Hilda P Hudson's books

Hilda P Hudson published two books, Ruler and Compasses (Longmans, Green and Co., London, 1916) and Cremona Transformations in Plane and Space (Cambridge University Press, 1927). Ruler and Compasses was reprinted in 1953 in a volume which also contained reprints of Squaring the Circle by E W Hobson, The Theory and Construction of Non-differentiable Functions by A N Singh and How to draw a Straight Line by A B Kempe. We give information on these books below such as the Introduction, the Preface, the Contents, and short extracts from some reviews.

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Ruler and Compasses (1916)

Ruler and Compasses, and other works (1953)

Cremona Transformations in Plane and Space (1927)

1. Ruler and Compasses (1916), by Hilda P Hudson.
1.1. Introduction.

At the beginning of his Elements, Euclid places his three Postulates:
Let it be granted

(i) that a straight line may be drawn from any one point to any other point:

(ii) that a terminated straight line may be produced to any length in a straight line;

(iii) that a circle may be described from any centre, at any distance from that centre;
and all the constructions used in the first six books are built up from these three operations only. The first two tell us what Euclid could do with his ruler or straight edge. It can have had no graduation, for he does not use it to carry a distance from one position to another, but only to draw straight lines and produce them. The first postulate gives us that part of the straight line AB which lies between the given points A, B; and the second gives us the parts lying beyond A and beyond B; so that together they give the power to draw the whole of the straight line which is determined by the two given points, or rather as much of it as may be required for any problem in hand.

The last postulate tells us what Euclid could do with his compasses. Again, he does not use them to carry distance, except from one radius to another of the same circle; his instrument, whatever it was, must have collapsed in some way as soon as the centre was shifted, or either point left the plane. The three postulates then amount to granting the use of ruler and compasses, in order to draw a straight line through two given points, and to describe a circle with a given centre to pass through a given point; and these two operations carry us through all the plane constructions of the Elements. The term Euclidean construction is used of any construction, whether contained in his works or not, which can be carried out with Euclid's two operations repeated any finite number of times.

In fact, Euclid gives only very few of the constructions which can be carried out with ruler and compasses, and probably every student of geometry has at some time or other constructed a figure which no one else had ever made before. But from very early times there were certain figures which everyone tried to make with rule and compasses, and no one succeeded. The most famous of these baffling figures are the square equal in area to a given circle, and the angle equal to the third part of a given angle; and it has at last been proved that neither of these can possibly be constructed by a finite series of Euclidean operations.

The set of figures which it is possible to construct with ruler and compasses is thus on the one hand infinite, and on the other hand limited. It is easy to see that it is infinite; even if we consider only the very simple type of figure consisting of a set of points at equal distances on a straight line, which can certainly be constructed with ruler and compasses, the figure may contain either three or four or a greater number of points without any upper limit, so that there are an infinite number of figures even of this one simple type; much more is the whole set of possible figures infinite. And yet the set is limited, for many figures can be thought of which do not belong to it, and require apparatus other than ruler and compasses for their construction; besides those mentioned above, there are for example the regular heptagon and nonagon; or the ellipse, which can be drawn as a continuous curve with the help of two pins and a thread, but of which we can only obtain an unlimited number of separate points by Euclidean constructions.

The first question treated in this book is the one which naturally arises here: what constructions can be built upon Euclid's postulates, and what cannot? or, in other words, what problems can be solved by ruler and compasses only? For centuries, vain attempts were made to square the circle and trisect an angle by Euclidean constructions, and these attempts were often of use in other ways, though their immediate object failed; but it was only through the growth of analysis that it was proved once for all that they must fail. The ancient or classical geometry lends itself curiously little to any general treatment; and even modern geometry lacks a notation or calculus by which to examine its own powers and limitations. Occasionally we can be sure beforehand that a certain class of problem will yield to a certain type of method, but as a rule each problem has to be taken on its own merits and a separate method invented for it. There can be endless variety in the methods, and it might seem a hopeless task to sum them up, and impossible to say of any one problem that no ruler and compass construction for it ever will be found by some ingenious person yet unborn. Yet this is just what can be said in the case of the circle-squarers, and the final word came not from a geometer but from an analyst. The methods of coordinate geometry allow us to translate any geometrical statement into the language of algebra, and though this language is less elegant, it has a larger vocabulary; it can discuss problems in general as well as in particular, and it can give us the complete answers to the questions: what constructions are possible with ruler only, or with ruler and compasses?

In the next chapter we shall show how each step of a rule and compass construction is equivalent to a certain analytical process; it is found that the power to use a ruler corresponds exactly to the power to solve linear equations, and the power to use compasses to the power to solve quadratics. Since each step of a ruler and compass construction is equivalent to the solution of an equation of the first or second degree, we consider what these algebraic processes can lead to, when combined in every possible way, and that enables us to answer the question before us and say that those problems and those problems alone can be solved by ruler only, which can be made to depend on a linear equation, whose root can be calculated by carrying out rational operations only; and that those problems and those problems alone can be solved by ruler and compasses, which can be made to depend on an algebraic equation, whose degree must be a power of 2, and whose roots can be calculated by carrying out rational operations together with the extraction of square roots only; and that those problems and those problems alone can be solved by ruler and compasses, which can be made to depend on an algebraic equation, whose degree must be a power of 2, and whose roots can be calculated by carrying out rational operations together with the extraction of square roots only.

This is a complete answer to our question, but it is stated entirely in algebraic language, and in the general form in which it stands it cannot be translated into the language of pure geometry, for the words are lacking. But they are not really needed; for if a definite problem is before us, stated geometrically, we can always apply the test to the algebraic equivalent of the problem, though the application in some particular cases may not be easy; and if the test is satisfied, we can from the analysis deduce a Euclidean construction for the geometrical problem. As an example, we shall consider, at the end of chapter II, what regular polygons are within our powers of construction. It is a curious and unexpected result that the regular polygon of 17 sides is included, and a construction for this is given on page 34.

When we have agreed that the set of possible Euclidean constructions is both infinite and limited, and when we have found out in some measure what its limitations are, it is natural to seek a clearer view of the set, and to ask what are the best ways of classification. The first main subdivision has already been brought to our notice; it consists of linear problems, which can be solved with ruler only, and chapter III is devoted to these. We try to show how the data of a problem control its construction, and how the properties and relations of the data fall into distinct classes, each of which allows a particular set of constructions to be carried out.

Now though Euclid's compasses can to some extent carry distances, we are not making use of compasses in the section referred to; and Euclid's ruler cannot carry distances at all. So we find that in general, that is, if the data of the problem do not have some exceptional relations, we are not able to carry a distance from one part of the figure to another, nor to compare the lengths of two segments, even of the same straight line, unless one is a part of the other, and then all that we can say is that the whole is greater than the part. We can never say that two segments are equal in length unless they coincide; for the very idea of comparing the lengths of two segments which have different positions, involves the idea of making them lie alongside of one another in order that we may compare them, and therefore of carrying one at least into another position; and this we cannot in general do with ruler only. Now the property of a parallelogram, that its opposite sides are equal in length, shows that if we can draw parallels, we can carry a distance from a straight line to a parallel straight line, so that there is a close connection between carrying distances and drawing parallels; and we shall show that, to a certain extent, if we can do either we can do both. But even so, it is only the parallel sides of a parallelogram that are equal, and not the adjacent sides, and drawing parallels does not help us to transfer the length of a segment into any direction other than that of the original segment. It is only in the special case in which the data allow us to construct a rhombus, whose adjacent sides are equal as well as its parallel sides, that we can obtain equal lengths on any of two different sets of parallel straight lines. We could hardly expect, with ruler only, to be able to turn a given length into any direction, for this is one of the chief uses of a pair of compasses.

In this way we get the usual classification of linear constructions according to the projective and metrical properties of the data, properties of length and properties of angle. The idea of cross-ratio is fundamental to them all, and when we have to compare the cross-ratios of two different ranges, we are led to the theories of homography and involution; but in connection with these we come upon several problems that require the common or double points; and the construction of these points is equivalent to the solution of a quadratic equation, and therefore impossible with ruler only.

In chapter IV therefore, in which we admit the use of compasses, the comparison is worked out between describing a circle and solving a quadratic, and then we are able to carry the treatment of cross-ratio and involution to a more satisfactory stage. A digression is introduced at this point to show that the ordinary modern instruments, dividers, parallel ruler and set-square, that are commonly used along with ruler and compasses, amount to short cuts in Euclidean constructions without extending the range of soluble problems. We also consider which ways of using these other instruments can completely replace the use of compasses.

But instead of classifying the data of a construction, we may classify the methods, and ask afterwards in what sort of problem each method is likely to be useful. Two fundamental ideas are put forward in chapter V, one or other or both of which are prominent in very many constructions: these are separation of properties and transformation. They give rise to half a dozen fairly well- marked lines of attack which are illustrated in that chapter. There is the method of loci, when some of the conditions to be fulfilled convince us that a required point must lie on a certain locus, which must be made up of straight lines and circles if the method is to succeed, and the other conditions convince us that the same point must also lie on another such locus, so that it can only be a point of intersection of the two loci. There is the method of trial and error, when from a finite number of unsuccessful attempts at a construction we are able to discover the way to begin which is bound to lead to success, if a solution exists; the method of projection, and several of its particular cases, in which the two principles of separation of properties and of transformation are both present; the method of inversion, which is very appropriate to ruler and compass constructions because of the way in which it relates circles and straight lines; and the method of reciprocation, which rests upon the principle of duality.

Besides these intrinsic ways of classification, which come from considering the constructions themselves, there are others that come from considering them from some quite external point of view, according to how far they avoid certain draw-backs which are practical rather than mathematical, and arise from faults in our actual apparatus. Yet there is some theoretical interest also in devising constructions that shall as far as possible get over the difficulties of a small sheet of paper and a blunt pencil (chapter VI). The idea of the last section of the same chapter is to make a numerical estimate of the length of a construction, by reckoning up all the different operations with ruler and compasses that it requires, so as to be able to say which is the shortest of different solutions of the same problem. This plan of "giving marks" is little more than a pastime, and the scale of marking is very arbitrary; but Lemoine's book on Geometrography deserves to be better known, and some account of the matter is given here in the hope of introducing more English readers to his original work.

The subjects of the last two chapters are also now mere curiosities, though one at least arose as a practical point in machine construction. Any Euclidean problem can be solved by drawing only one circle and the requisite number of straight lines, usually a large number; or else by drawing the requisite number of circles and no straight line at all. This is proved, and examples of the methods are given, in chapters VII and VIII.

Thus the connecting link throughout the book is the idea of the whole set of ruler and compass constructions, its extent, its limitations and its divisions. /But the matter of the following pages consists largely of examples, which have been brought in wherever possible, and it is hoped that those readers who are not attracted by general or analytical discussions, may yet find something to interest them among these problems and their geometrical solutions.

1.2. Table of Contents

I. Introduction
II. Possible Constructions
III. Ruler Constructions
IV. Ruler and Compass Constructions
V. Standard Methods
VI. Comparison of Methods
VII. One Fixed Circle
VIII. Compasses Only

1.3. Review by: M B Woods.
The American Mathematical Monthly 24 (6) (1917), 275-276.

This new volume of Longmans' Modern Mathematical Series is an attempt to collect from many sources solutions of problems and discussions of methods in which the Euclidean ruler and compasses are used as instruments; and to present them as part of a well-ordered development of the theory of such constructions. According to the author "the connecting link throughout the book is the idea of the whole set of ruler and compass constructions, its extent, its limitations, and its division." The reader will require no more advanced mathematics than college algebra and elementary analytic geometry, although a knowledge of projective geometry and of the theory of equations in general will be helpful. The development of the theme is carefully carried out and there are no breaks in the logic, although in one or two places the author quotes a theorem which is developed later. This may prove a bit annoying to the reader with a minimum of preparation, but otherwise it is not a serious fault. The subject matter of the text is presented as a whole in the introduction, which is rather well written, although it presupposes at times a rather full acquaintance with the material which is developed later.
On the whole, the teacher of analytic or projective geometry or of college algebra will find this a valuable source-book for many illustrative problems. Its development of fundamental theorems of projective geometry for the general conic starting from properties of the circle is carried out in an interesting way. The sources of many theorems are carefully indicated and a fairly complete bibliography is given facing page 1. In the way of criticism, the reviewer believes that some of the section headings are misplaced and that many of them are not illuminating. The chapter on methods of solution is an exception to this criticism.

1.4. Review by: Anon.
The Mathematics Teacher 9 (2) (1916), 129.

Miss Hudson in this little volume starts with Euclid's three postulates and shows that the use of the ruler corresponds with the linear equations and the use of the compasses with the quadratic equation. The answer then to the question as to what constructions are possible with ruler only and what constructions with the ruler and compasses, which answer geometry failed to give, is furnished by analysis. In other words, those problems and those alone can be solved by ruler only, which can be made to depend on a linear equation; and those problems and those alone can be solved by ruler and compasses, which can be made to depend on an algebraic equation, whose degree is a power of 2 and whose roots can be found by rational operations and the extraction of square roots only. The book contains very much of interest and profit for teachers of geometry.

1.5. Review by: Philip E B Jourdain.
Science Progress (1916-1919) 11 (44) (1917), 697.

It is well known that Euclid put at the beginning of his Elements the postulates that a straight line might be drawn between any two points, that such a straight line might be produced to any length in the same straight line, and that a circle might be described from any centre and at any distance from that centre; and that all the constructions used in the first six Books are made with these three operations only. The many attempts to solve the great problems of the duplication of a cube, trisection of an angle, and quadrature of a circle by what were thus called "Euclidean methods" failed, and such attempts were analytically proved in more modern times to be necessarily vain. On the other hand, certain constructions can be carried out by Euclidean methods which were not given by Euclid. It is the purpose of this excellent little book to investigate, both analytically and geometrically, how far Euclidean constructions can carry us, although no attempt is made to give an account of work on transcendental numbers. It is well remarked that "the ancient or classical geometry lends itself curiously little to any general treatment; and even modern geometry lacks a notation or calculus by which to examine its own powers and limitations"; thus it has to be by using coordinate methods that Miss Hudson attacks the general question as to the problems which can be solved by ruler and compass. The second chapter also contains an answer to the question as to what regular polygons are within our powers of construction, and Richmond's (1909) construction of one of seventeen sides is given on page 34. Chapters III and IV are devoted to ruler constructions and ruler and compass constructions respectively; Chapter V is concerned with a classification of methods; and Chapter VI is devoted to a comparison of methods. The last two chapters (VII and VIII) are concerned with the solution of Euclidean problems by drawing only one circle and the requisite number of straight lines (Poncelet and Steiner), or else by drawing circles only (Mascheroni and Adler).

This is an exceedingly competently written book, and there are only a few little criticisms that might be offered. In the first place, the way of printing analytical formulae does not strike one as wholly pleasant, though this is probably due to the fact that one is used to small italic letters for the purpose. In the second place, it is surely rather misleading to the student continually to speak as if the geometry of the ancient Greeks were intimately concerned with drawing instruments and the material on which figures were drawn: although the solution of the problem as to why Euclid stopped at a postulate that lengths may be carried about from place to place, when he had admitted that one end of a straight line may be carried about if the other end is fixed, may, perhaps, be answered historically by such considerations. This book is a very valuable addition to "Longmans' Modern Mathematical Series."

1.6. Review by: Virgil Snyder.
Bulletin of the American Mathematical Society 23 (8) (1917), 377-378.

This little book is a welcome addition to the literature on the boundary between elementary and advanced mathematics. While a number of excellent texts exist in other languages, either originally or by translation, heretofore we have had neither in English. The present claims only to be a compilation from other books and memoirs on the subject, yet it is really much more, as it is full of the ingenious devices peculiar to the British school of mathematics.

With the exception of the well known book of Enriques it is more systematic than any of the preceding treatises, and excels that one by being more practical and less verbose. In fact, in places the book is too compact to be of maximum service.

The problem is introduced by comparing the postulates and constructions of Euclid and showing that every construction is based on the interpretation of the solution of linear and quadratic equations. This excellent presentation is followed by a survey of the methods by which the various problems are to be attacked; it is unfortunate to bring in a considerable number of new words here, the meaning of which is given later.

The elements of cartesian geometry are developed and applied, but the explanations are too brief. A straight line is assumed tacitly to have a linear equation; the coordinates of the point common to two lines are shown to be expressible rationally in the coefficients in the equations of the lines. The section on the domain of rationality is well written.

The regular polygons are scarcely mentioned. It is stated entirely without proof that the inscription of a regular polygon depends on a binomial equation; the limitation on the exponent of the unknown is clearly derived. The entire construction of the heptadecagon is disposed of in ten lines and a figure.

The treatment of constructions by means of the ruler alone is prefaced by a discussion of projective properties, including vanishing lines, Desargues's theorem, cross ratio, harmonic section, involution, homography, and double elements. Many of these are given entirely without proof, and in no case is the proof full enough to be of much use. On the other hand, the application to elementary metrical cases is much more satisfactory.

Constructions by ruler and compasses bring in the cross ratio of four points on the circle; otherwise it keeps closely in touch with principles developed by Euclid. Then follows a chapter which employs the method of three trials, the idea of displacement, similarity, inversion, and duality, followed by another devoted to special devices-on account of limited space, etc.

The last two chapters are given up to constructions made by the ruler and one fixed circle, and by the compasses alone, respectively; each contains a scheme of comparison of the relative merits of different constructions.

To sum up, the immediate practical problem is everywhere well treated, but the foundations, taken from post-Euclidean ideas, are by no means so well done. Many of these ideas could have been dispensed with altogether, and the others easily proved for the purpose in hand, under restrictive hypotheses. It is true that this alternative also has objections; if a reader delves further into projective and analytic geometry, he would approach the fundamental theorems under false impressions. To one already familiar with these two subjects, the whole problem becomes an easy one, but to those who are familiar with elementary geometry and algebra only, the choice of the restrictive premises seems to the reviewer to be the wiser procedure.

The style is clear and the figures are well drawn; one unusual feature is that every algebraic equation is written in boldfaced type, making a formidable-looking page. The book is not provided with an index, nor are more than a very few exercises left for the reader to work out.
2. Squaring the Circle, by E W Hobson; Ruler and Compass, by H P Hudson; The Theory and Construction of Non-Differentiable Functions, by A N Singh; How to Draw a Straight Line, by A B Kemp.
2.1. Review by: Thomas Arthur Alan Broadbent.
The Mathematical Gazette 39 (327) (1955), 87.

"Can you lend me .. .?" "Where can I find a second-hand copy ... ?" Such requests about the books by Hobson and Miss Hudson have been distressingly frequent of recent years, since neither book has been easy to come by. They are related classics, for each deals essentially with what can and what can not be done by Euclidean construction. Styles differ, since Hobson sets himself a clear and well-defined goal, whereas Miss Hudson ranges over a wide field and exhibits a richness and variety of content, a freshness and vitality of treatment, hardly to be inferred from the austere title.

2.2. Review by: Cecil B Read.
The Mathematics Teacher 47 (2) (1954), 128.

The four monographs reprinted in one volume each originally appeared as a single publication, but recently have not been readily available. The selection seems a little odd, for the same groups will not be interested in the four monographs. Portions of the first two and the fourth will be of interest to high-school teachers and students; this will not be true of the third. Historical material on the circle squaring problem is exceptionally good; it is unfortunate that some notes on recent developments, such as the computation of π by modern calculating machines, could not have been included. Some interesting material for mathematics club programs is included. As a single example, the ruler and compass solution of the problem of the shortest path by which a spider in one corner of a room may reach a fly in the opposite corner of the ceiling.
3. Cremona Transformations in Plane and Space (1927), by Hilda P Hudson.
3.1. Preface.

Cremona transformations are powerful tools in many lines of research; the aim of this book is to bring together all that has so far been published on their construction and use, as regards points and loci in two and three dimensions. The most important application is to the resolution of singularities of curves and surfaces, which is treated fully in Chapters VII and XVI. The useful and interesting part of a transformation is the way in which it changes the neighbourhood of its fundamental elements, and the fundamental systems are first studied, both in general and in the examples which occur most often in the literature. Special attention is drawn to elements of contact.

A historical account of the subject is given in Chapter XVII; Cayley and Cremona are still leaders, though we British have fallen behind the rest of the world in their track. Pure and analytical methods are here used together; a curve or surface, a function and an equation are treated as the same thing; under different aspects, and certain liberties of language are taken in this connection, which make the sentences shorter but not less clear.

The bibliography shows how varied is the mass of material; yet the space theory is far from complete, and it is hoped that the list of outstanding problems on p.394 may attract more workers. The most promising line of advance is probably from space of higher dimensions: this I must leave to other writers.

By the untimely death of Miss Grace Sadd, a mathematician of promise, and my friend and fellow-worker, the book has suffered loss and delay. Chapter III is mainly her work, also the collection of material for Chapters V and VI, and much helpful criticism of all the MS. which then existed.

I offer grateful thanks to Mr Arthur Berry, who first introduced me to the subject; to Mr T L Wren, for his skilled and tireless work on the proofs; to many correspondents who have helped with the bibliography; and to the Cambridge University Press for their care in the printing.

3.2. Table of Contents.

Part I. Cremona Plane Transformations

I. Outline of the General Plane Theory
II. Clebsch's Theorem
III. The Quadratic Plane Transformation
I. Planes Distinct
II. Planes Superposed
III. Involutions
IV. Composition and Resolution of Plane Transformations
I. The Problem of Composition and Resolution
II. Construction of Tables
III. Properties of the Characteristic Numbers
V. Transformation in One Plane
I. Planes Superposed
II. Involutions
VI. Special Plane Transformations
I. The De Jonquières Transformations
II. Other Special Transformations
VII. Resolution of Singularities of Plane Curves
VIII. Noether's Theorem

Part II. Cremona Space Transformations

IX. Outline of the General Space Theory
X. The Quadro-Quadric Transformation
I. Spaces Distinct
II Spaces Superposed
XI. Postulation and Equivalence
XII. Contact Conditions
I. Points of Total Contact
II. Points of Partial Contact
III. Curves of Contact
XIII. The Principal System
XIV. Special Space Transformations
I. Transformations of Low Degree
II. The Bilinear T3-3
III. Monoidal Transformations
IV. Other Special Types
XV. A Cubo-Quartic Transformation
XVI. Resolution of Singularities of Surfaces
I. Composition of Space Transformations
II. Resolution of Singularities of Surfaces
III. Second Method of Resolution
IV. Classification of Transformations
XVII. History and Literature
XVIII. Bibliography

3.3. Review by: Virgil Snyder.
The American Mathematical Monthly 34 (9) (1927), 487-488.

The Cambridge University Press is famous for its comprehensive list of standard works on fundamental subjects of mathematics. Geometers will welcome this one to a worthy place in that distinguished company. The general plan is an elevated one; in the opening chapter the reader is introduced to homaloidal nets, postulation and equivalence of plane systems, the direct and inverse fundamental systems, followed in the second by Clebsch's theorem and its arithmetic consequences. Quadratic transformations are discussed under three headings; planes distinct, planes superposed, involutions, in 24 pages, yet nearly every known result is included. Then follows the discussion of series of composition, involution, and the application to the resolution of singularities of plane curves. This last chapter could have been made more useful and attractive by the use of figures and of more illustrative examples. The theory of plane transformations is well rounded out in the first third of the book.
It is particularly appropriate that an author who has enriched the field by so many important contributions has now put them in a proper setting by presenting this well-proportioned and carefully elaborated treatise on the whole subject.

3.4. Review by: F Puryer White.
The Mathematical Gazette 14 (194) (1928), 148-149.

The theory of Cremona transformations of the plane may be regarded as complete. Noether's theorem, first enunciated by him and by Clifford in 1869 and finally satisfactorily proved by Castelnuovo in 1900, that every such transformation may be resolved into the product of quadratic transformations, forms the coping-stone of the edifice. Miss Hudson then is able to present this part of the subject in a finished form. As she remarks, there are no very obvious gaps in the theory; though of course the consideration of particular cases can be continued as much as one wishes and may lead to results of geometrical interest. And the various problems enumerated by Coble in his interesting report to the American Mathematical Society (Bull. Amer. Math. Soc. 28 (1922), 329-364), with reference to the construction of an invariant theory of the Cremona group, and to various applications to algebra and modular functions, will no doubt repay attention. Hence, the account given in the first section of this book, the first with any claims to completeness, is of the greatest possible value. But with regard to Cremona transformations in three or more dimensions the situation is quite different. Here, again, Miss Hudson's treatment is invaluable, but for a somewhat different reason. The literature of the subject is so voluminous and so widely scattered that this authoritative and well-written presentation of what is already known will prove of incalculable use in furthering progress.

3.5. Review by: F P W.
Science Progress in the Twentieth Century (1919-1933) 22 (88) (1928), 705-706.

The most general (1, 1) correspondence between the points of two straight lines is, of course, the bilinear relationship, the projectivity. It is curious how long the notion remained that the same kind of thing was true for a plane and for space of three dimensions. Magnus and others thought that such relationships must be given by bilinear equations; which restricts us to linear and quadratic transformations for the plane and to linear, quadratic and cubic transformations for space. Fortunately it is not so, and, since the time of Cremona, transformations of all orders have been studied both for their own sake and for their important applications to the general theory of curves and surfaces. The literature of the subject is immense - Miss Hudson gives an annotated bibliography with 417 entries, and leaves three and a half pages blank for more - and at last we have a really comprehensive and scholarly treatise surveying the whole field. Doehlemann's second volume gives us merely an outline; Sturm is better, but far from exhaustive; Miss Hudson's book will undoubtedly remain the authority on the subject for years to come. English mathematics, Cambridge and, I think, we may say, considering the author's parentage, the dedication and a name in the Preface, St John's College, are to be congratulated.

The general theory of Cremona transformations in the plane may be said to be more or less complete; in particular, we have the theorem, due to Noether, but first completely proved by Castelnuovo, that every such transformation can be resolved into the product of quadratic transformations. But in space of three or more dimensions the position is very different. There is no extension of Noether's theorem; any set of fundamental transformations from which all others can be compounded must be an infinite set, but that is practically all that we know about it. The variety of fundamental elements is very great, the simple equations of condition in the plane case are replaced by equations whose complication is appalling, and we can no longer regard contacts as particular cases. Thus this book will serve, not only as a most interesting guide to what has already been done, but also as a jumping-off point for further research. Indeed, Miss Hudson finishes off with a list of outstanding problems, none of them easy, of course, to which students might well turn their attention.

It is churlish to grumble at omissions when one is given so much. Also it is perhaps not strictly included in the subject. But I should have liked from Miss Hudson some account of Castelnuovo's proof that all plane involutions are rational. And in the historical summary (Chapter XVII), while perhaps three British names, Kelvin, Cayley and Clifford, are more than one would expect, I miss a reference to MacLaurin and Braikenridge. It is true that most authors who mention them, such as Pascal and Severi in his new book, say that they really did not know what they were doing, but I happen to have a copy of Braikenridge, Exercitatio geometrica (1733), on my shelves, and there as plain as anything on page 24 is the quadratic transformation with the remark that it transforms a curve of order nn into one of order 2n2n.

3.6. Review by: A B Coble.
Bulletin of the American Mathematical Society 34 (1928), 373-375.

The appearance of a first exhaustive treatise in any field of mathematics is a matter of concern to those who pursue the particular subject. Books of that type frequently determine the trend of mathematical thought and progress for a considerable period. Perhaps no topic in algebraic geometry has been in greater need of such exposition than Cremona trans- formations. The earlier presentations are either elementary or incidental to some immediate geometric need. Existing encyclopaedic accounts are rather cursory. Thus a large body of researches on the subject, widely distributed in the journals, has been either inaccessible or unknown to those who might wish to become acquainted with the field.

To digest and to unify this mass of material was a task which demanded not merely a mastery of the subject but also an uncommon capacity for detail. This task Miss Hudson has accomplished in a most admirable manner in the book under review.

The book itself gives an impression of unity which is rather remarkable in view of the diversity of the transformations of which it treats. This doubtless is due to the wisdom of the author in selecting from the field a naturally related group of topics. Only transformations in the plane and in space are considered. For each case these are discussed first with reference to the properties common to all and secondly with reference to their division into various types. The single application considered is to the resolution of singularities of curves and surfaces and a treatment of this is practically inevitable since such singularities are present in the transformations. This is the division of the subject to which most of the author's own contributions have been made. No geometric applications are given except as they may be involved in the construction of a type, or as they may be inherent in the general class such as the isologues of a transformation in superposed planes or the complex associated with a transformation in space. Applications to other fields of mathematics are omitted. No account of Cremona groups, finite or infinite, appears.
In general the exposition is clear, brief, and effective. One may differ at times from the author with respect to methods of proof and emphasis on aspects of the theory.
It is in connection with the space theory that the book will be of greatest service to the advanced reader. The original articles, in widely different notations, with many obscurities and not a few errors, are here combined into an organic whole whose lapses from perfection are as a rule clearly indicated.

Last Updated December 2021