April '13, 191 1] 



NATURE 



207 



new arsenic compounds. Short chapters are devoted 

 to the therapeutics of tumours, inflammation, blood- 

 <liseases, gout, fever, and disturbances of the cir-' 

 . ulation and digestion so far as these have been deter- : 

 mined by experimental methods. It is, of course, ' 

 impossible to treat this extensive programme exhaus-; 

 lively in 174 pages, and the author seems rather to 

 liave aimed at giving a general idea of what is being 

 done to advance therapeutics experimentally with the 

 object of arousing the interest of the students and 

 \ ounger practitioners of medicine in the subject. The 

 ')()ok seems well fitted to attain this object, for it is' 

 written in an easy style, and deals with some of the' 

 nost interesting topics in medicine at the present! 

 lime. On the other hand, the chapters are very un- 

 qually written. In some instances pages are devoted- 

 .0 detailed description of surgical methods (pp. 34-36)* 

 ir of individual experiments, which seem out of . 

 place in an introductory handbook, while other sub- 

 jects are treated too briefly for anyone to follow 

 except the expert; and there is very little attempt' 

 made to differentiate the fundamental experiment ; 

 irom the less important or less generally accepted 

 result. ( 



The author tends too often to leave the solid ground 

 for speculations which are often based on experiments 

 A hich, to say the least, have not yet received general 

 i>sent. In a book primarily designed for German 

 medical students, perhaps it is right to direct their 

 attention chiefly to authors of their own nationality 

 whose works they can read, but we cannot help think- 

 ing that some of the chapters would have been im- 

 <roved by wider reading. For example, the chapter 

 on vaccines might have been rendered more intel- 

 ligible and also more up-to-date. 



The book is not free from serious errors ; for 

 •xample, where (p. 11) it is stated that Hunt found 

 I Icohol protects mice against the nitriles ; and the 

 .'intidotal effect of sodium sulphate in barium poison- 

 ing is surely due to the barium being precipitated, 

 and not to the restoration of the sodium, as the 

 author supposes (p. 13). 



PROJECTIVE GEOMETRY. 

 Projective Geometry. By Prof. O. Veblen and Prof. 

 J. W. Young. Vol. i. Pp. x + 342. (Boston and 

 London : Ginn and Co., 1910.) Price 155. net. 



IN ih( first page of their introduction the authors 

 say : 



"The starting-point of any strictly logical treatment 

 of geometry must be a set of undefined elements and 

 relations, and a set of unproved propositions involving 

 them ; and from these all other propositions (theorems) 

 are to be derived by the methods of formal logic." 



Here, in a nutshell, is the modern mathematician's 

 creed; and it is significant that it should thus appear 

 in a treatise on projective geometry, which at first 

 -ight would seem to be one of the most intuitive of 

 : he branches of mathematics. 



In accordance with the above dictum, the authors 

 .;ive a brief discussion of the axioms of geometry so 

 far as they are required for the purposes of this 

 volume, rightly, we think, deferring the more com- 



NO. 2163, VOL. 86] 



plete theory of order and continuity to a later stage. 

 Enough, however, is done to make the reader aware 

 of the numerous tacit, and often complex, assump- 

 tions made in the ordinary treatment of the subject. 

 For instance, we have an explicit statement of the 

 fundamental postulate : 



"If A, B, C are points not all in a line, and D, E 

 are distinct jjoints such that (B, C, D), (C, E, A) are 

 respectively colli near, then there is a point F such 

 that (A, B, F) and (D, E, F) are respectively collinear." 



With the help of this and a few other assumptions, 

 a plane is defined in such a way that it can be proved 

 that if A, B are any two points in a plane, every 

 point of the line AB is in .the plane. No one can 

 fail to see that this is an improvement on the 

 Euclidean definition of a plane, which is a question- 

 begging assumption, based no doubt on the practical 

 tests applied by masons and carpenters. 



After this the reader is introduced to the funda- 

 mental operations of projection and section, and to 

 the principle of duality. The latter is very properlv 

 stated, at the outset, with reference to three-dimen- 

 sional space : that is, point and plane are correlative 

 terms, not point and line. It is easy enough to 

 deduce the special laws of duality for two-dimensional 

 fields ; and the more general form of statement at once 

 brings home to the student the fact that, as a rule, 

 the propositions of projective geometry arrange them- 

 selves in sets of four, only one of which need be 

 formally proved. For instance, Pascal's theorem for 

 a conic in a plane leads at once to Brianchon's 

 theorem, and two corresponding theorems for a 

 quadric cone. 



Even yet it may be asserted that von Staudt is the 

 great master of projective geometry, much as Gauss 

 is the incomparable arithmetician. It is one of the 

 great merits of this work that the influence of von 

 Staudt's work is so apparent in it. For instance, 

 involution is treated at a comparatively early stage; 

 and this is important for several reasons. In the 

 first place, it simplifies the proofs of many funda- 

 mental properties of conies ; in the second, it shows 

 the existence of a polar system, in a plane or in 

 space, apart from the assumption of a quadric curve 

 or surface defining it. Ultimately, of course, the 

 best definition of a conic or quadric surface is that it 

 is the locus of self-corresponding points in a polar 

 system. This, with Staudt's theory of imaginary (or 

 complex) elements, permits of the inclusion of 

 " imaginary " conies and quadrics as actually existing 

 things. It is to be hoped that the second volume will 

 contain a sufficient account of Staudt's beautiful 

 theory, which, as a rule, seems to be very imperfectly 

 apprehended. As he unfolds it in the supplements to 

 his "Geometrie der Lage," it is purely geometrical, 

 though no doubt he was led to it by analysis — at 

 least, this seems the most probable assumption. 



Among the interesting pxiints of the present volume 

 there is a brief account of Staudt's theory of " throws " 

 (Wiirfc), and his constructions for addition and multi- 

 plication. In the latter there is a slight modification, 

 arising from a change in the order of deduction. 

 What is here shown is that if we take any three points 



