1. Leçons sur la composition et les fonctions permutables (1924), by Vito Volterra and Joseph Pérès.
Science Progress in the Twentieth Century (1919-1933) 21 (84) (1927), 711-713.
... step by step, the theory is developed in analogy with ordinary analysis, introducing the inverse of composition, logarithms of composition, and differential and integral calculus of composition. Included is an investigation of the functions which are permutable with a given function; they can be constructed by transformations applied to functions of the difference (y - x). M Pérès has discovered those transformations which conserve composition, and by use thereof he is able to find new results and simplify proofs. A final chapter contains suggestive remarks upon the connection between permutable functions and the summation of divergent series.
1.2. Review by: Horace Bryon Heywood.
The Mathematical Gazette 18 (227) (1934), 61.
The book under review contains another chapter of the work which is being done by the Volterra school. We may infer from the circumstance that the introduction is written by Professor Volterra, who offers his thanks to Professor Peres, that the book was made by the latter. But Peres, while piously setting out the work of his maître, has himself made substantial contributions. ... First the problem of finding the complete set permutable with a given function and then that of finding the group of transformations for which the permutability of any such set is conserved, are solved completely and in a manner masterly in the extreme. It will be seen that the solving of an equation in which processes of composition replace those of algebra gives the solution of an integral equation of a more or less complex nature. Indeed the same may be said for any inverse process. Thus the method affords the solution of a wide range of functional equations, some of very subtle complexity. The suggestiveness of this work is even greater than the work itself. It will be noted that all that is said is confined to integration of the Volterra type and to permutable functions at that. Beyond these boundaries lies new country.
Revue de Métaphysique et de Morale 38 (2) (1931), 4-5.
Excellent accurate history of mathematics. The author first endeavours to draw up a summary account of the knowledge which was transmitted, especially by the Arabs, to the men of the fifteenth century. It indicates the differences of perspective and method which characterize Greek science as opposed to modern science. He emphasizes the fact that the Greek founders of geometry were the inventors of the mathematical proof, whereas with arithmetic they were satisfied with mere calculation checks: "the practical knowledge of the Egyptians, reduced to a few empirical rules of surveying, constituted a very humble point of departure for inquiries which resulted in the work of the Greco-Alexandrian geometers. Nowhere, perhaps, would it be more legitimate to speak of a Greek miracle." The following chapters deal successively with the progress made up to Newton exclusively; from the period of Newton to Euler, which represents the science of the eighteenth century, and then of the period from 1780 to 1860, which was that of the great French geometers: Lagrange, Laplace, Legendre, Gauchy, Galois, and which is also brilliantly illustrated by the German school with Gauss, Jacobi, Riemann, Weierstrass, the latter being connected, however, rather with the contemporary movement. In a final chapter, M Pérès points out the importance of the work carried out during the second half of the nineteenth century and in the first years of the present century: renewal of analysis, the theory of functions accelerated by study of the sets, the progress of the functional calculus, the renewal of geometry by the theory of the transformation groups, and so on.
2.2. Review by: Anon.
Revue Philosophique de la France et de l'Étranger 112 (1931), 310.
This volume offers a good summary of the history of the exact sciences from antiquity to the present day. Indeed, the author did not wish to attempt a restitution of the modern development of these sciences without first defining the initial conditions, and for this he drew up a summary, but very accurate, account of the knowledge transmitted to the scholars of the Renaissance by Greek thinkers, Arab mathematicians and their Western successors. By the choice the author has made to limit his study to the framework imposed by the very nature of the collection, he has been guided above all by his desire to highlight the great discoveries and to show the progress of the knowledge; and this attitude, which is well justified in such a work, has contributed to determine its divisions. It also obliged M Pérès to concentrate upon the most characteristic scholars; but it would have been preferable, nevertheless, to reject the biographical details in the notes, which in the text, though not very developed, sometimes impede the unity of certain statements. The last chapter, in which some features of the contemporary development of the mathematical sciences are indicated, is particularly interesting, because it presents personal views of the author, which dominates this very delicate question with great ease. We can only point out some very important remarks which we would like to emphasize. The significance, in particular, of the work of M Volterra, on which M Pérès draws the attention of the reader, invites us to go beyond the scientific point of view, to consider, in a new light, certain problems of philosophy of science, for example, without restriction, of philosophy.
2.3. Review by: L G.
Isis 15 (1) (1931), 188-190.
J Pérès, a professor of mechanics at the University of Marseille, succeeded well in making in 180 pages the essence of the historical development of mathematics, and of mechanics; he speaks only incidentally of astronomy, of physics (here the principle of conservation of energy is not even mentioned) and of geodesy. The first half of the work leads us to Descartes and Fermat; The second to the work of Borel, Volterra, etc. But the desire to be as complete as possible in so few pages creates a fatal imbalance as the pages follow one another, the reader finds himself increasingly confronted with an accumulation of names which cannot mean anything to him if he is not a mathematician (the book is not intended for mathematicians), or names of mathematical arguments which say nothing to his imagination.
2.4. Review by: S R.
Revue Archéologique (5) 31 (1930), 391.
Although we already possess many histories of mathematics and astronomy, the work of M Pérès is not superfluous, for, in brevity, it goes back very far and leads the exposition to the present day. The non-mathematician reader will naturally learn more words than concepts; we would have liked, in connection with Archimedes for example, to have more thorough details, especially on the feeling that to him is rightly attributed the methods of differential calculus.
Science, New Series 88 (2286) (1938), 380-381.
The volume before us is the first in a series of three, the three together to give a resume of something over fifty years of the work of Volterra. Although the complementary studies of others are given extensively and a systematic treatment is constructed, thus portraying the whole development of the subject, the exposition would nevertheless demand the attention of mathematicians of this generation, even if it limited itself to the inventions and discoveries of the author himself. Moreover, the author has the good fortune of having as collaborator Professor J Pérès, who has made many contributions to the subject of functionals and integral equations. This volume and the other two in prospect, as well as the recent treatise, "Opérations inflnitésimales linéaires," by Volterra and Hostinsky, form an amplification and modernization of the two volumes on functions of curves and integral equations published in the same Borel series on the theory of functions, some twenty-five years ago.
L'Enseignement Mathématique 36 (1937), 127-128.
This is a new and remarkable work, at least in two respects. First, it is the theoretical course on fluids that now exists in most French universities, a course founded by the Ministry of Air for aviation and which corresponds to the Certificate of Advanced Studies in Fluid Mechanics but which, to my knowledge, had never before been written up and published. This course, which one would have thought very specialised and intended especially for aircraft manufacturers, is really an elementary exposition of hydrodynamics and aerodynamics with elegant borrowings from analytical functions as well as the simplest vector and vortex concepts. It is, in short, the problem of the fluid flow around a fixed obstacle, which in turn causes the obstacle to move, become an airplane, into an indefinite aerial mass, originally at rest. The paradox of D'Alembert and others, studied in particular by M Villat, do not prevent the plane from flying. What a wonderful incentive not to consider the science in question as blocked by paradoxes.
Étude Structurale d'une Aile D'avion Leger Essai en Flexion Statique d'un Longeron de Luciole Mc30, Université Paris-Sud 11.
The tables established by Joseph Pérès and Lucien Malavard make it possible to determine the distribution of lift by rheoelectric analogy (physical simulation of electric and aerodynamic phenomena by analogy). These tables are established for different wing families: elongation, taper, plane shape, twisting/rigging, steering control. They make it possible to take account of the deflections of flaps and/or ailerons etc. ... Very precise, they are however tedious to use.
The Mathematical Gazette 38 (326) (1954), 307-308.
The uniform excellence of French treatises on analytical dynamics if traditional, going back at least as far as Lagrange; at the present day, there are many topics on which no author need be ashamed of copying the master strokes of Appell. There is still room, however, for experiment in the presentation of principles, in the light of the needs of particular classes of students. Professor Pérès expects his readers to have had a sound first course in dynamics; for them he provides a bridge to the domain of advanced specialised studies. ... Although much of the material is classical, there are many points of novelty and interest in the exposition; the close examination of some details often dismissed very hastily in our own text-books can be recommended to those who lecture on dynamics in our universities, while it may be added that they will not find a single rod of mass m and length 2a. Where precisely this volume fits into the great French hierarchy of such books it may presumptuous to discuss; naturally it does not pretend to rival the massive comprehensiveness of Appell, but it digs more deeply into principles than Vallee Poussin's brilliant and readable two volumes of 'Mécanique analytique'.