Encyclopaedia Britannica [18, 3 ed.]

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ENCTCLOPJEDIA

BRITANNICA;

DICTIONARY O F ARTS,

SCIENCES, AND

MISCELLANEOUS LITERATURE; Conftrudled on a

PLAN,

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Diftindt

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HISTORY, THEORY,

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according to the Lateft Difcoveries and Improvements; AND FULLEXPLANATIONS GIVEN OF THE

VARIOUS DETACHED PARTS OF KNOWLEDGE^ WHETHER RELATING TO

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Including ELUCIDATIONS of the molt important Topics relative to RELIGION, MANNERS, and the OECONOMY of LIFE : TOGETHER

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DESCRIPTION

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of all the Countries, Cities, principal Mountains, Seas, Rivers, be. throughout the W o R L D ;

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PQCTI DISC A NT,

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ET d ME NT ME MINISS E

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EDINBURGH. PRINTED FOR A. BELL AND C. MACFARQJ1 HAR*

MOCCXCVIL

(ZEntereO in ©tationew Jpall in (ZTcrmsi of tfje aft of parliament.

ENCYCLOPEDIA BRITANNICA,

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Strength of QjTRENGTH OF MATERIALS, in mechanics, is a fubMatenak je(e^, importance, that in a nation fo emij uent as this for invention and ingenuity in all fpecies Importanceof manufaftures, and in particular fo diftinguifhed for of the iub. its improvements in machinery of every kind, it is fomeje°75

II

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II

Tire reader will furely obferve, tha} thefe numbers ex-Strength,of tt riat prefs fomething more than the utmoft cohefion ; for the ‘- Q weights are fuch as will very quickly, that is, in a minute ^ or two, tear the rods afunder. It may be laid in general, No fubthat two-thirds of thefe weights will fenfibly impair the jfance to ftrength after a confiderable while, and that one half is the utmoft that can remain fufpended at them without rifle f°rture above ever ; and it is this laft allotment that the engineer fhould rec-one half it* kon uporf in his conitrudtions. There is, however, confiderable ftrength, difference in this refpedt. Woods of a very ftraight fibre, fuch as fir, will be lefs impaired by any load which is not iufficient to break them immediately. According to Mr Emerfon, the load which may be fafely fufpended to an inch fquare is as follows : Iron Brafs Hempen rope Ivory Oak, box, yew, plum-tree Elm, afh, beech Walnut, plum Red fir, holly, elder, plane, crab Cherry, hazle Alder, afp, birch, willow Lead Freeftone -

76,40a

35>6o° 19,600 15,700 1^5° 6,070 5*36° 5,000

4>29':} 430

974

Fie gives us a practical rule, that a cylinder whofe diameter is d inches, loaded to one-fourth of its ablolute ftrength, will carry as follows : Iron Good rope Oak Fir

Cwt.

The rank which the different woods hold in this lift of Mr Emerfon’s is very different from what we find in Mufchenbroek’s. But precife meafures muff not be expected in this matter. It is wonderful that in a matter of fuch unqueftionahle importance the public has not enabled feme perfons of judgment to make proper trials. They are beyond the abilities of private perfons. ,IL

BOBIES

MAY

BE

CRUSHED.

46 It is of equal, perhaps greater, importance to know the ft is of finftrain which may be laid on folid bodies without danger ofportance to crufhing them. Pillars and polls of all kinds are expofed to^'!ow wfiat this Ilrain in itsfimpleft form; and there are cafes where the1 Itrain is enormous, viz. where it arifes from the oblique pofition of the parts; as in the ftuts, braces, and truffes, which occur very frequently in our great works. 11 is therefore moll dcfirable to have fome general knowledge of the principle which determines the ftrength of bodies in oppofition to this kind of flrain. But unfortunately we are much more at a lofs in tins than in the laft cafe. The mechanifm of nature is much more complicated in the prefent cafe. It muft be in fome circuitous way that compreffion can have any tendency to tear afunder the parts of a folid body, and it is very difficult to trace the fteps. If we fuppofe the particles infuperably hard and in conta£l, and difpofed in lines which are in the diredlion of the external preffures, it does not appear how any preffure can ' difunite the particles ; but this is a gratuitous fuppofition. There are infinite odds againft this precife arrangement of the lines of particles; and the compreffibility of all kinds of matter in fome degree fhows that the particles are in a fituation equivalent to diftance. This being the cafe, and the particles, with their intervals, or what is equivalent to in13 2 • tervals,

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Strength of tervals, being in fituations that are oblique with refpeft to ^a'e' ’1 ‘‘ the preffures, it muft follow, that by fqueezing them together in one direction, they are made to bulge out or feparate in other directions. This may proceed fo far that fome may be thus pufhed laterally beyond their limits of cohefion. The moment that this happens the refiftance tocompreffion is diminifhed, and the body will now be crufhed together. We may form fome notion of* this by fuppoling a number of fpherules, like fmall foot, flicking together by means of a cement. Comprefling this in ibme particular direction caufes the fpheruies to aCt among each other like fo many wedges, each tending to penetrate through between the three v/hich lie below it : and this is the fimpleft, and perhaps the only diftinCt, notion we can have of the matter. We have reafon to think that the conftitution of very homogeneous bodies, fuch as glafs, is not very different from this. The particles are certainly arranged fymmetrically in the angles of fome regular folids. It is only fuch an arrangement that is confiflent with tranfparency, and with the ^ free paffage.of light in every diredlion. Their If this be the conftitution of bodies, it appears probaflren^th ble that the flrength, or the lefiftance which they are caCr

n.er * i,*0 f all a force

Pable of making to an attempt to crufh them to pieces, is proportiona.1 to the area of the feftion whofe plane is perpendicular to the external force ; for each particle being limilarly and equally aCled on and refifted, the whole reiiftance mull be as their number; that is, as the extent of the feClion. Accordingly this principle is affumed by the few writers who have confidered this fubjeft ; but we confefs that it appears to us very doubtful. Suppofe a number of brittle or friable balls lying on a table uniformly arranged, but not cohering nor in contad, and that a board is laid over them and loaded with a weight ; we have no hefitation in faying, that the weight neceffary to crufh the whole colleiflion is proportional to their number or to the area of the feCtion. But when they are in contaCl (and ftill more if they cohere), we imagine that the cafe is materially altered. Any individual ball is crufhed only in confequence of its being bulged outwards in the direction perpendicular to the prelfure employed. If this could be prevented by a hoop put round the ball like an equator, we cannot fee how any force can crufh it. Any thing therefore which makes this bulging outwards more difficult, makes a greater force neceffary. Now this effeCt will be produced by the mere contaCt of the balls before the preffure is applied; for the central ball cannot fwell outward laterally without pufhiug away the balls on all fides of it. This is prevented by the friction on the table and upper board, which is at leaft equal to one third of the preffure. Thus any inlerior ball becomes itronger by the mere vicinity of the others ; and if we farther fuppofe them to cohere laterally, we think that its ftrength will be ftill more increafed. The analogy between thefe balls and the cohering particles of a friable body is very perfeCt. Wc fhould therefore expeCt that the ftrength by which it refifts being crufhed will increafe in a greater ratio than that of the feCtion, or the fquard of the diameter of fimilar feCtions ; and that a fquare inch of any matter will bear a greater weight in proportion as it makes a part of a greater feCtion. Accordingly this appears in many experiments, as will be noticed afterwards. Mufchenbroek, Euler, and fome others, have fuppofed the ftrength of columns to be as the biquadrates of their diameters. But Euler deduced this from formuke which occurred to him in the courfe of his algebraic analyfis; and he boldly adopts it as a principle, without looking for its foundation in the phyftcal affumptions which he had made in the beginning of his inveftigation. But

fome of his original affumptions were as paradoxical, or atStrenfi, leaft as gratuitous, as thefe refults : and thofe, in particular, from which this proportion of the flrength of columns was deduced, were almoft foreign to the cafe ; and therefore the inference was of no value. Yet it was reserved as a principle by Mufchenbroek and by the academicians . of St Peterfburgh. We make thefe very few obfervations, becaufe the fubjeft is of great praftical importance ; and it is a great obftaele to improvements when deference to a great name, joined to incapacity or indolence, caufes authors to adopt his carelefs reveries as principles from which they are afterwards to draw important confequences. It mult be acknowledged that we have not as yet eftablifhed the relation between the dimeniions and the ftrength of a pillar on folid mechanical principles. Experience plainly contradicts the general opinion, that the ftrength is proportional to the area' of the fettion ; but it is ftill more inconfiftent with the opinion, that it is in the quadruplicate ratio of the diame- ,^4^ ^ ters of fimilar fettions. It would feem that the iatio de-^ ^ pends much on the internal ftru£ture o! the body , and ex- 011[, ei periment feems the only method for afeertaining its general perm; .t, laws. . If we fuppofe the body to be of a fibrous texture, having the fibres lituated in the direction or the preffure, and (lightly adhering to each other by fome kind of cement, fuch a body will fail only by the bending of the fibres, by which they will break the cement and be detached from each other. Something like this may be fuppofed in wooden pillars. In Inch, cafes, too, it would appear that the refiftance muft be as the number of equally refilling fibres, and as their mutual fupport, jointly ; and, therefore, as fome function of the area of the feftion. The fame thing muit happen if the fibres are naturally crooked or undulated, as is obfeived in many woods, &c. provided we fuppofe fome fimilarity in thenform. Similarity of fome kind mult always be fuppofed,. otherwife we need never aim at any general inferences. In all cafes therefore we can hardly refufe admitting that the ftrength in oppofition to compreffion is proportional to a funftion of the area of the feftion. As the whole length of a cylinder or prifm is equally preffed, it does not appear that the ftrength of a pillar is at all affefted by its length. If -indeed it be fuppofed to bend under the preffure, the cafe is greatly changed, becaufe it is then expofed to a tranfverfe ftrain j and this mcreafes with the length of the pillar. But this will be confidered with due attention under the next clafs of ftrains. Few experiments have been made on this fpecies of ftrength and ftrain. Mr Petit fays, that his experiments, and thofe of Mr Parent, fhow that the force neceffary for crufhmg a body ts nearly equal to that which wiil tear it afunder. He fays that it requires fomething more than 60 pounds on every fquare line to crufh a piece of found oak, the rule is by no means general 1 Olafs, for mftance, will carry a hundred times as much as. oak in tins way, that is, refting on it; but will not Jufpend above foui or five times as much. Oak will fufpend a gteat deal more than fir; but fir will carry twice as much as a pillar. Woods of a foft texture, although confifting of very tenacious fibres, are more eafily crufhed by then load. I his foftnefs of texture is chiefly owing to their fibres not being ftraight but undulated, and there being confiderable vacuities between them, fo that they are eafily bent laterally and crufhed. When a poft is overftrained by its load, it is obferved to fwell fenlibly in diameter. Increafing the load caufes longitudinal cracks or fhivers-to appear, and it piefentry after gives way. This is called crippling. In all cafes where the fibres lie oblique to the ftrain the ftrength is greatly diminifhed, becaufe the parts can then be • made

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S'.reng'h of made to Aide ©n each other, when the cohefion of the ceMateiials. mcnt;ng matter is overcome. _ r • r r- xi ' ' Mufchenbroek has given Tome experiments on this iubjett; but they are cafes of long pillars, and therefore do not belong to this place. They will be conlidered afterwards. The only experiments ©f which we have feen any detail (and it is ufelefs to infert mere affertions) are thofe of _Mr Gauthey, in the 4th volume of Rozier’s Journal de Phyjique. This engineer expofed to great preffures fmall reftangular parallelepipeds, cut from a great variety of ftones, and noted the weights which crurtied them. The following table exhibits the medium refults of many trials on two very uniform kinds of freeftone, one of them among the hardeft and the other among the fofteft ufed in building. 49 Column ift expreffes the length AB of the fe&ion m Expo imeir.s for French lines or i2ths of an inch ; column 2d expreffes the this oiir- breadth BC ; column 3d is the area of thefeaion in Iquare pofe made lines; column 4th is the number of ounces required to cram on freethe piece ; column 5th is the weight which was then borne ftojie by each fquare line of the fedton ; and column 6th is the round numbers to which Mr Gauthey imagines that thofe in column 5th approximate. Hard Stone. Weight Force AB X BC AB BC 8

8

64

8 8

12

96

9 9 18 18

736

11>5

16

128

2625 4496

7>3 35>i

16 18 18 24

Soft Stone. 560 144 848 162 2928 324 5296 432

3>9 5>3 9

2

12,2

4 4>5 9 12

Tittle can be deduced from thefe experiments: The 1 ft and 3d, compared with the 5th and 6th, fhould furnifh iimilar refults; for the ift and 5th are refpedively half of the 3d and 6th : but the 3d is three times ftronger (that is, a line of the 3d) than the firft, whereas the 6th is only twice as ilrong as the 5th. It is evident, however, that the ftrength increafes much fafter than the area of the fedion, and that a fquare line can carry more and more weight, according as it makes a part of u larger an . larger fedion. In the feries of experiments on the foft ftone, the individual ftrength of a fquare line feems toincreafe nearly in the proportion of the fedion of which it makes a part. Mr Gauthey deduces, from the whole of his numerous experiments, that a pillar of hard ftone of Givry, whofe fedion is a fquare foot, will bear with perled fafety 664,000 pounds, and thafits extreme ftrength is 871,000, and the fmalleft ftrength obferved in any of Iris experiments wasqbo,000. I he loft bed of Givry ftone had for its fmalleft ftrength 187,000, for its greateft 311,000, and for its fafe load 249,000. Good brick will carry with fafety 320,000 ; chalk will carry only 9000. 'The boldeft piece of architedure in this refped which he has feen is a pillar in the church of All-Saints at Angers. It is 24 feet long and 11 inches fquare, and is loaded with 60,000, which is not 4th ol what is neceftary for cruftring it. We may obfe.rve here by the way, that Mr Gauthey’s ineafure or the fufpending ftrength of ftone is vaftly frnall in proportion to its power of fupporting a loadjaid above it. He linds that a prifm of the hard bed of Givry, of a foot iedion, is torn afunder by 4600 pounds ; and if it be lirmly fixed horizontally in a wall, it will be broken by a 50, weight of 56,000 fufpended a foot from the wall. It it reft Not fatison two props at a foot diftance, it will be broken by 206,000 failory. laid on its middle. Thefe experiments agree fo ill with each

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other, that little ufe can be made of them. The fubjed is Strength cf of great importance, gnd well deferves the attention of the Mater. 1i0,^ patriotic philofopher. 5x A fet of good experiments would be very valuable, be-Good excaufe it is againft this kind of ftrain that we mult guard by perimeuta judicious conftrudion in the moft delicate and difficult Pro"^”ued blems which come through the hands of the civil and military engineer. 'The conftrudton of ftone arches, and the conftrudion of great wooden bridges, and particularly the conftrudion of the frames of carpentry called centres in the eredion of ftone bridges, are the moft difficult jobs that occur. In the centres on wdrich the arches of the bridge of Orleans were built fome of the pieces of oak were carrying upwards of two tons on every fquare inch of their fcantling. All who faw it laid that it was not able to carry the fourth part of the intended load. But the engineer underftood the principles of his art, and ran the rifle : and the refult completely juftified his confidence ; for the centre did not complain in any'part, only it was found too fupple ; fo that it w'ent out of fhape while the haunches only of the arch were laid on it. The engineer correded this by loading it at the crown, and thus kept it completely in fhape during the progrefs of the work. In the Memoirs (old) of the Academy of Peteifburgh for 17"’8, there is a diftertation by Euler on this fubjed, but particularly limited to the ftrain on columns, in which the bending is taken into the account. Mr Fufs has treated the fame fubjed with relation to carpentry in a fubiequent volume. But there is little in thefe papers beftdes a dry mathematical difquifition, proceeding on affumptions which (to {peak favourably) are extremely gratuitous. The moft important confequence of the compreffion is wholly overlooked, as we ffiall prefently fee. Our knowledge of the mechanifm of coheiic5n is as yet far too imperfed to entitle' us to a confident application of mathematics. Experiments fnould be multiplied. 52 The only way we can hope to make thefe experiments How they tifeful is to pay a careful attention to the manner in which the fradure is produced. By difeovering the general refemblances in this particular, we advance a ftep in our power of introducing mathematical meafurement. Thus, when a cubical piece of chalk is ftowly crufhed between the chaps of a vice, we fee it uniformly fplit in a furface oblique to the preffure, and the two parts then Hide along the furface of fradure. This ftiould lead us to examine mathematically what relation there is between this furface of fradure and the nectfiary force ; then we fhould endeavour to determine experimentally the pofition of this furface. Having difeovered fome general law or refemblance in this circumftance, we ihould try what mathematical hypothefis will agree with this. Having found one, we may then apply our limpleft notions of cohefion, and compare the refult of our computations with experiment. We are authorifed to fay, that a feries of experiments have been made in this way, and that their refults have been very uniform, and therefore fatisfadory, and that they will foon be laid before the public as the foundations of iuccefsful pradice in the conftrudion of arches. III.

A

BODY

MAY

BE

BROKEN ACROSS..

53

The moft ufual, and the greateft ftrain, to which mate-ft i*5 of imrials are exoofed, is that which tends to break them tranfverfely. It is feldom, however, that this is done in a man-what {train ner perfedly fimple ; for when abeam projeds horizontally will break from a wail, and a weight is fufpended from its extremity,a the beam is commonly broken near the wall, and the inter-remediate part has performed the fundions of a lever. It iy. fometimes, though rarely, happens that the pin in the joint of a pair of pincers or feiffars is cut through by the ftrain ;

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n ; and this is almoftthe only cafe of a fimple tfanlverfe fradlure, Beiryf fo rare, vve may content ourfdves with faying, that in this cafe the ftrength of the piece is proportional to the.area of the fedfion. H Experiments were made for difcovering the refiftances Experimcnts made by bodies to this kind o! ftr£in in the following manner : ?nadr tf) I wo iron bark were diipofcd horizontally at an inch diitance; afcertain a third hung perpendicularly betv/een them, beiu p fupported it. by a pin made of the fubftance to be examined. This pin was made of a prilmatic form, fo as to lit exactly the holes in the three bars, which were made very exaft, and of the fame fize and fhape. A fcale was lufoended at the lower end of the perpendicular bar, and loaded till it tore out that part of the pin which filled the middle hole. This weight was evidently the meafure of the lateral cohefion of two fedtions. f he fide-bars were made to grafp the middle bar pretty itrongly between them, that there might be no diftance impofed between the oppolite preflures. This would have combined the energy of a lever with the purely tranfverfe preiTure. For the fame reafon it was neceflkry that the internal parts of the holes flionld be no fmaller than the edges. Great irregularities occurred in our lirft experiments from th is caufe, becaufe the pins were fomewhat tighter within than at the edges; but when this was corredted they were extremely regular. We employed three fets of holes, viz. a circle, a fquare (which was occafionally made a redfangle v/hofe length was twice its breadth), and an equilateral triangle. We found in all our experiments the ftrength exactly proportional to the area of the fedtion, and quite independent of its figure or pofition, and we found it confiderably above the diredt cohefion ; that is, it took confiderably more than twice the force to tear out this middle piece than to tear the pin afunder by a diredt pull. A piece of fine freeftone required 205 pounds to pull it diredtly afunder, and 575 to break it in this way. The difference was 55 Their re- very conftant in any one fubftance, but varied from yds to yds fuit. in different kinds of matter, being fmalleft in bodies of a fibrous texture. But indeed we could not make the trial on any bodies of confiderable coheiion, becaufe they required fuch forces as our apparatus could not fupport. Chalk, clay baked in the fun, baked fugar, brick, and freeftone, were the ftrongeft that we could examine. But the more common cafe, where the energy of a lever intervenes, demands a minute examination. Let DABC (fig. 5-n01.) be a veitical ftdfion of a prifma56 tic folid (that is, of equal fize throughout), projedting horiThe zontally from a wall in which it is firmly fixed ; and let a ftrength vf a lever weight P be hung on it at B, or let any power P adt at B in a diredfion perpendicular to AB. Suppofe the body of infuperable ftrength in every pari: except in the vertical fection DA, perpendicular to its length. It muft break in this fedfion only. Let the cohefion be uniform over the whole of this fedfion ; that is, let each of the adjoining particles of the two parts cohere with an equal force/i There are two ways in which it mlfy break. The part ABCD may limply Aide down along the furface of fiadture, provided that the power adting at B is equal to the accumulated force which is exerted by every particle of the fection in the diredlion AD. But fuppofe this effedlnally prevented by fomething that fupports the point A. The adb’on at P tends to make the body turn round A (or round a horizontal line palling thro* A at right angle? to AB) as round a joint. This it cannot do without feparating at the line DA. In this cafe the adjoining particles at D or at E will be feparated horizontally. But their cohefion refills this feparation. In order, therefore^ that the fracture may happen, the en-

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enry or momentum of the power P, adting by means of the ^tsength . lever AB, muft be fuperior to the accumulated energies ot Materia*‘ the particles. The energy of each depends not only on its cohefive force, but alfo on its fituation ; for the fuppofed infuperable firmnefs of the reft of the body makes it a lever turning round the fulcrum A, and the cohefion of each particle, fuch as D or E, ndls by means of the arm DA or LA. The energy of each particle will therefore be had by multiplying the force exerted by it in the initant of fiatfture by the arm of the lever by which it adts. Let us therefore firft fuppofe, that in the iuftant of frao ture every particle is exerting an equal force /. The energy of D will be/XDA, and that of E will be /"XEA, and that of the whole will be the fum of all thefe products. Let the depth DA of the fedtion be called d, and. let any undetermined part of it E A be called x, and then the fpace occupied by any particle will be jc. The cohefion of this fpace may be reprefented by/T, and that of the whole by f d. The energy by which each element x of the line DA, or d, refills the fradture, will be f x x, and the whole accucumulated energies will be fXy^v x. This we know to be y X 4 d~, or fdX 4 d. It is the fame therefore as if the cohefion yk/ of the whole fedtion had been adting at the point G, which is in the middle of DA. The reader who is not familiarly acquainted with this fluxionary calculus may arrive at the fame conclufxon in another way. Suppofe the beam, infte.-.d of projedting horizontally from a wall, to be hanging fr®m the ceiling, in which it is firmly fixed. Let us confiderhow the equal cohefion of every part operates in hindering the lower part from feparating from tire tipper by opening round the joint A. The equal cohefion operates jult as equal gravity would do, but in the oppofite diredtion. Now we know, by tire moft elementary mechanics, that the effedt of this will be the fame as if the whole weight were concentrated in the centre of gravity G of the line DA, and that this point G is in the middle of DA. Now the number of fibres being as the length d of the line, and the cohefion of each fibre being — f, the cohefion of the whole line is f X d or f d. 1 he accumulated energy therefore of the coheiion in the inftant of fradture is f d X\ d. Now this muft be equal or juil inferior to the energy of the power employed to break it. Let the length AB be called /; then P X / is the correfponding energy of the power. This gives us fd 4 T--/ / for the equation of equilibrium correiponding to the vertical fedtion ADCB. Suppofe now that the fradture is not permitted at DA, but at another fedtion ^ « more remote from B. The body being prifmatic, all the vertical fedtions are equal; and theiefore j d\ d is the lame as before. But the energy of the power is by this means increafed, being now zr P X B », inftead of P X B A : Hence we fee that wheivthe prifmatic body is not infuperably ftrong in all its parts, but equally ftrong throughout, it muft break clofe at the wall, where the ft rain or energy of the power is greateft. We fee, too, that a power which is juft able to break it at the wall is unable to break it anywhere elfe ; alfo an abfolute cohefion yV, which can withftand the power p in the fedtion DA, will not withftand it in the fedtion *, and will withfland more in the fedtion d! d. I his teaches us to diftinguifh between abfolute and relative ftrength. The relative ftrength of a fedtion has a reference to the ftrain adtually exerted on that fedtion. This relative ftrength is properly meafured by the power which is juft able to balance or overcome it, when applied at its proper

S Strercth rf proper place Materials.

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Now fince we had f d ' d, : p /, we have

fd\d

for the meafure of the ftrength of the fedlion DA, in relation to the power applied at B. It the folid is a reftangular beam, whofe breadth is l, it is plain that all the vertical fe&ions are equal, and that AG ox \d is the fame in all. Therefore the equation expreffing the" equilibrium between the momentum of the external force and the accumulated momenta of cohefion will be p l — f dl>y.\d. The produft db evidently exprefies the area of the fection of fvafture, which we may (all /, and we may exprefs the equilibrium thus, pi— f s 4 d, and 2 l: cl — J s : p. Now fs is a proper expreffion of the abfolute cohefion of the feftion of frafture, and / is a proper meafure of its ftrength in relation to a power applied at B. Wfe may therefore fay, that twice the length of a reBangular beam is to the depth as the abfolute cohefion to the relative Jlrmgth. Since the aftion of equable cohefion is iimilar to the action of equal giavity, it follows, that whatever is the ftp ure of the feftion, the relative ftrength will be the fame as if the ahfolute cohefion of all the fibres were adling at the centre of gravity of the feftion. Let g be the diilance between the centre of gravity of the fe&ion and the axis of fraflure, we fhall have pi — f sg, and l: g — f s : p. It will be very ufeful to recollect this analogy in words ; “ The length of a prifmatic beam of any faape is to the height of the centre 0/ gravity above the lower fde, as the abjolute cohefion to theJlrer.gth relative to this length."' Becaufe the relative ftrength of a re6tangular beam is ^—f—ox^-~, it follows, that the relative ftrengths of different beams are proportional to the abfolute cohefion of the particles, to the breadth, and to the fquare of the depth diredtly, and to the length inverfely ; alfo in priims whole fedlions are fimilar, the ftrengths are as the cubes of the diameters. 57 . Afterr. in Such are the more general refults of the mechanifm of ed on ri e this tranfverfe ftrain, in the hypothefis that all the pat tides

1 .

S

T

R

proportion cacli fibre is extended. It feems moll probable ’trength of that the extenlions are proportional to the dillances from A. We fhall fuppofe this to be really the cafe. Now recollect the general law which we formerly faid was obferved in all moderate extenfrons, viz. that the attractive forces exerted by the dilated particles were proportional to their dilatations. Suppofe now that the beam is lo much bent that the particles at D are exerting their utmofl force, and that this fibre is iutl teadv to break or aftually breaks. It is plain that a total frafture mull immediately enfue ; becaufe the f orce which was fuperior to the full cohefion of the particle at D, and a certain portion of the cohefion of all the reft, will be more than fuperior to the full cohefion of the, particle next within D, and a fmaller portion of the cohefion of the remainder. Now let F reprefent, as before, the full force of the exterior fibre which is exerted by it in the inftant of its breaking, and then the force exerted at the lame inftant bythe fibre E will be had by this analogy AD : AE, or S' x dix—f: and the force really exerted by the fibre E.

U/Xi. The force exerted by a fibre whofe thicknefs is K is therefore-'-E—; but this force refifts the ftrain by afting d Its energy or moby means of the lever EA mentum is therefore

and the accumulated momenta d _ of all the fibres In the line AiE. will be / X fum of —~ d This, when x is taken equal to d, will exprefs the momentum of the whole fibres in the line AD. This, therefore^ is/~ , or/|d\ orfdXj d.

Now/d expreffes the ab-

folute cohefion of the whole line AD. rI he accumulated momentum is therefore the tame as if the abfolute cohefion of the whole line were exerted at ^d of cvD from A. 59. From thefe premiles it follows that the equation expref- fG fing the equilibrium of the ftrain and cohefion is/> / — J ^afeerfainedX 4 d ; and hence we deduce the analogy, “ As thrice the un OC|ier length is to the depth, fo is the abfolute cohefion to the relative-^ inaples, frength.” This equation and this proportion will equally apply to reftangular beams whole breadth is b ; for we fhall then have pl—fbdx\d. We alfo fee that the relative ftrength is proportional to the abfolute cohefion of the particles, to the breadth, and to the fquare of the depth direftly, and to the lefigth inverfely ; for p is the meafure of the force with which it is. fb did fb d' , . n 1 T r refilled, and p — ■ ^ , — . In this refpeft there-