April 29, 1920] 



NATURE 



257 



Boutroux properly directs attention to the fact that 

 much of Apollonius's "Conies" is essentially 

 analytical, though, of course, there is no algebra, 

 strictly so called. In the sections on function- 

 theory, due reference is made to M6ray, who shares 

 in great measure with Weierstrass the credit of 

 laying the foundations of a sound theory of 

 analytical functions. It is fortunate that the great 

 similarity of the work of these two mathematicians 

 did not give rise to bitter polemic; there was at 

 least as much material for it as in the famous 

 Newton-Leibniz controversy. 



The author's critical remarks, we fancy, will 

 not meet with such unqualified acceptance. To 

 take one example, he says of Peano's symbolism : 

 " Unfortunately, it is not everyone who can read 

 with facility these combinations of signs, which 

 are often grotesque and repulsive, and unaccom- 

 panied by a single word of the vulgar tongue. 

 Moreover, M. Peano's symbolism cannot claim to 

 have made any contribution to the progress of 

 mathematics ; it remains a remarkable method of 

 scientific shorthand." As a criticism of the work 

 of Peano and his school, this is distinctly unfair. 

 Anyone who has the patience to become moder- 

 ately familiar with the notation is bound, we 

 believe, to admit that the alternative is either to 

 produce a text full of ambiguities and tacit 

 assumptions, or else one of intolerable prolixity. 

 The present reviewer has come to this conclusion 

 with very great reluctance; even the Cambridge 

 Press has not succeeded in making the " Principia 

 Mathematica " attractive to the eye ; and it is to 

 be feared that the first impression it is likely to 

 produce is that it is the work of a drunken com- 

 positor. Probably its use will be mainly, if not 

 wholly, confined to the logical foundations of 

 mathematics ; for this purpose we think its value 

 is indisputable. There are other controversial 

 statements scattered about the text ; they all 

 deserve careful attention, even if the reader is 

 inclined to disagree with them. 



There is one point, of frequent occurrence, 

 against which we feel bound to protest. Prof. 

 Boutroux repeatedly says that such an equation 

 as .-v^^ys^o represents a point. This is abso- 

 lutely untrue ; it may be said to represent a point- 

 circle (circle of zero radius), or a pair of isotropic 

 lines, according as we exclude or include complex 

 elements. But no single equation in point- 

 co-ordinates can represent a point; moreover, it 

 is fatal to ignore the degree of the equation. 

 Oddly enough, Halphen makes the same mistake 

 in his memoir on characteristics; he repeatedly 

 gives the name of "a single line" to what is, as 

 a degenerate quadratic locus, a double line with 

 two special points (or, exceptionally, one special 

 NO. 2635, VOL. 105] 



double point) upon it. Fortunately, this does not 

 affect Halphen 's conclusions, the reason (appar- 

 ently) being that he discusses point-equations and 

 line-equations simultaneously. 



We hope that this work will have a goods 

 circulation in England ; its virtues are precisely 

 those in which our text-books still leave some- 

 thing to be desired : elegance, breadth of view, 

 choice of topics, and regard to historical perspec- 

 tive. G. B. M. 



The Proteins. 



The Physical Chemistry of the Proteins. By 

 Prof. T. Brailsford Robertson. Pp. xv + 483. 

 (London: Longmans, Green, and Co., 1918.) 

 Price 255. net. 



THIS is not a new book. It first appeared in 

 the form of an edition in German published 

 at Dresden in 1912. The second edition, in 

 English, has, however, been so completely re- 

 written as to make it practically a new account of 

 the subject. 



There are four parts, of which the first deals 

 with the mode of preparation and estimation and 

 the chemical constitution of proteins ; the second 

 with their electro-chemistry ; the third with the 

 physical properties of their -solutions, such as 

 viscosity, refractive indices, etc. ; and the last 

 with what the author calls the chemical dynamics 

 of protein systems, by which, broadly, he means 

 their reactions with catalysts. It will be seen that 

 a complete survey of the subject has been 

 attempted, and it may be said at once that, as an 

 introduction to the literature, already extensive, 

 the book can be commended. 



It is now agreed that the proteins are chemically 

 a homogeneous group the molecules of which are 

 built up by the synthesis of amino-acids. The size 

 of the molecules so formed is still open to doubt. 

 Emil Fischer, than whom no one could speak with 

 more authority, refused to accept the molecular 

 weights of 15,000 to 20,000 commonly ascribed 

 to native proteins. The molecular weight, indeed, 

 varies widely from the 16,000 of haemoglobin, or 

 the 17,000 of edestin, to the values reckoned in 

 hundreds of the polypeptides. It certainly lies in 

 the thousands for native proteins, and is large 

 enough to upset the simpler stoichiometrical 

 relations. 



Consider, for example, the reaction with acids 

 and alkalis. Proteins, like amino-acids, are 

 amphoteric — that is to say, they will form salts 

 with either an acid or a base — but, according to 

 the author, when their combining equivalents are 

 determined by known methods, their combining 

 capacity is found to be much in excess of the 



