in this place, in endeavouring to make his escape from one who was going to correct him, opened the door of a cellar, and threw himself into it; but in his hurry so entangled his right thumb with the latch, that the whole weight of his body was suspended by it, until it gave way, and was torn off at the first articulation; the flexor tendon being at the same time pulled out in its whole length, having broke when it became muscular. Mr. H. was immediately sent for, found little or no hæmorrhage, and the bone of the 2d phalanx safe, and covered with its cartilage, but protruding considerably, occasioned by part of the skin belonging to it being irregularly torn off with the first joint. He was doubtful whether he should not be obliged at last to make a circular incision, and saw the bone even with the skin; but thought it proper to give him a chance for the use of the whole phalanx. He complained only for the first day of a pretty sharp pain in the course of the tendon; to which compresses, wrung out of warm brandy were applied: but his arm was never swelled; there was no ecchymosis; nor had he so much fever, as to require bleeding even once. The cure proceeded happily, no symptoms arising from the extracted tendon. At the third dressing the bone was covered; and no other application but dry lint was necessary during the whole time. No exfoliation happened; yet it was 12 weeks before it was entirely cicatrized, owing to the loss of skin and he seemed to enjoy the use of the stump as completely, as if that tendon had not been lost. LXXXIV. On the late Discoveries of Antiquities at Herculaneum, and of an Earthquake there. By Camillo Paderni, F.R. S. Dated Portici, Feb. 1, 1758. p. 619. Having been working continually at Herculaneum, Pompeii, and Stabiæ, since his communication of Dec. 16, 1756, the most remarkable discoveries made there are the following: February 1757, was found a small and most beautiful figure of a naked Venus in bronze, the height of which is 6 Neapolitan inches. She has silver eyes, bracelets of gold on her arms, and chains of the same metal above her feet; and appears in the attitude of loosening one of her sandals. The base is of bronze inlaid with foliage of silver, on one side of which is placed a dolphin. In July was found an inscription, about 12 Neapolitan palms in length, as follows: IMP, CAESAR, VESPASIANVS, AVG, PONTIF› MAX TRIB, POT› VII · IMP · XVII ·P · P ̊ COS • VII· DESIGN VIII After having found a great number of volumes of papirus in Herculaneum ; many pugillaries, styles, and stands with ink in them as formerly mentioned; at length, in the month of August, on opening a sinall box, was also found the instrument with which they used to write their manuscripts. It is made of wood, of an oblong form, but petrified, and broken into 2 pieces. There is no slit in it, that being unnecessary, as the ancients did not join their letters in the manner we do, but wrote them separate. In September were discovered 8 marble busts, in the form of terms. One of these represents Vitellius, another Archimedes; and both are of the finest workmanship. The following characters in a black tint, are still legible on the latter, namely, APXIME which is all the inscription that now remains. In October was dug up a curious bust of a young person, who has a helmet on his head adorned with a civic crown, and check-pieces fastened under his chin. Alsoanother very fine bust of a philosopher with a beard, and short thick hair, having a slight drapery on his left shoulder. Likewise two female busts; one unknown, in a veil; the other Minerva, with a helmet; both of middling workmanship. In November we met with two busts of philosophers of excellent workmanship, and, as may be easily perceived, of the same artist; but unfortunately, like many others, without names. In January was found a small but most beautiful eagle, in bronze. It has silver eyes, perches on a prafericulum, and holds a fawn between its talons. In the same month was discovered at Stabiæ, a term 6 palms high, on which is a head of Plato, in the finest preservation, and performed in a very masterly manner. Also divers vases, instruments for sacrificing, scales, balances, weights, and other implements for domestic uses, all in bronze.. Having finished the examination and arrangement of the scales, balances,, and weights, which were very numerous in the museum; it was remarkable that many of the former, with all the weights, exactly answer. those now in use at Naples. The whole day and night of the 24th of last month it seemed as if Mount: Vesuvius would again have swallowed up this country. On that day it suffered 2 internal fractures, which entirely changed its appearance within the crater, destroying the little mountain that had been forming within it for some years, and was risen above the sides; and throwing up by violent explosions, immense quantities of stones, lava, ashes, and fire.. At night the flames burst out with greater vehemence, the explosions were more frequent and horrible, and our houses shook continually. Many fled to Naples, and the boldest persons trembled. But the mountain having vented itself that night and the succeeding day, is since become calm, and throws out only a few ashes.. LXXXV. A Further Attempt to Facilitate the Resolution of Isoperimetrical Problems. By Mr. Thomas Simpson, F. R. S. P. 623. About 3 years before, Mr. S. laid before the R. s. the investigation of a general rule for the solution of isoperimetrical problems of that kind, wherein one only of the 2 indeterminate quantities enters along with the fluxions, into the equations expressing the conditions of the problem. Under which kind are included the determination of the greatest figures under given bounds, lines of the swiftest descent, solids of the least resistance, with innumerable other cases. But though cases of this sort do indeed most frequently occur, and have there. fore been chiefly attended to by mathematicians, others may nevertheless be proposed, such as actually arise in inquiries into nature, where both the flowing quantities, together with their fluxions are jointly concerned. The investigation of a rule for the solution of these, is what Mr. S. attempts in this paper by means of the following GENERAL PROPOSITION. Let a, R, S, T, &c. fig. 2, pl. 8, represent any variable quantities, expressed in terms of x and y, with given coefficients, and let q, r, s, t, &c. denote as many other quantities, expressed in terms of ¿ and ý: it is proposed to find an equation for the relation of x and y, so that the fluent of aq + Rr + SS + Tt, &c. corresponding to a given value of x or y, may be a maximum or minimum. Let AE, AF, and AG, denote any 3 values of the quantity x, having indefinitely small equi-differences EF, FG; and let EL, FM, and GN, perpendicular to AG, be the respective values of y, corresponding to them; and, supposing EF = FG = x, to be denoted by e, let cм and dN, the successive values of y, be represented by u and w. Also, supposing p'p' and "p" to be ordinates at the middle points P'P', between E, F, and F, G, let the former p'p' be denoted a, and the latter P"p" by ; putting AP′ = a and AP" = b. Then, if a aud a, the mean values of x and y, between the ordinates EL and FM, be supposed to be substituted for x and y, in the given quantity aq + Rr + ss + Tt, &c. and if, instead of i and y, their equals e and u be also substituted, and the said given quantity, after such substitution, be denoted by Q'q + R'r' + s ́s′ + T't', &c. it is then evident, that this quantity a'q' + R'r' + s's' + T't, &c. will express so much of the whole required fluent, as is comprehended between the ordinates EL and FM, or as answers to an increase of EF in the value of x. And thus, if b and ẞ be conceived to be written for x and y, e forr, and w for y, and the quantity resulting be denoted by Q"q" + R"r" + s"s" + "1" &c. this quantity will, in like manner, express the part of the required fluent corresponding to the interval FG. Whence that part answering to the interval EG will consequently be equal to o'q′ + R ́r &c. + a"q" +R""&c. But it is manifest, that the whole required fluent cannot be a maximum or minimum, unless this part, supposing the bounding ordinates EL, GN VOL. L.] to remain the same, is also a maximum or a minimum. Hence, in order to de-termine the fluxion of this expression oʻq' + n'r' &c. Q"q" + R"7" &c. which must of consequence be equal to nothing, let the fluxions of a' and q', taking « and u as variable, be denoted by a and qu; also let Ra and Fu denote the respective fluxions of R' and r'; and let, in like manner, the fluxions of a", 9", R", r", &c. be represented by Q, qw, RB, rw, &c. respectively. Then, by the common rule for finding the fluxion of a rectangle, the fluxion of our whole expression a'q + B'r' &c. + a"q" + "r" &c. will be given equal to o'qu + qʻqa + R'îu + r' ñà &c.. +a" qiv + 9" QB + R′′TW + ~ RB &c. = 0. But uw being = GN — EL, and ẞ—a = we therefore have, and g Ra (GN-EL) constant quantity, =a: also u being = 2 rp = 2 a — 2, EL thence will i = 2 a: which values being substituted above our equation, after the whole is divided by a, will become 20 g + g ́e + 2 RF + r', &c.—2.Q" g+q′′-2 R7 + r' &c. = 0; or, a′′7 — a'7 + R′′ 7 — R7 &C. = Q" զ -- But aqa'q, the excess of a" swering to the increment or fluxion ÷ (q′ē + 9′′ ☎) + ÷ (r' R + r" R, ) &c. above a'q, is the increment or fluxion (an) arising by substituting b for a, ß for a, and w for u. And, with regard to the quantities on the other side of the equation it is plain, seeing the difference of q' and q is indefinitely little in comparison of their sum, that qq may be substituted instead of (g+9′′) &c.. which being done, our equation will stand thus: . 9 1 Flux. RqRT &c. = qQ+r' R &c. α alone: But q'e + r'a &c. represents (by the preceding notation) the fluxion of q ́a' + r R' &c. or of ag + Rr &c. arising by substituting a for y, making variable, and casting off. If therefore that fluxion be denoted by v, we shall have flux. a'q. + RT &c. v, and consequently a'q + R ́7 &c. = v. But o'q + R &c. (by the same notation) appears to be the fluxion of a'g' + R'r' &c. or of aq + Rr &c. arising by substituting u for y, making u alone variable, and casting, off u. Whence the following. GENERAL RULE... Take the fluxion of the given expression (whose fluent is required to be a maximum or minimum) making y alone variable; and, having divided by ij, let the quotient be denoted by v: then take again the fluxion of the same expresssion, making y alone variable, which divide by y; and then this last quotient will be = v When y is not found in the quantity given, v will then be = 0; and conse-quently, the expression for, equal to nothing also. But if y be absent, then will=0, and consequently the value of va constant quantity. It is also easy to comprehend that, instead of y and y, and x may be made successively variable. Also, should the case to be resolved be confined to other restrictions, besides that of the maximum or minimum, such as having a certain number of other fluents, at the same time, equal to given quantities, still the same method of solution may be applied, and that with equal advantage, if from the particular expressions exhibiting all the several conditions one general expression composed of them all, with unknown, but determinate coefficients be made use of. In order to render this matter quite clear, let A, B, C, D, &c. be supposed to represent any quantities expressed in terms of x, y, and their fluxions, and let it be required to determine the relation of x and y, so that the fluent of A i shall be a maximum, or minimum, when the cotemporary fluents of вử, cử, Dr, &c. are all of them equal to given quantities. It is evident in the first place, that the fluent of ar + b Â¿ + cc¿ + dòi &c. (b, c, d, &c. being any constant quantities whatever) must be a maximum or minimum in the proposed circumstance: and, if the relation of x and y be determined (by the rule,) so as to answer this single condition, under all possible values of b, c, d, &c. it will also appear evident, that such relation will likewise answer and include all the other conditions propounded. For, there being in the general expression, thus derived, as many unknown quantities b, c, d, &c. to be determined, as there are equations, by making the fluents of B., cr, D., &c. equal to the values given; those quantities may be so assigned or conceived to be such, as to answer all the conditions of the said equations. And then, to see clearly that the fluent of the first expression, Ai, cannot be greater than arises from hence (other things remaining the same) let there be supposed some other different relation of x and y, by which the conditions of all the other fluents of Bở, cử, Di, &c. can be fulfilled; and let, if possible, this new relation give a greater fluent of Ar than the relation above assigned. Then, because the fluents bвx, ccx, dɔr, &c. are given, and the same in both cases, it follows according to this supposition, that this new relation must give a greater fluent of Ar + b br +cci+ ddv &c. (under all possible values of b, c, d, &c.) than the former relation gives: which is impossible; because whatever values are assigned to b, c, d, &c. that fluent will, it is demonstrated, be the greatest possible, when the relation of x and y is that above determined by the general rule. To exemplify, now, by a particular case, the method of operation above pointed out, let there be proposed the fluxionary quantity; where the relation of x and y is so required, that the fluent, corresponding to given values of x and y, shall be a maximum or minimum. Here, by taking the fluxion, making y alone variable (according to the rule) and dividing by ÿ, we shall have = 2. And, by taking the fluxion a second time, making y alone vaM-1 jp . Now from these equariable, and dividing by y, will be had many |