1.1. Review by: D T Whiteside.
British Journal for the History of Science 4 (1) (1968), 84-85.

John Wallis - otherwise government cryptographer, arch-enemy of Thomas Hobbes, a founding father of the Royal Society and many other things - achieved the rare distinction of having two sets of his Opera Mathematica published in his lifetime and has since been carefully accorded his place in the standard histories of mathematics. A cynic might add waspishly that, as Savilian Professor of Geometry at Oxford, Wallis was ex officio a controlling member of the Delegation which licensed the "typographus academicus" Lichfield to print the former in 1656-57 and, as Keeper of the Archives, was effective head of the infant Oxford Press which produced the latter in 1693-99, while none of his technical works has since been republished. None the less, Wallis's over-eagerness to rush into print and, in Collins's words, a penchant for "printing discourses that came in private Letters ... without permission or consent" did conceal a genuine mathematical talent, if not one of first rank. A number of scholars during the past half century, notably Adolf Prag and most recently F D Kramar, have closely examined the integration techniques propounded by Wallis in his Arithmetica Infinitorum of 1656 and allied works, the most celebrated outcome of which is his infinite product for 4/π. In his present monograph Christoph Scriba broadens this assessment, emphasising neglected aspects of his minor researches in algebra and number theory and, more generally, giving us the benefit of two years' undivided study of the unique collection of Wallisiana now in Oxford.

In common with many of his contemporaries, not least Newton, Wallis's introduction to algebra was effected (in the late 1640's) through an unaided reading of Oughtred's Clavis and consummated by his subsequent study of Descartes' Geometrie, Schooten's 1646 edition of Viète's Opera Mathematica and, later still, Schooten's own Exercitationes Mathematicae of 1657. Among his juvenilia we find, for instance, in indication of his incipient prowess and relative ignorance of standard published results, his independent "discovery" of Cardan's rule for solving the reduced cubic equation. His improved exposition (printed only long after in 1685) of Schooten's theory of combinations and the Viète-Alexander codification of angular sections is elegant if long-winded but represents no essential advance upon his sources. His unpublished researches on number theory pursued in the late 1650's (which Scriba here describes for the first time) are a little more rewarding. Of the topics touched upon, Wallis's discussion of Heronian triangles is a not markedly original reworking of passages in Book 5 of Schooten's Exercitationes, but a fresher response was evoked by two challenges from Fermat: to find perfect numbers other than those generated by Euclid's formula (Elements, Book 9) and "trouver deux nombres cubes, dont la somme soit cube" (see page 141 of Wallis's 1658 Commercium Epistolicum). He rightly guessed that all even perfect numbers are of Euclid's form and that "non potest numerus cubus rationalis in duos cubos rationales dividi", but his proof of neither conjecture is adequate and his attempt to demonstrate the latter merely skirts the central issue. Scriba works manfully to make Wallis's drab calculations interesting, ingeniously completing the former proof that all even perfects are Euclidean, but could have perhaps better conveyed the quality of the original text by a more liberal quotation from it.

The English reader of Scriba's (German) book will probably be most attracted by the Introduction and opening of Part I, where an informative précis of Wallis's earliest formative years is given, and especially the concluding Appendix, which outlines the content of existing autograph MS. (notably Bodleian Don.d.45) and lists some 200 books in the Savile collection which were probably once owned by Wallis. The latter in particular, with its copious explanatory notes, is at once an invaluable guide to primary sources and a revealing glimpse of Wallis's mathematical preferences. Scriba's crisp account is written with verve, erudition and authority, but in the central portion of his text I was left with a curiously flat impression. In part, no doubt, this was my own blindness, but it is hard to resist the inference that Wallis, too, in the topics discussed is often dull and monotonous. Certainly, he is not here at his mathematical best. discussed

1.2. Review by: Adolf Prag.
Studia Leibnitiana 2 (1) (1970), 75-77.

The author feels he must apologise to the reader from time to time because neither the works discussed are very profound, nor was the author truly eminent enough: but this excellent book requires no external justification. Wallis is known as the discoverer of an infinite product for π; otherwise, he was not a great creative mathematician. His significance stems from his central position in the scientific life of the second half of the 17th century. He was a professor of mathematics at Oxford for more than 50 years; he published frequently and readily, and was easily inclined to write or respond to polemical works. For scholars on the continent, be it Fermat or Hevelius, he was the natural representative of English mathematics; Huygens and Leibniz corresponded with him even in his old age. Incidentally, he came to mathematics relatively late, entirely as an autodidact - he had not been to Italy like Barrow or James Gregory, and he mentions the difficulty of obtaining a copy of Cavalieri's work in England. He was already nearly 40 when the Arithmetica Infinitorum appeared with the product for, and the large-scale Algebra was first published 30 years later. Scriba gives an account of Wallis's early development and then analyses some of his smaller works: one had to thoroughly examine even the less interesting ones. Wallis's reputation certainly prompted many a beginner to study all his works, as they were easily accessible. The details are often quite primitive, but not boring, because they reveal areas of interest: "that's all one can expect or offer." How much of this work appeared in lectures is unknown overall; no prominent mathematician was, for example, a "student" of Wallis, but for the large number of amateur mathematicians in England, the Oxford Savilian Professor was a well-known and helpful authority.

Although the treatise on angle sections was not printed until 1684/85, it originated as early as 1648. Despite its geometric framing, it is a study of specific algebraic equations, from which Wallis primarily derived the insight that higher-degree equations can even have more than two solutions. This early work in particular demonstrates Wallis's characteristic tendency to elaborate on everything in great detail and as completely as possible. Scriba significantly enhances the value of his presentation and elevates it above the level of a mere explanatory summary by taking it as the starting point for a retrospective history. He presents the development (from Ptolemy) through Viète, Kepler, Bürgi, Girard, and Oughtred, in whose Clavis Wallis had found his immediate inspiration, in detail and with full evidence, sometimes even preserving the original notation. Similarly, for the short treatise on combinations, the most interesting aspect is not only the model in Schooten's Exercitationes of 1657, but also a detailed analysis of a directly contemporary work by Thomas Strode, printed in London in 1678, before Wallis's essay (as an appendix to Algebra) appeared. To round things off, Scriba publishes some of Wallis's attempts at number theory from the manuscript collection held at the Bodleian Library: everywhere Wallis shows himself capable of quickly absorbing and developing suggestions from others, but he was not a mathematician who independently raised essential questions. His method repeatedly consists of attempting incomplete inductive inferences based on many tables; he contributed nothing to the progress of number theory.

Wallis became known through the Arithmetica Infinitorum, the Commercium Epistolicum (on Pell's equation), the Mechanica, and finally the Algebra secured him the recognition of his contemporaries, and thus he was a respected university professor whose work helped prepare the way for the resurgence of mathematics in England.

One must not fail to praise Scriba's didactic skill: first, a precise presentation of the text, a modern mathematical analysis - which sometimes goes beyond the textual material, so that one incidentally learns a bit more mathematics - then a brief retrospective, so that one can see the forest for the trees; to this are the aforementioned classifications within the overall history. Thus, for many readers, this little book will become a veritable intensive introduction to the history of mathematics in the 17th century, albeit only to a subfield, but in a very personal and concrete way.

In an appendix, Scriba provides a brief history of the Savile Library in the Bodleian Library, that is, the special library of the two Savilian professors (astronomy and mathematics), and, most importantly, a detailed list of the books from Wallis's own collection, as well as a preliminary overview of his handwritten papers. The book list is exceptionally valuable; it is rare for the mathematical library of a prominent scholar to be preserved so completely. Many of the books came to Wallis as a donum autoris, a testament to his international renown. In many of them, Wallis made corrections and other notes, which Scriba carefully records in the footnotes. Wallis is responsible for one error in both Bodleian's complete catalogue and Scriba's list: "Bartholinus, De harmoniae (instead of harmonice) proportionalibus, G 25 (3*)" should be deleted and replaced by "Dioristice, seu aequationum determinationes, Copenhagen, 1663, title lost, G 25 (1)", whereas the number G 25 (3*) belongs to "Selecta Geometrica, 1674". Wallis even wrote to Collins (on August 7, 1666): "Bartholinus' Dioristice, I do not remember that I have seen at all..." but we do not know, of course, when the treatise came into his possession!

1.3. Review by: Jean Itard.
Revue d'histoire des sciences et de leurs applications 21 (2) (1968), 185-186.

John Wallis has been the subject of several studies in recent years, by A Prag (Quellen und Studien zur Gesch. der Math., Astron., und Physik, 1931), J F Scott (The Mathematical Work of John Wallis, London, 1938), and F D Kramar (Istoriko-matemat. Issledovanija, 1961).

C J Scriba has already devoted several articles to him. The work we are analysing is number 6 in the "Boethius" series.

John Wallis was the Savilian Professor of Geometry at Oxford for more than half a century, from 1649 until his death in 1703. He also played a prominent role in the scientific community from which the Royal Society emerged in 1660.

C J Scriba's work is divided into three parts, supplemented by an appendix.

The first part is devoted to Wallis's mathematical beginnings. After studying Oughtred's Clavis mathematicae in 1647-1648, he independently rediscovered Cardano's formula and composed a Treatise on Angular Sections. The greatest benefit he derived from his early work in algebra was a deeper understanding of the structure of equations, and in particular the discovery that the number of roots corresponds to the degree of the equation. From this perspective, Descartes's Géométrie offered him little that was new when he undertook its study shortly thereafter.

In the second part, we see that Wallis's Discourse of Combinations, Alternations, and Aliquot Parts is, for the most part, a development of a theme from the Exercitationes mathematicae published by Schooten in 1657. The last chapter of the Discourse, however, deals with the methods used by the author in his polemic with Fermat concerning problems in number theory. These methods are elementary and offer nothing comparable to those of his great adversary.

The third part concerns some unpublished studies by the Oxford mathematician. An essay on rational triangles is again inspired by Schooten's Exercitationes. Another, on perfect numbers, fails in the proof of what will become Euler's theorem: every even perfect number is Euclidean. Other attempts to solve the problems proposed by Fermat in number theory are unsatisfactory but illustrate Wallis's characteristic mode of thinking in his mathematical research.

The appendix contains a short history of the Savilian Professors' Library, founded in 1619, and a brief description of the preserved Wallis manuscripts at the Bodleian Library, and finally an annotated list of the books that belonged to Wallis and were given to the Savile Library upon his death.

Apart from the results of his mathematical research, the most famous of which is his evaluation of 4π by an infinite product, Wallis's contributions must be seen in relation to the very low level of the exact sciences in England in the mid-17th century. Through his tireless activity as a teacher and author of numerous works, he, along with a few others, was at the origin of the development of mathematics in his country in the second half of the century.

The works of C J Scriba will, for their part, contribute to a better understanding of the state of mathematics during this essential period for the history of science.

1.4. Review by: F Krafft.
Mathematical Reviews MR0247982 (40 #1241).

The book under review presents the Habilitationsschrift (post-doctoral thesis) of the current holder of the Chair for the History of the Exact Sciences and Technology at the Technical University of Berlin. It is the first major study by Scriba to emerge from his two years of research on the manuscripts of the English mathematician John Wallis in English libraries, and it constitutes a significant contribution to the history of mathematics in 17th-century England. What distinguishes this book from the wealth of literature on the history of mathematics - including that concerning the 17th century - is not merely its findings regarding the history of mathematics in pre-Newtonian England; indeed, Wallis - whose contributions to the development of mathematics, when viewed from a continental perspective, do not nearly match those of figures such as Viète, Descartes, Fermat, and Bernoulli - would hardly have been the ideal subject for such a study. Rather, the book's merit lies in its successful attempt to outline and contextualise the state of mathematics in 17th-century England from the perspectives of cultural and scientific history, and to present Wallis's achievements against this backdrop. These achievements consisted essentially in the fact that, through his teaching as Savilian Professor of Geometry at Oxford (from 1649 until his death in 1703) and his extensive mathematical publications (beginning in 1655), he acted as a conduit for continental mathematics and its new methods, thereby elevating English mathematics to a level that made the emergence of an Isaac Newton possible.

John Wallis (b. 1616) himself - who had focused his broad-ranging studies on theology and served as a chaplain following his ordination (1640) - appears not to have come into direct contact with mathematics until relatively late, around 1647-48; it was then that he developed a keen interest in theoretical mathematics after reading the first edition of William Oughtred's Clavis mathematicae - a work that was rather meagre compared to the state of mathematics on the Continent. He remained self-taught throughout the ensuing period: while he studied ancient and modern mathematics thoroughly - albeit without always having access to the most suitable texts for the latter - his methods of discovery and proof were characterised more by boldness and ingenuity than by classical rigour. Consequently, even after his new results (such as the Wallis product for 4/π) gained recognition, he remained a target of criticism from Continental masters of classical, rigorous proof techniques. Towards the end of his life, he had to face the painful realisation that the younger generation - partly thanks to his own assistance - had far surpassed him. Yet, even if he lacked the creative power to transcend the very standards he had helped establish, he deserves credit for not dwelling on this realisation - painful as it was for an ambitious pioneer - but rather contributing significantly to the advancement and dissemination of Newtonian mathematics.

In contrast to the few earlier studies on Wallis, the author incorporates a vast amount of manuscript material alongside printed sources into his account; this material characterises Wallis's methods and documents their evolution even more effectively than the printed works. Following a detailed introduction, the first main section covers "Wallis's Early Mathematical Studies: The Treatise on Angular Sections and Its Predecessors" (Viète, Kepler, Oughtred), along with an outline of his education leading up to his appointment to the Savilian Professorship; the second main section addresses "The Treatise on Combinations, Permutations, and Divisors, and Its Predecessors" - each concluding with a comparative analysis that reveals Wallis to be a systematic, pedagogically skilled, and thoroughly original mathematician. The third main section presents Wallis's investigations into number-theoretic problems based on previously unpublished manuscripts; perhaps the most notable of these concerns perfect numbers, where his results came close - though characteristically, only almost close - to those of L Euler. This section concludes with a "Retrospective" summarising the findings of the preceding studies regarding Wallis and his circumstances. An appendix provides an overview of the books and manuscripts from Wallis's library that passed into the Savile Collection and subsequently to the Bodleian Library (Oxford), preceded by a brief history of the Savile Collection. The author is the first to have examined the collection of books and manuscripts formerly owned by Wallis; in doing so, he discovered various interesting details regarding signs of use and handwritten additions made by Wallis, all of which are noted in the footnotes. In future, this compilation cannot be overlooked when addressing questions regarding sources. Moreover, it offers interesting insights into the library of a 17th-century scholar with diverse interests, thereby acquiring broader significance for cultural history. The author has since provided a supplementary compilation of the extant letters to and from Wallis - documents from which he hopes to gain greater insight into the development of Wallis's mathematics [the author, Notes and Records Roy. Soc. London 22 (1967), 58-93].

1.5. Review by: J E Hofmann.
Sudhoffs Archiv 51 (1) (1967), 93-94.

This excellent habilitation thesis by the author draws on numerous printed works, his handwritten papers, and relevant secondary literature. The introduction provides a general overview of the state of mathematics in England during the first half of the 17th century, as well as Wallis's emergence, work, and influence. Part 1 deals with his origins, education, and early independent studies (cubic and biquadratic equations, angle sections, and similar results in Viète, Kepler, and Oughtred). Part 2 focuses on the contributions to the theory of aliquot parts, inspired by F van Schooten (Exercitationes mathematicae, Leiden 1657), and the related, hitherto largely overlooked treatise on combinatorics (London 1678), which includes a supplementary report on Th Strode's work. To these works, which later became part of Algebra (Oxford 1685), are added in Part 3 insightful communications concerning unpublished number-theoretical investigations (rational triangles, report on Schooten, Wren, and Pell's determination of isosceles rational triangles with equal perimeter and area, perfect numbers, Fermat's impossibility theorems), with a clear indication of Wallis's shortcomings. In retrospect, Wallis's personality and mathematical aspirations, his central importance for the intensification of mathematical studies in England, and the limits of his professional understanding are sharply outlined in brief. The appendix refers to Wallis's largely preserved library, which contains numerous rarities with informative marginal notes. The superbly produced volume (portrait, title photos, manuscript samples) is supplemented by a meticulously compiled index of names and a short subject index.

1.6. Review by: Michael S Mahoney.
Isis 64 (2) (1973), 265-266.

One's estimation of this book will depend in large part on what one takes to be its aim. It seems, in fact, to have several aims, not all of which are equally well served by its content and organisation. Entitled "Studies," the book is structured about three main chapters in which Scriba presents highly skilled exegeses of Wallis' research (both published and unpublished) on angular sections, combinatorics, and number theory. Within each chapter he includes summary accounts of the related work of some of Wallis' contemporaries and immediate predecessors: François Viète, Johann Kepler, and William Oughtred on angular sections; Franz van Schooten and Thomas Strode on combinations, permutations, and aliquot parts; and van Schooten and Christopher Wren on rational triangles. Rich in technical detail, these accounts offer welcome information about the work of men whose con-temporary importance is matched only by the paucity of historical studies devoted to them. But the accounts are also straight-forwardly exegetical, with little to link them together in a continuous historical or analytical narrative. And when the author attempts to make the links, the ambiguity of his intentions emerges clearly.

For example, of the three men discussed in connection with Wallis' Treatise on Angular Sections, only Oughtred, whose Clavis mathematicae was the source of the general problem, had demonstrable influence on Wallis' analysis. Scriba recounts the work of Viète and Kepler apparently only to show they did not influence Wallis. But why just Viète and Kepler, and why such detail for a negative conclusion? Similarly, why devote ten pages to Strode's Short Treatise of the Combinations, Elections, Permutations & Composition of Quantities only to inform the reader afterward that Wallis knew nothing of the work? Finally, why resurrect Wallis' feeble attempts in later life to prove two of Fermat's theorems in number theory, when, as Scriba himself admits, those attempts might best be left in oblivion?

To these questions Scriba has answers, but they are answers that belong in another sort of book. Wallis' demonstrable independence from Viète and Kepler and his demonstrably superior treatment of angular sections document at the same time the autodidactic nature of his training and the real spark of original talent he possessed. Similarly, Strode serves as a measure of Wallis' achievement in the late 1670s. Finally, Wallis' application to number theory of the inductive methods that had proved successful in his Arithmetica infinitorum, his cryptography, and especially his continued-product expression for 4/π illustrates the lack of flexibility that denied him any real chance for greatness.

Behind these justifications lies the con-fusion over aims that weakens Scriba's book. For he would clearly like to offer a picture of the developing state of mathematics in England during Wallis' lifetime, in order to support the claim that Wallis more than any of his compatriots (indeed, perhaps almost singlehandedly) was responsible for the renascence of mathematics in seventeenth-century England.

This is an eminently worthwhile goal. Whether or not one agrees exactly with Scriba about the initial low state of mathematics in England during the first third of the century, the pinnacle reached during the last third is undeniable, and Wallis as an active mathematician, holder of the first (and, until 1664, only) chair of mathematics, charter member of the Royal Society, and ambitious career man during the middle third seems the most likely candidate to explain the upsurge. To assess his influence, however, it will not suffice to analyse the details of his minor (or even his major) works. Much as Wallis' role as rejuvenator mathematicae depended on his knowledge and talent as a mathematician, it depended also (ultimately perhaps more) on his relations with other mathematicians, on his ability to use his positions at Oxford and the Royal Society to steer interest toward mathematics, on the self-image he entertained as critic and patron of the art, and on the extent to which others accepted him in his assumed role. Only an investigation along these lines can establish what Scriba seems most to want to show. Since he clearly knows the work and setting of both Wallis and his contemporaries extremely well, one can only hope he will someday carry out that investigation.

2.1. From the Preface.

The material which is presented here was collected for a graduate unit in education, which was first offered in the summer of 1961, by the Department of Graduate Studies of the Ontario College of Education. An attempt has been made to exemplify the title of the course "Historical Development of Mathematical Thought" in those sections of mathematics that are most closely related to the number concept: arithmetic, algebra, and elementary number theory. Particular emphasis has been given to topics likely to be of value to educators interested in the teaching of mathematics in the schools. The reference to the needs of the classroom is, at times, historically motivated: at other times, it is expressed in the choice of subject matter. It is my belief that the combination of these two points of view can lead to a profitable course for graduate students in education. The present text is merely the outcome of a first step in this direction. To what extent I have been successful in meeting the need, the reader will have to judge for himself. No attempt has been made to suggest where, or even if, portions of this material might be included in the school curriculum it is directed at the teacher, to assist him in obtaining a broader view of the development and implications of the subject that he is offering to his students.

The need for the preparation of this text arose from the fact that there seems to be no single book available which covers approximately the desired ground. Those which come nearest to some of the sections have been listed in the bibliography which as far as possible is restricted to books written in English. However, most of the content of the German "Zahlwort und Ziffer" by the late Dr Karl Menninger - a history of the development of the number concept in word and script in various cultures and during different periods - is not available in English. I have drawn from it heavily, and it is for this reason that Chapter I is considerably longer than any other chapter.

When I was preparing the course Professor Emeritus C E Phillips and Professor G E Flower (both then of the Ontario College of Education) and Professor D B DeLury of the University of Toronto showed great interest in this undertaking and gave helpful encouragement. Miss M E Dormer Ellis, B.A.Sc., B.A., Ed.D., P. Eng. revised the whole manuscript, added numerous ideas, contributed many of the problems, and typed all the stencils for the mimeographed copies which were made available as a textbook. She also taught the course after I had left Canada in 1962. Except for this slightly revised preface, the correction of a few misprints, and an addition to the bibliography the text is now published by photographic reproduction and reduced in size in a permanent form.

2.2. Review by: A S Saidan.
Isis 61 (1) (1970), 124.

Of the five chapters in this book, the first two cover 101 pages and present the language of numeration in ancient civilisations and primitive societies together with historical notes on number symbols and the cardinal arithmetical operations. The Babylonian sexagesimal system receives a passing mention, but the story of the development of the Hindu-Arabic numerals is given almost as much space as the idea of numbers as adjectives or counting as a stairway. The author, Christoph J Scriba, draws heavily from Menninger's Zahlwort and Ziffer.

"Higher Arithmetic" is the heading of the third chapter. With reference to Hilbert's Foundations of Geometry and Peano's axioms of natural numbers, the number system in the Gaussian plane is sketched. The reader is then briefed about the historical development of the system as well as fractions, square roots, and logarithms.

In the fourth chapter, one section is on historical approaches to the solution of linear and quadratic equations, one deals with cubic and quartic equations, and two cover divisibility, congruence, perfect numbers, Euclid's algorithm for finding the highest common factor, and Diophantine equations.

The fifth chapter is on contributions to the number concept since the nineteenth century. It deals with the fundamental theorems concerning roots of equations, Peacock's and Hankel's principle of permanence, Hamilton's number pairs and algebra of quaternions, and Cantor's study of infinity.

The preface states that The Concept of Number constitutes the subject matter of a course on the historical development of mathematical thought given to teachers of school mathematics at the Ontario College of Education. Opinions have differed - and for some time to come may continue to differ - as to what to include in a book made to serve such a purpose and the relative importance of the material included. No severe comment should therefore be made on the author's choice of topics.

But certainly the contents should reflect the best of the knowledge of the scholarly world. Regarding ancient and medieval mathematics, this knowledge is now developing rapidly, and monumental works are being superseded. We now know for example that finger-reckoning was the current practice in the Mediterranean world until Indian arithmetic was introduced, with the dust abacus as a necessary tool thereof, at which time the two systems shared a long coexistence and a slow modification and amalgamation which led to the best schemes of calculation known to, say, the sixteenth-century arithmeticians. We know as well that as early as 953 A.D. one Arab at least dealt with decimal fractions and used a relatively elegant decimal sign. In the light of such knowledge some statements in Mr Scriba's work should be corrected.

2.3. Review by: Margaret E Baron.
The Mathematical Gazette 54 (389) (1970), 316.

This book resulted from a course in the 'Historical Development of Mathematical Thought' which was given in the summer of 1961 at the Department of Graduate Studies, Ontario College of Education. The college is to be congratulated for sponsoring such a course and Dr Scriba for pioneering it.

The topic explored is the development of concepts of number through the ages and the choice of subject matter throughout is partly motivated by historical considerations and in part by the needs of the classroom. The intention is to provide teachers, or future teachers, of mathematics with a historical background to material they will be teaching. Algebra and geometry are considered in so far as they have influenced, or been influenced by, the invention of new or 'artificial' numbers, as Scriba calls them. This, indeed, is how the history of mathematics should be taught if it is to illuminate and guide the student through the confusing, and sometimes even conflicting, welter of concepts he encounters in his path from school to university - and, in the case of teachers of mathematics, back again to school.

The historical treatment is sound and careful throughout although Scottish readers may feel that Napier has been given something less than adequate credit for his invention of logarithms. In the later chapters Scriba has made a useful contribution in sorting out some of the sophisticated developments which took place in the nineteenth and early twentieth centuries. May one hope that the author will find time to develop this theme in another volume?

Scriba takes care to ensure that the mathematical concepts and methods discussed should be fully understood by the students and the exercises provided are, for the most part well chosen to provide insight and understanding of the techniques and modes of reasoning in use at a given time. Where the North American method of testing factual material by 'True-False' answers has been adopted the exercises seem less successful. The Bibliography is useful and I particularly like the timely warning as to the damage which may be done through the uncritical reading of out-of-date accounts of the history of mathematics. It is unfortunate that Boyer's History of Mathematics (John Wiley, New York, 1968) was presumably published too late for inclusion.

2.4. Review by: Galina Pavlovna Matvievskaya.
Mathematical Reviews MR0241232 (39 #2574).

The book deals with the historical and the logical development of the number concept and is addressed mainly to teachers. Chapter I commences with the description of the principles involved in our number system and then gives an account of the evolution of the number concept in word and script. Examples of number words and number scripts by various peoples are presented and discussed.

Chapter II describes the elementary operations with numbers which have been common in ancient Greece and Rome, and Central and Western Europe. The first section is on the various historical forms of the abacus: the sand or dust abacus, the table with loose counters, the sliding bead abacus. The second section discusses the methods of performing elementary arithmetic operations described in the medieval manuscripts and in the printed reckonbooks of the Renaissance. The concluding section of this chapter deals with the binary number system.

Chapter III, on higher arithmetic, is divided into three sections: the logical construction of a number system, the historical development of artificial numbers, and basic operations with artificial numbers.

Chapter IV is on algebra and number theory. The first and second sections contain a brief account of the history of the theory of equations from ancient times to the XVIIth century. Some of the problems connected with the divisibility of numbers and with Diophantine analysis are discussed in the next two sections.

With Chapter V, on the development of the number concept since the beginning of the XIXth century, the main body of the book closes. There are also two appendices (finger multiplication, method of extracting cube roots), a bibliography with critical remarks, and index. Each section of the book is followed by numerous exercises.

2.5. Review by: Jean Itard.
Revue d'histoire des sciences et de leurs applications 22 (4) (1969), 371.

Professor Scriba, of the University of Hamburg, is well known for his work in the History of Mathematics, and more particularly for his research on the English mathematician John Wallis (1610-1703).

The work we are referring to here bears the very significant subtitle: A Chapter in the History of Mathematics, with Applications of Interest to Teachers.

We are dealing with a course taught in the summer of 1961 at the Department of Graduate Studies of the Ontario College of Education. The author thanks, among others, Miss M E Dormer Ellis, who revised the manuscript, supplementing it in several places, and who continued the teaching he had begun after leaving Canada in 1962.

It is therefore a very successful introduction to the history of numbers, given to future teachers. Thus, we are not overwhelmed with biographies, but can follow the evolution of the fundamental concepts of Mathematics.

The documentation is very rich and reliable. The various spoken, mimed, or written numerals are carefully analysed, from the most remote periods and among the most diverse peoples. But the most modern periods are not neglected. Thus, on the last page, we find the names of Russell, Whitehead, Brouwer, Hilbert, and Gödel.

Numerous exercises are provided.

May French education draw inspiration from examples similar to this effective and unpretentious course.

2.6. Review by: Laura Guggenbuhl.
The American Mathematical Monthly 78 (2) (1971), 212.

This is one of a number of recent, small, inexpensive, paper-back mathematics books written primarily for use in graduate classes in a Teacher Education Program. The authors or editors of these books, who have been careful to respect the needs and the maturity of prospective readers, have skilfully integrated results of current mathematical research with traditional subject matter and treatment. Thus the student has a taste of the spirit of discovery, an attribute which does much to enliven courses in a Teacher Education Program.

The present book is devoted to a discussion of material which can be found in books on the History of Mathematics, Modern Algebra, Higher Arithmetic, Theory of Numbers, etc. As the author says, there seems to be no other book in the English language which has gathered together in a single volume his selection of subject matter. The student will find the book interesting and well written; he will find it a challenge, though not on too advanced a level. Since chapter headings are given in the telegraphic review cited above, we will refer here only to some of the important features of the author's work.

After discussing the generally more familiar topics of the development of number systems and the elementary operations with numbers, the author goes into a discussion in Chapter IV of algebra and number theory. In this chapter he discusses the theory of congruences, Euclid's algorithm, and Diophantine arithmetic. Chapter V, the final chapter, is devoted to a discussion of contributions to the number concept since the beginning of the nineteenth century.

Here the author discusses works of Hamilton and Cayley, Liouville, Hermite, Peano, Dedekind, and Cantor, Hilbert and others. What a pity, however, that the names of these mathematicians have been omitted from an otherwise carefully prepared index. The book is copiously illustrated by tables and diagrams; it contains a wealth of exercises and problems, with a section on answers to selected problems; and an extensive bibliography. With the exception of the classic references, Menninger's Zahlwort und Ziffer, Tropfke's Geschichte der Elementar Mathematik, and one or two others, the bibliography of forty titles is confined to books in the English language. An exceedingly helpful innovation is the brief description which follows the title of an individual item listed in the bibliography.

Unfortunately the process of photographic reproduction and reduction in size, by means of which the book has been published, has rendered the print so small that reading the text is difficult. One hopes that a wide acceptance of this fine book will warrant a more worthy form of publication of a new edition in the near future.

2.7. Review by: J E Hofmann.
Sudhoffs Archiv 53 (1) (1969), 109.

This is a reproduction of a deliberately simple, excellently researched introductory lecture that emphasises the historical aspect. The introduction deals with the historical development of counting (number words and digits) and arithmetic (basic operations, binary arithmetic); this is followed by the logical structure of the number system, the transition to algebra (calculating with letters), and indeterminate analytic analysis. Finally, we read more about the general treatment of equations and their solutions, and about trends within algebra since the beginning of the 19th century. The engaging presentation is complemented by attractive exercises at the end of each section, the more difficult ones of which are solved at the end of the book. The index is meticulously compiled, as is the concise but well-chosen bibliography.

3.1. Review by: Wilhelm Strube.
Sudhoffs Archiv 65 (3) (1981), 311.

H Schimank's lectures were not edited for publication by him, but rather "carefully edited" in their spoken form by C J Scriba. Schimank approaches his topics from the perspective of a historian who surveys the overall course of the natural sciences, compares the differences and similarities of various stages of development, and highlights connections and contrasts. His extensive knowledge of details serves him well, as does his ability to think in broader contexts. Above all, Schimank aims to stimulate thought, to question what is taken for granted, or to shed new light on it through new perspectives

Christoph J Scriba's "Memorial Lecture" on "Mathematics' Self-Understanding and Public Perception" complements Schimank's lectures well in that it also subjects general contemporary problems to a historically comparative analysis. The four lectures provide answers to the frequently asked question about the "usefulness of history" and demonstrate that and how problem situations in the natural sciences and mathematics recur throughout history.

4.1. Review by: Doru Ştefănescu.
Mathematical Reviews MR0762986 (86e:01011).

The author presents some important moments from the history of the arithmetic of the rational points on elliptic curves. This book covers a very long period, from Diophantus to the present. As an illustration a problem of the Islamic mathematician Behā-Eddīn 'Amūlī (1547-1622) is expounded.

The problem of Behā-Eddīn 'Amūlī is the following: to find the rational solutions of the system of equations x2 + x + 2 = y2, x2 − x − 2 = z2. This is a standard Diophantine problem. In 1700 this problem was thought to have no solutions, but in 1843 it was noticed that there is a solution: (x, y, z)=(−17/16, 23/16, 7/16). (Behā-Eddīn 'Amūlī accepted only positive rational solutions.) To the problem of 'Amūlī is associated a cubic plane curve. The curve associated to 'Amūlī's problem is an elliptic curve.

The author presents the solution given by Nesselmann in 1843 and the complete solution of the problem obtained by É Lucasin 1877. He also remarks that a Persian translation of the work of 'Amūlī describes a positive solution of the problem.

The book also contains a history of the genesis of the elliptic curves. The contributions of Viète, Fermat, Descartes, Newton, Euler, Jacobi, Poincaré, Hilbert, Hurwitz, and Mordell are presented. The results obtained by H G Zimmerin 1982 with the help of computers, concerning the rational points on elliptic curves over algebraic number fields, are also discussed.

5.1. From the Publisher.

This is the first volume of a six volume collection of the complete correspondence of John Wallis (1616-1703). Wallis was Savilian Professor of Geometry at Oxford from 1649 until his death, and was a founding member of the Royal Society and a central figure in the scientific and intellectual history of England. In addition to serving as a decipherer on the Parliamentary side during the Civil Wars, he prepared the ground for the discovery of infinitesimal calculus by Newton and Leibniz and played a decisive role in the modernisation of English mathematics.

This volume provides fascinating insight into the life of Wallis through his correspondence with intellectual and political figures of the mid-17th century and will be of interest to historians of science and mathematics at all levels, from undergraduates to researchers.

5.2. From the Preface.

'Your letters,' wrote the German philosopher and mathematician G W Leibniz to his then regular correspondent John Wallis in 1697, 'always communicate something which serves to increase our knowledge.' Indeed, the letters of Wallis have long been valued as an important source of our understanding of scientific and theological developments in the seventeenth century, as one would expect from one of the leading figures of the time. However, using this source has previously presented scholars with considerable problems. Apart from those letters which Wallis published himself, most notably in the third volume of his monumental Opera mathematica (Oxford 1693-1699), or which later appeared in major editions such as the Oeuvres complètes of Christiaan Huygens or the Correspondence of Henry Oldenburg, much of his epistolary commerce is not easily accessed, being widely scattered in libraries and archives, mainly in Great Britain and on the Continent, but also in the United States.

It was the eminent historian of mathematics Joseph Ehrenfried Hofmann (1900-1973) who first drew Christoph J Scriba's attention to the Wallis papers and who, by securing the support of the Deutsche Forschungsgemeinschaft, enabled him to carry out investigations for his Habilitationsschrift in Oxford in the 1960s. Alongside the Studien zur Mathematik des John Wallis (Wiesbaden 1966), one of the fruits of Scriba's stay was a preliminary list of all the Wallis correspondence then known to exist, though largely excluding those letters which were concerned with theological topics, university politics, and deciphering or which were connected directly with his post as Keeper of the Archives of the University of Oxford. Building on this foundation, the edition, of which this is the first volume, is intended to publish for the first time in as complete a form as possible the entire correspondence of the Savilian professor of geometry.

In creating this edition, the editors are indebted once again to the Deutsche Forschungsgemeinschaft, which has born most of the material and personnel costs involved. Without their generous support and their willingness to accept necessary changes to the scope of the project once it was already underway, the edition could not have been realised in the form it now has.

Thanks are due also to the University of Hamburg, which has not only generously supplied the editors with rooms to house the edition itself, but also placed considerable technical resources and expertise at their disposal. The editors gladly record their gratitude in particular to the Institute for History of Science, Mathematics, and Technology for its unwavering support and co-operation over the years. They should also like to express their appreciation of the close ties which they were able to forge with the Graduate College for Text Transmission, headed by Dieter Harlfinger.

From the outset, the editors have been fortunate in being able to draw on the dedicated efforts of students of the University of Hamburg. In particular, they should like to record their thanks to Dmitri Abramov, Christian Alberti, Sergine Dupont, Alexandra Grigat, Christine Maack, Uwe Mayer, and Athina Pantoula. They wish to express special gratitude to Uwe Mayer, who has now assumed an editorial role, for his careful and tireless assistance in the final stage of producing the manuscript for this volume.

The editors gladly acknowledge their debt to Siegmund Probst, who was responsible for much of the editorial work in the initial stage of the project. After taking up a permanent editorial post at the Leibniz Archives in Hannover he has continued to work closely with the editors in Hamburg.

Thanks are due also to Ralf Haubrich (Göttingen), Christian Tapp (Munich), and Gerhard Brey (Munich) for their readiness to provide assistance on technical problems connected with LATEX. The editors are thankful also for assistance provided by Menso Folkerts (Munich) and Peter Schmitt (Vienna), who readily supplied them ITEX macros without which progress in the editorial work would have been considerably impeded.

A project of this nature can only succeed with the close co-operation of others working in the same field both at home and abroad. The editors have greatly benefited from their exchanges with friends and colleagues, not a few of whom were able to direct them to hitherto unknown papers of the Savilian professor. They should like to convey their warm thanks to David Cram (Oxford), Mordechai Feingold (Pasadena), Alan Gabbey (New York), Frances Harris (London), Robert Hatch (Miami), Michael Hunter (London), Adolf Prag (Oxford), and Jackie Stedall (Oxford). They acknowledge particular gratitude to the Keeper of Oxford University Archives, Simon Bailey, as well as to the Assistant Keeper, Alice Blackford.

The editors are indebted to the staff of innumerable libraries, archives, and county record offices for help in tracing material. In particular, they should like to express their thanks to the following for granting them permission to publish material held in their possession: The Syndics of Cambridge University Library; the Librarian of the Bibliotheek der Rijksuniversiteit Leiden; the British Library Board, the Director of Dr Williams's Library on behalf of the Trustees; the Librarian of the Bodleian Library, University of Oxford; the Keeper of the Archives, University of Oxford; the Bibliothèque Nationale de France; the Director of the Bibliothèque de l'Observatoire, Paris; the University Librarian, the University of Sheffield; the Director of the Osterreichische Nationalbibliothek, Vienna. The editors are particularly grateful to Adam Perkins and his colleagues in the Department of Manuscripts at Cambridge University Library for enabling them to obtain at short notice copies of the two earliest of numerous Wallis letters contained in the former Macclesfield collection, now in the Library's possession.

5.3. Review by: Paolo Mancosu.
Studia Leibnitiana 38/39 (1) (2006/2007), 124-127.

See below where Paolo Mancosu's review of both Volume 1 and Volume 2 are both given.

6.1. From the Publisher.

This is the second volume of a six volume compendium on the correspondences of John Wallis (1616-1703). Wallis was Savilian Professor of Geometry at Oxford from 1649 until his death, and was a founding member of the Royal Society and a central figure in the scientific and intellectual history of England. Along with his role as decipherer on the Parliamentary side during the Civil War, he prepared the ground for the discovery of infinitesimal calculus by Newton and Leibniz and played a decisive role in modernisation of English mathematics. This volume provides fascinating insight into the life of Wallis through his correspondences with intellectual and political figures of the latter part of the 17th century.

6.2. From the Preface.

The second volume of the Correspondence of John Wallis (1616-1703) covers a decisive period of political and scientific reorganisation in England following the eleven years of the Interregnum. It begins just a matter of weeks before the triumphal return of Charles II from the Netherlands and his accession to the throne on 29 May 1660 and sees the emergence of the Royal Society, soon to become the leading organisation of its kind in Europe, from two main scientific circles which had flourished in London and Oxford during the revolutionary years. Wallis, who had been appointed Savilian professor of geometry at the University of Oxford largely in recognition of his services to parliament as a decipherer during the Civil Wars was in this period establishing himself as one of the central figures in the growth of modern science in England. Fittingly, his name stands alongside those of Robert Boyle, Kenelm Digby, Robert Moray, John Wilkins, and Christopher Wren as one of the original members of council, appointed on 15 July 1662, when the first charter 'for the incorporation of the society under the title of the Royal Society' was passed.

The years encompassed by the present volume witness the Savilian professor consolidate his reputation both in London and in Oxford. At the Royal Society he was a close associate of its secretary Henry Oldenburg, with whom he corresponded regularly and in whose journal, The Philosophical Transactions of the Royal Society, he published articles on a wide range of themes from the fields of mathematics and the physical sciences. Within the walls of the University of Oxford he continued to fulfil his professorial duties and at the same time succeeded in acquiring considerable authority and influence through the office of keeper of the archives, to which he had been elected under somewhat questionable circumstances in 1657/8. Although not a trained lawyer, Wallis displayed considerable legal acumen and an unswayable commitment to defending the University's rights in the face of numerous attempts to infringe these on the part of city authorities.

The volume contains a total of 232 existent or identified letters exchanged between Wallis and over 30 correspondents at home and abroad between February 1660 and September 1668. More than sixty of these letters have never appeared in print before, while many of the rest are only to be found in publications which are rare or not widely accessible. Eighteen of the letters printed are part of the former Macclesfield Collection, now in the possession of Cambridge University Library. Although these letters have been previously available in S J Rigaud's Correspondence of Scientific Men (1841), they are published here completely in the form of a critical edition for the first time.

As in the first volume, additional letters and material have been included where these form an essential part of exchanges between Wallis and other scholars or where they are able to throw light on correspondence for which no other documentary evidence has survived. In this way it has been aimed to provide as complete a picture as possible of the literary activity of the Savilian professor during the nine years covered.

Work on the edition has been made possible through funding from the Deutsche Forschungsgemeinschaft. The editors should like to express their sincere thanks to this organisation for its continued generosity.

An enterprise such as the Wallis correspondence project can only succeed through close international cooperation of scholars working in the field. The editors are therefore once more glad to record their debt to numerous friends and colleagues who through advice or assistance have contributed significantly to realising the volume in the present form.

The editors are once again grateful to Robert Hatch for providing information on manuscript sources in France and the United States. David Cram and Jackie Stedall are to be thanked not only for sharing their profound knowledge of Wallis and his contemporaries but also for their willingness to provide help in almost every conceivable way. Further, the editors should like to express their gratitude to Iordan Avramov for useful discussions on questions concerning Henry Oldenburg and Robert Moray. They are grateful also to Jason Rampelt for kindly allowing them to read parts of his as yet unfinished Cambridge dissertation on the life and work of Wallis. Special thanks go to Mordechai Feingold who read the manuscript at an early stage of completion and who as always provided many helpful suggestions on commentaries and general presentation.

On the technical side the editors have once again had the good fortune to be able to fall back on the sound advice of Menso Folkerts and his colleagues in Munich. They are particularly grateful to Albert Krayer for giving vital assistance on questions concerning the LATEX system used for editing.

The editors have continued to enjoy the generous support of the Institute for History of Science, Mathematics, and Technology at the University of Hamburg. They should like to express their deep gratitude in particular to Karin Reich in her capacity as director of the Institute for ensuring that the correspondence project continues to flourish. The University of Hamburg has generously provided rooms to accommodate the project, and the Department of Mathematics has made sure that those employed on the project receive all the necessary technical assistance. Without this help much of the work on the edition would not have been possible.

A personal note of thanks goes to Siegmund Probst who has continued to share generously in his knowledge of seventeenth-century mathematics. The editors are also grateful for the dedication shown by Ines Harrie and Oliver Leistert while working on the project in the course of their studies.

The editors' greatest debt goes to Uwe Mayer who has taken on a substantial part of the workload over the last years. His editorial efforts have been enormous. Without his energy and commitment the second volume could not have been realised in the short length of time it has taken.

Staff at numerous libraries and archives have provided often invaluable help in preparing the present volume. The editors should like to express their particular gratitude to Adam Perkins of Cambridge University Library, Rupert Baker of the Library of the Royal Society, Frances Harris of the British Library, William Hodges of Duke Humfrey, Bodleian Library, and Peter Rau and staff of the Staatsbibliothek Hamburg. Once again, the editors are especially indebted to Simon Bailey, the Keeper of the Archives of the University of Oxford, and to Alice B Millea, the Assistant Keeper.

The editors are grateful to the following persons and institutions for granting permission to publish copyright material held in their possession: The Syndics of Cambridge University Library; the Librarian of the Bibliotheek der Rijksuniversiteit, Leiden; the British Library Board; the Librarian of the Bodleian Library, Oxford; the Keeper of the Archives, University of Oxford; the Koninklijke Bibliotheek, The Hague; the Bibliothèque Nationale de France; the Royal Society; the National Archives, Kew; and the Governors and Guardians of Marsh's Library, Dublin. Quotations from the Devonshire Manuscripts, Chatsworth House, are published by permission of the Chatsworth Settlement Trustees.

Finally, the editors should like to thank editorial staff at Oxford University Press for their continued help in realising the edition, their care in proof reading, and their patience in the light of often considerable delays in the submission of material.

6.3. Review by: Paolo Mancosu.
Studia Leibnitiana 38/39 (1) (2006/2007), 124-127.

At the end of the Autobiography written towards the end of his life (1697), John Wallis wrote:

"It hath been my lot to live in a time, wherein have been many great Changes and Alterations. It hath been my endeavour all along, to act by moderate Principles, between the Extremities on either hand, in a moderate compliance with the Powers in being, in those places, where it hath been my Lot to live, without the fierce and violent animosities usual in such Cases, against all, that did not act just as I did, knowing that there were many worthy Persons engaged on either side. And willing whatever side was upmost, to promote (as I was able) any good design for the true Interest of Religion, of Learning, and the publick good; and ready to do good Offices, as there was Opportunity; And hereby, (thro' Gods gracious Providence) have been able to live easy, and useful, though not Great" (C J Scriba: "The Autobiography of John Wallis, F. R. S.", in: Notes and Records of the Royal Society 25 (1970), 17-46).

Religion, learning, and the "publick good": these are indeed the three areas which capture the multifaceted life of John Wallis who was not only Savilian Professor of Geometry at Oxford and founding member of the Royal Society but also a divine and an important player in the public arena. Born in 1616, John Wallis was educated (1632-34) at Emmanuel College in Cambridge as a theologian. He was ordained minister in 1640. His contact with mathematics seems to have been fortuitous and to date to 1631 when, still a student, coming back home for some school vacations he was taught by his older brother, who was being instructed in arithmetic, a certain amount of practical mathematics. In his Autobiography Wallis asserts that he received no further teaching in mathematics thereby implying that he was self-taught in this area. While at Emmanuel he not only studied theology but also languages (Greek, Latin, Hebrew), linguistics, logic and pursued in addition his interests in the mathematical sciences and experimental philosophy. Wallis' first book, Truth tried (1643), was philosophical in nature.

The 1640s were indeed a period of "great Changes and Alterations". The civil war was raging and in 1642 Wallis deciphered his first piece of Royalist correspondence. Cryptography will from now on play a major role in Wallis' life and his services to Parliament were later rewarded with the Savilian Chair in Geometry in 1649. His "moderate Principles, between the Extremities on either hand", exemplified by his opposition to the execution of Charles I in 1649, also accounts for the fact that he did not suffer any consequences with the restoration of the monarchy in 1660.

After serving in different positions, in 1645 Wallis moved to London where

"beside the Conversation of divers eminent Divines, as to matters Theological; I had the opportunity of being acquainted with divers worth Persons, inquisitive into Natural Philosophy, and other parts of Humane Learning; And particularly of what hath been called the New Philosophy or Experimental Philosophy" (Scriba (1970), 39).

These very famous lines point at Wallis' theological activities during this period and to the meetings that will later give rise to the Royal Society. The theological activities were also related to his involvement in the 1640s with the Westminster Assembly, an assembly of divines whose role was that of revising the standards of the English Church. This connection was also influential in Wallis' later theological publications (such as Mens Sobria (1657). On the theological work of John Wallis see chapters 4 and 6 of Jason Rampelt's unpublished PhD thesis "Distinctions of Reason and Reasonable Distinctions: The Academic Life of John Wallis (1616-1703)", Cambridge University, 2005). The scientific meetings, which involved Goddard, Ent, Wilkins and others, continued into the 1650s although Wallis could not continue his regular attendance as he moved to Oxford in 1649 as Savilian Professor of Geometry.

Wallis' major mathematical publications occur in the 1650s, which also sees publication of Arithmetica Infinitorum (1656), a central text in the development of seventeenth century mathematics (now accessible in English: The Arithmetic of Infinitesimals: John Wallis 1656. Translated from Latin to English with an introduction by Jacqueline A Stedall, New York 2004). The 1650s also witnessed the vitriolic polemic with Thomas Hobbes and the mathematical exchange with Brounker, Frenicle and Fermat, published by Wallis as Commercium Epistolicum. In addition to the theological, scientific and mathematical interests, Wallis' position as Savilian Professor of Geometry and, from 1658, as Custos Archivorum of the University of Oxford, propelled him into public life. Indeed, this latter position was extremely important for providing support from the University Archive in all kinds of issues affecting the University's rights.

This brief summary of Wallis' career up to the end of the1650s is meant only to highlight the variety and richness of Wallis' activities. The edition of Wallis' correspondence by Philip Beeley and Christoph J Scriba provides us with an invaluable tool for appreciating in its full extent Wallis' extraordinary contribution to the theological, academic and political life of his period. The two volumes here reviewed cover the period from 1641 to 1659 and from 1660 to 1668, respectively. Each one is prefaced by a rich introduction detailing the most important aspects of Wallis' activities during the period followed by a thematic analysis of the correspondence. For instance, the thematic analysis for the first volume covers a) Matters concerning the Savilian Lands; b) University affairs; c) Family correspondence; d) Correspondence on theological topics; e) Poole's scheme for the support of young scholars intended for ministry; f) Cryptoanalysis; g) Scientific correspondence. The scientific correspondence covers not only mathematical topics but also astronomical topics and other issues in natural philosophy. In the second volume a significant part of the scientific correspondence is devoted to discussions and observations concerning Wallis' theory of the tides. The first volume contains correspondence with more than 30 scholars including Brouncker, Digby, Fermat, Hevelius, Ch Huygens, and Schooten. The second volume contains also more than 30 correspondents, including among the new correspondents, John Collins, James Gregory, and Henry Oldenburg.

The edition follows the principles of publication used for the publication of the Sämtliche Schriften und Briefe von G W Leibniz. Each letter is presented by a 'Transmission' which gives a full account of how the letter was handed down. This is often followed by relevant information about the context or the content of the letter. The criteria for inclusion have been generous, including for instance dedicatory letters found in the prefaces of Wallis' books. Moreover, letters containing important information about Wallis are also included even when they were not written by or addressed to Wallis. The first volume contains 199 items and the second 254 items.

The critical apparatus is outstanding. In footnotes much care has been devoted to the identification of people, books, and events mentioned in the correspondence. The annotations detail variations in different versions of the item under discussion, when applicable. Succinct biographies of all correspondents together with a list of all the letters are included at the end of each volume. Both volumes contain a useful and accurate index.

This is a first rate edition and a major scholarly accomplishment. The correspondence of John Wallis is a trove for the historian of seventeenth century social and scientific life. The scholarly world is indebted to Philip Beeley and Christoph J. Scriba (and their assistants Uwe Mayer and Siegmund Probst) for having made available this correspondence which was scattered around the world either unpublished or published in often obscure and inaccessible publications. The books are beautifully produced but are unfortunately very expensive. Hopefully a paperback edition will follow. This edition is a must for every library and for every scholar whose interests touch upon the seventeenth century. We eagerly await the remaining four volumes.

7.1. From the Preface.

Geometry (from the Greek word for 'measuring the Earth', the modern scientific discipline of which is now called geodesy), branch of science which deals with regular patterns, shapes and solids, was one of the first human attempts, after counting, to concern themselves with the emerging science mathematics. This is evident from the spirals on megalithic graves, incisions in stone and patterns on clay fragments.

In this book, you will learn how geometry has developed over the millennia from these earliest origins in distant times and much more. Geometry is an indispensable aid for building and surveying, and became an axiomatic science of plane and spatial shapes in Ancient Greece. It served as a basis for astronomical observations and calculations, for Islamic decorative art, and the building of medieval Christian cathedrals. Furthermore we will look at the discovery of perspective and its application in Renaissance art, at the disputes regarding the Euclidean parallel postulate, the discovery of non-Euclidean geometries in the 19th century, and, finally, the theory of infinite-dimensional spaces and contemporary computer graphics.

This book is edited by the project group "History of Mathematics" at the University of Hildesheim as part of the series Vom Zählstein zum Computer (From Pebbles to Computers). Other titles in this series published by Springer Publishing Heidelberg are: 4000 Jahre Algebra (4000 Years of Algebra) [Alten et al. 2003], and 6000 Jahre Mathematik (6000 Years of Mathematics) [Wussing, in two volumes 2008/09]. To the series 'From Pebbles to Computers' two video films have been produced (University of Hildesheim): 'Mathematik in der Geschichte Altertum' (Mathematics in History Antiquity) [Wesemüller-Kock/Gottwald 1998] and 'Mathematik in der Geschichte - Mittelalter' (Mathematics in History Middle Ages) [Wesemüller-Kock/Gottwald 2004]. Following multiple reprints and the second edition in 2004 we now present the third edition of 5000 Jahre Geometrie including new research results on circular ditches in the Stone Age and the Nebra Sky Disk, as well as many illustrations in colour.

In this book, we will reflect on the development of geometry as part of our cultural history over the course of five millennia. Both authors have succeeded in portraying the origins and growth of this branch of mathematics, which is often thought of as dry and jejune, in a tremendously lively manner. They uncover the origins and impulses for the development of geometric notions and methods, and present how they are related to historical events and personal fates. Moreover, they describe the applications of geometrical knowledge and methods in other areas and the interdependencies that resulted from them. Finally, they emphasise their importance for other disciplines.

At the heart of this book series is portraying the history of mathematics as an integral part of the history of mankind, particularly as a fundamental part of our cultural heritage. Both authors have done justice to this task in an impeccable manner. They have depicted the genesis of geometry and its interlacing with cultural developments in other areas, such as literature, music, architecture, visual arts and religion, by a standard far higher than usual in mathematical-historical presentations. They also describe the implications of geometrical findings and methods for other areas. As such, the authors also deal far more extensively than usual with the development of geometry in other cultures, mainly in the ancient oriental cultures, in Islamic countries, as well as in India, China, Japan and the old American cultures. Tables at the beginning of each chapter give an overview of important political and cultural events of each cultural area and era dealt with. Tables at the end summarise the main geometrical contents of each chapter in note form.

Moreover, the authors compare views of ancient and medieval mathematicians with modern mathematical findings and link those to contemporary mathematics and related sciences, for example, references to computer sciences regarding the description of Euclid's "algorithmic accomplishment". Furthermore, they highlight the specifications of geometrical examinations of different eras and cultural areas and the changes in content, methods and approaches geometry has faced as a proto-physics within three-dimensional or even infinite-dimensional spaces. They discuss the relationship of geometry with other branches of mathematics, for instance with algebra, analysis, and stochastics. Refreshing asides with biographical highlights and references to unexpected relations, as well as text excerpts in the appendix, bring this book to life.

Chapters 1 through 4, with the exception of sub-chapter 2.3 (Euclid), were written by Dr Christoph J Scriba, professor emeritus for the history of the natural sciences in the former Institute for History of Natural Sciences, Mathematics and Engineering at the University of Hamburg. Euclid's accomplishments and the development of geometry in modern times from Chapters 5 through 8 were described by Dr Peter Schreiber, professor for geometry and the foundations of mathematics at the University of Greifswald.

We are also grateful to the authors for numerous illustrations and the texts for the appendixes. The figures that have been added to support geometrical theorems that are not referenced were drawn by the authors themselves.

They also thought of the summarising problems for every sub-chapter at the end of each chapter (cf. Introduction). They often differ from ordinary tasks in regard to type and size and also vary in level of difficulty. Thus, solving them requires very different background knowledge, as well as the use of secondary literature at times. Hence, to solve some of the problems of Chapters 1 through 4, you will mainly need knowledge gained in junior high school, while other problems will require high school knowledge, whereas some problems to Chapters 5 through 8 demand insight into notions and methods taught at university. This is due to the nature of the subject, since mathematics has grown more and more complex and difficult over the course of the centuries and understanding modern mathematics usually assumes knowledge of the mathematics of past eras. Therefore, you will occasionally find hints to solutions within the text and also the literature. However, the solutions themselves have not been included in the appendix to avoid the following: first, we do not want you to look up the solutions too quickly; second, the solutions most often are not the result of calculations, but require the description of approaches for solving the problem at hand or retracing more or less extensive considerations.

All this has been done intentionally in order to attract as large a readership as possible. Cursory readers or those that are in a hurry should not simply skip the problems, since they include many interesting historical remarks and additions to the text, which is why reading the problems carefully will benefit everyone. The extensive bibliography and index of names invite the reader to study further.

I thank both authors sincerely for the multifaceted and intensive work in particular their dedication to setting new accents with this book integrating geometry in cultural history and composing many interesting problems.

I further express my gratitude to my colleagues Dauben, Flachsmeyer, Folkerts, Grattan-Guinness, Kahle, Lüneburg, Nádeník und Wussing for their scholarly advice and critical reviewing and thank H Mainzer for advice on historical details and Lars-Detlef Hedde (University of Greifswald), Thomas Speck and Sylvia Voss (University of Hildesheim) for converting the manuscripts, illustrations and figures into printable electronic formats.

Moreover, I wish to thank media educator Anne Gottwald, who helped us clear the licensing for printing the illustrations, and each publisher for authorising the printing rights.

I also remain grateful to the director of the Centre for Distance Learning and Extension Studies (ZFW), Prof Dr Erwin Wagner, the present and former directors of the Institute for Mathematics and Applied Computer Science, Prof Dr Förster and Prof Dr Kreutzkamp, the deans Prof Dr Schwarzer and Prof Dr Ambrosi and the administration of the University of Hildesheim.

Last but not least, I wish to thank the members of the project group "History of Mathematics" of ZFW: the historian of mathematics Dr Alireza Djafari Naini and the media expert and sociologist Heiko Wesemüller-Kock, for the great and intensive teamwork while planning and preparing this book. I express my gratitude to Springer Publishing Heidelberg for taking my requests into account and the excellent design of this book.

I hope that this volume will inspire many readers to study the history of mathematics more intensively, and to learn about the background of the origins and incredibly exciting development of geometrical notions and methods. Hopefully, this will result in the reader viewing geometry not just as a mathematical discipline or as an indispensable aid for architects, robot engineers and scientists, but also as a valuable part of our culture that we encounter everywhere and that makes the world in which we live so much richer.

On behalf of the project group

Hildesheim, August 2009

Heinz-Wilhelm Alten

8.1. From the Publisher.

Containing many previously unpublished letters, this third volume of a six volume collection of the complete correspondence of John Wallis (1616-1703), documents an important period in the history of the Royal Society and the University of Oxford. By providing access to these letters, this painstakingly crafted edition will enable readers to gain a deeper and richer awareness of the intellectual culture on which the growth of scientific knowledge in early modern Europe was based.

Wallis was Savilian Professor of Geometry of Oxford from 1649 until his death, and was a founding member of the Royal Society and a central figure in the scientific and intellectual history of England. In the period covered Wallis is engaged in scientific debates on techniques for determining areas contained by curves (quadratures) and figures (cubatures), as well as on the theory of motion and the nature of the tides. He also continues to attack the mathematical undertakings of Thomas Hobbes and to respond to attacks which the philosopher in turn levels against him. We also find evidence for the consolidation of mathematics as an academic discipline in the University of Oxford just fifty years after the establishment of the first mathematical lectureships. Wallis is called upon more than once to deliver ceremonial lectures on mathematical topics to foreign dignitaries visiting the University.

At the same time the volume allows us to witness the beginnings of a remarkable development in mathematical publishing. Many of Wallis's letters to Henry Oldenburg, secretary of the Royal Society, on a variety of topics in the mathematical and physical sciences, are transformed into articles and published in Oldenburg's journal, the Philosophical Transactions. Part of the reason for this development also becomes clear in the letters: the long and costly process of publishing mathematical books such as Wallis's three part Mechanica: sive de motu. This volume not only signals the modernisation of mathematics in the second half of the seventeenth century but we also see two new figures emerge for the first time, whose careers are in different ways closely associated with Wallis: Isaac Newton and Gottfried Wilhelm Leibniz.

8.2. From the Preface.

The period covered by this third volume of The Correspondence of John Wallis (1616-1703) was one of remarkable political stability and economic prosperity. To an unprecedented degree, England was at peace with itself and with its neighbours. Scientific activity also prospered with the Royal Society and especially its industrious secretary Henry Oldenburg at the core. The institutional role of the Royal Society reflected London's increasing importance as a centre of intellectual and material commerce, of coffee houses and meeting places, of trade and shipping, of government and court, of finance, and of a postal service spreading out like so many arteries to the provinces.

The Oxford-London axis is of central importance to the present volume. Oldenburg regularly sent Wallis news of the weekly meetings of the Royal Society in London, to which Wallis replied as a matter of course. Through the medium of correspondence, the Savilian professor was a key participant in Royal Society debates such as those on the laws of motion or the theory of tides. With ordinary post between the English capital and the university city usually requiring at most one day, Wallis was able to make his presence felt, even when not in London.

The very nature of intellectual commerce and scientific debates conducted over distance poses difficulties when compiling an edition such as this. Many letters sent to Oldenburg by his correspondents around the country and abroad were actually intended for Wallis, even if they were not addressed to him. Such letters are of course included in the present volume together with letters between third parties which throw light on Wallis letters which cannot be gained from elsewhere. In this way, the editors seek to provide as complete a picture as possible of Wallis's intellectual activity during the period concerned.

The volume contains a total of 254 existent or identified letters exchanged between Wallis and thirty-two correspondents in England, Scotland, Ireland, France, Italy, Poland, the Dutch Republic, and the Spanish Netherlands. More than thirty-five of these letters have never appeared in print before, while many of the others have only been published in part or are to be found in publications which are either rare or not widely accessible. Only a very few of the letters have previously appeared in the form of a critical edition.

Much of the work on the volume has been generously funded by the Arts and Humanities Research Council (AHRC), and was carried out in the framework of the Oxford-based project on John Wallis entitled 'Harmony and Controversy in Seventeenth-Century Thought' between 2007 and 2010. The editors would like to thank the AHRC for its generous support throughout the duration of that project. Since September 2010 the edition of the correspondence of John Wallis has become an integral part of the Cultures of Knowledge project a collaboration between the Bodleian Libraries and the Humanities Division of the University of Oxford funded by The Andrew W. Mellon Foundation. The editors are deeply indebted to the Mellon Foundation and to the Director of the Cultures of Knowledge project, Howard Hotson, for enabling this transfer to take place and for providing necessary financial support for the conclusion of this volume.

The editors would like to express their gratitude also to David Cram for his enthusiasm and assistance in bringing the edition to Oxford, and to the Faculty of Linguistics, Philology, and Phonetics for accommodating it before its move to the Faculty of History.

The excellent facilities of the Oxford Faculty of History, and the congenial and frequently inspirational environment provided by friends and colleagues in the Cultures of Knowledge project have enabled this volume to be brought to a rapid conclusion under near-ideal conditions. For the wonderful support given to them, the editors should like to thank James Brown, Miranda Lewis, Kim McLean-Fiander, Leigh Penman, Anna Marie Roos, and Helen Watts. Howard Hotson has been a tower of strength in all respects, and is to be especially thanked. Without James Brown's steady hand in coordinating and organising matters and his ready willingness to find solutions to problems at short notice, work on the volume would not have moved so smoothly. The editors are extremely grateful to him for his steadfast efforts.

One of the joys of working on a major correspondence edition is the scholarly collaboration such work engenders. Such assistance can take many forms, ranging from information on previously unknown manuscripts or printed versions of letters to willingness to read edited material and suggest ways in which improvements might be made. The editors are extremely conscious of the help given to them in the course of their work and take great pleasure in acknowledging this. Specifically, they should like to express their gratitude to Michael Hunter, Noel Malcolm, Per Landgren, Kate Bennett, Robert Hatch, Kelsey Jackson Williams, Alexander Häussler, Katerina Oikonomopoulou, Jason Rampelt, Richard Sharpe, William Poole, and Charles Webster.

Editorial assistance during 2008 was provided by Ellie Stedall. The careful way in which she made transcriptions from manuscripts and the energy she brought to the project were sorely missed after her departure for Cambridge.

Invaluable advice on transcriptions has been generously provided by Jutta Heinle. The patience with which she has followed the conclusion of the volume is more deeply appreciated than she can know.

The Vogel Foundation kindly provided financial support for the editors in order for them to obtain digital images of Wallis material formerly contained in the Macclesfield Collection and now housed in Cambridge University Library. The editors are extremely grateful to Menso Folkerts for his help and advice in this regard.

The editors would like to express two special debts of heartfelt gratitude. They are indebted to Jackie Stedall, who has provided invaluable support, advice, and encouragement throughout the course of work on the volume. She has also kindly enabled work on the edition to be presented to colleagues in Oxford on a number of occasions. The editors are also indebted to Pietro Corsi, who has been a constant source of sound advice and moral support since the edition arrived in Oxford. His tireless work as head of the steering committee of the AHRC project has been particularly appreciated.

Initial transcriptions of a substantial number of letters were made by Uwe Mayer. The editors are grateful to him for his lasting and conscientious input into the edition. Siegmund Probst has continued to share generously in his knowledge, and has been constantly willing to provide assistance wherever he can.

The editors' greatest debt is to Kim McLean-Fiander, who has taken time off preparing the Calendar of the Correspondence of John Wallis in order to proof-read and check a substantial part of the book manuscript. Not a few of her suggestions have found their way into the commentaries of the letters. At the same time her industry and care in preparing the Calendar, for which they are immensely grateful, will ensure that future work on the edition will be considerably facilitated.

Staff at numerous libraries and archives have provided often invaluable help in preparing the present volume. The editors should like to express their particular gratitude to Adam Perkins of Cambridge University Library, Rupert Baker and Keith Moore of the Library of the Royal Society, Frances Harris of the British Library, and Östlund Krister of the Universitetsbibliotek, Uppsala. Once again, the editors are especially indebted to Simon Bailey, the Keeper of the Archives of the University of Oxford, not only for pointing them in the direction of new material but also for ready support on details of University history.

The editors are grateful to the following persons and institutions for granting permission to publish copyright material held in their possession: The Syndics of Cambridge University Library; the Librarian of the Bibliotheek der Rijksuniversiteit, Leiden; the British Library Board; the Librarian of the Bodleian Library, Oxford; the Keeper of the Archives, University of Oxford; the Director of the Bibliothèque Nationale de France; the Royal Society; the Director of the Bibliothèque de l'Observatoire, Paris; the National Archives, Kew; the Librarian of the Universitetsbibliotek, Uppsala; the Director of the Biblioteca Nazionale, Florence.

Finally, the editors should like to thank editorial staff at Oxford University Press, especially Keith Mansfield, Elizabeth Hannon, and, more recently, Clare Charles, for their continued help in realising the edition, their care in proof reading, and their patience in the light of often considerable delays in the submission of material.

9.1. From the Publisher.

The Correspondence of John Wallis (1616 -1703) is a critically acclaimed resource in the history of early modern science. Volume IV covers the period from 1672 to April 1675 and contains over eighty previously unpublished letters. It documents Wallis's role in the crucial debate over the method of tangents involving figures such as Sluse, James Gregory, Hudde, Barrow, Newton, and Christiaan Huygens. In this way it illuminates further an important part of the history of the calculus.

Wallis's letters also provide valuable new insights into mathematical book production and the importance of the international exchange of books in the growth and dissemination of mathematical knowledge. We learn more about the part played by the intelligencer John Collins and the astronomer royal John Flamsteed in the edition of Jeremiah Horrox's Opera posthuma, published by Wallis in 1673. There are also new insights on the background to Wallis's early work on equations, and the reasons why he criticised Gaston Pardies's proposed tract on motion. The causes of the breakdown in Wallis's epistolary relation to Christiaan Huygens following the publication of the Horologium oscillatorium in 1673 are also revealed.

Many letters reflect Wallis's active involvement in the Royal Society. Through the medium of correspondence the Savilian professor participated in numerous debates such as those over the anomalous suspension of mercury in the Torricellian tube or Hevelius's use of plain sights in positional astronomy. The volume allows us to gain a deeper understanding of the background to these debates. Furthermore, the volume throws important new light on the history of the University of Oxford and of the University Press in the early modern period. As keeper of the University Archives, Wallis was one of the institution's highest officers. Scarcely any event of note concerning the University did not require his involvement in some way, and this is reflected in numerous letters and documents which the volume publishes for the first time.

9.2. From the Preface.

Intellectual commerce in the 17th century, no differently from trade in commodities such as wine, silk, or sugar, was subject to forces dictated by geopolitical interests and rivalries, alliances, and conflicts. For more than half of the period covered by the fourth volume of The Correspondence of John Wallis (1616-1703), the Third Anglo-Dutch War, which saw England fight alongside France, was waged on land and at sea. The conflict disrupted epistolary exchange between the British Isles and continental Europe. Longer times than usual elapsed between posting and receipt and more letters than usual failed to reach their destination at all.

Conflicts took place also through the medium of correspondence, sometimes mirroring those between nations. The publication of Huygens's Horologium oscillatorium, in which the Dutch mathematician reiterated his earlier claim that Hendrik van Heuraet had been the first to rectify any curve successfully, prompted Wallis to enlist the help of friends and fellow scholars to prove that this honour should go to the English mathematician William Neile instead. Neither Wallis nor Huygens was prepared to back down and thus reaching an impasse it was not long before their epistolary relationship, which had survived lesser moments of turbulence in the past, was finally ruptured.

Throughout his life, Wallis used correspondence to exchange information and engage in scholarly debates, and the first half of the 1670s is no exception. In order to maintain a degree of objectivity, letters were often addressed not to the intended recipient, but instead to an intermediary who was able to ensure that the communication reached its true destination. Thus the scientific amateur Francis Jessop, who lived near Sheffield, sent his reflections on Wallis's hypothesis of tides enclosed in a letter to his friend Martin Lister in York who forwarded them with a letter of his own to Henry Oldenburg in London. The secretary of the Royal Society in turn conveyed Jessop's reflections to Wallis in Oxford enclosed in the latest offering of his own. The Savilian professor followed suit by addressing his reply to Oldenburg, not to Lister or to Jessop himself. Moreover, this format was maintained throughout the correspondence of the two men.

Such epistolary practices make it necessary to extend the concept of what constitutes an item of Wallis's correspondence. The person to whom a letter is addressed is not always the true or intended recipient. This means, for example, that it is occasionally necessary to include letters sent to others, but which were in fact intended for Wallis. Nor is this the only circumstance where we consider an inclusive policy justified. Sometimes the existence of a letter can be established only through a reference found in an exchange between third parties. Occasionally such third-party exchange is our only source of a letter's content. In some cases, we have found it necessary to include letters in order to provide context to others or to make them comprehensible at all. This is especially true of Wallis's correspondence as Custos archivorum of the University of Oxford. Letters exchanged in the course of defending the University's ancient rights and privileges often assume far more than they themselves reveal. Not to provide epistolary context would mean rendering the background to Wallis's letters unnecessarily opaque.

The present volume contains in total 252 letters, of which 113 are from Wallis, while 124 are either addressed to him or he was the intended recipient. Of the letters contained in this volume, eighty-one have not been published before. All existent enclosures which were integral to Wallis's correspondence have been edited and are printed here. A further twelve letters have been included for contextual reasons or because they throw light on other letters for which no further evidence exists. In addition, three letters have been included in accordance with editorial policy to print all those letters neither from nor addressed to Wallis but which went through his hands at some stage and which were first published by him. The Savilian professor thus contributed decisively to their contemporary reception. Most letters in this category were originally published in the third volume of Wallis's Opera mathematica.

Part of the work in preparing this volume was carried out during the course of the Oxford-based project on John Wallis entitled 'Harmony and Controversy in Seventeenth-Century Thought', funded by the Arts and Humanities Research Council (AHRC) between 2007 and 2010. The editors should like to thank the AHRC for its generous support throughout the duration of that project. From September 2010 up to January 2013 the edition of the Correspondence of John Wallis was an integral part of the Cultures of Knowledge project, a collaboration between the Bodleian Libraries and the Humanities Division of the University of Oxford funded by The Andrew W Mellon Foundation. Without the Mellon Foundation's generosity and farsightedness in supporting editorial projects of this nature the timely realisation of the present volume would not have been possible. To all concerned the editors wish to express their sincere gratitude.

A more congenial and intellectually stimulating environment in which to produce an edition of this nature than the History Faculty of the University of Oxford is scarcely imaginable. The editors should like to thank the Faculty for providing the edition with superb technical and administrative support. The proximity to the treasures of the Bodleian Library is an added but greatly valued advantage.

As with earlier volumes, the editors are in this case, too, aware that the success of their endeavours is dependent to a considerable extent on friends and colleagues who have been prepared, often at short notice, to check transcriptions, to assist with understanding historical context, suggest new commentaries or variant readings, and so on. Without the help and generosity of others, this volume could not have been produced to the standard it has achieved. Of course, any deficiencies which remain are wholly the responsibility of the editors.

We should like to thank particularly our close colleagues and friends at Cultures of Knowledge, namely Miranda Lewis, Kim McLean-Fiander, James Brown, and Anna Marie Roos, all of whom have followed the emergence of the volume at Oxford closely. Miranda Lewis has helped tirelessly in checking bibliographical details, proof-reading, and supplying all sorts of encouragement. Without her dedication to the aims of the Cultures of Knowledge project a large part of our editorial work would have been so much more difficult. Anna Marie Roos is to be thanked many times over for sharing generously with the editors her transcriptions of the correspondence of Jessop and Lister. Additionally, the editors would like to thank Moti Feingold, Michael Hunter, Vivienne Larmine, Noel Malcolm, Leigh Penman, Will Poole, Jason Rampelt, and Richard Sharpe for their advice and help on specific points. Per Landgren deserves a special word of thanks for the warm spirit of collaboration he brought to Oxford and for the assistance he has given on checking transcriptions.

The editors are especially grateful to Howard Hotson, the director of the Cultures of Knowledge project, for his tireless efforts in supporting the Wallis edition and for ensuring that it continues to flourish. Particular thanks go to Pietro Corsi, who has been ready to give thoughtful advice at all times, and whose moral support has been immeasurable. Jackie Stedall deserves special mention, for her steadfast support and for continuing to be such an inspirational voice in the history of mathematics in Oxford.

Siegmund Probst has, as always, been willing to share his profound knowledge and expertise on seventeenth-century mathematics with the editors. A considerable amount of the early stages of editing was carried out by Uwe Mayer. More recently, Ellie Stedall has contributed to the progress of the edition by meticulously transcribing letters from manuscripts. The editors should like to thank all of these collaborators for their lasting contributions to the edition.

Once again, the editors take pleasure in thanking the Vogel Stiftung for its generous support in enabling us to obtain high quality digital images of letters from Cambridge University Library. In this respect particular thanks are due to Menso Folkerts who sponsored the editors' approach to that foundation.

The editors should like to express their gratitude and appreciation to the staff of the two libraries they have used most intensively during the preparation of the present volume: the Staats- und Universitätsbibliothek Hamburg and the Bodleian Library in Oxford. They wish to thank in particular Hans-Walter Stork of the manuscript department in Hamburg and Colin Harris, of the Bodleian Library's department of Special Collections.

This volume could not have been realised without the assistance and cooperation of librarians and archivists at numerous other institutions. The editors should like to thank especially Clare Hopkins, who made available for consultation the papers of Ralph Bathurst held in the archives of Trinity College, Oxford. Östlund Krister of the Universitetsbibliotek, Uppsala, has been willing always to provide expert help on manuscripts in that library's holdings. Jackie Cox, deputy Keeper of the Archives of the University of Cambridge has readily supplied us with information on that archive's seventeenth-century holdings. Andrew Mussell, archivist at Lincoln College, Oxford, has been extremely helpful in identifying former undergraduates of that college. Robert Athol has kindly provided access to the archival holdings of Clare College, Cambridge. Adam Perkins, curator of scientific manuscripts, has been generous in sharing his expert knowledge of the Newton and Collins papers held in Cambridge University Library. An special word of gratitude goes to the library of the Royal Society in London. Without the generosity of that marvellous institution in making its rich manuscript holdings available to scholars, publications like the present volume would not be possible. We wish to thank in particular Keith Moore and Rupert Baker for the thoughtful and qualified assistance they have given at all times.

Readers of this volume will note that substantial numbers of the letters and other documents it contains are housed in the archives of the University of Oxford. Without the unwavering support given by the Keeper of the Archives, Simon Bailey, the edition would not be able to appear in the form and breadth which it has. His generosity in placing the entire seventeenth-century resources of the University Archives at the editors' disposal in order that nothing be missed, is truly remarkable. Not only have he and his assistant archivist Alice Millea been accommodating at all times, often at short notice, but they have also allowed the editors to draw on their palaeographical expertise and profound knowledge of the history of the University. The editors' indebtedness to the University Archives is quite simply immense.

The editors wish to express their gratitude to the following persons and institutions for granting permission to publish copyright material held in their possession: the Syndics of Cambridge University Library: the Librarian of the Bibliotheek der Rijksuniversiteit, Leiden; the British Library Board; Bodley's Librarian, University of Oxford; the Keeper of the Archives, University of Oxford; the Director of the Bibliothèque Nationale de France; the Royal Society; the National Archives, Kew; the Librarian of the Universitetsbibliotek, Uppsala; the Director of Det Kongelige Bibliotek, Copenhagen; the Director of the Koninklijke Bibliotheek, The Hague; the President and Fellows, Trinity College, Oxford, and the Librarian, University of St Andrews Library, St Andrews.

Finally, the editors should like to thank the editorial staff at Oxford University Press, especially Keith Mansfield, Clare Charles, and Victoria Mortimer, for their continued support in realising the edition, their care in supervising production, and above all their patience and understanding in the light of often considerable delays in the submission of material.

9.3. Review by: William Poole.
British Society for the History of Mathematics Bulletin 30 (2015), 174-175.

Three prior volumes in this edition of the correspondence of the mathematician John Wallis (1616-1703) have appeared to date, covering the years 1641-59, 1660-68, and 1668-71 respectively, and now this fourth volume carries the series to April 1675. There is some way to go: Wallis held the Savilian Chair in Geometry from 1649 until his death in 1703, and so there are almost three decades of letters to come. Scholars need not despair: the Early Modern Letters Online project (http://emlo.bod leian.ox.ac.uk) has, courtesy of the present editors, listed the complete correspondence as it is currently known, and readers are encouraged to consult this resource. This new volume contains 252 letters, although that number includes many items now missing, and of the remainder, only eighty-one have not previously appeared in print. Several are not strictly to or from Wallis, but are included for contextual understanding, or because, as was common in the period, an intellectually sensitive correspondence was conducted through an intermediary. In theory this is a particularly welcome editorial policy, albeit in practice it is one exercised on occasion perhaps overgenerously (for instance, do we really need certain Leibniz-Oldenburg letters published yet again?). Eighty-one fresh letters may seem like slim pickings for a thick volume, but it is vital for such editions to be complete, and the letters re-edited here are treated with the editors' customary meticulousness, including full notes on the transmission of letter manuscripts, identification of prior and succeeding letters in a given correspondence 'thread', and a brief contextualisation of each letter, often citing parallel sources for the earliest reception of the letter. The volume also contains an essential historical introduction, brief biographies, and the necessary bibliographies and indices.

This period of Wallis's life witnessed several major mathematical disputes, of which one of the most violent was that between the philosopher Thomas Hobbes and Wallis himself. The two men had been skirmishing since 1655, and Hobbes rekindled their quarrel by publishing in 1672 a restatement of their contention. Wallis wrote a crushing review; Hobbes responded acidly to the Royal Society's secretary, Henry Oldenburg; and Oldenburg, usually the most urbane of correspondents, replied to Hobbes with his patience worn particularly thin. This period also witnessed the demise of relations between Wallis and the Dutch mathematician Christiaan Huygens. In 1673, Huygens published the second, much expanded edition of his Horologium oscillatorium, and in one new section he claimed precedence for the rectification of the semi-cubical parabola. (This was important for potential regulation of clock pendulums.) Wallis believed that the crucial work had been done here by the Englishman William Neile in the summer of 1657. As ever, Wallis was keen to protect the priority of English mathematicians, and in the absence of a dated, printed account, he had to rely on the testimony of those who had been present at discussions that had taken place around a decade and a half ago. Huygens was offended, and their friendship did not survive this disagreement, Wallis angrily peppering later letters with accusations of Huygens's 'Frenchified' behaviour.

Wallis maintained an interest not just in mathematics but in the history of it, in this period notably the achievements of the Caroline mathematician and astronomer Jeremiah Horroх (1618-41). Wallis edited and presented to the Royal Society Horrox's posthumous Opera in late 1672, although the imprint of the book is dated to the following year; Wallis's lengthy epistolary dedication to his fellow mathematician William Brouncker is, following the conventions of this series, reprinted in its entirety here. Another foreign presence, this time a new one, was the young Gottfried Wilhelm Leibniz, who visited the Royal Society in 1673 and demonstrated to them an early model of his calculating machine.

This section of Wallis's correspondence also displays work in several non-mathematical areas, for instance Wallis's hypothesis of tides, which drew him into a lengthy and decreasingly amicable exchange with the gentleman naturalist and mathematician Francis Jessop, conducted largely through their intermediary, Martin Lister. Finally, many of the new letters in this volume concern Wallis's role as Keeper of the Archives of Oxford University. As such, Wallis was quite frequently asked to compile reports on the university's past legal behaviour, to the end of settling disputes over jurisdiction. The most important of these cases concerned the licensing of taverns (that is, places where wine and not just ale was sold), and Wallis prepared several very lengthy reports on this legal issue.

As we have come to expect of this fine series, this is a near-immaculate edition, and it includes an errata and addenda table for the earlier volumes. I have space to add one addendum, as it seems colourful: the unidentified Chinese almanac perused by Wallis and reported on by him in a letter in the previous volume of this series is in fact still in the Bodleian, shelf-mark Sinica 57, inscribed by its donor, Robert Boyle; cryptographer Wallis did a creditable job deciphering it, although his report for Boyle was never sent.

One slight criticism of the volume is that the index appears to have been fixed before the final pagination of the letter texts, as the indexed pagination is often out by one or even two pages (check, for instance, the pages referred to for Robert Hooke, Thomas Hyde, or Francis Willughby). Secondly, it is an established convention of this series that letters are presented in their original languages without translation. This is still to be regretted, even if it is not practical to hope for a policy reversal at this time. The community to which this edition is addressed is simply no longer up to it, as the publishers of comparable cycles of letters, notably those of Henry Oldenburg, acknowledged decades ago. Over thirty letters in this volume - and perforce the letters dealing with mathematics on the international stage - are in untranslated Latin. Oxford University Press should concede that presenting such materials without translation is foolhardy, and facilitate translations, which may then be offered either in print at some later date, or online.

10.1. From the Publisher.

The present volume provides a fascinating overview of geometrical ideas and perceptions from the earliest cultures to the mathematical and artistic concepts of the 20th century. It is the English translation of the 3rd edition of the well-received German book "5000 Jahre Geometrie," in which geometry is presented as a chain of developments in cultural history and their interaction with architecture, the visual arts, philosophy, science and engineering.

Geometry originated in the ancient cultures along the Indus and Nile Rivers and in Mesopotamia, experiencing its first "Golden Age" in Ancient Greece. Inspired by the Greek mathematics, a new germ of geometry blossomed in the Islamic civilisations. Through the Oriental influence on Spain, this knowledge later spread to Western Europe. Here, as part of the medieval Quadrivium, the understanding of geometry was deepened, leading to a revival during the Renaissance. Together with parallel achievements in India, China, Japan and the ancient American cultures, the European approaches formed the ideas and branches of geometry we know in the modern age: coordinate methods, analytical geometry, descriptive and projective geometry in the 17th an 18th centuries, axiom systems, geometry as a theory with multiple structures and geometry in computer sciences in the 19th and 20th centuries.

Each chapter of the book starts with a table of key historical and cultural dates and ends with a summary of essential contents of geometry in the respective era. Compelling examples invite the reader to further explore the problems of geometry in ancient and modern times.

The book will appeal to mathematicians interested in Geometry and to all readers with an interest in cultural history.

10.2. From the Preface.

This is the first volume of our series 'From Pebbles to Computers' that appears in English. It is a translation of the 3rd edition (2010) of 5000 Jahre Geometrie, again updated, supplemented and enriched with many illustrations by the author P Schreiber and the editors H-W Alten, K-H Schlote, and H Wesemüller-Kock.

Meanwhile the book 4000 Jahre Algebra appeared 2014 in its 2nd edition, in 2011 3000 Jahre Analysis was published by Springer Berlin Heidelberg, and we are now preparing its translation 3000 Years of Analysis to be published by Springer Basel. Some other volumes of this series will also be published in English. Besides the film 'Mathematik in der Geschichte - Mittelalter' has now been produced in English as History of Mathematics - Middle Ages.

All of us have been affected by the death of our author Prof Dr C J Scriba in 2013. We are grateful for his support over many years and glad to be able to present Chapters 1 through 4 of this book with the supplements he wrote before his death as part of his scientific legacy. We shall miss his advice in future.

The translation of this book was done by Jana Schreiber, the daughter of the author P Schreiber. We are very grateful to her because she has done this with great efficiency and commitment in a short time.

After the corrections and supplements of the editor, language copy editing by the publisher, and proof reading by the author and editor we now present this volume.

I thank the members of the project group for their intensive teamwork: A K Gottwald for clearing the licenses (now world-wide) for printing the illustrations, H Wesemüller-Kock for his involvement inserting new illustrations with his comments and the index of illustrations, proof reading and preparing the graphic design and layout for the whole book, the historian of mathematics Dr K-H Schlote for many comments and for transferring the index of names and the subject index, Prof Dr K-J Förster and Prof Dr E Wagner for providing financial support.

We are grateful for the help of our secretaries B David and R Falso, the students J Schönborn and N Westphal for preparing the text, illustrations and indexes ready for printing.

Last, but not least we thank Springer Publishing Basel AG and its editor Dr A Mätzener for her kindly support and the excellent layout of this book.

I hope that this book will please, inspire and benefit many readers all over the world.

On behalf of the project group

Hildesheim, August 2014

Heinz-Wilhelm Alten

10.3. Review by: James J Tattersall.
Mathematical Reviews MR3362344.

This book is a translation of the third edition of 5000 Jahre Geometrie, by Christoph J Scriba and Peter Schreiber, published in 2010 [Vom Zählstein zum Comput., Springer, Berlin]. The first German edition appeared in 2001 and the second in 2005. The main additions to the third edition are new information on Stone Age cultures and on the discovery of the Nebra sky disk near Halle, Germany. The book was edited by the project group History of Mathematics at the University of Hildesheim. It is part of the series Vom Zählstein zum Computer (from Pebbles to Computers). Other works in the series include 3000 Jahre Analysis [T Sonar, Springer, Heidelberg, 2011], 4000 Jahre Algebra [H-W Alten et al., Springer, Berlin, 2003], and 6000 Jahre Mathematik [H Wussing, Springer, Berlin, 2008]. With the exception of the section on Euler, Scriba authored the first four chapters and Schreiber authored the section on Euler and the remaining chapters. The main aim of the book is to portray the history of geometry as an integral part of the history of civilisation and to describe how the subject developed and relates to other branches of mathematics. The book is highly readable and richly illustrated. This book contains a wealth of geometric and historic information. It also contains a list of references, a list of figures together with their provenances, an index of names, and an index of subjects. A more mathematically oriented book with somewhat similar goals but much more limited in scope is J Heilbron's Geometry civilized [Oxford Univ. Press, New York, 1998].

The first chapter covers the geometry of Stone Age civilisations and that of Egypt, Babylonia, and the Indus River Valley. African geometric patterns such as those found in Claudia Zaslavsky's Africa counts [Prindle, Weber & Schmidt, Inc., Boston, MA, 1973] are briefly mentioned. The second chapter concentrates on the developments of Greek geometry from the Hellenistic era to late antiquity. The geometrical investigations and accomplishments of Thales, Pythagoras, Eudoxus, Euclid, Aristarchus, Archimedes, Apollonius, Heron, Pappus, Proclus, Ptolemy, and Menelaus are discussed. The third chapter covers the geometric discoveries of the Chinese, Japanese, Indian, Islamic, Northern American and Meso-American cultures. Chapter 4 covers the European Middle Ages and the accomplishments of the Venerable Bede, Gerbert, Boethius, Fibonacci, Sacrobosco, Bradwardine, and others.

The fifth chapter, on new impulses for geometry during the Renaissance, contains an extensive section on projective geometry in Renaissance art. Chapter 6 is devoted to the development of geometry in the 17th and 18th centuries and includes a section on the discovery and impact of analytic geometry. The seventh chapter deals with the influence of descriptive geometry and differential geometry. There is also an extensive section on non-Euclidean geometry. The last chapter includes sections on the foundations of geometry and geometry's relation to computer science.

Except for the first chapter, each chapter begins with an introduction and a list of major historical events for the period under discussion, and ends with a table of the major geometric developments of the period. Each chapter also contains a set of problems. As with the problems in V J Katz's A history of mathematics [HarperCollins Coll. Publ., New York, 1993], solving them adds immensely to the understanding of the geometric concepts covered in the chapter.

The translator has done a masterful job. However, a few quirks have crept in. For example, "pen" is used for "pin", "musaeum" for "museum", "ration" for "ratio", "Platon" for "Plato", for angle measure "" is used for "", and the term "zeroizing", and the phrase "spooked through math literature" appear, which may vex or confuse some English readers. However, such incidents are few and far between and in no way detract from the authors' goal. Nevertheless, the book would have benefited from a good copyeditor. There are several instances where words are run together, and on page 360, almost a whole page of text is repeated. That said, the book is a well researched history of geometry reference manual and would make a welcome addition to any mathematician's bookshelf.