December i, 1904J 



NA TURE 



99 



roots of equations. Moreover, there is the paper on 

 the contacts of lines and surfaces of the second order, 

 \vhere the invariant factors of a matrix are recognised, 

 and the system of two quaternary quadratics is con- 

 sidered in detail with reference to the simplest simul- 

 taneous reduction of the forms. 



Appreciations of Sylvester's character and of the 

 value of his mathematical work have been written by 

 able hands, and it is unnecessary to enlarge upon 

 them here. His egotism was obvious and often 

 amusing, but never offensive ; his enthusiasm was re- 

 freshing, and though his temper was touchy, he was 

 very generous and kind. As a master of formal 

 analysis he has few equals ; the birth of the calculus 

 of invariants occurred just at the right time to attract 

 his attention, and his contributions to this subject alone 

 are enough to make him famous. He had the instincts 

 of an architect, and it is well, on the whole, that he 

 did not always trouble to clear away the chips. The 

 casual remarks scattered about his papers and the 

 fragmentary nature of some of them, help to make 

 the reading of them very stimulating ; he takes us into 

 his confidence, shows us how his ideas arose, and gives 

 us hints of unexplored regions. He was eminently 

 original, and spent little time in studying the works 

 of his contemporaries ; thus he did not even realise 

 that his theory of reciprocants had been more than 

 anticipated by others, especially by Lie. But any mis- 

 understanding arising from this source must have been 

 long since dissipated, and his place among the great 

 mathematicians of his time is quite secure. 



Sylvester's occasional notes on the theory of 

 numbers and his lectures on partitions suggest 

 problems to those who are interested in arithmetic. 

 The present volume, for instance, contains three notes 

 on cubic Diophantine equations, a subject not yet ex- 

 hausted, though Sylvester's own theory of resideration 

 throws much light upon it. The late Henry Smith 

 once referred to this problem as being one which might 

 be hopefully attacked with the engines of modern 

 analysis ; perhaps the appearance of this edition of 

 Sylvester's works may lead to the discovery of a com- 

 plete theory. 



A good example of Sylvester's power of illuminating 

 and drawing general conclusions from the simplest 

 mathematical problem is the note (p. 392) on an 

 elementary geometrical theorem for which no direct 

 proof had been discovered. He observes that the 

 proof may be made to depend on showing that a certain 

 analytical equation has no real root, and suggests that 

 in all such cases where the analytical proof consists 

 in demonstrating the non-existence of roots, the 

 geometrical proof must necessarily be indirect, while 

 in other cases the reductio ad absurdum may be con- 

 venient, but is not necessary. This observation re- 

 minds us at once of Gauss's discussion of the division 

 of the circle, and if Sylvester's conjecture is true it 

 gives another case of the curious points of contact 

 that exist between analysis and geometry. 



It is not to be expected, or even desired, that many 



should share Sylvester's keen delight in the beauty of 



formal analysis ; but it is a mistake to discourage 



those who are inclined to enjoy it, however Unpractical 



NO. I 83 I, VOL. 71] 



parts of the subject may be. Quite apart from other 

 reasons, the study of pure mathematics may be de- 

 fended, like that of music or chess or painting, from 

 the merely aesthetical side, and this Sylvester does in 

 terms both vigorous and quaint. For example : — 



" The fortunate proclaimer of a new outlying planet 

 has been justly rewarded by the offer of a baronetcy 

 and a national pension, which the writer of this wishes 

 him long life and health to enjoy. In the meanwhile, 

 what has been done in honour of the discoverer of a 

 new and inexhaustible region of exquisite analysis? " 

 the latter reference being to Cayley's discovery of the 

 calculus of invariants. Fortunately Cayley was saved 

 in another way from the cares of money-making, and 

 he lived long enough to realise to the full his great 

 reputation among those who would appreciate his 

 work. Sylvester in his early life suffered unjustly from 

 the current prejudice against his race ; so far as it was 

 possible this was afterwards atoned for, and it is to 

 be hoped that no bitter feeling was left behind. 



G. B. M. 



MENTAL AND SOCIAL MEASUREMENTS. 

 An Introduction to the Theory of Mental and Social 

 Measurements. By Edward L. Thorndike, Pro- 

 fessor of Psychology in Teachers' College, Columbia 

 University. Pp. xii + 212. (New York: The 

 Science Press, 1904.) Price 1.50 dollars net. 



AMERICAN colleges seem more awake than our 

 own to the fact that the newer methods of 

 statistics have made it possible to deal with facts with 

 which they are directly concerned, and to discuss them 

 with far more completeness than was practicable a few 

 years ago. They are making in consequence large 

 collections of anthropometric data to serve as tests of 

 health and development, and for comparisons between 

 colleges. Again, there are more teachers in America 

 than in this country who, appreciating the fact that 

 the above methods have far wider applicability, extend 

 the range of their measurements to psychophysical 

 subjects. They are also eager to deal with purely 

 psychical matters that elude direct measurement but 

 admit of being arranged by mutual comparison into 

 their proper class places, or to utilise a third and still 

 more general method, which deals with such objects 

 as can be sorted into a few distinct classes without re- 

 gard to their internal arrangement. The author is 

 fully justified in saying that 



" The obscurest and most complex traits, such as 

 morality, enthusiasm, eminence, efficiency, courage, 

 legal ability, inventiveness, can be made material for 

 ordinary statistical procedure, the one condition being 

 that the general form of distribution of the trait in 

 question shall be approximately known." 



In these circumstances a system of elaborate - 

 measurements has come into vogue in many American 

 colleges. Whether the authorities have always planned 

 their measurements wisely, and whether they discuss 

 them adequately and accurately, will not be considered 

 here. The volume is written to direct and to warn, 

 in doing which it reveals some grave blunderings. 

 Unfortunately, it is composed chiefly for those persons 

 who are ignorant of even simple mathematics. The 



