TC TO ROBERT MITCHELL; 13 November 1816; DOI: 10.1215/lt18161113TCRM01; CL 1:8587.
TC TO ROBERT MITCHELL
Mainhill, 13th Novr 1816—
My Dear [Mitchell]
I shall set out for Edinr tomorrow morning; and before going, I have begun (as in duty bound) to give you an account of my procedure.— I have done nothing at all since I saw you, but put off my time. I was sick two or three days; and went over to Allonby to recreate myself. I returned from Allonby in three days—and remained mostly at home—waiting with patience for the day of my departure—which at length is near at hand—
I am glad you get on so well with your Mathematics. Your demonstration of that theorem of West—about the triangle—which you sent me—is simple & neat—much better than mine, as far as I can recollect. ‘Samuel Cowan LandSurveyor’1 seems not to be such a dull person—as from the confused dropping of perpendiculars, drawing of lines &c with which he made his first appearance, I had conjectured that he was. I tried his problem, but was able to make out only a very bad solution. I will not trouble you with it at this time. I did not see the ‘Courier’2 last week; but I suppose there was nothing in it, that I do not know about. I have got nothing to send you—of any use—unless you think of inserting the following problem; which notwithstanding the technical jargon, in which it is enveloped—is after all a silly enough piece of work. You will perceive that it is a general solution of the problem, concerning a particular case of which Mr White was so facetious, above two years ago. It is [MS torn] the only thing I have done, since I saw you: and as I now write it out [for the fi]rst time, I am not without apprehensions of errors in the computation [tho' sure] enough of the principle. But if you propose it—I shall have time [enough to] give it a revisal before a solution is required.
To determine the nature of the curves; extending between the opposite corners of a given rectangle (each having one of the sides for its axis), that shall divide it into any given number (n) of equal portions.
Since the rectangle and the number, n are given, it is plain, that the area contained by each curve must be a given part of the rectangle under its absciss & ordinate[.] The problem, therefore includes this—To find the equation of the curve, the area of which is given. Let x represent the absciss & y the ordinate, being a given proper fraction.
Then = the fluxion of the area = flux. .
Therefore . From which it appears that x is a simple function of y, or that some multiple or submultiple of x is equal to one of the pow[ers] of y. Since x and y are of the same dimensions, put , or and substituting this value of x in the former equation, . Consequently , and : and the equation of the curve, the area of which is , or .
Now it is evident that if a curve be described having the base of the rectangle for its absciss & the altitude for its ordinate, and containing an area equal to the th part of the rectangle it will leave [of the rectangle] on its convex side; and [illegible] equation (according to the general formula [illegible] be . Again if another curve having the same absciss x [illegible] containing the [illegible] part of the rectangle—it will cut off [illegible] of the rectangle; and its equation, as before, will be a [illegible.] Let the division be continued till the number of curves described is the g[reatest] integer in ; and if the same operations be repeated on the other side of the diagonal—there will remain at last a space inculded between the opposite interior curves, a space equal to the n ——.
——But here is Johnson with intelligence that my intended comp[a]gnon de voyage cannot go tomorrow; and I must off in the Coach tonight at six o'clock— ‘Night thickens—& the crow makes wing to the rooky wood.’3 I have not a moment to lose— Good b'ye my Dear Mitchell—
You shall hear of me ere long.
Yours ever— /
Thomas Carlyle
