# Hamilton and De Morgan discuss Buée's 1806 paper

We present below a few extracts from the correspondence between Hamilton and De Morgan in which they discuss Adrien-Quentin Buée's

*Mémoire sur les quantités imaginaires*which was read at the Royal Society in 1805 and was published in 1806. The quotations we present do not make too much mathematical sense without the several texts which the correspondents discuss but we give them here to give a feel for their attitude towards Buée.**Hamilton to De Morgan: 8 January 1853.**

... if you take the trouble to refer me to any paragraph or paragraphs in Buée's Paper, of which I possess a carefully collated copy, partly written out by myself from the Philosophical Transactions for 1806, which paragraphs, in your opinion, contain or shadow out the rule for multiplication of lines within the plane, I shall, in justice to him, and prudence to myself, examine them.

**Hamilton to De Morgan: 26 January 1853.**

I suppose that I and others must just submit to whatever you may at any time choose to make your own definition of Double Algebra; and as you once were pleased to admit that my old doctrine of Algebraic Couples was a sort of Double Algebra, so I must grant that something double was involved in Buée's Paper; for he did, no doubt, in his No. 35 for example, represent a directed diagonal of a square by the expression $2 (\cos 45° + \sin 45° √-1)$, or by $1 + √-1$.

....

Wallis's phraseology, what we call Real, is very remarkable. I think Buée's construction of imaginary roots more elegant; but can you point to any passage in Buée's Paper (120 years later than Wallis's Treatise) which contains even the germ of the method illustrated by the figure in page 119 of your lately cited book? And can you explain away Buée's own paragraph 46, in which, when he proceeds to state in what sense he regards his geometrically constructed roots, AE, EB, of Carnot's quadratic, $x(a - x) = a^{2}/2$ as having the required product, $a^{2}/2$, he expressly and formally excludes the consideration of position (or of direction of the factor-lines), and expressly and formally limits the proposed multiplication to that of the arithmetical values? I have no more of personal interest in the matter than you can have in a question about Newton and Leibniz; but I wish that justice should be done, and wish you to help me in doing it (namely, to Argand, who was, in my opinion, the inventor of Warren's rule of multiplication).

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Wallis's phraseology, what we call Real, is very remarkable. I think Buée's construction of imaginary roots more elegant; but can you point to any passage in Buée's Paper (120 years later than Wallis's Treatise) which contains even the germ of the method illustrated by the figure in page 119 of your lately cited book? And can you explain away Buée's own paragraph 46, in which, when he proceeds to state in what sense he regards his geometrically constructed roots, AE, EB, of Carnot's quadratic, $x(a - x) = a^{2}/2$ as having the required product, $a^{2}/2$, he expressly and formally excludes the consideration of position (or of direction of the factor-lines), and expressly and formally limits the proposed multiplication to that of the arithmetical values? I have no more of personal interest in the matter than you can have in a question about Newton and Leibniz; but I wish that justice should be done, and wish you to help me in doing it (namely, to Argand, who was, in my opinion, the inventor of Warren's rule of multiplication).

**De Morgan to Hamilton: 27 January 1853.**

As to Buée, I agree with Peacock, p. 228 (Report, &c.), Buée had no completely formed double algebra, and yet he was the first formal maintainer of geometrical interpretation of √-1.

**Hamilton to De Morgan: 31 January 1853.**

As to Buée, he is again gone pretty much out of my head, although I read nearly through a carefully collated copy of his Paper not long ago. Are you sure that we do not read him by our own lights, and see in his Essay, more clearly than he saw himself, the rules of addition and subtraction of lines? My attention was directed rather to the question whether he possessed what we may for the moment call Warren's rule of multiplication, from which, in his No. 46, Buée has formally cut away all sort of claim from himself. Without re-examination I could not venture to assert but to you I shall just throw it out as a hint for investigation, that Buée perhaps did not see the general rule for addition more clearly than Wallis. Observe that I refer to chapter LXVIII of the latter (London, 1685, according to my memorandum of transcription; J T Graves has cited the book to me as Oxford, 1683; probably you can decide that question), and not to the preceding chapter, which Gompertz has referred to (in his second tract on Imaginaries, &c., London, 1818). Wallis seems to me to consider $AB$ and $Ba$ as roots of his quadratic, because their sum (or at least their quasi-sum) is the straight line $Aa = b$, and because their semi-difference (or quasi semi-difference) is = $BC$ = the imaginary radical realised, where $C$ is the middle point of $Aa$. Buée's seeming "proportion," in his No. 35, is in fact (as he himself remarks) an identity. You are likely to know much better than I do but I doubt whether Buée's Paper, from its vagueness, and what I own, between ourselves (for if I said it publicly, I should be greeted with the Quis tuterit Gracchos), appears to me its intolerable licence of interpretation, may not have done more harm than good in England, to the cause of our friend $√-1$.

**De Morgan to Hamilton: 6 December 1857.**

Have you got Argand's tract (1806), or can you borrow it for me? Airy has lent me an old paper of his (MS.) written when he was a Freshman in 1820. He has $√-1 (±)^{(1/n)}$ well explained. But though he knew nothing of Argand or Buée (1806, 1815), he is more of 1815 than of 1806. There is an epidemic in the air about mathematical thoughts.

Last Updated January 2020