65 Harley S^{t} ^{<1>}

Tuesday April 11^{th}

Dear Sir

I think that Sir W. Hamilton’s ^{<2>} valuable memoir would find an appropriate place in the Transactions of some learned Society – I observe that he is dissatisfied with some parts of Abel’s proof ^{<3>} of the impossibility of solving the 5^{th} degree; but that he considers the principal theorem upon which Abel relies, to be capable of proof in other ways that are more satisfactory–

He therefore agrees with Abel with respect to the main point, the impossibility of a solution.

I must say that I am not persuaded of this – I found some time ago a method of reducing the general equation of the 6^{th} degree to the 5^{th} (unless my process involves some error, which I would not guarantee not having had leisure enough to attempt a __numerical__ example)

Now would not Abel’s reasoning lead us to infer that this reduction was impossible? I do not say that it would or would not but it appears to me that the question merits examination.

Believe me

Yours most truly

H. F. Talbot

I should be glad to read Sir W. Hamilton’s memoir again at another time, if you can lend it to me – I return it with this note^{<4>}

#### Notes:

**1**. Harley Street, London.

**2**. Sir William Rowan Hamilton (1805–1865), Irish mathematician, ‘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.

**3**. The theorem of Niels Henrik Abel (1802–1829), mathematician was written in 1824, lost in Paris and re-discovered in 1828. It proved the impossibility of solving equations of the fifth degree (quintic equations) through the use of radicals.

**4**. No enclosure.