866 



SCIENCE. 



[N. S. Vol. XXII. No. 574. 



grade appears in America. Its standards of 

 rigor, its terminologies, even its styles of 

 typography, are destined to leav^ a relatively 

 lasting influence upon future treatises of the 

 same class. Such an influential book is 

 Bolza's ' Lectures on the Calculus of Varia- 

 tions.' It will appear from what follows that 

 this influence is to be great in many a subject 

 besides the ' Calculus of Variations,' and that 

 the quality of its influence is excellent in 

 practically every particular. 



The calculus of variations is one of the very 

 first of the old (or formal) developments of 

 the infinitesimal calculus; it is one of the 

 newest conquests of the modern (or critical) 

 school. The history of the older calculus of 

 variations is almost trite from reiteration; 

 to select among many, the works of IMoigno- 

 Lindelof, Diegner, Todhunter, Caroll, Jordan 

 and (more recently and more perfectly) Pas- 

 cal, have made known the achievements of 

 the old school from Newton to Jacobi to 

 mathematicians throughout the world. The 

 modern theory — the critical revision of the 

 older theory — has appeared so recently that 

 its existence is still unknown to many scien- 

 tific men. Since the book under discussion is 

 practically the first exponent of this newer 

 work which is reasonably easy to follow, it 

 seems fitting to state here the characteristic 

 differences between the older theory and the 

 new. 



The essential problem may be stated here as 

 follows : Given an integral of the form 



where 





y = -^ 



dx 



that integral will take on a fixed numerical 

 value whenever a curve G is given by an equa- 

 tion of the form y = <j>(x). It is proposed 

 to find that curve C for which this numerical 

 value is at a minimum (or maximum). The 

 finesse of the whole modern theory, and the 

 exquisite exactness of the present treatment, 

 lies in so refining this crude statement that 

 an accurate solution of the problem becomes 

 possible. Thus the first question which arises 



is the following: Taking a supposed solution 

 y=^<^{x), are we to presume that the cor- 

 responding integral value is less than (or 

 greater than) the integral value which corre- 

 sponds to absolutely any other curve? — or to 

 those which lie reasonably near to the sup- 

 posed solution? — or to those which lie reason- 

 ably near and also have reasonably small dif- 

 ference in inclination ? Are the ' comparison ' 

 curves to be reasonably continuous and have 

 a respectable number of derivatives, or are we 

 to admit all kinds of outlandish curves to our 

 considerations — curves with corners — curves 

 with no tangents — curves with no radius of 

 curvature — curves which misbehave themselves 

 in the variety of ways which are now known 

 to be possible? If not, then in what sense 

 may we say that the supposed solution really 

 gives the integral a smaller value than any 

 other curve near it? These questions are 

 fundamentally important; their consideration 

 gives to the modern theory an exactness which 

 the older theory lacked completely; it is this 

 attitude which renders Bolza's book a modern 

 work. 



The results of the modern theory are very 

 satisfying in their evident accuracy, and they 

 are none the less satisfactory in the simplicity 

 of the conclusions reached. From a practical 

 standpoint, it may be said that any curve 

 which is a true solution of the problem must 

 satisfy the very same conditions which were 

 derived in the older books', viz., (1) the condi- 

 tion discovered by Euler (sometimes known as 

 Lagrange's condition), (2) that discovered by 

 Legendre, (3) that discovered by Jacobi. In 

 so far there is no change, save for a greater 

 accuracy in statement and a very material 

 advance in the exactness of the proofs. These 

 considerations practically exhaust the first two 

 chapters of the book, and their presentation 

 here can not fail to arouse lively enthusiasm 

 in the intelligent reader on account of irresist- 

 ible force of the precise reasoning and the 

 consequent indisputable truth of the conclu- 

 sions. 



A curve may satisfy the preceding condi- 

 tions, however, and still fail to render the in- 

 tegral a minimum (or maximum). This fact 

 was discovered by Weierstrass, who succeeded 



