490 Sir W. R. Hamilton on Quaternions^ 



same mathematical subject, many of wliich are hitherto un- 

 published. 



I have the honour to be, Gentlemen, 



Your obedient Servant, 

 William Rowan Hamilton. 

 Observatory of Trinity College, Dublin, 

 November 20, 1844. 



{Copi/ of a) Letter from Sir William It. Hamilton to John 

 T. Graves, Esq, on Qiiatcrnions. 



Observatory, October 17, 1843. 



My dear Graves, — A very curious train of mathematical 

 speculation occurred to me yesterday, which I cannot but 

 hope will prove of interest to you. You know that I have 

 long wished, and I believe that you have felt the same desire, 

 to possess a Theory of Triplets, analogous to my published 

 I'heory of Couplets, and also to Mr. Warren's geometrical 

 representation of imaginary quantities. Now I think that I 

 discovered* yesterday a theory of quateTi'nions, which includes 

 such a theory o{ triplet s'\. 



My train of thoughts was of this kind. Since V —\ is in 

 a certain well-known sense, a line perpendicular to the line 1, 

 it seemed natural that there should be some other imaginary 

 to express a line perpendicular to both the former ; and be- 

 cause the rotation from 1 to this also being doubled conducts 

 to — I, it also ought to be a square root of negative unity, 

 though not to be confounded with the former. Calling the 

 old root, as the Germans often do, /, and the new one J, I in- 

 quired what laws ought to be assumed for multiplying toge- 

 ther a + ih-\-jc and x-\-iy+jz. It was natural to assume the 

 product :=^ax—by-'CZ-\-i[ay + bx)-\-j[az + C3c)+ij {bz + cy) ; 

 but what are we to do with ijl Shall it be of the form a-i-|3/ 

 + 77? Its square would seem to be =1, because i^=^j^=z — 1 ; 

 and this might tempt us to take /j=l, or ij=:^ — 1 ; but with 

 neither assumption shall we have the sum of the squares of the 

 coefficients of 1, /, and^' in the product = to the product of 

 the corresponding sums of squares in the factors. Take the 

 simplest case of a product, namely, the case where it becomes 

 a square ; we have a^ — b'^ — c'^ -\- 2 i a b + 2j a c + 2 ij b c .'. and 

 (a2_Z,2_^oj2^ {2abf + {2acf={a^ + b'^-\-c^f', the condition 



* The reader is requested to pardon this expression for the reason men- 

 tioned in another note. 



■\ Prof. De Morgan has most obligingly given the writer a sketch of a 

 Memoir on Triple Algebra, which ho has lately presented to the Cambridge 

 Philosophical Society, and which he is pleased to say was not begun till 

 after the publication of the first part of the paper on Quaternions in this 

 Magazine. 



