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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 because the rotation from 1 to this also, being doubled, 

 conducts to- 1, it also ought to be a square root of negative unity, 

 though not to be confounded with the former." 



Starting thus with the conception of triplets involving two dis- 

 tinct square-roots of negative unity, and endeavouring to frame laws 

 for their algebraical treatment, analogous to those which hold in the 

 case of couplets, he was soon led to perceive that the existence of the 

 two imaginaries, just alluded to, necessarily involved the existence 

 of a third, which was also a square-root of negative unity, distinct 

 from either of the former. He was thus led to the conception of 

 quaternions, or quadrinomials whose real parts, or constituents, 

 are multiplied, the first by unity, and the other three by the three 

 imaginary roots of negative unity just referred to ; and he deter- 

 mined the conditions which must subsist amongst these new imagi- 

 nary coefficients, in order that the resulting quadrinomials should 

 be subject to the same algebraical laws as the ordinary imaginary 

 binomials, or couplets. 



I may here observe, in passing, that one of these laws, namely, 

 the law of the moduli, is equivalent to a celebrated theorem of Eu- 

 ler ; viz. : that the sum of four squares, multiplied by the sum 

 of four squares, is also a sum of four squares. An extension of 

 this theorem to sums of eight squares has been efiected, independ- 

 ently, by Mr. John Graves and Professor Young; and the latter 

 writer (whose paper on the subject is published in the last part 

 of the Transactions of the Academy) has proved that the property 

 cannot be extended to higher numbers. 



To return to the Quaternion, — we have seen that it is made up 

 of a real part, and an imaginary trinomial, using the terms real 

 and imaginary in their ordinary acceptation. The latter of these 

 represents a right line in space, drawn from the origin to the 

 point whose co-ordinates are the three constituents of the trinomial, 

 and it is accordingly designated by Sir William Hamilton by the 

 term vector. The real part of the quaternion, on the other hand, 

 designates number alone, whether positive or negative, without 

 direction in space ; and, accordingly, although real inXhealgehraical 



