August 6, 1908] 



NATURE 



315 



ton is equally gifted, and there is quite remarlcable 

 merit in tine drawings by Mr. A. K. Maxwell. 



One of Mr. Budgett's discoveries was a fresh-water 

 medusa in the delta of the Niger. .'According to Mr. 

 E. T. Browne this is referable to the species 

 {Limnociiida tanganicae) that has been found in Lake 

 Tanganyika, and its occurrence in these widely- 

 separated localities becomes intelligible if we regard it 

 as a relic of the fauna of the Lutetian Sea, which the 

 geologists believe to have stretched across Africa in 

 Middle Eocene times. Mr. Browne gives an interest- 

 ing account of this remarkable type, which remains 

 unclassifiable. There is probably a gymnoblastic 

 hydroid stage awaiting discovery. 



The volume is an eloquent tribute to the esteem 

 in which Mr. Budgett was held by his friends, for 

 besides the editorial labours and Mr. Shipley's beau- 

 tifully executed sketch, the working-up of the embryo- 

 logical material — especially that which fell to Prof. 

 Graham Kerr and Mr. Assheton — must have involved 

 much time and thought. Those who have by 

 their own masterly work thus completed Budgett's 

 have certainly been generous in their friendship, and 

 the memorial volume, which is a valuable contribu- 

 tion to embryology, will surely not fail of its highest 

 purpose, of perpetuating by its stimulus the tradition 

 which Budgett's life and work e.\pressed. 



HIGHER ALGEBRA AT HARVARD 

 UNIVERSITY. 



Introduction to Higher Algebra. By Maxime Bochcr. 

 Prepared for publication with the cooperation of 

 E. P. R. Duval. Pp. xi + 321. (New York : The 

 Macmillan Co.; London : Macmillan and Co., Ltd., 

 1907.) Price Ss. net. 



WITH the cooperation of a former pupil, Mr. 

 E. P. R. Duval, Prof. Bocher has made into 

 a book part of the subject of his lectures during the 

 last ten years. Of course, it would be unfair to try 

 to reconstruct the lectures from the book, but we do 

 get an impression from this volume, and especially 

 from the preface, of the problems which a Harvard 

 professor has to face, and how he deals with them. 

 We are told that 



" .\n American student approaching the higher 

 parts of mathematics usually finds himself 

 unfamiliar with most of the main facts of algebra, 

 to say nothing of their proofs. Thus he has only a 

 rudimentary knowledge of systems of linear equations, 

 and he knows n-ext to nothing about the subject of 

 quadratic forms." 



For a student of this kind Prof. Bocher provides a 

 book which contains the elementary theory of poly- 

 nomials, determinants, systems of linear equations, 

 matrices, invariants, and symmetric functions, besides 

 a discussion of quadratic forms which goes into con- 

 siderable detail, and is based on the work of .Smith, 

 Sylvester, Weierstrass, and Kronecker. 



The author makes special mention of Kronecker 

 and Frobenius, and, in fact, it is easy to see how 

 these great analysts have influenced the writer both 

 in his choice of topics and mode of treatment. What- 



NO. 2023, VOL. 78] 



ever may be thought about the subjects discussed, 

 there is no doubt about the soundness and thorough- 

 ness of the treatment; and it would be very interest- 

 ing to know what is the effect of a course of this kind 

 upon a class of American students, and what would 

 be the result of giving it to an English class of a 

 similar grade. It ought at least to arouse interest, 

 and would probably induce the best students to read 

 some first-rate memoirs, and possibly produce some- 

 thing of their own — though it must be confessed 

 that the theory of quadratic forms, which is the 

 terminus ad quern of this book, has been so thoroughly 

 worked out now that there does not seem much 

 chance of developing it further. 



It should be stated that Prof. Bocher by no means 

 confines himself to abstract analysis ; on the contrary, 

 like Klein, he takes every opportunity of giving geo- 

 metrical interpretations, and although he does not say 

 so, it is likely enough that he has illustrated by dia- 

 grams and models the different ways in which two 

 proper or degenerate quadric surfaces may intersect. 

 Some of his groups of examples are calculated to 

 suggest methodical lines of research in other direc- 

 tions; for example, on pp. 260-61 we have an outline 

 of the theory of the binary quartic. 



Generally' speaking, the style of the book is very 

 clear and simple. One exception occurs on p. 120, 

 theorem 2, where it is not sufficiently emphasised that 

 in the first case 5:(?y.ri_)0 = o is to be made an identity 

 in the .\-'s bv giving suitable fixed values to the y's. 

 .\nd although the notion of a domain of rationality is 

 explained (pp. 175, 212, 216), it would have been 

 better to do this at an earlier stage. 



The printing, done at the Norwood Press, is very 

 good on the whole, but the suffixes are often un- 

 pleasant to look at, owing to want of gradation in 

 type. On the other hand, we do find .r,= printed 

 nearly as it is written— a very rare experience in Eng- 

 lish-printed books. It would serve a useful purpose 

 if a congress could be held between mathematicians 

 and mathematical printers and compositors; the 

 former often forget that what is easy to write may be 

 difficult to print, and the latter are very conservative, 

 besides failing to sympathise with a mathematician's 

 way of writing formulae, especially in matters of pro- 

 portion and spacing. G. B. M. 



^rETEOROLOGIC.AL OBSEKV.iTIONS. 



The Observer's Handbook, Meteorological Office, 



1908. A new and revised edition of Dr. R. H. 



Scott's " Instructions in the use of Meteorological 



Instruments." (London: H.M. Stationery Office.) 



THIS book consists of two parts, and appendices. 

 Part i. follows somewhat in the line of Dr. 

 Scott's handbook, but part ii. is practically new, and 

 deals with recording instruments, several of which 

 have come into use since the previous book was pub- 

 lished. 



The issuing of a new official guide for its observers 

 by the Meteorological Office gives an opportunity of 

 comparing the state of the science of meteorolom- 

 now and twentv vears ago, in so far as instruments and 



