board still may be described. NICHES Are hollows sunk into a wall for the reception of statues, having their bottoms planned ac- cording to any segment of a circle or ellipse, or an ellipse, and their tops terminating or formed into a kind of canopy. Niches are sometimes made square, but are then entirely destitute of elegance or beauty. Let us suppose, in the first place, the plan of a niche to be in the form of the segment of acircle, and its head a semicircle to describe the ribs for its top. Let a, B, g, Fig. 1, Plate 21, represent the sill, and a, b, ¢, d, e, f, the front ribs ; then it is evi- dent that the head of the niche forms one half of the segment of a sphere, the base of which portion is the semicircle a b c d ef. Hence then any person who is conversant K 2 182 THE CARPENTERS AND with the elementary principles of spheriecs may easily perceive that each of the required ribs of the niche will be of the same length, and possess the same degree of curvature as aB, the one half of the sill of the niche; therefore by it the several ribs can be readily obtained. - We deduce therefore from the preceding, that though the sill of the niche and its front rib also should form a semicircle, yet ithe ribs could have been obtained in the 'same manner by merely applying one half of the sill. This might have been otherwise accom- plished, for the very same form of a niche, as in Fig. 2; the principle for spherical headed niches is, that whatever be the position of the ribs, they will always be portions of circles. In Fig. 2, since the head is a semicircle, the ribs will be portions of semicircles; but here the ribs are supposed to stand in parallel planes perpendicular to the great circle, which is the base of the spherical head ; therefore the ribs will be of different lengths ; these are shown described in plans on the plan; however, as the two halves are the same, it will be only neces- sary to show one of the halves. The principle of sphericity being kept in JOINERS' INSTRUCTOR. 133 mind, the following problem will be easily un- derstood. Lastly, having given both the plan and ele. vation of a niche, in the form of the segment of a circle, less than a semicircle, dissimilar, having their chords equal, but their versed sines unequal. In order to accomplish this with the least trouble to the workman, we shall suppose all the ribs to be in vertical planes, making equal angles with each other ; and therefore, though of different lengths, they will be all portions of great circles. From what has now been ob- served, a bare inspection of the lines in Fig. 3 is sufficient for an intelligent person or an ex- perienced workman ; but for those who are not yet instructed, in order to awaken them to the principle, we shall proceed to a farther ex- planatidn. Let the arc Boo be the plan of the spherical surface,; and let this are be continued forward until it become a semicircle, of which ar is the diameter. Let tuy be the segment which ter- minates the under edge of the front rib, tv being the chord, and au its versed sine; pro- duce qu to r, and find the centre r of the are Tuy; bisect ar by a perpendicular or, and 134 THE CARPENTERS AND make or equal to the cosine ar, and join ar and Ep; with the radius ar or Er describe the are AFE, which will be the curve of the under edge of all the ribs, and when the ribs are arranged, these under edges will be in the spherical surface. The portions to be used will be found in all the ribs in the same manner; therefore if the principle is shown for any par- ticular rib, it will be sufficient for the whole for this purpose. Let cn, be the plan of a rib; from o describe the are 1, meeting ar in 1; draw Ir perpendi- cular to ar, then ar is the portion of the rib over GH. BRIDGE (FROM PRICE'S CARPENTRY.) ¢ I have inserted a bridge that may be more acceptable than the foregoing ones, because it is adapted to public and private uses, by being so formed of small parts, that it may be carried to any assigned place, and there put together at a short notice. " This bridge u, Plate 22, I suppose to con- sist of two principal ribs, as i, k, made thus: the width of the place is spanned at once by an arch rising one sixth part of its extent ; its curve is divided into five parts, which I propose to be JOINERS' INSTRUCTOR. 135 of good seasoned Engilsh oak plank, of three inches thick, and twelve broad, their joint or meeting tends to the centre of the arch ; within this rib is another, cut out of plank as before, of three inches thick, and nine broad, in such sort as to break the joints of the other. In each of these ribs, are made four mortises, of four inches broad, and three high, and in the middle of the said nine inch plank, (these mor- tises are best set out with a templet, on which the said mortises have been truly divided and adjusted): lastly, put each principal rib up in its place, driving loose keys into some of the mortises, to hold the said two thicknesses together; while other help is ready to drive in the joists, which have a shoulder inward, and a mortise in them outward; through which keys being drove, keeps the whole together : on these joists lay your planks, gravel, &c. so is your bridge complete, and suitable to a river, &c. of thirty-six feet wide. « In case the river, &c. be forty or fifty feet wide, the stuff should be larger, and more par- ticularly framed ; as is shown in part of the plan enlarged, as 1; these planks ought to be four inches thick, and sixteen wide; and the inner ones that break the joints, four inches thick, 136 THE CARPENTERS' AND and twelve broad; in each of these are six mortises, four of which are four inches wide, and two high ; through these are drove keys, which keep the ribs the better together ; the other two mortises are six inches wide, and four high; into these are framed the joists, of six inches by twelve; the tenons of these joists are mortised to receive the posts, which serve as keys; as is shown in the section x, and the small keys are shown. as in 1; all which inspection will explain. That of m, is a method whereby to make a good butment in case the ground be not solid; and is by driving two piles per- pendicularly, and two sloping; the heads of both being cut off, so as to be embraced by the sill, or resting-plate; which will appear by the pricked lines drawn from the plan 1, and the letters of reference. " All that I conceive necessary to be said farther, is, that the whole being performed without iron, it is therefore capable of being painted on every part, by which means the timber may be preserved ; for though in some respects iron is indispensably necessary, yet, if in such cases where things are, or may be often moved, the iron will rust and scale, so as that the parts will become loose in process of JOINERS' INSTRUCTOR. 137 time; which, as I said before, if made of sound timber, will always keep tight and firm together. It may not be amiss to observe, that whereas some may imagine this arch of timber is liable to give way, when a weight comes on any par- ticular part, and rise where there is no weight, such objectors may be satisfied that no part can yield or give way, till the said six keys are broke short off at once, which no weight can possibly do." Mr. Price, speaking of the circular domes, observes, that " of what has been hitherto de- scribed, nothing appears so beautiful when done, as domes, or circular roofs; and as far as I can perceive, nothing has appeared so difficult in doing, therefore it will be proper to speak something of them." Plate.-** Let s represent a plan, in which let b, b, b, be the plate on the supposed wall ; and let c, c, c, be the kirb on which stands a lantern, or cupola; also let a, a, a, represent the principal ribs. « From the plan B, make the section a ; in which the kirb, or plate b should be in two thicknesses; as also that of c, by which it is made stronger; and indeed the principal ribs would be much better to be in two thicknesses. 158 THE CARPENTERS' AND The best timber for this use is English oak ; because abundance of that naturally grows crooked. As to the curve or sweep of this dome a, it is a semicircle; although in that point, every one may use his pleasure; and in it are described the purlins d, e, from which perpendiculars are dropped to the plan B; so that f is the mould the lower purlins are to be cut out by, before they are shaped or squared for use; and that of & is the mould for the upper purlins. I rather show it with purlins, because under this head may be shown the manner of framing circular roofs in form of a cone. "'To shape these purlins, observe in a, as at d and e, they are so squared, that the joints of the supposed small ribs are equal. Ob- . serve, as at c, the corners of the purlin, from which the perpendiculars ate let fall to the plan s. - So that your purlin being first cut out to the thickness required, as appears in e, and also to the sweep f; so that k is the mould for the bottom, and / the mould for the top; by which, and the lines for the cornets of the said purlin e, the same may be truly shaped and squared. «N. B. This particular ought to be well di- JOINERS' INSTRUCTOR. 139 gested, it being a principal observation in a cir- cular roof. «* And from the purlin d, in the section a, per- pendiculars are dropped to the plan s; and in which it appears that 2 is the mould for the top, and ? the mould for the bottom ; so may this be squared, which completes the performance. As to other particulars, due inspection will explain them. If any should say, a dome cannot be done so safe without a cavity as usual, let them view St. Stephen's, Walbrook, Stock's-market, built by that great architect, Sir Christopher MTroent' « N. B. In all roofs of a great extent, the wind is to be prepared against, as strictly as the weight of the materials which cover it, because it has so great a force in storms of wind and rain; that is, it acts with more violence than the materials do, they being, (what we may call) a steady pressure." to mil -r‘kk%f. ( oust a SECOND DIVISION. JOINERY, Wr have already defined to be, the art of working in wood, or of fitting up various pieces of timber to each other, for the purpose of ornamental appendages to certain parts of edi- fices, which are called by the French menwiserie, small work. ENUMERATION OF THE MOST USEFUL JOINERS' TOOLS, Figure 1. represents the Jack Plane. 4+ Trying Plane. &_ --------- Smoothing Plane. Plane Irons. wiz m Cross cut Saw. Compass Saw. Keyhole Saw. 3 4 5 --- _ 6, --- 7. 8 Square, 9 Gauge. Mortice chissel. 142. THE CARPENTERS' AND Figure 11. represents the Turn Screw. --- 12, Plough. --= 18, ----- Back Rebating Plane. 14, Rebating Plane. --» 15. Brad Awl. 16. Stock and Bit. 17. Side Hook. 18. Work Bench. ---=- 19, --- Tanor Saw. m- 20, _----- Sash Saw. --- 21], ~-----»---- Hammer. 22, Mallet. Besides these, Joiners make use of a variety of other tools, whose general forms are nearly similar to those exhibited in the Plate; and which consist of the Long Plane, Jointer, Com- pass Plane, Fork-staff Plane, Straight Block, Sinking rebatting Planes, Skew mouthed re- batting Planes, Square mouthed rebaiting Planes, Side rebatting Planes, Bed Planes of various sizes, Snipes' Bill, Hollows and Rounds, Moulding Planes of various kinds (which would be endless to enumerate), Centre Bit, Counter Sink, Rimers, Taper Shell Bit, Drawing Knife, Ripping Saw, Half Ripper, Hand Saw, Dovetail Saw, Mortice Gauge, Mitre Box, Shooting Block, Straight Edge, Winding Sticks, Mitre Square; and se- veral others, which are likewise in common use, both with the Carpenter and Joiner. JOINERS' INSTRUCTOR 148 STAIR CASES. Palladio, after observing that "great care ought to be taken in the placing of stair cases," so "that they may not obstruct other places, nor be obstructed by them," says that " three openings are required in stair cases; the first is the door through which one goes up to the stair case, which the less it is hid to them that enter into the house, so much the more it is to be commended. And it would please me much, if it was in a place, where before that one comes to it, the most beautiful part of the house was seen; because it makes the house (although it should be little) seem very large; but, how- ever, let it be manifest and easily found. « The second opening is the windows that are necessary to give light to the steps ; they ought to be in the middle, and high, that the light may be spread equally, every where alike. The third is the opening through which one enters into the floor above; this ought to lead us into ample, beautiful, and adorned places." Stair cases ought to be proportioned in width and commodiousness, to the dimensions and use of the building in which they may be placed. The height of a step ought not to exceed seven 144 THE CARPENTERS' AND inches, nor in any case should be less than four ; but six inches is a general height. The breadth of the steps should not be less than twelve inches, if it can possibly be avoided ; nor should they ever be more than eighteen ; and to render the ascent free from the interruption of persons descending, their length should not exceed twelve, nor be less than four, except in common and small buildings, whose area will not admit of a stair case of more than three feet. That the ascent may be both safe and agreeable, it is requisite also to introduce some convenient aperture for light, which ought to be as nearly opposite to the first entrance to the stairs as the nature of the building will permit. An equal distribution of light to each flight of stairs ought to be particularly regarded ; for which reason, the apertures or windows are commonly placed at the landings or half spaces ; though some- times the whole is lighted from a dome. Stair- cases are of various kinds; some wind round a newel in the middle, while the risers of the steps are straight, and sometimes curved; others are of a circular plan, but form a well in the centre. The same may be observed of those whose plans are elliptical; the most common, however, are those whose plans form a square or parallelogram. JOINERS' INSTRUCTOR. 145 The ancients entertained a singular notion, that the number of steps ought to be uneven, in order that, when the right foot was placed on the first stair in ascending, the ascent might terminate with the same foot. This was con- sidered as a favourable omen, on most occasions, and they imagined, that, when they entered a temple in this way, it produced greater and more sincere devotion. § Palladio, apparently actuated by this super- stitious motive, allows the stair case of a dwell- ing-house eleven or thirteen steps to each flight. When a stair case winds round a newel or co- lumn, whether its plan be circular or elliptical, the diameter is divided into three parts, two of which are set apart for the steps and one for the column. But in cireular or elliptical stair cases, which are open, or form a well in the middle, the diameter is divided into four equal parts ; two of which are assigned for the steps, and two for the well or void space in the centre. Modern stair cases, however, have often a kind of well of a mixed form ; straight on each side, and circular at the returns of each flight. The openings of these wells vary in the point of width, but seldom exceed eighteen or twenty inches. £. 146 THE CARPENTERS' &c. To most stair cases it is absolutely necessary, both for convenience and ornament, to affix hand-rails; these generally begin from the ground by a twisted scroll, which produces a very good effect. 147 JOINERY. on GEOMETRICAL STAIRS. A geometrical stair is that of which the steps have only one of their ends supported by a wall. Plate 24, Fig. 1, is a geometrical stair, with a semi-circular half space. Fig. 2, a geometrical stair with winders in the semi-cylindric part. Fig. 3, a geometrical stair, with winders in two quarters, separated by a flight of steps. Fig. 4, an elliptical stair case. The hand-rail of a geometrical stair is that which is used for preventing the passenger from falling over. The well-hole of a stair is an open space within the hand-rail ; through which any thing may de- scend from the top to the bottom. A stair is said to have one or more revolutions when the hand-rail is continued from the bottom, so as to meet a vertical straight line from the - 1.2 148 THE CARPENTERS' AND lowest point of the hand-rail, in one or more points. The riser of a step is the vertical part of that step connecting the horizontal parts for stepping upon. The tread of a step is the breadth of the horizontal board for stepping upon in the act of ascending or descending. To find the section of any prism, supposing three of its sides at least to be a plane surface, and the other sides or side either planes or a curved surface of any kind, and that two of the plane surfaces form the ends or base, and the third to connect the other two; so that the section shall pass through three given points on the surface. § Let as cp, Plate 25, Figs. 1 and 2, be the con- necting plane, and ars the base, and let a, £, B, be any three given points in the base ; but, for the sake of an easy description, we shall suppose the two extreme points a and ms of the are or curve, to be also in the chord line as. Draw an, E1, and sc, perpendicular to as. Make a » equal to the line ®, ® 1 equal to the line a, and s c equal to the line u. Join a ® and p 1, and produce them to meet each other in x. Join p c, and produce a B and » c to meet each other in t. "idoin'k Ly andin 's 1 take any point m. JOINERS' INSTRUCTOR. 1409 Draw »» perpendicular to sL, meeting ®L in » ; also draw mr perpendicular to cr. From t, with the distance u», describe an arc inter- secting mp in the point p, and join uP. To find any ordinate in the required section : take any point g, in the curve, and draw gr parallel to x1, meeting the chord line as in r; draw rs parallel to a p, meeting pc in the point s, and draw s¢ parallel to up. Make st equal to rq, then will ¢ be a point in the section re- quired ; a sufficient number of points being found in this manner, will complete the section cp tuvwc, as required. The hand-rail of a stair is a fence or guard for preventing persons from falling over the steps into the well-hole of the stair. To understand this subject, and to execute a hand-rail in the best manner, and in the most agreeable form, is reckoned a master-piece in joinery. From some convenient point c, Fig. 1, Plate 26, in the straight line K1, draw car, perpendicular to x1. Inca equaltothe radius of the plan of the well-hole, and draw da parallel to x1; from the point o with the distance oc, describe the semicircledca. Inommakethedistanceoe equal to one and three quarters of the radius. Join e@& 150 THE CARPENTERS AND and e d, and produce e d to meet x 1 in n, and ea to meet x1 in r. Make ux and ru each equal to the breadth of a step, and draw KR and r J perpendicular to x1. Make xr equal to the height of eleven steps, that is, equal to the height of nine winders and two flyers, one at each end of the winders. Make xe equal to the height of ten steps, and rJ equal to the height of one step; then ar as well as r 3 will be the height of a step. Join J 1, 1 P, and p r. Then rr;1 will be the regulating line of the falling mould of the hand-rail. Draw the line tx parallel to r 3, so as to be distant from rJ about three inches ; then r f u 1 will be the middle line of the falling mould ; set off one inch on each side of this line and complete the falling mould by reducing the angles at the junction of the winders, and the straight parts, so as to be everywhere about two inches broad. Let f a gh ik be half the plan of the semi- circular part of the rail, including a small por- tion of the straight part of the rail, which we shall suppose to be about three inches. The radius of on the plan, Fig. 2, is equal to od or 0 a, Fig. 1. Make r 1, Fig. 1, equal to the straight part of the plan ; then c 1, Fig. 1, will be equal to JOINERS' INSTRUCTOR. 3 151 the stretch out of fa g / i, the convex side of the plan Fig. 2. Draw the chord line c & to pass through the vertices c f, at the ends of the concave side. In «Fig. 1, bisect c 1 at e, and draw e v and 1 w perpendicular to x L, meeting the upper edge of the falling mould in the points v and w ; let the perpendicular c m intersect the upper edge of the falling mould in the point v. In Fig. 2, set off the arc [g, equal to the de- velopement of c a, Fig. 1; then the arcs f g and f g i, will be equal to the straight lines c c and c 1, No. 1. Draw c u, g v, and i w, per- pendicular to c %, and make c u, g v, and iw, respectively equal to c v, c v, and 1 w, Fig. 1. Join c & and xv, and produce them to meet in m; join x a, and produce u w and ¢ ? to meet in n. Join m », and produce m n and c X to meet in p. Find p q, and trace the concave and convex sides of the figure u y d z a, ex- actly in the same manner as in finding the sec- tion of a prism, and the figure « y d & a will be the face mould required. To describe the scroll of a hand rail. From any convenient point, 0, Fig. 1, Plate 27, as a centre describe a circle, ¢ f # A, and describe 152 THE CARPENTERS' AND the square a b c d, of which the centre is the same, point 0, and of which the sides a b, b c, c d, d a, are each one third of the diameter of the circle. Divide each side of the square into six equal parts, (see also fig. 2, drawn at large,) and through the points of division draw lines parallel to the sides ; taking the distance 0, 1, equal to the side of one of the lesser squares ; the distance from 1 to 2, equal in length to twice the side of one of the little squares ; the distance from 2 to 3, equal to three times the side of one of the little squares, and so on increasing each line by the side of one of the lesser squares, for the other distances, from 8 to 4, from 4 to 5, and from 5 to 6, in such a manner, that the distance from 1 to 2, from 2 to 3, from 3 to 4, from 4 to 5, from 5 to 6, may be respectively perpendicular to the distances from 0 to 1, from 1 to 2, from 2 to 3, from 3 to 4, and from 4 to 5. See the centres constructed to a larger scale, Fig. 2. Let the distance between 1 and 0 be pro- duced to meet the circle f. Then from the centre 1, with the radius i/; describe the quad- rant f i; from the centre 2, with the distance 2 i, describe the quadrant i /; from 8, with the distance 3 X, describe the quadrant 47; from 4, with the distance 4 Z, describe the quadrant l m ; JOINERS' INSTRUCTOR. 153 from 5, with the distance 5 m, describe the quad- rant m n, and lastly from 6, with the radius 6 n, describe the quadrant n p. In the radius 6 p, make p q equal to the breadth of the hand-rail, say two inches ; then, from the centre 6, describe the quadrant q r, meeting the radius 5 n in r; from the centre 5, with the radius 5 r, describe the are r s, which will complete the scroll. The shank of the scroll is drawn from the points p and q, parallel to the radius 6 n. Curtail step is the lowest step of the stair, one end being formed into a spiral, agreeable to the scroll of the hand rail. Let the balusters be so placed, that the dis- tances between every two nearest may be all equal, that two balusters may stand upon every step, and that the front of a baluster may be in the plane of every riser. Likewise that the middle of every baluster, may be in the middle of the breadth of the hand-rail, and that the middle of the balusters which support the scroll, may also be in the middle of the breadth of the scroll, and that the outer edges of these ba- lusters, may be all at the same distance from the curved edge of the curtail step, and lastly, that none of the distances between those which support the scroll, be greater than those which rise from the flyers to support the rail, and that 154 THE CARPENTERS AND the distances of the balusters which rise from the curtail step to support the scroll, may be less and less, as the radius of curvature for de- scribing the plan of the scroll is less. Hence the plan of the rail will be in the middle of the plan of the curtail step, and be- cause the projection of the nosing from the out- side of the baluster, is greater than the pro- jection of the outside of the rail, the breadth of the scroll of the hand-rail, will be less than the breadth of the scroll worked on the end of the curtail step. To describe the curtail step. Let the balusters be properly placed in the order now described, and let k be the angle of the baluster in the front of the second riser, and let n o be the breadth of a baluster in the breadth of the rail. Make o n each equal to the projection of the nosing, and through the point » describe the spiral o ® r G 1, to follow the nearest spiral in the place of the rail ; also through the point r describe the spiral P M L, so as to be every where equidistant from that which forms the other edge of the plan of the scroll. These two last described spirals may be drawn from the same centres, as those from which the edges of the scroll were drawn. To JOINERS' INSTRUCTOR. 155 render this description evident to workmen, B c is a section of the nosing of the front of the cur- tail step, and c n is a plan of the nosing attached to the straight part of the curtail step. - Likewise 1 x is the plan of the nosing which projects from the strong board. To describe the face mould for the wreathed part of the scroll. Draw the straight line u v, parallel to the shank of the scroll, to touch the interior part nearest the centre, and let it meet the outside of the scroll in v. Make v u equal to the breadth of one of the flyers, and draw x w, per- pendicular to v w, and make w w equal to the height of a step, and join v w. - From the curved edge of the scroll, draw ordinates perpendicular to u v,. Transfer the several distances from w, where the ordinate or ordinates produced, in- tersect v w, to the line v w, Fig. 5, and draw the ordinates in Fig. 5, and make them respectively equal to the ordinates both of the outside and inside curves of Fig. 5, and the ends, then the shadowed figure of Fig. 5, will be the face mould. To find the fulling moulds. Let a be that point in the outside spiral 156 ~THE CARPENTERS' AND which separates the wreathed part of the scroll from the level next to the centre. In Fig. 4, describe the right-angled triangle aA Cc B, identical to the right-angled triangle u vw, Fig. 1, a B being equal to the height of a step. Produce a c to E, and make » ® equal to the developement of the next line, t p v » q. Divide a s into six equal parts, and let » be the first point of division. Draw » r parallel to a E, and E r, parallel to a p, intersecting B c in c. Describe a parabola B x r, to touch G F and a B at B and r; this may be conveniently formed by the intersections of lines ; also draw another curve, 1 K 1, under the other, and every where 2 inches distant from it, which distance must be the depth of the rail, and the falling mould, Fig. 4, for the convex side will be com- plete. Fig. 3, inside falling mould. The method of describing hand rails as now practised, is the sole invention of Mr. P. Nichol- son ; before his time, several unsuccessful at- tempts had been made, but, for want of true geometrical principles, they were found not to succeed. Of doors and windows. In forming the apertures of doors, whether arched or quadrangular, the height should, in JOINERS' INSTRUCTOR. 157 # general, be about their breadth, or a little more. It was necessity, most probably, that gave birth to this proportion, which habit has confirmed and rendered absolute. The disposition of doors and windows, and assigning to them their proper dimensions, according to the purposes for which they are intended, are not the busi- ness of the Joiner, but of the Architect; for which reason we shall here advert only to the com- mon method of decorating doors and windows, the former of which have an architrave, around the sides and top of the aperture, with a regular frieze and cornice upon it. In some cases, the cornice is supported by a console, on each side of the door, and sometimes, besides an architrave, the aperture is adorned with columns, pilasters, &e. which support a regular entablature, with a pediment, or with some other termination, either in architecture or sculpture. Front doors, in- tended to be ornamented with any of the orders, should not be less than three feet six inches wide ; the height should be twice the width and one sixth part more, which might also be the height of the column; the abacus may be then taken out of that dimension, in order to separate the door from the fan light. The windows of the principal floor are generally most enriched. The simplest method of adorning them is, with an ar- chitrave surrounding the aperture, and crowned 158 THE CARPENTERS' AND with a frieze or cornice. The windows of the ground floor are sometimes left entirely destitute of any ornament ; at other times are surrounded with rustics, or a regular architrave having a frieze or cornice. The windows of the second floor, have generally an architrave carried en- tirely round the aperture ; and the same method is adopted in adorning attic and Mezzanine windows ; but the two latter seldom possess either frieze or cornice; while the windows of the second floor are often crowned with both. With regard to the hanging of doors, shutters, or flaps with hinges, care should al ways be taken to place the centre of the hinge in the middle of the joint; but, as in many cases there is a ne- cessity for throwing back a flap to some distance from the joint, the distance between the joint and the intended point must be divided into two equal parts, which point of division will de- note the situation of the centre of the hinge. Sometimes doors are required to be hung in such a manner, that when folded back, they shall be at a certain distance from each other, as is frequently desirable in Churches and Chapels ; this may be easily effected by hinges, with knees projecting to half that distance. In all elegant rooms, it is necessary to con- trive, that the doors when opened, should pass clear over the carpet ; now, it is evident, that JOINERS' INSTRUCTOR. 159 this cannot be the case, if the jamb on which the door hangs, is truly perpendicular, and the bottom of the door is close to the floor, as the bottom of doors commonly are. An inconsi- derate observer might recommend a part of the bottom of the door to be.cut off, in order to permit its free passage over the carpet, but still, when the door is shut, an open space will in- tervene between it and the floor, unless, as in some cases, the carpet is continued through the opening of the door to an adjoining passage or room. - When this is not the case, the room will be rendered cold and uncomfortable ; and the necessity of contriving some method to re- medy the defect, becomes immediately obvious. This remedy may always be found by hanging the door with rising hinges, constructed for. the purpose, with a spiral groove, which , winding round the knuckle as the door opens, gives it a free passage over the carpet. - Hinges, however, thus constructed, require that the door should be bevelled at the top next to the ledge or door catch, in proportion to their rise at one quarter of their revolution. This is an effectual mode of enabling a door to clear the carpet; but a combination of the following modes recommended by Nichol- son are less objectionable. Raise the floor under the door, as much as the thickness of the 160 THE CARPENTERS' AND carpet might require. - Make the knuckle of the bottom hinge project an eighth of an inch beyond the perpendicular direction of the top hinge,-fix the jamb to which the door might be hung about the eighth of an inch out of the perpendicular; and place a common butt hinge at the top, and one with a projecting knee at the bottom. The introduction of rising hinges requires a notch to be cut out of the door where the hinged edge and the top edge meet, and since this can- not be concealed on both sides of the door, this method is considered as defective ; besides the hinges are liable to go out of order. A jib door is one, which is intended to be con- cealed in the side of a room, and therefore par- takes of the same surface and finish as the wall in which it is inserted. - Therefore, the face of a jib door, and the face of the wall from which the aperture is made to receive the jib door, are in the same surface. Fig. 1, Plate 28, exhibits the elevation of a jib door, having the same moulding as the base and surbase of the room. _ a is a section of the base moulding to a large scale, and B that of the sur- base to the same scale. A portion of the plan of the door and of the surbase, as also a part of the jamb, is shown at Fig. 2. In order to make the most complete job, the JOINERS' INSTRUCTOR. 161 door should be hung with centres, and not with hinges, and the centres should be inserted within the solid of the jamb lining. Let o be the centre of the hinge o », a portion of the inner edge of the surbase in contact with the door, and c B a portion of the outer edge ; let c be some point on the outside of the perpendicular 0 s opposite the jamb. Join 0 c, and draw c » perpendicular to o c; then c p will be the plan of the joint, in order that it may be a vertical plane. Though there is no absolute necessity, it is usual to make the distance s c equal to s o. The object of this is, to make the distance o p the least pos- sible, so that the strength of the jamb may not be impaired by cutting away more wood than is necessary to effect the purpose. To clear the jamb lining of a door, in the act of opening and shutting. Let o, Plate 28, be the centre of the hinge, and a s the opposite edge of the door, the point A being the inner edge, which comes in contact with the side of the rabbet, in the jamb that stops the door. Draw o a and a s perpendicular to 0 a, then a B is the bottom of the rabbet. To make folding doors open and shut freely at the meeting joint. Let a B, a b, Figs. 2 and 3, Plate 29, be the meet- M 162 THE CARPENTERS' AND ing joint 0, the centre of the hinge. The part a B of the joint, intersecting thé inside of the door at A, and the part a b the outside of the door at b. These two parts a B, «a b, are connected together by the part a B, parallel to the front or back of the door. Join o a, Figs. 2 and 3, and draw a B per- pendicular to o a, also join o a, and draw a b perpendicular to 0 a. The joint may also be formed as in No. 4, by describing the arcs de and f g from the cen- tre 0. In the execution of shop fronts, the cornices are frequently formed in segments of circles, which have their sagittas very small, compared to the chords of the ares. Such is the design Fig. 4, Plate S0, here presented, No. 1 being the plan exhibiting the circular cornice, and No. 2 the elevation. To draw the circular arcs. Let a c be the chord, and B p the sagitta. Make each of the angles s a r and B c G, equal to the angle » B a or D B c, and produce E B and F a to u and 1; set off the projections of the cornice in the lines B x and a 1, describe the several arcs, and these arcs will be all concentric. JOINERS' INSTRUCTOR. 163 A pediment is a triangular cornice, of which one of the sides is horizontal, and the other two inclined and of equal length, such as Fig. 1, Plate 31, or a pediment is the segment of a circle, Fig. 2, with a circular and straight cor- nice. s In consideration that no pediment can be executed without two kinds of cornices, except it is kneed at the bottom, or springing, which is reckoned a kind of defect ; therefore to give each of the cymas such a shape or curve as shall agree in their mitre, we must first describe the level cornice a 4 c d e, Figs. 3 and 4, and through the points a, b, c, d, ¢, draw lines ag, bh, ci, dk, el, agreeable to the rake or angle which one of the inclined sides makes with the horizon, or from a centre agreeable to the are re- quired to form the segment. Draw f o' perpendicular to a g, and draw o' p perpendicular to f o'. From the points a, 6, c, d, e, draw lines perpendicular to a g, to meet the line » », and from o', in o' p, set off distances equal to the distances from m, in the line m », where perpendiculars meet it, and draw perpen- diculars from the points in 0' p, and let these per- pendiculars intersect the lines ag, b/, ci, &e. in the points f; 0, p, g, r; then the curve £ 0 p g r is mM 2 164 THE CARPENTERS' AND the section of the inclined moulding. In like manner draw & s, perpendicular to a q : from s, in the line s 7, set off the several distances from », in the line #» », and draw lines from the points of di- vision parallel to s g, to meet the lines parallel to a g, in the points 2, A, i, k, I, then g A i%Akl is the section of a moulding, returning parallel to othe monlding: of o which ia ibe d" e®is 'the section. Figs. 5 and 6 are described in a similar man- ner, as is evident. Fig. 7 shows the manner of drawing angle bars for shop fronts, supposing the angle bars to be the same thickness as the intermediate ver- tical and horizontal bars. These bars being duly placed according to the plan of the window t u v, through c draw c a, meeting r s in a, and the side of the angle bartin 5: from''c, wth the distance & a, describe an are intersect- ing ap ag. in «d. , Draw é i, f k,.oparallel "to ut, meeting r s in i, %, and draw e g, f A, parallel to ¢ d, meeting p q in g and A.- Make i I, km, &c. equal to e g, # /, &c. finding a sufficient number of points ; in the same manner draw the curves and complete the angle bar as required. Let it be required to bend a cornice round the external surface, or round the internal sur- JOINERS' INSTRUCTOR. 105 face of a cylinder, so that the edges of the mouldings may be in planes perpendicular to the axis of the cylinder. j Let a b c d, Figs. 1 and 2, Plate 82, be a sec- tion of the cornice, placed to the spring, and let e f be the axis of the cylinder. Produce a d to meet the axis of the cylinder in ¢; from e with the radii e d, e a, describe the ares a f g, and d hi; these two ares will be the bounding of the pieces of wood, which is to form the cornice upon the spring. This depends upon the covering of the frus- tum of a cone, since the general surface of the mouldings is that of a cone frustum. Laying of floors. The chief excellence of a floor consists in its being perfectly level, and no higher in any one part than another; but experience teaches, that this desirable object must frequently be sacrificed to considerations of convenience. The mode recommended for hanging doors furnishes a sufficient proof of the correctness of this observation. A frequent defect is observ- able in floors, (the origin of which may be at- tributed to the carpenter) arising from their sinking in the middle, or in those parts that are unsupported ; which circumstance demonstrates 166 THE CARPENTERS' AND how necessary it is that every floor should pos- sess a certain degree of camber, in order that when it settles or the moisture has exhaled, it may be as near to a plane as possible.-When the joists are depressed in the middle, it will be proper to fur them up; on the contrary, when they project in the middle, they ought to be reduced by the adze ; the former, however, is most generally the case, The joints of flooring boards are either square, plowed and tongued, rebatted or dowelled ; the boards being nailed on each edge, when the joints are square, or plowed and tongued ; but when the dowel work is executed in a proper manner, the outer edge only is nailed, and this is effected by driving the brad, in an oblique direction through the edge, without suffering it to pass through the upper surface of the board, - The headings are in some cases square ; in others splayed or plowed, and tongued. Grounds are pieces of wood fixed to the wall around doors, windows, &e. for the purpose of re- ceiving the architraves ; they are also fixed in different positions in rooms, in order to receive the various kinds of mouldings required in or- namenting the same such as bases, surbases, JoINERS' INSTRUCTOR. 167 chimney-pieces, &e. &e. Now as nothing has a more disagreeable effect to the eye than the untrue appearance of any work of this kind, it is absolutely necessary, if the workman would avoid deformity, to fix all grounds in a true vertical position, both on their edge and face, and in a firm and solid manner to the wall. Of gluing up the base, shaft, and capital of columns. To each order belongs a particular kind of base, and the first operation required is that of gluing up the base. The shaft of a column should be glued up in eight or more staves, according to its intended dimensions; but care should be always taken to have the joint in the middle of a fillet, and not in a flute, which would impair its strength very much. Figs. 8 and 4 show a plan of the upper and lower ends, or the horizontal section at top and bottom. If eight pieces are sufficient to form the column, let an octagon be described round the ends, and let lines be drawn from each angle of the octagon to the centre ; when the bevel of the edges of the staves will be given for the joints, which must be quite straight from top to bottom, though the staves be narrower at the 168 THE CARPENTERS' AND top, as shown in fig. 3. These staves must be of sufficient thickness, because their outsides have to assume a curvature proportioned to the swell of the column by means of a diminishing rule ; next glue the pieces together one after the other as the glue dries ; block them well at the corners in the inside, which will greatly strengthen the joints ; and proceed in this manner to the last stave ; but all the blocks must be glued on and dried, before the last stave can be fastened. Pieces, however, may be glued quite across for the last stave, and fixed to the inside of the two adjoining staves, or they may be fixed by screws to each stave, in which case the under side of the last stave must be planed so as to rub well on the cross pieces. % When the stave is put in, and glued upon the cross pieces, it may be driven tight home, like a wedge, and the whole will be firm and sub- stantial throughout ; great care, nevertheless, ought to be taken as to preparing the staves and blocks out of wood thoroughly dry, because, after the lapse of some time, if the wood be moist, the column will be in danger of falling to pieces at the joints. _ It will be ne- cessary also to make each piece according to - the plan, for a trifling error in any one piece JOINERS' INSTRUCTOR. 169 will make a very material difference in the column after gluing. When the glue used in combining the column is dry, the angles must be regularly worked off all round : and the column will then have double the number of sides, or cants, bearing a proportionable regu- larity to each other. Proceed in a similar manner to work off the angles as before, so as to make the sides, or cant of the column, quite regular. - Lastly, let a plane be formed, in order to fit the curve of the column at the bottom, or render it rather flatter, then round off all the angles, until the surface of the column is per- fectly smooth. - One thing to be observed, with respect to the moulds employed in jointing the staves together, is, that they cannot be con- sidered as exactly true when applied in a di- rection perpendicular to the joint. The most correct mode is that made use of in finding the backing of a hip rafter, which has been al- ready noticed ; but this exactness, nevertheless, is not always attended to, in consequence of the difficulty of discerning the deviation in some in- stances. - When the column is quite finished, it should be well painted, by way of protection from the effects of the weather. Sometimes columns are glued up in two halves, in which cases those two halves are glued 170 THE CARPENTERS' AND together, and the blockings are introduced a considerable way by hand ; but if the column be too long, a rod of sufficient length may be used. Both these methods have some incon- veniences, which cannot be avoided ; by the former method, the last joints cannot be rubbed together from the obstacle presented by the ta- pering of the stave, but if this be glued quickly it will be pretty sound ; by the latter method there will be an uncertainty of the blockings being sound. - In all cases, however, care should be taken to place the grain of the blocking piece in the same direction as the grain of the column, so as that they may both expand and shrink alike, when affected by the weather. Of gluing up the Ionic capital. Fig. 5 represents the mode practised in glu- ing up a capital. The parts denoted by B B, &c. are triangular blocks of wood, glued upon the front, in order to complete the angular square : upon them the pieces a, a, A, &c. are glued, and this is considered the best method of gluing up the capital. - Another method is, to glue the tri- angular blocks c, c, at the angle of the abacus, then the four sides of the abacus as D, ®, r, may be made of one entire length, and mitred at the horns, or they may have a joint in the middle of JOINERS' INSTRUCTOR. 171 the abacus, where the rose is placed, as the work- man shall think fit : this method will do either for a column, or a pilaster. Fig 6, exhibits the method of gluing up the base of a column. The mode of mitring the bottom course to- gether, which must be effected on a perfectly flat board, and by fitting all the joints as close as possible :-When the course has been well glued together, and secured on the inside with blocks - at the several angles, the top of the course must be planed quite smooth and out of winding ; after this, the next course must be glued on, and the joint must be broken in the middle of the under course, (as shown by the dotted lines in the plate) by which means as many courses can be glued down as may be required. - When the whole is thoroughly dry, the operations of the turner may commence. THIRD DIVISION. STRENGTH OF MATERIALS. StrENGTH denotes, that particular force or power, with which any mass or body resists a breach or change in its state, endeavoured to be produced by a stroke or pressure. 17¢ THE CARPENTERS' AND Stress or Strain, may be defined as the force exerted upon a body in order to break it. Thus, every part of a pillar is equally strained by the load which it sustains ; and it is evident that no structure can be considered fit for its purpose, unless the strength prevailing in all its parts be at least equal to the stress laid on, or the strain excited in those parts : from hence may be perceived the necessity of becoming ac- quainted with the nature of the resistance made by various bodies, since it will teach us to pro- portion the materials in a machine or structure of any kind, so as that there shall be neither a surplus nor a deficiency. It has been justly observed by an excellent writer, * that in a nation so eminent as this for invention and ingenuity in all species of manu- factures, and in particular, so distinguished for its improvements in machinery of every kind, it is somewhat singular that no writer has treated it in the detail, which its importance and dif- ficulty demands. The man of science who visits our great manufactories, is delighted with the ingenuity which he observes in every part, the innumerable inventions which come from indi- vidual artisans, and the determined purpose of improvement and refinement which he sees in every workshop. - Every cotton-mill appears an JOINERS' INSTRUCTOR. 1783 academy of mechanical science ; and mecha- nical invention is spreading from these fountains over the whole kingdom. - But the philosopher is mortified to see this ardent spirit so cramped by ignorance of principle; and many of these original and brilliant thoughts, obscured and clogged with needless and even hurtful ad- ditions, and a complication of machinery which checks improvement by its appearance of inge- nuity. - There is nothing in which this want of scientific education, this ignorance of principle, is so frequently observed, as in the injudicious proportion of the parts of machines and other mechanical structures ; proportions and forms of parts in which the strength of position are nowise regulated by the strains to which they are exposed, and where repeated failures have been the only lessons." " It cannot be otherwise," the same author continues : " We have no means of instruction, except two very short and abstracted treatises of the late Mr. Emerson, on the strength of ma- terials. We do not recollect a performance in our language from which our artists can get in- formation. - Treatises written expressly on dif- ferent branches of mechanical arts, are totally silent on this, which is the basis and only prin- ciple of their performances. - Who would ima- gine that Price's British Carpenter, the work of 194 THE CARPENTERS' AND the first reputation in this country, and of which the sole aim is to teach the carpenter to erect solid and durable structures, does not contain one proposition, nor one reason, by which one form of a thing can be shown to be stronger or weaker than another? We doubt very much if one carpenter in a hundred can give a reason to convince his own mind, that a joist is stronger when laid on its edge, than when laid on its broad side. - We speak in this strong manner, in hopes of exciting some man of science to publish a sys. tem of instruction on this subject." The strength of materials arises immediately, or ultimately, from the attraction of cohesion which is observable in almost every natural object, and is, in reality, that which holds their component parts together. Now as cohesion admits of various modifica- tions, in its different appearances of perfect softness, elasticity, hardness, &c. and has a great influence on the strength of bodies, it will by no means admit of the application of mathematical calculations, with that precision and certain success which are desirable in a point of so much importance. The texture of materials is a subject of no less importance, and experiments in this re- spect have been made by Couplet, De la Hire, ~ Pitot, and Du Hamel : but the same remark is JOINERS' INSTRUCTOR. 175 applicable to them as to the experiments on the cohesion of bodies, and consequently being so limited, that information is not to be obtained from them which we could wish. Buffon, how- ever, carried on some experiments on a more extensive and proportionately useful scale, and from him only are to be obtained those measures which may be relied on with safety and success. Our countrymen, Emerson and Banks, have, it is true, made several experiments on the strength of bodies, but their researches too have been so limited and imperfect, as to pre- clude the student from placing any particular confidence in their results. It may not be irrelevant to observe here, that experiments may be considered as nothing less than a narration of certain detached facts, if some general principles are not established, by which we can generalise their results. Some idea, for instance, may be entertained of that medium or cause, by the intervention of which an external force applied to one part of a lever, joist, or pillar, occasions a strain on a distant part. - This can be nothing more or less than the cohesion existing between the parts which are brought into action, or as we more shortly express it, excited. In order properly to comprehend the nature of cohesion, it will be necessary to take a view of its laws, or rather 176 THE CARPENTERS' AND of those general facts which are observable in its operations. In doing this, however, it will be sufficient to notice such general laws only as seem to present the most immediate inform- ation of the circumstances required to be attended to by mechanics in general, if they would wish to unite strength with simplicity and economy;, in their several constructions. 1st. - We have presumptive evidence to prove, that all bodies are elastic in a certain degree, that is when their form or bulk is changed by certain moderate compressions, it requires the continuance of the force producing the change, in order to continue the body in its altered state, and when the compressing force is removed, the body recovers its original form and tension. 2d. That whatever may be the situation of the particles composing a body, with respect to each other when in a state of quiescence, they are kept in their respective places by the balance of opposing forces. 3d. - It is an established matter of fact, that every body has some degree of compressibility, as well as of dilatability ; and when the changes produced in its dimensions are so moderate, that the body completely recovers its original form on the cessation of the changing force, the ex- tensions or compressions bear a sensible pro- portion to the extending, or compressing forces ; JOINERS' INSTRUCTOR. 1957 and therefore, the connecting forces are propor- tioned to the distance at which the particles are diverted, or separated, from their usual state of quiescence. - 4th. -It is universally observable, that when the dilatations have proceeded to a certain length, a less addition of force is afterwards sufficient to increase the dilatation in the same degree. For instance, when a pillar of wood is overloaded, it swells out, and small crevices appear in the direction of the fibres. After this, it will not bear half of the previous load. 5th. That the forces connecting the par- ticles composing tangible or solid bodies, are altered by a variation of distance, not only in degree, but also in kind. Having now enumerated the principal modes in which cohesion confers strength on solid bodies, we proceed to consider the strains to which this strength may be opposed. These strains are four in number, viz.- 1st. A piece of matter may be torn asunder, as is the case with ropes, king posts, tic beams, stretchers, &c. &c. 2d. It may be crushed, as is the case with pillars, truss beams, &c. &c. 3d. It may be broken across, as happens to a joist or lever of any kind, or 178 THE CARPENTERS' AND 4th. - It may be wrenched or twisted, as is the case with the axle of a wheel, the nail of a pressp &orike, With respect to the first strain, it may be ob- served, that it is the simplest of all strains, and that the others are but modifications of it ; it being directly opposed to the force of cohesion, without being much influenced, except in a slight degree in its action, by any particular circumstances. When a prismatic; or cylin- drical body of considerable length, such as a rope, or a rod of wood, or metal, has any force exerted on one of its ends, it will naturally be resisted by the other, from the effect or opera- tion of cohesion. When this body is fastened at one end, we may conceive all its parts to be in a similar state of tension, since all experi- ments on natural bodies concur to prove, that the forces which connect their particles, in any way whatever, are equal and opposite. Since all parts are thus equally stretched, it follows, that the strain in any transverse section, as well as in every point of that section, is the same. - If then, the body be of any homogene- ous texture, the cohesion of the parts is equable, and from every part being equally stretched, the particles are diverted or separated from their usual state of quiescence to equal distances : JOINERS' INSTRUCTOR. 179 of course the connecting powers of cohesion thus excited, and now exerted in opposition to the straining force, are also equal. - It is evident therefore that this external force may be in- creased by degrees, so as gradually to separate the parts composing the body more and more from each other, and that the connecting forces of cohesion will bear a relative proportion to the increase of distance, till finally some par- ticles weaken, then the rest are overcome by the pressure or tension, when a fracture ensues, and the body itself is soon crushed, or broken in all its parts. If the external force be insuf- ficient to produce any permanent change on the body, and that body recovers its former dimen- sions when the operating force is withdrawn, it is clear that this strain may be repeated whenever desired, and that the body which has withstood it once will always be equal to the task of withstanding it. .This circumstance should not only be attended to in constructions of every kind, but kept constantly in view in every investigation of the subject. Bodies of a fibrous texture exhibit very great varieties in their modes of cohesion. In some, the fibres have no lateral connecting force, as in the case of a rope. - The only way in which all x £ 180 . THE CARPENTERS' AND the fibres composing a piece of matter can be made to unite their strength, is by twisting them together, which has the effect of bending each to each, so fast, that any one of them will rather break than be separated in a perfect state from the remainder. - In timber, the fibres are held together by some glutinous cement, which is seldom, however, as strong as the fibre ; and for this reason timber is much easier pulled asunder when operated on in a direction trans- verse to the fibres ; but, nevertheless, there is every possible variety in this particular. In stretching and breaking fibrous bodies, though the visible extension is frequently very considerable, it does not solely arise from the increasing the distance of the particles com- posing the 'cohering fibre, but is chiefly oc- casioned by drawing the crooked fibre straight. In this respect a great diversity prevails, as we'll as in the powers required to withstand a strain. - In some woods, such as fir, the fibres on which the strength most depends are very straight; and woods of this nature, it should be remarked, are generally very elastic, and break abruptly when overstrained ; others, as oak, have their resisting fibres very crooked, and stretch very sensibly when subjected to a strain. - These JOINERS' INSTRUCTOR. 181 kinds of woods do not break so suddenly, but ex- hibit visible signs of a derangement of texture. The absolute attraction of cohesion,. or strength, is proportioned to the area of the section which stands at right angles with the extending force. - This will be readily admitted in the case of fibrous bodies, if we suppose the fibres composing them to be equally strong and dense, and to be disposed similarly through the whole section ; there is a necessity for admit- ting this, or else the diversity must be stated and the cohesion must be measured accord- ingly. ‘ The following observation may be admitted as a general proposition in this respect ; the ab- solute strength in any part of a body, which en- ables it to resist being pulled asunder, or the force which must be employed to tear it asunder in that part, bears a proportion to the area of the section which stands at right angles with the extending force. Hence, then, we may deduce that cylindrical and prismatic rods are equally strong in every part, and will break alike in any part, and that bodies formed into unequal sections will always break in the most slender part. - The length of the prism or cylinder produces no effect on the 182 THE CARPENTERS' AND strength ; and the vulgar notion that a long rope may be broken more easily than a short one, is altogether absurd.-It may be further observed, that the absolute strengths of bodies whose sections are similar to each other, bear a relative proportion to the squares of their di- ameters, or homologous sides of the section. The weight of the body itself may, in some instances, be employed to strain and to break it; as is the case with a rope, which may be so long as to break by its own weight.-When the rope hangs in a perpendicular direction, although its strength is equal in every part, the fracture will take place towards the upper end, since the strain on any part is equal to the weight of all below it, or in other words, its relative strength in any part, or power of withstanding the strain to which it is subjected, is inversely as the quan- tity below that part. f When the rope is stretched horizontally, the strain arising from its weight often bears a very sensible proportion to its whole strength. Let a 5 B, Fig. 1, Plate 33, represent any portion of such a rope, in which case the curve a ® B will be the catenaria ; and if the tangents a c, B C, be drawn through the points of suspension, if the parallelogram a ® c n be completed, and if JOINERS' INSTRUCTOR. 183 the diagonal » c, be drawn ; p c will be to a c, as the weight of the rope a E B, is to the strain exerted at a, and the strain exerted at B, will be found by a similar process. When a suspended body is required to be so strong throughout as to carry its own weight in any part, the section in that part must be proportioned to the solid contents of all below it. If a a e, (Fig. 2.) be supposed to represent a section of a conoidal spindle, we must have asotivia o* :clat avs ib sol. c a a e is known among mathematicians by the name of the logarithmic curve, of which c c is the axis. These are the chief general rules, which can with safety be deduced from our clearest, though imperfect congeptions of the nature of that co- hesion which connects bodies together ; and in order to make a practical use of these, it is ne- cessary that we should be acquainted with those modes of ascertaining the attraction of cohesion in solid bodies which are most commonly em- ployed by practical mechanics. As the cohesion of bodies of the same kind is known to differ in innumerable circum- stances, we will take for the measure of cohe- sion the weight of pounds avoirdupois which 184 THE CARPENTERS' AND suffice to tear asunder a rod, or bundle, of one inch square. - From this, it will be easy to com- pute the strength corresponding to any other dimensions. With regard to the tenacity, or strength of wood : 1st. The wood which surrounds immedi- ately the pith, or heart of the tree, is sup- posed to be the weakest, and this weakness is greater, as the tree is older. We give this as the result of experiments made by Muschen- broék ; but Buffon says, his experiments proved to him that the heart of a sound tree is the strongest ; for which assertion, however, he as- signs no authority. - It is certain from accurate observations which have been made on very large oaks and firs, that the heart is much weaker than the exterior parts. 2d. The fibres next the bark, commonly called the white or blea, are also weaker than the rest, and the wood gradually increases in strength as it recedes from the centre to the blea. 3d. The wood is stronger in the middle of the trunk than at the springing of the branches, or at the root ; and the wood forming a branch, is weaker than that of the trunk. Re JOINERS' INSTRUCTOR. 185 4th. - The wood on the northern side of all trees which grow in Europe is the weakest, while that on the south-eastern side is the strongest ; this difference is most remarkable in hedge-row trees, and such as grow singly. The heart of a tree never lies in its centre, but always towards its northern side, and the an- nual coats of wood are thinner on that side.-In conformity with this, it is a general opinion of carpenters, that timber is stronger in propor- tion to the thickness of its annual plates. The trachea, or air vessels, being the same, in di- ameter and number of rows, in trees of the same species, occasion the visible separation between the annual plates ; for which reason when these are thicker, they contain a greater portion of the simple ligneous fibres. 5th. All woods are most tenacious while green ; but after the trees are felled, that tenacity is considerably diminished by their drying. Muschenbroék is the only author who has given us an opportunity of judging minutely in this respect. The woods which he selected for experiment were all formed into slips, part of each of which was cut away to a parallelo- piped, of 1-fifth of an inch square, and therefore 1-twenty-fifth of a square inch in section. The 186 THE CARPENTERS' AND absolute strengths of a square inch were as follows :- Pounds. Locust tree s s i 3 s . 20100 ep " Mi Sul c Hund? 1 v= Beech and oak . & f A R « 17300 Orange _ . a & f & 15500 Alder + % * & A a . 18900 Elm . & . o * & A . 13200 Mulberry . F + o f s ~ 12500 Willow -. a 5 f + < C "12500 Ash . F 3 ; : A 2 (12000 Plum e > F ? § t . 11800 Elder ¥ R s A * P . 10000 Pomegranate _. s : ( f 0790 Lemon .,. f + R . A -. 9250 Tamarind C o 6 s s "~ 8700 Cir - . e € f f A # .." 8880 Walnut _. 4 6 f ® Pitch pine e f » i # A 7650 Quince _. s g * o % -% 6750 Cypress |. : p ; F $ .~ 6000 Poplar -. f f . a 7 :- 5500 Cedar 4 s o f F ? . 4880 Muschenbroék gives a very minute detail of his experiments on the ash and walnut, in which he states the weights required to tear asunder slips taken from the four sides of these trees, and on each side, in a regular progression from the centre to the circumference. The JOINERS' INSTRUCTOR. 187 numbers in the foregoing table corresponding with these two woods, may be considered, there- fore, as the average of more than fifty triais of each. He mentions also that all the other numbers were calculated with the same care. For these reasons some confidence may be placed in the results; though they carry the degrees of tenacity considerably higher than those enumerated by some other writers. Pitot and Parent observe, that a weight of sixty pounds will just tear asunder a square line of sound oak, but that it will bear fifty pounds with safety. - This gives 8640 for the greatest strength of a square inch, which is much inferior to Muschenbroék's calculation. To the fore- going table may be added :- Pounds. ory. .oo ot us Yan erty Bone x . F %, A § . 0290 Horn o 8 $ 4 ¢ . f- 8750 Whalebone k 4 s s o -_ 7500 Tooth of sea calf $ 4 § l '. 4075 These numbers express something more than the utmost attraction of cohesion ; the weights are such as will very quickly, (that it is in a minute or two,) tear the rods asunder. In general it may be observed, that two-thirds of these weights will greatly impair the strength 188 THE CARPENTERS' AND after a considerable time, and that one-half is the utmost that can remain suspended at them without incurring the risk of their demolition ; and on this calculation of one-half of the no- minal weight, the engineer should reckon in all his constructions; though, even in this respect, there are great shades of difference. Woods of a very straight fibre, such as fir, will suffer less injury from a load which is not sufficient to break them immediately. Mr. Emerson mentions the following as the weights, or loads, which may be safely sus- pended to an inch square of the several bodies hereafter enumerated. Pounds. Tron s ¥ 3 % X A . 76400 Brass .. . s s A 7 a .. 05600 Hempen Rope s : s a .- 19600 Iyory - - : -.: 15700 Oak, Box, Yu\, and Plum tree A Elim, Ash, and Beech . F t « ~, 6070 Walnut, and Plum _. o +, ©8060 Red fir, Holly, Elder, Plane, and Crab >5000 Cherry, and Hazel . F s s 4760 Alder, Asp, Birch, and Willow . . | ~4290 Lead i s f A . § a 430 914 Freestone This ingenious gentleman has laid down as a practical rule, that a cylinder whose diameter is JOINERS' INSTRUCTOR. 189 d inches, will carry, when loaded to one-fourth of its absolute strength, as follows : Iron a A u s * ' Good rope | . . % P F 22 oy gaily 84> vi. toa Pad S Owt. Fir 9 It is necessary to remark that the ranks which the different woods hold in Mr. Emerson's list, in point of tenacity, differ materially from those assigned to some of them by Muschen- broék. Secondly, we observe that bodies may be crushed.-It is an object of the first importance to ascertain the weight, pressure, or strain, which may be laid on solid bodies without the danger of crushing them. - Posts and pillars of all kinds are exposed to this strain in its most simple form ; and there are some cases where the strain is enormous, as, for instance, where it arises from the oblique position of the parts, which is the case with struts, braces, and trusses, and frequently occurs in our great works. Some general knowledge of the principle which determines the strength of bodies in op- position to this strain, must be allowed to be 190 THE CARPENTERS' AND desirable. Unfortunately we are much more at a loss in this respect, than in the preceding. It is the opinion of some eminent men, that the resistance which bodies are capable of making to an attempt to crush them, bears a proportion to the external force; for as each particle composing the body is similarly and equally acted upon, the aggregate resistance of that body, must correspond with the extent of the section. This principle, however, is considered as ill founded, by others no less eminent for their scientific and experimental knowledge. But as it must be acknowledged, that the relation existing between the dimensions and the strength of a pillar has not been established on solid mechanical principles, and experience plainly contradicts the prevalent opinion that the strength is proportional to the area of the section, it would appear that the required ratio depends much on the internal structure of the body, and experiment seems to be the only method of ascertaining the general laws of co- hesion. If a body be of a fibrous texture, with its fibres situated in the direction of the pressure, and slightly connected with each other by some JOINERS' INSTRUCTOR 191 kind of cement, such a body will fail only when the cement connecting them gives way, and they are detached from each other. Something like this may be observed in wooden pillars, in which it would appear, that the resistance must be as the number of equally resisting fibres, and as their mutual support jointly, or as some function of the area of the section. Precisely the same thing will happen, if the fibres are naturally crooked, provided some similarity in their form be supposed. We must imagine always that some similarity of kind exists, or otherwise it will be absurd to aim at any general inferences. In all cases, therefore, we can hardly hesitate to admit, that the strength exerted in opposition to compression, bears a relative proportion to a certain function of the area of the section. It does not appear that the strength of a pillar is at all affected by its length, as the whole length of a cylinder or prism is equally pressed. If, indeed, these bodies may be supposed to bend under the pressure, the case is materially altered, because then they are subject to the influence of a transverse strain, which, it is well knewn, increases with the length of the pillar. This, however, will be considered under the next class of strains. ~ 192 THE CARPENTERS' AND Parent has shown that the force required to crush a body is nearly equal to that which will tear it asunder. - He observes, also, that it re- quires something more than sixty pounds on every square line, to crush a piece of sound oak ; but this rule is by no means general. Glass, for instance, will carry a hundred times more on it than oak in this way, but will not bear suspended above four or five times as much. Oak will suspend a great deal more than fir, but fir will carry twice as much as a pillar. - Woods of a soft texture, although they may be com- posed of very tenacious fibres, are more easily crushed by the load upon them. - This softness of texture is chiefly owing to the crooked nature of their fibres, and to the existence of considerable vacuities between each fibre, so that they are more easily bent in a lateral di- rection and crushed. - When a post is over. strained by its load, it is observed to swell sen- sibly in diameter. In all cases where the fibres lie oblique to the strain, the strength is considerably diminished ; which may be ascribed to the circumstance, that the parts in such case slide on each other, and the connecting force of the cementing matter is for that reason easier overcome. Mr. Gauthey, in the fourth volume of Rozier's JOINERS' INSTRUCTOR. 193 Journal de Physique, published the result of some experiments which he had made on small rectangular parallelopipeds, cut from a great variety of stones. The following table exhibits the medium results of several trials on two very similar sorts of freestone, one of which was among the hardest, and the other among the softest kinds used in building. The first column expresses the length a B, of the section, in French lines, or 12ths of an inch; the second points out the breadth s c; the third shows the area of the section, in square lines; the fourth exhibits the number of ounces required to crush the piece,; the fifth represents the weight then borne by each square line, or twelfth of an inch of the section; and the sixth displays the round numbers, to which Mr. Gauthey imagines that those in the fifth column approximate. HARD STONE. AB BC _ ABx BC - Weight,. Force. 1 8 8 64 736 11:5 12 2 | 8 12 l 96 2625 24 3 8 16 128 4496 35°1 36 soFT sToNE. 4 9 16 144 560 39 4 5 9 18 162 848 53 4°5 6 18 18 324 2028 9 9 7 18 24 432 5296 122 12 It may be proper to observe, that the first and third columns, compared with the fifth and o 194 THE CARPENTERS' AND sixth, ought to furnish similar results, because the first and fifth respectively form half of the third and sixth ; but the third, it will be re- marked, is three times stronger than the first, while the sixth is only twice as strong as the fifth. It is evident, however, that the strength increases in a much greater ratio than the area of the section, and that a square twelfth part of an inch can carry more and more weight, in proportion to the increased dimensions of the section, of which it forms a part. In the series of experiments on the soft stone, the individual strength of a square line seems to increase nearly in the proportion of the section of which it is a part. Mr. Gauthey deduces from the whole of his numerous experiments, that a pillar, formed of hard stone from Givry, whose section is a square foot, will bear with perfect safety 664.000 pounds, that its extreme strength is 871000, and that the most inferior instance of strength is 460000. The soft bed of Givry stone had for its smallest strength 187000, for its greatest 311000, and for its safe load 249000. This gentleman's measure of the suspending strength of stone, is very small in proportion to its power of supporting a load laid above it. He found that a prism, of the hard bed of Givry stone, the section of which was one foot, A JOINERS' INSTRUCTOR. 195 was liable to be torn asunder, when subjected to a weight of 4600 pounds, and when firmly fixed horizontally in a wall, that it was broken by a weight of 56000 pounds, suspended a foot from the wall. If the prism rests on two props, separated a foot from each other, it will be broken by 206000 pounds when that weight ope- rates or is laid on its middle. These experiments differ so very widely from each other, in their several results, that they cannot be deemed of much advantage to us. A judicious series of experiments on this most interesting subject, would be exceedingly valuable, and its usefulness cannot be too highly estimated. In the construction of wooden bridges, centers, &c. this species of strain is very frequently found, and therefore it is particularly entitled to the attention of the engineer. But how few engineers can find sufficient leisure, in the hurried operations of their business, for prosecuting experiments with that coolness and patient investigation, which the subject demands ? It is singular that, in an empire like this, and in a matter of such unquestionable importance, some person of sufficient judgment and abilities has not been appointed to institute the necessary in- quiries, on an extensive and liberal scale, into o 2 196 THE CARPENTERS' AND the various strains to which materials in general are subject. The only way in which we can effect any good, during the absence of these essential ex- periments, is by paying a careful attention to the manner in which fractures are produced. By attending to this, there may be some prospect of introducing a degree of accuracy, by mathe- matical measurement, which is an object " de- voutly to be wished for," in matters of this kind. BODIES MAY BE BROKEN ACROSS. The strain which most commonly acts on materials of any nature, is that which tends to break them in a transverse direction. This species of strain, however, is but seldom effected, or rather tried in that simple manner which the subject apparently admits of ; for when a beam projects horizontally from a wall, and a weight is suspended from its extremity, the beam is most commonly broken near the wall, in which case the intermediate part has performed the operation of a lever. It sometimes, though rarely, happens, that the pin in the joint of a pair of pincers or scissars, is cut through by the strain; and this is almost the ouly instance of a simple transverse fracture. In consequence JOINERS' INSTRUCTOR. 19% of its being so rare, we shall content ourselves with remarking, that in this case, the strength of the piece bears a proportion to the area of the section. Experiments have been made in the following manner, for discovering the re- sistances made by bodies to this species of strain. Two iron bars were disposed horizon- tally, at the distance of an inch from each other; a third bar was then hung perpendicularly be- tween them, being supported by a pin made of the substance intended to be examined. This pin was made in the shape of a prism, so as to accommodate itself to the holes in the three bars, which were made very exact, and of similar size and shape. A scale was next sus- pended at the lower end of the perpendicular bar, and loaded till it tore out that part of the pin which occupied the middle hole, which load was, evidently, the measure of the lateral co- hesions of two sections. The side bars were made so as to grasp the middle bar pretty strongly between them, and that no distance might intervene betwen the conflicting pres- sures. This would have combined the energy of a lever, with the purely transverse pressure ; for which reason it was necessary that the internal parts of the holes should not be smaller than the edges. Great irregularities occurred in the 198 THE CARPENTERS' AND first experiments, in consequence of the pins being somewhat tighter within, than at the edges; but when this had been corrected, the trials became extremely regular,. Three sets of holes were employed on this occasion ; viz. a circle, a square, and an equilateral triangle, though the square was occasionally converted into a rectangle, the length of which was equal to twice its breadth. In all the experiments the strength was found to bear an exact pro- portion to the area of the section, to act per- fectly independent of its figure or position, and to rise considerably above the direct cohesion ; that is, it required the operation of considerably more than twice the force to tear out this middle piece, than to rend the pin asunder by a direct pull. A piece of fine freestone re- quired 205 pounds to rend it directly asunder, while 575 were required to break it in this way. The difference was very constant in any one substance, but it varied from four-thirds to six- thirds in bodies of different kinds, and was smallest in those of a fibrous texture. But the more common case, where the energy of a lever intervenes, demands a strict con- sideration. Let a s c p, Fig. 3, be supposed to represent tne vertical section of a prismatic solid, pro- JOINERS' INSTRUCTOR. 1099 jecting horizontally from a wall in which it is firmly fixed ; and let a weight p, be hung on it at B, or let any power p, act at B, in a direction perpendicular to a this body also be considered to possess insuperable strength in every part, except in the- vertical section D 4, perpendicular to its length, in which section only it must break.-Let the cohesion be uni- form throughout the whole of this section ; that is, let each of the adjoining particles of the two parts cohere with an equal force f- There are two ways in which it may then break. The part a B c D, may simply slide down along the surface of the fracture, provided the power acting at B, be equal to the accumulated force which is exerted by every particle composing the section, in the direction a p. But let this be supposed as effectually prevented by some- thing supporting the point a. The action at P, tends to make the body turn round a (or round a horizontal line passing through a at right angles with a B) in the same manner as round a joint. - This it cannot do, without separating at the line » a, in which case the adjoining particles at p, or at ®, will be separated hori- zontally. _ But their attraction of cohesion resists this separation. In order, therefore, that the fracture may happen at the place in- 200 THE CARPENTERS' AND tended, the energy of the power r, acting by means of the lever a B, must be superior to the accumulated energies of the component par- ticles. The energy of each depends not only on its cohesive, or connecting force, but also on its peculiar situation; for the supposed in- superable firmness of the rest of the body, renders it a lever turning round the fulcrum a, and the individual cohesive power of each particle, such as » or r, acts by means of the arm p» a. The precise energy of each particle will consequently be ascertained by multiplying the force individually exerted by it at the moment of fracture, by the arm of the lever which enables it to act. Let us then suppose that, at the moment of fracture, every individual particle exerts an equal force £. The energy of Dn, will be p» a X f, that of ® will be E a K f, and that of the whole will be the sum of all these products. Let the depth » a, of the section, be called d, and let any undetermined part of it, as a ®, be called a, then the space occupied by any particle will be &. - The cohesion of this space may be represented by., and that of the whole by f d. The energy by which each element a, of the line » a, or d, resists the fracture, will be f @ a, and the whole accumulated energies will be £ JOINERS' INSTRUCTOR. 201 X f az. This is well known to be J X 4 d:, or fd X 4 d. It is the same thing, therefore, as if the cohesion £4, of the whole section, had been concentred together at the point c, which is in the middle of » a. A similar conclusion may be deduced from other principles. Suppose the beam, instead of projecting horizontally from a wall, to be suspended from a ceiling, in which it is firmly fixed. Let us next consider what effect the equal or accumulated cohesion of every part, has in preventing the lower part from separating from the upper, by opening round the joint a. The equal cohesion operates just in the same manner as equal gravity would do, but in a direction diametrically opposite. We know that the effect of this will be the same as if the whole weight were to be concentrated in the centre of gravity c, of the line p a, and that this point e will be- in the middle of » a. Now the number of fibres being as the length d of the line, and the cohesion of each fibre being == f, the cohesion of the whole line is £ K d or fd: The accumulated energy, therefore, of the cohesion in the instant of fracture, is f d X $ d. Now this must be equal, or just inferior to the energy of the power employed to break it. Let the length a s, be called /; then r X Z, is the 202 THE CARPENTERS) AND corresponding energy of the power. - This gives us £ d X 4 d==p I, for the equation of the equi- librium corresponding to the vertical section «' B'¢'p. Let us suppose, however, that the fracture is not permitted at D a, but at another section m n, more remote from s. From the body being prismatic, all the vertical sections are equal ; and, therefore, fd 4 d is the same as before: but the energy of the power is nevertheless in- creased, it being in this instance, =r X B n, instead of rp XK B a. Hence, we may see, that when the prismatic body is not insuperably strong, in all its parts, but only moderately, though equally strong throughout, it must break close at the wall, where the strain or energy of the power exerts itself with the greatest effect. We may see likewise, that a power which is just able to break it at the wall, is unable to break it any where else; and that the absolute cohesion fd, which withstands the power P in the section pn a, will not withstand it in the section m n, though it resists more in the section o p. This example affords a criterion for distin- guishing between absolute and relative strength. The relative strength of a section has a reference to the strain actually exerted on that section ; JOINERS' INSTRUCTOR. 203 and this relative strength is properly measured by the power which is just able to balance, or overcome it, when applied at its proper place. Now since we had £4 we deduce p= fd d o/ section D A, as it relates to the power applied at B. - If the solid be a rectangular beam, whose breadth is b, it is evident that all its vertical sections will be equal, and that a a or 4 d is precisely the same in all. Therefore the equa- tion expressing the equilibrium existing between the momentum of the external force, and the accumulated momenta of cohesion, will be p I =f a b y { d. The product dob evidently expresses the area of the section of fracture, which we may call s; the equilibrium may be expressed thus : p /== f's $ d, and 21: d :f s :p. Now fs properly expresses the absolute cohe- sion of the section of fracture, and p is a proper measure of its strength, as it relates to a power applied at B. We may, therefore, say, that twice the length of a rectangular beam, is to the depth as the absolute cohesion is to the relative strength. Since the action of equable cohesion is similar to that of equal gravity, it follows, that what- ever may be the figure of the section, the for the measure of the strength of the 204. THE CARPENTERS' AND relative strength will be the same, as if the absolute cohesion of all the fibres were exerted at the centre of gravity of the section. Let & represent the distance between the centre of gravity of the section and the axis of fracture, we shall have p i= fg s, and therefore I: g :: .f 8: p. This analogy in words is not unworthy of the reader's recollection, and may be thus stated : ** The length of a prismatic beam of any shape is to the height of the centre of gravity above the lower side, what the absolute cohesion is, to the strength that bears relation to this length." Since the relative strength of a rectangular d f 21 strengths of different beams not only bear a proportion to the absolute cohesion of the particles, and to the breadth, but to the square of the depth directly, and to the length in- versely ; in prisms, also, whose sections are similar, the strengths are as the cubes of the diameters. 'This investigation has been con- ducted on the hypothesis of equal cohesion, a law not exactly conformable to the operations of nature. We know, for instance, when a force is applied transversely at B, that the beam bending downwards, becomes convex on the upper side; and that that side is, consequently, beam is ; it; follows,, that . the relative JOINERS' INSTRUCTOR. 205 on the stretch. - The particles at o are farther removed from each other than those at E, for which reason they exert greater cohesive forces. It is impossible to ascertain with certainty in what proportion each fibre is extended : but we will suppose, for example, that their remoteness from each other is proportioned to the distance from a, a supposition which is by no means improbable. Now, recollecting the general law, that the attractive forces exerted by dilated" particles are proportioned to the extent of their being dilated ; let us suppose the beam to be so much bent, that the particles at o are compelled to exert their utmost energy, and that this fibre is just ready to break, or even actually breaks ; it is plain, in this instance, that an absolute fracture must ensue, since the force originally superior to the full cohesion of the particle at p, and a certain portion of the cohesion of all the rest, will become more than superior to the full cohesion of the particle next within », and a smaller portion of the cohesion of the re- mainder. Let r represent, as before, the full force of the exterior fibre », exerted by it in the moment of breaking, when the force exerted at the same instant by the fibre r, will be shown by this 206 THE CARPENTERS' AND analogy, viz. d : a :: f : f and the force really exerted by the fibre 5, is I; The force exerted by a fibre whose thickness is &, is thereforefz l strain through its being enabled to act by means of the lever E a, or a ; its momentum therefore is fax ax rg=s and the aggregate momentum of all the , but this force resists the fibres in the line ar, will beff%. This, when a is taken equal to d, will express the momen- tum of the whole fibres in the line a p, which is f Bald orf d X id. Now fd expresses the absolute cohesion of the whole line a p. The accumu- lated momentum is consequently the same as if the absolute cohesion of the whole line were exerted at the distance of one-third of a p, from a. From the preceding, it follows, that the equation expressing the equilibrium of the strain and cohesion, is p i=f d X 4 d, from whence the following analogy may be deduced, viz. " As thrice the length is to the depth, so is the absolute cohesion to the relative strength." JOINERS' INSTRUCTOR. 207 This equation and proportion apply equally to rectangular beams, whose breadth may be 4; since we shall then have p 4 d %K $4. We see, also, that the relative strength is not only proportioned to the absolute cohesion of the particles, and to the breadth, but to the square of the depth directly, and to the length in- versely : for p is the measure of the force with which it is resisted, and pzib—ilfi—d- _—_f {Cg—ff In this respect, therefore, the hypothesis coin- cides with that of Galileo, except that it assigns to every beam a smaller proportion of the abso- lute cohesion in the section of fracture, in the proportion of three to two. Galileo supposes that this section has a momentum equal to one- half of its absolute strength, while in our hypo- thesis it is only one-third. In beams of a different form, the proportion may be different. The consideration of the intricate problem of the elastic curve, which was first investigated by the celebrated James Bernoulli, is too deep and profound to be discussed in a publication like this ; and we shall therefore briefly observe, in the first place, that the elastic curve cannot be a circle, but becomes gradually more in- curvated, in proportion as it recedes from the point where the straining forces are applied. t 208 THE CARPENTERS' AND At this point it has no curvature, and if the bar were extended even beyond this point, still there would be no curvature. -In conformity with this principle, when a beam is supported at the ends, and loaded in the middle, the curvature is greatest in the middle: but at the props, or beyond them, if the beam extend farther, there is no curvature. Therefore, when a beam projecting 20 feet from a wall, is bent to a certain curvature at the wall, by a weight suspended at the end, and a beam of the same size, projecting 20 feet, is bent to the very same curvature at the wall, by a greater weight, at 10 feet distance, the figure and the mechanical state of the beam in the vicinity of the wall is different in these two cases, though the cur- vature close to the wall is the same in both. In the former case every part of the beam is incurvated ; in the latter, all beyond the 10 feet is without curvature. - In the former case the curvature at the distance of five feet from the wall, is three-fourths of the curvature at the wall ; in the latter, the curvature at the same place is only one-half of that at the wall. This circumstance must tend to weaken the long beam, throughout the whole interval of five feet, because the greater curvature results from the greater extension of the fibres. JOINERS' INSTRUCTOR. 209 In the next place, we may remark, that a certain determinate curvature being suitable to every beam, it cannot be exceeded without breaking it} since two adjoining particles are thereby separated, and an end is put to their cohesion. A fibre can be extended only to a certain degree of its - length. The ultimate extension of the outer fibres must bear a certain proportion to its length, and this proportion is similar in the point of depth to the radius of ultimate curvature, which is, therefore, deter- minate. - Consequently, a beam of uniform breadth and depth is most incurvated where the strain is greatest, and will necessarily break in the most incurvated part. - But by changing its form, so as to render the strength of its dif- ferent sections in the ratio of the strain, it is evident that the curvature will be the same throughout, or that it may be made to vary according to any law. A Again, since the depth of the beam is thus proportioned to the radius of ultimate curvature, this curvature is inversely as the depth, and may be expressed by lg We may observe also, that when a weight is suspended on the end of a prismatic beam, the curvature bears a very near proportion to the T 210 THE CARPENTERS' AND weight, and the length directly, and to the breadth and the cube of the depth inversely ; for the strength is known to be 1)ng , and let us suppose that this produces the ultimate cur- 1 a # vature -p when, if the beam be loaded with a smaller weight w, and if the consequent cur- vature be represented by c, we shall have bd: f . Sh the extreme and mean terms, and reducing the 31 w bd This may be said also of a beam supported at its ends, and loaded between the props ; by the same method, the curvature may be determined in its different parts, whether it arises from the load, from its weight, or from the united ope- ration of both. f When a weight operates either at one end, - or in the middle of a beam, the point where this weight is applied is necessarily bent down, and the distance through which it descends has been termed the deflection ; this may be con- sidered as the versed sine of the arch into which the beam is bent, by the operation of the weight, and, therefore, is as the curvature when the & wiro:e; consequently, by incorporating resulting equation, we shall deduce c == JOINERS' INSTRUCTOR. B11 length of the arch is given (admitting the flexure to be moderate), or as the square of the length of the arch when the curvature is given. The deflection consequently is as the curvature, and as the square of the length of the arch 1 w 313m b: f ° Id bP f The deflection from the original shape is as the bending weight and the cube of the length directly, and as the breadth and the cube of the depth inversely. We may further observe, that in beams just ready to break, the curvature is proportioned to the inverse depth, and that the deflection bears a proportion to the square of the length divided by the depth; for the ultimate curvature at the breaking part is constantly the same whatever may be the length; and in this case the deflection is as the square of the length. From this subject may be deduced various theorems, which afford excellent methods of inquiry into the laws of corpuscular action. James Bernoulli, however, called this law (which was originally laid down by the cele. brated Dr. Hooke) into question. Mariotte corrected it; but yet it does not properly ex- plain the mechanism of transverse strains, as has been fully proved by various experiments. r» 2 jointly ; that is, as' I: X 212 THE CARPENTERS' AND Du Hamel made assiduous researches into the compressibility of bodies, which tended to confirm the observation of an eminent philo- sopher, "that the power of resisting a transverse strain is diminished by compressibility, and so much the more diminished as the stuff is more compressible." Du Hamel took 16 bars of willow, 2 feet long, and 4 an inch square, and after supporting them by props under the ends, he subjected them to the operation of weights suspended at the middle.. Four of them were broken by weights of 40, 41, 47 and 52 pounds; the mean of which is 45lbs. He then cut through one third of four of them, on the upper side, and filled up each cut with a thin piece of harder wood, stuck in tolerably tight. These several pieces were then broken by weights of 48, 54, 50 and 52 pounds; the mean of which is 51lbs. Four others were then cut through one half, and broken by 47, 49, 50 and S6lbs ; the mean of which is 48lbs. The other four were cut through two-thirds, and their mean strength was 42lbs. At another time Du Hamel took six battens of willow 36 inches long, and 14 square ; after suitable experiments, he found that they were broken by 525 pounds at a medium. JOoInk®rs' InstRUCToR. 13 Six bars were next cut through one-third, and each cut was filled with a wedge of hard wood stuck in with a little force: these were broken by 551 pounds on the average. Six other bars were broken by 542lbs. on the medium, when cut half through, and the cuts were filled up in a similar manner. Six other bars were cut three-fourths through, and broken by the pressure of 530 pounds on a medium. A batten was cut three-fourths through, and loaded until nearly broken ; it was then unloaded, and a thicker wedge was introduced tightly into the cut, so as to straighten the batten, by filling up the space left by the compression of the wood, when the batten was broken by 577 pounds. From these experiments, we may perceive that more than two-thirds of the thickness, we may perhaps with safety say, nearly three- fourths, contributed nothing to the strength. From hence we see also, that the compressibility of bodies has a very great influence on their power of withstanding a transverse strain. - We may observe likewise, that in this most favour able supposition of equal dilatations and com- pressions, the strength is reduced to' one half of the value of what it would have been, had 214 THE CARPENTERS' AND the body been incompressible; and although this may not seem obvious at first sight, yet it will readily appear when the case is considered. In the instant of fracture, a smaller portion of the section exerts its actual cohesive forces, while a part of it serves only as a fulcrum to the lever, by whose means the strain on the section is produced ; and we may further per- ceive, that this diminution of strength does not depend so much on the sensible compressibility, as on the proportion it bears to the power of being dilated by equal forces. The foregoing experiments on battens of willow, moreover, show, that, its compressibility is very nearly equal to its dilatability. Experiment alone can render us efficient aid, in investigating the degree of proportion that exists between the compressibility and dilata- bility of bodies ; and the nature of the strain we have just been considering is peculiarly adapted to guide us in the research. Thus, if a piece of wood an inch square requires 12,000 pounds to tear it asunder by a direct pull, while 200 pounds will break it transversely, by acting 10 inches from the centre of fracture, we may conclude that the attractive and repulsive forces are equal. By the ideas we entertain con- cerning the particular constitution of such fi- JOINERS' INSTRUCTOR. 215 brous bodies as timber, we are led to conceive that the sensible compression, which arises from the bending up of the compressed fibres, are much greater than the real corpuscular exten- sions. This circumstance, however, will be better comprehended, after we have considered what must happen during the fracture. An undulated fibre can be drawn straight only, when the corpuscular extension begins ; but it may be bent up by compression to any degree, the corpuscular compression being but little af- fected all the time. - This fact is of an import- ant nature. - Though the forces of corpuscular repulsion may be, deemed almost inseparable by any compression we can employ, a sensible compression, nevertheless, may be produced, by forces not enormous, but sufficient to crlpple the beam. The proportional strengths of different pieces follow the same ratio ; for, although the relative strengths of a prismatic solid have been consi- dered as extremely different in the foregoing hypotheses, yet the proportional strengths of different pieces follow the same ratio, that is, the direct ratio of the breadth, the direct ratio of the square of the depth, and the inverse ratio of the length. We derive, also, from this im- 216 THE CARPENTERS' AND portant fact the useful information, that the strength of a piece depends most on those di- mensions which lie in the direction of the strain ; or, to use other words, it depends more on its depth than on its thickness. The strength of a bar of timber, two inches in depth and one in thickness, is four times more than that of a bar of an inch square; while at the same time, it is twice as strong as a bar two inches broad, and an inch deep. The manner in which cohesion opposes itself to a strain may be farther exhibited and applied, by supposing a triangular beam to be fixed firmly by one end in a wall, with its other end unsupported, and to be acted on by a cer- tain weight; in which position it will bear three times more weight, when one of its sides is uppermost, as it would if it were undermost. Thus, for example, the triangular beam deli- neated at Fig. 4 is three times as strong, when the side a s is uppermost, and the edge p c is undermost, as it would have been, if the edge p ¢ were uppermost, and the side a s under- most. Hence, also, we may find, that the strongest rectangular beam, which can be cut out of a given cylindrical tree, is not that which con- tains the greatest quantity of timber, but that JOINERS' INSTRUCTOR. $17 the product of whose breadth, by the square of its depth, is a maximum, or the greatest pos» sible. The following solution will show, that the squares of the breadth and depth, with the square of the diameter, are respectively as the numbers 1, 2, and 8. In Fig. 5, let a B, the diameter of the cylin- drical tree, be designated by d; let the depth a c, of the beam, be shown by d, and the breadth sc by a; then, when s c is horizontal, the lateral strength will be truly represented by d* x, which, agreeably to the conditions of the problem, must be a maximum ; but we know, from the nature of the figure, that a c*== a B* -- B c', or d'= d'- &: hence, then (de-az>*) expresses the maximum : this put in fluxions, is d* t-& a® i420, or de a =3 a a"; whence 3 s' =i@#', and therefore d- d' -_" . == Suto a' @i as;} consequently x" ~ d" :y: :: 1 : 2 : 8, as before observed. From this solution we deduce the following very easy mode of construction, which every practical carpenter may apply with the greatest facility. Divide the diameter a m into three equal parts, at the points E r; erect the per- pendiculars £ », r c, and join the points c, D, to the extremities of the diameter, when a B c p will be a section of the rectangular beam re- 218 THE CARPENTERS' AND quired. For, in consequence of a ®, a », and a B, being in continued proportion, we have A E: AB:: a b' :a B'; and similarly a r : AaB tila Cia b's : ap wp" :' NC sax B's: ts 2 : 9: The ratio of ¥ to d, is very nearly that of 5 to 7, or still more nearly, that of 12 to 17. 'The strength of a sc nis to that of a a s b, as 10,000 to 9186, and the weight and expense are as 10,000 to 10,607 ; so that a B c p is pre- ferable to a a B 6, in the proportion of 10,607 to 9186, or nearly as 115 to 100. A square beam from the same cylinder would have its side == dy 4 == 4 dy 2. - Its solidity would be to that of the strongest beam, as 4 d* to # PV 2, orf as 4 to +/ 2, or as 5 to '4714 ; whlle its strength would be to that of the strongest beam as (dy $)] to dy 4% # d', Or as 4/2 to . 3, or -as 3560 to 3849. We may further remark, in conformity with the observation just now made, that either of these beams will be enabled to exert its greatest lateral strength when the diagonal part of one of its ends is placed in a vertical position ; since, from the area of the section being the same in both positions, the strength is known to vary in the same manner as the distance of the centre of gravity varies from the base of fracture; but JOINERS' INSTRUCTOR. 219 when one of the sides is vertical, as in Fig. 6, the distance of the centre of gravity of the end will be =a 1 or 1 », that is, equal to half the side; whereas in the case where the diagonal is vertical, as in Fig. 7, that distance will be c E or E D, that is, half the diagonal. By the application of the same principle, we may discover, that a hollow tube is stronger than a solid rod containing the same quantity of matter. Let the diagram, delineated at Fig. 8, re- present the section of a cylindrical tube, of which a r and B ® are the exterior and interior diameters, and c the centre; draw s p perpen- dicular to s c, and join o c; then because -e B pois the measure of the radius of a circle, which contains the same quantity of matter as the ring. - If the strength be estimated by the first hypothesis, the strength of the tube will be to that of the solid cylinder, whose radisis bD,as «COK 60, to Bb % BD, of as a C to B D by division of ratio. Otherwise, let a B z, g 1 x, as in Figs. 9 and 10, represent the ends of two cylinders of equal length, and containing equal quantities of matter, the former of which, however, is sup- posed to form the section of a tube, composed of cylinders with a common axis. We know @20 THE CARPENTERS AND that the lateral strengths are conjointly, as the areas and the distances of the centres of gravity of the sections, from a or from s, accordingly as the fractures may terminate at the one, or the other point ; but the areas of the annulus in Fig. 9, and of the circle in Fig. 10, are equal, and the centres of gravity of both are at their centres of magnitude, for which reason, since the radii vary in the same manner as the dia- meters, the strengths, in this case, also vary in a similar ratio. When the area of a circular section is given, its diameter is greater if the section form an annulus, than when it is a circle without any cavity ; and since the power, with which the parts of the cylinder resist the operation of ex- traneous force, is greater in the same proportion, it follows, according to the theory thus stated, that the strength may be increased indefinitely, without increasing the quantity of matter. The absurdity of this conclusion becomes manifest, when we inquire, will not the tube be rendered flaccid after the diameter exceeds a certain limit, and therefore bend under the smallest additional weight ? The fact is simply this, the foregoing theory is founded on the supposition, that the figure of the section will constantly remain circular; but this suppo- JOINERS' INSTRUCTOR. 221 sition does not apply, except under those cir- cumstances, where the pressure or stroke upon the tube will not cause its section to dege. nerate from its circular shape to an elliptical, or any other figure. By way of illustration, let a hole be bored, lengthwise, through a cylinder of half its dia- meter, then the strength in this instance is di- minished 4th, while the quantity of matter is diminished ith. Galileo, from a consideration of this subject, justly concludes, that nature, in a thousand - operations, greatly augments the strength of substances without increasing their weight ; as is manifested in the bones of animals, and the feathers of birds, as well as in most tubes, or hollow trunks, which though light, greatly re. sist any effort made to bend or break them. * Thus (says he) if a wheat straw, which sup- ports an ear that is heavier than the whole stalk, were made of the same quantity of matter, but solid, it would bend or break with far greater ease than it now does. And with the same reason art has observed, and experience con- firmed, that a hollow cane, or tube of wood or metal, is much stronger and more firm, than if, while it continued of the same weight and 9909 THE CARPENTERS' AND length, it were solid, as it would then, of con- sequence, be not so thick; and therefore art has contrived a method to make lances hollow within, when they are required to be both light and strong:" in this instance, as in many others, imitating the wisdom of nature. In all such instances, however, there is an obvious distinction between the works of nature and those 'of art: «< In the former" (as M. Girard remarks, when treating of the same subject), " the cause and effect essentially agree; the one cannot undergo any modifi- cation, without the other's experiencing a cor- respondent change ; or to speak more precisely, anew effect always resultsfrom a new cause.-In the productions of human industry, on the con- trary, there is no necessary proportion between the effect and cause : if, for example, a deter- minate weight is to be raised, it is indifferent whether we use the thread which has precisely the adequate force, or the cable which has a superabundant one; while, if the same weight had rested naturally suspended, it would have done so by means of fibres peculiarly appropri- ated, in their organization, to the object, and whose disposition would have presented the most advantageous form. - Perfection resides in JOINERS' INSTRUCTOR. 223 a single point, at which nature arrives without effort ; while man is obliged, by repeated trials, to pass over an immense space which separates him from it." Our best engineers have wisely begun to imitate nature, by making many parts of their machinery hollow, such as the axles of cast iron, &c. . In the supposition of homogeneous texture, the fracture happens as soon as the particles on the upper sides » a, Fig. 11, are separated ber yond their utmost limit of cohesion ; this is a determined quantity, and the piece bends until a similar degree of extension is produced in the outermost fibre. - It follows as a very necessary consequence, that the smaller we suppose the distance to be, between the upper part of the beam and the centre of fracture c, the greater will be the curvature acquired by the beam be- fore it breaks. We may perceive, therefore, that an increase of depth not only renders a beam stronger, but stiffer ; however, if the pa- rallel fibres are supposed to slide on each other, the degree of strength and stiffness will be di- minished. - Instead of one beam, let us, by way of illustration, suppose a B c p, and c D E r, to- represent two equal beams, which do not co- here, but whose aggregate magnitude shall be 904 THE CARPENTERS" AND equal to the former beam. - In this instance it is plain that each of them will bend, and that the extension of the fibres c p of the under beam will not by any means prevent the com- pression of the adjoining fibres c » of the upper beam. - The two beams, therefore, instead of being four times as strong as a single beam, will only be of twice the strength; and they will moreover bend as much as a single beam would be affected by half the load. This, un- doubtedly, could be prevented, if it were pos- sible to unite the two beams firmly in the joint c », so as to prevent one from sliding on the other. | In smaller works, however, it may be effected by gluing them together with a cement, proportioned, in point of strength, to the natural lateral cohesion of the fibres. But as this desideratum cannot be obtained in large works, the sliding may be prevented by joggling the beams together ; various methods for which have been already exhibited. It is, nevertheless, possible to combinestrength with pliability, by forming a beam of several thin planks laid on each other till they form the required depth, and afterwards leaving them at full liberty to slide on each other. Coach springs are formed after this mode, as is shown in Fig. 12. Neither joggles nor bolts of any JOINERs' INSTRUCTOR. 225 kind should be introduced among the planks ; but they must be kept together solely by straps contrived so as to surround them, or by some- thing else of a similar nature. From long experience, practical men have been enabled to introduce into their construc- tions many principles which sound theory does not decline to sanction. This, for instance : when a mortise is required to be cut out of a piece exposed to a cross strain, it should be taken from that side which becomes concave by the strain, as in Fig. 13, but by no means as in }, Farther, when a piece is to be strengthened by the addition of another, the piece to be added should be fixed to the side which grows convex by the strain, as in Figs. 15 and 16. We shall next consider the analogy that ex- ists between the strain on a beam projecting from a wall, and loaded at the extremity, and a beam supported at both ends, and loaded at some intermediate part. Let a c B, Fig. 16, represent the beam sup» ported by the props a and B, and loaded at its middle point c, with a weight w; when it is evident that the beam will receive the same support, and become subject to the same strain, as if, instead of the supports a and B, the ropes a 926 THE CARPENTERS' AND A a E, B b r, which pass over the pulleys a, b, were to be substituted, and have the proper weights x, r, fastened to them. These weights are equal to the support afforded by the points of support, while their sum is equivalent to the weight w ; and on whatever point w may be hung, the weights E and r are to the weight w, in the proportion of » B, and D a, to a m. From hence, it appears, that the strain on the section c », arises immediately from the upward action of the ropes a a, and B b, from the pressure exerted upwards by the points of support a and s, and the office of the weight w, is obliging the beam to oppose this strain. The beam has a tendency to break in the section c p, because the ropes pull it up- wards at ® and c, while the weight w, confines it down at c. It inclines to open at D, and c becomes the centre of fracture. The strain therefore, is the same as if the half a p were fixed in the wall, and a weight equal to the one- fourth of w were applied at c. From these circumstances we may conclude, that a beam supported, but not fixed at both ends, and loaded in the middle, will bear four times as much weight as it would be capable of supporting at one extremity, when the other is fast in a wall. JOINERS' INSTRUCTOR. g97° _ The strain occasioned at any point 1, by a weight w, suspended at any other point D, is w D X L. ts XB I XE. For it is known, that aB: apn :: K w: the pressure occasioned at s. This would be balanced by some weight r, acting over the pulley b, which tends to break the beam at 1, by acting on the lever 1s. The pressure at ® is D: a w yx o and therefore the strain at 1 is w % DjA Als In a similar manner, when the strain occa- sioned at the point », by the weight w, is w % XXI B. -- x» B, which is equal to 4+ w, when » forms the middle point. Hence then, we deduce, that the general strain on a beam arising from a particular weight, bears a proportion to the rectangle of the parts composing the beam, and is greatest when the load is laid on the middle of the beam ; which latter circumstance is confirmed by daily experience. Further, the strain at 1, by a load at c, is equal to the strain at D, by the same load at 1, and the strain at 1, from a load at D, is to the strain by the same load at 1, as D a is to 1 a. If we now a 2 G28) - THE CARPENTERS' AND suppose the beam to be firmly framed at the two ends 1 », into the upright posts L N, a 0, placed beyond the former points of support A B, then it will carry twice as much as when its ends were free ; for admitting the beam to be sawn through at c », the weight w, suspended there, will be but just sufficient to break it at A and B, while, by restoring the connexion of the fibres composing the section c D, it will require another weight w, to break it there at the same time. It should be observed, therefore, that when any piece of timber is firmly connected at three fixed points B, A, 1, it will bear a greater load between any two of them, than if it had no con- nexion with the remote point ; and if it be firmly fastened at the four points L, A, B, M, it will be twice as strong in the middle part, as it would be when deprived of the two remote connexions. It may be thought, perhaps, from the pre- ceding observation, that the joist of a floor, or a girder, will derive an increase of strength from being firmly built in the wall. However plau- sible this idea may appear, the fact is, that it de- rives but little additional strength, for the hold thus afforded to it is too circumscribed in its effect to render much essential service, and JOINERS' INSTRUCTOR. £20 farther, it tends greatly to shatter and crack the wall, when the beam is pressed by any con- siderable load, since it forces up the wall with all the energy of a long lever. For this reason, those builders who are most eminent for their practical knowledge never allow the ends of their joists or girders to be bound tight in walls ; but when the joists of adjoining rooms lie in the same direction, they justly consider it a great advantage gained to have them in one piece, because, in that form, they are twice as strong as when composed of two lengths. Having taken a view of the circumstances which affect the strength of any section of a solid body, when strained transversely, it may next be proper to take notice of some of the chief modifications of the strain itself, that occur most frequently in our constructions. This strain depends on the operation of ex- ternal force, and also on the lever on which it acts; for since the strain may be produced in any section, by means of the cohesion of those parts which intervene between the section (under consideration) and the point of appli- cation of the external force, the body must have sufficient energy in all those intervening parts, to excite the strain in the remote section, 230 THE CARPENTERS' AND and in every part it must be able to resist the strain excited in that part. The body, there- . fore, ought to be equally strong, and it is use- . less to have any one part stronger; because the piece will nevertheless break when it is not stronger throughout, and it is useless to make it stronger with regard to its strain, for it will, nevertheless, equally fail in the part that is too weak. If the strain arises from a weight suspended at one extremity, while the other end is sup- posed to be fixed firmly in a wall ; or if each transverse section of the beam be rectangular ; there are several ways of forming the beam so as to render it equally strong throughout. Let Fig. 17 represent the intended beam, which is fixed to the vertical wall s ®, and has a weight w, suspended at a, its extremity. It is obvious that the effort made by the weight w, upon any point n of the beam, will, by the common properties of the lever, be as the rect- angle w X a D, or as a p, since the weight w is constant and invariable. The strength also at any point D, is as the breadth into the square of the depth at that place, or as the breadth c », the depth being constant. - Consequently, when the beam is equally strong throughout, the JOINERS' INSTRUCTOR. 2381 strength and stress are in an invariable ratio, and we shall have c », constantly as a c; and therefore a c o must be a rectilineal triangle, and the form of the beam a wedge. Again, if we suppose the beam to be of uniform breadth, its length must be propor- tional every where to the square of its depth, if it be fixed horizontally by one end in a wall, and a weight operate on the extremity of the other. The following solution will exhibit this most interesting particular in a clear point of view. On referring to Fig. 18, we observe that the stress is as the length a nin the same manner as in the preceding, and that the strength is as the breadth into the square of the depth, or because the breadth is constant by the con- ditions of.the problem, the strength is as c. 1. But the stress and strain must remain in a con- stant ratio ; wherefore a D must vafty, as. 0.D# : this law, it may be observed, acts invariably throughout the figure, and is the well-known property of a parabola whose vertex is a. The following circumstance, which is worthy of notice, may be immediately deduced from the preceding solution, by way of corollary. Since the parabola is 4 of the parallelogram which circumscribes it, it follows, that parabolic beams require 4 less matter than prismatic 2854 THE CARPENTERS) AND ones ; this circumstance may be beneficially attended to, in cases where iron is used. It is also deserving of remark, That the beams of balances intended to sup- port very great weights, may be constructed of a parabolic shape, which will have the effect of saving materials without any diminution of use- ful strength. To pursue this subject still farther, let us suppose that another beam has one end fixed to a wall, and is diminished gradually towards the other end, where a weight, if suspended, so that all its vertical sections, such as circles, squares, similar polygons, &c. may be similar ; in this case, in order to render the beam equally strong throughout, the bounding curve must be in the form of a cubic parabola. If we refer to Fig. 19, we find that the stress or effort of the weight which operates upon any point r, will be as a r; and, since the sections are all similar, the strengths will vary as the cubes of the depths. Hence, in this case, a r is as p c, which is a well known property of the cubic parabola. All these modes of forming beams render them equally strong in all their parts, and they are all supposed to have the same section at the front of the wall, or at the fulerum. They are JOINERS' INSTRUCTOR. 238 not, however, equally stiff. The beam repre- sented in Fig. 17, will bend the least, upon the whole, while that in Fig. 19, will bend the most, but the curvatures at the fulcrum will be pre- cisely the same in all the beams. The same principles, and the same con- struction, apply to beams when supported at their ends, and loaded at some intermediate part. We have hitherto confined our remarks to the supposition that the external straining force acts only in one point of the beam. - But, it is proper to observe, that this may be uniformly distributed all over the beam. - To form a beam equally strong under such circumstances, the shape must be contrived very differently from the former. If we suppose a beam to project from a wall, and to be of equal breadth throughout, with its sides forming vertical planes parallel to each other and to the length, the vertical section, in the direction of its length, must be a triangle instead of a common parabola ; since the weight uniformly distributed over the part, from lying beyond any section, is as the length beyond that section ; and since also this way it may be considered as collected at its centre of gravity, which of course is the middle of that 234 THE CARPENTERS' AND length, the lever by which this load strains the section, of course bears a proportion to the same length. The strain on the section is as the square of that length, and the section must have strength in the same proportion. From its strengths being as the breadth, and the square of the depth, and from the breadths being constant, the square of the depth of any section must be as the square of its distance from the end, and the depth must be as that distance ; and, therefore, the longitudinal ver- tical section must form a triangle. But if all the transverse sections are supposed to be squares, circles, or any other similar figures, the strength of every section must be proportioned to the square of the lengths be- yond that section, or the square of those sec- tions, distance from the end ; in which case, the sides of the beam must be a semicubical parabola. If the upper and under surfaces are supposed to be horizontal planes, it is evident that the breadth must be proportioned to the square of the distance from the end ; and the horizontal sections may be formed by arches of the com- mon parabola, having the{length for their tan- gents at the vertex. We shall next direct our attention to the JOINERS' INSTRUCTOR. 235 proper form of a beam intended to be fixed at one end, and uniformly loaded throughout its whole length, so as it may be rendered equally strong in all its parts. The vertical sides of this beam we will suppose to be parallel planes, in which case the beam will be of equal thick- ness throughout, and, therefore, the strength at any» part noe, will be aso oipt; on accordingly as a p B, or a c B, represent the bottom of the beam (Fig. 20). - Now the stress at the point n is as the rectangle a D X D B ; for which reason c 1° or c c* must vary as a D x p B, in order to ensure equal strength throughout.> This, it is well known, is the fundamental principle of the ellipse, the vertices of which are a and s. If the transverse sections be similar, we must make c n° as a c K CB. If the upper and under surfaces are parallel, the breadth must be as a c X c B. If, however, the beam is necessarily loaded at some given point c, and we would have it equally able, in all its parts, to resist the strain arising from the weight at c, we must pro- portion the strength of every transverse section between c, and either end, to its distance from that end ; for which reason, if the sides are 236 THE CARPENTERS' AND parallel vertical planes, we must make c p*: BF C!: :A ig. If the sections are similar, then c »: : ® r :c 2C sais. If the upper and under surfaces are parallel, then 'the breadth at ic : breadth at ®: : ac: ip! The same principles lead to the conclusion, that all circular plates, whether large or small, provided they be of the same matter and thick- ness, and supported all round on the edges, will bear equal weights. This conclusion applies also to square plates, or any other ones of a similar figure. The weight, moreover, which a square plate will bear, is to that able to be borne by a bar of the same matter and thickness, as twice the length of the bar to its breadth. There is yet another modification of the strain which tends to break a body transversely, and which occurs very frequently ; this is the strain arising from its own weight, and it requires some consideration in many instances. When a beam projects from a wall, every section is strained, by the weight of all that projects beyond it. 'This weight may be con- sidered as operating at its centre of gravity. JOINERS' INSTRUCTOR. 237 Hence, the strain on any section is in the joint ratio of the weight of the part which pro- jects beyond it, and the distance of its centre of gravity from the section. The determination of this strain, as well as of the strength required to withstand it, is more difficult of attainment than the former, because the mode in which the piece may be formed to meet or adjust the strain, has a considerable in- fluence on the strain itself. It may be admit- ted, perhaps, as a general principle, that the strength of cohesion in every section must be as the product of the weight beyond it multiplied by the distance of its centre of gravity. The result of the application of this general princi- ple is, that the depth must be as the square of the distance from the extremity, and the curve will then form a parabola touching the horizon- tal axis of the figure. We may perceive, therefore, that a conoid formed by the rotation of this figure round its axis, will have sufficient strength in every sec. tion to bear its own weight. A projecting beam becomes less able to bear its own weight, in proportion to the extent of its farther projection ; and whatever may be the strength of the section, the length, nevertheless, 238 THE CARPENTERS' AND may be such as to render it liable to break by its own weight. For instance, if we suppose two beams to be composed of similar matter, with their diameters and lengths in equal pro- portion, but that the shorter beam can only just bear its own weight, then the longer beam will not be able to do the same ; 'since the strengths of the sections are as the cubes of the diameter, while the strains are as the fourth powers of the same. From these considerations, we may take it for granted, that in all cases where a strain is produced by the weight of the parts composing a machine, or structure of any kind, the smaller bodies are more capable of withstanding it than the greater. - Indeed a limit seems to be set by the hand of nature to the size of machines, of - whatever materials they may be constructed ; for, even when the weight of the parts com- posing a machine is not taken into the account, we cannot enlarge it so as to produce a similar proportion in all its parts. A limit is evidently set by nature to the size of animals and plants, when formed of the same matter. The attrac- tion of cohesion in an herb could not support it, if it were increased to the size of a tree, neither could an oak support itself, if it were forty or JOINERS' INSTRUCTOR. 230 fifty times larger than it is, nor could an animal resembling in its make a long legged spider, be augmented to the size of a man. -From the preceding deductions it follows, that greater beams and bars must be in greater danger of breaking, than the less similar ones ; and that, though a less beam may be firm and secure, yet a greater and similar one may be made so long, as necessarily to break by its own weight. Hence, Galileo justly concludes that what appears very firm, and succeeds well in models, may be very weak and unstable, or even fall to pieces by its weight, when it comes to be executed in large dimensions, according to the model. From the same principles he argues, that there are necessarily limits in the works of nature and art, which they cannot surpass in magnitude ; that immensely great ships, palaces, temples, &c. cannot be erected, their vands, beams, bolts, &c. falling asunder by reason of their weight. - Were trees of a very enormous magnitude, their branches would, in like manner, fall off. Large animals have not strength in proportion to their size; and if there were any land animals much larger than those we know, they could hardly move, and would be perpetually subjected to most danger- 24.0 THE CARPENTERS) AND ous accidents. As to the animals of the sea, indeed, the case is different, as the gravity of the water sustains those animals in great measure, and in fact these are known to be sometimes vastly larger than the greatest land animals ; it is, says Galileo, impossible for nature to give bones for men, horses, or other animals, so formed, as to subsist, and proportionally to per- form their offices, when such animals should be enlarged to immense heights, unless she uses matter much firmer, and more resisting than she commonly does ; or should make bones of a thickness out of all proportion, whence the figure and appearance of the animal must be monstrous. - This he supposes the Italian poet hinted at, when he said, « Whatever height we to the giant give, He cannot without equal thickness live." And this sentiment being suggested to us by perpetual experience, we naturally join the idea of greater strength and force with the grosser proportions, and that of agility with the more delicate ones. - The same admirable philosopher likewise remarks, in connexion with this subject, that a greater column is in much more danger of being broken by a fall, than a similar small JOINERS' INSTRUCTOR. -. 241 one; that a man is in greater danger from accidents than a child ; that an insect can sus- tain a weight many times greater than itself ; whereas a much larger animal, as a horse, could scarcely carry another horse of his own size. The ingenious student may easily extend these practical remarks to any cases which may come before him. The compression of materials is another ob- ject that demands our most serious consider- ation. - In adverting to the operation of strains of this kind, it is absolutely impossible to con- ceive how a piece of timber that is perfectly straight can be bent, crippled, or broken, by the application of any force whatever at the ex- tremes. But if a very small force be supposed to act in the middle, in a direction at right angles with the length, it will be sufficient to give it some certain small degree of curvature ; and if a powerful force be likewise supposed to act at the ends at the same time, so as both the greater and the lesser force shall press on the timber in the direction of its length, these forces will conjoin together in producing the effect of a fracture. The first author who considered the com- pression of columns with any degree of proper R 242 THE CARPENTERS' AND attention was the ingenious and learned Euler, This eminent philosopher, in the Berlin Memoirs for 1757, published his " Theory on the Strength of Columns." - The general proposition endea- voured to be established by this theory is, that the strength of prismatic columns is in the di- rect quadruplicate ratio of their diameters, and the inverse ratio of their lengths. He prose» cuted his subject in the Petersburgh Commen- taries for 1778, where he confirms his former theory. Muschenbroék has compared Euler's theory with the results of his own experiments, but the comparison has produced nothing that is satisfactory ; since the variation existing be- tween the experiments and the theory is so enormous, as to offer no argument for the cor- rectness of the latter. Still, however, the ex- periments do not contradict it, though they are so very anomalous, as to lead to no conclusion or general rule whatever. In consequence of our intimation, that the theory of Euler may be deemed erroneous, it may be asked, what is the true proportion in the strength of pillars or columns? We have not the means of giving a satisfactory answer, which could proceed only from the result of a previous experience of the proportion existing . JOINERS' INSTRUCTOR. 243 between the extensions and compressions pro- duced by the operation of equal forces ; that is, from knowing accurately the absolute compres- sions produced by a given force, as well- as the degree of that derangement of parts which is termed crippling. Unfortunately very little is known on these points, and consequently a wide field of experimental inquiry lies before us. It may be considered fortunate, however, that the force required to cripple a beam is prodi- gious, and that a very small lateral support only is sufficient to prevent that bending, which places the beam in imminent danger of destruction. A judicious mechanic will always employ transverse bridles (as they are termed), in order to stay the middle of long beams in- tended to perform the office of pillars, truss beams, struts, &c. and exposed, from the nature of their peculiar position, to immense pressures in the direction of their lengths ; but such stays should be arranged in a judicious, as well as economical manner. As experiments on the transverse strength of bodies are easily made, they have been ac- cordingly very numerous, particularly on tim- ber; but amid this great variety of experi- ments, few have afforded that practical in- R 2 249 THE CARPENTERS' AND formation which is so desirable. - The generality of them have been made on very small scant- lings (in which the unavoidable natural in- equalities bear too great a proportion to the strength of the whole piece) ; for which reason, the results of the experiments of different per- sons have varied considerably ; andeven between those made by the same person, great irregu- larities have existed. Belidor has presented us, in his " Science des Ingenieurs,"' with the most complete series of experiments that has come under our notice.-- His results appear in the following table ; and the pieces on which he made his several trials were sound, even-grained oak. The column s comprises the breadth of the pieces in inches ; the column p contains their depths ; the column 1 includes their lengths ; p demonstrates the weight (in pounds) which broke them when hung on their middles ; and the column a points out the mediums. In order to obtain the respective strengths of pieces of different dimensions with more cer- tainty, three pieces of each dimension were tried, under the expectation that the medium would be better shown by repeated trials than by a single experiment. JOINERS'. INSTRUCTOR. B D L P Experiment 1st, ends loose 18 400 415 405 406 Experiment 2d, ends firmly fixed 600 600 624 608 Experiment 3d, ends loose 810 795 812 805 Experiment 4th, ends loose 18 1570 1580 1590 1580 Experiment 5th, ends loose 36 185 195 180 187 *Experiment 6th, ends fixed 36 285 280 285 283 Experiment 7th, ends loose 36 1550 1620 1585 Experiment 8th, ends loose to to- 36 1665 1675 1640 16860 By comparing the first experiment with the third, the strength appears proportional to the breadth, while the length and depth of each piece are the same. By comparing the first and fourth experi- 216 THE CARPENTERS' AND ments together, the strength appears as the square of the depth nearly, while the breadth and length are all the same. By comparing the first and fifth experiments together, the strength appears to be nearly as the lengths, inversely, while the breadth and depth of each piece are the same. By comparing the fifth and seventh experi- ments together, the strengths appear to bear a near proportion to the breadth, multiplied by . the square of the depth, while the length is the same in both. ’ By comparing the first and seventh experi- ments together, the strengths are shown to be as the square of the depth, multiplied by the breadth, and divided by the length. Experi- ments the first and second show the increase of strength acquired by fastening the ends to be in the proportion of 2 to 3. Experiments the fifth and sixth demonstrate the same thing. This irregularity in the result of experiments may be ascribed to the fibrous or plated texture of timber, which, as is well known, consists of annual additions, whose cohesion with each other is much weaker than that of their own fibres. Let the diagram, denoted by Fig. 21, represent an horizontal section of a tree, and the parallelograms a s c », a b c d, exhibit the JOINERS' INSTRUCTOR. 247 section of two battens cut out of the tree, for the purpose of making experiments. In these parallelograms we intend a D, & d, to point out the measure of their depths, and » c, d c, to re- present that of their breadths. It is evident that the fibres composing the section a B C D, may be considered as an assemblage of planks set edgeways, while those which form the sec- tion a b c d, may be considered as laid flat- ways ; but we know, both from theory and ex- perience, that the former is stronger than the latter, and the reason of this may be easily ex- plained. A series of planks set edgeways will form a stronger beam than planks laid on each other, like the plates of a coach spring. - Buffon made some experiments on oak, in order to as- certain the ratio of strength in these parallelo- grams ; after many trials, he found that the strength of a s c p was to a b c d, nearly as 8 to 7. Buffon, however, (like other experimen- talists) did not take care to have the plates of the battens disposed in a similar manner with respect to the strain ; still, had this precaution been taken, his experiments would not have furnished sure grounds of computation for con- structing works in which large timbers are re- quired ; and it should be observed that, - 28 large timbers occupy a great deal, if not the 248 THE CARPENTERS AND whole of the section of a tree, their strength is proportionably less than that of a small lath or batten. Here, again, we feel ourselves embarrassed and at a loss for the want of an extensive series of suitable experiments on large timbers. To unite the principles of accurate theory with the results of judicious and extensive experiments would, ultimately, tend to promote the arts and manufactures of this country. Besides, as an excellent writer most justly observes, ," a for- bidding distance, and awkward jealousy, seem to subsist between the theorists and the practical men engaged in the cultivation of mechanics in this country." < And, therefore, it is a. most laudable task to endeavour " to shorten this di- stance, and to eradicate this jealousy." For, while we prize the deductions of sound theory, and rely firmly upon their results, we should nevertheless recollect, that, " as all general prin- ciples imply the exercise of abstraction, it would be highly injudicious not to regard them in their practical applications as approximations, the defects of which must be supplied, as in- deed the principles themselves are deduced from experience." - Theoretical as well as prac- tical men would not only greatly promote their mutual interests by blending and uniting their JOINERS' INSTRUCTOR. 240 efforts, but render an essential service to me- chanical science. There are few persons who do not enjoy sufficient occasional leisure and opportunity for making some experiments on the strengths of bodies : the results of their several efforts would tend to elucidate and ad- vance the subject. But since these experi- ments, when made on an extensive scale, are both laborious and beyond the means of most individuals who may be inclined to inquire into the subject, it is singular, in an empire like this, which is certainly the most eminent in the world for its extensive mechanical structures, that a judicious series of suitable experiments . has not been made, on a liberal and extensive scale, with a view to the knowledge of those laws which regulate the strengths of different mate- rials, according as different strains may operate. Amid this deplorable want of information in our own country, we must have recourse to the Continent, and solicit the aid of those philoso- phers who have investigated the subject with the most attention. Buffon and Du Hamel were supplied by the old government of France with ample funds and extensive apparatus for carrying on the necessary experiments. A de- scription of these is to be found in the Memoirs of the French Academy for 1740, 1741, 17492, 250 THE CARPENTERS' AND and 1768; as well as in Du Hamel's ingenious performances sur /' Eaploitation des Arbres, et sur la Conservation et le Transport de Bois. Our readers may not be dissatisfied with an abstract of M. Buffon's experiments. This ingenious philosopher prosecuted, during two years, a variety of experiments on small battens of oak. - He found, however, from these experiments, that the variation in a single layer, or in part of a layer, either more or less, or even a different disposition of them, had so much in- fluence, that he was under the necessity of aban- doning the method, and proceeding to operate on the largest beams that he could possibly break, The annexed table shows a series of experiments on bars of sound oak, four inches square, and free from knots. 1 2 3 4 5 60 | 5350 | 3+5 29 26 1 5275 1.45 yA 68 | 4000 | 375 |. 15 63 | 4500 | 47 13 77 | 4100 | 4585 14 7 P { $950 ° 105'5 12 84 | 3625 | 5°83 | 15 82 | 3GO0 | 6-5 15 100 | 7 98 | 2025 | s el Pu acad o N~ ~*~ ro) JOINERS' INSTRUCTOR. 251 The first column exhibits the length of the bar, in clear feet, between the supports. The second expresses the weight of the bar in pounds, on the second day after it was felled, as evinced by experiments performed on two bars of each sort. - Each of the first three pairs consisted of two cuts off the same tree. The one next the root was always found to be the heaviest, and Buffon uniformly observed, that the heaviest was constantly the strongest, and recommends this particular as a sure rule for the choice of timber. He observes, also, that this always proved to be the case when the timber, by growing vigorously, had formed very thick annual layers : but this circumstance takes place only during the advances of the tree to a state of maturity, because the strengths of the different circles ap‘proach in a gradual man- ner to equality, during the tree's healthy growth, when they decrease in these parts in a contrary manner. The third column represents the number of pounds required to break the tree in the course of a few minutes. The fourth column points out the number of inches in which a tree bends down before breaking. 252 THE CARPENTERS' AND The fifth column shows the time at which it broke. The experiments made on other sizes were conducted in a similar way. All the beams were formed square, and their sizes in inches are signified at the head of the columns in the following table. In the first column are ex- pressed their lengths in feet. A to l 4 5 6 7 | 7 | 11525 | 18950 | 32200 | 47649 | a1s25 8 | 4550 9787 | 15525 | 26050 | 39750 | 10085 9 | 4025 8308 | 13150 | 22350 | 32800 8964 10:1 :S8612 1125. |; 11250 | 19475 .1 27750 8068 12 | 2987 6075 9100 | 16175 | 23450 6723 14 5300 | 7a7s | rsge5 | 1077s | s763 16 4350 | 6s62 | 11000 | 16375 | so42 18 3700 | 5562 | 9245 | 13200 | 4482 20 8225 | 4950 | ss7s | 11487 [ 4034 22 2975 3667 24 2162 3362 2s 1775 2881 M. Buffon, in order to effect uniformity in his experiments, had all his trees felled in the same season of the year, squared the day after, and operated on the third day, when he found also, that the strength of oak timber dimi- nished much in the course of drying. After a piece of this green timber had been placed in the situation required for the experiment, and weights nearly sufficient to break it were ap- JOINERS' INSTRUCTOR. 253 plied with briskness, a very sensible smoke was perceived to issue from its two ends, with a sharp hissing noise, which continued during the whole of the time the tree was bending and cracking. - This result undeniably proved that the whole length of the tree was strained (which may be inferred, indeed, from its bend- ing through its whole length); and nothing, per- haps, could evince in a stronger manner the powerful effects of compression. The experiments made by our philosopher on the five inch bars, he considered as his stand- ard of comparison, and he accordingly pro- secuted his inquiries to a greater extent, on pieces of this dimension. The deductions derivable from the theory which has been adopted would make us think that the relative strength of bars of the same section is inversely as their lengths ; but Buf fon's experiments (excepting those in the first column) deviated very considerably from this rule. For instance, by referring to the last table, we perceive that the strength of the bar, 28 feet long and 5 inches square, is 1775 ; and that the strength of a bar 5 inches square, and 14 feet long, is by the same table 5800 ; but. we know that the strength of the 14 feet bar ought to be double that of 28 feet ; in which $54 THE CARPENTERS AND case the strength of the latter bar in this would be 2650, whereas it is only-1775. Again, the bar of 7 feet ought to possess double the strength of that of 14 feet ; but this is not the case, for the strength of the latter is 5800, whereas half the strength of the former is 57625. In like manner, {the strength of the 8 feet bar ought to be treble that of the 24 feet, whereas the strength of the latter is 2162, while one-third of the strength of the former is 8262-8. So also the strength of the 7 feet bar ought to be four times that of the 28 feet bar, but the strength of the latter we perceive is only 1775, while one-fourth of the strength of the former is 2881. - The column a specifies the strength which, by the theory, each of the five inch bars ought to have exhibited. The foregoing defect seems to prevail in all the experiments from the first to the last; it may be observed also in the experiments of Be- lidor, as well as in all those we have noticed ; from whence we may conclude that it is a law of nature depending on the true principles of 'cohesion, and the invariable operations of me- chanics. But still the difficulty is inexplicable; for the only effect produced by the length of a beam is an increase of the strain at the section JOINERS' INSTRUCTOR. - 255 of fracture, arising from the operation of the in- tervening beam as a lever; though we cannot see clearly how the mode of action of the fibres in this section is effected, so as to change either their cohesion or the situation of its centre of effort; and yet something of the kind must happen. f There are certain circumstances, however, which must contribute to render a smaller weight sufficient (as in the experiments of Buf- fon), to break a long beam, than in the exact inverse proportion of its length ; for the weight of the beam itself increases the strain as much, as if half of it were added to the strain, and- operated on it as a farther weight. The weight of every beam on which Buffon performed his experiments was very nearly 74 pounds per each cubic foot ; but these beams were by far too small to account, in a satisfactory way, for the deviation from the theory. Even the half weights of the 5 inch beams, whose respective lengths were 28, 14, and 7 feet, were only 182, 92, and 45 pounds, which rendered the actual strains in the experiments 11560, 5890, and 1956 ; but these, it is evident, deviate consider- ably from the before-mentioned proportions of the beams 4, 2, and 1. Buffon observes, that healthy trees are uni- 256 'THE CARPENTERS' AND versally strongest at the root end; of course when a long beam is made use of, its middle point, where the fracture takes place in the ex- periment, is situated in a weak (perhaps the weakest) part of the tree. The trials, neverthe- less, of the 4 inch beams proved that the differ- ence arising from this cause is almost insensible. Again, it is probable that the relative strength of beams decreases faster than the inverse ratio of their lengths. We have already observed, that when a weight operates on the middle of a beam so as to break it, its whole length is af- fected, and therefore a certain definite curvature of a beam, of a given form, is always accompa- nied by rupture. Let us suppose two beams, whose lengths are respectively 10 and 20 feet, to be bent to the same degree, at their places of fixture in a wall ; and we shall find that the weight operating on the former is nearly double that which hangs on the latter.-But the form of any portion of these two beams (say for 5 feet immediately adjoining to the wall) differs considerably, since the curvature of the first beam at this distance is only one half of its curvature at the wall, while the curvature of the latter in the corresponding part is three- fourths of the same curvature at the wall. Hence, therefore, through the whole of the in- JOINERS' INSTRUCTOR. 257 termediate space of 5 feet, the curvature of the former is less than that of the latter, and conse- quently the latter beam must be weaker through- out. - It likewise occasions the fibres of the beam to slide more on each other, whereby their lateral union is effected ; and therefore those possessed of greater degrees of strength will not render assistance to those that have less. In addition to this, the force with which the fibres of shorter beams are pressed laterally on each other is double, and must of course impede the mutual sliding of the fibres. In fact this lateral compression is not only calculated to change the law of longitudinal cohesion, but to - increase the strength of the very surface of fracture. It is much to be desired, that the engineer would carefully remember, that a beam of qua- druple length, instead of possessing one-fourth of the strength, has only about one-sixth; and the ingenious theorist should inquire into the nature as well as the cause of this diminution, in order that he may be enabled to furnish the mechanic with a more accurate rule for com- putation. Our want of an intimate acquaintance with the law by which the cohesion existing between particles is changed by an alteration of distance, s 258 THE CARPENTERS' AND deprives us of the means of discovering the pre- cise relation which exists between the curvature and the momentum of cohesion, and in order to obtain some rules whereby the strengths of dif- ferent solids may be calculated, it will be ne- cessary to multiply experiments. The experi- ments of Buffon furnish us, however, with con- siderable assistance in this particular,. If we . select, for instance, any number in the column of the & inch beams, and add to it the sum of half the weight of the beam, with the constant number 1245, a set of numbers will be given very near the reciprocals of the lengths. From this we may readily deduce a convenient for- mula, very easy to be remembered. Let the length of the beam of 5 inches square be designated by a, the number 1245 by m, and the weight known to break the beam by w, then (w+m) a weight required to break the 7 foot bar is 11525, (@+{+m)a A we shall have -m» - p.. Thns,. the and let / be 18, then we shall have g- f 1245 i (115Q51-2 1 )7__ 189415 = 8720 == 1 . differs about ere from the result furnished us by that experiment. - This formula may be applied JOINERS' INSTRUCTOR. 250 with success to all the other lengths, those of 10 and 24 feet. - Although this formula cannot be admitted as universally true, yet it will be found to be tolerably correct in a great variety of lengths. We will next consider the relation that exists between the strength and the square of the depth of the section. By comparing the num- bers in any horizontal row of the table, we shall perceive on trial, that the numbers of the five inch bars are uniformly greater than those of the rest. If the numbers, however, in this column be omitted, or uniformly diminished about one-sixteenth, as to their strength, the different sizes will be found to differ but little from the ratio of the square of the depth, as de- termined by theory ; though we should observe that a small deficiency takes place in the larger beams. Our next inquiry will be directed to the ab- solute cohesion and the relative strength. The values deducible from experiments on absolute strength must be confined to very small pieces, in consequence of the very great force which is required to teat them asunder. The whole we can furnish on this head, for the consideration of our readers, are two passages extracted from Muschenbroék's Essais de Physique. In one g 2 A 260, THE CARPENTERS' AND of these passages he observes that a piece of sound oak $6760!” an inch square was torn asunder by 1150 pounds; and in the other, that an oak plank one inch in thickness, and twelve inches broad, will suspend about 189162 pounds. - We may conclude from these passages, that the cohesion of a square inch is 15755 and 15763 pounds. - Bouguer, another experiment- alist, observes, that a rod of sound oak, one- > fourth of an inch square, will be torn asunder by 1000 pounds, which furnishes the round num- ber 16000, for the cohesion of a square inch. It may not be unprofitable to the reader to compare those circumstances with the experi- ments made by Buffon on four inch beams. The absolute cohesion of the before-mentioned section is 16% 16000=256000. - Were every fibre to exert its entire energy, at the moment in which the fracture takes place, the effect of the momentum of cohesion would be the same as if it had entirely acted at the centre of gravity of the section, at the distance of two inches from the axis of fracture, and therefore it is 256000 x2=512000. We shall find by reference to the last table, that the beam 7 feet long and 4 inches square was broken by a weight of 5812 pounds, suspended on its middle ; but if it had JOINERS' INSTRUCTOR. 261 been suspended at its extremity, projecting 42 inches from a wall, it would have been broken by one-half of the foregoing weight, viz. 2656 pounds. The momentum of this strain is 42% 2656 = 1115592; being in equilibrio with the actual momentum of cohesion, which is 111552 instead of 512000 ; consequently the strength is diminished in the proportion of 512000 to. 111552, or nearly as 4 59 : 1. The ignorance that prevails relative to the particular situation of the centre of effort, renders it altogether useless to consider the full cohesion that employs its energies, at the cen- tre of gravity, and produces the momentam 512000; we may, however, convert the whole into a simple multiplier » of the length, and say, that n times the length is to the depth, as the ab- solute cohesion of the section is to the absolute strength. j If, therefore, we represent in inches the breadth by 6, the depth by d, the length by /, and the absolute cohesion of a square inch by s ; the relative strength, or the external force p, which bd s 9 [ Wecannot attribute this diminution of strength to any inequality of those cohesive forces which may be exerted at the instant of fracture; since balances it, is in round numbers 202 THE CARPENTERS' AND we must know, from the centre of effort in a rectangular beam being situated at one-third of the height, that the relative strength would be 9 5218, and that p would be 8127 instead of 2056. This great diminution may be ascribed to the compression of the under part of the beam ; and we have before observed, that the forces actually exerted by the particles of a body, when stretched or compressed, are very nearly pro- portioned to the distances to which the particles are drawn from their natural positions ; and though in cases of great compression the forces may increase in a faster ratio, yet this increase will produce no sensible change in the present question, because the body is broken before the compressions have proceeded so far ; in fact, we may conceive that the compressed parts are crippled before the extended parts are torn asunder, - Muschenbroék asserts this with a con- siderable degree of confidence, and says that although oak will suspend half as much again as fir, yet it will not- support, as a pillar, two- thirds of the load which fir will support in that form. The experiments of Buffon furnish us with a useful practical rule, without our being obliged JOINERS' INSTRUCTOR. 263 to rely on any value of the absolute cohesion of - oak. - From knowing that the strength is nearly as the breadth, and the square of the depth, and the inverse of the length; and taking for the sake of convenience the length of the beam in feet, and its breadth and depth in inches; and also knowing from the table, that a beam four inches square, and seven feet between the supports, is broken by 5312 pounds, we may conclude that a batten, one foot in length be- tween the supports, and one inch square, will be broken by 581 pounds. Hence the strength of any other oak beam, or the weight barely re- quired to break it, when hung on its middle, is 3$1 97d; b, d, and I, respectively representing the breadth, depth, and length. In some of our former inquiries, we have found a considerable deviation from the inverse proportion of the length, and we must neces- sarily accommodate our rule to it. When the number 1245 was added to each of the numbers in the column of the five inch bars, a set of numbers was produced, which were very nearly reciprocals of the lengths ; if we make a simi- lar addition to the other columns, but in pro- portion to the cubes of their dimensions, we shall have nearly the same result. Hence, to 264 THE CARPENTERS' AND find the necessary number, say, as 5° : 4° : : 1245 : 647, the required number ; this added to 5312, gives 5959, the 64th part whereof we may call 93, which answers to a bar 7 feet long, and an inch square. Hence, 98%K7 will be the reci- procal corresponding to a bar of one foot ; this is 651, and after taking from it the present cor- rection, which is at? == 10, 641 will remain for the strength of the bar. From this result, we 651 b d* obtain the general rule -10 bd*'=p, which we may otherwise state in words as follows, in order to render it clearer to those who are not versed in Algebra. Multiply the breadth of the beam, in inches, twice by the depth, and this again by 651, and di- vide the whole by the length in feet. From the quotient, take 10 times the product (which arises by multiplying the breadth of the beam, in inches, and the square of the depth), and the remainder is the number of pounds required to break the beam. Example.-Required the weight necessary to break an oak beam, 16 feet long between the props, and seven inches square. Here p= 651 % L§Of——10><7x7°=~_10526, whereas the experiment gives 11000. JOINERS' INSTRUCTOR. 265 the weight necessary to break an oak beam, 12 feet long between the Props and six inches square. ; Here i pz=b651 xfix -10 x 6 yx 6° =: 9558, while the experiment furnishes us only with 9100. Example.-It is required to determine the weight barely necessary to break an oak beam 20 feet long between the props, and five inches square. Here p= 651 Xijz-(ég— 10. % 5 % while the table exhibits only 8225. We may compare, in like manner, any other dimension ; but we shall find that the rule is most deficient when applied to the five inch bars, which, we have before observed, appear stronger than the rest. -- The sure way of applying the foregoing rule is to suppose the beam square, by increasing or diminishing its breadth, till it becomes equal to its depth ; then find the strength, by this rule, and increase or diminish it agreeably to the change which has taken place in its breadth, when there can be no doubt that the strength of the beam, given as an example, will be double 266 THE CARPENTERS' AND that of a beam of the same depth, and half the breadth. It may be necessary, perhaps, to remind the reader, that the whole of the preceding calcula- tions and observations are founded on the sup- position, that the weight applied is the greatest which a beam will bear for a very few minutes. Buffon observes, that two-thirds of this weight will sensibly impair the strength of the beam, and indeed will frequently break it, if per- mitted to operate continually, for two or three months. - One-half bent the beam when applied only for a few minutes ; but though, as he ob- serves, this weight may be borne by it for any length of time, still the beam will contract a certain curvature, from which it will not easily get into its pristine state. One-third seemed to produce no permanent effect on the beam, which recovered its original shape, even after it had been kept loaded for several months. But this depends on the pieces being seasoned ; for a piece just felled will break under one-fourth .. of a given weight, while a third of the same weight may be laid on the well seasoned piece, for any length of time, without giving the beam a set. We are destitute of experiments on the JOINERS' INSTRUCTOR. 267 strength of other kinds of timber. - M. Buffon says, that fir possesses about $ths of the strength of oak. - Parent, however, asserts, that it pos- sesses 4ths ; while Emerson says it has 4ds. Before we conclude our inquiries into this particular strain, we would wish to impress on: the minds of practical men in general the ne- cessity of avoiding transverse strains as much as possible, since the injury that many struc- tures have sustained from their influence is in- calculable. Bodies may be wrenched or twisted. This species of strain operates on all axles which connect with the working, or moveable parts of machines. The resistance occasioned by this species of strain must be proportioned to the number of particles, when all the particles act alike. In proof of this, let £ r c n represent a body possessing insuperable strength, but cohering in a weaker manner in the common surface a B which separates the body into two parts, viz. A B C D, A B F E ; then, if one part, as a B c D, is supposed to be pushed laterally in the direction A B, it becomes evident that it can only yield there, and that the resistance produced will be proportioned to the surface. ’ 208 THE CARPENTERS' AND The same result will take place, if we suppose a thin cylindrical tube to be twisted in contrary directions, in which case it will undoubtedly fail first in that section where the cohesion of the fibres is the least. This section forms the circumference of a circle; for which reason the particles composing the two parts contiguous to this circumference will be drawn from each in a lateral direction, and the total or absolute resistance will be as the number of particles exhibiting an equal degree of resistance, which is in fact the circumference. - Within the cir- cumference to which we have just alluded, let us conceive a series of tubes till they reach the centre. Now, if the particles composing each of these tubes exerted a similar degree of force,. the resistance offered by eachwring of the section would be as its circumference and its breadth, (which we will suppose to be indefinitely small) the entire resistance would be as the surface ; and this would represent the resistance of a solid cylinder. But the external parts of a cylinder when twisted by the application of an external force applied to its circumference, will suffer a greater circular extension than the in- ternal, and it appears that this extension will bear a proportion to the distance of the parti- cles from the axis. This proportion would JOINERS' INSTRUCTOR. 2650 seem to be very probable, and if it really exist, the forces simultaneously exerted by each par- ticle will be as their several distances from the axis. - Consequently, the entire force exerted by each ring will be as the square of its radius, and the accumulated force actually exerted will be as the cube of that radius. By referring to Fig. 28, we shall have the ac- cumulated force exerted by the whole of a cy- linder, whose radius is a c, is to the accumu- lated force exerted at the same by the part whose radius is c ®, as a C to c r'. The whole cohesion exerted in this instance is just two-thirds of what it would be if all the particles exerted the same attractive forces as are exerted by the particles in the external circumference. - This will appear evident, if we first suppose the rectangle a c ¢ a to be erected in a position perpendicular to the plane of the circle, along - the line a c, and then, that it is permitted to re- volve round the a c. 'This rectangle, by its motion, will generate a cylinder, whose height will be denoted by ce or a a, which will have the circle 1am fof its base. Next, if the tri- angle c c a be likewise permitted to perform a revolution around the line c c, it will describe the surface of a cone. - Now, the cylindrical sur- » 270 THE CARPENTERS AND face supposed to be generated by a a, will re- present the whole cohesion exerted by the cir- cumference a x ®; the cylindrical surface ge- nerated by ® e, will show the cohesion exerted by the circumference ® 1 m; the solid gene- rated by the triangle c a a will display the co- hesion exerted by the entire circle a u x; and the cylinder generated by the rectangle a cc c will exhibit the measure of the cohesion ex- erted by the same surface, supposing each par- ticle to suffer the extension a a. It is evident, in the first instance, that the solid produced by revolution of the triangle c E e, is to that generated by a a c, as ® c" to Aa C. - In the next instance, the solid generated by a a c is two-thirds of the cylinder, because the cone generated by c ca forms one-third of at. It may now be supposed, that the cylinder may be twisted so powerfully, that the particles situated in the exterior circumference must lose their cohesion, when little doubt can be entertained that it will be wrenched asunder, in consequence of all the inner circles giving way in succession. If we admit this, then a body, the texture of which is homogeneous, will resist a simple twist with two-thirds of the force with which it would resist it, if an attempt were JOINERS' INSTRUCTOR. #71 made to force one part laterally from the other, or with one-third part of the force which will cut it asunder by a square-edged tool. When two cylinders are wrenched asunder, we must of necessity conclude, that the exter- nal particles of each are placed just beyond their limits of cohesion, that they are extended equally, and operate with equal forces; from whence it follows, that in the instant of fracture the entire sum of the forces actually exerted is as the squares of the diameters. The real strength of the section, and the re- lation it bears to its absolute lateral strength, being now ascertained, our next business is to inquire into its strength, as it relates to the external force which may be employed to break it. The straining force and the cohesion oppose each other, on the principle of levers ; and the centre of the section may be the neutral point, the position of which is not disturbed. Let r represent the radius of the cylinder, f the force exerted laterally by an exterior particle, v the intermediate distance of any circumference, and a the infinitely small interval between the con- centric arches. Now, the forces being as the extensions, and they being as their distances from the axis, the cohesion, actually exerted at 272 THE CARPENTERS' AND any part of the ring, will De sz s. Ae Aoree I 7‘ J' # exerted by the entire ring ; and the mo- s mentum of cohesion (which, in any ring, is as f) the force multiplied by its lever) will be --- Consequently, the accumulated momentum wxll i as /t be/ yI and when x becomes equal to 1tw1Hbe——- _f’ = if 9%, From hence we leam that the particular strength of an axle, by which it resists being wrenched asunder, by a force acting at a given distance from the axis, is as the cube of the diameter. Again, the expression £5 75, may be decom- posed into the factors fr° & 4 r, the former of which f° expresses the full lateral cohesion of the section ; while the momentum thereof is the same as if the full lateral cohesion were ac- cumulated at a point, distant from the axis, one- fourth of the radius of the cylinder. Let d denote the measure of the diameter of the cylinder in inches, £ the number of pounds which measure the lateral cohesion of a circular inch, / the length of the lever, by which the JOINERS' INSTRUCTOR. 276 straining force represented by p is supposed to f 4 ; act, and we shall have a> rE I, from which 3 may be deduced, 17:45;- In general, therefore, the particular strength which enables an axle to resist its being wrenched asunder by twisting is as the cube of its diameter. s The interior parts do not act so powerfully as the exterior ; for if a hole be bored out of an axle, equal in size to one-half of its diameter, the strength will be diminished only one-eighth, while the quantity of matter will decrease one- fourth ; from whence it follows, that hollow axles are stronger than solid ones containing the same quantity of matter. The propriety of engineers introducing this very important improvement into their ma- chines becomes now very obvious, since the parts so constructed have not only the advan- tage of being much stiffer, but they furnish much better means for fixing the flanches made use of to connect them with the wheels or levers by which they are turned and strained. We shall now introduce to the notice and observation of the student a variety of pro- T 274 THE CARPENTERS' AND positions and deductions, chiefly selected from Emerson's excellent treatise of Mechanics. * Proposition.-If a beam of timber, (Fig. 1, Plate $4,) be supported at c and B, lying upon the wall, a c ®, with one end, and if c be the centre of gravity of the whole weight sustained ; and the line ra n be drawn perpendicular to the horizon, and c r, and B n, to c B, and B F, drawn ; I say, The weight of the whole body 4: Pressure at the top c, B H, Thrust or pressure at the base} ® B, and in these se- veral directions." If the beam support any weight, the beam and weight must be considered as one body, whose centre of gravity is e. Then the end c is supported by the plane s c ®; and the other end s may be supposed to be sustained by a plane perpendicular to s r ;- therefore the weight and forces at c and B are respectively as r n, B H, and 'B F. B, are respectively as * Cor. 1.-Produce r s towards a, then B a is the direction of the pressure at B ; and the pressures at B in the directions B q, F D, D B, are as F B, F D, D B.' Cor 2-Draw o r perpendicular to B ¢, and » MMCO site msr JOINERS' INSTRUCTOR. 875 draw c D, then the weight, pressure at the top, direct pressure at bottom, and horizontal pres- sure at bottom, are respectively as c B, B D, D c, and p »." «" For since the angles B c F, B D F, are right, a circle described upon the diameter s r will pass through c o. cp= B r D standing on the same arch s v, and because the Le B u and zs at o are right, Bu r= CB Dj; therefore, the triangles r m r and coB n» are similar, and the figure s ux n r similar to the figure p B r c, whence r :B H :B F :B D: are aig bxip.D.: p-:elzand p 1.!! * Cor. 8.-All this holds true for any force instead of gravity, acting in direction a p." «* Proposilion. -If s ¢, (Fig. 2,) be any beam bearing any weight, « the centre of gravity of the whole ; and if it lean against the perpendicular wall c a, and be supported in that position : draw s a, c F, parallel, and r a p perpendicular to the horizon ; and draw r B, then The whole weight, 1 F D, Pressure at the top ¢, B D, Thrusts or pressure at the} r B, and in the same: directions." «< For the end c is sustained by the plane a c; and if the end s be supposed to be sustained by TA bottom s, are respectively as, 276 THE CARPENTERS' AND a plane perpendicular to r B; then the weight, and pressure at top and bottom, are as D F, D B, r B. If you suppose the end s is not sustained by a plane perpendicular to r B, the body will not be supported at all." © Cor. If r s be produced to a, then B q is the direction of the pressure at B; and the per- pendicular pressure at B (r ») is equal to the weight ; and the horizontal pressure at B (B D) is equal to the pressure against c." ** Proposition.-If a heavy beam, or one bear- ing a weight, be sustained at c, (Fig. 3,) and moveable about a point c ; whilst the other end s lies upon the wall s s, and if u o r be drawn through the centre of gravity G, perpendicular to the horizon, and s r, c n, perpendicular to s c, and c r be drawn ; then The whole weight, H F, Pressure at B, 'c, Force acting at ¢, CB, and in these di- are respectively as _ rections." © For the end s is sustained hy the plane c B, and the end c may be supposed to be sustained by a plane perpendicular to r c, or by a cord in direction cr. - Then since x c is parallel to s r, the weight, force at c, pressure at s, are respect- ively as u F, C F, 4 c." * Cor. But if, instead of lying upon the in- JOINERS' INSTRUCTOR. 277 clined plane at B, the end s be laid upon the horizontal plane a B, then the weight and the pressure at B and c are respectively as B c, c c, and B ¢; and in this case there is no lateral pressure." «For s r will be perpendicular to B a, and parallel to _ r, and consequently c r is also pa- rallel to u r, therefore the forces at c, G, B, are as 6 C, s c, and c G." « Proposition.-If a heavy beam s c, (Fig. 4,) whose centre of gravity is c, be supported upon two posts B a, c p, and be moveable about the points 4a, i8,:¢, D.;, and if a .B, p ¢, produced, . meet in any point g, of the line a r, drawn per- pendicular to the horizon ; and if from any point r, in the line a r, r x be drawn parallel to a £ ; I say, The whole weight, H F, Pressure at G, HEB, Thrust or pressure at B, E F, and in these di- Atections." For the points a, B, ¢, », being in a plane per- pendicular to the horizon, the body may be sup- posed to be supported by two planes at B, c, perpendicular to a B, D c; or by two ropes B #, c x; and in either case, the weight in direc- tion # c, the pressure at B c, in directions H B, H C, afe as H -P,E-F, :and n-. are respectively as 278 THE CARPENTERS' AND Cor. Hence, whether a body be sustained by two ropes, B H, C H, Of by two posts a B, C D, or by two planes perpendicular to s a, c n ; the body then can only be at rest, when the plumb- line g a r passes through o, the centre of gra- vity of the whole weight sustained, or, which is . the same thing, when a B, p c, intersect in the plumb.line u a r, passing through the centre of gravity. Scholium. «* By the construction of these last four pro- positions, there is formed the triangle of pres- sure, representing the several forces. - In which, the line of gravity (or plumb-line passing through the centre of gravity) always represents the ab- solute weight, and the other sides, the corre- sponding pressures." «* Proposition. -If several beams a B, B C, C D, &c. (Fig. 5,) be joined together at B, c, D, &c. and moveable about the points a, B, c, &c. be placed in a vertical plane, the points a r, being fixed, and through s, c, D, drawing r ?, 8 m, ? p, perpendicular to the horizon, and if several weights be laid on the angles s, c, p, &e. so that the weight on any angle c may be as - s-, then all the beams. wil be s. mCcByXs.m c b E kept in equilibrio by these weights." JOINERS' INSTRUCTOR. - 270 Produce oc tors Then s. waBo:is. 2ABr:: BX S%: A B9 § . A BC a and s.s c o :s.D c s:: weight : force in direc- C: J€ 8S:~BD;C s $ BC. D the equilibrium, must be equal to the force in di- weight s : force in direction s c- % fon °C" B= ;, which, to preserve B ¥ s. als n cs g4. abo. winos {iste mi' $.AOriew sik ono whence, B : ¢ : : -- : ----. _ And by f sit wipn|redsalin es { $'¢ 8C the same way of reasoning, c : D : e acs. rection sc; that is, ng Therefore, ea equo, weight s : weight 8A BOC _ sa cipis» orf. gs g wk Bir ¥ ai$i. bi Cis X tuk b ¢ 5 ; § CGD £ S AK Siz CGB £:) 8): C bpX®® . Ep Cor. 1.-Produce c p, so that » w may be equal to c r, and draw w a, parallel to » p, cutting n E, in w, then the weight at c, the forces in directions c B, and c D, are as r B, C B, and c r, respectively, and the weight c is to the weight », as B r to w a. *©Cor. 2.-The force or thrust at c, in direc- tion Cc B, or at B, in direction B c, is as the secant of the elevation of the line s c, above the horizon." ¢. 280 THE CARPENTERS' AND «* For, force in direction c s : force in direc- C B :C 7: : 1 8...0.7.B, Of 1C 10, Of. 8 C Db :s .r B c:: cos. elevation of c » : cos, ele- vation of cs :: sec. elevation of c B : sec. ele- vation c »; because the secants are reciprocally as the cosines." " Cor. 8.-Draw c p, » m, parallel to D E, c B, then the weights on c and p, to preserve the equilibrium, will be as c m to » p, and therefore, if all the weights are given, and the position of two lines c », » E, then the posi- tions of all the rest, c B, B a, &c. will be suc- cessively found. For let the force in direction c 1, or pc, be c b, then c p is the force in di- rection » ® and » » in direction cs. And o p, or the weight », is the force compounded of » c, c p 3; and c m, or the weight c, is the force compounded of c p, p m." " Cor. 4.-If the weights lie not on the angles B, c, D, &c. let the places of their centres of gravity be at g, A, k, LL and let g, A, k, I, also express their weights. And take the weight s A & ko ) Jib) # k p Cok o-- --A. C- 4+ pus --- $# ws 8 A.tlC 0 Bc+cD’ C p I E ) ( +51, &c. then B, c, D, &¢. will be the weights lying upon their respective angles." « Cor. 5.-If the weights were to act upwards, JOINERS' INSTRUCTOR. @81 in the directions » c, p D, &c. or, which is the same thing, if the fignre a, B, C, b, B, r, was turned upside down, and the weights remain the same, and the points a r be fixed as before ; all the angles at B, c, D, &c. and consequently the whole figure, will remain the same as be- fore; and that whether the lines a B, B ¢, C D, &c. be flexible or inflexible cords or timbers. " This will easily appear by the demonstra- tion of the prop. For the ratio of the forces at any angle c will be the same, whether they act towards the point c, or from it ; that is, it will be the same thing, whether the weight at any angle c, acts in direction c m, or c s; and as the forces were supposed before to thrust against c, the same forces now do pull from it." " Scholium.-If D a, B r, (Fig. 6,) be a semi- circle, whose diameter is D r, draw a c perpen- dicular to » r, then the force or weight at any place a, to preserve the equilibrium, will be re- ciprocally as a G, or directly as the cube of the secant of the arch s a." < Likewise it follows from Cor. 5, that if any cords of equal lengths be stretched to the same degree of curvature, the stretching forces will be as the weights of the cords." «* Proposition.-If the distance of the walls 282 THE CARPENTERS' AND a b and B c be given, (Fig. 7,) and a B, a ¢, be two beams of timber of equal thickness, the one horizontal, the other inclined ; and if two equal weights r, aq, be suspended in the middle of them, the stress is equal in both, and the one will as soon break as the other, by these equal weights." ( A B ® "*For a cc: as:: weight p i--t =-- pressure against the plane, or part of the weight the beam a c sustains. And the stress upon a c is —A~%PXA c or aA BXP ; and the stress on a B is axa s, which is equal to a BXP, because the weights r, q are equal. - Therefore, the stress being the same, and the beams being of equal thickness, one will bear as much as the other, and they will both break together." «* Cor. 1. - If the beams be loaded with weights in any other places in the same perpendicular line as r, G, they will bear equal stress, and one will as soon break as the other." « For they are cut into parts similar to one another ; and therefore stress at r : stress by P 2 6 A B Ceo -_1_ 20 . % * # % ® ..AFC.4AC..AGB.4..StreSSbyB. stress by a or stress by p. Therefore stress at P m-=-stress:i at B.!" FJOINERS' INSTRUCTOR. 283 " Cor. 2. ~If the two beams be loaded in pro- portion to their lengths, the stress by these weights, or by their own weights, will be as their lengths, and therefore the longer that stands aslope, will sooner break." * For the stress upon a C was A BXPr, and the stress on a B was a BXq@; but since p and qare to one another as a c and a B, therefore the stress on a c and a B will be as a s X a Cc and a 6 x a n': that is, as' a oto 4 5s. ~Andan regard to their own weights, these are also pro- portional to their lengths." € Propo$ition.—Let A B, a c, (Fig. 8,) be two beams of timber of equal length and thickness, the one horizontal, the other set sloping ; if c D - be perpendicular to a B, and they be loaded in the middle with two weights, P, q, which are to one another as ac to a p, then the stress will be equal in both, and one will as soon break as the other." A D ades n ..A .D, 4: T zx—Epzpressure of p in the middle of a c. And, by supposition, a c : A D f A, Di; :,p.:1:0 3 therefore wp the weight in the middle of a s:. Therefore the forces in the middle of the two beams are the same, and the lengths of the beams being the same, there- 84 THE CARPENTERS' AND fore the stress is equal upon both of them ; and being of equal thickness, if one breaks the other will break." « Cor. If the weights r, q, be equal upon the two equal beams a B, a c, the stress upon a B will be to the stress upon a c, as a B or a C to a p. The same holds in regard to their own weights." For the weight aq is increased in that pro- portion. * Proposition.-If several pieces of timber be applied to any mechanical use where strength is required, not only the parts of the same piece, but the several pieces in regard to one another, ought to be so adjusted for bigness, that the strength may be always proportional to the stress they are to endure." «This proposition,"* Mr. Emerson observes, «is the foundation of all good mechanism, and ought to be regarded in all sorts of tools and instruments we work with, as well as in the se- veral parts of any engine ; for who that is wise will overload himself with his work tools, or make them bigger and heavier than the work requires? Neither ought they to be so slender as not to be able to perform their office. In all engines, it must be considered what weight every beam is to carry, and proportion the JOINERS' INSTRUCTOR. o 285 strength accordingly. - All levers must be made strongest at the place where they are strained the most ; in levers of the first kind, they must be strongest at the support; in those of the second kind, at the weight ; in those of the third kind, at the power, and diminish propor- tionally from that point. The axles of wheels and pulleys, the teeth of wheels which bear greater weights or act with greater force, must be made stronger, and those lighter that have light work to do. Ropes must be so much stronger or weaker as they have more or less tension ; and, in general, all the parts of a ma- chine must have such a degree of strength as to be able to perform its office, and no more ; for an excess of strength in any part does no good, but adds unnecessary weight to the ma- chine, which clogs and retards its motion, and makes it languid and dead. And, on the other hand, a defect of strength where it is wanted will be a means to make the engine fail in that part, and go to ruin. So necessary it is to ad- just the strength to the stress, that a good me- chanic will never neglect it, but will contrive all the parts in due proportion, by which means they will last all alike, and the whole machine will be disposed to fail all at once. - And this will ever distinguish a good mechanic from a bat 286 THE CARPENTERS, AND one, who either makes some parts so defective, imperfect, and feeble, as to fail very soon, or makes others so strong or clumsy, as to outlast all the rest." -** From this general rule follows : ; © Cor. 1.--In several pieces of timber of the same sort, or in different parts of the same piece, the breadth, multiplied by the square of the depth, must be as the length, multiplied by the weight to be borne ;- for then the strength will be as the stress." " Cor. 2.-The breadth multiplied by the square of the depth, and divided by the pro- duct of the length and weight, must be the same in all." 4 « Cor. 3.-Hence may be computed the strength of timber proper for several uses in building. As, « First.-To find the dimensions of joists and boards for flooring. Let b, d, l, be the breadth, depth, and length, of a joist, » == the number of them, a == their distance, g == the depth of a board, w== the weight ; then n b d: == the strength of all the joists, and t == the stress on them, also, » / g* == the strength of the boards, and w a, their stress ; therefore, n boo ntg d * et OB 22 for the distance =- _ and # -= t I tw a C b JOINERS' INSTRUCTOR. 287 of the joists, or the length of a board between them. Or b _l”g*, or d' _l”_g_2_, and so on,‘ ; 4 & b &a according to what is wanted." Secondly.-To find the dimensions of square timber for the roof of a house. Let r, s, l, be the length of the ribs, spars, and lats, so far as they bear ; a, y, &, their breadth or depth, » the distance of the lats, w == the weight upon a rib, c == the cosine of elevation of the roof.- f hey: a*" Then by reason of the inclined planeT-Xcz $ w the weight upon a spar. And = the weight upon a lat ; for the ribs and lats lie hori- f e £ 2 & zontally. Therefore, --= Jl e w P s Itw l In w o iG yon 1 are 3 2 3 ' 7 v2 sz Whence, ° == , and a" == -_-, cls E a Hence, if any one, a, y, or z, be given, and all the rest of the quantities, the other two may be found. Or, in general, any two being un- known, they may be found from having the rest given. * For example : Let r= 9 feet, s = 4 feet, [z= 15 inches,. # ~: 11. mches, 70}, the.cosinge of 45° the pitch of the roof. And assume y -= 21 288 THE CARPENTERS AND inches, then a==24 -= inches, and <3 '585 (543 =~ 13 inches." * Proposition. -If any weight be laid on the beam a B, as at c, (see Figs. 9 and 10,) or any force applied to it at c, the beam will be bent through a space c p, proportional to the weight or force applied at c ; and the resistance of the beam will be as the space it is bent through < nearly." "In order to find the law of resistance of beams of timber, or such like bodies, against any weights laid upon them, or straining them, I took a piece of wood planed square, and sup- porting it at both ends a, B, I laid successively on the amddle ofit at c;.1,; 2, 3; 4, 5, 6,7, and 8 pounds; and I found the middle point c to descend through the spaces 1, 2, 3, 4, 5, 6, 7, and 8 respectively. And repeating the same experiment with the weights 3, 6, 9 lbs. they all descended through spaces, either accurately or very nearly as the numbers 1, 2, 3. I tried the same things with springs of metal, and found the space through which they were bent pro- portional to the weight suspended. I also tried several experiments of this kind with wires, hairs, and other elastic flexible bodies, by hanging JOINERS' INSTRUCTOR. 280 weights at them, and I found that the increase of their lengths, by stretching, was in each of them proportional to the weights hung at them, except when they were going to break, and then the increase was something greater. - It may be observed, that none of these bodies regained their first figure, when the weights were taken off, except well tempered springs ; so that there are no natural bodies perfectly elastic. And even springs are observed by experience to grow weaker by often bending, and, by remain- ing some time unbent, will recover part of their strength, and are something stronger in cold than in hot weather. - But at any time, a spring, and all such bodies, observe this law, that they have the least resistance when least bent, and in all cases are bent through spaces nearly pro- portional to the weights or forces applied. And, therefore, I think this law is sufficiently esta- blished, that the resistance any of these bodies make, is proportional to the space through which it is bent, or that it exerts a force pro- portional to the distance it is stretched to. " The knowledge of this property of springy bodies is of great use in mechanics ; for by this means a spring may be contrived to pull at all times with equal strength, as in the fusee of a U 200 THE CARPENTERS' AND watch ; or it may be made to draw in any p10- portion of strength required. ¢ The action of a spring may be compared to the lifting up a chain of weights, lying upon a plane, or to the lifting a cylinder of timber out of the water endwise. This author farther observes, that " the pro- positions before laid down concerning the strength and stress of timber, &c. are also of excellent use in several concerns of life, and particularly in architecture ; and upon these principles a great many problems may be re- solved, relating to the due proportion of strength in several bodies, according to their particular positions and weights they are to bear, some of which I shall briefly enumerate. * If a piece of timber is to be holed with a mortise the beam will be stronger when it is taken out of the middle, than if it be taken out of either side. And in a beam supported at both ends, it is stronger when the hole is taken out of the upper side than the under one, provided a piece of wood is driven hard in to fill up the hole. «* If a piece is to be spliced upon the end of a beam, to be supported at both ends, it will be stronger when spliced on the under side of a JOINERS' INSTRUCTOR. 201 beam, than on the upper side. -But if the beam is supported only at one end, to bear a weight on the other, it is stronger when spliced on the upper side. « When a small lever, &c. is nailed to a body. to remove it, or suspend it by, the strain is greater upon the nail nearest the hand, or point, where the power is applied. " If a slender cylinder is to be supported by two pieces, the distance of the pins ought to be $858 parts of the length of the cylinder, that is 3 its length, the pins equidistant from its ends, and then the cylinder will endure the least bend- ing or strain by its weights. * By the same principles, if a wall faces the wind, and if the section of it be a right angled triangle, or the foreside be perpendicular to the horizon, and the backside terminated by a sloping plane intersecting the other plane in the top of the wall, such a wall will be equally strong in all its parts to resist the wind, if the parts of the wall cohere strongly together ; but if it be built of loose materials, it is better to be convex on the backside, in form of a para- bola. " If a wall is to support a bank of earth, or any fluid body, it ought to be built concave, in form of a semi-cubical parabola, whose vertex v 2 202 THE CARPENTERS' AND is at the top of the wall ; this is when the parts of the wall stick well together ; but if the parts be- loose, then a right line or sloping plane ought to be its figure. Such walls will be equally strong throughout. " All spires of churches in the form of cones or pyramids are equally strong in all parts to resist the wind ; but when the parts cohere not together, parabolic conoids are equally strong throughout. ¢ Likewise, if there be a pillar erected in the form of the logarithmic curve, the asymp- tote being the axis, it cannot be crushed to pieces in one part sooner than in another, by its own weight. And if such a pillar be turned upside down, and suspended at the thick end in the air, it will be no sooner pulled asunder in one part than another by its own weight. And the case is the same if the small end be cut off, and instead of it, a cylinder be added, whose height is half the subtangent." The same author states also his having found by experience, that there is a great deal of dif- ference in strength, in different pieces of the same tree. Some pieces I have found would not bear half the weight that others would do : the wood of the boughs and branches is far weaker than that of the body ; the wood of JOINERS' INSTRUCTOR. 208 the great limbs is stronger than that of the small ones; and the wood in the heart of a sound tree is the strongest of all. I have also found by experience, that a piece of timber which has borne a great weight for a small time, has broke with a far less weight, when left upon it for a longer time. Wood is like- wise weaker when it is green, and strongest when thoroughly dried, and should be two or three years old at least. If wood happens to be sappy, it will be weaker upon that account, and will likewise decay sooner. Knots in wood weaken it very much, and this often causes it to break where a knot is. Also when wood is cross grained, as it often happens in sawing, this will weaken it more or less, according as it runs more or less cross the grain. And I have found by experience, that tough wood, cross the grain, such as elm or ash, is 7, 8, or 10 times weaker than staight; and wood that easily splits, such as fir, is 16, 18, or 20 times weaker. And for common use, it is hardly possible to find wood, but it must be subject to some of these things. Besides, when timber lies long in a building, it is apt to decay, or to be worm. eaten, which must needs very much impair its strength. From all which it appears, that a large allowance ought to be made for the 204 THE CARPENTERS' AND strength of wood, when applied to any use, especially where it is demgned to continue for a long time. We come next to some observations and ex- amples of that ingenious experimental philo- sopher Mr. John Banks, contained in his useful treatise on the Power of Machines ; which work we strongly recommend to the attention of mechanics in general. This gentleman commences with " Rules and observations respecting the form and strength of beams of wood and iron for supporting weights, working engines, &c. " If the materials of which different beams are made be equally good, the comparative strength under any regular form may easily be investigated. But we find by experiment that the same kind of wood, and of the same form and dimensions, will break with very different weights ; or, one piece is much stronger than another, not only cut out of the same tree, but 'out of the same rod ; or, a piece of a given length, planed equally thick, and cut in two or three pieces, these pieces will be broken with different weights,. Iron also varies in strength, and not only from different furnaces, but from the same furnace, and the same melt- ing ; but this seems to be owing to some im» JOINERS' INSTRUCTOR. 205 perfection in the casting, and in general iron is much more uniform than wood. - The resist- ance which any beam of wood or iron affords, will be as the sum of the products of all the fibres between the top and bottom, multi- plied by their respective distances from the top. For if a= length, b== breadth, and &= depth, we shall have # X #, and divided by g; | 3* b x* the fluent of & z =J j hence, ——Q—u=the whole resistance, which, when the weight is suspended from the middle of the beam, must be divided by half the length, or by %,which will be equal bx ¢ to --; which expresses the strength of the a beam. - From which we have the following- * Rule, -Multiply the breadth in inches by the square of the depth in inches, and divide that pro- duct by the length in inches, the quotient is a frac- tion, or whole number, &c. which expresses the comparative strength of the beam. "© The dimensions may be taken in feet, or the breadth and depth in inches, and the length in feet, but to compare one piece with another they must all be taken in the same manner. From a great number of experiments which I 2096 THE CARPENTERS' AND have made on the strength of wood, and that on pieces of various lengths and breadths, &c. I found that the worst or weakest piece of dry heart of oak, 1 inch square and 1 foot long, did bear 660 pounds, though much bended, and two pounds more broke it. - The strongest piece I have tried of the same dimensions broke with 974 pounds. * The worst piece of deal I have tried bore 460 pounds, but broke with 4 more. The best piece bore 690 pounds, but broke with a little more. - These pieces were 1 inch square, and 1 foot long. "© Example 1.-Given a piece of oak 6 inches square, and 8 feet in length, to find what weight suspended from the middle will break it. «* Solution.-In the worst piece of oak, 1 inch square, and 12 inches long, the strength is 1 squared, viz. the depth squared and multiplied by the breadth, and divided by the length, {ya at Ad P & which is 19: In the given piece we have 6 the depth squared equal 36, which multiplied by the breadth 6, gives 216, which divided by the Uh Z1 124 length 96, glves——6 0&2; hence as 17 is to 96° 660 pounds, so is g—to 17820 pounds, JOINERS' INSTRUCTOR. 207 " From the above, we may compute the fol- lowing weights, placed opposite to the fraction or whole number, which is obtained by the rule before given, and the dimensions taken in inches; in the second column, in feet; in the third, the breadth and depth in inches, and the length in feet. 44 | § ¥ $ o bu. 2 . 14 <2 21 & eg 41 'a .S "3 Ball 2A |g 2 YI (tsa |. f g | 3 'sa s{ Eg] g Lire a: tnlpg d ag 8 mul -R "20 = | tp gA Rp) - 3 | a ha 4 3] 25 | 8 F "g i (J =a o |. P RB k: $ |8 A Ag- £- vo ebo i 1: D600 To 7021 gs | 14250! 4 | sso § 990] z; | 19008|| 4 (1650 3 (| 1920 {4 | 283191 3 (1986 1 { 1980| 43 | 38016] 4 [2640 £ | 3960| «4 | - 57024) 5 13300 1 { 7920] 7, | 114048] 6 (3960 2 - 1 [1140480] 8 [5280 4 123760 0 (6600 4 - |3168s0 " Example.-Given the length of an oak beam 16 feet, breadth 15 inches, and depth 18 inches; required its strength ; or the weight which, suspended from the middle, will nearly break it. 208 THE CARPENTERS AND © 1. - Let the dimensions be taken in inches, 18 15 and we have T§§_ =< 259125 ; thon ftom the first column of the table we say as 4; is to 660 pounds, so is 25:38125 to 200475 pounds, the answer. © 2. Let the dimensions be taken in feet, and 1:5 x 1:5 y 1-25 16 from the second column in the table we may we have 45 omued 1757 s E56 3 and I # 45 6 1s --, A75. ivas" to 660 pounds, so § 575 to 200475 * 3. The breadth and depth in inches, and the length in feet, and we shall have the breadth multiplied by the square of the depth, equal say, as 4860, which, divided by 16, gives 192145 or 30375, and as in the third column, as 1 is to 660 pounds, so is 30375 to 200475 pounds, the answer. « A beam of the above dimensions is com- monly used for working a steam engine, the cylinder of which is from 20 to 24 inches _ diameter; suppose 22 inches, then the greatest pressure that can possibly act upon the beam will not exceed 10000 pounds ; hence the beam would require above: 20 times the force of the engine to break it ; nevertheless, if it was much weaker, the engine might bend it, and in time break it. JOINERS' INSTRUCTOR. 200 Suppose we take the above beam for a standard, or conclude that every beam ought to be able to bear 20 times as much as it is employed to do; then what must be the di- mensions of a beam 20 feet long, to work a cylinder 86 inches diameter ? « Solution. -The weight suspended from the ends of the beam will be as the squares of the diameters of the cylinders, viz. as 22 squared, and 36 squared, or as 484 to 1296 : the last di- vided by the first, gives 2:6777, or $3; so that the strength of the new beam must be 26777 times as great as the other. The strength of the first, when the breadth and depth are taken in inches, and the length in feet, is ex- pressed by 12—12, which, multiplied by §gi gives 4 121 393660 484 ' no regard is paid to the ratio of the breadth and depth, the problem is simply answered by assuming the breadth what we please ; suppose 18 inches, and z == the depth, we shall have 1—2—0? -== §139:34 ; and &" will equal———813'358X RC -= 90371 ; the square root of which is = z, or the depth = 30 inches nearly. equal 813384 for the strength. If 300 THE CARPENTERS" AND «" Otherwise, let the depth be taken at 27 inches and 6 = breadth, then our theorem will 2 O hecunie "*+" 84) j 7.2 81884 % 20 20 729 -= 22+31 inches. IN worps. « Rule.- Multiply the expression for the stren oth by the length of the beam in feet, and divide that by the square of the depth in inches, the quotient will be the breadth in inches. «*N. B. - Though the above two beams are equally strong, yet the second contains about 8+ feet more wood than the first. PROBLEM 2. « Let it be required to make a beam equally strong with the last, and of the same length, but that the ratio of the breadth to the depth be as 2 to 3, or in any other proportion. Let a == length in feet, b = breadth, s= depth, and s== $1334, the expression for the strength. q ¢ 6%: R «+_ FThen ' will gi. = 813°34 ; and s* -= 813:34 X a a ; also, by the problem, as 2:8 : : 4: z: Q 8 6 3 hence, 2 *= S 6, and * =—g—, which squared, ach JOINERS' INSTRUCTOR. 301 | 9 b* s a 9 D gives x* = also p ==: from 4sa __ 409° 813534 %€ 20 whence we get /' =- - 9 3 -7220 7, the cube root of which is 19:337;, the breadth ininches ; and as 2: 8 :: 19-387: 20-005, the depth in inches. « If m is to n as the breadth to the depth, we shall have the following general theorem, viz. m* § .a zS" 712° PROBLEM 3. « Required to make a beam 24 feet long, to work a cylinder 24 inches diameter, and that the breadth be to the depth as 3 to 7. «*To find an expression for the strength of the beam, we may say, from the last problem, as 484, the square of the diameter of the cylinder, is to the expression for the strength of its beam, so is 576, the square of the diameter of the pre- sent cylinder, to a number which will express its strength, viz. as 484 : 1—212 : 301 4187, the number required. " But when the length is taken in feet, and the breadth and depth in inches, we know that 302 THE CARPENTERS' AND an oak beam which will just break with 660 pounds has its strength expressed by 1; and if we wish to have a beam which will bear 20 times as much as it is intended to load it with, we may take 7;th part of the load, viz. 7th of 660, or 33, and say as 38 : 1 :: twice the area of the intended cylinder, multiplied by 14, to the strength of the beam ; in this case, as 33 : 1 : : 12672 : 384, the strength. - But this is much more than the real load of a common 2 feet cylinder ; if it should amount to even 10 pounds per inch, the whole would be only 864.0 pounds ; and as 38 : 1 : : 8640 :261:8, the strength. - Let a beam be constructed by both expressions, viz. by 384 and 262. In the first case, by the ge- . m s a neral theorem, & == ; in the problem, m== 3; and r»=.7, also a =-=+44, and s == 884;~ there. 9 x 384 x 24 fore I " tad | root of which is the breadth = 11°918 inches ; and as : 7: < 11'918: 27-808 inches, the depth. I have known a beam of deal of this size used for a 2 feet cylinder ; but in a few years it was much bended, though there was no danger of its breaking. -== C& 16927, the cube " Secondly, when the strength is expressed # JOINERS' INSTRUCTOR. 303 by 262, we have every thing the same as before, 9 % 202. 4.24 49 -= 1155, and the breadth = 10492 inches, and as 3 : 7 : : 10492 : 24648 inches, the depth. « But suppose the engineer wishes it to sup- port any greater weight, for instance, 25 times its common load, he may divide 660 by 25, and the quotient will be 2654; then as 264 :; 1 :: 8640 : 3272, the expression for the strength, and 3C@J-BY¥ 24 % O 49 root is 11:29 == bs And as 317 :: 11:29 : 2634, the depth. «< By the above process,; we find the strength of a beam to work a 12 inch cylinder ex- pressed by 96, from which we have the fol- lowing- "* Rule.-Square the diameter of the cylinder in feet, and multiply the product by 96 ; the last product will express the strength. «< N. B. The depth and breadth are taken in inches, and the length in feet, which I think is the most convenient in general. The length of the beam will make no difference, for the di- mensions will turn out so as to make it equally strong of any length ; for example : except s=262 ; therefore =A ==ih 2s 14112:851 ;. its cube 304 THE CARPENTERS' AND PROBLEM 4. * Required a beam 16 feet long, to work a cylinder of 80 inches diameter, and that the breadth be to the depth as 3 to 5. © Solution.-25 feet squared, is 625 ; which by the rule, multiplied by 96, gives 600 for the strength. m* sa n* 9 x 600 % 16 25 == & - 8456; the cube root of which is 15119 inches ; and as 8 is to 5, so is the breadth 155119 to the depth 25198 inches. * Let it be required to make a beam for the 9 x 60o x 24 25 == b -= the breadth =. 5184 ; the cube Loot of which is 17-307, from which we find the depth 28:845 inches. This is equally strong with the first, but contains much more wood. «< Then by the theorem, /* == s m ==9.; 1'==@95; $=600; a- 16; therefore same cylinder of 24 feet, then will PROBLEM 5. «Required a deal beam 16 feet long, to work a cylinder of 1 foot diameter, the breadth to the depth as 3 to 5. «* Solution.-If we take the pressure at 14 JOINERs' INSTRUCTOR: 305 pounds pert inch, the weight upon the centre will be about 3168 pounds ; also, the worst picce of fir, one inch square, and one foot long, bore 460 pounds, and broke with 7 pounds more ; hence, if we take glow part of 460, viz. 23 pounds, and say, as 23 : 1 (the expression for its strength), so is 3168 : 1877, the strength. - Then from the theorem we multiply the length, the strength, and the square of the ratio of the breadth together, and divide by the square of the ratio 16 % 25 the cube root of which is 9:9 inches for the breadth ; then as 3 : 5 : : 0:0 : 16:5, inches, the depth. f : " Hence, if the square of the diameter -of any cylinder in feet is multiplied by 138, for the strength of a deal beam, it will, without break» ing, support more than 20 times as much weight as the engine can ever exert upon it. of the depth, viz. is-073"1597, Eaperiments on the strength of oak. «' These experiments have been made on pieces of one inch square, and one foot long, and from that size to 2 inches square, and five feet long. To mention all the trials would take up time and room to no great purpose. -It may r 306 THE CARPENTERS' AND be sufficient to observe, that when the compu- tations were made on the different pieces, and applied to pieces 1 inch square and 1 foot long, that the worst would bear 660 pounds, and the best not more than 974. From similar experi- ments on deal of the same dimensions, I found the worst which I used would just break with 460, and the strongest with 690 pounds. " But in all the computations, I have taken the worst pieces to compute from, and at the same time have made them to bear 20 times as much as the load they have to sup- port ; not taking notice of their own weight, which would have made the process much more troublesome. 6. " Required the length of a piece of oak one inch square, so that it may just break with its own weight. - Solution.-** Let x == the length in feet, and one foot in length weigh %ths of a pound. #: 660 Then as 1 : 660 :: N4) ~- the weight which will ¥ break it ; but the bar only acts with half its own r 4a s weight at the centre, therefore 10 the weight, JoINERS' INSTRUCTOR. 307 of the bar, must be equal to £529 y 2; viz. 4a * 1s}. 8 19200 == $:" "and" r =' ¢ 3800 =z 5744 feet. Eaperiments on the strength of cast fron. " Of late, cast iron has been used in various cases in place of stone or wood, as in bridges, engine beams, pillars, railways, or roads, &e. and is still likely to be more in use. The fol- lowing experiments were given to me by Messrs. Reynolds, of Ketley, at the same time request- ing me to make them as public as I could, for the advantage of others. Eaperiments on the strength of cast iron, tried at Kelley, March, 1795. "N B. The different bars were all cast at one time out of the same air furnace, and the iron was very soft, so as to cut or file easily. Experiment 'Two bars of cast iron, one inch. square, and exactly 8 feet long, were placed upon an horizontal bar so as to meet in a cap at the top, from which was suspended a scale; these bars made each an angle of 465° with the base plate, and of consequence at the top so as to form an angle of 90°; from this cap x 2 308 THE CARPENTERS' AND was suspended a weight of 7 tons, which was left for 16 hours, when the bars were a little bent, and but very little. Eaperiment 2.-** Two more bars of the same length and thickness were placed in a similar manner, making an angle of 221° with the base plate; these bore 4 tons upon the scale; a little more weight broke one of them, which was ob- served to be a little crooked when first put up. In this case the pressure would be as the sines of the angles of elevation, viz. as 3826 to 7071 ; and 'as 3826 4 tons ; 27071 #7:6 tons, that is, if the second bars broke with 4 tons, the first ought to have taken 7:6 tons to break them ; and as they\were not broken, it is likely that would, if tried, have been the case. f Experiment 3.-** Another bar was placed horizontally upon two supporters, exactly 3 feet distant ; it bore 6 cwt. 3 grs. but broke when a little more was added. Experiment 4.-*"* The same experiment re- peated with the same result. Experiment 5.-* The bearings were 2 feet 6 inches apart, the bar bore 9 ewt. and broke. This was perceptibly bent with 1 cwt. but bore two safely. Three more experiments were tried the next day with the prisms 3 feet distant: the average result was 6 cwt. 2 grs. 74 pounds. JOINERS' INSTRUCTOR. 309 Experiments tried at Colebrookdale, on curved bars or ribs of cast iron, April, 1795. " Rib 29 feet 6 inches span, and 11 inches high in the centre ; it supported 99 cwt. 1 qr. 14 lbs. ; it sunk in the middle 8Zths, and rose again 4ths when the weight was removed. The same rib was afterwards tried without abut- ments, and» broke with ©55 . ewt. O |i grs.i-4 pounds. 6.-*" Rib 29 feet 3 inches in span, a segment of a circle 3 feet high in the centre ; it supported 100 cwt. 1 gr. 14 lbs. and sunkl—l—Gm the middle. The same rib was afterwards tried without abutments, and broke with 64 ewt. 1 qr. 14 pounds. «* N. B. _The thickness of these ribs is not mentioned, but the experiments show that they are much stronger with abutments, as little more than half the weight which they support breaks them when the abutments are removed. « The following experiments on cast iron I made at Messrs. Aydon and Elwell's foundry, at Wakefield : the iron came from their furnace at Shelf, near Bradford, and was cast from the air furnace ; the bars one inch square, and the props exactly one yard distant; one yard in length weighs exactly 9 pounds, or one was 310 THE CARPENTERS' AND about half an ounce less, and another a very little more ; they all bended about one inch before they broke. ** 1. - The first bar broke with - 963 pounds. t @. broke with =:.) {:< $58 pounds. t* 8. - Bar broke with ~. . . 994 pounds. " 4. - Bar made from the cupola brokewith . giiy 86@ pounds: «5. Bar equally thick in the mid- dle, but the ends formed into a pa- rabola, and weighed 6 lbs. 3 ozs., proke -with # ista» g9% pounds. «¢ Other experiments were made by giving the same quantity of iron a different form, (see Fig. 11.) The top and bottom of this beam were each 1 inch broad, and half an inch thick, till they joined at a and b, where they were one inch square; the piece c, d, in the middle from which the weight was suspended, increased the weight of this to 10+ pounds. The length from prism to prism, viz. from prop to prop, was an exact yard ; the depth in the middle from top to bottom 4+ inches. The first piece bore 29 cwt. 20 lbs. and broke with a little more. A second piece bore 23 cwt. 1 qr. but broke with another half ewt. j "* A second form is represented in Fig. 12, where the bar a c 4 is 1 inch broad, and 44 inch JOINERS' INSTRUCTOR. 311 deep ; the bar a d b is also 1 inch broad, and 7; inch thick, so that it contains no more iron then the straight bar, except the: piece: at c, from which the weight was suspended ; the weight of these was 10 pounds each, the depth c, d, 4+ inches, but there was no connexion betwixt the two, except at the ends where they were in one piece at c, and the lower at the props. * Trial 2. 48 cwt. 2qrs."7 lbs. broke this in the same manner as the other. A gentleman present wished to have the upper and lower part connected, as at the dotted lines ; one was cast in this form, and broke with 31 ewt. 2 rs. another bore above 40 ewt. * Another beam of the same length and depth at the centre, but in the form of a para- bola, and weighed 104 pounds ; the flat, part of the beam was 4 of an inch thick, and was sur- rounded by a moulding + of an inch thick, and on the outside 1 inch broad : first trial broke with 50 ewt. 8 grs. 25 lbs. ; a second piece, or beam, broke with 44 cwt. 3 qrs., but on ex- amining the fracture, it was full of pores at the gate, or place, where the metal entered the mould. See Fig. 18. " From the above experiments, it appears that cast iron is from 3+ to 4} stronger than 312 THE CARPENTERS' AND oak of the same dimensions, and from 5 to 64 times stronger than deal. « Iron is much more uniform in its strength than wood, yet it appears that there is some difference in different kinds of ore or iron stone ; there is also a difference from the same furnace, perhaps owing to the degree of heat which it has when poured into the mould. If we take iron upon an average to be 4 times as strong as oak, and 5+ as strong as deal, we may proceed to make a comparison betwixt wood and iron, in respect of magnitude, weight, expense, &c. «It is proved by the experiments, that the worst or rather weakest cast iron, 1 inch square, and 3 feet long, will break with about 730 pounds ; and as 4 (viz. the breadth and square of the depth multiplied together, and divided by the length in feet) is to 730 pounds, so is 41 or 1 to 2190 pounds, the weight which would break a bar 1 inch square, and 1 foot long. But I have computed the beams of wood to bear 20 times as much as the intended load.. Let the iron be made to support 6 times the weight, which I presume, from observation, will be sufficient to keep them from bending or vibrating. But if the engineer thinks differently, he may make them of what strength he pleases by the following ~ JOINERS' INSTRUCTOR. ° 813 "© Rule- Divide 2190 by the number of times you wish to increase its strength above the load, and say, as the quotient is to 1, so is the load of your new beam to the number which expresses its strength. « For example, the load upon a 12 inch cy- linder will be about $168 pounds. One-sixth part of 2190 will be 865, and 3168 divided by 865 quotes 8:6; which expresses the strength for cast iron. PROBLEM 11. "* Let it be required to make a cast iron beam for a 12 inch cylinder, so that the breadth may be to the depth as 1 to 6, and the length A4 feet. © Here we have a =14 ; m= 1;'m== A m* s/d 1204 6.3 8G; then 5 == m soa si $ ; then ~a 36 3344, and b == 1+4954 inches, which multiplied by 6 gives 8:9724, the depth. On the strength of beams or poles of wood or iron, when used in the form of triangles, to support weights, load waggons, raise stores upon build- ings, &c. "* Fig. 14.-Let B s c be the triangle ; then as the sine of the angle which B s makes with the horizon is to the radius, so is the whole 314 THE CARPENTERS AND weight, suspended from the top s, to the pres- sure against s s and s c; for if they stand in the same position, they will support equal parts of the weight. * Those who have not tables of natural sines by them, may proportion, by saying, as s D, the altitude of the pin which supports the weight, is to the length of a pole or leg of the triangle, so is the weight suspended to double the pres- sure against one pole, that is, when the triangle consists of two poles. If there be three poles, as for loading carriages, &e. we may say, as the perpendicular altitude of the pin is to the length of one pole, so is one-third of the weight to the pressure against one pole. According to Messrs. Reynold's experiments, we easily infer, that a bar of cast iron one inch square, and one foot long, will bear a pressure against the ends of about 15 tons ; and it appears from other experi- ments, that deal, alder, and other soft wood of the same dimensions, will bear about 2:3 tons ; but suppose we call it only two tons. * From which we get the following pro- portions. "* For iron. - As 1 : 15 : : the expression for its strength to the weight which it will bear. * For wood. As 1 : 2: : the expression for its strength to its load. JOINERS' INSTRUCTOR. 315 PROBLEM 20. «® Examples.-Given two pieces of cast iron, two inches square, and 16 feet long, making an angle with the horizon of 60% each ; what welght will they support ? * Solution. -The expression for the strength is the cube of a side in inches divided by the length in feet, viz: #, == 41; and the natural sine of 60°, when the radius is 1, is '86560254 ; for the weight which the bar can bear against its end, Say, as T : 18 9% J+ 74 tons."Next,(gs the length of the bar is to its perpendicular al- titude, viz. as I+ 8660054 :: 7:5 : which doubled is 12:99 tons for the whole load. PROBLEM 21. "* Suppose the same bars make each an angle of 30° with the horizon, what weight will they bear ? * Solution.-The strength as before is 4; and the pressure which they can support is 74 tons ; the sine of 30° is '*5% and asl 1:5 4 838""5 tons for each bar, and the whole load is twice as much, or 74; hence, it appears that the less the angle which they make with the horizon, the less weight will they support. 316 THE CARPENTERS'. AND " N. B. In round and square props, poles, &e. of equal length and weight, the round will be stronger nearly in the ratio of 81 to 29. But if the lengths are equal, and the diameter of the round equal to a side of the square pole, the strength of the round pole will be to that of the square one as 21 to 20, nearly. PROBLEM 22. * Three poles, 4 incltes diameter and 10 feet long, make each an angle of 60° with the horizon ; what load may be suspended from them ? "* Solution. wfizfiw forthe strength ; then as I:;@: : 0:4.:.19-8 tons, the strength of one pole. - And as radius is sine of 60%, so is 12 8 to 11 tons, the weight which will produce a pressure of 128 against one pole: the whole three will therefore support 33 tons. PROBLEM 23. © Required a triangle with 2 legs, 20 feet high, and making an angle of 70° with the horizon ; what must be their diameter to sup- port 3 tons, that is each 1+ ? ¢ Solution. -As the sine of 70° +98396926 : JOINERS' INSTRUCTOR. $17 radiug 1 :~ 14 half the weight : the x3 pressure upon one pole. And $5 1-59063, ofr x* =1°5968 :y 20 == 81-0@0; ;n '= (81-:020)7 == 317 inches, the diameter of a pole. PROBLEM 24. ** Required (a 'triangle with 8 legs, each making an angle of 60° with the horizon, and . 12 feet long, to support scales for weighing 4 tons, viz. the whole load will be 8 tons. ** Solution. --First, as the sine of 60° +866, &e. : radius 1 :; : 2:666, one-third of the weight, to 3077 tons, the pressure on one pole. Then $3 12 (36:924)4 = 3:298 inches, the diameter of a pole. '' N. B. In the above problems, the legs will do no more than bear the weight ; but if we want a triangle to support eight tons, we ought by all means to make it strong enough to bear 3 times as much, and then we should say, as 866, &C. 1 :s 0:23 tons ; -and: a*. will equal f43.%x 125 110 76.;, (110-76) == 4+8016 inches. «If wood, metal, &c. intended for bows, springs, &c. be formed of the above process, == 8077.; and a = S6 004; henee, a ~ 318 THE CARPENTERS' AND they will be equally strong from end to end ; but if one side is to be flat and equally broad through the whole length, then the other side is formed into a parabola, by the rule given for engine beams. f « When shafts are placed in an upright posi- tion, they are only in danger of being broken by the twist, between the wheel which drives them, and the resistance they have to overcome. A cast iron bar 1 inch square, fixed at one end, and 631 pounds suspended by a wheel of 2 feet diameter, fixed on the other end, will break by the twist. The strength of square bars in re- sisting the twist is as the cube of a side ; in round bars or spindles, as the cube of the dia- meter. - Hence, if a bar of 1 inch square re- quires 631 pounds to break it, one of 2 inches square will require 8 times as much ; and one of 3 inches, 27 times, or ©7037 pounds. " I have made experiments on some bars of the same size, and the power or force applied in the same manner, which have required 1008 pounds to break them by twisting, but have mentioned the worst I have tried, that I might not deceive. < It has been observed, that there is no weight or pressure to break these upright spin- dles besides the twist; yet it may be noticed, that weight on the top, the shaking of the ma- FJOINERS' INSTRUCTOR. 319 chinery, &c. tend to make them vibrate, on which account it may be well to make them something thicker at the middle than at the ends.. PROBLEM €5. " If a shaft 3 inches square will just bear the force acting upon it, how much must a side of the square be, when it is strong enough to bear 5 times as much ? «* Solution.-The strength is here represented by the cube of 3 = 27, which multiplied by 5 gives 135, the cube root of which is 5:13 inches for the answer. \ PROBLEM 26. " Fig. 15.-A weight w is suspended from r, the arm of a crane a B c D E; what is the pres- sure against the end of the spur p s ? f cm XG * Solution. - .: the pressure at Dp, in the direction » a ; but the pressure at p, in the direction D B, will be as o G to D B; that is, C E°X w o Rn XE Ck w : - Also, C P D the pressure against the upright post a c at B, in the direction aB, will be'as s c: c p, viz. as as - D:G-:-D.B.!.l \ 320 THE CARPENTERS! AND CEXW,CDXCEXW_ECXW &C; C p":I =e= c:D 5. C¢:0¢C. D BG EXAMPLE 1. " Given c c- 16 feet, sa c or o a'==7 fest, pco-7 fect, w -=Bitonsypa- 9090 :. then from pB X ®ECcXw_99 % 16 x 3 P: G:0KC.Dicnormu?]}> 7 the above, we have 4752 =" the spur p B. * For the perpendicular pressure against the ; EG. ) 3 upright axle a c, we have -- mea --- B.¢ J 36 == 6:8571 tons, the force tending to break the said piece at B. == 9:6979 tons, for the pressure against EXAMPLE 2. "Given £ .C == 12 BK =yw6 FC -42 67 P wis! :d. © Required the pressure on the spur, and the horizontal pressure against the upright. p B X® cX w __9 X 12 X4 DoD cl lif «bax 617» id 5 pemnnoned 1071. JOINERS' INSTRUCTOR. 321 the pressure against the end of the spur. The s ~s12 y< 4 pressure against the post is Tana ta" = 8. In this example, let a c and c r be oak beams, each 10 inches square, and the spur D B be six inches square. The strength of ® c is 1000 iof " 94%; which, multiplied by 660, gives 31132 pounds, which, suspended at r, would break the beam cor at p.. The length of the upright a c is 12 feet, and has its strength ex- pressed by 1ng9 ; which, multiplied by 660, pro- duces 55000 pounds, the weight which would 3 31 132 12 break it at s. . But ———6l—— == 62204, the pressure at B, which is 7264 pounds more than the beam a c can support. The strength of 6 xX 6 x 6 the spur B » is -= 24. winch multi: plied by 2 gives 48 tons for the strength, or a f »" 'X <% wl. 107520 pounds. - But FEX ns = 9 x 12 x s31182__ss62256 __ —6—><67~———4—6§—— h 83638 pounds, which is 23882 pounds less than the force re- quisite to break the spur. From ' the above it appears that the upright a c is the weakest C 329 THE CARPENTERS' AND part ; but, from the principles already explained, the ingenious mechanic will easily proportion the parts so as to be equally strong. I will add one example. In the above crane the horizon- tal beam bears 31132 pounds ; the length of the spur and upright being given; what must be their dimensions, that is, how much square, to be equally strong as the above horizontal beam ? ** First for 1the spur. - Let z = a side of the square, then —J— y 4480 -= 31182 pounds, or 2 9 X $1132 n 39694. inches. Second..-The strength of the upright is expressed by ilfgfi—G— which must be equal to = 62542: and z= (62542) i= ore 62264 ; hence 3*== (fig—(1563512 =-" 1182:0006 ; its cube root is 10422 == z, the side of the square post. « Let it be required to make a crane of cast iron to bear 4 cwt.; but that it may be perfectly safe, let it be calculated for 10 cwt. and let a c es @'F GE 8 feet, alsoon 6 =='cip -= 1+ foot. the thickness of the iron be half an inch, and put &= depth of c r. Then JOINERS' INSTRUCTOR. 328 as 1 : 2190 : ¥ $3 : 1120, from which we find 2 1—162;— == 30685; the square root of which is the depth = 1.75 inches. The pres- sure upon the spur at p, in the direction D G, e 1120 pounds; the length of the spur is 212 feetsiand -as D c (15).10.0 (2:18) : + 1120;; 1588, for the pressure in the direction » B. As a bar one inch square, and one foot long, will bear 15 tons, or $3600 pounds, we say, as 1 : 33600 231 g : 1583; from which we find z the side of the prop or spur = .46385 of an inch square. Next, for the upright, we have , Of 2-22? == 1120 pounds the pressure against , then;as 1 : 2100 4: the square of the w! S. i2: breadth to 1120 pounds, the same as c ®, as they are of the same length, and the breadth will be the same, that is 1'75 inches." We shall now present to the attention of our readers a few interesting problems, collected from various sources. In the sixth number of that valuable publi- cation, the Gentleman's Mathematical Companion, ¥ 2 324 THE CARPENTERS' AND we find the following question proposed by the truly ingenious Mr. Joseph Edwards, private teacher of the mathematics, at Hoxton, and most elegantly resolved by him in the succeed ing year. PROBLEM. « Let s denote the strength of a parallelo- pipedic beam of timber, at any given place, found according to the common principles. Required the strength at the same place, supposing the expansion and contraction of the fibres, when at the greatest, to have a given ratio. SOLUTION, «* Let the parallelogram a B c », (Fig. 16.) represent the given beam supported at the middle of B c, by the fulcrum r, and strained by two equal weights suspended to its ends ; -and parallel to which let E r be a section. << In the investigation of the lateral strength of timber, upon the common principles, the common fulcrum of the bended levers, B r 5, c r E, is supposed to be in the surface of the beam at r, and consequently that all the longi- tudinal fibres resist (in various degrees) the ten- dency ot the force of the weights to overcome the cohesion at the section, in the direction of JOINERS' INSTRUCTOR. 325 their lengths; but when the fibres are susceptible both of contraction by pressure and expansion by tension, it is evident that the beam will, as soon as it is submitted to the action of sufficient weights, turn (until it breaks) about that point (0) in ® r, which divides the expanded from the contracted fibres, and of the latter, an equal number in each part, r A, r p, may be imagined to press upon each other at o r, and form a common finite base, upon which the said levers are supported when they strain the fibres be- tween o and E. Now, from hence it is evident, that the magnitude of o r depends upon the ratio e e to r r, if these lines represent the in- crement and decrement of an expanded and con- tracted fibre respectively ; which being given, that of E 0 : 0 r is also given by sim. a s ; and E F being given, E 0, 0 F, are each given. " Again draw J; o K || B c, then (by prop. 57. Emerson's Mech.) the strength of the beam A 1 0 K p, of which the depth is E o, and ful- Ht o" P ord crum 0, is as ---, when o x is given, and the breaking tension of a fibre at o = 1; also 92—0 - sum of all the tensions at E 0 ; therefore it is evident that all the fibres in the beam a'r would be broke by a force at x, which is as 326 THE CARPENTERS AND £393, if the arm o r of the double lever x o ®, k o r, were not prevented from turning about the fulcrum 0, by the resistance of those fibres at o r, which belong to the part 1 r of the beam, E 0° Q E o + 2 r E from what has been quoted above, and the property of the lever. Moreover, because the resistance at o r is equal to the sum of all the © o .E 0°), E 0 E o . TN L8.) (E 0 + 0 F), the greatest weight that the given beam will support, or the least that will break which resistance == , as appears tensions at E 0 2 its." But Egg is as the strength of the beam, according to Emerson's principles; therefore 2 F555?XEF::EF:E0::S:S XS—gfihe strength of the given beam, as required. - Hence it follows, if the beam be soft and elastic, (as fir, yew, &e.) that it will acquire additional strength by cutting a piece out of the under part on each side of o r, and substituting a similar one of a firmer texture in its stead." In the 14th number of the Mathematical Companion, we find the following interesting question, proposed by Mr. John Surtees, of JOINERS' INSTRUCTOR. 327 Houghton Le Spring, in the county of Dur- ham. E " A beam of a given length, having its per- pendicular section every where a plane triangle vertex upwards, projects horizontally from a wall. Compare its strength, when whole, to support a weight at its end, with the strength of the remaining piece, after {th of the section from the vertex is cut away parallel to the ho- rizon ; also with the strength of the remaining piece, after {ths, 4ths, $ths, &e. is cut away, the weight of the beam itself not being con- sidered. f «* Note.-In Emerson's Mechanics, page 114, it is asserted, that, when 4th is cut away, the remaining piece is stronger than when whole, which is a paradox in mechanics. But I pre- sume that a true solution to the above will prove that to be a mistake." The correctness of this assertion, the ingeni- ous proposer fully demonstrated (on the suppo- sition that the beam was destitute of weight), in the 15th number of the same publication ; to which the reader is referred for a very ex- cellent solution of the foregoing question. The writer of this article has received the following very elegant and masterly solution of this interesting problem, from Mr. William 328 THE CARPENTERS' AND Watts, of Plymouth ; a gentleman whose emi- nent scientific attainments are surpassed only by his ingenuous modesty. «* Let the triangle a B c (Fig. 17.) represent a perpendicular section of a beam, supposed to project horizontally from a wall, and let a weight, w, be applied at the other extremity to break it ; also let a B c represent the section when the fracture takes place, the fulcrum being at p. «Put c »=o,1 s= o, » E=v, and E r=y. Let 1 also denote the length, and w the weight of the whole beam, and let Z and t be considered as the length and weight of any trapezoid or hori- zontal section of the beam. Then by sim. tri- angles c n (a) :B » (b):: ¢c® (a-v) : ® r (y) 7-2? (a-v), and by substituting this value of y { 95 b in the known formula ~ , it will become gz (a x' v+-v" v), the fluxion of the strength of the beam; and by integrating each term, we shall iit a* have 37 ~ *~ the strength of the trape- zoid B » E r, which expression when n = a becomes 77 b a, the dlrect lateral stxength of the entire beam. 6 These two strengths are to each other in the JOINERS' INSTRUCTOR. 320 f be pe Avs ratnoofT—‘gthba;%—4Ta—2,orof1:—a§—— 8 v* a4’ (I)‘ «© This ratio expresses only the relation ex- isting between the strengths or efforts that tend to preserve the adhesion of the fibres. - If, therefore, we would take into consideration the efforts that tend to destroy their adhesion, these efforts, being in the inverse ratio of the strengths, will be found to vary both as to the weights of the respective beams, and the dis- tance at which those weights act; therefore, by incorporating the direct and inverse ratios, we readily find that the strengths are to each other in the Hatio of 1: i”%_fl). C() . a a a' lw ar of1:(?aT”S—fiaff). (C) "y. tein when the lengths are the same. " Example 1.-If v== $ a, then, by the first form, the ratio for the direct strengths becomes 1 : Lésg—gl—Q— 3——6>f—5—64T09—6, or as 2187 to 2048. " Example 2.-If v= 4 a, then the ratio of 4«_X_ 343 the direct strength (I1) becomes = 1 : 7ag _'3—)—gg§ifgl-, or as 2187 to 1716. 3380 THE CARPENTERS' AND << Example 3.-If v = $& a, then, by form (1), the ratio of the direct strengths becomes = 1 : 4 % 216 s x 1296 S Far 72? "~. Nef -~" as 2187 to 1296. << Example 4.-If v = $ a, then by form (1), the ratio of the direct strengths becomes = 1 : 40} 125.1 Sik 625 -————792— ~ ~ C30f "2 or as 2187 to 875. «* If we continue the examples by supposing 3 $ 4 tv as 6 5 4 &e. &c. we shall obtain a series of results, the whole of which, with the pre- ceding, are exhibited in the following table, where the several ratios are arranged in their proper order. [2048 when 1 ninth is cut away. 1715 . ..@ ninths: . "; ditto 12906 - . \s38" \ iv ... 875° | . ditto S18] A01 .u 8 -. - " difto $48 mans thous oilt 10 . viditto 80 4 o i> ditto 11%. ~.~ ; ditto ts { =x.39 L.o-.S mdifto. ~ Remark.-It appears by the preceding solu- tion, that the whole triangular beam is stronger JOINERS' INSTRUCTOR. 351 than any remaining trapezoid whatever, after any triangular prism, similar to the given beam, is cut off from the top parallel to the horizon, and that Emerson was mistaken (at least in theory) in asserting that when $th of the al- titude of the triangular prism is cut away, the remaining piece is stronger than when whole. But it should not be forgotten that the entire - beam, being in the form of a triangular prism, and having a sharp edge at the vertex, is deemed by practical men much more liable to spring, than it would be if the upper edge were cut off parallel to the base of the prism ; and it is probable that this circumstance may more than compensate for the loss of strength occa- sioned by 4th of the altitude of the beam being cut off from the top; but this I presume can only be determined by suitable experiment." On comparing this solution with the one given by the proposer in the Mathematical Companion, they will be seen to present the same results; from which cireumstance it is evident that theory contradicts the assertion of Emerson, and all that remains now, is to insti- tute a series of experiments, in order to ascer- tain the conclusion which may be deduced from them. 332 THE CARPENTERS' AND The writer has already commenced some ex- periments on the subject, which will most pro- bably be communicated to the public in some form or other at a future day. As his sole ob- ject, however, in mentioning this, is to excite an emulation in others to undertake the task, it will be no mortification to him to be deprived of the honour of being the first who shall either confirm or invalidate the assertion of Emerson, by correct experiments. Although the preceding solution, as well as that of Mr. Surtees, have. been conducted on the supposition that the beam is devoid of weight, yet, even on this principle, Emerson's assertion will be found altogether erroneous. In the mathematical department of the En- quirer, is the following problem, an answer to which has not yet been published ; though we shall now attempt it in the annexed solution. PROBLEM. * Two prismatic beams of equal length, whose sections are respectively a triangle and a trape- zoid, the latter being cut from a triangular beam similar to the former, have their ends fixed in an upright wall, with their bases downwards ; putting s, s, for the strengths of the beams c, g, JOINERS' INSTRUCTOR. 393 for the distance of their centres of gravity from their bases, wershall have s @ c:g pre- quired the investigation. " Solution.-Let amo», ®ron &, (Fig. 18.) be two beams of equal lengths and of the same materials, fixed in horizontal positions in a ver- tical wall ; and let the latter be a prism cut off from a triangular beam which is equal and simi- lar to oB c. t+ Pitt a z=arearof the section n a c-; c = dis. tance of its centre of gravity from the base B c j mis- its length; ow: == mits weight, and s:lits strength ; and let a, g, l, w and s represent cor. responding particulars in the other beam. Then it is well known that s :s :: *C : 4E Lw - lw Is w L. wisi walle. uas: w s and .snp- posing we consider, for example, that the tra- pezoid represents 44 of the triangle » B c, (as Emerson has done, vide the foregoing question), fi; -As G+ 80 A 80 w we shall have a == AG and i == —8—1—— hences S::80A.G.W:80A.g.W:. aig: so > : 81 81 G :g 3; as is required to be demonstrated." This solution confirms the assertion made in the preceding problem ; for as s and s denote 384. THE CARPENTERS' AND the respective strengths of the triangular and trapezoidal prisms, and these strengths bear an invariable ratio to c and &, (as has been just de- monstrated), and knowing from the common properties of the centre of gravity in the beams, that a is greater than g, we may be considered as correct, when we assert that the strength of the former beam is greater than that of the latter ; and this, too, without any limit or regard to the size of the small triangular prism, sup- posed to be cut off from the whole triangular prism, in order to form the trapezoidal beam E F G H K. j To determine the position of the two posts A D and B E supporting the beam a B, so that the beam may rest in equilibrio. Solution.- Through the centre of gravity a of the beam (Fig. 19.), draw c G perpendicular to the horizon; from any point c in which draw c A D, C B E, through the extremities of the beam ; then a » and s - will be the positions of the two posts or props required, so as a B may be sustained in equilibrio; because the three forces sustaining any body in such a state must be all directed to the same point c. Cor. -If a ® be drawn parallel to c o; then the quantities of the three forces balancing the JOINERS' INSTRUCTOR. 985 beam will be proportioned to the three sides of the triangle c a r, viz. c a will be as the weight of the beam, cor as the thrust or. pressure in B r, and r G as the thrust or pressure in A D. The strength of a beam, a s (Fig. 20.) being given, it is required to find its strength, when a hole (a ¢) is cut out of the middle, and another equal hole (r ») in the side. By the principles of mechanics, the strength of beams, the thicknesses of which are d b, d a, d c, will be as d it; 4a; and d &.} Now, as the strength of all the particles between b and d is denoted by d b, and the strength of all the par- ticles between a and d is expressed by a d*; consequently the strength of all the particles between b and a (the point D being fixed) will be d b-d a; add to the same the strength between. c and d, which is c d, and the strength of b a and c d, when the strength of the hollow beam will be d b -d a_ 4+ ¢ d; but at the section r, the strength will be f. Whence if ninoz a o, theo strength at 6, to the strength at r, is as b d' - d a -}- c d' to (d b- ¢ &); that is, as d / - 2*d c X om +- Ado d:lR -'2 dob oic a p cut. d iP be the strength of the whole beam (2 de -+ c a) 386 THE CARPENTERS' AND X c a will be the deficiency in strength of the hollow beam, when it breaks at b; and (2 d 4 - C4) % ¢ a, will be the defect of strength when it breaks at » or f; which is greater than the former. - For the same reason, the deficiency in strength required to break it at d will be (2 b d 4+-a4 c) X a c. Let. a b: (Fig. 21.) . be a, beam. in. a hori- zontal position, supported at the end a by the upright piece a E, it is required to find the position of another piece, B c, of a given length, so as it may support a n with the greatest force possible. Let B c denote the absolute strength of the beam s c; when, agreeably to the principles of the resolution of forces, c r will express that part of it which is employed in supporting a D ; consequently, by the property of the lever, a c % c r is to be a maximum. But it is well known -that the rectangle of two quantities forms a maximum, when those quantities are equal; therefore s c is in the best position for supporting a n, when a c = a B, or when the angle a B c is equal to the angle a c B. From these data, we learn that the cross bars of gates should not be placed diagonally, as JOINERS' INSTRUCTOR. 337 they most commonly are; because the bar in that position counteracts, in a great measure, what it is intended to remedy. Having now treated of the article Carpentry, not only as a mechanical art, but as a science, we trust that it will prove a useful companion to the reader, as the subjects introduced are those which most frequently occur in the prac- tice of building. THE END. LONDON : PRINTED BY THOMAS PAPISON, WHITEFRIARS.