NATURE 



^01 



THURSDAY, DECEMBER 35, 1873 



QUATERNIONS 



A MATHEMATICIAN is one who endeavours to 

 secure the greatest possible consistency in his 

 thoughts and statements, by guiding the process of his 

 reasoning into those well-worn tracks by which we pass 

 from one relation among quantities to an equivalent rela- 

 tion. He who has kept his mind always in those paths 

 which have never led him or anyone else to an incon- 

 sistent result, and has traversed them so often that the act 

 of passage has become rather automatic than voluntary, 

 is, and knows himself to be, an accomplished mathemati- 

 cian. The very important part played by calculation in 

 modern mathematics and physics has led to the develop- 

 ment of the popular idea of a mathematician as a calcu- 

 lator, far more expert, indeed, than any banker's clerk, but 

 of course immeasurably inferior, both in resources and in 

 accuracy, to what the " analytical engine " will be, if the 

 late Mr. Babbage's design should ever be carried into 

 execution. 



But though much of the routine work of a mathemati- 

 tician is calculation, his proper work — that which consti- 

 tutes him a mathematician — is the invention of methods. 

 He is always mventing methods, some of them of no 

 great value except for some purpose of his own ; others, 

 which shorten the labour of calculation, are eagerly 

 adopted by all calculators. But the methods on which 

 the mathematician is content to hang his reputation are 

 generally those which he fancies will save him and all 

 who come after him the labour of thinking about what 

 has cost himself so much thought. 



Now Quaternions, or the doctrine of Vectors, is a 

 mathematical method, but it is a method of thinking, and 

 not, at least for the present generation, a method of 

 saving thought, It does not, like some more popular 

 mathematical methods, encourage the hope that mathe- 

 nraticians may give their minds a holiday, by transferring 

 all their work to their pens. It calls upon us at every 

 step to form a mental image of the geometrical features 

 represented by the symbols, so that in studying geometry 

 by this method we have our minds engaged with geo- 

 metrical ideas, and are not permitted to fancy ourselves 

 geometers wlien we are only arithmeticians. 



This demand for thought — for the continued construc- 

 tion of mental representations — is enough to account for the 

 slow progress of the method among adult mathematicians. 

 Two courses, however, are open to the cultivators of (Quater- 

 nions : they may show how easily the principles of the 

 method are acquired by those whose minds are still fresh, 

 and in so doing they may prepare the way for the triumph 

 of Quaternions in the next generation ; or they may apply 

 the method to those problems which the science of the 

 day presents to us, and show how easily it arrives at those 

 solutions which have been already expressed in ordinary 

 mathematical language, and how it brings within our 

 reach other problems, which the ordinary methods have 

 hitherto abstained from attacking. 



Sir W. R. Hamilton, when treating of the elements of 

 the subject, was apt to become so fascinated by the meta- 

 physical a:;pccts of the method, that the mind of his 

 disciple became impressed with the profundity, rather 

 Vol.. i,x. — No. 217 



than the simplicity of his doctrines. Professors Kelland 

 and Tait in the opening chapter (II.) of their recently 

 pubhshed work* have, we think, successfully avoided 

 this element of discouragement. They tell us at once 

 what a vector is, and how to add vectors, and they do 

 this in a way which is quite as intelligible to those 

 who are just beginning to learn geometry as to the 

 most e-xpert mathematician. 



The subject, like all other subjects, becomes more in- 

 tricate as the student advances in it ; but at the same time 

 his ideas are becoming clearer and more firmly esta- 

 blished as he works out the numerous examples and ex- 

 ercises which are placed before him. 



The technical terms of the method— Scalar, Vector, 

 Tensor, Versor— are introduced in their proper places, 

 and their meaning is sufficiently illustrated to the begin- 

 ner by the examples which he is expected to work out. 

 The pride of the accomplished mathematician, however 

 (for whom this book is not written), might have been 

 somewhat mollified if somewhere in the book a few pac^es 

 had been devoted to explaining to him the differences 

 between the Quaternion methods and those which he has 

 spent his life in mastering, and of which he has now be- 

 come the slave. He is apt to be startled by finding that 

 when one vector is multiplied into another at right angles 

 to it, the product is still a vector, but at right angles to 

 both. His only idea of a vector had been that of a line, 

 and he had expected that when one vector was multiplied 

 into another the result would be something of a different 

 kind from a line, such, for instance, as a surface. Now 

 if it had been pointed out to him in the chapter on vec- 

 tor multiplication that a surface is a vector, he would be 

 saved from a painful mental shock, for a mathematician 

 is as sensitive about " dimensions " as an English school- 

 boy is about " quantities." 



The fact is, that even in the purely geometrical appli- 

 cations of the Quaternion method we meet with three 

 different kinds of directed quantities : the vector proper, 

 whiqji represents transference from A to B ; the area or 

 " aperture," which is always understood to have a positive 

 and a negative aspect, according to the direction in which 

 it is swept out by the generating vector ; and the versor, 

 which represents turning round an axis. 



The Quaternion ideas of these three quantities dift'er 

 from the old ideas of the line, the surface, and the angle 

 only by giving more prominence to the fact that each of 

 them has a determinate direction as well as a determinate 

 magnitude. When Euclid tells us to draw the line A B, 

 he supposes it to be done by the motion of a point from 

 A to B or from B to A. But when the line is once gene- 

 rated he makes no distinction between the results of these 

 two operations, which, on Hamilton's system, are each 

 the opposite of the other. 



Surfaces also, according to Euclid, are generated by the 

 motion of lines, so that the idea of motion is an old one, 

 and we have only to take special note of the direction of 

 the motion in order to raise Euclid's idea to the level of 

 Hamilton's. 



With respect to angles, Euclid appears to treat them as 

 if they arose from the fortuitous concourse of right lines ; 



* " Introduction to Quaternions, with numerous Examples." by P. Kel- 

 land. F.R.S., formerly Fellow of Queen's College, Cambridge ; and P. G. 

 Tait, formerly Fellow of St. Peter's College, Camljridge : Professors in the 

 Department of Mathematics in the University of Edinburgh. (Macmillan, 



