Lacock Abbey, near Chippenham

Nov^{r} 25. 1834

My Dear Sir,

I have received a Report signed by yourself and M^{r} Peacock ^{<1>} upon a paper which I submitted to the R. Society on the Integral Calculus. ^{<2>} I am now drawing up a few observations which I will transmit to you and M^{r} P. – but I wished in the first instance to enquire whether you ever received a letter (or I believe two letters) which I wrote to you on this subject last spring ^{<3>} – They were with reference to a former paper of mine “on the arcs of the Equilat^{l} Hyperbola” ^{<4>}– and in reply to a letter in which you informed me that you had tried Abel’s theorem ^{<5>} upon this example, but that you had obtained a result different from mine.

In proof that my result was the true one I transmitted to you two numerical examples which came out very satisfactory, and at the same time I expressed a desire that some calculator could be met with who would have the patience to extend the calculation to 8 or 10 places of decimals – I you did not receive the letter, I could easily write out these examples again.

As the two papers I sent to the R. Society are closely connected in subject, I have considered them as one.

In your letter you referred me to Chap X of Legendre’s Traité des Fonctions Elliptiques, ^{<6>} but on attentive perusal of that Chapter I can find nothing in which he anticipates me.

It relates to the sum of __two__ arcs, my theorem to the sum of __three__: and I believe these results are quite different in their nature, inasmuch as I have never been able to reduce one of them to the other.

To put this in a clearer light. It is an easy thing to find the algebraic sum of two arcs in several curves, but not of three of more arcs.

You indeed get the sum of __four__ arcs by repeating the first operation twice, but you cannot in that way obtain the results which I find by my method. The arcs must be considered as generated all at the same time. So on the other hand I find curves in which 3, 4, or 5 arcs possess an algebraic sum, while no such sum can be found for __two__ arcs. This is what entirely escaped the attention of Fagnani ^{<7>} whom I regard as the inventor of this branch of the Integral Calculus, and of Euler: ^{<8>} and Legendre also, and is a point which I shall endeavour to elucidate as far as I can when I have time to draw up a more enlarged paper on the subject. –

I remain Dear Sir Yours very truly

H. F. Talbot

1834 Chippenham November twenty five __H.F. Talbot__

J. W. Lubbock

29 Eaton Place

London

#### Notes:

**1**. Prof George Peacock (1791–1858), mathematician.

**2**. WHFT, ‘On the rectification of parabolic curves’, first appears in the minutes of the Committee of Papers on 3 July 1834. It was referred at this time but subsequently delayed, until WHFT withdrew the paper, recorded officially in the minutes of the meeting of 28 January 1836.

**3**. This sequence of correspondence begins with the letter Doc. No: 02999, no earlier correspondence has been located.

**4**. WHFT, ‘On a new property of the Arcs of the Equilateral Hyperbola’, *Proceedings of the Royal Society of London*, v. 3 no. 15, 1833–1834, p. 258.

**5**. Niels Henrik Abel (1802–1829), mathematician. His theorem, written in 1824, lost in Paris and then rediscovered in 1828, proved the impossibility of solving equations of the fifth degree (quintic equations) through the use of radicals.

**6**. Adrien Marie Legendre (1752–1833), *Traité des fonctions elliptiques et des intégrals eulériennes avec des tables pour en faciliter le calcul numérique* (Paris: Huzard Courcier, 1825–1828).

**7**. Giulio Carlo Fagnano dei Toshci (1682–1766), mathematician.

**8**. Leonhard Euler (1707–1783), mathematician.