65 Harley St
Monday Evening
Dear Sir
I left Sir W. Hamilton’s memoir <1> at your house today. He seem persuaded of the truth of Abel’s demonstration <2> –
I mentioned to you some time ago that I had found a method of solving equations of the 6th degree i.e. reducing them to the 5th degree, whereas it had been previously supposed that this was impossible – The common method leads to an eqn of the 15th degree, which of course is of no use –
I have at length found time to apply this method to a numerical example and the result appears to me sufficiently satisfactory. The errors do not seem greater than may be accounted for by the length of the process – If the principle were erroneous I should expect the results of the calculation would vary altogether from the truth – To effect the calculations rigorously would require the assistance of practised calculators, such as those who compute the Nautical Almanac, <3> and when several persons are employed at once on the same computations, their results serve to check and verify each other at every step of the process, and therefore enable it to be conducted much more quickly and satisfactorily.
Now it appears to me that Abel’s argument would, if applied to eqns of the 6th degree, most probably not show the possibility of this reduction, and therefore I think that there still may be a latent fallacy in some part of his reasoning – However his demonstration, and Sir W. Hamilton’s confirmation of it weigh with me so far, that I cannot expect this method of mine will answer for the 5th degree – The reasoning indeed applies, and I cannot make out where the fallacy lies, but I suppose an arithmetical example would detect it. This however I have not time to undertake without assistance at present.
Believe me
Yours very Truly
H. F. Talbot
Notes:
1. William Rowan Hamilton, ‘On the Argument of Abel, respecting the impossibility of expressing a root of any general equation above the fourth degree, by any finite combination of radicals and rational functions’, read in 1837, published in Transactions of the Royal Irish Academy, v.18, 1838, pp. 171–259.
2. Niels Henrik Abel (1802–1829), mathematician wrote his theorem in 1824. It was lost in Paris and re-discovered in 1828, and proved the impossibility of solving equations of the fifth degree (quintic equations) through the use of radicals.
3. Nautical Almanac of Astronomical Ephemera (London: published by order of the Lords Commissioners of the Admiralty).