1.1. From the Preface.

This monograph intends to give a general survey of the different branches of the geometry of linear displacements which so far have received attention. The material on this new type of differential geometry has grown so rapidly in recent years that it is impossible, not only to be complete, but even to do justice to the work of the different authors, so that a selection had to be made. We hope, however, that enough territory is covered to enable the reader to understand the present state of the theory in the essential points.

The author wishes to thank several mathematicians who have helped him with remarks and suggestions; especially Dr J A Schouten of Delft and Dr N Hansen Ball of Princeton.

1.2. Review by: Patrick Du Val.
The Mathematical Gazette 21 (245) (1937), 299-300.

This is a brief but fairly thorough exposition of the modern tensor algebra and its geometrical applications which have arisen in connection with the Relativity theory and more lately with the theory of spinors. The whole treatment is very abstract; algebraic ideas are fundamental, geometry is explicit but of the most formal, physical applications are not hinted at. The work is rather on the scale and in the style of a Cambridge Tract, and gives a comprehensive survey of the topic; but it must be said that it seems too condensed to be tolerably easy reading to anyone who does not know a good deal of the matter already. The literary style, and the mathematical style, have the concise formality of German mathematicians, and the notation is very complicated, perhaps unavoidably so in view of the range of different geometries covered; suffixes appear in four positions instead of the familiar two, and dashes precede as well as follow the symbols which they affect. The page or less at the end headed "Notations used" is not very helpful except as a reminder to one who has already read most of the book.

On the other hand few are likely to want to master this subject who have not already some knowledge of at any rate Riemannian geometry, and the affine connections, and with this much preparation Dr Struik should be more or less intelligible. He leads the reader on from here to the Hermitian and projective connections, by means of Veblen's "geometry of paths", and deals also with the geometry induced in a manifold by being embedded in another whose geometry is known.

There are nearly eleven pages of bibliography, chronologically arranged and, as far as I can see, very far reaching and carefully constructed.

1.3. Review by: W Fenchel.
Matematisk Tidsskrift. B, tematisk tidsskrift B (1938), 88-89.

This booklet gives an overview of the recent development of differential geometry in the direction founded by Riemann. This subject became particularly topical through its application in the general theory of relativity, and many of the investigations presented here have their origin in attempts to interpret the electrodynamic field theory and Dirac's relativistic quantum mechanics as "space structures" just as Einstein did for gravitation. The general concepts of space that have thus been arrived at can be described, according to Cartan, in the following way. One starts from a "a certain number of times differentiable", but otherwise structureless n-dimensional space X_n. To every point x in X_n one now imagines a k-dimensional space S_k(x) attached, and all these spaces S_k(x) are assumed to have a structure, since they carry a geometry in the sense defined by Klein's Erlanger Program, i.e., it is assumed that a transformation group in S_k(x) is given, but in such a way that all spaces S_k(x) are mutually isomorphic. For example, the S_k s can be Euclidean spaces, projective spaces, unitary (i.e., complex Euclidean) spaces, etc. The first and most important example is obtained by taking for S_k(x) the "tangent space" of the space X_n corresponding to the point x, which is an n-dimensional vector space; its group is the homogeneous affine group. Furthermore, it is assumed that for each x and dx there is a unique mapping of the space S_k(x) onto the space S_k (x + dx) corresponding to the neighbouring point x + dx. Such a mapping in dependence on x and dx is called a connection. Depending on the structure of the spaces S_k(x), the connection is called Euclidean, projective, etc. The author gives a summary treatment of the most studied special cases, several of which are related to the mentioned topics in physics. Finally, a list of references is given.

2.1. Review by: C E Weatherburn.
The Mathematical Gazette 20 (237) (1936), 60-61.

The work before us is the first volume of the second edition of a book that was published in one volume some years ago. Though appearing under the same title this new edition is, as the authors affirm, really an entirely new book. The volume under consideration is largely the work of Professor J A Schouten. It is concerned mainly with the mathematical preliminaries of the subject, developing the calculus employed and the theory of Connections. It takes the place of the first two chapters of the first edition, but contains a great deal more than those, which were confined to manifolds determined by quadratic metrics.

The volume is divided into two Parts (or Chapters), the first of which deals with the Algebraic portion of the subject in five sections, while the second is concerned with the theory of Connections and embraces seven sections. The first section treats the introductory notions of coordinate systems, geometrical objects and transformation groups, and mentions the characteristics of Kleinian geometry. The second section introduces the simple but important case of the affine group, the corresponding affine space being devoted by E_n where n is the number of independent variables. This section, which is the longest in Part I of the volume, treats such fundamental concepts as invariants, directions, covariant and contravariant vectors, affinors, densities, Hermitian quantities, projection and various types of indices. The section closes with a brief consideration of a few special cases corresponding to certain restrictions on the affine group. The third section is concerned with affinors of valency 2 in E_n and their corresponding matrices. Besides meeting sums and products of affinors we are introduced to the reciprocal, the isomer, the complex conjugate, and the isomer of the complex conjugate. Hermitian, symmetric and alternating affinors are seen to play an important part. The fourth section treats the algebraic geometry of an affine space R_n, which is associated with a fundamental tensor of constant components. The angle between two vectors is defined, and the meaning of the term orthogonal as applied to vectors or to groups of transformations is made clear. In particular infinitesimal orthogonal transformations are discussed, and also the principal directions of a tensor. Part I of the volume is brought to a close in section five, which treats the algebraic geometry of an affine space, U_n, with which is associated as fundamental tensor a covariant Hermitian tensor of order n with constant components. Results are obtained corresponding to those derived in the case of an R_n.

The second part (or chapter) of the volume, dealing with the theory of Connections, begins by introducing the local affine space at any point of an n-dimensional manifold, and then defines the gradient of a scalar field. Systems of reference, both holonomic and anholonomic, are considered, and there is a brief discussion of the Pfaffian problem. The seventh section is devoted to the covariant differential and the covariant derivatives of a vector, along with the idea of pseudoparallel displacement of the vector. The concepts of the divergence and the rotation of a vector field are also introduced. These ideas are further developed in the next section, which treats the general linear connection, as well as metric and half-metric connections, and deduces the conditions that a connection be Riemannian. Section 9 is concerned largely with the connection induced in a subspace of the manifold considered; and here we are introduced to the D-symbolism of Van der Waerden and Bortolotti, by means of which the results obtained are expressible in very compact forms. The tenth section is devoted to geodesics, geodesic manifolds and geodesic systems of reference. A geodesic is defined as a line whose direction undergoes a pseudoparallel displacement along the curve and a geodesic system of reference with pole at Pas one in which all the coefficients of the linear connection vanish at P. A proof is given of the existence of holonomic systems of this kind when the connection is symmetric, and of the extended theorem, due to Fermi and Eisenhart, according to which the coefficients of connection can be made to vanish at all points of a given curve. Section 11, which deals with curvature properties, is the largest in this volume, and contains some very interesting geometrical applications. The Riemann-Christoffel curvature tensor is introduced, the identities which it satisfies are proved, and their geometrical significance is discussed. Manifolds of constant Riemannian curvature are considered, as well as a generalisation of the theorem of Stokes. In the final section various problems in small deformations are discussed; and it is proved that a geodesic in V_n is a line whose length is stationary for small deformations of the curve.

This new edition thus represents a much wider treatment of the subject than that contained in the first edition, both in generality and in the number of topics undertaken. Still, as the authors mention in their Preface, they have intentionally limited the range of the work so as to exclude the geometries of Finsler and Berwald, as well as projective and conformal differential geometries. In the matter of notation they have discarded the double formulation of the first edition, and here confine themselves to that of the Ricci Calculus.

...

A large number of important and useful examples are given throughout the text, and their solutions are indicated at the end of the volume. Many of the examples are really additional theorems. There is a Bibliography of ten pages which, though not intended as a complete one, will prove very useful to the reader. An index of seven pages should also be very helpful; and a careful reading of the proofs has made it possible to print a short list of errata at the end of the book. The dedication is very appropriately to Dr Tullio Levi-Cività, who is one of the outstanding figures in this field of work.

A book, written by two men who have contributed so much to the development of the subject, is obviously a valuable addition to its literature. According to the Preface, the authors believed, or at least hoped, that they were writing an elementary treatise. The word "elementary" is of course a relative term. The book is not suitable for a beginner. But it will be a great help to the student who is already familiar with the elements of the subject and who wishes to extend his knowledge. We confidently recommend it to every mathematician who is interested in Differential Geometry.

2.2. Review by: H Hornich.
Monatshefte für Mathematik und Physik 45 (1936), A22.

Of this textbook, which has been significantly revised compared to the first edition, the first volume is now available. Written by J A Schouten, it is dedicated to algebra and transfer theory, while the second volume covers applications. The book is not only very suitable as an introduction but will also serve the expert well as a reference work due to the wealth of material it contains, especially the numerous exercises (with solutions) that supplement the text. Only a few aspects of the content will be highlighted here: In algebra, the consistent application of the kernel-index method, the treatment of Hermitian quantities, and the inclusion of elementary divisor theory and matrix theory. The text covers the theory of transmission, which is limited to linear transmissions, the use of anholonomic reference frames, the application of the D-symbols of van der Waerden and Bortolotti, normal coordinates, curvature theory, and a section on variation and deformation, as well as Lagrangian derivation. The needs of the physicist are always taken into consideration.

3.1. Review by: C E Weatherburn.
The Mathematical Gazette 23 (253) (1939), 98-100.

The work before us is the second part of the second edition of a book which was published in one volume some years ago; but the plan has been so altered and extended that this edition is really an entirely new book. The volume under consideration is the work of D J Struik. It is devoted entirely to geometry, the mathematical analysis employed having been developed by J A Schouten in the first volume, which appeared in 1935. (See Math. Gazette, Vol. XX, p. 60.) Though Riemannian geometry occupies a prominent place in the discussion, the treatment embraces also linearly-connected (or affine) spaces L_n. The geometrical objects introduced are not necessarily real, neither are the coordinates employed. A Riemannian space is denoted as usual by V_n; and, if the metric is positive definite, the author describes the space as "ordinary".

The volume embraces four chapters, of which the first is devoted to curves. This begins by considering curves in Euclidean 3-space, finding the equivalent of Frenet's formulae in any system of coordinates; and then extends the results to real or imaginary curves in Riemannian space of n dimensions. Points of inflection of any order are discussed, and the theorem is proved that an ordinary real curve in a V_n is uniquely determined when the positions of its tangent and normals are known at one point, and its n-1 curvatures are known functions of the arc-length measured from that point. An application of preceding results is then made in considering evolutes of any order of real curves in Euclidean space of n dimensions. A section is devoted to the treatment of curves in a linearly-connected space L_n, results being found which are analogous to the formulae of Frenet, though not exactly generalisations of them; and Hlavaty's theorem on the determination of a curve by its affine curvatures is established. Another section - the longest of the chapter - discusses congruences of curves, chiefly in Riemannian space. The usual topics arise including geodesic and normal congruences, orthogonal ennuples, Ricci's coefficients of rotation, congruences canonical with respect to a given congruence, and rectilinear congruences in Euclidean n-space. The chapter closes with a discussion of the properties of a system of paths (bahn-system), i.e. a system of curves such that, through two specified points in a given region, passes one and only one curve of the system. The chief emphasis is given to a "natural" system of paths, i.e. one which can be transformed into a system of geodesics by a conformal transformation of the fundamental tensor.

Chapter II is devoted to hypersurfaces of a Riemannian space. It covers the usual ground, beginning with the first and second fundamental tensors for the hypersurface, and then passing on to proofs of the theorems of Meusnier, Euler and Dupin. In connection with normal curvature, geodesics and asymptotic lines of the hypersurface are considered; and the discussion then proceeds to conjugate directions, lines of curvature and principal curvatures of the hypersurface at any point, and to asymptotic lines of any order. The chapter concludes after the equations of Gauss and Codazzi have been established, and their special forms for a hypersurface of a flat space Rn.

Chapter III deals at some length with a general subspace V_m of a Riemannian V_n. By considering the curvature of a curve in V_m relative to the enveloping space, the author is led to the discussion of normal curvature and curvature relative to V_m, and thence to geodesics of V_m and the definition of geodesic subspaces. The mean curvature vector of V_m is then introduced, leading to the mention of minimal subspaces and some of their properties. The discussion then turns to congruences of curves in a subspace, and in particular to conjugate systems and asymptotic systems. As subspaces of some importance, surfaces (V_2) are considered in some detail, and their properties at axial, planar and spatial points are examined. Special mention is given to Steiner surfaces and Veronese surfaces. The author then discusses curvatures of any order of a subspace of V_n, and proves the Frenet formulae for the subspace. The equations of Gauss, Codazzi and Ricci are also established for the subspace, and a number of applications are made. Some attention is given to questions concerning small deformations of a subspace V_m, and also to the different kinds of scalar curvature of the subspace. The last part of the chapter deals fairly fully with subspaces of a space of constant curvature; and in this connection the class of a Riemannian space is discussed.

Chapter IV, the longest in the book, is devoted to a number of selected topics, among which the linearly-connected space L_n occupies a prominent place. The chapter opens with a discussion of various curvature properties of an L_n, and proves the Gauss-Codazzi-Ricci equations for an Lam. It then passes to small deformations of a manifold, among which infinitesimal deformations which leave the metric of a space unaltered are given prominence. A fairly lengthy section discusses transformations of an affine space A_n of symmetric connections, such that the paths of A_n correspond to those of the transformed space, and proves among others the theorems that two Riemannian spaces S_n of constant curvature can be so transformed into each other, or an S into a flat space R_n; while a V_n In can be so transformed into an S_n only when the former is also an S_n. This section closes with a discussion of projective connections, which are the basis of projective differential geometry. The author then proceeds to consider the conformal representation of one Riemannian space upon another. Incidentally it is shown that, for a conformal transformation of the fundamental tensor of a V_n, umbilics of the space remain umbilics, and the principal directions of a hypersurface of the space are invariant. A good deal of attention is devoted to conformal-euclidean Riemannian spaces, i.e. spaces which may be conformally transformed into flat spaces. The discussion is also extended to affine spaces by the consideration of conformal connections; and the chapter closes with a fairly long section dealing with Hermitian connections.

A large number of important and useful examples are given throughout the text; and at the close of the book about fifty pages are devoted to hints and methods for their solution. Many of the examples are really additional theorems. There is a bibliography of twenty-two pages which, though not intended as a complete one, will prove very useful to the reader. An extensive index of sixteen pages should also prove a great convenience. The dedication is given very appropriately to Dr Tullio Levi-Cività, who is one of the outstanding figures in this field of work.

A book, written by two men who have contributed so much to the development of the subject, is clearly a valuable addition to its literature. As the treatment is by no means elementary, the reader is advised to acquire some knowledge of Riemannian geometry from a more introductory work before beginning the study of this one. The mathematician who wishes to extend his knowledge of Differential Geometry, and is prepared to take the trouble necessary to master the notation employed by the writers, will find these two volumes a mine of information and, we hope, a source of inspiration. To such a one we confidently recommend the book.

4.1. From the Introduction.

Mathematics is a vast adventure in ideas; its history reflects some of the noblest thoughts of countless generations. It was possible to condense this history into a hook of less than three hundred pages only by subjecting ourselves to strict discipline, sketching the unfolding of a few main ideas and minimising reference to other developments. Bibliographical details had to be restricted to an outline; many relatively important authors - Roberval, Lambert, Schwarz - had to be by-passed. Perhaps the most crippling restriction was the insufficient reference to the general cultural and socio-logical atmosphere in which the mathematics of a period matured - or was stifled. Mathematics has been influenced by agriculture, commerce and manufacture, by warfare, engineering and philosophy, by physics and by astronomy. The influence of hydrodynamics on function theory, of Kantianism and of surveying on geometry, of electromagnetism on differential equations, of Cartesianism on mechanics and of scholasticism on the calculus could only be indicated in a few sentences - or perhaps a few words - yet an understanding of the course and content of mathematics can only be reached if all these determining factors are taken into consideration. Often a reference to the literature has had to replace an historical analysis. Our story ends by 1900, for we do not feel competent to judge the work of our contemporaries.

We hope that despite these restrictions we have been able to give a fairly honest description of the main trends in the development of mathematics throughout the ages and of the social and cultural setting in which it took place. The selection of the material was, of course, not exclusively based on objective factors, but was influenced by the author's likes and dislikes, his knowledge and his ignorance. As to his ignorance, it was not always possible to consult all sources first-hand; too often second- or even third-hand sources had to be used. It is therefore good advice, not only with respect to this book, but with respect to all such histories, to check the statements as much as possible with the original sources. This is a good principle for more reasons than one. Our knowledge of authors such as Euclid, Diophantos, Descartes, Laplace, Gauss, or Riemann should not be obtained exclusively from quotations or histories describing their works. There is the same invigorating power in the original Euclid or Gauss as there is in the original Shakespeare, and there are places in Archimedes, in Fermat, or in Jacobi which are as beautiful as Horace or Emerson.

* * * * * * *

Among the principles which have led the author in the presentation of his material are the following:

1. Stress the continuity and affinity of the Oriental civilisations, rather than the mechanical divisions between Egyptian, Babylonian, Chinese, Indian, and Arabian cultures.

2. Distinguish between established fact, hypothesis, and tradition, especially in Greek mathematics.

3. Relate the two trends in Renaissance mathematics, the arithmetical-algebraic and the "fluctional," respectively, to the commercial and the engineering interests of the period.

4. Base the exposition of Nineteenth Century mathematics on persons and schools rather than on subjects [Here Felix Klein's history could be used as a primary guide. An exposition by subjects can be found in the books by Cajori and Bell, or with the more technical details, in the "Encyklopaedie der mathematischen Wissenschaften" (Leipzig, 1898-1935, 24 vols.) and in Pascal's "Repertorium der höheren Mathematik" (Leipzig 1910-29, 5 vols.).]

4.2. Review by: M Dehn.
The American Mathematical Monthly 56 (9) (1949), 643-644.

To write on the history of mathematics has been, from old times to our days, a very attractive task. To write a history of mathematics is a tremendous undertaking because it is identical with a presentation of mathematics itself immersed in its human surroundings. As an approximate completeness is here even more unthinkable than in other historic fields, it is laudable to write a concise history as this is presented.

This work has the great advantage of being written by an accomplished mathematician as well as a reliable historian. Of course, the choice of what is described and what is omitted depends on individual taste. In the beginning we find, caused by recent discoveries about pre-Greek mathematics, a relatively detailed report on the beginnings. The report on the nineteenth century, on the other hand, is obviously influenced by F Klein's inspiring work on the development of mathematics during this period.

The reviewer regrets that the author does not mention the beginning of Axiomatics and does not give the name of the basic work of the "Father of Axiomatics," M Pasch. This trend in the last quarter of the nineteenth century is important because of its continuation in the present century, particularly in this country.

There are, of course, many such points, more or less important, with which some people might disagree. However, this cannot change the impression that it is a very valuable book, especially for three reasons. First, the reader will find many interesting details, difficult to find in other, much more voluminous treatises. Second, the author gives ample text references. Also, there are many portraits, drawings and text reproductions. It is only regrettable that, apparently, it was not possible to print the reproductions on a more adequate paper.

4.3. Review by: E J Dijksterhuis.
Mathematical Reviews MR0026572 (10,173a).

A very condensed survey of the history of mathematics covering the whole development from the earliest times up to 1900. The conciseness of the work imposed the necessity of a strict limitation to the main currents of thought. This, however, did not prevent the author from paying due attention to what always has been his favourite topic, viz., the relations of mathematics to the general cultural and sociological background of the period. In this way the two trends which can be distinguished in Renaissance mathematics, the arithmetical-algebraic and what may be called the fluctional one, are related, respectively, to the commercial and to the engineering interests of these centuries. In treating of Greek mathematics, that most important, but very imperfectly known, phase of the development of mathematics, the author takes care to distinguish between established facts, plausible theories, wild hypotheses and traditional ideas. The exposition of 19th century mathematics is based more on persons and schools than on subjects. The work is richly illustrated.

4.4. Review by: T. A. A. B.
The Mathematical Gazette 39 (328) (1955), 175.

This volume, a British edition of an American work reviewed in the Gazette, December 1953, is a marvel of compression. The author has concentrated on main lines of development, stopping at the end of the nineteenth century, in order to compress what he has to say into 300 pages; he also assumes that the reader is moderately familiar with the elements of mathematics. Personal details of biography are suppressed, but something of the historical background is supplied. There are 47 plates, mainly portraits of great mathematicians.

The teacher who wisely obtains this book for the school library should be prepared to answer questions which the young reader may raise. What does the author mean by saying that the problem of the quadrature of the circle is "to find the square of an area equal to that of a given square", or that Euler "simply did not always use some of our present tests of convergence or divergence as a criterium for the validity of his series"? But errors of fact are rare, though I think that Cardan is given less than his due; and the statement about De Morgan in the footnote on p. 182 is unjustifiable.

4.5. Review by: M S Bartlett.
Journal of the Royal Statistical Society. Series C (Applied Statistics) 4 (2) (1955), 131.

To compress the history of mathematics into three hundred pages, even when the last fifty years is omitted, is to attempt the impossible, as the author admits; but this said, the reviewer found himself quite fascinated at times by the material presented in this book, especially by the illustrations. These include portraits of many of the great mathematicians, and also reproductions of pages from some of the mathematical classics (for example, Euclid's Elements, and Leibniz on the calculus). If one adds that the history extends back to the number language of primitive races and traces the influence of Greek and Oriental mathematics on the later developments in Western Europe, some indication may be conveyed of the impression left with readers of this book, of mathematics as one of the supreme cultures and techniques, intimately related at each stage of its progress both to particular individuals and to the societies in which they lived.

Obviously the statistician interested in the use of mathematics in his own field should not expect to find very much, for statistical mathematics is in any case a mere three hundred years old. References to the Fermat-Pascal correspondence on probability, to Huygens's book De ratiociniis in ludo aleae, to de Moivre's discovery of the normal distribution, to Laplace's monumental work Théorie analytique des probabilités, to Bayes's theory of inverse probability, to Gauss's method of least squares are all there, but brief. (The reader must not be mystified by a reference on p. 192 to the use of the principle of virtual velocities in statistics; for the latter, read 'statics'.)

As partial compensation for the brevity of much of the text, a list of some of the most important books on the history of mathematics is given in the introduction, as well as further reference lists at the end of each chapter.

5.1. From the Publisher.

An original and absorbing book about the tremendous upsurge in science and technology which marked the first century of American Independence and established the patterns of investigation which have brought us to our present stage of development.

5.2. Review by: Frederick B Tolles.
Pennsylvania History 16 (2) (1949), 150-151.

New Englanders, observed a nineteenth-century Frenchman, "are engineers as Italians are musicians and Germans are metaphysicians - by birth." The popular stereotype of the "ingenious Yankee" is amply documented in this book by a professor of mathematics at the Massachusetts Institute of Technology. Professor Struik introduces us to several hundred inventors, engineers, physicians, mathematicians, astronomers, geologists, botanists, and whatnot, who flourished on the rocky soil of New England chiefly between the Revolution and the Civil War. He is more successful in demonstrating that science and technology prospered in this region than in explaining why it did so, although he does venture tentatively into the sociology of science in an effort to find an answer to that question, too. He points, for example, to the significant fact that most of the scientists came from the coastal towns where a mercantile society had special reasons to be interested in such subjects as astronomy and meteorology, whereas many of the inventors and manufacturers came from among the whittling boys and Yankee peddlers of the inland towns.

In the course of his researches, Professor Struik has uncovered some fascinating and unappreciated Americans. Here, for example, is the self-taught astronomer and instrument-maker Walter Folger, who constructed a clock which tells seconds, minutes, hours, days, months, years, and centuries (not forgetting leap years), and also indicates the time of the rising and setting of the sun and moon and of high tide at Nantucket for each day. Henry D Thoreau figures in these pages in the unexpected rôle of a limnologist - a student (and apparently a good one) of the behaviour of ponds. Here is the Howe family which produced the inventors of the spring bed, the Howe bridge truss, and the sewing machine. Here are Drs Edward Jarvis and Lemuel Shattuck, the neglected pioneers of the public health movement; Loammi Baldwin, Jr., the father of civil engineering in the United States; and George Washington Whistler, who built the Boston and Worcester Railroad through the Berkshires and then went to Russia to build a railroad for the czar. Not overlooked is that irascible genius, Dr Charles T Jackson, who made important discoveries in chemistry and geology and claimed (not without a measure of justice) to have invented the telegraph and to have discovered anaesthesia. Here, too, set in their proper context, are those more familiar figures, Benjamin Thompson (Count Rumford), Nathaniel Bowditch, Eli Whitney, Benjamin Silliman, Louis Agassiz, Asa Gray, and James Dwight Dana.

In execution the book doesn't quite live up to its possibilities. It has an air - perhaps inevitable in such a comprehensive survey - of having been compiled rather than written. It lacks the verve and narrative flow of books like James T Flexner's Steamboats Come True or S E Morison's Maritime History of Massachusetts, to cite two works somewhat akin in subject-matter. In a laudable effort to set scientific activity in its social, political, and cultural matrix, Mr Struik occasionally allows the background to dominate and obscure the foreground. Irrelevant and unimportant details sometimes clog the narrative: is it helpful, for example, to know that the son of Jared Mansfield, the Connecticut mathematician, became an editor in Cincinnati, or that his daughter married a West Point professor? What these criticisms come to is simply that the book is not written with the artist's shaping sense of form. But it is written with the scientist's curiosity, diligence, and thoroughness, and we have reason to be grateful for the story of this segment of our cultural heritage which Mr Struik has uncovered for us.

One final query: who will do for the scientists and inventors of Pennsylvania and the Middle States what Mr Struik has done for Yankeedom? It would be a rewarding piece of research, and the results, one can prophesy, would be quite as impressive.

5.3. Review of the 1991 reprint by: P Weimerskirch.
Archives of Natural History 21 (1) (1994), 133.

This inexpensive paperback edition of Professor Dirk Struik's Yankee science in the making is a reprint of the 1961 edition to which the author has added a brief postscript and a "supplementary bibliography". The first edition appeared in 1948 and so Professor Struik, a 99 year old Dutch mathematician with Marxist leanings who taught for many years at MIT, saw his book appear in a new form 43 years later.

The book is a very readable and well-balanced social history of science and engineering in New England before the American Civil War. The author says in the new postscript that the idea for writing it came to him after he read The flowering of New England, 1815-1865 by Van Wyck Brooks, a literary history of New England.

The first part of the book covers the Colonial period from 1620 until about 1780. The second treats the Federalist period, from after the Revolution until about 1830. The third and much the longest describes the Jacksonian period, the 31 years between 1830 and the American Civil War. The main text is followed by 30 pages of notes, a 48-page bibliography, including the 1991 "supplementary bibliography", which, however, only adds seventeen references, and an index.

Although there is much here on natural history, most of it comes in the third and last section. During the Colonial period and for most of the Federal period, New England produced few naturalists of any stature. Things began to change in the early nineteenth century when many New England scientists studied and travelled in Europe, when good textbooks on natural history began to be published and when money was made available for the endowment of chairs in science, the establishment of botanical gardens and cabinets of minerals, and the purchase, usually from abroad, of scientific books and instruments.

The last chapter on the Federal period, "Lecture Hall and Textbook," describes the introduction of formal science teaching at New England universities, particularly at Harvard, where Benjamin Waterhouse began to teach botany and mineralogy. He also raised money to endow a professorship of botany and to establish a botanical garden. At Yale, Benjamin Silliman, chemist and mineralogist, inaugurated scientific education and obtained a large collection of minerals. Among other colleges that appointed outstanding naturalists around this time were Amherst, Bowdoin, Dartmouth, and Williams.

In the Jacksonian period natural history flourished in New England, largely through increased government support. In 1830 the Massachusetts Legislature established the first state-supported survey of natural resources, and this example touched off a flurry of activity in other states. In the same year the Boston Society of Natural History was founded. The last section of the book has chapters on "The Surveys" (pp 237-260), "The Great Naturalists" (pp 335-372) and "Science and Religion" (pp 373-397), which is largely about the reception of Darwin's ideas in America.

Struik shows how progress in science and engineering in New England was promoted or hindered at different times by various social, political, religious and economic forces. His book is a pleasure to read and it is based on a wide variety of primary and secondary sources as the extensive bibliography attests.

5.4. Review of the 1991 reprint by: Darwin H Stapleton.
Isis 88 (1997), 505-511.

Yankee Science in the Making, Dirk J Struik's survey of science and technology in New England from the colonial era up to 1861 that was first published in 1948, is one of the earliest scholarly surveys of American science and technology and one of a small number of regional studies of the subject. Few works in the history of science and technology read as well fifty years after their creation and continue to fill a niche in the literature of the field. Yankee Science has been reprinted three times in its five decades of service, most recently in 1991.

Why has Yankee Science in the Making been so durable, and what has been its contribution to scholarship? Its strengths include Struik's clear, literate style, his focus on individuals and their relationships to institutions and each other, his careful assessments of New England's achievements and contributions to American science and technology, and his efficient and effective coverage of an entire region and a considerable epoch in one volume.

When John C Greene concluded American Science in the Age of Jefferson (1984) by attempting to locate his work in the context of other major surveys of early American science, he found that there were only two worth critiquing: Yankee Science in the Making, which he described as "stimulating"; and Perry Miller's The Life of the Mind in America, from the Revolution to the Civil War (1965). Greene gave far more attention to Struik's book, arguing that Struik's attempt to show that the colonial merchant classes were interested in the more abstract sciences of astronomy and mathematics, and that the rising Jeffersonian middle classes of the new republic tended toward the more applied sciences of chemistry, geology, and natural history, was a Marxist-rooted thesis unsupported by the preponderance of evidence. Greene conceded, however, that Struik's viewpoint had much in its favour and noted that Struik himself recognised that his thesis was constrained by the anachronistic concept of "applied science," which had limited applicability before the mid-nineteenth century. In all, Greene demonstrated considerable appreciation for Struik's pioneering work, and the appearance of Yankee Science in his book's first footnote may have been a gesture of respect.

It is worth reviewing briefly the background and accomplishments of the scholar that Greene honoured. Struik is a native of the Netherlands (born in Rotterdam in 1894), where he received a broad education, including training in English, French, and German. Recommended to the University of Leiden for his skill in mathematics, he studied with the theoretical physicist Paul Ehrenfest as an undergraduate and in 1922, under the direction of W van der Woude, took a Ph.D. at Leiden in the area of tensor calculus. That same year he met Ruth Ramler, a Czech with a Ph.D. in mathematics from Prague, at a German mathematical congress; they married in July 1923. Struik had by that time become deeply involved in a radical socialist group in the Netherlands, and the Struiks became known for their commitment to marriage, mathematics, and Marxism.

Before and after receiving his doctorate Struik served as an assistant to J A Schouten at the technical institute in Delft, contributing to several of Schouten's works in the field of tensor analysis. In the spring of 1924, at an international conference on technical mechanics held in Delft, he met Tullio Levi-Civita of the University of Rome, who was doing fundamental work on differential geometry and tensor analysis, and Richard Courant of the University of Göttingen; they recommended him for a traveling fellowship from the International Education Board, a new Rockefeller philanthropy dedicated to stimulating graduate education and research, particularly in the sciences. Acting quickly, the IEB granted Struik a fellowship sufficient for him and his wife to travel to Rome and Göttingen, as well as to the international mathematics congress in Innsbruck in September 1924.

The next eighteen months on the fellowship redirected Struik's career. Not only did he collaborate with Levi-Civita and Courant, as planned, but in Rome Struik was introduced to the history of mathematics and the history of science in general, particularly though Giovanni Vacca and Federigo Enriques. Reading deeply in fifteenth- and sixteenth-century mathematics in what he called "the great old libraries of Rome" and later in the rich collections at Göttingen, Struik published studies of the Dutch mathematician Paulus van Middelburg (1445-1534) and began work "on the first theories about vertices and spherical triangles." Thus kindled, the love of the history of mathematics and science remained a constant in Struik's academic life. After completing his fellowship Struik briefly returned to Delft, and in the fall of 1926 he joined the mathematics department at the Massachusetts Institute of Technology, an appointment he held until his retirement in 1960. At MIT Struik continued his historical studies, although his teaching and primary scholarship were in mathematics. In 1948 he published an epic work, A Concise History of Mathematics, which was issued in a revised edition as recently as 1987 and has been published in eighteen languages.

Appearing in print at the same time - 1948 was Struik's annus mirabilis - was Yankee Science in the Making, a work with roots in various aspects of Struik's career and experience. The immediate stimulus for the book itself was Struik's reading of Van Wyck Brooks's The Flowering of New England, 1815-1865 (1936). As Struik recalled: "Brooks' book was about the literary scene in the period of Prescott and Emerson; [I thought,] was it possible to write a parallel account of the science, and related technology, of the same period, for the early years of republican New England?" Indeed, Brooks's stirring account of New England's cultural ferment provides only a few hints of the powerful industrial developments occurring at the same time.

Struik brought to his new task exceptionally broad interests (e.g., he was an amateur botanist) and an ability to read the modern and classical languages (he had to learn Greek and Latin in order to be admitted to postgraduate study) that gave him ready access to the sometimes obscure literature of New England's antebellum science and technology. He soon found that he needed to extend his search into the colonial era to create a complete picture of what had happened in New England. The result, as his contemporaries recognised, was one of the earliest reliable general surveys of American science and technology. Four decades later it was still regarded as a standard work.

Yankee Science was shaped by certain of Struik's predilections. One was an interest in and appreciation of the New England countryside that led him and Ruth Struik to visit many of the sites, particularly those associated with technological change. This first-hand experience, expressed in remarks ranging from topographic commentary on the historic and modern appearance of various locales (especially in Massachusetts) to references to artifactual collections, gives Yankee Science a kind of verisimilitude it would not have if Struik had relied only on printed history. Technics and Civilization, which benefited from Lewis Mumford's visits to museums throughout Europe prior to writing, and Svante Lindqvist's work on the introduction of the steam engine into Sweden, prepared for by his careful study of surviving buildings and drawings, are other examples of histories of technology and science that have drawn on knowledge of site and object as well as literature.

Marxism also informed Struik's work. It gave him a certainty about the results of industrialism, a process firmly established just as Yankee Science reaches its chronological conclusion. Struik has no doubt that the factory is a force for degrading and exploiting labour, and he readily asserts not only that there was a "ruling class" in the Federalist era but that it controlled most industrial production. His point of view shapes his commentary on broader matters of politics and economics as well: he characterises Joel Barlow's switch to Jeffersonianism as a defection to "the people's camp" and judges the American Revolution as "a vast popular movement," quoting Lenin as an authority. Struik is on occasion economically deterministic, such as when he restates the hoary argument that Eli Whitney's cotton gin created the economic conditions that led to the Civil War. This Marxist framework sometimes led his work to be taken lightly during the Cold War era.

Still, Struik's work is fundamentally based on his reading of the sources and is not significantly limited by his Marxist viewpoint, as Greene recognised. Struik is careful to note that science and technology were rather distinct enterprises throughout much of the period that he describes, a difficult position for a doctrinaire Marxist to take. Although in general he argues that science usually received an impetus from economic problems - in New England the problems of navigation and agriculture drew particular attention from the scientifically inclined, according to Struik - he states flatly that "colonial science was not exclusively utilitarian," and his numerous biographical sketches recognize the importance of intellectual excitement in the generation of scientific knowledge.

Struik's approach in Yankee Science thus is consistent with his published view that historical materialism cannot be applied "in a too narrowly economic manner," and indeed in Yankee Science his approach is essentially cultural and social. Clearly it was broad enough to encompass a strong interest in the interrelationship of religion and science. When Leo Marx's review of the first edition criticised Yankee Science for being insufficiently cognisant of the importance of the colonial Puritan legacy in the development of science in nineteenth-century New England, Struik accepted the criticism as valid.

A central but unexamined premise of Yankee Science is that a regional focus can sustain a scholarly approach to the history of science and technology. Today it seems that a global perspective is important to any work in the field, particularly as we deepen our understanding of the international diffusion of science and technology as one of the most important phenomena of the last few centuries and recognise that the history of any area must account for substantial flows of people and information. Struik acknowledges that by the nineteenth century it was routine for New Englanders deeply interested in the most advanced science or technology to travel to Europe - or at least to Philadelphia, which was the primary American node of communication with leading European scientists and institutions. (As a young man the nineteenth-century New England scientist Benjamin Silliman went to Philadelphia to obtain the best scientific education the United States had to offer, and he twice went overseas to hear from the leading figures - once early and once late in his career.) Struik describes these travels in individual cases but offers no general view of how outside scientific and technological innovations affected New England or of what role New England played in those processes. In the lack of any sustained discussion of transfer and diffusion Yankee Science shows its age.

From another perspective, however, Struik's regionalism was prescient. Even though he offered no considered justification for his approach, his firm belief that scientists carry out their work influenced by the culture in which they live gave him the confidence to take a regional frame for a study of the history of science and technology when few others had done so. Only in the last dozen years have historians of American science and technology begun to grapple with regionalism as more than a matter of geographic or political boundaries. For example, Judith McGaw's collection of essays in Early American Technology: Making and Doing Things from the Colonial Era to 1850 was intended in part to demonstrate that the Mid-Atlantic states had created a regional technological style that was distinct from that of New England. Thomas P Hughes's Networks of Power identified the region as the proper level of analysis for the development of electrical systems. Looking at the mid-twentieth century, Robert Kargon and Stuart Leslie recently have argued that "the respective paths of Princeton and Stanford Universities to distinction in science were profoundly affected by their respective conceptions of their relations to the region or territory in which they worked." It is perhaps ironic that in the globalism of the late twentieth century there is an increased acceptance of the region as a useful framework for historical analysis of certain aspects of science and technology.

In sum, Yankee Science in the Making bears rereading. Its contributions are substantial, and it could provide a new generation with inspiration for new directions in the history of American science and technology.

6.1. From the Publisher.

Excellent brief introduction presents fundamental theory of curves and surfaces and applies them to a number of examples. Topics include curves, theory of surfaces, fundamental equations, geometry on a surface, envelopes, conformal mapping, minimal surfaces, more. Well-illustrated, with abundant problems and solutions. Bibliography.

6.2. From the Preface.

This book has developed from a one-term course in differential geometry given for juniors, seniors, and graduate students at the Massachusetts Institute of Technology. It presents the fundamental conceptions of the theory of curves and surfaces and applies them to a number of examples. Some care is given to historical, biographical, and bibliographical material, not only to keep alive the memory of the men to whom we owe the main structure of our present elementary differential geometry, but also to allow the student to go back to the sources, which still contain many precious ideas for further thought.

No author on this subject is without primary obligation to the two standard treatises of Darboux and Bianchi, who, around the turn of the century, collected the result of more than a century of research, themselves adding greatly to it. Other fundamental works constantly consulted by the author are Eisenhart's Treatise, Scheffers' Anwendung, and Blaschke's Vorlesungen. Years of teaching from Graustein's Differential Geometry have also left their imprint on the presentation of the material.

The notation used is the Gibbs form of vector analysis, which after years of competition with other notations seems to have won the day, not only in the country of its inception, but also in many other parts of the world. Those unfamiliar with this notation may be aided by some explanatory remarks introduced in the text. This notation is amply sufficient for those more elementary aspects of differential geometry which form the subject of this introductory course; those who prefer to study our subject with tensor methods and thus to prepare themselves for more advanced research will find all they need in the books which Eisenhart and Hlavaty have devoted to this aspect of the theory. Some problems in the present book may serve as a preparation for this task.

Considerable attention has been paid to the illustrations, which may be helpful in stimulating the student's visual understanding of geometry. In the selection of his illustrations the author has occasionally taken his inspiration from some particularly striking pictures which have appeared in other books, or from mathematical models in the M.I.T. collection. The Art Department of Addison-Wesley Publishing Company is responsible for the excellent graphical interpretation and technique.

The problems in the text have been selected in such a way that most of them are simple enough for class use, at the same time often conveying an interesting geometrical fact. Some problems have been added at the end which are not all elementary, but reference to the literature may here be helpful to students ambitious enough to try those problems.

The author has to thank the publishers and their adviser, Dr Eric Reissner, for their encouragement in writing this book and Mrs. Violet Haas for critical help. He owes much to the constructive criticism of his class in M 442 during the fall-winter term of 1949-50, which is responsible for many an improvement in text and in problems. He also acknowledges with appreciation discussions with Professor Philip Franklin, and the help of Mr F J Navarro.

6.3. Review by: J A Todd.
The Mathematical Gazette 35 (312) (1951), 136.

This is an elementary textbook covering much the same ground as the well-known work by Weatherburn, and uses the vector calculus in much the same way. The most striking feature of the book is the way in which the author emphasises that the subject-matter is geometry and not vector algebra, and the explanations which accompany the exposition keep the geometrical point of view well to the front. The text is enriched by a splendid series of plates and diagrams, which not only enhance the appearance of the volume but make the visual aspects of the subject unusually clear. The exposition is lucid and accurate, and the book would be eminently suitable for an undergraduate course in our own universities. The reviewer has noticed only one slip: the statement on p. 55 that the vanishing of two of the determinants from the 2 × 3 Jacobian matrix implies the vanishing of the third, admits a trivial exception. The printing and layout of the book are excellent.

The five chapters deal respectively with curves, elementary theory of surfaces, the fundamental equations (Gauss theory), geometry on a surface (geodesics), and, finally, some special surfaces (ruled surfaces, minimal surfaces, mappings). There are numerous exercises throughout the book, with answers and hints for solution at the end. This book can be warmly welcomed.

6.4. Review by: M P Gaffney.
The Mathematics Teacher 44 (8) (1951), 593.

This is an excellent account of the classical theory of differential geometry. The book is an outgrowth of a course which the author teaches at the Massachusetts Institute of Technology and is designed as a textbook, with an ample supply of problems. There is also a large collection of diagrams and photographs of models.

Anyone who is thoroughly grounded in advance calculus has adequate preparation to read the book; a knowledge of the rudiments of vector analysis would be highly desirable, but it can probably be picked up from the text.

The subject of the book is the application of calculus to the study of geometry of curves and surfaces. The presentation differs from that of many textbooks in English in that vector methods and vector notation are used throughout; the reader who has studied differential geometry without the use of vectors will be startled by the simplification that results.

The author has included numerous thumbnail biographies of the men who have contributed to the development of the subject.

6.5. Review by: Paul Vincensini.
Mathematical Reviews MR0036551 (12,127f).

This work comprises five chapters. The first chapter deals with the theory of space curves, outlining fundamental concepts while addressing both the imaginary perspective (isotropic planes, cones, and curves) and the global perspective (ovals and related key theorems of integral geometry). The second chapter addresses the general theory of surfaces through a study of the first and second fundamental forms and associated concepts (metric, curvature, conjugate nets, orthogonal triple systems, elliptic coordinates, etc.). Whenever appropriate, the author highlights the true geometric origins of the concepts introduced. For instance, regarding lines of curvature, he notes that Monge [Application de l'analyse à la Géométrie, 5th ed., Bachelier, Paris, 1850] introduced them in a famous paper concerning the cut-and-fill problem, the precise formulation of which he restates. The notation used - as in the first chapter and throughout the rest of the book - is that of vector analysis; however, the transition to tensor calculus notation and methods is carefully prepared through a series of particularly well-chosen problems. Chapter III focuses on the fundamental second-order equations of surface theory, beginning with a comparison of the perspectives of Monge and Gauss. The Gauss-Weingarten and Codazzi equations follow immediately, presented in the form best suited for the transition to tensor methods. The author provides the necessary background on surface deformation invariants and applies them to various cases, such as the continuous deformation of minimal helicoids or the characterisation of the sphere - either as a surface where every point is an umbilical point or as a closed surface of constant positive curvature without singularities. This same chapter introduces the basic concepts of curvilinear coordinates in space and concludes with the fundamental theorem regarding the existence of a surface possessing two given fundamental forms. Chapter IV deals with the geometry of surfaces - that is, properties involving only the line element ds2 and thus invariant under deformation of the surface. Geodesics play a central role here. The geodesic form of the ds2 leads to surfaces of constant total curvature and to Minding's theorem concerning their isometry for a given value of curvature. The geometry of surfaces of constant curvature offers an initial glimpse into non-Euclidean geometries; the author provides a well-documented history of this subject, notably highlighting the importance of G. Saccheri's contributions. The chapter concludes with an in-depth study of the Gauss-Bonnet theorem and its topological implications. Chapter V addresses specialised topics intended to prepare the reader for major treatises or original research papers: envelopes, conformal mappings, isometric mappings, geodesics, and so forth. The general problem of deformation is presented in the two forms formulated by Bour and Minding; regarding the problem of geodesic mapping, mention is made of Beltrami's theorem on the planar geodesic representation of surfaces of constant total curvature, as well as Liouville's or Lie's pairs of associated surfaces. The study of minimal surfaces, begun in Chapter II, is resumed from a variational perspective and extended by drawing on general results concerning isometric or conformal mappings. A section is devoted to the role of imaginary quantities in surface theory, and the work concludes by outlining a number of open questions.

6.6. Review by: C E Springer.
The American Mathematical Monthly 59 (2) (1952), 119-120.

Here is a welcome addition to the books in English on metric differential geometry of curves and surfaces in ordinary space. Among the features which distinguish it as outstanding are the many interesting historical and biographical notes, the wealth of useful references to the original papers, the clear and concise presentation, and the numerous well drawn figures which will no doubt be effective in carrying out the author's purpose of "stimulating the student's visual understanding of geometry."

The comprehensiveness of the book is amazing for so small a volume. It is designed as a first course in the subject although it contains much more than most classes can assimilate in one semester. There are many well graded exercises ranging from those which require routine manipulation illustrating the text to those which require penetrating thought. Answers or hints for solution for all exercises are found at the back of the volume.

For the most part the notation used is that of the Gibbs form of vector analysis, which, according to the author, "after years of competition with other notations seems to have won the day." This is probably a matter of personal taste and experience, and a matter to be decided according to the future needs of the students. For those who will take only one course in the subject the vector analysis is quite adequate and probably advisable. However, for those who wish to continue to more advanced stages of Riemannian geometry of higher dimensions or to study relativity theory the tensor notation would be a valuable tool in the first course.

After treating the fundamental theory of space curves in the fifty-four pages of Chapter I, the author introduces in Chapter II the two fundamental quadratic forms to portray the elementary theory of surfaces. He is careful to point out the distinction between singular points of a surface and singular points of its parametric representation. For a model of clarity, one should note the article on the indicatrix of Dupin on page 83. The equations of Gauss and Weingarten are discussed in Chapter III. The distinction between intrinsic and extrinsic geometry of a surface is forcefully brought out. A treatment of geodesics, geodesic curvature, geodesic coordinates, and the Gauss-Bonnet theorem is found in Chapter IV. Transformation of coordinates receives little attention. Differential parameters appear in an exercise but their invariance is not mentioned. In the last chapter the author begins with a discussion of envelopes of one parameter families of surfaces, continues with a study of conformal, isometric, and geodesic mapping, and concludes with minimal surfaces, ruled surfaces, and imaginaries in surface theory. On page 169 it is stated that equiareal mapping will be discussed but the concept appears only in three exercises.

Remarkably few typographical errors were noticed. A few minor lapses in the characteristic care with which the book was written might be mentioned. ...
...
This is a useful volume which should be at hand for every student and teacher of differential geometry, and it will surely become a popular text book for students who can devote only one semester to differential geometry.

7.1. From the Publisher.

Based on a historic approach taken by instructors at MIT, this text is geared toward junior and senior undergraduate courses in analytic and projective geometry. Starting with concepts concerning points on a line and lines through a point, it proceeds to the geometry of plane and space, leading up to conics and quadrics developed within the context of metrical, affine, and projective transformations. The algebraic treatment is occasionally exchanged for a synthetic approach, and the connection of the geometrical material with other fields is frequently noted.

Prerequisites for this treatment include three semesters of calculus and analytic geometry. Special exercises at the end of the book introduce students to interesting peripheral problems, and solutions are provided.

7.2. From the Preface.

The extension of the mathematical curriculum in our colleges has not infrequently been at the expense of some of the most valuable fields of more "old-fashioned" mathematics. Among the victims we find elementary, projective, and algebraic geometry, fields which used to stir the enthusiasm of an older generation. This decline into relative neglect not only means that mathematicians grow up poorly acquainted with one of the most attractive parts of their science, elegant in form and in results, but also means a loss of more fundamental values. This "modern" geometry of the nineteenth century was to a considerable extent responsible for the whole revolution in mathematical thinking typical of that period; out of it came the concepts of non-Euclidean geometry and geometry of more than three dimensions, and it contributed substantially to the formation of such concepts as transformation, group, invariant, and oriented quantity. Moreover, it has profoundly influenced axiomatics. Neglect of this part of mathematics therefore tends to stifle the understanding of some of the most important notions of modern mathematics, physics, and engineering. The task is set to find the legitimate place for this field inside our present mathematical curriculum, and to stress those fundamentals which are most vital for the understanding of our science as a whole.

The present book represents an approach to this task by means of a one-term course, mainly for juniors and seniors, given at the Massachusetts Institute of Technology. Those students have passed through a three-term course of the now traditional "analytical geometry and calculus" type, so that they have some understanding of Cartesian coordinate work in the plane and in space applied to straight lines, conies, and some other curves, have had some vector algebra, and enough of space geometry to recognise a quadric surface. This knowledge is presupposed in our text, so that we can start with affine and projective concepts concerning points on a line and lines through a point. Then the geometry of plane and space, leading up to conies and quadrics, is developed inside the triple frame of metrical, affine, and projective transformations. The algebraic treatment is occasionally exchanged for the synthetic one. At many places the connection of the geometrical material of the text with other fields is indicated, notably with axiomatics, algebra, descriptive geometry, vector analysis topology, and statics. Further developments serve the purpose of allowing the student to find his way in the literature. Care is bestowed on the figures as well as on the problems for further exercise. The Special Exercises at the end are not necessarily meant as a further elaboration of the material of the text; they are rather aimed at allowing the student to take notice of interesting problems on the periphery.

The transliteration of Greek and Russian names should follow some consistent rule, for instance, that adopted in Sarton's "Introduction to the History of Science" and in the "Mathematical Reviews." The names usually spelled Apollonius, Pappus would then be transliterated as Apollonios, Pappos, and the name of the co-discoverer of non-Euclidean geometry as Lobacevskii. Since the spelling Apollonius and Pappus is customary in mathematics, we have compromised on these names, just as Sarton has compromised on Euclid instead of Eucleides, since the word Euclid has become part of common language.

In preparing the manuscript I have had the advice of Mr L Cross, Dr Ruth Struik, and especially of Professor H S M Coxeter, to whom my thanks. I also take this opportunity to pay respect to the memory of Professor P Zeeman of Leiden University, who introduced me to the charms of analytical and projective geometry, and whose lecture notes, after all these years, have still been of use to me in preparing my own text.

7.3. Review by: T K Pan.
The American Mathematical Monthly 61 (2) (1954), 131-133.

Projective geometry may be developed either projectively or metrically. In the former case, no metric idea is used so that a projective mentality is established without Euclidean prejudices. In the latter case, the concepts are approached from the metric side so that the treatment can be based on students' previous training in metric geometry. This book belongs to the second category.

The avowed purpose of the book is "to find the legitimate place for this field inside our present mathematical curriculum and to stress those fundamentals which are most vital for the understanding of our science as a whole." It is to be covered in one semester and only enough knowledge of elementary analytic geometry to recognise conics and quadrics is presupposed. Algebraic methods are occasionally exchanged for synthetic ones. The book is built around the Erlanger Program and contains well selected materials. Of the well known texts on this subject perhaps it most nearly parallels Graustein's Higher Geometry in subject matter and in method. But it enlarges the horizon by the introduction of historical remarks and connections with other fields while being only a little over half as long - an advantage for those colleges whose curriculum is so crowded that only a one-term course can be allotted to analytic geometry beyond the traditional course. It is certainly a very suitable and long-awaited text.

There are eleven chapters in the book. The first seven chapters cover the projective geometry of the plane and the last four chapters cover that of space. The titles of the chapters are: (1) Point Sets on a Line; (2) Line Pencils; (3) Line Coordinates. Homogeneous Coordinates; (4) Transformations of the Plane; (5) Projective Theory of Conics; (6) Affine and Euclidean Theory of Conics; (7) Projective Metric; (8) Points, Lines, and Planes; (9) Projective Theory of Quadrics; (10) Affine and Euclidean Theory of Quadrics; (11) Transformations of Space. Cross ratio is defined in terms of division ratios which are expressed in terms of Cartesian coordinates of points. Projectivity is defined as one-one algebraic correspondence between division ratios. Projective coordinates of a point are defined in terms of its trilinear coordinates and those of a line by means of its equation in projective point coordinates, which are shown to be expressible as cross ratios. That the projective plane is nonorientable and closed is illustrated with great clearness by its topologically equivalent figures. Einstein's summation convention is used to simplify the notation for projective transformation. Transformations of projective coordinates are called projective transformations when they are passive or collineations when they are active. Elementary divisors are used for the classification of collineation. A conic is defined as the locus of intersections of corresponding lines in two projective pencils. The concepts of point, line and plane in space are developed without metric ideas. Several sections are devoted to non-Euclidean geometries, vectors, force systems, central projection, perspective, parallel projection, oblique projection, and orthogonal projection.

The book is provided with collateral reading, with answers to exercises and with an index. The exercises are well selected, and the special exercises at the end of the book are most interesting.

There are several inconsistencies with regard to symbols and terminology used in the book. ...

7.4. Review by: Ruth Moufang.
Mathematical Reviews MR0057552 (15,246f).

In the author's view, modern 19th-century geometry is often neglected in current mathematics curricula, to the detriment of a deeper understanding of other mathematical disciplines. This book aims to remedy that deficiency through a one-semester course. Readers are expected to have completed the standard three-semester course in analytic geometry and calculus. The book treats classical projective geometry in one, two, and three dimensions using an algebraic approach, though synthetic methods are also employed occasionally. Related fields such as algebra, vector analysis, and statics are touched upon. Readers are introduced to the topological models of projective geometry and the fundamentals of descriptive geometry. The book includes historical notes and numerous exercises. The material is clearly organised and the exposition is accessible, complemented by excellent figures. Chapter headings: Point sets on a line; Line pencils; Line coordinates. Homogeneous coordinates; Transformations of the plane; Projective theory of conics; Affine and Euclidean theory of conics; Projective metric; Points, lines and planes; Projective theory of quadrics; Affine and Euclidean theory of quadrics; Transformation of space.

8.1. From the Publisher.

The Origins of American Science: New England is a book written by Dirk Jan Struik that delves into the history of science in America, with a particular focus on its origins in New England. The book explores the intellectual and cultural context in which science emerged in America, tracing its roots back to the Puritan settlers who arrived in the region in the 17th century. It examines the role of institutions such as Harvard College and the American Academy of Arts and Sciences in fostering scientific inquiry, as well as the contributions of individual scientists such as Benjamin Franklin and Joseph Priestley. The book also looks at the impact of the Enlightenment on American science, and the ways in which science helped shape American identity and culture. Overall, The Origins of American Science: New England provides a fascinating and in-depth look at the early history of science in America, shedding light on the intellectual and cultural forces that helped shape this important field.

8.2. Review by: Editors.
Mathematical Reviews MR0090512 (19,825g).

A study of the growth of science, and of the corresponding sociological changes, in New England between the Revolution and the Civil War. Mathematics is discussed chiefly in connection with navigation and astronomy.

9.1. From the Preface.

What we usually call the Golden Age or Golden Century of the Netherlands starts around 1580-1590 with the earlier years of the Republic and comes to an end with the last years of the seventeenth century. We know it as the time of Oldenbarnevelt and De Witt, the time of the great princes of the House of Orange, the time of the admirals Tromp and De Ruyter, the time of Frans Hals and Rembrandt. It is also a period of important contributions to science, and in this respect may be characterised by the names of Stevin and Huygens, the representative figures at the beginning and the end of the period. Compared to what has been written about the politics, the art of warfare and the arts in general of the Golden Century, the literature on the science (and technology) of the period is of very modest dimensions. There exists a small number of special studies about certain definite persons and fields of science, supplemented by the monumental edition of the works of Huygens and by equally valuable editions of the works of Beeckman, Stevin and van Leeuwenhoek. But there exists, as far as I know, no book that attempts to give an overall picture of the whole scientific development. This is a pity, not only because the image of the Golden Century is distorted in this way, but also because Dutch science of this period is so rich. In the small world of Amsterdam and surrounding cities representatives could be found of almost all sciences cultivated at that time; moreover, many of these representatives were leaders in their field, be it the study of mathematics, mechanics or physics, chemistry, biology or medicine, the work with map, telescope or microscope. The history of science in the Republic is a good introduction to the history of the formative years of our present-day science in general.

What we offer here is not such a book. For this we need more study of sources than is possible in a domain so little explored as Dutch science in the Golden Age. But we hope that our survey can give a not too inadequate picture of the different forms of science and technology cultivated in those interesting years. Such a picture may show the importance of this development and introduce to a more general public some of the principal actors. They certainly merit our attention, these gentlemen called Waghenaer, Plancius, Stevin, Beeckman, van Schooten, Huygens, Tulp, Ruysch, Swammerdam, van Leeuwenhoek and so many others, among whom such better known men as De Witt, Descartes and Maurice of Orange stand out in their own way, but presented here in a somewhat different light.

We believe that the work of these men can be better understood, if it is sketched against a background illuminated by facts from the economic, political, religious and general cultural life in the Republic.

We have only mentioned a minimum of sources; a book like this should not become top-heavy with footnote erudition. Those who look at the bibliography at the end usually can reach the primary sources without much trouble - and in the process discover many more interesting facts and persons than have been presented here in this book: the Golden Century was rich in interesting and picturesque figures. What a creative vitality existed in the Dutch people of those days! What a promise for the future when the whole Dutch people will have a chance to develop its creative powers!

Let us end with a remark made by an older contemporary of Huygens and Newton, a naturalist in his own way, and, if not the most learned of the ichthyologists, then certainly the most pleasant one:

I might say more, but it is not fit for this place; but if this Discourse which follows shall come to a second impression, which is possible for slight books have been in this Age observed to have that fortune; I shall then for thy sake be glad to correct what is faulty, or by a conference with any to explain or enlarge what is defective: but for this time I have neither a willingness nor leisure to say more than wish thee a rainy evening to read this book in.

Farewell.

From: Izaak Walton, The Compleat Angler, 1653.

9.2. Review by: Editors.
Mathematical Reviews MR0091901 (19,1030a).

A selective history of Dutch science (including, e.g., the work of Leeuwenhoek) but particularly of mathematics as represented by Stevin and Huygens. There are ten chapters, on the topics: the new republic, the old science, the period of transition, calculation and navigation in Holland, Simon Stevin, the new science, Descartes, Christiaan Huygens, biology, colonial science.

9.3. Review by: D Burger.
Isis 50 (3) (1959), 280-281.

In the history of the Northern Netherlands the period from about 1580 till the end of the seventeenth century is usually called "The Golden Age." It was the period of the statesmen Van Oldenbarnevelt and De Witt, the naval commanders Tromp and De Ruyter, the painters Hals and Rembrandt, and also the period of great scientists. In first place among these we should name Simon Stevin and Christiaan Huygens. There are publications on the scientific work of these scholars, but a historical survey of the period, in which the history of the sciences is treated in connection with general history, has not yet appeared. Dr Struik is to be praised for having given us a little sample of this type of work.

The first chapter deals with the "Young Republic," where trade and shipping gave rise to prosperity and an efflorescence in all fields that aroused the admiration of many countries of Europe.

In the following chapters the author describes the essential methods of the "Old Science" and explains that mankind had grown away from it because of the demands made by society. Arabic numerical notations superseded the abacus and Roman numerals. Vesalius and Copernicus had given the impulse to the revival of anatomy and astronomy. The exigencies of technics could no longer be satisfied by Aristotelian concepts, and the works of Archimedes were studied to provide a foundation for new investigation. Dutch scholars such as the mathematicians and astronomers Gemma Frisius and Snellius, the cartographers Mercator and Ortelius, the chemists Van Helmont and others are discussed. The teachers of arithmetic and navigational science are amply dealt with, after which a chapter on the "New Science" explains the relationship between science and society in the period in question.

Separate chapters are devoted to Simon Stevin, who was born in the Southern Netherlands but spent a great deal of his life in the North as an adviser of Prince Maurits, to the Frenchman Descartes who lived twenty years in the republic, and to Christiaan Huygens. After this, the investigators of the "Living World" are discussed: the physicians Tulp, Ruysch, Van Foreest, Van Deventer, Regius, Sylvius, Coiter and Reinier de Graaf, the herbalists Dodonaeus and Clusius, and, very fully, the biologists Swammerdam and Van Leeuwenhoek.

In the last chapter, "Colonial Science," we find the description of scientific research in the tropics. Here we read of Keizer who observed the constellations of the southern hemisphere, Jacob Bontius, the founder of tropical medicine, Maria Sibylla Merian who described the insects of Surinam and drew them with the utmost accuracy, and the biologists Van Rheede and Rumphius.

A book like this, which is not just a dry summary of facts but rather a graphically sketched outline of a society in which science is seen in relation to the other components of life, naturally must have a personal cachet. Hence it will be clear that the author's strongly marked inclination to Marxism will be conspicuous in several places. There is no denying that it was a great asset to science that persons of the lower classes rose and helped to expand the territory of scientific knowledge. It is no wonder that the author accentuates this side, but still he cannot deny that even in that period important contributions to science were made by others who came from court circles, Christiaan Huygens being the greatest of them all. It seems a little exaggerated that the author, prompted by his "hatred and contempt for the colonial system," should label the research of the scientists in the tropics "Colonial Science." And readers with a different viewpoint will think it odd that the name of Marx is dragged in in many places. But all this does not disturb the train of thought in any way. The book is fluently written in a pleasant style.

Only a few small mistakes struck us. On page 33 the title of Vesalius' work is given incorrectly; on page 36 an annoying printer's error slipped into the Latin quotation from Ramus. Bering Strait cannot be well identified with the Strait Anian (page 53). This hypothetical strait was thought to be much more to the West. The name of Van Oldenbarnevelt is spelled wrongly on page 9; elsewhere we find the correct spelling. The name of Volcher Coiter is spelled on pages 110 and 126 in different ways, but both times incorrectly. The Cyfferinge (not Cijfferinghe) of Willem Bartjens (not Bartjes) appeared in 1604 (not 1608). These are all little slips. They do not interfere with the reading. An index of names would have been useful, but we are happy to find a short list of recommended books and a vocabulary for the general reader in which some details on scientists mentioned in the text are also inserted. There are many illustrations, among them seven portraits of scholars, viz. Plancius, Stevin, Descartes, Huygens, Leeghwater, Van Leeuwenhoek and Rumphius.

This is not a penetrating scientific work, but it was not meant to be. Much original research would have been needed for that. We can regard this book as a first step in this direction and perhaps it will stimulate others to undertake a thorough study on particular points. However limited in scope it may be, this book has much to offer, and not only to the layman, since it gives a coherent survey of a specific period. For that we must thank the author.

10.1. From the Preface.

The first edition of this book appeared in 1948. There have been several translations, and in some of them the translators have added material of special interest to their readers. Thus we find a section on Russian mathematics in the Ukrainian translation [Rjadjansʹka Škola, Kiev, 1960] and in the Russian one [Izdat. "Nauka", Moscow, 1964]. I myself, when preparing the Dutch version of the book [Aula-Boeken, Het Spectrum, Utrecht, 1965], added items of interest to Dutch readers. These editions, as well as the German ones [VEB Deutsch. Verlag Wissensch., Berlin, 1961; second edition, 1963], also contain bibliographical data of particular interest to the respective countries. When these translations were in progress, I used the opportunity to add amendments or corrections to the text, so that the book, through its different editions, has undergone a gradual, although not fundamental, evolution. Much of this has been beneficial in preparing this new English edition.

One day, one of my friends in Peking discovered a Chinese translation [Science Publishing House, Peking, 1956] which he forwarded to me. The translator of this edition, in his preface, praised the book but objected to its treatment of Chinese mathematics. Since I already had some misgivings, I rewrote the section on this subject. In this edition ancient Chinese mathematics now appears, as it should, as an integral part of medieval and pre-medieval mathematics, and not as a phenomenon outside of the main current of scientific development.

This new edition follows the history of mathematics to the end of the nineteenth century. In 1966 this is quite far away. It is time that the history of mathematics from 1900 to 1950 be written, if only in the form of a "concise history." Nobody seems to have attempted it, although some monographs exist. This is the more remarkable since the market teems with histories of twentieth-century physics. Although this history has the advantage of being more spectacular and (at any rate in some important aspects) more easily understood, the period that began with Poincaré, Hilbert, Lebesgue, Peano, Hardy and Levi-Civita offers a wealth of material for a fascinating history of mathematics, both in its own right and in relation to logic, physics, and engineering. Who of you, gentle readers, is going to take the initiative?

10.2. Review by: Telegraphic Reviews.
The American Mathematical Monthly 74 (9) (1967), 1163.

The additional material on China, based on the work of Needham and others, makes this new edition of interest even to those who are completely familiar with its predecessors, which have been translated into more languages and been more widely read than any other contemporary history of mathematics. The bibliographies have been extended.

11.1. From the Preface.

During the later Middle Ages, the Low Countries, or Netherlands, were divided into a number of feudal principalities headed by such lords as the counts of Holland, the counts of Flanders, the counts, later dukes, of Gelderland (Guelders), the dukes of Brabant and of Hainaut and the bishop of Utrecht. The principalities were known as the seventeen Netherlands. During the fifteenth century most of them became part of the domain of the dukes of Burgundy with their splendid court at Brussels. These dukes began a system of centralisation, continued with some success by their successors of the House of Habsburg. It was the Habsburg emperor Charles V who, in 1543, succeeded in incorporating all seventeen Netherlands as provinces of his vast empire, extending from America to Spain, Austria, Germany and the Netherlands.

This was the time of the Reformation which came to the Netherlands first as Lutheranism, then as Calvinism. The persecutions, together with attempted inroads into the jealously guarded rights and privileges of towns and provinces by the Brussels administration, fostered feelings of resentment. This lead to actual rebellion in the days of Philip II, Charles' son and successor, who tried to rule from Spain and to enforce the Catholic religion much more strictly than his father. In 1567, he sent the duke of Alva with an army to the Netherlands. Armed resistance began in 1568, opening the Eighty Year War. At first Alva could book some successes, but the tide turned on the first of April, 1572 (a date remembered in the Northern Netherlands like the nineteenth of April, 1775, in the U.S.A.) with the capture of the town of Den Briel in Holland by rebellious anti-Catholic pirates (called Geuzen after the French gueux, beggar's). Many towns now shook off what was felt as the Spanish yoke. Alva's counter-offensive, consisting of a series of sieges, was at first successful, but met defeat at the walls of Alkmaar (1573) and Leiden (1574). The Spanish had more success in the Southern Provinces, a success culminating in the recapture of Antwerp (1585). In 1579 the Northern Provinces had already built a loose federation in the Union of Utrecht, resulting in what became known as the Union of the Seven Provinces (Holland, Zeeland, Utrecht, Gelderland, Overysel, Friesland, Groningen plus the territory of Drente). The leading genius was William of Orange, dit William the Silent, probably on account of his diplomatic tact. The Southern Provinces eventually returned to Spain and the Catholic Church. The Northern Union, often called the Dutch Republic, was dominated by the Calvinists, organised in the Reformed Church, although a large number of its inhabitants remained faithful to Catholicism.

The formal abjuration of Philip II by the North occurred in 1581. William the Silent was assassinated in 1584, but the war against Spain continued successfully under the strategic leadership of his son Maurice, and, after his death in 1625, of his brother Frederic Henry (the grandfather of William III, king of England). These princes bore the title of stadhouder (lit.: lieutenant). The constitutional structure of the Republic and the creation of its powerful position in the world was the work of the jurist-statesman Johan van Oldenbarnevelt.

There was a truce from 1609 to 1621, troubled by the religious conflict between Remonstrants or Arminians, representing in the main patrician merchants with a section of the more liberal intellectuals one of the most famous being Hugo Grotius - and the Counter-Remonstrants, orthodox Calvinists with their strength in the popular masses led by their preachers and supported after 1617 by Maurice. This struggle led to the execution of van Oldenbarnevelt (1619). Despite a temporary victory of the Counter-Remonstrants, the Republic continued to be ruled by a dominant section of the merchant patriciate, the so-called Regents (Regenten), of whom those at the head of Amsterdam were the most powerful. They exercised their power, usually successfully, through the States General meeting at The Hague, composed of delegates sent by the States of the provinces, among whom the States of Holland were again the most influential.

The official religion was Dutch Reformed (Calvinist), but other Protestant sects, Lutherans, Baptists, Collegiants and others, were usually not molested. Neither were the Jews, concentrated in Amsterdam, and as a rule even the Catholics could meet, but supposedly in secret ("Our dear Lord in the attic", a hidden Catholic church in Amsterdam, now a tourist attraction).

Peace with Spain came in 1648 with the peace of Westphalia. The Republic emerged as a major power in the Atlantic world. In art we speak of this period in its existence as the age of Rembrandt. In science, as we hope to show, we may likewise speak of the age of Stevin and Huygens.

* * * * * *

From what has been sketched above it will be clear that the word Holland in this book is not used, as so often in the present, for what is now the kingdom of the Netherlands. Holland, for us and in accordance with history, is a territory, first county, then province, in the Western region of the Low Countries near the North Sea. It is the richest part of the North; in it are many of the leading cities, such as Hoorn, Enkhuizen, Alkmaar, Amsterdam, Haarlem, Delft, Rotterdam, Dordrecht as well as The Hague, the centre of government of the Republic. This Holland has been, since 1840, divided into the two present provinces North and South Holland.

The Southern Netherlands have been called Belgium from time to time in this book, because its territory is essentially that of the present kingdom with that name, established in 1830. Before that time they belonged first to Spain, then to Austria and then to the United Kingdom of the Netherlands, founded in 1815 and split in two by the revolution of 1830.

The word 'Dutch', derived from an ancient Germanic word meaning people, will be used for the language as well as the people of the Northern Netherlands.

* * * * * *

The translation from the original Dutch edition (1958, revised ed. 1979) is by the author, who takes all responsibility for the text. This text, though at most places a direct translation from the original, has been amended at places by some corrections and additional material, as in the last chapter on colonial science. In the translation I have been assisted by Mrs. Renate Hanauer and by Dr Ruth Struik and have been encouraged by Professors Robert S Cohen of Boston University and Erwin N Hiebert of Harvard University. To all these friends I express my appreciation for their assistance.

11.2. Review by: Yvonne Dold-Samplonius.
Mathematical Reviews MR0631852 (84a:01020a).

The English edition of this book is a translation by the author from the original Dutch edition [Pegasus, Amsterdam, 1958], amended in places by some corrections and additional material. The period treated, from around 1580–1590 until the last years of the seventeenth century, is the Dutch Golden Age. This is the time of Frans Hals and Rembrandt but also a time of important contributions to science, characterised by the names of Simon Stevin (1546–1602) and Christiaan Huygens (1629–1695). The author gives a vivid picture of the different forms of science and technology cultivated, and of the general historical background. As reasons for the prosperity and influence of the Northern Netherlands he sees two main factors: the Republic was a "pure burgher community", and secondly, the Dutch "knew how to use their old superiority in vessels and crew to all possible advantages, now that the heart of European commerce had been moved from the Mediterranean". Science was stimulated, as it was needed to make sea traffic safer. Also there existed a general atmosphere of religious tolerance (Erasmus!), which attracted many refugees. At first the Southern Netherlands under the Burgundians and the Habsburgs were preferred by many distinguished scientists from the North (e.g., Gemma Frisius). After 1567, when the Spanish occupation started its reign of terror, a general exodus set in: Mercator and Bernoulli went to Germany, others fled to England, but many moved north to the new republic (e.g., Stevin, Plancius, Lansbergen, van Schooten).

Stevin is treated in a special chapter (V) as are Descartes (VII) and Huygens (VIII). In the chapter on Descartes, Frans van Schooten and Jan de Witt also get special attention, as does Spinoza in the chapter on Huygens. The merits and achievements of Petrus Plancius and Philips (van) Lansberge(n) as well as those of Willem Barentsz, Willem Jansz Blaeu, Ludolf van Ceulen, Willebrord Snellius and of minor scientists are recorded in Chapter IV, entitled Teachers of mathematics and navigation. Chapter IX, called The living world, deals mainly with Antoni van Leeuwenhoek, but also with such well-known scientists as Nicolaas Tulp, Franciscus Sylvius, Reinier de Graaf, Frederik Ruysch, Carolus Clusius and Jan Swammerdam. Following an interesting chapter (X), On colonial science, the book ends with a short chapter, not present in the first edition, Into the age of Boerhaave. This age, in which we find a shift from Cartesianism to experimental philosophy, succeeds the Golden Age. The Republic was not a leading power anymore, but was still regarded with great respect.

12.1. From the Publisher.

This compact, well-written history - first published in 1948, and now in its fourth revised edition - describes the main trends in the development of all fields of mathematics from the first available records to the middle of the 20th century. Students, researchers, historians, specialists - in short, everyone with an interest in mathematics - will find it engrossing and stimulating.

Beginning with the ancient Near East, the author traces the ideas and techniques developed in Egypt, Babylonia, China, and Arabia, looking into such manuscripts as the Egyptian Papyrus Rhind, the Ten Classics of China, and the Siddhantas of India. He considers Greek and Roman developments from their beginnings in Ionian rationalism to the fall of Constantinople; covers medieval European ideas and Renaissance trends; analyses 17th- and 18th-century contributions; and offers an illuminating exposition of 19th century concepts. Every important figure in mathematical history is dealt with - Euclid, Archimedes, Diophantus, Omar Khayyam, Boethius, Fermat, Pascal, Newton, Leibniz, Fourier, Gauss, Riemann, Cantor, and many others. For this latest edition, Dr Struik has both revised and updated the existing text, and also added a new chapter on the mathematics of the first half of the 20th century. Concise coverage is given to set theory, the influence of relativity and quantum theory, tensor calculus, the Lebesgue integral, the calculus of variations, and other important ideas and concepts. The book concludes with the beginnings of the computer era and the seminal work of von Neumann, Turing, Wiener, and others.

12.2. From the Preface.

This new edition incorporates into the text numerous amendments and corrections, and includes copious additions to the bibliographies. A new chapter has been added on the mathematics of the first half of the present century; in the Preface to the previous edition [1967] I expressed the desire that such a history be written. I now have tried my hand at it myself. The late Professor P B Pogrebysskiĭ, translator of the book into Russian, had treated the subject in an appendix to the second edition of the Russian translation ["Nauka", Moscow, 1969] which can also be found, translated, in the fifth edition of the German translation [VEB Deutscher Verlag Wissensch., Berlin, 1972]. My own approach to this complicated subject has been based in part on Pogrebysskiĭ and on different monographs and chapters in such books as those of C Boyer, M Kline, J M Dubbey, H Wussing, and others, but also on my own experience. My version is, again, a 'concise' history; there is still room for a full-scale history of mathematics from the beginning of the century to the Second World War.

After the original publication of this book, an edition appeared in Great Britain and translations were published in several languages. In some of these translations, paragraphs or sections were added dealing with the mathematics of the country in which the book appeared. (See "Preface to the Third Revised Edition," below.) A few more translations of the book have appeared since 1966. For the Spanish and Italian translations (by Professors P Lezama and U Bottazzini, respectively) I have written prefaces with a sketch of the history of mathematics in Spain and Italy. The Italian translation, moreover, is enriched by a 64-page monograph by Professor Bottazzini on the Italian mathematics of the nineteenth century. The bibliographies in these books, as in some other translations, contain titles of interest to the readers in those particular languages.

12.3. From the Introduction.

Mathematics is a vast adventure in ideas; its history reflects some of the noblest thoughts of countless generations. It was possible to condense this history into a book of two to three hundred pages only by subjecting ourselves to strict discipline, sketching the unfolding of a few main ideas and minimizing reference to other developments. Bibliographical details had to be restricted to an outline; many relatively important authors - Roberval, Lambert, Schwarz - had to be bypassed. Perhaps the most crippling restriction was the insufficient reference to the general cultural and sociological atmosphere in which the mathematics of a period matured - or was stifled. Mathematics has been influenced by agriculture, commerce, and manufacture, by warfare, engineering, and philosophy, by physics and by astronomy. The influence of hydrodynamics on function theory, of Kantianism and surveying on geometry, of electromagnetism on differential equations, of Cartesianism on mechanics, and of scholasticism on the calculus could only be indicated in a few sentences - or perhaps a few words - yet an understanding of the course and content of mathematics can be reached only if all these determining factors are taken into consideration. Often a reference to the literature has had to replace an historical analysis. Our story ends around 1945, for we feel that the mathematics of the last decades of the twentieth century has so many aspects that it is impossible - to this author at any rate - to do justice even to the main trends.

It is hoped that, despite these restrictions, we have been able to give a fairly honest description of the main trends in the development of mathematics throughout the ages and of the social and cultural setting in which it took place. The selection of the material was, of course, not based exclusively on objective factors, but was influenced by the author's likes and dislikes, his knowledge and his ignorance. As to his ignorance, it was not always possible to consult all sources first-hand; too often, second- or even third-hand sources had to be used. It is therefore good advice, not only with respect to this book, but with respect to all such histories, to check the statements as much as possible with the original sources. This is a good principle for more than one reason. Our knowledge of authors such as Euclid, Diophantus, Descartes, Laplace, Gauss, or Riemann should not be obtained exclusively from quotations or histories describing their works. There is the same invigorating power in the original Euclid or Gauss as there is in the original Shakespeare, and there are places in Archimedes, in Fermat, or in Jacobi which are as beautiful as Horace or Emerson.

Among the principles which have led the author in the presentation of his material are the following:

1. Stress the continuity and affinity of the Oriental civilisations, rather than the mechanical divisions between Egyptian, Babylonian, Chinese, Indian, and Arabian cultures.

2. Distinguish between established fact, hypothesis, and tradition, especially in Greek mathematics.

3. Relate the two trends in Renaissance mathematics, the arithmetical-algebraic and the "fluxional," respectively, to the commercial and engineering interests of the period.

4. Base the exposition of nineteenth-century mathematics on persons and schools rather than on subjects. (Here Felix Klein's history could be used as a primary guide. An exposition by subjects can be found in the books by Cajori and Bell, or with the more technical details, in the Encyklopædie der mathematischen Wissenschaften [24 vols., Leipzig, 1898-1935], and in Pascal's Repertorium der höheren Mathematik [5 vols., Leipzig, 1910-29].) For the twentieth century a more mixed method has been followed.