companied by the advantages of complexion, and by that young fresh. nefs which defies the imitation of art. Sterne even confiders them a indicative of moral qualities the moft amiable, and afferts that they de note exuberance in all the warmer, and confequently in all the better icelings of the human heart. The tranflater does not wifh to deem this epinin as wholly urfounded. He is, however, aware of the danger to rich fuch a confefton cxpefes him; but he flies for protection to the temple of AUREA VENUS. Note to p. 56. There we shall leave him, nor prefume to violate with unhal lowed locks the afylum of the goddefs: but, whilt we earnestly hope that he may there meet with fome prophetie prieftefs who will kindly fatisfy his doubts, and relieve his anxiety as to his fu ture destiny, we must inform our readers that the canzon, as well as the note in queftion, appear to have been written for the ex prefs parpofe of conveying to the world the very interefting parti Clard which they contain with regard to the noble author; there hot being to be found in the original, from which it profeffes to Le draws, any mention whatever of blue eyes, auburn hair, young freshness, amorous difpofition, or any other of thole advantages which the noble writer either poffelles, or thinks he has the prof- | pect of poffeffing over the rest of the world. It is almost unneceffary to add, that the work is neatly printed upon wire-wove paper, and hot-prefled. A portrait of Camoens is prefixed to it, in which we fufpect that his perfon (which was in reality disfigured by the lofs of an eye in an expedition to the coaft of Africa) has been as fancifully embellithed by the graver of the artift, as his writings have been by the pen of Lord Strang ford. ART. IV. Nuova Seluzione d'un Problema Statice Euleriano. Di Gregorio Fontana. From Memorie di Matematica è Fifica della Societa Italiana di Scienze. 1802. pp. 626. THE THE problem difcuffed in this paper, was fuggefted to the celebrated mathematician whofe name is affixed to it, by a paffage in Euler's memoir upon the law of equilibrium, pointed out by Maupertuis, about the middle of the last century. We fhall preface our account of Signor Fontana's fpeculations by a few remarks upon the hiftory of that law. It was maintained by Maupertuis, that, in preferving the repose of the universe, and in performing its various motions, na ture ture uniformly employs thofe means which require the fmalleft expence of force. A body falls to the ground in the vertical line; and this is the shortest route which it can take to arrive at the furface of the earth. Light is reflected at an angle always equal to the angle of incidence. If the force of incidence is divided into two, one parallel and one perpendicular to the fpeculum, it is clear that the force of reflexion must be either greater or lefs than those two forces of incidence, unlefs the angle of reflexion is precifely equal to the angle of incidence. In the former cafe, more force would be required to reflect the light; in the latter, a portion of the force of incidence would be loft. Therefore Maupertuis contended, that the actual proportion of equality is the only one which neither creates any loss of the original force, nor requires any increase of it. This theory was illuftrated by various applications. Confiderable ingenuity of demonftration and elegance of arrangement was mingled with a large portion of metaphyfical reafoning, and a variety of fubtleties, formerly unknown in this department of fcience. No trifling portion of error was introduced by the undif ciplined talents and prefumptuous imagination of the inventor. The whole was paraded in a manner peculiarly obnoxious to men of real fcience, from its vanity and doginatifin. The author was elevated to the chair of the Berlin academy; an inftitution remarkable, at that time, for the violence of its ariftocracy, and fubmiffive, even in its opinions, to the pleafure of the court of Potsdam. He obtained,. in this affociation, a kind of political fupport, not unmingled with perfecution, againit thofe who ventured to attack his doctrines. The wit of Voltaire, and the more fober expofitions of M'Laurin, were repelled for a feafon by this most unfcientific combination. But all was in the end ineffectual. And, as if to punifh, by a fignal fate, fuch an undue method of defence, the name of Maupertuis is now only known by his expedition to Lapland, in the company of thofe very French academicians who afterwards attacked his theory; while, of that theory, there does not remain a veftige in the fcience of Dynamics as at prefent established, although, unquestionably, it contained much valuable matter. It will scarcely be credited that fuch fhould have been the premature end of a doctrine, to prove, illuftrate, and extend which, all the talents of the first of analyfts were exhaufted. The il luftrious Euler appears to have been, from the firft, peculiarly captivated with the fimplicity and elegance of Maupertuis's general law. In a fingle volume of the Berlin Memoirs, we meet with no lefs than three elaborate papers in its defence. Eulernot only adopted the doctrine, but followed it through a thou D 2 fand fand confequences, which were far above the reach of the inventor himself. He became its warmeft eulogift, and undertook its defence against the French and German mathematicians, with all the keennefs of controverfy. The terms which he uses to defcribe its merits, are inferior to none of thofe which the univerfal confent of mankind have almoft confecrated to the fervice of the Newtonian philofophy. Yet the very remains have vanished, etiam periere ruina,' of a fyftem which was praised with the following encomiums by the difinterested zeal of the firft mathematician of the age. We extract these eloges with which Euler concludes two several papers on the fubject, because the whole fact is extremely curious in the hiftory of the fcience. There cannot remain any doubt that this great principle contains, as it were, the effence of all our knowledge in the fcience of statics. It muft be regarded as the true principle of dynamics; as the most sacred law of nature. It is evidently the moft happy and most important difcovery that has ever been made in this fcience; unfolding to us at once the general law which all cafes of equilibrium obey, and difplaying the genuine plan of nature, to operate at all times with the leaft poffible expence of force. (Mem. de l'Acad. des Sciences et Belles Lettres de Berlin. Tom. VI. p. 183. In a reply to fome attacks made upon the doctrine of the minimum, he concludes as follows: This principle is far fuperior to all former difcoveries in dynamics." Its application embraces the whole range of that science,' &c. p. 217. It is worth while to ftate more precifely the nature of a doctrine thus defcribed by one who himself extended, by various difcoveries, this very fcience, and to examine what peculiar merits he could difcover in it. He certainly has given a much more scientific explication and developement of it than can be found in the writings of the original author; and whatever value we may be difpofed to allow the materials, the excellence of the fabric cannot be called in queftion. The fundamental propofition of the doctrine is this: Let there be a system of bodies, of whatever number, maffes, and pofition, attracted in any directions by forces acting as any conceivable powers of the distances from the centres of thofe forces. Call the maffes of the bodies, M, m, μ, &c.; their distances from the centres of attraction, Z, z, %, &c. refpectively; the forces of attraction, F, f, 4, &c.; and the powers of the distances to which thofe forces are proportional, N, n,, &c. Then, in order that an equilibrium may take place, and that the system may remain at reft, it is neceffary that N+1 MFZ +mfa be either a maximum or a minimum; and therefore, the fluxion of this quantity being puto, will give the condition effential to the equilibrium fuppofed. There are therefore two kinds of equilibrium contemplated in this theory; the one when the above forces (denominated by Maupertuis the quantity of action) are a maximum, the other when they are a minimum. Thefe two kinds of equilibrium Euler defcribes by the example of a cone. If it is required to place a cone at reft on a plane, two pofitions will equally fatisfy the conditions; the cone may be placed on its bafe, or vertically on its apex. In the former cafe, the variation of its pofition to any fide does not permanently derange the equilibrium; in the latter, the fmalleft movement is fatal to the rest of the body. In Euler's memoir, above quoted, we find various demonflrations and applications of this general principle. The author, in each of the particular cafes, first propofes to find the conditions of equilibrium by this doctrine. He eafily deduces the fluxional equation according to the method of maxima and minima. By exterminating from thence the fluxions of the variable quantities, he obtains a finite folution; and he fhews that this is the fame with the folution obtained by the ordinary procefs. Thus he finds, by the doctrine of Maupertuis, that two forces drawing a body can only keep it at reft when they act in the fame line, and are equal and oppofite; and that the lever can only be in equilibrio when its oppofite weights are inverfely as their diftances from the fulcrum. He applies the doctrine in the very fame manner to the general and very difficult cafe of curvilinear levers, as the catenarian curve, and velarian curve; inveftigates the law of the compofition of forces, and of the different mechanical powers (which indeed are all diflinctly refolvable into the cafe of the lever). He alfo deduces, from the fame doctrine, the general propofition relative to the operations of machines, or combinations of the mechanical powers, and extends the whole to the principle of the fpring. Nothing can be more elegant than the whole of these investigations. We are at every step delighted with the discovery of exact coincidence between the deductions from the theory and the known pofitions of dynamical fcience, deduced from principles diametrically oppofite. We experience the fame pleafure as when the doctrine of the fluxional calculus is proved to us by the coincidence of its refults with thofe of common geometry, in all the cafes which the latter can reach; and, at each step of the inquiry, we acknowledge the hand of a master, in the fimplicity and rapidity of the touches by which the effect is produced. After giving thefe illuftrations of the principle, which are all cafes of the minimum of force, Euler proceeds to confider one in which the other fpecies of equilibrium is exemplified. This is the problem which Signor Fontana has undertaken to investigate by common analyfis, and upon the ordinary principles, in the paper now before us. Euler ftates that the folution, by fuch principles, would not be eafy, and appears to draw from hence an inference in favour of the utility of Maupertuis's fyftem. It is, however, extremely evident that he was in this refpect mistaken; for the folution demands no very long or intricate investigation; and Signor Fontana has given a very elegant and fimple inveftigation of it, both in the eafier cafe, which Euler folved, and in the more general and abftrufe conditions, which that great analyst did not take into confideration. The problem which Euler fays was propofed to him for folution, long before Maupertuis's theory occupied his thoughts, but which he feems rather unaccountably to have failed in folving until that theory attracted his notice, may be enunciated as follows: On a given fulcrum to place a given rod, loaded at one end by a given weight, fo that the other end may remain at reft upon a vertical plane (or wall) given in pofition. Euler only confidered the cafe when the rod was deprived of all weight, and when no friction interfered. In thefe circumftances, putting the distance of the fulcrum from the wall; a= the length of the red; z = the force applied to its extremity; and x = the length of the part intercepted between the fulcrum and the wall, he cafily deduced the equation, The folution was therefore reduced to finding a line whofe cube was equal to a given paralellipiped; and the problem was thus refolved by Maupertuis's law. In fact, a very fimple geometrical conftruction is obvious, and might be eafily defcribed by the interfection of a cubic parabola, whofe parameter is equal to the length of the rod. Signor Fontana, doubting the folidity of Maupertuis's law, investigates the problem by the arithmetic of fines. He eafily ob tains an equation, involving the fame quantities as thofe in Euler's, together with the fine and cofine of the angle, which the rod forms with the perpendicular from the fulcrum to the wall. From thence he exterminates the angular expreflion, juft as Euler does |