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 two internal fractures, which intirely 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. For my own part, I resolved to abide the event here at Portici, on account of my family, consisting of eight children, and a very weak and aged mother, whose life must have been lost by a removal in such circumstances, and so rigorous a season. But it pleased God to preserve us; for 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._ [Read April 13, 1758.] ABOUT three years ago I had the honour to lay before the Royal Society the investigation of a general rule for the resolution of isoperimetrical problems of that kind, wherein one, only, of the two 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 altho’ cases of this sort do, indeed, most frequently occur, and have therefore been chiefly attended to by mathematicians, others may nevertheless be proposed, such as actually arise in inquiries into nature, wherein _both_ the flowing quantities, together with their fluxions, are jointly concerned. The investigation of a _rule_ for the resolution of these, is what I shall in this paper attempt, by means of the following GENERAL PROPOSITION. _Let_ Q, R, S, T, &c. _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_ ẋ _and_ ẏ; _It is proposed to find an equation for the relation of_ x _and_ y, _so that the fluent of_ Qq + Rr + Ss + Tt, &c. _corresponding to a given value of_ x (_or_ y), _may be a_ maximum _or_ minimum. [Illustration]