1.1. Note.

Information about the second edition published in 1953 is given below.

2.1. Review by: Nathan Altshiller-Court.
The American Mathematical Monthly 39 (10) (1932), 597.

In the first three chapters, about a half of the book, the author summarises the basic ideas on geometry that have become current since the beginning of the present century: the relation of geometry to the notion of a group of transformations, the various interpretations of the non-Euclidean geometries and their interrelation, etc. Both the analytic and the synthetic approaches are used. The presentation is clear, orderly, and comprehensive, considering the space.

The last two chapters are devoted to algebraic geometry and the geometry on an algebraic variety. Little has been done to make the fundamental ideas of these subjects accessible to wider mathematical circles, and these contributions are quite welcome. The author thought it preferable to omit all bibliographical references from the text, a procedure which has its disadvantages. He compensates for the omission by a bibliography at the end of the book, in which he includes several American titles. The book may be read with profit by any student who is familiar with projective geometry.

3.1. Review by: Joshua Irving Tracey.
Bulletin of the American Mathematical Society 40 (1934), 520.

This volume is the outgrowth of the course of lectures on projective geometry given by the author at the University of Liege.

In the opening chapter the primary propositions regarding points, lines, and planes are given, also the fundamental forms and operations, postulates on the ordering of elements, and the principle of duality. From these the author develops in an extremely logical manner the fundamental theorems of projective geometry. Chapters II to VI inclusive deal with perspective figures, harmonic pairs, Dedekind's postulate of the continuum, and theorems on projectivities and involutions. The conies are taken up in the next two chapters; these include the construction and classification of the conies and the important theorems of Pascal, Brianchon, and Desargues. The remaining chapters, IX to XII inclusive, treat in a similar manner the projective properties of planes, of quadrics and their classification, the space cubic curve, collineations and correlations of space, and null systems.

The synthetic method is used throughout the text. The author however is careful to give the analytic representation of each type of projection considered, for which a knowledge of projective homogeneous coordinates is assumed. The text contains a very few figures, and a limited number of bibliographical references. A brief historical sketch of the development of projective geometry is included in the introduction.

The logical development and clear presentation of the subject is evidence of the care exercised in the preparation of the text. The appearance of the book is attractive, and it is unusually free of errors. It will be a valuable reference for students in projective geometry, and may help to revive interest in a subject which has been neglected in recent years.

4.1. Review by: Hans Hornich.
Monatshefte für Mathematik und Physik 41 (1934), A18.

The author draws attention to some unsolved problems relating to the algebraic manifolds of three dimensions, in particular to how such a manifold can be rationally represented by three parameters.

4.2. Review by: Virgil Snyder.
Bulletin of the American Mathematical Society 40 (1934), 519.

This little pamphlet contains an accurate and well written account of the present state of three famous unsolved problems:

Involutions in space.

Conditions for rationality of a three-dimensional variety.

Demonstration of the irrationality of the general cubic variety in four-way
space.

The bibliography contains titles of 12 books and of 71 recent articles.

5.1. Review by: Virgil Snyder.
Bulletin of the American Mathematical Society 41 (1935), 170.

The most important invariant of an algebraic curve is its genus. A necessary and sufficient condition that a non-composite curve shall be rational is that its genus be zero. An algebraic surface has an infinite number of genera, each a direct and natural analogue of the genus of a curve. These are the arithmetic genus $p_{a}$, the geometric genus $p_{g}$, and the series of multigenera $P_{i}$. It was shown by Castelnuovo, immediately after Enriques had discovered the invariants $P_{i}$, that a necessary and sufficient condition for rationality is $p_{a} = 0, P_{2} = 0$. Later, each of these scholars actually constructed irrational surfaces having $p_{a} = 0, p_{g} = 0, P_{2} ≠ 0$.

The present pamphlet clearly states the problem, shows that these illustrations are birationally distinct, gives the further case found by Campedelli, $P_{2} = 2$, and adds a new one for which $P_{2} = 2, P_{3} = 4$. The method of proof is entirely algebraic, using only the well known theorems concerning grade, genus, and laws of combination for linear systems of curves on an algebraic surface. An excellent introduction to this memoir is the paper by Roberta F Johnson, Involutions of order two associated with the surfaces of genera $p_{a} = p_{g} = 0, P_{2} = 1, P_{3} = 0$ (American Journal of Mathematics 56 (1934), 199-213).

6.1. From the Introduction.

A surface can be considered either as the locus of its points, or as the locus of its tangent planes, or finally as the locus of its tangents. In this last form, it constitutes a complex of lines. In a series of notes published since 1927, we have systematically used the complex of tangents to a surface to obtain properties of it. Our starting point is on the one hand the profound research of M Demoulin on the Lie quadric, on the other hand a theorem obtained by MM Bompiani and Tzitzeica, namely that the tangents at a point to the asymptotics of a surface correspond in a Laplace transformation. A kind invitation from M Hadamard allowed us to present our research in the session of 6 December 1932 of his Seminar at the Collège de France; it is the development of this presentation which is the subject of this work.

It was Darboux who, in his masterly Lectures on the Theory of Surfaces, was the first to highlight the projective character of the Laplace transformation. Based on this transformation, our research therefore belongs to differential projective geometry. The reader will find the properties of the Laplace sequences that we use either in Darboux's Lectures or in M Tzitzeica's Geometry of Networks.

We have also used the properties of ruled space and the representation of this space on a hyperquadric belonging to a five-dimensional linear space. All the necessary information can be found in the fine work that G Koenigs has devoted to these questions. It is fair to add here that, with his memoir On the infinitesimal properties of ruled space, G Koenigs has done pioneering work in Differential Projective Geometry.

At the end of this work, we have given the list of the papers in which the surfaces are considered as loci of their tangents; we have attached to this list the titles of certain papers where this concept is not used, but whose results are necessary for our presentation. Let us point out that the Introduction to Differential Projective Geometry by MM Fubini and Cech, contains a very complete list, stopped at 1929, of the publications relating to Differential Projective Geometry.

We still have to say a few words about our notation. We have represented by $f^{ik}$ the partial derivative of a function $f(u, v)$ taken $i$ times with respect to $u$ and $k$ times with respect to $v$. We indicate by $f^{ik}$ a point whose homogeneous projective coordinates are the derivatives, according to the preceding notation, of those of a point $x$ depending on $u, v$. These notations were used more than twenty years ago by G Segre; they do not conform to those of the Absolute Differential Calculus which, since then, has undergone considerable development. We ask, however, the reader's permission to use them, no confusion being possible.

6.2. Review by: Virgil Snyder.
Bulletin of the American Mathematical Society 41 (1935), 14.

In this little pamphlet the author has collected and blended those papers in projective differential geometry which regard the line as the space element. At its end is a bibliography consisting of four treatises and eighty memoirs, of which thirty-eight are from the pen of the author.

The starting point is that of the four solutions of two partial differential equations of the second order in terms of two variable parameters $u, v$ as developed by Wilczynski. These are interpreted as homogeneous point coordinates $x$ in $S_{3}$ which generate a surface $(x)$.

The tangents to the $u$ and $v$ curves are interpreted as point coordinates of a quadric primal $Q$ in $S_{5}$. The line joining these points $u, v$ lies on the primal. It is the image of the pencil of tangent lines to the surface $(x)$ at $x$. The points $u, v$ are consecutive in a series of Laplace (Bompiani) which in the general case is unlimited in both directions. It is self polar as to the primal $Q$. To this series corresponds in $S_{3}$ a series of quadrics having the property that any two consecutive ones touch in four points.

Associated with each point $x$ is a Lie quadric $\Phi$, which belongs to the series of quadrics just mentioned. As $u, v$ vary $\Phi$ has an envelope. Apart from the four points of contact, the Lie quadrics have at most five distinct characteristic points. Demoulin has considered the case in which they have just two; this case is featured extensively. It gives rise in $S_{5}$ to a series of Laplace of period six. If the Lie quadrics have three characteristic points there is conservation of the asymptotes on the three nappes of the envelope. The Lie quadrics of the envelope have also three characteristic points. Cases in which the Lie quadrics have four or five characteristic points are treated in less detail.

The pamphlet is well written, is free of typographical errors, and furnishes a good insight into this fascinating problem.

We have presented, in issue XXII of the Mémorial des Sciences Mathématiques, the current state of the theory of birational transformations of the plane; we are currently pursuing the same goal with regard to birational transformations of space. The plan of this new issue is, in its broad outlines, the same as that of the first: properties of birational transformations; singular points; algebraic geometry; continuous groups.

After having indicated the properties of the linear systems of algebraic surfaces which are necessary for us, we define birational transformations and study the fundamental elements; we present in particular the recent results of Montesano. We then move on to the study of singular points of algebraic surfaces and to the decomposition of singularities by means of birational transformations. In the following Chapter we define the algebraic geometry of space and report on the results obtained in this area. We conclude with the presentation of the fundamental work of Enriques and Fano on finite continuous groups of birational transformations.

The birational transformations of space were studied by Cremona, Cayley, Noether, etc. as soon as the first of these geometers had created the birational transformations of the plane. However, their study has not yet reached the same level of advancement as that of the latter. While the theory of birational transformations of the plane can be considered complete, in the sense that each of these transformations is the product of a finite number of quadratic transformations, general results are lacking in the corresponding theory of space. We have basically limited ourselves to studying particular birational correspondences, without managing to identify, with rare exceptions, general properties common to all these transformations. There is therefore a vast field of research which, although much neglected in recent years, nevertheless seems to deserve the attention of geometers. The questions which arise are, moreover, difficult and doubtless of the same order as those which one encounters in the study of three-dimensional algebraic varieties, the theory of which progresses so slowly.

8.1. Review by: Virgil Snyder.
Bulletin of the American Mathematical Society 42 (11) (1936), 797-798.

The same scheme is used in the present pamphlet as in the preceding ones, namely, to present a brief and fairly popular account of the status of a problem, but accurate and including a suggestion for the direction of most probable further development. Proofs are not given, but the bibliography of fifty-eight titles is cited as occasion arises. This is fairly full, but a number of important recent contributions are not included.

The problem is very clearly stated ; it is the study of rational or irrational $(1, p)$ correspondences between the points of two algebraic surfaces, the $p$ points being the successive images of a given one in a birational correspondence of period $p$. A projective model of each surface is constructed in hyperspace, such that the birational correspondence becomes a collineation. Involutions are then classified according to the number of invariant points. An isolated invariant point may be perfect or imperfect, according as every direction through it does or does not remain fixed. In the case of imperfect points the two invariant directions are examined further. This process is continued until the form of contact at each point is completely accounted for. The procedure is illustrated by a detailed discussion of a plane cyclic collineation.

A regular surface can not have irregular involutions, but the converse is not true. Each combination is discussed, and the criterion obtained in order that a surface shall represent an involution. Finally the theory is applied to surfaces having a canonical curve of order zero.

The booklet is excellently printed on stiff paper, making an attractive page. It furnishes a welcome resumé of this interesting theory.

8.2. Review by: John Arthur Todd.
The Mathematical Gazette 20 (240) (1936), 290-291.

This monograph is essentially a report on some researches of the author, and its subject-matter is highly specialised. The topic under discussion is that of cyclic involutions on an algebraic surface, in particular those which have only a finite number of points of coincidence. A periodic birational transformation $T$ of a surface $F$ into itself, whose period is a prime number $p$, defines a cyclic involution $I$ of sets of $p$ points on $F$, and these sets may be mapped on the points of another surface $\Phi$. Of special interest are the coincidence points of $I$, namely those points of $F$ which coincide with their transforms under $T$. These points may be either perfect or imperfect. A perfect coincidence point is such that all directions through it are invariant under $T$; at an imperfect coincidence point, on the other hand, $T$ determines a homography among the tangents to $F$ which has two invariant lines. Such a point thus has two coincidence points infinitely close to it, and these in their turn may be perfect or imperfect. To each coincidence point of $F$ corresponds a singular point of $\Phi$ whose structure depends on the nature of the coincidence. After these generalities the author proceeds to consider some special problems, e.g. the condition that a surface must satisfy in order that it should represent the sets of an involution on another surface; the relations holding between $F$ and $\Phi$ when they have the same irregularity, and the cases in which $F$ and $\Phi$ belong to the class of surfaces with pluricanonical curve of order zero. There is and excellent bibliography down to 1934.

9.1. From the Foreword.

The Greeks left us elementary geometry and we will begin by recalling their discoveries. The development of Greek geometry is marked by stages: Pythagoras, the Eleatics, Euclid, Apollonius. We have added to this first chapter some indications on the theorems of Quételet and Dandelin relating to conics, theorems that one would not be surprised to see appear in the Treatise of Apollonius.

After the Greek period, we have to wait until the 17th century to see new methods appear in geometry. First, it is the coordinate method of Descartes and Fermat, analytic geometry, which is the subject of the second chapter. Second, it is the method of projections, which appears, with Desargues and Pascal, at the same time as analytic geometry. But the development of projective geometry will be much slower: it is only at the beginning of the 18th century, after Monge and Carnot, that it will be established as an autonomous doctrine by Poncelet and largely developed by Chasles. This geometry is the subject of the third chapter.

We have devoted the fourth chapter to the presentation of research on the principles of geometry and the fifth to the introduction into geometry of the notion of group. We know that this notion, due to a twenty-year-old mathematician, Évariste Galois, whose life was as short as it was tormented, allowed Sophus Lie and Félix Klein a rational classification of geometries; we give its broad outlines. The elementary framework that we have drawn for ourselves has not allowed us, other than by a brief allusion, to indicate the remarkable extension given very recently to the ideas of Lie and Klein by Élie Cartan.

The last chapter deals with topology. We have introduced this one, following an idea of ​​Federigo Enriques, starting from elementary geometry and successively abstracting the notion of straight line, then that of distance. Thanks to a broad appeal to intuition, we hope that the reader will be able to get an idea of ​​the nature of this geometry.

We would still have many points to deal with. We have briefly indicated hyperspatial geometries, algebraic geometry, and classical infinitesimal geometry.
We have tried to write the book that we would have wanted to read when we were twenty years old. May it be of some service!

9.2. Review by: Albert A Bennett.
The American Mathematical Monthly 45 (7) (1938), 465.

The little volume by an author well known for his studies in algebraic geometry, admirably achieves its modest aim of giving a brief general view of the history and nature of elementary geometry in language that may be significant to the non-professional reader. The trained mathematician need not expect to find here new theorems or unfamiliar philosophy. The work falls into six chapters: 1. Elementary geometry, II. Analytic geometry, III. Projective geometry, IV. Principles of geometry, V. Geometry and the theory of groups, VI. Topology. Chapter IV discusses postulates, starting with the parallel postulate, and then taking up the postulate of Archimedes, the "fundamental theorem" of projective geometry, and duality. In Chapter V Euclidean affine and projective transformations are considered in the light of group theory. An Arguesian space is defined as obtained from cartesian space by adjoining points at infinity. This chapter gives the theorem: The ruled projective geometry of ordinary space is equivalent to the geometry of a hyperquadric of five-dimensional projective space having as principal group the group of homographies of this space transforming this hyperquadric into itself. In the chapter on topology such topics as knots, the Peano curve, infinitesimal geometry, and the polyhedral formula are discussed. This book is not a textbook. It offers much not usually given in courses. It reads fluently, and avoids vague allusions to more difficult fields. This work should prove of service and inspiration to the inquiring student, particularly if he is seeking material for reporting in a mathematics club meeting and in any case if he is planning to teach in the elementary schools.

9.3. Review by: Nathan Altshiller Court.
Books Abroad 12 (3) (1938), 321.

"We tried to write the book we would have wanted to read when we were twenty," says the author, himself a mature and renowned geometrician. He offers the reader a bird's eye view of what geometry has been through the ages, beginning with the elementary geometry of the ancients and ending with modern analysis situs, or topology. Of each historical phase the main character is underscored and the salient points are singled out, with much skill and good judgment. Very limited demands upon the reader's mathematical training are made by the contents of the book. But even if all mathematical formulas are omitted, a considerable insight into the nature of geometrical thought may be gleaned from these pages. This little volume belongs to the admirable Collection Armand Colin and comes up to its exacting standards.

9.4. Review by: T G.
Nature 143 (1939), 424.

Prof Godeaux's addition to this excellent series of popular monographs, will be found of interest to the general reader and to the specialist. The former will be made familiar with the fundamental concepts of the various types of geometry; and both will learn much about their historical and methodological background. The six chapters of this work deal respectively with elementary geometry, analytical geometry, projective geometry, the principles of geometry, the theory of groups and topology. The mathematical notation used is quite simple, and the book makes easy reading owing to the explanations covering all the difficult intuitions of the geometricians.

10.1. Review by: Pierre Brunet.
Revue d'histoire des sciences et de leurs applications 1 (1) (1947), 82-86.

It is almost with Stevin (previously Gemma Frisius, Mercator, Ortelius and Stadius) that the 'Sketch' of Lucien Godeaux begins, who does not fail to regret precisely that the Geometric Problems are so little known. Among the mathematicians whose work is then quickly examined, let us mention Michel Coignet (1549-1623), Adrien Romain (1561-1615) and especially three scholars who "certainly have a marked place among the precursors of infinitesimal calculus", Grégoire de Saint-Vincent (1584-1667), Jean-Charles délia Faille (1597-1652), whose treatise De centro gravitatis, of 1632, is three years earlier than that of Guldin on the same question, and André Tacquet (1612-1660), whose research, similar to that of Wallis, is quite contemporary with it. "Some have wanted to see in Grégoire de Saint-Vincent an emulator of Descartes and Fermat in the creation of analytical geometry, arguing that the notion of coordinates was familiar to him. This is perhaps going a bit far, because the method used by the Bruges geometer to study conics is very similar to that of the ancients and algebra is little used." This precision is all the more interesting, as it clarifies a delicate question. One of the most remarkable Belgian mathematicians of the 17th century is certainly the Liège-born René-François de Sluse (1622-1685); and, after the torpor of the 18th century, the 19th century opens more brilliantly with Adolphe Quételet. One will find on the whole period that follows up to ours very clear insights in this small volume with precise lines, whose documentary value is incontestable.

10.2. Review by: Jean Pelseneer.
Ciel et Terre 61 (7-8) (1945), 1-3.

It is with great pleasure that we have read the little book by M Lucien Godeaux. This 'Sketch' will render the most useful service not only to the student, but to the specialist in the history of science and, more generally, to the cultivated man. The Sketch of a History of Mathematical Sciences in Belgium, which could also be entitled, to a certain extent, "Sketch of the History of Physical Sciences", constitutes a substantial contribution to the already considerable historical work of the eminent geometer of the University of Liège; moreover and above all, it represents the most extensive overall work devoted to the subject since the publication of Quetelet's two volumes some eighty years ago.

The valuable "Alphabetical List of Belgian Mathematicians" which ends the work (and which unfortunately did not retain the names of the people mentioned incidentally), shows that the author studied 79 mathematicians; Quetelet, in his own History of Mathematical and Physical Sciences among the Belgians, cites about 600 people, a good number of whom, it is true, are far from being mathematicians or Belgians; the difference in these numbers is therefore hardly conclusive if we want to compare the two works, and it is better to refer to the National Biography; by examining this, we see that only 183 notices out of 9,932, or 1.8%, concern scholars who cultivated the physical and mathematical sciences, as well as cartographers and engineers insofar as they made original contributions to these sciences. It therefore appears a priori, from this second comparison, that M Godeaux is not far from having exhausted the subject; one can grant without hesitation that he has given a very sufficiently representative idea of ​​it, because the few dozen scholars that he neglected could without inconvenience be omitted in the first approximation that is a sketch, since they were either non-mathematician scholars, or mathematicians about whom there is very little to say; let us hasten to add that we have not noted any regrettable omission.

M Godeaux preferred to abstain from speaking of living scientists; this is the consequence of a point of view that can certainly be defended; however, we cannot be prevented from finding it unfortunate that the vast public to whom the National Collection is addressed must thus remain in ignorance of the often brilliant names that our country has today in the field of mathematical sciences; however delicate a selection may be, we refuse to think that one can claim the title of honest man and not suspect the existence of a de La Vallée Poussin for example. On the other hand, it is above all an internal, that is to say technical, history of mathematics that M Godeaux tells us, in the sense that the general external circumstances - those which have conditioned the progress of science in our country - and the biographical data have been relatively left in the background; circumstances and biography are certainly not negligible, however, but we readily acknowledge that the greater or lesser importance that one is inclined to grant them depends essentially on the temperament of the writer, and the author - this is obviously not a reproach that would be singularly out of place on our part - has little taste for what quickly becomes polemic.

M Godeaux's story actually begins quite normally with Gemma Frisius; we are happy to see thus confirmed the result that we had obtained in our note "Statistical aspect of the progress of science in Belgium, through the centuries," namely that it is futile to speak of a history of science in Belgium before about 1525 in the field of physical and mathematical sciences; the erudite Biographie Nationale itself has only preserved the memory, in this field, of 9 characters prior to the period 1525. We believe however that a line could have been fairly granted to the mention of the work of the translators; it is not an exaggeration, in fact, to say that the greater part of the activity of Christian scholars in the 12th century and up to the middle of the 13th century consisted in translating. M Godeaux does not speak of the builders of mathematical instruments, whose life and work M Henri Michel has recently attempted to resuscitate; but it would be improper to blame him for this, given the currently very fragmentary state of our knowledge of a school that was undoubtedly quite brilliant. Let us note a typo: for the date of birth of Ortelius, it should read 1527 instead of 1594. As M Paul Ver Eecke, the eminent historian of Greek mathematics, has kindly agreed to point out to us, it is preferable not to say that "we owe ... to Stevin ... the translation of four books of Diophantus;" despite the title of the work by Stevin, it is in reality a paraphrase of an insufficient text, or better, adaptations that are involved, as M Ver Eecke has shown in his Diophantus of Alexandria (1926). Mr Ver Eecke's opinion is reproduced by M Robert Depau in his recent Simon Stevin, 1942; it was also the opinion of Bosmans, who wrote: "It is much more a commentary than a translation properly speaking." As for Stevin's description "of the ellipse by a point on a straight line whose two fixed points run through two rectangular straight lines", M Ver Eecke points out to us that this construction is already reported in Proclus. Regarding the English Jesuit College in Liège, we would have liked to see Anthony Lucas (1633-1693), who corresponded with Newton, mentioned after Linus.

Now a few trifles, which we record here in the margin of our reading and which the present bibliographical note gives us the opportunity to gather. Stevin: A Dutch author, M E-J Dijksterhuis, informed us of the publication of his book on Simon Stevin (1 volume, 1943); circumstances have not yet allowed us to become acquainted with this work. Grégoire de Saint-Vincent: Let us draw attention to the recent and substantial contribution of M Jos E Hofmann: "Das Opus Geometricum des Gregorius a S Vincentio und seine Einwirkung auf Leibniz" (1941). Chasles: Let us point out to researchers that there is a Belgian chapter in the career - so insufficiently studied as yet - of Chasles, which would merit investigation. Incidentally, let us note in praise of the Royal Academy of Belgium, regarding the famous Aperçu historique sur l'origine et le développement des méthode en géométrie, this flattering assessment by Joseph Bertrand: "The volume of the Brussels memoirs, devoted entirely to Chasles' memoir, will remain the glory of the company. It is, perhaps, the most consulted and the most avidly sought in the academic collections of all countries." Dandelin: Bertrand, speaking of Dandelin's acceptance of the Legion of Honour after his return from the island of Elba, writes this: "... by opting shortly afterwards for Belgian nationality, he escaped the very serious consequences" of the disapproval of his comrades at the École Polytechnique. We wanted to reproduce this detail, which is not found in any Belgian source relating to Dandelin. Here is also the beginning of an undated letter from Brialmont to Stas: "My dear friend. I am writing to you to find out if Colonel Dandelin is as ill as is said. Is his situation hopeless? Is he dying? I like to believe that it is a mistake, an invention of his enemies who find it convenient to pass him off as dead before he is. May God grant that this amiable philosopher who, all things considered, is an excellent man, may remain among us for a long time! Gilbert and Le Paige: There are some rather interesting letters - one from Ph Gilbert, two from C Le Paige - in the library of the Royal Observatory of Belgium, under the reference number 708 g.

We would have been tempted to extend somewhat the Bibliography that one finds at the end of M Godeaux's work. Thus, to stick to the publications after the war of 1914-1918, we would have willingly found the mention of: The Royal Academy of Belgium since its foundation (1772-1922), Brussels, 1922; articles by the permanent secretary and Paul Stroobant. - The Belgian Homeland 1830-1930, the illustrated editions of Le Soir, Brussels, 1930; articles by M Th De Donder and Paul Stroobant. - Ministry of Public Instruction. Royal Library of Belgium. History of Science in Belgium until the end of the 18th century. Exhibition (1 vol., Brussels, 1938).

We cannot express enough gratitude to M Lucien Godeaux for the trouble he has taken in writing his precious and timely little book. Let him be assured of the deepest gratitude of a very wide audience. We have taken the liberty of making one or two reservations or critical remarks above, but no one is more convinced than we are of their insignificance. It is a great honour for the National Collection to be able to present a text of such value on a subject of unusual importance and also difficulty.

11.1. Note.

We have been unable to find any details of the three earlier editions. Even the Belgium Royal Academy only holds the this 4th edition. It is a book with 304 pages.

12.1. Review by: Temple Rice Hollcroft.
Mathematical Reviews MR0050914 (14,401e).

This textbook is designed for an intermediate course in algebraic geometry to follow elementary projective geometry. As stated in the preface, the topics covered are intended to be studied by all mathematical students in the University of Liège and to serve as a basic course for those specialising in geometry. The fundamental concepts of projective space of one, two and three dimensions are treated.

The first chapter contains a detailed discussion of series of groups of points, correspondences, and polar groups on a line. Then follows a chapter on the properties of plane curves of any order and one on cubics and quartics. The study of cubics includes elliptic cubics, singularities, invariants and methods of generation. Non-singular quartics only are treated.

Almost half of the book is devoted to three-space. In chapter four the author deals with the properties of algebraic surfaces of any order, including singularities, polar surfaces and ruled surfaces. This chapter also contains a brief treatment of space curves. In chapter five, curves on a quadric are studied. The quadric $Q$ and its curves are projected from a point of $Q$ on a plane. Curves of order $n$ on $Q$ are classified for $n ≤ 6$. In the final chapter, cubic surfaces are treated in considerable detail, including representation on a plane, singularities, curves on cubics, generation of cubics, and ruled cubics.

The book is well written. The arrangement and printing leave nothing to be desired. It is an excellent introduction to algebraic geometry.

13.1. Note.

This work is based on Godeaux's lectures in the Faculty of Science at the University of Liège. The three volumes contain a total of 862 pages and have 92 figures. Volume 1 covers Differential Calculus, Integral Calculus, and Series. Volume II covers Geometric Applications. Volume III covers Differential equations, Calculus of variations, and Conformal representation.

The review of the second edition of Volume 1 in 1954 (given below) gives details equally relevant to this first edition.

14.1. Review by: Patrick Du Val.
Mathematical Reviews MR0036533 (12,124d).

Each of the two volumes of this work is divided into two parts, dealing respectively with (I) birational transformations, (II) hyperspatial projective geometry, (III) geometry on an algebraic curve, and (IV) algebraic plane geometry. A third volume, on algebraic surfaces, appears to be in contemplation.

Part I contains a lot of theory, much of which one feels would be more at home in some of the other parts, or at least could be read more intelligently by a student who already knew a good deal of the matter in the other parts. It deals with Cremona transformations in the plane and in three dimensions, some of the familiar types of low order being analysed in great detail, with special consideration of the fundamental elements, of which the arithmetical properties are also given with considerable generality. These are used to resolve singularities of algebraic plane curves, and surfaces in space; for the former, the Noether-Chisini transformation into a curve with only multiple points with distinct tangents, and the Halphen transformation into one with only double points, are given, and for the latter, biplanar and uniplanar double points are classified (somewhat defectively as regards the uniplanar, since only the three types found by Cayley for cubic surfaces would appear to exist). The genus of a plane curve is also defined in terms of order and singularities, and it is shown that this cannot be negative, and that if it is zero the curve is rational.

Part II begins with a brief study of projective space of n dimensions (defined, of course, by means of complex homogeneous coordinates) and its subspaces, and of projective and dual transformations. Hypersurfaces and their elementary polar properties and dual treatment as envelopes come next, followed by varieties of lower dimensions which are treated by the Cayley-Halphen representation as the intersection of a cone and a number of monoids residual to a set of generators of the cone. Rational curves receive a chapter to themselves; those of order ≤ 6 are treated in detail and linear series on a rational curve studied (under the odd name of "involutions," whatever their dimensions). Rational surfaces also receive a chapter, containing the general theory of the plane mapping, and detailed study of the rational ruled surfaces and the Veronese and Steiner surfaces. A further chapter is given to Segre varieties, representing the sets of $k$ points, each varying independently in a linear space. The $V_{4}^{6}$ representing the pairs of points of two planes is treated at length. The final chapter deals with the representation of three-dimensional line geometry on a quadric in five dimensions.

Part III (in the second volume) is devoted to the classical theory of curves; linear series, the canonical series, and the Riemann-Roch theorems are treated in an elementary way which should be intelligible to beginners; the canonical curves of the first few genera are described projectively (it is odd, however, that in dealing with the curve of genus 6 the author appears unaware that the most general type is the quadric section of a quintic del Pezzo surface, as he treats the plane sextic with four and the septimic with nine double points as essentially distinct cases). Correspondence theory is treated first by classical algebraic methods; then Riemann surfaces and Abelian integrals are introduced and discussed in a manner that is both simple and thoroughgoing, and applied to correspondence theory.

In part IV linear systems of plane curves are studied from the point of view of Cremona-invariance; the general theory of the Jacobian and adjoint systems, and of successive adjoints is given. The continuous groups of Cremona transformations are considered, and the cyclic and involutary transformations are classified. Finally, the plane involutions of order 2 are studied in detail. Much of this last part consists of the amplification, from a more sophisticated point of view, of topics already studied in part I.

15.1. From the Introduction.

The study of correspondences between points of algebraic curves starts with the Chasles correspondence principle (1864). We know that Chasles considered a correspondence of indices α, β between two superimposed rectilinear points and that his principle stated that the number of united points is $\alpha + \beta$. The extension of Chasles' principle to correspondences between points of a rational curve is immediate. It was Cayley (1866) who sought to extend this principle to correspondences between points of an algebraic curve of genus $p$ greater than zero. Cayley supposes that, given on a curve $C$ of genus $p$, a correspondence $(\alpha, \beta)$, there exists a curve passing through the $\beta$ homologous points of a point $P$, having a contact of order $\gamma - 1$ with the curve $C$ at $P$, variable in a linear system when $P$ describes the curve $C$. The number $\gamma$ is the valence (valenza, Wertigkeit) of the correspondence. By induction, Cayley managed to evaluate the number, $\alpha + \beta + 2\gamma p$, of the united points of the correspondence. A first demonstration of Cayley's formula was given by Brill a few years later; a second is due to Zeuthen, who gave the formula the name of Cayley-Brill correspondence principle.

In 1888, Hurwitz published a fundamental work on the theory of correspondences between the points of an algebraic curve. Using the properties of the Abelian integrals attached to the curve, he brought to light the meaning of the valence of a correspondence, showed that there can be correspondences devoid of valence and stated a general principle of correspondence. In 1903 Severi constructed a geometric theory of correspondences; he had been led to it by the study of surfaces which represent the pairs of points of one or two algebraic curves. The works of Hurwitz and Severi form the basis of modern research on the theory of correspondences. It is mainly in Italy that this theory was developed, but it is worth mentioning an important work by Lefschetz, where this geometer applies his profound research in topology to correspondences.

This booklet is devoted to the presentation of the research which originated in the works of Hurwitz and Severi; we have mainly focused on making known the methods used by these geometers.

A first paragraph is devoted to the Jacobi variety attached to an algebraic curve. This variety is introduced immediately by the inversion of the Abelian integrals; it seemed useful to us to introduce it in a geometric way, due to Castelnuovo.

In the second paragraph, we consider the involutions belonging to an algebraic curve, that is to say the correspondences $(1, n)$ between two algebraic curves. The theory of these involutions leads to a difficult problem, the solution of which still seems distant: that of the existence of curves containing an involution of order $n$ whose image curve is given a priori. We point out the work to which this problem has given rise.

We have divided the presentation of research on the correspondences between two algebraic curves, distinct or coinciding, into three paragraphs, where we successively consider the geometric point of view, the transcendent point of view and the topological point of view. It is not obvious that there are watertight partitions between these three points of view; on the contrary, they are often closely mixed in many Memoirs. But it seemed to us that this subdivision allowed us more clarity in the presentation of the methods used.

The geometric point of view is at the base of the important Memoir of Severi to which reference was made above; it was mainly used by this geometer and by his students.

It is at the transcendent point of view that Hurwitz had placed himself. A happy interpretation of Hurwitz's formulas allowed Rosati to associate with a correspondence between the points of one or two algebraic curves, a hyperspatial homography with rational coefficients. This idea of ​​Rosati has proven to be very fruitful; it is closely related to the point of view adopted by Scorza and Cotty in the study of Abelian functions.

R Torelli's theorem on the birational identity of two curves having the same table of periods of normal integrals of the first kind is directly related to the transcendental point of view. We have limited ourselves to citing this theorem, having had the opportunity to present elsewhere the work that led to it.

The topological point of view was mainly considered by Lefschetz, as we said above. It was also used by Chisini in interesting research.

The list of works devoted to the theory of correspondences between algebraic curves, placed at the end of the booklet, is, we believe, complete. The numbers, in bold, placed after a name, refer to this list.

15.2. Review by: David Bernard Scott.
Mathematical Reviews MR0040031 (12,632d).

The author, with admirable clarity, gives in about 60 pages an account of the results and methods of the theory of correspondences between algebraic curves in the complex domain. There is no room for proofs, but adequate references are given and the bibliography is complete up to 1940 when this tract was written. The work is in five chapters. The first introduces the Jacobi variety of a curve $C$ of genus $p$, following a brief summary of the notions of linear system, the canonical system, and the two types of correspondence between the sets of $p$ points of a curve. The next section deals with an involution on $C$ whose sets are represented by the points of a curve Γ, with the relation between linear series on $C$ and on Γ, and with the correspondence between the Jacobi varieties of these curves. The section ends with a discussion of the problem of enumerating the possible birationally distinct curves $C$ containing an involution of order $n$ represented by a given curve Γ with assigned branch points. Chapter III discusses, by algebro-geometric means, correspondences between two algebraic curves, and the effect of such correspondences on linear systems of the two curves, and deals also with the corresponding results for their Jacobi varieties. Correspondences of valency on a single curve are then discussed, followed by the notion of dependence of correspondences and Severi's discovery of the base for correspondences between two (possibly distinct) curves. A number of applications of the general theory are mentioned. The fourth chapter discusses the transcendental theory of correspondences, beginning with the theory of correspondences on a single curve and their relation to the periods of the simple integrals of the first kind. An account is given of Rosati's geometrical interpretation of Hurwitz's formulae, and of his generalisations of the notions of valency. Hurwitz's application of the theory of theta-functions is mentioned but theta-functions are not employed at any place in the text and the author confines himself to what can be done by more elementary methods, and there is no space for an account of the theory of Riemann matrices. The final chapter of the book discusses the contributions of Chisini and Lefschetz to the topological theory of correspondences. The order of presentation followed by the author is based on the historical one, and the necessity of compressing the work into such small compass doubtless made it inevitable that it be followed. Nevertheless it is a pity that it was not possible to invert the order of the last two chapters so as to use the topological treatment to illuminate the transcendental theory. The subject matter of this tract already forms a very complete and well rounded theory, and nobody who reads it can fail to be impressed by our enormous debt to the masters of the Italian school with first their German and later their American collaborators. The difference in precision and finality between the theory expounded here and what is known even for correspondences between algebraic surfaces, must ensure that this tract will serve both as a guide and a challenge to further efforts.

16.1. Review by: Louis Millet.
Les Études philosophiques, Nouvelle Série 14 (2) (1959), 221.

The author addresses readers with the culture that the baccalaureate represents. He outlines the evolution of geometry since the Greeks; this history provides him with the opportunity to emphasise the work of the 17th century (Descartes and Fermat, Desargues and Pascal), of the 18th century (protective geometry: Monge, Carnot, etc.). A study of the principles of geometry and some insights into axiomatics precede the two chapters devoted to group theory and topology. This excellent introductory work is completed by a bibliography.

17.1 Review by: Harold Scott MacDonald Coxeter.
Mathematical Reviews MR0047339 (13,861l).

This well-written textbook is a synthetic treatment of real projective space in the manner of Enriques [Lezioni di geometria proiettiva, 2nd ed., Zanichelli, Bologna, 1904]. Assuming the reader's familiarity with Euclidean geometry, the authors introduce ideal elements by an appeal to the analogy between points and directions. They define projective space as consisting of all the points, lines and planes, proper and ideal. After this motivation, they state nine axioms: six for incidence, one for order, one for sense, and one for continuity (Dedekind). They deduce the classical properties of duality, harmonic sets, ordered correspondences, projectivities, collineations and correlations, homologies and polarities, conics, cones, quadrics, and twisted cubics.

In the plane, they define a polarity to be an involutory correlation, and a conic to consist of the self-conjugate elements (if any) of a polarity. In space, they call an involutory correlation a null system or a polarity according as there does or does not exist a plane containing three self-polar lines. (Rather confusingly, they call polar lines "conjugate" lines.) Following Seydewitz, they define a quadric to be the locus of the point of intersection of a corresponding line and plane of two correlated bundles ("gerbes") in general position. As compared with von Staudt's treatment, this has the advantage of including cones as well as non-degenerate quadrics. It is proved later that the non-degenerate quadrics can alternatively be derived from the self-conjugate points and planes of certain polarities, and that a general linear complex consists of the self-polar lines of a null system. Noteworthy features are the synthetic classification of collineations and the account of various ways of determining a null system.

This systematic development is interspersed with digressions of two kinds: analytic digressions, in which a knowledge of projective coordinates is assumed, and metrical digressions, in which the plane at infinity and the absolute polarity are used to supplement the reader's previous ideas of parallelism and orthogonality. No attempt is made to segregate the affine results.

Diagrams are scarce throughout most of the book, but are plentiful in the final chapter, added for this edition by the second author. Here we find a generous supply of worked exercises: projective, descriptive, affine and Euclidean. These include many problems of construction, both linear and quadratic.

18.1. From the Introduction.

The theory of Irrational transformations was created by Cremona in 1863-1865 and for this reason these transformations are often called Cremonian transformations. Before Cremona, the only known birational transformations were homography and quadratic transformation (inversion). In a Memoir that remained unpublished, de Jonquières had also considered, around 1864, certain transformations to which his name remained attached. A birational transformation of order $n$ makes curves of order $n$ correspond to the straight lines of the plane, forming a homaloidal network, that is to say a network in which two variable curves meet at a single variable point. Knowledge of this network implies knowledge of the transformation and this leads to a first problem: constructing the homaloidal networks of the plane. Many geometers have tackled this problem; it seems that it was Montesano who obtained the most precise results. These were also recently found by B Segre, using a simpler method.

On the other hand, Cremona, in order to arrive at the general concept of birational transformation, had carried out products of quadratic transformations and the question arose whether any birational transformation could be obtained by this process. The answer is affirmative and was stated simultaneously around 1870 by Clifford, Noether and Rosanes. The latter two gave demonstrations in extended cases, but it was not until 1901 that Castelnuovo gave a complete demonstration.

The concept of a birational transformation led geometers to construct a geometry: Algebraic Geometry, whose principal group, in Klein's sense, is the group formed by birational transformations. This geometry appeared for the first time in 1877 in a work by Bertini. Two figures are considered identical when one can pass from one to the other by a birational transformation. In each family of two-by-two birationally identical figures, we set out to find a figure satisfying specific projective properties, which determines the family to which it belongs.

In this pamphlet, after recalling the composition of multiple points of plane algebraic curves by means of the notion of infinitely neighbouring multiple points and indicating the first properties of linear systems of plane curves, we present the theory of birational transformations, using in particular the method that we have recently introduced. We then give the decomposition of birational transformations into products of quadratic transformations, using a method due to Chisini. We conclude with a presentation of the results obtained in plane algebraic geometry.

In the bibliography that ends the Work, we have cited most of the works relating to birational transformations, even when we have not had to refer to them in our text. This is how we will find mention of Fano's work on birational contact transformations and of Villa's work on the approximation of point correspondences by birational transformations.

Liège, 6 July 1951.

18.2. Review by: Patrick Du Val.
Mathematical Reviews MR0054282 (14,898b).

This monograph appears to be intended as a fairly easy and elementary introduction to the classical theory of plane Cremona transformations. The first brief chapter contains a résumé of Enriques' treatment of superlinear plane curve branches by their Puiseux expansions, the second an equally brief summary of the general properties of linear systems of plane curves, leading up to the characters of a homaloidal net. Chapter III contains a simple treatment of the general plane Cremona transformation, and the numerical relations between the characters of the homaloidal net and the exceptional curves fundamental to it are obtained rather lucidly by considering the projective model of the sum of the homaloidal net of the transformations with that of lines -- a surface which has the merits of being symmetrically related to the given transformation and its inverse and of containing all the fundamental curves of both explicitly as curves. The possibility of some base points being in the neighbourhoods of others, and of some fundamental curves being reducible, is touched on. Chapter IV is a careful and fairly clear study of the Noether-Chisini theorem. The remaining two chapters are devoted to the elementary properties of Cremona-covariance of such things as the Jacobian and adjoint systems of a given one, to the search for minimum-order models of systems of given genus, etc., and to finite continuous groups of Cremona transformations. In these connexions the classical results are stated without much proof, but with adequate references. The book ends with what seems to be a very copious bibliography.

19.1. Note.

This work is based on Godeaux's lectures in the Faculty of Science at the University of Liège. The three volumes contain a total of 862 pages and have 92 figures. Volume 1 contains 373 pages and covers Differential Calculus, Integral Calculus, and Series. Volume II covers Geometric Applications. Volume III covers Differential equations, Calculus of variations, and Conformal representation.

19.2. Review by: Anon.
Mathesis (1953).

With a very clear order and writing, this course designed in the classical mode should attract the attention of all those who wish to initiate themselves into the possibilities of investigation offered by differential calculus and integral calculus.

Young people who have acquired in secondary education the mathematical training targeted by the official programme can approach these lessons of Analysis with confidence. They will easily cross the transitional step between their knowledge and their new efforts; along the way, they will be grateful to the author for having illustrated the abstraction of arithmetic reasoning by the vision of its geometric meaning and by the application to simple examples. It is also pleasant to underline the moderate conciseness of the text, the careful arrangement of the mathematical writings, the precise clarity of the figures, the impeccable typographical presentation.

Here are the titles of the chapters. I. Preliminaries, variables, limits (one paragraph, pp. 34-42, is devoted to arithmetic continued fractions). - II. Real functions of real variables. - III. Study of some particular functions (inverse, exponential, logarithmic, circular, hyperbolic). Derivatives and differentials of the first order of explicit functions. V. Successive derivatives and differentials of explicit functions. - VI. Derivatives and differentials of implicit functions. VII Changes of variables. VIII. Taylor's formula. - IX. True values ​​of indeterminate expressions. X. Extrema of functions. XI. General or indefinite integrals. - XII Definite integrals. - XIII. Extension of the notion of definite integral. - XIV. Curvilinear integrals and total differentials. XV. Multiple integrals. XVI. Study of series. XVII. Notions on Fourier series.

20.1. Review by: Bernard d'Orgeval.
Mathematical Reviews MR0155221 (27 #5160).

In 1912, Enriques proposed to the author to study the cyclic involutions belonging to a surface whose genera are all one. This first study led the author to that of the cyclic involutions belonging to any algebraic surface and to that of the image surfaces of these involutions. This research, pursued for more than fifty years, led the author to publish numerous works devoted both to the general study of the problem and to the resolution of particular cases and to particularly interesting applications. It is in short the summary of this very important work that the published volume gives us in a more didactic form, thus taking up in chapter I the theory of singularities of algebraic curves and surfaces. Chapter II leads to the structure of the diramation points of the image surface of a cyclic involution belonging to an algebraic surface in its most general form. Chapter III, particularly interesting, summarises the main applications of the theory with the construction of regular and irregular surfaces of any Severi divisor, of non-rational surfaces $p_{a} = p_{g} = 0$, of surfaces deduced from surfaces whose canonical system is either a sheaf or an isolated curve, of involutions belonging to the surface of pairs of points of an algebraic curve and to their representations, to certain notable particular cases of involution, to surfaces possessing a canonical system endowed with non-exceptional fixed component curves, finally to a remarkable property of irregular surfaces containing a regular involution. The appendix devoted to the extension of the theory to 3-dimensional varieties gives only partial results. There is a vast field of research to pursue, doubtless very difficult, but of great interest, especially if one seeks to extend the results to any dimension.

21.1. Note.

This book has 83 pages.