1.1. Preface by Václav Hlavatý.

I would like to express my heartfelt gratitude to everyone who in any way facilitated the publication of this work:

To Prof Dr B Bydžovský and Associate Professor Dr V Jarník for their dedicated and careful reading of the manuscript and for the advice they provided to the formal and substantive editing of the book, to Associate Professor Jarník also for his help in reading the proofs, to Professor Dr L Berwald for kindly supplementing the literary information, to Assistant Technician M Mikan for carefully drawing the figures according to my sketches, to Professor Fr Balad for carefully compiling the subject index;

To the Union of Czechoslovak Mathematicians and Physicists for their care and dedication in publishing this monograph, and to the printing house "Politika" for the exemplary typesetting.

In Prague in December 1925.

V. H.

1.2. Preface by B Bydžovsky.

There are few mathematical theories that can match non-Euclidean geometry in terms of the richness of its relations to other branches of human thought, the inner charm of its structure, and at the same time the uniqueness of its position within the overall organism of mathematical sciences, especially when we look at its development. Modern mathematics has its roots in a period that lies about 300 years behind us, and to correctly understand the development of most modern mathematical disciplines, it is not necessary to go further back than this period at most. However, the significance of non-Euclidean geometry, which in its entire concept belongs to the modern way of thinking, will not be fully understood unless we place it in connection with the great period of Alexandrian mathematics, more than 2000 years away; the bridge that connects the period of the famous Euclid with the beginnings of non-Euclidean geometry is among the most interesting pages of the historical miniature of mathematics. And both historically and philosophically, this geometry has an extraordinary position. It arose at the time of Kant's belief in the apriorism of spatial perception as a rebellion against this belief, perhaps unintentional, but still full of noetic and psychological consequences. From the logical point of view, it meant a thought structure of a completely new kind and of an unusual character. It is not surprising that a theory so different from everything that geometry had brought up to this point did not immediately meet with universal understanding, but it gradually began to make an impression with the novelty of its ideas. It began to determine a new path for geometry. It contributed most to the fact that the significance of opinion for geometry and mathematics in general was precisely defined. This is related to the fact that it was non-Euclidean geometry that initiated research into the foundations of geometry, which is one of the distinctive features and glory of the critical 19th century. At the same time, however, the discovery of non-Euclidean geometry gave the first impetus to another direction in modern geometry, the fruit of which has only been coming to fruition in recent decades and represents almost a revolution in geometric thinking: to the gradual generalisation of the mathematical concept of space, which from the narrow limitation of three dimensions with Euclidean metrics gradually expanded to the concept of $n$-dimensional manifolds with very general forming principles.

I must limit myself to these few terms in my attempt to characterise non-Euclidean geometry. I would only add that, due to the influence of the theory of relativity, this purely mathematical discipline has recently come into sharp relation to reality, thanks to various attempts at a new interpretation of the structure of the universe, which has spread interest in it beyond the narrow circle of experts.

If Dr Hlavatý undertook the difficult but meritorious task of writing the first Czech textbook of non-Euclidean geometry, he thereby met not only the needs of specialists, but also the interests of that part of the educated public that is eager to get to know the scientific component of worldview closely. In interpreting this geometry, one can proceed in various ways; I think that the method chosen by the author of this book best corresponds to the current state of science and the current position of non-Euclidean geometry in the overall structure of geometry. I wish his book that the benefit derived from it will be proportional to the importance and beauty of the subject under discussion and the effort he has made to explain the difficult material in a clear and accessible way.

In Prague in November 1926.

B Bydžovsky.

2.1. Description of the book

This book on Differential Geometry of Curves and Surfaces and Tensor Calculus was significant for bringing together the traditional, intuitive, "visual" differential geometry of curves and surfaces (studied by Monge, Gauss, etc.) with the abstract "tensor calculus" developed by Ricci and Levi-Civita. It covers the intrinsic geometry of surfaces, moving frames, curvature, and geodesic lines, using modern tensor notation to analyse them.

2.2. Reviews.

For reviews see the reviews of the German edition by J A Todd (Review 3.2 below), D J Struik (Review 3.3 below), and F A Ficken (Review 3.4 below).

3.1. Note.

This book is a translation of the Czech book Diferenciální geometrie křivek a ploch a tensorový počet (1937) into German.

3.2. Review by: J A Todd.
The Mathematical Gazette 23 (257) (1939), 491-492.

This work is a systematic treatise on differential geometry in ordinary Euclidean space of three dimensions. Its scope is roughly that of the existing standard treatises on the subject, but it makes extensive use of modern methods and uses throughout vectors and tensors, the necessary properties of these being developed in the text. The treatment of curves follows the conventional lines, though with more precision than in many works on the subject. When we come to the theory of surfaces the conventional order of treatment is somewhat modified. A very extensive chapter, amounting to some two hundred pages, is devoted to those properties of surfaces which depend only on the first fundamental form: conformal representation, geodesics and the theorems of Gauss and Bonnet. Only then is the second fundamental form introduced, leading to a study of lines of curvature, asymptotic lines, and so on. A final chapter discusses particular types of surfaces (ruled, minimal, etc.).

This book may be warmly recommended; the text is precise and clear, and the order of development of the subject is logical. The printing is beautifully done, and there seem to be comparatively few misprints even in the most complicated formulae.

3.3. Review by: D J Struik.
Mathematical Reviews MR0000161.

This textbook of differential geometry of curves and surfaces in ordinary space is a translation from the Czech. It represents an attempt to present the classical material with the full use of vector and tensor methods, and as such has some similarity to the first volume of Duschek-Mayer's textbook of 1930. Hlavatý's book contains (generally speaking) more material, and also differs in presentation, method and notation from its predecessor in this field.

The book is divided into four chapters. The first chapter, of 84 pages, is devoted to plane and space curves. It contains the customary topics, including natural equations and minimal curves. The notation is mainly vector analysis.

The second chapter, from page 85 to page 313, deals with the first fundamental form of surface theory. We find here an introduction into the algebra and analysis of tensors, which may have its independent use. It is applied to conformal mapping, geodesics, Gaussian curvature and applicability of surfaces.

The third chapter, from page 314 to page 434, discusses the second fundamental tensor. This leads to spherical representation, Dupin indicatrix, lines of curvature, asymptotic lines, and construction of surfaces with given fundamental tensor or tensors.

The last chapter, pages 435-560, contains special surfaces, as ruled, Weingarten, minimal, spherical, pseudospherical, Monge and Joachimsthal surfaces. A carefully composed index closes the book.

Among the many points of interest we may mention the discussion of geodesic mapping, the construction of surfaces from the spherical image of conjugated and asymptotic curves, Bonnet's theorem on the determination of a surface with given fundamental tensors, and a formula of all surfaces with given metrical tensor. A trivalent symmetrical tensor stands godfather to ruled surfaces. The concluding articles deal with surfaces with one congruence and with two congruences of plane lines of curvature, and end with Dupin cyclids.

3.4. Review by: F A Ficken.
Bull. Amer. Math. Soc. 46 (7) (1940), 597-598.

This treatise presents a large portion of the classical differential geometry of one and two dimensional subspaces of ordinary Euclidean space. Just enough vector and tensor analysis is given to enable the reader to manage profitably the abbreviated symbolism. Definitions and results are stated in such a way as to generalise readily to higher dimensions.

The first chapter is devoted to curves. After the theory is developed in terms of a general parameter, an account is given of the various specialisations arising from the use of the arc length as parameter. In particular, the construction of a curve from its curvature and torsion is treated carefully.

The second chapter concerns those properties of a surface which depend only on its metric tensor. The absolute differential is used systematically. There is an unusually full discussion of the applicability of surfaces, including explicit equations for developing a surface on a plane or on a surface of revolution.

The normal to a surface leads, on differentiation, to a tensor associated with the behaviour of the surface toward the ambient space. In the third chapter, those properties are discussed which depend on this (second fundamental) tensor. A feature of this chapter is the discussion of the explicit construction of surfaces having prescribed first or first and second fundamental tensors.

The fourth chapter contains applications to a variety of special surfaces, notably ruled surfaces and minimal surfaces. There is no treatment of quadrics nor of congruences of lines.

The book is carefully written, and a high level of explicitness is maintained in the details of the argument and in the treatment of special cases. The reviewer feels that two exceptions to this statement may confuse readers unfamiliar with tensor analysis. "Tensor" is so defined that there is no distinction between a tensor and the set of its components in any particular coordinate system. This distinction - analogous to that between the number two and a couple of apples - is the key to the geometric significance of tensors, and this point seems to have been obscured by the form given to the definition.

In the second place, "differential invariant" is defined in the algebraic sense - as a function which is transformed by substituting for the old variables the appropriate functions of the new variables and multiplying by a certain power of the Jacobian. It is then stated that "The problem of metric differential geometry ... consists in the study of the invariants ... which can be obtained from the equations of the surface (or curve)." As examples, two invariants for each dimension are constructed, the whole treatment covering about five pages. No further mention is made of the concept. This procedure appears open to several objections. First, no explicit connection is established between the remainder of the text and "the problem of differential geometry." Second, it seems pedagogically unsound to introduce such a basic concept without giving the reader more definite occasion to become familiar with its applications and to grasp at least one precise significance of a word having a confusing variety of meanings. Finally, the term has been so used in other books that tensors are "invariants." If a practice opposed to this usage is adopted, a word of caution to readers seems desirable.

The book has an ingenious index which greatly enhances its usefulness as a source for reference. There is no bibliography, nor exact reference to sources, nor exercises. In dealing with surfaces, roman numerals I and II are used as indices for quantities pertaining to the surface ; the gain in emphasis seems offset by the visual hesitation in distinguishing I II from II I.

Chiefly for its thoroughgoing use of tensor methods, the book is a valuable complement to the available treatises. It is distinguished from them by numerous interesting details in the presentation

4.1. Review by: F Vyčichlo.
Mathematical Reviews MR0031741.

This textbook deals with one-parameter families of linear and quadratic figures (of points on a straight line, of lines in a pencil, of points on a conic, of conics with four points common or with four tangents common, of straight lines on a quadric, etc.). A projective property is defined as an invariant property relative to the groups of projective transformations in a figure. The construction of the theory is based on a projective scale on the real straight line; this scale is constructed by means of the quadrangle. The proofs are not always rigorous. The complex points of the figures are also considered; the book contains many elementary synthetic projective constructions of figures, e.g., the constructions of conics. The dual considerations, theorems, and constructions are also included.

5.1. Review by: A Schwartz.
Mathematical Reviews MR0018936.

By using Plücker coordinates, a straight line can be made to correspond to a point on a four-dimensional quadric in a five-dimensional projective space (called the $K$-quadric in $K$-space). The goal of this book is to study systematically the differential geometry of the $K$-quadric and its submanifolds and to carry the results back to the corresponding line-manifolds. For instance, a $K$-curve on the four-dimensional $K$-quadric, and with it the corresponding ruled surface, is determined by three curvatures. This point of view for differential line geometry has been investigated before in papers by W Haack, K Takeda and J Kanitani [references on p. vii], but nowhere in as great detail as here. It is especially valuable in studying properties of ruled surfaces, line congruences and line complexes, and allows the author to carry through projective, affine and metric studies at the same time. However, the basic correspondence is hard to use in situations where the line is not considered the primary element, and questions which lead to point set studies are left practically untouched. The tensor calculus is used throughout the book; those parts of the tensor calculus which are needed in the text are presented concisely in a thirty-one page appendix.

Here we can only touch a few of the subjects covered in this detailed, thorough, clearly written text. In part one, pages 1-50, the basic correspondence is introduced and the theory of linear line-manifolds is studied. The concept of projective angle between two nonspecial complexes which determine a nonparabolic congruence is presented. In part two, pages 50-132, ruled surfaces are studied. They correspond to $K$-curves on the $K$-quadric. The Chasles correlation and Lie osculating quadric are presented, as are five specially selected complexes which satisfy Frenet equations containing three projective curvatures. Separate additional sections go further into the affine and metric aspects of the theory. In part three, pages 132-299, line congruences are studied. They correspond to $K$-surfaces on the $K$-quadric. First, fundamental concepts like elementary surface and focal surface are presented. Then, studying the $K$-surfaces without reference to their imbedding, tensors $a_{bc}$ and $g_{bc}$ are introduced and their algebraic properties interpreted in affine and metric geometry. A metric, asymmetric, integrable connection is then considered and Frenet formulas for $K$-curves on the $K$-surfaces set up. Finally, the $K$-surfaces are studied as submanifolds of the f-space and a conformal connection is introduced in a natural way. The algebraic properties of congruences whose $K$-images have five, four, or three-dimensional second osculating spaces are studied, as are the fundamental equations of Gauss, Weingarten, Kühne, and Codazzi-Mainardi.

In part four, pages 300-469, complexes are studied. They correspond to three-dimensional manifolds on the $K$-quadric. The principal correlation for a line in a given complex is considered. Then a conformal connection is introduced, leading again to the construction of Weingarten, Gauss, Codazzi-Mainardi, and Kühne fundamental equations. There follow detailed separate chapters on ruled surfaces in complexes, on line congruences in line complexes, and on ruled surfaces in line congruences in line complexes. In part five, pages 470-501, the line space is studied. Again fundamental tensors and a conformal connection are introduced, and the fundamental equations of Weingarten, Gauss, and Codazzi-Mainardi are derived. The line space is shown to be conformal Euclidean and its metric properties are studied in detail.

There was an attempt to make the various parts of the book independent, and hence there is some repetition. There are well chosen problems of varying difficulty distributed throughout the text, all illustrating and carrying further the material under consideration. There is a fine index at the end.

6.1. Review by: F Vyčichlo.
Mathematical Reviews MR0031742.

The second volume of this text deals with the projective geometry of two-parameter families of linear and quadratic figures. The projective punctual and lineal fields in a plane are studied by means of the projective transformations from Klein's point of view and many synthetic examples are constructed. Further, the non-Euclidean hyperbolic (or elliptic) plane with elementary geometry is studied. The foundation of projective geometry in the plane is based on the theorems proved in the first volume and on the analytic method. The complex points of the real straight line are defined as double points of an elliptic involution. The classification of the point-projective transformations is based on the double points; that of the reciprocities is based on the classification of the adjoint projective transformations. In the elementary Euclidean and non-Euclidean geometry in the plane distance and angle are defined by Laguerre's method. Some considerations, theorems and constructions seem to be superfluous, because they follow from analogous ones by the principle of duality.

7.1. Review by: F Vyčichlo.
Mathematical Reviews MR0036999.

This book for the layman contains only nonessential changes and complements to the first edition (1926). The book does not contain the axiomatics of geometry. It starts from the elementary notions of the analytic geometry of the metric plane and deals with non-Euclidean geometry as the study of properties invariant with respect to a certain group of linear transformations. From this point of view it deals with the one-dimensional non-Euclidean variety and the varieties of the hyperbolic, elliptic, and parabolic plane.

8.1. From the Preface.

In this book we study the differential geometry of line space and of its line manifolds. For this purpose, we use an algorithm that can be characterised briefly as follows: line space can be mapped on a four-dimensional quadric in five-dimensional projective space. (We call this quadric the $K$-quadric in $K$-space). Under this mapping, ruled surfaces, congruences of lines, and complexes of lines correspond to curves, surfaces, and three-dimensional spaces, respectively, in the (four-dimensional) $K$-quadric. The purpose of this book is the study of the differential geometry of these $K$-manifolds, and the interpretation of their various differential invariants for the corresponding line manifolds. For example, a $K$-curve, which is the image of a ruled surface, possesses three curvatures which in turn determine the $K$-curve, and therefore the surface, uniquely (to within initial conditions). A $K$-surface, the image of a congruence of lines, possesses a scalar Gaussian curvature, which vanishes for a linear congruence. For a three-dimensional subspace of the $K$-quadric, the image of a complex of lines, we can find, from its fundamental equations, a scalar Gaussian curvature that vanishes for a linear complex (in terms of a natural normalisation). Line space itself appears as a four-dimensional conformal-euclidean space.

This conception of line geometry has its advantages and disadvantages. On the one hand, we can investigate very fully the properties of line manifolds (see, for example, Chapters VII, VIII, IX, which deal with complexes and their subspaces); on the other hand, the above mentioned algorithm is unprofitable whenever we wish to conceive of a line, not as the primary element of space, but rather as a secondary, derived element. This is apparent in Chapter III, where we have brought together basic classical results from the differential geometry of the focal envelopes of a congruence. Because of this difficulty, we have almost without exception limited ourselves to the study of properties of line manifolds that remain unchanged along a given line and omitted from consideration properties of point sets on a line. Thus, for example, the reader will find no reference to the point geometry of various surfaces of a congruence, nor to anything like Darboux's derived congruences.

Lastly, the personal taste of the author has influenced the selection of material. The methods used in this book are based on the tensor calculus, as is to be expected from the fact that we are really concerned with a four-dimensional curved space. Since this space is itself embedded in a five-dimensional space, the problem becomes somewhat complicated. We have not regarded it as suitable to limit ourselves to the study of $K$-loci on the $K$-quadric without taking into account that the $K$-quadric is itself embedded in a five-dimensional space. Such a limitation would have greatly simplified the whole arrangement of the book, but it would not have corresponded to the actual state of affairs, even though line manifolds are frequently given without relating them to the four-dimensional line space. We have not used Study's transformation principle, which is so useful in metric geometry, because we have preferred a unified development for projective, affine, and metric geometry. The unified point of view of which we have spoken nearly always permits us to carry out the proofs of known theorems in a suitably generalised form. This work consists of five books and an appendix, subdivided into eleven chapters.

8.2. Summary.
Mathematical Reviews MR0057592.

A translation of the author's Differentielle Liniengeometrie. Minor corrections and modifications have been made in preparing the translation.

8.3. Review by: Alice T Schafer.
Bull. Amer. Math. Soc. 61 (4) (1955), 348-351.

This book was first published in Czech in 1941 and was later translated into German and published by Noordhoff in 1945. The present edition is a translation from the German edition with the author's collaboration. The translator himself has added to the text by suggesting changes in some theorems and by adding a few new ones.

The book is meant to be a definitive work on three-dimensional differential line geometry, where line geometry in 3-space is studied as point geometry on a 4-dimensional quadric in a projective point space of 5 dimensions. Klein discovered the mapping of line geometry onto the 4-dimensional quadric, and for this reason the quadric is called the Klein quadric (or $K$-quadric) and the 5-dimensional space the Klein- (or $K$-) space. This viewpoint of three-dimensional line geometry has been used by authors before, but never as extensively as in this text. Both the classical material on the subject and new contributions by the author have been included.

The text is necessarily quite long and the author has tried to over-come some of the difficulties of length by dividing his work into "books," each of which can be read without the others. This naturally leads to some repetition of material. The books are divided into chapters which are numbered consecutively throughout the text. There are five books: the first (Chapter I) an introduction to line space; the second (Chapter II) on ruled surfaces; the third (Chapters III, IV, and V) on congruences; the fourth (Chapters VI-IX) on complexes, and the last (Chapter X) on line-space. Tensor calculus is used at all times, and simplifies the notation. For those readers who are un-familiar with the tensor calculus the author has included in an appendix a straightforward, well written account of that part of the subject necessary for a reading of the text.

In the first book the author defines Plücker coordinates, Klein points, and Klein 5-dimensional space. He states that all topics, as far as possible, will be discussed from the viewpoints of projective, affine, and metric geometry. Unless explicitly stated otherwise, all theorems and definitions are given for projective geometry.

The standard material on ruled surfaces in projective differential geometry is given in the second book, ruled surfaces appearing in $K$-space as $K$-curves on the $K$-quadric. Also five particular linear complexes are introduced, as are three projective curvatures which determine a $K$-curve (hence the corresponding ruled surface) uniquely to within initial conditions. By use of one of the five special complexes, many of the familiar metric properties of ruled surfaces are derived.

More or less well known material on congruences, which are represented by $K$-surfaces in $K$-space, is presented in Chapter III. Chapter IV deals with congruences in affine and metric geometry, from both the algebraic and analytic viewpoints. Two uniquely defined metric tensors are introduced and used in the study of the congruences. Scalars representing the first and second mean curvatures of a congruence are defined in terms of these tensors. Normal congruences are studied, as are principal surfaces of congruences, distribution curvature, central points, and central planes. Pseudo-parallelism, autoparallelism, and teleparallelism are discussed and Frenet equations are developed for surfaces on congruences.

In Chapter V the relations between the $K$-surface and $K$-space are studied. For example, the fundamental equations of Weingarten, of Mainardi-Codazzi, and of Gauss are derived for congruences whose second osculating $K$-spaces are 5-dimensional (4-dimensional, 3-dimensional).

The complexes studied in the fourth book are not necessarily linear. Chapter VI deals with foundations and with fundamental equations concerning 3-dimensional manifolds in $K$-space (the images in f-space of complexes in 3-space). Algebraic complexes are studied, as well as the nature of a complex in the neighbourhood of its singular and special lines. The fundamental equations are derived.

Chapter VII is concerned with surfaces of a complex ($C$-surfaces). Elemental $C$-surfaces and elemental $C$-congruences are discussed, and a normal curvature for nondevelopable $C$-surfaces is defined. Definitions are given for two $C$-surfaces to be autoparallel, and for two nondevelopable $C$-surfaces to be extremal surfaces. It is shown that a necessary and sufficient condition that two nondevelopable $C$-surfaces be autoparallel is that they be extremals. A non-extremal surface has two curvatures which satisfy generalised Frenet equations and which determine the surface uniquely to within initial conditions. Absolute pseudoparallelism, absolute autoparallelism, and absolute teleparallelism are discussed briefly.

Under the mapping onto the $K$-quadric a congruence of a complex is represented by a $K$-surface contained in the 3-dimensional $K$-manifold which is the image of the complex. A normal surface of a congruence is defined in Chapter VIII, and it is shown that this surface is projectively orthogonal to every surface through the same line as the normal surface. Generalised fundamental equations are derived and used in determining whether there exists a $C$-congruence with two particular preassigned tensors.

Chapter IX deals with surfaces of a congruence of a complex (called $CC$-surfaces). Elemental principal $CC$-surfaces, characterised by extreme values of the outer curvature, are studied. Spheroidal lines, which are lines on which the elemental principal $CC$-surfaces are not determined uniquely, are discussed. Inner and outer curvatures of a $CC$-surface are defined. The inner curvature determines the $CC$-surface in the congruence to within initial conditions. These curvatures are closely related to the first curvature of the surface considered as a surface of the complex. The concept of torsion of a $CC$-surface leads to an equation connecting the invariants of the $CC$-surface. Brief mention is made of asymptotic surfaces of $C$-congruences.

The last book is devoted to a study of the entire line space represented by the $K$-quadric. This space is shown to be conformal-euclidean. A section is devoted to metric line space and there it is shown that the curvature tensor can be made to vanish. A short section is devoted to the resulting analagmatic geometry.

There are naturally advantages and disadvantages to the exclusive use of the mapping of line-space onto the $K$-quadric. Questions concerning line manifolds can be answered fully but, as the author himself points out, questions of point geometry must be left unanswered.

The exposition is clear throughout. Although this book has been written as a textbook, it is highly specialised and will probably be used more as a reference than as a text. The author has placed problems in the text, not at the end of sections, but where they arise naturally in the theory. The proofs of some of the simpler theorems have been omitted and left as exercises.

There is no bibliography, and this is a serious defect. Only a few references to other works are given. There is too much "organisation" in the writing. For example, there are Remark I, Agreement (I, 1), Theorem (I, 1), Definition (I, 1), etc.

There are a few minor misprints, but on the whole the book has been very well printed and proofread - this latter not always being the case today. The book is a valuable addition to the literature on line geometry, and a good translation into English makes it available to many more readers.

8.4. Review by: T J Willmore.
The Mathematical Gazette 39 (328) (1955), 160.

The original Czech edition of this book was published by the Czech Academy of Sciences in 1941. A German translation by M Pini was published by P Noordhoff in 1945. The present English translation by Professor H Levy is based upon the German text. Apart from numerous minor corrections and modifications throughout the whole book, the main difference between this and the German edition occurs in sections of chapter 9 dealing with irregular lines of congruences, and in certain sections of chapter 10 dealing with anallagmatic geometry.

It is well-known that by using its Plücker coordinates a straight line can be represented by a point on a four-dimensional quadric $K$ in a five-dimensional projective space. This book is essentially a treatise about the differential geometry of the quadric $K$ and its submanifolds, and its application to the study of properties of the corresponding line-manifolds. This method of investigation has been used previously in papers by W Haak, K Takeda and J Kanitani; but the method is developed here in great detail and in a manner which allows the simultaneous study of projective, affine and metric properties. The methods of tensor calculus are used throughout the book, and a carefully written appendix of 28 pages summarises those parts of the subject most frequently used.

The book is written in five parts, each of which can be read independently of the others. The first part (Chapter I) deals with the representation of lines by points of the quadric $K$. The second part (Chapter II) is concerned with ruled surfaces-these correspond to curves on the quadric $K$. The third part (Chapters III, IV and V) is devoted to the study of congruences. The fourth part (Chapters VI, VII, VIII and IX) deals with complexes. The fifth part is concerned with the study of line-space, including the derivation of the fundamental equations of Weingarten, Gauss and Mainardi-Codazzi.

The book is clearly written and contains a number of exercises which illustrate the text. There is a very useful index at the end of the volume. There is no doubt that the book will remain one of the standard works on this rather specialised subject for many years to come.