TRANSACTIONS OF THE SECTIONS. ^ 



And this is but one of the claims of Quaternions to the attention of physicists. 

 When we come to the important questions of stress and strain in an elastic solid, 

 we find again that all the elaborate and puzzling machinery of coordinates com- 

 monly employed can be at once comprehended and kept out of sight in a mere 

 single symbol — a linear and vector function, which is self-conjugate if the strain 

 be pure. This is simply, it appears to me, a proof either that the elaborate machinery 

 ought never to have been introduced, or that its use was an indication of a com- 

 paratively savage state of mathematical civilization. In the motion of a rigid solid 

 about a fixed point, a quaternion, represented hy a single symbol which is a func- 

 tion of the time, gives us the operator which could bring the body by a single 

 rotation from its initial position to its position at any assigned instant. In short, 

 whenever with our usual means a result can be obtained in, or after much labour 

 reduced to, a simple form. Quaternions -ndll give it at once in that fomi ; so 

 that nothing is ever lost in point of simplicity. On the other hand, in numberless 

 cases the Quaternion result is immeasurably simpler and more intelligible than 

 any which can be obtained or even expressed by the usual methods. And it is not 

 to be supposed that the modern Higher Algebra, which has done so much to sim- 

 plify and extend the ordinary Cartesian methods, would be ignored by the general 

 employment of Quaternions ; on the contrary. Determinants, Invariants, &c. 

 present themselves in almost everj' Quaternion solution, and in forms which 

 have received the full benefit of that simplification which Quaternions generally 

 produce. Comparing a Quaternion investigation, no matter in what department, 

 with the equivalent Cartesian one, even when the latter has availed itself to the 

 utmost of the improvements suggested by Higher Algebra, one can hardly help 

 making the remark that they contrast even more stronglj^ than the decimal 

 notation with the binary scale or with the old Greek Arithmetic, or than the 

 well-ordered subdivisions of the metrical system with the preposterous no-systems 

 of Great Britain, a mere fragment of which (in the form of Tables of Weights and 

 Measures) forms perhaps the most effective, if not the most ingenious, of the many 

 instruments of torture employed in our elementary teaching. 



It is true that, in the eyes of the pure mathematician, Quaternions have one 

 grand and fatal defect. They cannot be applied to space of n dimensions, they are 

 contented to deal with those poor three dimensions in which mere mortals are 

 doomed to dwell, but which cannot bound the limitless aspirations of a Cayley or a 

 Sylvester. From the physical point of view this, instead of a defect, is to be re- 

 garded as the greatest possible recommendation. It shows, in fact, Quaternions 

 to be a special instrument so constructed for application to the Actual as to have 

 thrown overboard everything which is not absolutely necessary, without the 

 slightest consideration whether or no it was thereby being rendered useless for 

 applications to the Ineonceivahle. 



The late Sir John Herschel was one of the first to perceive the value of Qua- 

 ternions ; and there may be present some who remember him, at a British Asso- 

 ciation Meeting not long after their invention, characterizing them as a " Cornu- 

 copia from which, turn it how you will, something valuable is sure to fall." Is it not 

 strange, to use no harsher word, that such a harvest has hitherto been left almost 

 entirely to Hamilton himself? If but half a dozen tolerably good mathematicians, 

 such as exist in scores in this country, were seriously to work at it, instead of 

 spending (or rather wasting) their time, as so many who have the requisite leisure 

 now do, in going over again what has been already done, or in working out mere 

 details where a grand theory has been sketched, a very great immediate advance 

 would be certain. From the majority of the papers in our few mathematical 

 journals one would almost be led to fancy that IBritish mathematicians have too 

 much pride to use a simple method while an unnecessarily complex one can be 



" those who adopt them (or even who admit that they may be reasonably adopted by other 

 " persons) to consider, or to admit that others may usefully inquire, what common grounds 

 " can be established for conclusions common to quaternions and to older branches of ma- 

 " thematics." 



" Could any thing be simpler or more satisfactory? Don't jonfeel, as well as think, 

 " that we are on a right track, and shall be thanked hereafter ? Never mind when." 



1* 



