i9^ 



NATURE 



[April 15, 1920 



minants, but a kind of entity of a very abstract 

 and comprehensive type. A large part of group- 

 theory and many complex linear algebras can be 

 expressed in terms of matrices, and this absorp- 

 tive property of matrix-theory will probably 

 become more evident in course of time. Matrices 

 occupy a special section in the International 

 Schedule, and the Royal Society Index contains 

 about sixty titles under that heading. Papers 

 under other headings (especially determinants) 

 have also more or less bearing on the subject. 

 Prof. Baker's works on Abelian functions show 

 the importance of matrices in the general theory 

 of theta functions. Some knowledge of the subject 

 is becoming essential in connection with various 

 branches of pure mathematics. 



Prof. Cullis's second volume, if we understand 

 the author aright, seems to be a continuation 

 rather different from that which he originally 

 planned. This is not to be regretted, 

 because in this portion we have statements 

 and proofs of well-known and important 

 theorems in the author's own notation, and 

 a large number of illustrative examples. Among 

 the subjects treated are ranks of matrix 

 products and factors, equigradient transforma- 

 tions, certain matrix equations of the second 

 degree, and various properties of a pair of matrices 

 ("paratomy," "orthotomy," and so on). Much of 

 the argument is put into a quasi-geometrical form. 

 The outstanding feature of the work, which the 

 author properly emphasises, is the detailed dis- 

 cussion of rectangular, as distinguished from 

 square, matrices. For this reason alone the work 

 ought to give a great stimulus to the subject, and 

 we hope that the publication of the whole treatise 

 will not be long delayed. Until it is finished, it 

 will be difficult, if not impossible, to give a proper 

 appreciation of it, especially as the author intro- 

 duces so many new symbols and technical terms. 

 One thing, however, is certain : we now have the 

 outlines of a calculus of matrices in which the 

 operations of addition, subtraction, and multiplica- 

 tion are definite. It may be conjectured that some 

 of the most important applications will be to 

 problems connected with a compound modulus, 

 arithmetical or algebraical as the case may be. 



As a matter of curiosity it may be noted that 

 one or two of the very first problems in the theory 

 of rectangular matrices occur in Gauss's "Dis- 

 quisitiones Arithmeticae " ; for instance, in connec- 

 tion with the theory of composition of quadratic 

 forms, we have the problem of finding a matrix 

 (02,4) the six determinants of which are to be six 

 assigned integers, subject to a certain relation. 



G. B. M. 

 NO. 2633, VOL. 105] 



The Chemistry of Animal Products. 



The Essentials of Chemical Physiology : For the 

 Use of Students. By Prof. W. D. Halliburton. 

 Tenth edition. Pp. xi -1-324. (London: Long- 

 mans, Green, and Co., 1919.) Price 75. 6d. 

 net. 



THE fact that this well-known and appreciated 

 text-book has reached its tenth edition is 

 sufficient evidence that it satisfies adequately the 

 need for a short practical course in the chemistry 

 of the substances found in and produced by the 

 activity of living tissues. This object is excellently 

 attained. It is not to be expected that a detailed 

 account of the chemical processes occurring during 

 the life and functional action of the organs of the 

 body is to be found therein. Indeed, it would be 

 impossible to separate the chemical from the 

 physical aspects of any of these physiological pro- 

 cesses. Such a separation appears to be an un- 

 fortunate necessity in a great part of the teaching 

 of the subject, but a more intimate union between 

 the chemical side and what is sometimes called 

 the "experimental" side of the student's work is 

 very desirable, and migh{ be arranged without 

 much difficulty. 



There are some Important questions which are 

 apt to fall out in the present arrangement; such 

 are those of permeability, osmotic pressure, 

 hydrogen-ion concentration, and the properties of 

 colloidal solutions. In a future edition Prof. 

 Halliburton might find it possible to include a 

 few simple exercises in these problems. An intelli- 

 gent grasp of the principles involved is not to be 

 attained by the mere reading of statements about 

 them, while even a small number of experiments 

 have great value. On account of its importance 

 in physiological phenomena, some experiment 

 showing the synthetic aspect of the action of 

 enzymes might well be inserted. 



A general criticism which applies to most text- 

 books on practical chemistry, especially to those 

 on biological chemistry, is that a number of the 

 tests given suggest cookery recipes rather than 

 scientific experiments. A student is very little the 

 better for performing Molisch's sugar test if he 

 is ignorant of what the result is due to. And 

 how many understand, when they make the tests, 

 why some sugars reduce copper salts, while others 

 do not; or why tartrates are added to Fehling's 

 solution? It would often be better to curtail the 

 recital of what is to be found in genfei:al text-^books 

 in order to explain the reasons for the results of 

 the actual experiments made. 



In the book before us the theoretical part is, on 

 the whole, brought well up to date. In view of 



