NA TURE 



THURSDAY, JUNE 8, 1905. 



.4 UAliVAL OF QUATERNIONS. 

 A Manual of Quaternions. By Prof. Charles Jasper 

 Joly, F.R.S. (London: Macmillan and Co., Ltd., 

 1905.) Pp. xxvii + 320. Price los. net. 



PROF. C. J. JOLY'S " Manual of Quaternions " is 

 an important addition to tlie literature of the sub- 

 ject. It at once takes ranli with Tail's " Treatise " as 

 an eminently serviceable exposition of Hamilton's 

 great calculus. 



Hamilton's own works, the " Lectures " and the 

 ■" Elements," are in their way inimitable. Unfortu- 

 nately, their style is not suited to the average student 

 eager to acquire a working knowledge of the mathe- 

 matical method developed in them. Tait alone of the 

 younger contemporaries of Hamilton seemed to have 

 been able to appreciate the " Lectures "; but he him- 

 self used to relate how, as he laboriously read through 

 the first six, he began to despair of his own powers. 

 There seemed to be such diffuse discussion, and withal 

 so little apparent progress. But the seventh lecture 

 came like a transformation scene. Every page dis- 

 played new beauties, every paragraph disclosed the 

 marvellous power and variety of the method. From it 

 Tait drew his inspiration, and proceeded to enlighten 

 the world as to the meaning and purpose of the 

 quaternion. 



To the student who has grasped the essentials of 

 the method Hamilton's second volume, the " Ele- 

 ments," will always prove a happy hunting ground; 

 but experience has shown that its very completeness 

 acts as a deterrent. In the much smaller treatise 

 written by Tait, the important practical aspects of 

 quaternions are more rapidly though less logically de- 

 veloped, and the chief value of Tait's work lies in his 

 characteristic treatment of dynamical and physical 

 problems. It has been long felt, however, that a good 

 working manual of quaternions was needed, by use 

 of which the mathematical student could come quickly 

 into touch with all that is essential in the calculus. 

 This is what Prof. Joly has endeavoured to supply. 



For reasons clearly explained in the preface, the 

 author has (reluctantly, he confesses) forsaken the 

 Hamiltonian approach. Instead of developing the cal- 

 culus logically from the definition of a quaternion as 

 the ratio of two vectors, he defines independently the 

 quantities Sa/3 and Vo/S, and then writes the product 

 a;8 as equal to the sum of these two. The student 

 must, of course, take on trust that there is some good 

 reason for defining Sa/3 as minus the product of the 

 length of one vector into the length of the projection 

 of the other upon it. This is, at root, the peculiarity 

 of Hamilton's system which troubled O'Brien nearly 

 sixty years ago, and has not ceased to trouble occasional 

 critics since. There is a kind of notion hovering about 

 in some minds that the positive sign in algebra is more 

 natural than the negative sign, the truth being, of 

 course, that the one necessarily implies the other. It 

 is to be feared, however, that this apparently arbitrary 

 assumption of the negative sign in translating S<«0 into 

 ordinary trigonometrical notation (Clifford calls it a 

 O. 1858, VOL. 72] 



convention) will puzzle many a student. Prof. Joly 

 soon gives the reason for the negative sign, though 

 not quite so definitely as might be desirable ; but it is 

 questionable if its full significance will be appreciated 

 until considerable progress has been made in acquiring 

 quaternionic skill. The reader is advised to exercise 

 a strong faith, and to proceed nothing fearing. If he 

 persevere he will soon get out of the valley of the 

 shadow of the negative sign. 



It is possible that some critic may regard this for- 

 saking of Hamilton's logical basis as a confession of 

 weakness. But this is not so. The weakness is in the 

 average student, for whom a somewhat simple intel- 

 lectual diet must be prepared in the hope that the 

 mental digestion may be strengthened sufficiently to 

 assimilate the strong Hamiltonian food which Prof. 

 Joly serves up a little later. The truth is that very few- 

 students are able to appreciate to the full an absolutely 

 logical argument until they have a certain amount of 

 practical knowledge imparted to them more or less by 

 authority. 



So far as the principle of the method is concerned, 

 Prof. Joly ranges himself at first on the side of those 

 vector analysts who neglect the quaternion. But it 

 is only for a couple of pages at the beginning of 

 chapter ii. On p. 8 the important formula 

 {al3 = SaS + Vci0) 



is introduced as a definition of the quaternion, and the 

 quaternion is never afterwards lost sight of. Its 

 fundamental importance and analytic value are in 

 evidence on every page. It must be admitted that by 

 this line of approach the reader is rapidly brought 

 into touch with the essential elements of the subject. 

 There is, nevertheless, a certain arbitrariness which 

 is not satisfying to the mind, nor is it clear when 

 all -is done what is really fundamental. A critically 

 minded student might possibly be inclined to say. Why 

 not define Sa^ as plus the product of the lengths of the 

 vectors into the cosine of the angle between them, and 

 then define the quaternion a0 by the formula Vo5— Sa^ ? 

 At first sight it seems to amount to the same thing, and 

 yet, as will be found on trial, it leads to a system 

 clothed in quaternion garments, but more like the 

 fabulous ass in the lion's skin than the real lion. 



Having thus established in chapter ii. the funda- 

 mental properties of the quaternion. Prof. Joly rapidly 

 runs over certain important transformations of vector 

 products and ratios (chapters iii. and iv.), and simple 

 applications to the geometry of the straight line, plane 

 and circle (chapters v. and vi.). Then follow, treated 

 in separate chapters, differentiation, linear vector 

 functions, quadric surfaces, and the geometry of curves 

 and surfaces. Here the poiver of the calculus asserts 

 itself strongly. Numerous examples are supplied 

 throughout for the student to work upon and develop 

 his analytical skill. In subsequent chapters dynamical 

 problems of various kinds are taken up — such as as- 

 tatic equilibrium, screws and wrenches, strains, central 

 forces, constrained motion, motion of a rigid body, and 

 the like. A valuable and well arranged chapter on the 

 operator v treats of heterogeneous strain, spherical 

 harmonics, hydrodynamics, elasticity, electromag- 

 netic theory, and wave propagation generally. The 



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