Lacock Abbey, near Chippenham
I hope you received the letter <1> which I sent, containing some numerical examples, and that they were satisfactory to you.
I believe I mentioned a theorem of Legendre’s <2> of some interest, to which I think I can advance an indisputable claim of priority, and as I can now refer you to volume and page, I should be glad if, when you have leisure, you would look at it.
In the Exercises de Calc. Intégral <3> published in 1811 tom.1. p.39, he says the Arcs of the Lemniscate represent Elliptic functions of the first kind in the case where the modulus = √½. And that it would be curious to enquire whether any other algebraic curve exists, which represents this function for other values of the modulus, but that such an enquiry is difficult. “Cette recherche ne laisse pas d’être difficile.” <4> In point of fact he does not attempt any solution of it.
Having turned my attention to this problem, I found a solution of it which I published in Gergonne’s Annales des Mathématiques tom.14 p.380 (published in 1824) <5>
In M. Legendre’s new work (published in 1825) tom 1. p.36. <6> he gives a solution of the problem (art.30). He again recurs to it tom.2 p.590. (art.79) and describes it as “un problème que nous avons regardé comme fort difficile.” <7>
That being the case I think I ought to claim to have my lost property restored to me, more especially as Legendre is so rich that he can well bear such a minute subtraction from the number of his discoveries –
Gergonne, to whom I sent several articles, has a custom of altering them considerably, which he calls rédaction, <8> but which I found so perplexing and in all respects inconvenient that I was obliged to discontinue his correspondence.
I remain Dear Sir Yours vy Truly
H. F. Talbot
1. Letter not located.
2. Adrien Marie Legendre (1752–1833), mathematician.
3. Legendre, Exercices de Calcul intégral sur divers ordres de transcendantes et sur les quadratures (Paris: Courcier 1811–1817).
4. This research allows of no difficulty.
5. WHFT, ‘Questions Résolues: Addition à l’article de la page 207 du présent volume’, Annales de Mathématiques Pures et Appliqués, v. 14 no. 12, 1 June 1824, pp.380–381.
6. Legendre, Traité des fonctions elliptiques et des intégrals eulériennes avec des tables pour en faciliter le calcul numérique (Paris: Huzard Courcier, 1825–1828).
7. A problem which we regard as very difficult.