ART. II. A Differtation on the Conftruction and Properties of Arches. By G. Atwood, Efq. 4to. 47 pp. 7s. 6d. Lunn, Egerton. 1801. THE problem treated by Mr. Atwood is the most important in the whole art of architecture, and has attracted the attention of the first mathematical geniufes in Europe, fince they begun to apply their fcience to mathematical problems. On this account we fhall, for once, deviate from our ufual practice, and, in reviewing the Differtation before us, make references, not only to fome of the author's diagrams, but alfo to two diagrams of our own. Without thefe, we could fay nothing on the fubject that would be intelligible; and though fuch of our readers as poffefs not Mr. Atwood's work, may not, even thus, fully understand our criticism, we trust that they will perceive the force of much of our reasoning, and pardon the unavoidable obfcurity of the remainder, for the fake of thofe who are employed in a moft ufeful art. The folution of the problem under review has been given in various forms, adapted to almoft every condition of things. that can occur. They are all to nearly the fame purpose, and indeed are all contained in that enigmatical enunciation in which Dr. Robert Hooke firft fhowed, in 1676, that the catenarean curve was the proper form for an arch of equal thicknefs; "ut pendet continuum flexile, fic ftabit folidum contiguum erectum. The properties of this curve have been demonftrated by many mathematicians; but, we believe, by none with greater perfpicuity than by Profeffor Robinfon of Edinburgh, under the title Roof, in the Encyclopædia Britannica. Referring our mathematical and mechanical readers for fuller information to that work, we fhall here only observe, that the general form into which the problem was brought, was deduced from the fuppofition, that the arch flones touched each other in furfaces fo narrow, that they might be confidered as evanefcent elements of curved furfaces, having a common tangent plane. This circumfiance was neceflary, in order to obtain a determined direction of the preffure which each pair of contiguous blocks mutually exerted. The problem then was to find the curve which would pafs perpendicularly through all these planes in the points of contact, fo that blocks of determined weights fhould balance each other; or, converfely, the curve being given, to determine the weights of the blocks, fo that they hall all balance. It is evident that this would be a tottering equilibrium, as when an egg ftands on one end. But if, inftead of touching in points, they touch in plane furfaces, of fome extent, and having the direction of the above-mentioned tangent planes, the figure will have ftability, that is, will bear certain moderate changes of form, and certain moderate changes of weight in the different blocks, without falling down; even though the joints be without cement, and perfectly fmooth and flippery. After the problem had been treated in this moft general and accurate manner, feveral mechanicians endeavoured to adapt it. more to the information ufually poffeffed by practical men. This they did, chiefly by confidering the arch ftones as so many wedges. But, by introducing this idea, they depart fo widelyfrom the real ftate of the cafe, that their demonftrations become exceedingly obfcure, perplexed, and often infufficient.. Mr. Couplet, of the Royal Academy of Sciences at Paris, has been the most fuccefsful, and has deviated leaft from the true ftate of the cafe. He has given folutions of the most usual cafes, of very beautiful fimplicity, and abundantly exact for all common purposes. The Differtation now before us, aims at deducing every. thing from the principles of the wedge, and pays no regard to the internal curve of preffures, which occupied the whole attention of former mechanicians. The ingenious author fays, indeed, exprefsly, in pp. 29, 30, and 31, that this is a matter of indifference, and that every thing depends on the proper adjustment of the weight and the angle of the wedges, and the inclination of their joints to the horizon. This, however, with all deference to fuch mathematical authority, feems to lead to rules of conftruction, that are inconfiftent with the principles adopted by the moft eminent mechanicians of Europe, and acquiefced in by the author himself. We shall venture therefore to make a few obfervations on his manner of treating the problem, and to point out fome confequences that seem to us undeniable, and yet are incompatible with ftability. We make them ftill with fome degree of diffidence, fearing that, notwithstanding the care with which we have perused the Differtation, we may have overlooked fomething of confequence; for we are by no means willing or ambitious to impute error to an author, whofe reputation ftands fo juftly high, and who has given undoubted proofs of eminent knowledge in phyfico-mechanical fubjects. It seems necellary to begin by remarking, that Mr. Atwood's whole procefs terminates in folving the elementary problem of determining the relations between the weight of each arch ftone, and the angles which its fides make with each other and with the horizon. This is the problem folved by Couplet, Emerfon, Emerfon, and every elementary writer on mechanics. Mr. Atwood's manner of folving it, differs indeed exceedingly, but his refults are precifely the fame, in every inftance; for this reason, we fay that Mr. A. acquiefces in the principles adopted by thofe writers. Now it certainly appears to us, as one general objection, that Mr. A.'s method is extremely ircuitous, and his mathematical procefs unneceffarily comlex and intricate; employing, on every occafion, compofiions of ratios, the arithmetic of fines, and others of the more abftrufe modes of inveftigation. Of this, we have a ftriking example in pp. 8 and 12, and a ftill more remarkable one in p. 25, &c. where the whole might be difcuffed in two lines. Much of this effect, though by no means the whole, has arifen from the author's keeping a feparate account of the three forces HA (Diff. fig. 2) A a, and H a. This method alfo leads to calculations equally complicated and intricate, requiring frequent references to the logarithmic tables. The whole process may, therefore, be greatly fimplified; and it is of importance to fhow, that the refults will be precifely the fame. (Differtation, p. 7, fig. 2.) The preffure м x being perpendicular to the joint D B, must be oblique in refpect to the joint O A. Therefore it pushes the key ftone upwards, along the joint o A, with a force which may be reprefented by R X. This must be balanced by the tendency which the weight of the key ftone gives it to flide down along o A. This determines the magnitude of Aa (which reprefents the weight of the key ftone); A a must be of fuch a length, that the perpendicular a H to OA fhall cut off HA equal to X R. All this is perfectly exact, and agreeable to the common rules for the refolution of forces. But furely it might have been more easily done, and without fo much difplay of mathematical knowledge. The figure A, annexed to our obfervations, is adapted to Mr. Atwood's fig. 3, and will ferve to exprefs the obfervations we have to make on the conftruction in the Differtation, and its confequences. Through x draw the vertical x u, cutting MR in r. It is evident that xr is equal to Mr. Atwood's A a, and R r to his н a. In the fame manner, drawing the vertical v w, cutting Q w in w, v w is equal to Mr. Atwood's в b, and w w equal to his Hb. It is alfo evident that qv is equal to Mr. We might proceed in the fame way with the refl. But the equality of Qv and Mr fuggefts a fill greater, and more inflructive fimplifi cation. Draw MS parallel to q w, or perpendicular to the joint K R. It is evident that Ms is equal to go, and reprefents the whole preffure |