REVIEWS 323 



method of distinguishing maxima from minima (p. 4). When Lagrange came to 

 consider the maxima and minima of functions of more than one variable, he made 

 an error in which he was followed by all writers until Peano, in his Calcolo of 1884, 

 founded on the lectures of Gennochi and translated into German in 1899, pointed 

 out this error (pp. iv-v, 33). The objection contained in Peano's work showed 

 that the entire former theory of maxima and minima needed a thorough renovation, 

 and in the main this work was the original source of the theories developed by 

 Scheeffer, Stolz, von Dantscher, and others, of which the volume under review 

 gives a detailed and excellent account. It may be noticed that Peano drew 

 attention to the mistaken criterion given by Serret and his followers ; the author 

 shows that this criterion goes back to Lagrange (pp. v, 33). 



The present volume arose from an outline of the theory of maxima and minima 

 founded on lectures of Weierstrass which was published by Prof. Hancock in 

 1903. It treated the cases where the functions are everywhere regular and where 

 the algebraic forms are either definite or indefinite. The fuller treatment of the 

 cases where only one-sided differentiation enters led ultimately to the exposition 

 given here (p. iv). The chapters are devoted to functions of one variable, 

 functions of several variables, functions of two variables, the theories of Scheeffer • 

 and others, functions of three variables, maxima and minima of functions of 

 several variables that are subjected to no subsidiary conditions, theory of maxima 

 and minima of functions of several variables that are subjected to subsidiary 

 conditions, relative maxima or minima, special cases, certain fundamental con- 

 ceptions in the theory of analytic functions. 



There seems to be no reason for regarding Weierstrass's methods as founded 

 on the work of Bolzano (cf. p. 12). In the interesting discussion of Gauss's 

 mechanical principle as an application of the method of maxima and minima 

 (pp. 1 5 1-3), the reason for the fact that the accelerations, and not the positions or 

 velocities, are to be varied might have been given in detail : it is rather puzzling 

 to a student — as it was to Lipschitz. This is a very valuable monograph. 



Philip E. B. Jourdain. 



Les Sciences Mathematiques en France depnis un deini-siecle. By Emile 

 Picard. [Pp. vi + 25.] (Paris : Gauthier-Villars et Cie., 1917. Price 

 2 francs.) 

 THIS short and well-written account of the progress in pure mathematics made in 

 France from the time in the middle of the nineteenth century when Fourier, 

 Cauchy, and Galois had opened up fruitful paths which led, to a great extent, to 

 the views of the country of mathematics obtained in the first few years of the 

 twentieth century, contains chapters on the theory of analytic functions, dif- 

 ferential equations, theory of numbers, algebra and geometry, and the theories of 

 functions of real variables and aggregates. Although the account is, of course, 

 largely of the nature of a catalogue of names, the names are chosen, with the skill 

 one would expect from the eminent author, so as to give a really good idea, to one 

 who knows something of the work of those men, of the essential features of the 

 progress of mathematics in France. Perhaps the dismissal of the paradoxes of 

 the theory of aggregates as "having made floods of ink to flow," the com- 

 parison of these difficulties with those which gave rise to the quarrels of the 

 schoolmen (pp. 20-21), and the complimenting of French mathematicians on their 

 limitation of view to certain parts of mathematics and their avoidance of " excessive 

 symbolisms " (pp. 23-4) seem rather superficial. 



Philip E. B. Jourdain. 



