The Collected Letters, Volume 1


TC TO ROBERT MITCHELL; 13 November 1816; DOI: 10.1215/lt-18161113-TC-RM-01; CL 1:85-87.


Mainhill, 13th Novr 1816—

My Dear [Mitchell]

I shall set out for Edinr to-morrow 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 Land-Surveyor’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, s / t being a given proper fraction.

Then xdot;y = the fluxion of the area = flux. s/t xy = s/t xy + s/t xdot;y.

Therefore t–s/t xdot;y = s/t x ydot;. 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 a<SUP>p–1</SUP>x = y<SUP>p</SUP>, or x = y<SUP>p</SUP>/a<SUP>p–1</SUP> and substituting this value of x in the former equation, t–s/t p. y<SUP>p</SUP>/a<SUP>p–1</SUP> ydot;= s/t y<SUP>p</SUP>/a<SUP>p–1</SUP> ydot;. Consequently t–s/t p = s/t, and p = s/t–s : and the equation of the curve, the area of which is a<SUP>2s–t/t–s</SUP> x = y<SUP>s/t–s</SUP>, or a<SUP>2s–t</SUP>x<SUP>t–s</SUP> = y<SUP>s</SUP>.

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 n-1/nth part of the rectangle it will leave 1/n [of the rectangle] on its convex side; and [illegible] equation (according to the general formula [illegible] be a<SUP>n–2</SUP>x = y<SUP>n–1</SUP>. Again if another curve having the same absciss x [illegible] containing the [illegible] n–2/n part of the rectangle—it will cut off [illegible] 1/n 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 n–1/2 ; 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 to-morrow; and I must off in the Coach to-night 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