- The Development of the Foundations of Mathematical Analysis from Euler to Riemann (1970), by I Grattan-Guinness.
1.1. Review by: T A A Broadbent.
The Mathematical Gazette 56 (395) (1972), 57-59.
Bell's chapter on Euler in Men of Mathematics is headed "Analysis Incarnate", and Boyer, in the American Mathematical Monthly (1951) made a strong claim for Euler's 'Introductio in analysin infinitorum' as the foremost textbook of modern times. Thus Mr Grattan-Guinness has selected a natural starting point, while the upper time limit is not so much an end with Riemann as a new beginning. The word "foundations" in the title indicates a quite reasonable restriction to real variables; the study of complex function theory must be regarded as superstructure rather than foundations. To mention these restrictions is far from suggesting that Mr Grattan-Guinness has undertaken an easy task; on the contrary, even a cursory inspection of his text reveals the enormous amount of labour involved. He has had not only to search the vast field of relevant literature, he has had to dissect and evaluate arguments frequently obscure and not seldom invalid. Littlewood in his Mathematician's Miscellany remarks that he finds much written before 1870 "very far from a model of lucidity"; it is exasperating to see superb technical skill vitiated by obscure definitions, or, quite frequently, by no definitions at all. The author picks his way through this jungle with considerable skill.
1.2. Review by: William C Waterhouse.
Bull. Amer. Math. Soc. 78 (3) (1972), 385-391.
Grattan-Guinness claims, confessedly without documentation, that Cauchy saw Bolzano's 1817 paper on the intermediate value theorem and took from it both the theorem and the definition of continuity. The argument for this is weakened by the admission (in a footnote) that a similar definition appeared in the most popular French calculus book before Cauchy. In any case it begs the question to simply describe Cauchy's proof as "a version of Bolzano's." Bolzano in fact first argues for the existence of least upper bounds by repeated bisection, constructing the l.u.b. as a convergent series of reciprocal powers of 2; he then finds the smallest zero of the function by taking the l.u.b. of the numbers where it is negative. Cauchy (beginning a section on numerical solution of equations) divides the interval into m equal pieces, chooses a subinterval on which the function changes sign, and iterates the process, getting convergent sequences of upper and lower bounds for a root together with estimates for the accuracy at each stage. Grattan-Guinness goes on to assert that the whole idea of basing analysis on the modern concept of limit was taken by Cauchy from Bolzano. The tone of the argument can be judged from a sample: "Needless to say, the name of Bolzano appears nowhere in the Cours d'analyse; Cauchy would have had more sense than to make Bolzano's work known to his rivals." The assertion in fact seems unlikely, since (1) the idea of quantities approaching but not achieving limits is clearly stated in the famous Encyclopédie article Limite, which gives the same example (inscribed polygons approaching a circle) that Cauchy uses, and (2) Bolzano's paper does not mention the concept of limit. There are several other serious mistakes.
1.3. Review by: Carl B Boyer.
Science, New Series 172 (3987) (1971), 1017.
This compact volume is an austere critique, addressed to those with background in advanced calculus, of certain specific problems truly pertinent to foundational questions in analysis. It is as difficult to read as it is rewarding, for it offers few facile generalities, concentrating instead on the details of deep theorems. Topics treated include definitions (for functions of a real variable) of limit, continuity, and the convergence of infinite series.
1.4. Review by: Michael Bernkopf.
Isis 62 (4) (1971), 532-533.
We are in the midst, it would appear, of the springtime of the history of nineteenth-century mathematics. The Grattan-Guinness book is the third to pass across my desk in as many months, and the three represent a welcome increase in activity. It is clear that a standard of judgment for such books is now overdue. A history of "recent" mathematics (post-Newtonian) is different from other histories of science; there is a level of sophistication inherent in the subject matter which requires extreme care on the historian's part if the non-specialist reader is to avoid becoming lost in the technicalities. Histories cannot be composed in the obfuscating and telegraphic style which has unfortunately become traditional in current mathematical writings. They should be presented in a manner which is called - pejoratively - "leisurely": definitions must be repeated often, key concepts gone over, technical vocabulary avoided whenever possible, and most important, emphasis supplied by pointing up the highlights. Also the author must keep the purposes of writing a mathematical history in mind, that is not only to present a chronology of events, but also to indicate the problems which gave rise to his particular topic and to show how it connects with the rest of mathematics. Ivor Grattan-Guinness, by avoiding all the pitfalls, proves that he understands clearly what is involved in the making of a good history, and he has succeeded in giving us a book which treats the subject well.. ... There are many gems in this book.
1.5. Review by: R Millman.
Amer. Math. Monthly 79 (3) (1972), 315-316.
There are three questions that must be answered when choosing a text for a course: (1) Does it follow the course which the instructor has outlined in his mind? (2) Are the students equipped both to understand and to enjoy the hook? (3) Is it a scholarly and honest book? In planning my course in the history of mathematics, I decided that in lieu of teaching a survey course, I would pick one rather general historical topic and do it in some depth. I felt that in doing this I would give the class (which consisted of eleven senior mathematics majors all of whom would be teachers) an opportunity to read the masters as well as a reminder of how hard mathematics is, even to the innovators of the subject. The course revolved around the history of analysis, with particular emphasis on the concepts of function, continuity, differentiability and integrability. I used the Grattan-Guinness book as a sequel to Boyer's History of the Calculus and Its Conceptual Development, supplementing both with some original source material. For the course outlined above, the Grattan-Guinness book is excellent because: (1) the development of mathematical analysis is a subject with a great deal of mathematical meat and the author treats it as such; (2) the author either quotes directly from the masters (e.g., Fourier, Cauchy, Bolzano, Riemann, Euler) or paraphrases what they have said; (3) the students learned a great deal of mathematics from this book. The answer to the first question is, therefore, an emphatic yes. The answer to the second question is also yes but not emphatically. Grattan-Guinness claims in the preface that "(the book) could be read by students with a knowledge only of solution by functional methods and separation of variables of linear partial differential equations, and of the general principles and development of classical analysis itself." My students had at least this much background (i.e., four semesters of calculus and at least one semester each of linear algebra, analysis, and modern algebra) and yet they found the going very rough in spots (especially Chapter 5 and parts of Chapter 2). ... The answer to the third question is also yes. The book is written in a fair manner which presents both sides of an argument and then comes to a conclusion. Moreover, it is very entertaining especially with regard to the Cauchy-Bolzano priority dispute.
1.6. Review by: J M Dubbey.
The British Journal for the History of Science 6 (1) (1972), 88-89.
All the ingredients of a typical undergraduate course in Real Analysis are here: continuity, differentiability, limits, the convergence and uniform convergence of infinite series, integrability, Fourier series, etc. Unlike the normal textbook covering these topics, Dr Grattan-Guinness presents these concepts as they actually arose, and demonstrates how remarkable advances in rigorous thinking evolved during the hundred years or so between Euler and Riemann. Instead of a static and usually forbidding body of knowledge unified only by epsilon mysticism, we meet the inspirations, the controversies, the guesses, and the errors to be expected of a creative process. So this book attempts not only to cover a relatively unknown period of mathematical history, but also a new approach to the teaching of Real Analysis as a dynamic conceptual development. Does it succeed? The reader is continually impressed by the soundness of both the mathematical and historical treatment, the mastery of some quite difficult material which has been gathered together and put in order from an impressive array of primary and secondary sources, and the depth of criticism and logical analysis. The author has clearly made a most thorough study of some widely scattered material and turned it into a narrative which is above all exciting and highly readable. Even so abstract a study as Real Analysis turns out to have a striking human story to it, and the leading characters, Euler, D'Alembert, Lagrange, Bolzano, Cauchy, Abel, Fourier, Dirichlet, Weierstrass, and Riemann are all presented in both their mathematical and personal relationships. Cauchy is in many ways the central character, by no means the hero, and the author, while clearly refuting the myth that he virtually invented Real Analysis single-handed, has worked hard to indicate the nature of his relationship with other mathematicians, the possible sources of his ideas, and the order of his thinking on these matters.
1.7. Review by: Einar Hille.
American Scientist 61 (2) (1973), 244.
The author, a senior lecturer at the Enfield College of Technology, in England, gives an excellent survey of ideas which stirred analysis from 1747 to about 1870. Grattan-Guinness stresses that progress came from working on specific problems, and these problems are his subject matter. The textual material is backed by heavy scholarly apparatus: a bibliography of some four hundred items, and profuse foot notes with abundant references. Skip the footnotes and enjoy the drama of steady conflicts between old concepts and new ideas. ... The book is well written, brimful of in formation, and presented in an attractive form .
- Joseph Fourier 1768-1830. A Survey of His Life and Work, Based on a Critical Edition of His Monograph on the Propagation of Heat, Presented to the Institut de France in 1807 (1972), by I Grattan-Guinness and J R Ravetz.
2.1. Review by: Thomas L Hankins.
Isis 64 (3) (1973), 424-425.
In this book we see more clearly than before how Fourier came to his famous series. It is not intended as a complete biography; in fact, it is primarily a publication (for the first time) of Fourier's monograph on the propagation of heat presented to the Institut de France in 1807. From this first treatise Fourier developed all his subsequent writings on the theory of heat and on trigonometric series. By careful analysis of the paper, and by comparing it to Fourier's later papers of 1808, 1809, 1811, and the 'Théorie analytique de la chaleur' of 1822, I Grattan-Guinness and J R Ravetz show the development of Fourier's theory in response to criticisms from other mathematicians.
2.2. Review by: C Stewart Gillmor.
Technology and Culture 14 (3) (1973), 501-503.
J B J Fourier left a great legacy to broad and diverse areas of science: mathematics, physics, and engineering. He is referred to among the French, for example, as the first mathematical physicist (excepting rational mechanics as a separate science from physics). The central document of Fourier's career, and various early drafts of it, has heretofore existed only in manuscript form. This work is his 1807 memoir on heat diffusion, "Sur la propagation de la chaleur." A much-expanded and altered version of this eventually was published in 1822 as "Théorie analytique de la chaleur," and previous historical interpretation of Fourier's work has often depended on this for the sequence of his researches. The 1807 memoir, however, is one of the great papers in the history of science which led to fundamental advances in the study of heat flow, mathematical analysis, and almost an unlimited number of applications in science and engineering. In a volume of over 500 pages Ivor Grattan-Guinness gives us the 1807 memoir in print, accompanied by an extensive bibliography of -Fourier manuscript materials and of published material by and about Fourier, and enlightens the reader with some 150 pages of close, reasoned commentary on the 1807 memoir and on related aspects of Fourier's life and career. Grattan-Guinness bases this critical edition on study of over 6,000 pages of Fourier manuscripts in Paris archives. In itself, the publication of an accurate text of the 1807 memoir with appropriate references would have been a formidable task, for many of Fourier's drafts and notes bear no date. In addition, physical ailments which eventually crippled Fourier caused his hand- writing during much of the last fifteen years of his life to be virtually illegible. Grattan-Guinness does a thorough job of sorting out Fourier's career from 19th-century reminiscences, friendly biographies, and published correspondence. The great value of this work, however, lies in the editor's elucidation of internal, substantial issues concerned with the study of heat diffusion and with developments in linear partial differential equations.
2.3. Review by: Robert Fox.
American Scientist 61 (2) (1973), 212.
The "Memoire sur la propagation de la chaleur," which forms the core of the book, has not previously been published. Submitted in December 1807 to the First Class of the French Institute (the revolutionary successor of the Academie des Sciences), it contained much that was original, both in its physics and in the range of novel mathematical techniques (including "Fourier series" and "Bessel functions") that it dis played. Dr Grattan-Guinness sees the paper, with some justice, as Fourier's "masterpiece," but understandably the 1807 paper has always been overshadowed by Fourier's later writings, notably by the revised and extended paper that won the Institute's competition of 1811 for mathematics and by the third version of his work on heat diffusion, which was published as the "Théorie analytique de la chaleur" in 1822. In this edition the text of the "Memoire," in the original French and with the original notation, is broken into chapters and interwoven with commentary. The book also contains biographical chapters, in which Dr Grattan-Guinness and his collaborator Dr Ravetz have produced a nice synthesis of externalist and internalist history, using biography as an integral part of their study of Fourier's mathematical and scientific work. There is much evidence of painstaking scholarship, and the book maintains the generally high standards of the M.I.T. Press History of Science series.
- Dear Russell - Dear Jourdain (1977), by I Grattan-Guinness.
3.1. Review by: R M Sainsbury.
Mind, New Series 88 (352) (1979), 604-607.
Philip Jourdain and Bertrand Russell corresponded on a wide range of mathematical and logical subjects from 1902 until Jourdain's death in 1919. In Dear Russell - Dear Jourdajn excerpts from the letters are reprinted, interwoven with commentary by Mr Grattan-Guinness, who provides biographical background, indicates connections between the letters and their author's publications, and offers critical elucidation of some aspects of the contents of the letters. In addition, Grattan-Guinness gives a synoptic account of the relationship between Russell and Jourdain; provides a translation of Russell's 'Sur les axiomes de l'infini et du transfini' which was read to the Société Mathématique de France in 19II and first published in the Society's Comptes Rendus ; reprints some of Jourdain's humorous contributions to the Cambridge University magazine The Granta; and prints for the first time some of the notes and comments in the margins of Jourdain's copy of Principia Mathematica. There are good bibliographies and indices.
3.2. Review by: Michael Hallett.
The British Journal for the Philosophy of Science 32 (4) (1981), 381-399.
Since Russell and Jourdain met infrequently (Jourdain was crippled with 'Friedrich's ataxia') contact was maintained via a substantial and detailed correspondence. Indeed, according to Grattan-Guinness, it seems that Russell did not correspond with anyone else in such detail. The correspondence ceased shortly before Jourdain's death in 1919. Very little of the Jourdain correspondence survives in the Russell Archives at McMaster University in Canada. The main source of the correspondence is therefore Jourdain's two letter notebooks which were acquired by the Swedish mathematician Gosta Mittag-Leffler in 1922, and were found by Grattan-Guinness in the Institut Mittag-Leffler, Mittag-Leffler's former home near Stockholm. The bulk of Grattan-Guinness's book is a series of extracts from this correspondence, in chronological order, linked by his own commentary. The result is a book of great value for anyone interested in the history of logic and set theory. It also contains the editor's translation of Russell's 'Sur les Axiomes de l'Infini et du Transfini' which was originally published rather obscurely in French. Its appearance here is a welcome addition to the Russell corpus in English. (One wonders why it was omitted from the various relevant collections of Russell papers.) There is also an excellent and extensive bibliography, invaluable on the works of Russell and Jourdain, as well as some amusing selections from Jourdain's contributions to the Cambridge magazine 'The Granta'.
3.3. Review by: G T Kneebone.
The Journal of Symbolic Logic 44 (2) (1979), 277-278.
Jourdain and Russell corresponded with each other on foundational matters, and parts of this correspondence survive in the notebooks kept by Jourdain that are now at the Mittag-Leffler Institute. It is this material that is presented, in an edited form, by Grattan-Guinness. The letters are of considerable historical interest, since they extend over the years from the completion by Russell of The principles of mathematics to the publication of Principia mathematics, affording glimpses into how his ideas were developing during that period. The book is by no means easy reading, on account of the nature of the surviving material. In his notebooks Jourdain kept copies of his own letters, interspersed with pasted-in replies from Russell; but the copies were often written by other hands, because of Jourdain's disability, with only the mathematical symbols inserted by him, and sometimes the symbols were not put in at all. This makes editing very difficult; and since, moreover, there is little overall continuity in the original correspondence, there cannot be much in the published extracts. The present book must therefore be taken more as a collection of material for study by readers who are already interested in the history of the formative years of logicism than as a connected exposition, to be read for its own sake. In any case, the discussions belong to an age that is now past, and they will be read more for the insight they give into the nature of mathematical and philosophical inquiry than for any specific results that came out of them. The editing of the correspondence has been well done, with restrained but helpful commentary, and an excellent bibliography is provided.
3.4. Review by: S Körner.
The Mathematical Gazette 63 (423) (1979), 62.
The interest of this book is threefold. It contains, first of all, a great deal of historically important, hitherto unpublished material from Russell's letters, Jourdain's notebooks and his copy of Principia mathematica, as well as some published contributions by Russell and Jourdain which are not easily accessible. To this there is, secondly, added a highly illuminating account of Russell's development as a logician and of the intellectual climate in which this development took place. Lastly, the book throws some light on Russell's and Jourdain's characters. The correspondence and the republished essay s are concerned with some crucial aspects of logicism, in particular the logicist analysis of real numbers, of transfinite cardinals and ordinals, of Russell's and Burali-Forti's paradoxes, and of various alternatives to the theory of types.
3.5. Review by: Dale M Johnson.
The British Journal for the History of Science 12 (2) (1979), 232-233.
This book is an edition of a correspondence between Bertrand Russell and Philip Jourdain which extended from March 1902 until the middle of 1919. The most significant part of the correspondence covers the crucial first decade of this century leading up to the publication of 'Principia mathematica', so that one may expect some new historical insights into the development of Russell's logical ideas. Jourdain attended Russell's lectures on mathematical logic during the winter of 1901-02, and their correspondence began immediately afterwards. Although Jourdain must be regarded as a minor figure in the history of logic and the foundations of mathematics, he possessed a vigorous and critical mind, and his questions elicited some interesting and illuminating responses from the master of logical type theory. The surviving Russell-Jourdain correspondence runs to some 116 letters or drafts, most of which Dr Grattan-Guinness found in a notebook kept by Jourdain (now held in the collection at the Institut Mittag-Leffler, near Stock- holm, Sweden), but a few of which he discovered in the Russell Archives (McMaster University, Ontario, Canada). This edition of the correspondence is not a complete one with all the letters printed, but rather consists of extensive extracts concerned with the most important logical and mathematical points, together with a substantial commentary by the editor. Apparently it was not worth while publishing the correspondence in its entirety, and, moreover, there would have been technical problems in doing so. Jourdain's draft letters are often difficult to read (he had a paralysing disease which affected his writing-but not his mind!) and they frequently lack the mathematical symbols. Dr Grattan-Guinness's comments are in an important sense incomplete, as he admits: 'My interpolated commentaries have been restricted to the contents of the correspondence, as I abhor as impracticable the simultaneous presentation of extensive amounts of documentary material and of detailed analyses of other matters.... Thus more conclusions can be drawn from these texts than I have explicitly discussed'.
- From the Calculus to Set Theory, 1630-1910: An Introductory History (1980), by I Grattan-Guinness (ed.).
4.1. Review by: Robin E Rider.
The American Mathematical Monthly 92 (3) (1985), 225-227.
Six historians of mathematics, all of whom have participated in this resurgence of interest and activity, have joined forces in 'From the Calculus to Set Theory, 1630-1910'. The book covers more than the standard territory of the history of calculus. It is intended as an historical introduction to a set of interrelated developments in mathematics: the origins of the differential and integral calculus in the 17th and 18th centuries; the elaboration of mathematical analysis in the 19th and early 20th centuries; and the re-examination of the foundations of mathematics, especially the progress of set theory and mathematical logic in the latter part of the 19th century. The level of exposition, sometimes heavily mathematical, presupposes considerable mathematical knowledge. As the editor cautions, "this is not the place from which the mathematics can be learnt." The book is thus directed at readers with some training in mathematical analysis and set theory, especially upper-division undergraduates and graduate students.
4.2. Review by: Roger Jones.
Philosophy of Science 51 (3) (1984), 519-522.
The goal of these historicist books on calculus then, and of this book in particular, is to provide an account of the problem background of the subject, an account accessible to the vast number of users of calculus for whom it is solely a collection of algorithms rather than a subject at all, an account that will enable them to appreciate the canonical definitions and theorems as having the kind of inevitability, the kind of intuitive necessity associated with discoveries within a large and coherent subject matter, and, peculiarly, a mathematical subject matter at that. This goal may sound a little overstated, but it is an ideal to which writers of this kind of mathematical conceptual history doubtless aspire. And I think it is worthwhile to judge this book by this goal, because it comes off rather well. Grattan-Guinness has assembled a very appropriate set of co-authors to complement his own formidable expertise in eighteenth and nineteenth century analysis ... Though the book is a history book, it does avoid much of the "this is the way it was" history which, for the goal sought, is even worse than the "this is the way it is" textbook accounts that the authors seek to supplement. The book also avoids lingering on life and times remarks ... Beyond this, Grattan-Guinness provides in his section another sort of remark which I find to be extremely helpful to calculus students. He is, as he indicates in his remarks in the Introduction, specifically interested in the relation between the textbook tradition and the actual propagation of mathematical understanding. And he exhibits that interest in a series of remarks explicitly directed to the textbook tradition
4.3. Review by: Charles V Jones.
American Scientist 70 (4) (1982), 436.
This book differs from most other introductory histories of mathematics by being confined to a very few topics which are thematically linked. ... This is not introductory mathematics, since the discussions are detailed and technical, at times demanding. ... As an introductory history, the book succeeds in describing kinds of problems and concatenations of ideas, especially within each chapter.
4.4. Review by: Morris Kline.
Isis 72 (4) (1981), 661-662.
This book is a useful collection of articles by six authors who sequentially treat the subject stated in the title. The authors are knowledgeable and the factual content is accurate. The subject is a fascinating one because in all of the development of mathematics the history of the calculus provides a superb lesson in how mathematics develops. Like other mathematical creations, it passed through a period when the greatest of mathematicians blundered and groped while proposing its foundations; it subsequently found a woefully defective foundation, and finally an acceptable one. One dare not say rigorous foundation because the question of ultimate or absolute rigor is open today. Full-length books are available on the several subjects treated in the articles and indeed are written by some of the authors of these articles. However, the concentration on the history of the calculus serves a purpose for those who wish to learn the history of that topic. Of course articles written independently by six authors will inevitably repeat each other to some extent, and the editor is well aware of this. This is a minor defect. However, the level of the articles does seem questionable. The subtitle describes the book as an introductory history, and in his introduction the editor states that the book is intended for supplementary reading by undergraduate students, who should learn how mathematics develops as opposed to the final text versions that deliver only results and proof.
- Psychical Research: A Guide to Its History, Principles, and Practices (1982), by I Grattan-Guinness.
5.1. Review by: Trevor Pinch.
Isis 74 (3) (1983), 439.
This book, published in celebration of that most curious of British scientific societies, the Society for Psychical Research, is intended as an introductory guide to the area. Its purpose is well served by thirty-four original contributions, which, cover most aspects of psychical research and include a glossary of terms and a list of relevant organisations. ... Grattan-Guinness has brought together a fascinating collection, but what has been achieved by this queer subject?
- History in Mathematics Education (1988), by Ivor Grattan-Guinness (ed.).
6.1. Review by: Joan L Richards.
Isis 81 (3) (1990), 546-547.
In the preface Ivor Grattan-Guinness argues for the value of including historical material in the teaching of mathematics. He is critical of what he sees as the usually dismissive or trivializing attitude of mathematicians and mathematical educators toward history. He argues that the subject has a critical role to play both in their teaching and in their understanding. In his view, the interplay of knowledge and ignorance found in historical development can be used to enrich and deepen our understanding of the parallel movement in students of mathematics. Most of the essays in this volume present historical materials with suggestions and discussion about how these can be incorporated into mathematics courses.
- Selected Essays on the History of Set Theory and Logics (1906-1918) (1989), by Philip E B Jourdain and Ivor Grattan-Guinness.
7.1. Review by: Joan L Richards.
Isis 85 (2) (1994), 354-355.
In a recent article ("Does History of Science Treat of the History of Science? The Case of Mathematics," History of Science, 1990 , 28: 149-173), Ivor Grattan-Guinness bemoaned the lack of interest in the history of mathematics among historians of science. There he characterized the history of mathematics as a "ghetto subject: too mathematical for historians and too historical for mathematicians". It certainly is the case that the history of mathematics is sadly isolated and loo often ignored within the history of science, but it is worth considering where the problem lies; whether historians of science are excluding or being excluded by those pursuing the history of mathematics. Grattan-Guinness's edition of Philip Jourdain's Selected Essays on the History of Set Theory and Logics brings this issue to the fore. ... Grattan-Guinness's contribution to this volume is a thirty-page introduction whose purpose is, in his words, "to be historical about both Jourdain's essays and their subject matter." Grattan-Guinness immediately clarifies, however, that he will include little historical content, for fear of upstaging Jourdain's historical remarks; he intends, therefore to confine himself to context and "meta-history."
- Convolutions in French Mathematics, 1800-1840: From the Calculus and Mechanics to Mathematical Analysis and Mathematical Physics (1990), by Ivor Grattan-Guinness.
8.1. Review by: James R Hofmann.
Isis 83 (2) (1992), 291-297.
Grattan-Guinness has been a rare voice objecting to the lack of attention to mathematical developments. In Convolutions he makes a major contribution toward the correction of that condition. Stylistically Convolutions is, in a word, encyclopaedic. The majority of its pages are dedicated to summaries and explications of published and manuscript sources. Fortunately, however, a plethora of additional information ensures that no one will mistake Convolutions for three more volumes of Todhunter. Biographical sketches are supplemented by thorough attention to institutional and educational context. ... although Grattan-Guinness claims to avoid thematic interpretation in order to concentrate on "what happened", several thematic strands do lend cohesion to Convolutions. Naturally, as he has done in other publications, Grattan-Guinness describes the unification of the relatively disjointed eighteenth-century differential and integral calculus techniques as limit theory and the algebra of inequalities gradually brought an understanding of the convergence and divergence of infinite series. Casual references to "any function" thus became increasingly qualified, with direct impact on acceptance criteria for potential solutions to differential equations.
- George Boole: Selected Manuscripts on Logic and Its Philosophy (1997), by George Boole, I Grattan-Guinness and G Bornet.
9.1. Review by: Graham Priest.
Studia Logica: An International Journal for Symbolic Logic 63 (1) (1999), 143-146.
The editors have done a thorough and commendable scholarly job. A 60 page introduction gives an account of Boole's life, the context of his work, the fate of his Nachlass, and provides an analysis of some of his philosophical ideas, notably his psychologism. This is rounded off with an account of the selection of the papers and the difficult job of dating them. There is also a set of helpful textual notes, a bibliography and indexes. ...
9.2. Review by: Theodore Hailperin.
The Journal of Symbolic Logic 63 (1) (1998), 332-333.
The book under review consists of a lengthy Editors' introduction, followed by selected material from Boole's Nachlass comprising about 40 per cent of the total. With regard to the Editors' introduction, Part 1, by Ivor Grattan-Guinness, is a richly detailed summary of Boole's life, career, and accomplishments. It also includes a history of the Nachlass, originally left as a confused and chaotic collection of undated items, and the various attempts at organizing it for publication. Grattan-Guinness also describes and evaluates their selected excerpts, in particular an uncompleted book 'The philosophy of logic', envisioned by Boole as a sequel to 'The Laws of Thought'.
- The search for mathematical roots, 1870-1940: Logics, set theories, and the foundations of mathematics from Cantor through Russell to Gödel (2000), by I Grattan-Guinness.
10.1. Review by: William Ewald.
Bull. Amer. Math. Soc. 40 (1) (2002), 125-129.
A comprehensive intellectual history is much needed, and Grattan-Guinness announces his intention to improve on the existing literature in two important ways: first, by considering the links between foundational research and research in mainstream algebra, geometry, and analysis; second, by calling attention to numerous minor or forgotten figures who were influential at the time but whose contributions have been overshadowed by subsequent developments. Much of the existing literature has been philosophically motivated and preoccupied with the exegesis of individual thinkers, notably Frege and Russell, who are widely (and rightly) viewed as founding giants of analytical philosophy. But the wider mathematical context has in the process often been lost from sight. Grattan-Guinness's insistence that Peano and Schröder, Grassmann and Peirce be given their due and that the mainstream developments in nineteenth-century mathematics be treated as central to the foundational story is surely correct, and the list of some fifty major archival sources gives an indication of how vast are the resources for such a study.
- Landmark Writings in Western Mathematics 1640-1940 (2005), by Ivor Grattan-Guinness.
11.1. Review by: Josipa G Petrunic.
The British Journal for the History of Science 40 (4) (2007), 608- 610.
It was not until the seventeenth century that doing mathematics became an identifiable profession, such that historians today can legitimately talk of 'mathematicians' as a community of experts producing mathematical works for one another in various institutional settings. This is the starting premise for Ivor Grattan-Guinness's mammoth edited volume 'Landmark Writings in Western Mathematics 1640-1940' - a book that takes the reader from René Descartes's 'La Géométrie' (1649) to David Hilbert's and Paul Bernays's 'Grundlagen der Mathematik' (1934-9). Written by a Who's Who? of authors in the history of mathematics, including Grattan-Guinness, Niccolò Guicciardini, June Barrow-Green, Jeremy Gray and Tony Crilly, to mention only a few, the book is composed of seventy-seven articles dealing with eighty-nine pieces of 'writing' spanning a wide range of mathematical topics. Importantly, the book does not contain the original texts under discussion. It is not a compendium of primary sources. It is, rather, a compendium of succinct survey articles that provide the social and educational background of the mathematicians or writers in question, descriptive accounts of the most salient aspects of the piece of writing being analysed, and a brief discussion of the 'impact' or effect the piece had on mathematical thought. Thus we get articles on aspects of geometry, calculus, functions and differential equations, algebras, number theory, real and complex analysis, set theory (and its foundations), mechanics, astronomy, probability and statistics, dynamics, mathematical physics and topology, as well as articles in the history of mathematics, including Jean-Etienne Montucla's 'Histoire des mathématiques' (1799-1802), and articles in the 'social and life sciences', including pieces on W S Jevons's 'Theory of Political Economy' (1871) and Vito Volterra's book on mathematical biology (1931).