Sir J. Herschel
31 Sackville St <1> London
Sept. 16. 1844
Dear Sir
I sent a copy of my Memoir to Mrs Somerville. <2> I am truly obliged for the “Note” on Spence’s Work <3> which you have sent me, especially as it was your last copy. In my memoir, I think I have placed in the hands of Analysts an instrument of great power, and of the readiest application. As proof of it I send you the demonstration of your pretty theorem in page 169 – which I think you will find very simple, considering always that the example was not selected by myself, and that I employ no previous reasoning about it. I have written it out on a separate leaf of paper <4> for more convenience. And now to other matters. –
If a Cone of the second order is defined to be one whose base is a conic section, I think I can demonstrate that there is no such thing; that is to say, there is no difference between such a surface, and a common oblique Cone. But I should be unwilling for obvious reasons to publish such an assertion unless I were sure of its correctness, and I think it possible I may be mistaken as to the definition of such a surface.
So much has been written, as I observe, about such Cones, that it seems rash to suppose there can be any fundamental error, yet on the other hand, the following theorem (by M. Chasles <5> I think) in Quetelet’s Annals <6> puzzles me exceedingly – He says
“Every Cone of the second order can be so cut by a plane that the section shall be a circle.”– admitting ye truth of his demonstration which follows, I say,
I take the circular section which you have obtained, and consider it to be the base of the Cone, and now how do you show that any difference exists between this cone and a common oblique cone, having same base and vertex? They must be identical. But then it may be said, this is self evident & could not escape the penetration of a geometer like M. Charles. So that I fairly at a loss what to think of it.
Yours truly
H. F. Talbot
Notes:
1. 31 Sackville Street, London residence of the Feildings, often used as a London base by WHFT.
2. WHFT, ‘Researches in the Integral Calculus, Part One’ Philosophical Transactions of the Royal Society of London, v. 126 part 1, 1836, pp. 177–215 And WHFT, ‘Researches in the Integral Calculus, Part Two’ Philosophical Transactions of the Royal Society of London Mary Somerville, née Fairfax (1780–1872), writer on science, was collecting mathematical papers on the subject of integral calculus.
3. John Frederick William Herschel, Note on an application of the inverse theory of functions to the integral calculus. (From the works of the late W. Spence) (London, 1819).
4. Enclosure not located.
5. Michel Chasles (1793–1880), mathematician.
6. Lambert Adolphe Jacques Quetelet (1796–1874), Belgian mathematician, editor of Annalen der Mathematik.