THE ROYAL IRISH ACADEMY, 1848. 471 



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 exist- 

 ence 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 imaginary roots of negative unity just referred to ; 

 and he determined the conditions which must subsist amongst these 

 new imaginary coefficients, in order that the resulting quadrino- 

 mials 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 

 Euler, 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 effected, inde- 

 pendently, 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 pro- 

 perty 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 tri- 

 nomial ; 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, with- 

 out direction in space; and, accordingly, although real in the 

 algebraical sense of the term, it is in some sense imaginary, when 

 contemplated on the geometrical side. This part of the quaternion 

 is denominated by Sir William the scalar. 



