270 PETER GUTHRIE TAIT 



Tait contributed important reviews on two works by W. K. Clifford. 

 The first of these, which appeared in Nature (Vol. xvm) on May 23, 1878, 

 referred to the Elements of Dynamic, Part i, Kinematic, which was particularly 

 interesting to Tait because of the use the author made throughout of 

 quaternion methods. I give the review in full : 



Though this preliminary volume contains only a small instalment of the subject, 

 the mode of treatment to be adopted by Prof. Clifford is made quite obvious. It is 

 a sign of these times of real advance, and will cause not only much fear and trembling 

 among the crammers but also perhaps very legitimate trepidation among the august 

 body of Mathematical Moderators and Examiners. For, although (so far as we have 

 seen) the word quaternion is not once mentioned in the book, the analysis is in great part 

 purely quaternionic, and it is not easy to see what arguments could now be brought 

 forward to justify the rejection of examination-answers given in the language of 

 quaternions especially since in Cambridge (which may claim to lay down the law on 

 such matters) Trilinear Coordinates, Determinants, and other similar methods were long 

 allowed to pass unchallenged before they obtained formal recognition from the Board of 

 Mathematical Studies. 



Everyone who has even a slight knowledge of quaternions must allow their 

 wonderful special fitness for application to Mathematical Physics (unfortunately we 

 cannot yet say Mathematical Physic !) : but there is a long step from such semi-tacit 

 admissions to the full triumph of public recognition in Text-Books. Perhaps the first 

 attempt to obtain this step (in a book not ostensibly quaternionic) was made by 

 Clerk Maxwell. In his great work on Electricity all the more important Electrodynamic 

 expressions are given in their simple quaternion form though the quaternion analysis 

 itself is not employed : and in his little tract on Matter and Motion {Nature, Vol. XVI, 

 p. 119) the laws of composition of vectors are employed throughout. Prof. Clifford 

 carries the good work a great deal farther, and (if for this reason alone) we hope his 

 book will be widely welcomed. 



To show the general reader how much is gained by employing the calculus of 

 Hamilton we may take a couple of very simple instances, selecting them not because 

 they are specially favourable to quaternions but because they are familiar in their 

 Cartesian form to most students. Every one who has read Dynamics of a Particle 

 knows the equations of non-acceleration of moment of momentum of a particle, under 

 the action of a single centre of force, in the form 



xy yx = o\ 

 yz zy =o\ 

 zx xz oj 



with their first integrals, which express the facts that the orbit is in a plane passing 

 through the centre, and that the radius-vector describes equal areas in equal times. 

 But how vastly simpler as well as more intelligible is it not to have these three 

 equations written as one in the form 



