TC TO ROBERT MITCHELL; 19 April 1816; DOI: 10.1215/lt-18160419-TC-RM-01; CL 1:73-76.
TC TO ROBERT MITCHELL
Annan, 19th April 1816.
My Good lad—You must be aware, that I have a right to rate you with considerable acrimony. Last time I saw you, you promised to send me a letter in a few days; yet several weeks have elapsed, and no letter has appeared. This is very blameworthy. But I need not scold you; for, in all probability, by the time you have studied the following disquisition—you will think yourself punished to the full. It is one of the theorems, we were speaking of. I found it out some weeks ago; and an odd fancy struck me of turning the demonstration into the Latin tongue. My dialect of that language is, I doubt not, somewhat peculiar—and you may chance to find some difficulty in interpreting it.
Si per basis [AC] terminos [A, C] et punctum quodvis [O] perpendiculae [BOD] (e vertice [B] in basin [AC] trianguli demissae); agantur lineae rectae [AOG, COF,] adversis occurrentes lateribus; [BC, AB,] dico quòd rectae; [DF, DG] intersectionem [D] perpendiculae et basis, cum intersectionibus [F, G] laterum [AB, BC] rectarumque [CO, AO] jam jam ductarum, jungentes—aequales facient angulos [ADF, CDG] cum basi [AC].
Sit ABC triangulum. Ad basim AC demittatur perpendicula BD; in qua assumatur punctum quodvis O: dum[?], actis COF et AOG, si D, F et D, G jungantur, d[ico] quòd angulus FDA angulo [GDC] aequatur. demitto enim perpendiculas FE et GH. Nun[c] quoniam CE:ED : : EF:EF — OD et AE:ED : : EF:BD — EF, patet quod CE:AE : : BD — FE:EF — OD, vel componendo (scil. in triangulo acuto) AC:AE : : BD — DO:EF — OD. Pari quoque ratione, liquet quod AC:CH : : BD — DO:GH — DO, atque ea de causa quod AE:CH : : EF — DO:GH — DO. —Sed porro CD:DO : : DE:EF — OD, inde DO = ; et AD:DO : : DH:GH — DO, ideoque DO = : itaque = . Sed ut supra demonstratum AE:CH : : CF — DO:GH — DO; quamobrem
View larger version:
[in a new window]
The two geometrical figures alluded to in "Theorema"
. Nunc ob parallelas FE et BD, AE:EF : : AD:DB, indeque . Eodemmodo, autem, CH: HG : : CD:DB et igitur ; unde perspicuum est quod : : : : ; vel, quod perinde est, : : : : , adeoque vel, EF:DE : : HG:DH; suntque triangula EFD & GDH similia (scil. quippe rectangula), et angulos FDE, GDH habent aequales.— Simili ratiocinio, in triangulo obtuso—necnon cum punctum O vel supra verticem vel infra basim assumitur, mutatis tantum mutandis, eandem attingemus finem.1
Such is the result of this investigation.— Your Theorem about the hepttagon does not answer exactly; it is only a very good approximation, as I perceive, and as you may perceive also, by consulting a table of sines and tangents.— I have forgot West's theorem2 about the circle and curves of the second order— It is no great matter surely, since I should not be able to demonstrate it tho' I had it. I wish you would send it me notwithstanding.— I hope you have sustained no injury from your excursion to Edinr.—I hope too that you spake to Mr. Leslie concerning the books.3 If you have not procured [me] one, I must request you to lend me yours immediately for a short time, if you can do without it: for one of the boys has begun conics & I have lent him my book—and should like to have another copy of it by me, to consult at home.— You must send me the theorem & book by Mr Johnson of Hitchill4—who I understand is to be at Ruthwell today and who can easily bring them to Annan.— A letter I am expecting with impatience—
Yours ever, /
Mr Glen's desires me to bid you present his compliments to Mr. Duncan, & to remind him that Mr. Mc Whir is to preach here tomorrow.—