Lacock Abbey, near Chippenham

Dear Sir

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

#### Notes:

**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.

**8**. Editing