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84
Also [[image - mathematical equation]]

Then 1,082 X 0,0786 = 0,084 feet, the fall sought

Which is about 1 inch; and is about half an inch more than the greatest fall observed by Mr. Labeleye.

[[image - line extending across the page]]

A general method for Isoperimetrical Problems. The paper mentioned is at p. 102.

LXXXV. (of the Philo. Trans. Vol. 50 Part 2. & for 1758) [[underlined]] A further Attempt to facilitate the Resolution of Isoperimetrical Problems. By Mr. [[/underlined]] Thomas Simpson, F. R. S.

Read April 13th. 1758.

ABOUT three years abo 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 fluxion, 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, where in [[underlined]] both [[/underlined]] the flowing quantities, together with their fluxions, are jointly concerned. The investigation of a [[underlined]] rule [[/underlined]] for the resolution of these, is what I shall in this paper attempt, by means of the following

GENERAL PROPOSITION. [[underlined]] Let [[/underlined]] Q, R, S, T, &c. [[underlined]] represent any variable quantities, expressed in terms of X and Y (with given coefficients), and let [[/underlined]] q, r, s, t &c. [[underlined]] denote as many other quantities, expressed in terms of [[/underlined]] X^[[dotted]] [[underlined]] and [[/underlined]] Y^[[dotted]]; [[underlined]] It is proposed to find an equation for the relation of [[/underlined]] X [[underlined]] and [[/underlined]] Y, [[underlined]] so that the fluent of [[/underlined]] Qq + Rr + Ss + Tt, &c. [[underlined]] corresponding to a given value of [[/underlined]] X ([[underlined]] or [[/underlined]] Y), [[underlined]] may be a [[/underlined]] maximum [[underlined]] or [[/underlined]] minimum.

[[image - mathematical diagram with points marked A, L, M, N, E, F, G, r, c, d, p', P', p", P"]] This is also made Fig. 8.

Let AE, AF, and AG, denote any three values of the quantity [[underlined]] X [[/underlined]], having indefinitely small [[underlined]] equi-differences [[/underlined]] EF FG; and let EL, FM, and GN, (perpendicular to AG) be the respective values of [[underlined]] Y [[/underlined]], corresponding thereto; and supposing EF (= FG = [[underlined]] X^[[dotted]] [[/underlined]]) to be denote by [[underlined]] c [[/underlined]], let [[underlined]] c [[/underlined]] M and [[underlined]] d [[/underlined]] N (the successive values of [[underlined]] Y^[[dotted]] [[/underlined]]) be represented by [[underlined]] U [[/underlined]] and [[underlined]] W [[/underlined]]. Moreover, supposing P' [[underlined]] p' [[/underlined]] and P" [[underlined]] p" [[/underlined]] to be ordinates at the middle points P' P", between E, F, and F, G, let the former (P' [[underlined]] p' [[/underlined]]) be denoted by a, and the latter (P" [[underlined]] p" [[/underlined]]) by B; putting AP'= [[underlined]] a [[/underlined]] and AP"= [[underlined]] b [[/underlined]]. Then, if [[underlined]] a [[/underlined]] and a (the mean values of [[underlined]] X [[/underlined]] and [[underlined]] Y [[/underlined]], in the given quantity Q [[underlined]] q [[/underlined]] + R [[underlined]] r [[/underlined]] + Ss + Tt, &c. and if, instead of [[underlined]] X^[[dotted]] [[/underlined]] and [[underlined]] Y^[[dotted]] [[/underlined]], their equals [[underlined]] c [[/underlined]] and [[underlined]] u [[/underlined]] be also substituted, and the said (given) quantity, after such substitution, be denoted by

Transcription Notes:
The section beginning "GENERAL PROPOSITION" includes mathematical notations I'm unable to transcribe. "X" and "Y" sometimes have dots above them, but not always. I've used the notation ^[[dotted]] to mark each "X" or "Y" that has a dot above it in the original text.