## H T H Piaggio's Treatise on Differential Equations

 In 1920 Henry Thomas Herbert Piaggio published An Elementary Treatise on Differential Equations and Their Applications. We give an extract from the Preface of this book and also extracts from reviews of the first edition and second edition of this work.

1. Preface.

"The Theory of Differential Equations," said Sophus Lie, "is the most important branch of modern mathematics." The subject may be considered to occupy a central position from which different lines of development extend in many directions. If we travel along the purely analytical path, we are soon led to discuss Infinite Series, Existence Theorems and the Theory of Functions. Another leads us to the Differential Geometry of Curves and Surfaces. Between the two lies the path first discovered by Lie, leading to continuous groups of transformation and their geometrical interpretation. Diverging in another direction, we are led to the study of mechanical and electrical vibrations of all kinds and the important phenomenon of resonance. Certain partial differential equations form the starting point for the study of the conduction of heat, the transmission of electric waves, and many other branches of physics. Physical Chemistry, with its law of mass-action, is largely concerned with certain differential equations.
The object of this book is to give an account of the central parts of the subject in as simple a form as possible, suitable for those with no previous knowledge of it, and yet at the same time to point out the different directions in which it may be developed. The greater part of the text and the examples in the body of it will be found very easy. The only previous knowledge assumed is that of the elements of the differential and integral calculus and a little coordinate geometry. The miscellaneous examples at the end of the various chapters are slightly harder. They contain several theorems of minor importance, with hints that should be sufficient to enable the student to solve them. They also contain geometrical and physical applications, but great care has been taken to state the questions m such a way that no knowledge of physics is required. For instance, one question asks for a solution of a certain partial differential equation in terms of certain constants and variables. This may be regarded as a piece of pure mathematics, but it is immediately followed by a note pointing out that the work refers to a well-known experiment in heat, and giving the physical meaning of the constants and variables concerned. Finally, at the end of the book is given a set of 115 examples of much greater difficulty, most of which are taken from university examination papers. .... An appendix gives suggestions for further reading. The number of examples, both worked and unworked, is very large, and the answers to the unworked ones are given at the end of the book.

2. First Edition (1920).

2.1. Review by: E H N.
Mathematical Gazette 10 (149) (1920), 184-195.

A book for undergraduates and their teachers. With a skill as admirable as it is rare, the author has appreciated in every part of the work the attainments and needs of the students for whom he writes, and the result is one of the best mathematical text-books in the language. The critic has nothing to do but to call attention to a few of the most valuable features and to one or two unlucky slips. It is good to find the first chapter to be a geometrical discussion, all too brief, of a kind which in England has been associated to their common loss rather with the lecture-room than with the text-book. In devoting an early chapter to some simple partial differential equations, Prof Piaggio has put teachers and students alike under a debt which the latter cannot realise. Is it credible that some of us became acquainted with the equations of wave motion and with their simpler solutions surreptitiously in treatises on sound, because in pure mathematics a partial differential equation of the second order, however simple, was expected to yield precedence to the twenty-four solutions of the hypergeometric equation and to an abstract discrimination between general integrals, complete primitives, and the like?

2.2. Review by: J E J.
The Monist 31 (4) (1921), 637.

There are many physicists and engineers today, who have often longed for a short and readable treatise on Differential Equations ; but who have found themselves repelled by the ponderous and academic volumes that have been produced hitherto upon the subject. Differential Equations, as it were, have up till now existed but for the mathematician, and have been jealously guarded by the exhaustive nature of the treatises published. Prof Piaggio has supplied this long-felt want. In his work we have, condensed into approximately two hundred pages, an account of Differential Equations, suitable for those with no previous knowledge of the subject, and yet amply sufficient to cover the course required for an Honours degree. One feature of the book which especially appeals to us is the superabundance of excellent examples, to which are added explanations of their physical significance. The engineer will be as delighted to find how soon he can understand the phenomenon of resonance (p. 36), as will the physicist to find himself (on p. 48) solving the differential equations used by Sir J J Thompson in a recent important contribution to the Philosophical Magazine. The miscellaneous examples at the end of the book with their interesting notes - as, for example, on the Zeeman effect, on Hamilton's dynamical equations, or on Einstein's equations of Planetary Motion - will be especially welcome. Graphical methods and Numerical Approximations form an important feature of the book and are quite up-to-date, including methods hitherto unpublished. In conclusion, we congratulate the author on producing a book which is concise without being obscure, and scholarly without being academic.

3. Second Edition (revised and enlarged) (1928).

3.1. Review by: T A A B.
Mathematical Gazette 14 (201) (1929), 465.

It is a pleasure to welcome the second edition of Professor Piaggio's admirable textbook. A revision of the text of the first edition has been made, and a new chapter, some forty pages long, on Miscellaneous Methods, has been added. The merits of the earlier edition have been so universally recognised that further praise would be superfluous; it is only necessary to consider the additional chapter. This is divided into six parts, the second of which is concerned with Riccati's Equation, a subject which was dismissed by a few exercises in the first edition. The remaining five parts are in the nature of addenda to preceding chapters and deal with
(1) the theory of singular solutions,
(3) Mayer's method for integrating Pdx + Qdy + Rdz = 0,
(4) solutions in series,
(5) the Equation of Wave Motion,
(6) numerical solutions.

3.2. Review by: W R Longley.
Bull. Amer. Math. Soc. 35 (2) (1929), 267.

Professor Piaggio's Differential Equations was first published in May, 1920, and was reprinted four times during the next six years. "The object of this book is to give an account of the central parts of the subject in as simple a form as possible, suitable for those with no previous knowledge of it, and yet at the same time to point out the different directions in which it may be developed." The only previous knowledge assumed is that of the differential and integral calculus. The style is admirably adapted to a text for beginners and the large number of examples with answers furnishes adequate drill material. The usual standard forms of ordinary and partial differential equations occupy the greater part of the book: one chapter is devoted to numerical approximations and another to existence theorems. In the revised edition more examples have been included and a new chapter "Miscellaneous Methods" has been added, dealing with a number of disconnected topics, which may be regarded as in the nature of supplementary reading. The author calls particular attention to some difficulties in the theory of singular solutions, for which he is indebted to some unpublished work by Mr H B Mitchell, formerly Professor at Columbia University.

JOC/EFR March 2014