REVIEWS 155 



fitude geometrique des Transfonnations birationnelles et des Courbes planes. 



Par Henri Malet. [Pp. viii + 262.] (Paris : Gauthier-Villars & 

 Cie, 192 1. Price 32 fr. net.) 



This book is an attempt to establish a theory of algebraic curves and in 

 particular of their birational transformations without the use of algebra. 

 It is not altogether successful. 



The first part deals with the ordinary theory of projectivities between 

 ranges of points on lines ; but it is not at all satisfactory. There seem, in 

 fact, to be only two reasonable methods of procedure, either to assume co- 

 ordinates frankly, as is done by Duporcq and Darboux, or else to examine 

 with some degree of logical completeness the axiomatic foundations of the 

 subject (see, for example, the treatment in Whitehead's Axioms of Projective 

 Geometry, or in the recent book by Baker : Principles of Geometry, vol. i. 

 Foundations). The " fundamental theorem," that a projective correspon- 

 dence between two lines is completely determined when the correspondents 

 of three distinct points of one line are determined on the other, requires for 

 its proof the introduction of relations of order among the points of a line 

 and the assumption of the Dedekind axiom (or else the assumption of Pappus 

 theorem). M. Malet proves it by vague considerations of continuity and 

 completeness. 



The introduction of complex elements too, though in a shadowy way 

 reminiscent of von Staudt (who is not mentioned anywhere in the book), 

 lacks precision and omits propositions which are logically necessary unless 

 algebra is used. 



Chapter II contains an interesting treatment of plane cubic curves 

 obtained as " courbes jumelaires " in the quadratic transformation. With 

 this definition the author is able to prove a number of residuation 

 theorems. 



In the next chapter we get on to algebraic curves in general and, as was 

 to be expected, on to very treacherous ground. On p. 128 we arrive at this 

 remarkable definition : Une courbe algebrique est une courbe telle que par k 

 points du plan il en passe une et une seule. This is, of course, nonsense. As it 

 stands it would mean that only one algebraic curve passes through k points 

 taken in the plane. That is clearly ridiculous, but can the definition be 

 satisfactorily amended ? In the first place, k must be of the form i^n{n-\- 3), 

 where n is a positive integer. The " definition " is evidently meant as a 

 generalisation of " two points detennine a line," which cannot be regarded 

 as defining a line, but merely as expressing one of the axioms of incidence 

 assumed between the undefined entities points and lines. But, even if we 

 interpret it as follows : " Just as two points determine a unique straight 

 line, so five points determine a unique ' conic,' nine points a unique ' cubic,' 

 and so on ; all curves so determined are called algebraic," it is clear that 

 this is no definition of algebraic curves. It is impossible to define conies 

 simply as the class of curves of which one and one only passes through five 

 points ; it is easy to construct classes of transcendental curves satisfying 

 this condition. 



However, the author somehow proceeds from this definition to deduce 

 general properties of algebraic curves, and then gets on to birational trans- 

 formations . There is a lot of useful and interesting stuff in these last chapters, 

 but it is difficult to be sure that any proposition has really been proved. The 

 diagrams are numerous and carefully drawn, but are often so complicated 

 that they do not greatly help in the elucidation of the text. 



It is interesting to notice what M. Malet, himself an engineer, is careful 

 to point out in his footnotes, namely that many of the French geometers, 

 Brianchon, Desargues, Poncelet, Peaucelher, were engineers. There is, 

 however, a regrettable absence of references to German works, 



F. P. W. 



