preffure (oblique) on the joint F C, confifting of the preffure ow on the joint K B, which is propagated through the block B to the block C, and the preffure ww (Mr. Atwood's H b) arising from the weight of the block B. It is alfo evident, that r s is equal to v w (Mr. Atwood's в b) and therefore may reprefent. the weight of the block B; and thus, the lines м r, мs reprefent the preffures mutually exerted at the joints G A and KB, and r s represents the weight of the block B.

This fuggells a very fimple conftruction for the rest, and indeed for the whole. Draw MT perpendicular to the joint FC, and м u perpendicular to the joint 1 D. Then M s is the whole preffure on KB, MT is the whole preffure on F c, and MU is the whole preffure on I D: alfo xr, rs, ST, TU, are the refpective weights of the blocks A, B, C, and D.

This conftruction, which is, virtually, the fame with Mr. Couplet's, is not more fimple than the calculation which may be deduced from it. Draw the horizontal line M z, cutting the verticle x U in z: about the centre M, with the radius Mz, describe the circle z N, cutting MT in N: draw N☀

and TP perpendicular to M S.

Then, we have MN: NOMZ: No, for M N = M Z but NO: TPMZ:M T,

and T PTS M Z: M S, by fimilarity of the

triangles z M s and P T S,-therefore,

M Z3: NO X MTX MS MN (or м z): TS

Now, if the line м z be confidered as radius, it is plain that NO is the fine of the angle contained between the fides K B and FC, of the block c. Alfo M s is the fecant of the angle z M s, or PT S, which the joint K B makes with the vertical. In like manner, MT is the fecant of the angle which F C makes with the verti cal. We may now confider the quantity

NOX MSX MT

M 23

as a number, accounting M z unity. M z is alfo unity. Hence we have the weight Ts of any block c equal to the horizontal thruft multiplied by the fine of the angle of the wedge, and by the product of the fecants of the inclination of its fides to the vertical. For TSM ZX fin S M T X fec. S M Z X fec. TM Z. The horizontal thruft may eafily be had, it being to the weight of the key itone A as M z to x r. Mr. Atwood,. in p. vi. of the Preface, feems to think the difcovery of this value a defideratum in the art; and, in p. 18, he thinks it myfterious that it should depend folely on the weight of the key ftone. But, we conceive, there is no myftery in the matter, and it is equally dependent on, or deducible from the weight of any other block, when the arch is balanced. If it is not ba

lance,

lanced, the horizontal thruft does not depend folely on the weight of the key flone, nor can it be deduced from it, when the unbalanced arch ftands by the help of cement, or by the friction of the joints.

This value of TS is, virtually, the fame with that given by Emerfon, but investigated by him in a much lefs intricate manner. The intelligent reader will fee that it coincides precisely with the refult of Mr. Atwood's laborious procefs.

This con

Unwilling to fuppofe that Mr. Atwood took fuch circuitous methods without fome good reafons, we imagined that we fhould meet with them in the converfe of the problem, where the weights of the blocks are given, and the angles are required. We were the more difpofed to expect this, in confequence of the elaborate conftruction for this purpose, given in pages 25, 26, 27, which requires fome of the moft refined theorems in trigonometry, although it might have been difcuffed in three lines: but we found nothing new. verfe may be thus conftructed. Take xr to represent the weight of A, and then make r s, s T, TU, in the given proportion of the intended weights of the other blocks to the block A. Then, fince the angle of the key flone is affumed in both cafes, let the angle X MR be equal to that contained between the joints GA and DC. This determines the point M. Draw M X, Mr, MS, M T, M U. This gives the angle of each wedge, and the inclination of its fides to the vertical. The angles found in this manner are precifely thofe furnished by Mr. A.'s procefs, and contained in his different tables. It is needlefs to demonftrate this; for it is found in the Rules given by the author himfelf in page 19, particularly Rule 3. If that be juft, all the reft are involved in it.

Since the refults of this procefs agree with all Mr. Atwood's deductions, it may now be afked, what objections lie against his method, befides fome want of fimplicity, and confequently of elegance? We have this objection; that it is by no means fufficient for equilibration and ftability, that the weights of the blocks, and the directions of their fides, be properly adjufted to each other. It is further neceffary, that the preffures which are balanced in thefe propofitions be actually exerted, and fo combined, by the meetings of their directions in each block, that a balance may take place in it. That an egg may fland on one end on a table, it is not enough that the table be horizontal. It is true, that if it be, the fupport given to the egg is directed vertically upward through the point of contact, and the weight of the egg acts vertically downward through its centre of gravity. Thefe directions are indeed oppofite: yet the egg will fall down, unless the centre of gravity be directly above the point of contact. In like manner, the three

forces,

forces, namely, the weight of the block, and the preffures on its two fides, muft act in lines which meet in fome one point in that block.

The common way of treating this problem fhows what is wanting for this purpose. Let a be fome point in the vertical paffing through the centre of gravity of the block A. Draw a м perpendicular to D C, and a fb perpendicular to GA. Let us fee what fupports this key flone. Take a a to reprefent its weight. It is urged downwards with a force a a. It preffes on the joints DC and G A, in the directions a м and ab; and they react in the directions м a and b a. That it may be fupported, and no more than fupported, a a must be the diagonal of a parallelogram a e af, of which the two fides a e and a freprefent the reactions, or fupports given by the adjoining blocks. Thefe, when combined, balance the downward force of the gravity a a.

That the next block в may be supported (and no more than fupported) obferve that a preffes on the joint G A with a force af. Therefore produce a ftill it meets, in b, with the vertical drawn through the centre of gravity of this block B. Make bg equal to a f; and, drawing bc, cutting K B at right angles, draw g 8 parallel to 6 c, and complete the parallelogram

Bhb. It is plain to any mechanician, that 6 B will reprefent the weight of this block, and bh will be the preffure which it exerts on the block c; for the two preffures gb and hb will just balance the weight b B.

We may proceed in the fame manner with the reft, and obtain cx for the weight of c, and d for that of D, and c k and dm for the preffures on F C and I D.

But obferve what is required for effecting thefe balances of force. The line af muft pafs through fome point b of the vertical bß, drawn through the centre of gravity of the blocks; and this point must be in the matter of the block; and, moreover, it must be poffible to draw the perpendicular b c through fome part of the joint K B. It is not enough that it may be drawn perpendicularly on fome point of the mathematical line KB produced: it must be a point of the material furface, otherwife there is no reaction or fupport. Without this, the parallelogram of forces cannot be formed*.

We may now perceive the general requifite condition. It must be poffible to begin at one extremity, fuppofe the abut

Note, that by drawing e a, eß, ey, ed parallel to a f, b b, c k, and dm, we juft transfer the triangles g b B, icy, and id, to the vertical ad; and thus its portions a a, a ß, By, y, are the very weights determined for the different blocks. This first fuggested the figure м x U, employed in the preceding obfervations.

ment

ment I D, and to draw n d perpendicular to I D, meeting the vertical through the centre of gravity of this block in d, and from d to draw dc, with the fame conditions, and in like manner, c b, ba, and a м*; and all this must be within the folid matter of the arch ftones. If only one fuch line of equili brium can be drawn, the figure will ftand, but will not bear the smallest change of form or weight in any part. If more can be drawn, it will have ftability, and will bear certain moderate changes of form or preffure. Thefe lines will be parallel; and the more remote they can be drawn from one another, the figure will have the greater ftability.

When the arch ftones are taken indefinitely thin, this line of equilibration coincides with the curve which the mechanicians have investigated for every fuppofition of preffure on its different parts. The joints of the ftones are always fuppofed perpendicular to the curve.

Mr. Atwood pays no regard to this curve, and fays exprefsly, in page 29, &c. that he makes the problem more ge neral by keeping clear of this condition. In page 31, he indeed hints, that too great liberty must not be taken with the extent of the bafes of the wedges; faying, that if this be too great for their depth, they lofe their property of a wedge; and adds, that the due limits are better learned by experience than by mathematical investigation. We confefs that we do not understand this kind of expreffion, in a problem purely mathe matical; nor do we fee the force of proofs drawn from models and from calculations. The latter are no proofs, but merely another manner of expreffing the conftruction.

That no doubt may remain of his general principle, that no conditions are neceffary but the adaptation of the weights to the angles of the wedges, and inclination of their fides to the horizon, Mr. A. gives an example of an arch in figure 11, which he transfers to figures 12 and 13, by making the joints of the two laft portions of the radii, which form the joints in figure 11, the weights of the corresponding arch ftones being in the fame proportion to each other; which being secured, he fays, that the figure of the under and upper fide of the arches. is a matter of indifference. But were this the case, it would follow that, having made a balanced arch, A B C, (fig. в) another, Ab c, having the fame abutments, and joints formed by the fame radii, and the fame proportion of the weights of the correfponding blocks, should also be in equilibrio, although

The points a, b, c, d, being each in the vertical drawn through the centre of gravity, of the block.

its convexity should be downward, or though its form should be a waving curve, or any equally incongruous figure. No limitation is mentioned by Mr. Atwood; yet all this is plainly impoffible. The arch, fig. 15, may have fucceeded in a model; but it is certain, that if it were reduced to half the thick-' nefs (fill preferving the fame joints, and the fame proportion of the weights of the different blocks) it would not stand; becaufe, in fuch cafe, a line drawn perpendicular to the joint TM ort m, would fall below the abutment Q or q'. All that is between T M and Q, could then turn round the point Q, even' although the rest of the arch were held up. It was improper to fay, in p. 30, that the ftability required a certain proportion' between the bafe and the depth of the wedge, otherwise they would lofe their property of a wedge; for in fig. 15, when its flability has been deftroyed by making it too thin, this is not because the wedges have now too large bases for this depth; for the due proportion (whatever this may be) can be reftored by increafing the number of joints. But it is evident that this fubdivifion cannot make any change in its flability.

Upon the whole, we are firmly of opinion, that the adjustment of the weights to the angles of the arch ftones, and the inclination of their joints to the horizon, will not secure the ftability of an arch, unless a series of parallelograms of equilibration can be traced, without interruption, in the folid matter. of the arch; or a curve of equilibration can be drawn through the furfaces of contact. We have only further to remark, that the prodigious friction which obtains in the joints of an arch, the blocks of which are preffed together with fuch enor mous force, introduces an agent which is altogether overlooked by the mathematicians. This enables an overloaded wedge not only to retain its fituation, without being pushed through the arch, but also, when too much preffed, to drag inward with it the adjoining wedges. Thefe, in like manner, drag in those fituated beyond them, on both fides of the overloaded wedge; and thus extend the action of this unbalanced preffure to diftant parts of the arch, and thus tend to break it acrofs. Upon the whole, however, the effect of this friction tends greatly to ftrengthen the arch; bringing the adjoining parts, on each fide of the overloaded point, to its affiftance. It is owing to this that arches fland, and are exceedingly strong and durable, that are conftructed in a way altogether inconfiftent with the theories of the mathematicians: but no rules, tolerably precife, have yet been obtained for their conftruction. The obfervations on this fubject, in the fupplementary volumes of the Encyclopædia Britannica (article Arch) we have

B

BRIT. CRIT, VOL. XXIII. JAN. 1804.

reafon