#### by I. A. Barnett

This review might very properly begin with the quotation of the first paragraph of the author’s preface which explains the origin and purpose of these monographs. “This book is the first of a series of monographs on mathematical subjects which are to be published under the auspices of the Mathematical Association of America and whose publication has been made possible by a very generous gift to the Association by Mrs. Mary Hegeler Carus as trustee of the Edward C. Hegeler Trust Fund. The purpose of the monographs is to make the essential features of various mathematical theories accessible and attractive to as many persons as possible who have an interest in mathematics but who may not be specialists in the particular theory presented, a purpose which Mrs. Carus has very appropriately described to be the ‘diffusion of mathematical and formal thought as contributory to exact knowledge and clear thinking not only for mathematicians and teachers of mathematics but also for other scientists and the public at large.’ ”

The principal aim of this review is to give an appraisal of this work as to the extent to which it has fulfilled the purposes of these monographs, leaving to others to give a detailed description and criticism of the subjects discussed. I might say at the outset that the publication committee could have made no wiser choice for the subject of the first monograph than that of the calculus of variations. The beautiful geometrical and mechanical properties of curves and surfaces which unfold themselves so naturally in this subject are sure to arrest the attention of the reader who is even only mildly interested in mathematics. Coupled with this inherent beauty of the subject, we have here an exposition by a man whose ability both as a teacher and scholar is of such outstanding character as to insure the success of any mathematical work, let alone a subject with which he has such an intimate acquaintance and in which he has such a deep interest. Those of us who have always regarded mathematics merely as a science find in these pages an artistic structure and a beauty of technique that lead us to think that mathematics in this country is at last taking on the form of an art also.

The book opens with an introductory chapter giving a statement and historical sketch of some of the typical problems of the calculus of variations. This chapter is so clear and vivid that it must surely arouse the interest and curiosity of the reader. For those who have no acquaintance with this subject it may not be amiss at this place to give a brief statement of one of the most interesting of these problems, viz., the brachistochrone problem which was first proposed by John Bernoulli in 1696. For some reason a rivalry had sprung up between him and his older brother James and this problem was proposed as a sort of a challenge. It is required to pick out of all the possible paths joining two points in a vertical plane that one down which a particle will fall in the shortest time. It is clear that this problem is one which is quite distinct from the maxima and minima problems solved in the differential calculus. In fact we have here to find not merely a value, or values, of a variable which will give an assigned function the largest or smallest value, but a whole curve or function which, when substituted in the integrand of a certain definite integral, will give the latter a value smaller than would be obtained by the substitution of any other function. The brothers both solved the problem in 1697; and they found much to their surprise that the curve down which a particle will fall in the least time is the same curve which will be generated by a particle on the rim of a wheel as the wheel rolls along a straight line, and more surprising still it is the same curve on which a particle starting at rest will fall to the lowest point in the same time no matter at what point of the curve the particle is started. In other words, a pendulum constrained to move along a cycloid, as this curve is called, would have a period independent of the arc of swing, a property not enjoyed by the so-called simple pendulum. We can feel with John Bernoulli when he says (see page 54), “With justice we admire Huygens because he first discovered that a heavy particle falls on a cycloid in the same time always, no matter what the starting point may be. But you will be petrified with astonishment when I say. that exactly this same cycloid, the tautochrone of Huygens, is the brachistochrone which we are seeking.”

In the next three chapters this and two other problems are taken up in great detail with a view to explaining and illustrating the modern methods of the calculus of variations. These modern methods of studying this subject are here presented for the first time in a form which it is possible for the intelligent layman with a thorough training in advanced calculus to understand. I do not mean to imply that the reader will always find it easy to follow every statement at a first glance. In fact I doubt if any one but a specialist could read everything in the book that way. But I am sure that it is possible for a reader with a good training in elementary mathematics to verify every statement made in this book, provided he has sufficient patience and concentration. If we recall that mathematicians had been working on this subject for over 300 years before it was finally put on a rigorous foundation by Weierstrass, we can see why a beginner should not expect to understand the calculus of variations in all its aspects at a first reading. In fact, I would suggest to the beginner to omit on a first reading all the difficult proofs in the book and come back to them only after he has seen the underlying general ideas of the subject. This, by the way, is I think the best and most efficient method of reading all mathematics, even technical articles. But, in spite of the inherent difficulty of the subject, Professor Bliss has given us an exposition which I think will be considered a model for many years to come. Whenever he needs to use some result from an advanced portion of mathematics, as for example the theory of implicit functions, he either substitutes for it a clear geometric proof (see for example the proof that a cycloid can always be drawn through two points, page 56) or he gives a proof using nothing more than what one should know after a thorough course in calculus.

The book ends with a chapter on the general theory and we have to marvel that any one can in the space of fifty pages give such a clear and complete treatment of what is known as the simplest problem of the calculus of variations. Yet we actually find that the necessary conditions as well as the sufficient conditions for a minimum or maximum are stated and proved with all the rigor that one could desire. This chapter should be excellent reading for one who has already had some acquaintance with the calculus of variations, and even the specialist will find here a form which is not lacking in novelty. The chapter closes with an interesting historical account of the whole subject. It may be seen from this brief description that this is an interesting and useful book for a large group of readers whose mathematical equipment varies all the way from a course in the differential and integral calculus to one in the calculus of variations. In fact; I should think it would make an excellent introduction for a course in this subject if not as a text, at least for collateral reading.