## Reviews of W V D Hodge's books

Below we present short extracts of reviews from W V D Hodge's two books. We say two books, but one of these, namely Methods of algebraic geometry written with Daniel Pedoe, is a three volume work which we list as three separate publications. A second edition of

*Harmonic integrals*was published in 1952 and reprinted in 1989. Volumes 1 and 2 of*Methods of algebraic geometry*were republished as paperbacks in 1969 and all three volumes were reprinted in 1994.**1. The theory and applications of harmonic integrals (1941), by W V D Hodge.**

**1.1. Review by: P Fraser.**

*The Mathematical Gazette*

**25**(267) (1941), 320-323.

It is about a century since Riemann wrote on algebraic integrals, founding his method on the theory of potential or harmonic functions of two real variables. It must have occurred to every student in this field since his day that an analogous method for many variables would yield fruitful results, but, although many wide extensions have been made, Professor Hodge has been the first to give substantial meaning to harmonic functions in general and to apply them to the study of integrals of complex variables on an algebraic variety. This is therefore a remarkable book and much may be expected from the development of the devices which it puts in the hands of future students. The exposition is lucid, straightforward and free from pedantry, a freedom which is especially noticeable in the choice of notation. The terseness of the style does not make the book more difficult to read, but a more discursive paragraph occasionally might have led readers to a higher appreciation of the results.

**1.2. Review by: Dirk J Struik.**

*Bull. Amer. Math. Soc.*

**50**(1) (1944), 43-45.

This is one of those books which everyone who specializes in a particular branch of group theory, of the theory of algebraic surfaces, of the theory of Riemann surfaces, of topology or of the tensor analysis should consult. It shows how all these different fields are connected, and not connected in some superficial way or in the form of an analogy, but in an essential manner, so that interesting and profound theorems in one field cannot be understood without a thorough knowledge of other fields. In reading this book one is reminded of books like Klein's "Ikosaeder," which is also a blend of several important fields. The task of the reviewer of such a book is hard, because he has seldom the enviable mastery of the different branches of mathematics which the author possesses. At the same time he must praise the author for the beautiful exposition of so many and different fields.

**1.3. Review by: R J Walker.**

*National Mathematics Magazine*

**16**(8) (1942), 417-418.

The ideas developed in this stimulating book are essentially an outgrowth of the theory of Abelian integrals on algebraic curves. This theory, which proved to be extremely useful in the investigation of birationally invariant properties of curves, was soon generalized to algebraic varieties of higher dimensions. Many of the problems arising in this generalization have not yet been solved, and it was while working on some of these that Hodge developed his more general treatment of integrals. A harmonic integral, or, rather, the associated harmonic tensor, is a generalization of the familiar harmonic function of two- or three-dimensional Euclidean space. The general harmonic integral is defined over a much more general type of space known as a Riemannian manifold, and the first chapter of Hodge's book is devoted to a concise account of the topology and differential geometry of these spaces. This treatment is self-contained, and provides an excellent review for a person who has some acquaintance with these subjects. A reader totally unfamiliar with them would probably have some difficulty in supplying all the details of the arguments. ... The ideas and the techniques developed here are already beginning to influence the work of other mathematicians, and it seems very likely that they will have a significant effect on the future development of topology and differential geometry, especially in the fields where these subjects overlap. Hence for a person interested in modern developments in geometry this book is practically required reading.

**1.4. Review by: Oscar Zariski.**

*Science, New Series*

**95**(2457) (1942), 124-125.

The author of this monograph is one of the outstanding geometers in England, and in his special field - algebraic geometry - he is known, above all, for his important contributions to the theory of algebraic integrals attached to an algebraic variety (abelian integrals). This theory, inaugurated by the classical investigations of Abel and Riemann on the integrals attached to an algebraic curve (1826, 1857) has been ever since the object of intensive study by analysts and geometers alike (Poincaré, Picard, Castelnuovo, Enriques, Severi, etc.) who were attracted to it either by its profound analytical content or by its power as a tool for the discovery of geometric properties of algebraic varieties. The theory gives rise to many difficult questions, and the difficulties increase with the dimension of the variety .... The mathematical reader will find in Hodge's monograph a very well-written and highly stimulating account of a young and active mathematical theory. He will find ample evidence of the diversity of methods which we have mentioned above. As an ingenious blend of algebraic geometry, analysis, topology, tensor calculus and differential geometry, the monograph should prove of considerable interest to specialists in these various fields.

**1.5. Review by: Oscar Zariski.**

*Mathematical Reviews*, MR0003947

**(2,296d)**.

The theory of harmonic integrals on a Riemannian manifold was developed by Hodge in the course of his investigations on the integrals attached to an algebraic variety. It is possible indeed to put a finger on the exact spot of the transcendental theory of algebraic varieties which, in last analysis, is responsible for the appearance of this stimulating book: it is the question of whether or not there exist double integrals on an algebraic surface which are everywhere finite and which have all their periods equal to zero. This question remained unsolved for a long time until Hodge proved that such integrals do not exist (subsequently it has been recognized that a similar theorem holds, more generally, for integrals of closed forms on a topological manifold, as a corollary of results obtained previously by de Rham). However, the new theory, although an outgrowth of specific problems in algebraic geometry, goes far beyond algebraic geometry proper. ... Since the original investigations of Hodge are scattered in many notes in which the topic is at times presented in a somewhat condensed fashion, the present book will be welcome by those who wish to gain a better understanding of the central points of the theory of harmonic integrals. The book is exceedingly well written, there is great emphasis on rigour, and the original material is presented anew in a systematic and unified form.

**Review of 2nd edition by: George Seligman.**

*American Scientist*

**79**(3) (1991), 281.

This is a reissue of the 1941 classic. The extent of its influence on modem algebraic and differential geometry, and on new geometrical aspects of physics, is sketched in a 1988 foreword by Michael Atiyah.

**2. Methods of algebraic geometry. Vol. I (1947), by W V D Hodge and D Pedoe.**

**2.1. Review by: Arthur B Coble.**

*Science, New Series*

**107**(2785) (1948), 511-512.

The volume before us, which, it is announced, will be followed shortly by a second volume devoted to the theory of algebraic varieties and to the study of certain loci which arise in many geometric problems, is divided into two books. Book I is devoted to Algebraic Preliminaries, and Book II, to Projective Space. As the title implies, no attempt has been made to build up a body of geometric theorems. Though the projective group is necessarily fundamental, no discussion of its invariants is given except as these may appear incidentally in reductions to canonical forms. The polar operator is mentioned, but its invariance is not stressed. Yet the necessarily restricted choice of material is excellent, and the volume is a very welcome addition to the literature in this field.

**2.2. Review by: H S M Coxeter.**

*Bull. Amer. Math. Soc.*

**55**(3, Part 1) (1949), 315-316.

This work by two disciples of H F Baker naturally retains some of the flavour of the latter's

*Principles of geometry;*but in keeping with the modern trend it is more algebraic and less geometrical. The spirit of the book is indicated by the fact that there is no mention of order or continuity. The first four of the nine chapters are concerned with algebraic preliminaries, chiefly in preparation for vol. II, and are so clear and concise that they would serve very well as an introduction to modern algebra, quite apart from their application to geometry. The topics treated in this part include groups, rings, integral domains, fields, matrices, determinants, algebraic extensions, and resultant forms. The theory of linear dependence is developed without assuming commutativity of multiplication, and there is a neat algebraic treatment of partial derivatives and Jacobians.

**2.3. Review by: J H C Whitehead.**

*The Mathematical Gazette*

**32**(300) (1948), 213-214.

This first volume is subdivided into "Book I", which is on pure algebra, and "Book II", on n-dimensional projective geometry. Book I opens with a general account of groups, rings and fields. It includes a discussion of integral domains and of polynomial rings. The term field includes non-commutative fields, or division rings as they are often called. Chapter II is on linear dependence. It gives a good account of linear sets over fields, which may be non-commutative. ... Chapter III deals with algebraic extensions of a commutative field, K, and with algebraic function fields. ... The fourth and final chapter in Book I deals with algebraic equations and polynomial ideals. ... Book II opens with an algebraic definition of projective space. ... In the next chapter n-dimensional projective space, S_n, is re-defined, starting with the axioms of incidence, together with Desargues' Theorem as a separate axiom if n = 2. ... these two chapters give what is, so far as I know, the most compact account of the foundations of projective geometry, including the logical equivalence of the algebraic and synthetic definitions and, in particular, of the fact that Pappus' Theorem is equivalent to multiplication being commutative. The next chapter gives an account of Grassmann coordinates ... The two final chapters are concerned with collineations and correlations, which are discussed in terms of their elementary divisors.

**2.4. Review by: Oscar Zariski.**

*Mathematical Reviews*, MR0028055

**(10,396b)**.

The material covered in this book contains the essential prerequisites for a first course in algebraic geometry, namely certain well-defined topics in modern algebra and the general theory of projective spaces. Accordingly this volume is divided into two parts, entitled respectively "Algebraic preliminaries" and "Projective space." The book is presented by the authors as the first part of a treatise on algebraic geometry and is intended to clear the way for a forthcoming second volume which will be devoted "to algebraic varieties and to the study of certain loci which arise in many geometric problems."

**2.5. Review of 1969 reprint by: W L Edge.**

*The Mathematical Gazette*

**54**(388) (1970), 184.

The republication of these two familiar volumes deserves a warm welcome. Their appearance in paperbacks testifies to their dissemination and assures us that their merits are widely recognised. ... The covers of each volume display the figure portraying the uniqueness of the harmonic conjugate, and thereby direct attention to the most geometrical chapter. Faultless and clear though the argumentation is throughout both volumes, readers will specially remember the care and skill which piloted them through this chapter.

**3. Methods of algebraic geometry. Vol. II (1952), by W V D Hodge and D Pedoe.**

**3.1. Review by: H S M Coxeter.**

*Bull. Amer. Math. Soc.*

**58**(6) (1952), 678-679.

The book closes with a few pages of Bibliographical Notes, giving due credit to Cayley, Castelnuovo, Enriques, Severi, Macaulay, Lefschetz, van der Waerden, Chevalley, Zariski, Weil and others. The authors have skilfully blended the work of these illustrious men with many original ideas of their own. Their generality of outlook inevitably makes the book somewhat difficult to read, and there are no diagrams. However, an excellent index enables the reader to find much of interest without going through all the details.

**3.2. Review by: E C Thompson.**

*The Mathematical Gazette*

**37**(319) (1953), 61-62.

The considerable developments in algebraic geometry in the last twenty-five years have not yet percolated through to the textbooks of the subject. Indeed, the assimilation of this new material is not yet by any means complete, and the attempt to write a truly modern textbook on algebraic geometry requires a certain degree of courage as well as mastery of the subject. The authors of the book under review deserve our admiration for making the venture, no less than our congratulations on the success with which they accomplish it. This second volume of the series enables one to see more clearly the plan of the whole work, although a final appreciation must wait until the appearance of the third and final volume. Volume I is now seen to be mainly preparatory, and some of the material such as, for instance, Grassman spaces, which appeared to have undue prominence in that volume, now falls into perspective. The work naturally owes much to other writers, and in this volume the influence of van der Waerden is particularly apparent. The reader will also notice the affinity with A. Weil's Foundations of Algebraic Geometry. Those interested in the sources can have no better guide than the authors' own Bibliographical Notes. The book is however in no sense a summary of, or even an introduction to, recent original work. The authors' avowed intention is an exposition of method, and not an account of results. Perhaps it would be incorrect to describe the approach as new, but a systematic development of the foundations of the subject on these lines has not before been given in a form accessible to the general reader.

**3.3. Review by: Oscar Zariski.**

*Mathematical Reviews*, MR0048065

**(13,972c)**.

While in the first volume of this treatise the authors dealt with algebraic preliminaries and projective spaces, in this second volume they initiate a systematic exposition of algebraic geometry proper. The first three chapters (Book III, chapters X-XII) develop respectively the general concept of an algebraic variety, the theory of algebraic correspondences and intersection theory, while the remaining two chapters (Book IV, chapters XIII and XIV) deal respectively with the theory of quadrics and Grassmann varieties and serve also the purpose of illustrating the general methods developed in the preceding chapters.

**4. Methods of algebraic geometry. Vol. III. Birational geometry (1954), W V D Hodge and D Pedoe.**

**4.1. Review by: Pierre Samuel.**

*Bull. Amer. Math. Soc.*

**61**(3, Part 1) (1955), 254-257.

At a time when abstract algebraic geometry is becoming more and more popular among mathematicians, many of us are confronted with the problem of how to learn (or teach !) it. Some good books are now available, and, for the time being, no two of them have the same purpose. ... With Hodge and Pedoe's

*Methods*we now see a quite different type of book. In its three volumes we find a leisurely description of an impressively extensive part of algebraic geometry, and of the algebraic methods which are used nowadays. Motivations are given. Examples of significant and useful varieties are numerous. All the algebra which is needed is given, and, what is more, these books tell how to translate geometry into algebra, and conversely. ... the authors should be commended for having given the facts, many useful facts, in a straightforward way. They should also be commended for having successfully steered a course which is equally remote from bashfulness about using algebra and from over sophistication in its use. In writing their books they have rendered an invaluable service to the mathematical community.

**4.2. Review by: Oscar Zariski.**

*Mathematical Reviews*, MR0061846

**(15,893e)**.

In this third volume of their treatise on algebraic geometry the authors propose - and we quote from the preface - "to provide an account of the modern algebraic methods available for the investigation of the birational geometry of algebraic varieties". Thus, while in the preceding two volumes of this treatise the algebraic methods used belonged primarily to linear algebra, elimination theory and field theory, in the present volume the authors are chiefly concerned with some relatively advanced questions in birational geometry the solution of which requires a generous dose of ideal theory and valuation theory. The treatment of these questions is taken largely from the work of the reviewer, and this is the first time that such material, scattered in many original memoirs, and of recent vintage, has been gathered together and presented in book form. The result is a very readable and self-contained account of a modern phase of algebraic geometry.