The Concepts of the Calculus (1939), by Carl B Boyer. **1.1. From the Preface.**Some ten years ago Professor Frederick Barry, of Columbia University, pointed out to me that the history of the calculus had not been satisfactorily written. Other duties and inadequate preparation at the time made it impossible to act upon this suggestion, but my studies of the past several years have confirmed this view. There is indeed no lack of material on the origin and subject matter of the calculus, as the titles in the bibliography appended to this work will attest. What is wanting is a satisfactory critical account of the filiation of the fundamental ideas of the subject from their incipiency in antiquity to the final formulation of these in the precise concepts familiar to every student of the elements of modern mathematical analysis. The present work is an attempt to supply, in some measure, this deficiency. An authoritative and comprehensive treatment of the whole history of the elementary calculus is greatly to be desired; but any such ambitious project is far beyond the scope and intention of the dissertation here presented. This is not a history of the calculus in all its aspects, but a suggestive outline of the development of the basic concepts, and as such should be of service both to students of mathematics and to scholars in the field of the history of thought. The aim throughout has therefore been to secure clarity of exposition, rather than to present a confusingly elaborate all-inclusiveness of detail or to display a meticulously precise erudition. This has necessitated a judicious selection and presentation of such material as would preserve the continuity of thought, but it is to be hoped that historical accuracy and perspective have not thereby been sacrificed.

**1.2 Review by: W. M. M.**

*Philosophy of Science***6**(3) (1939), 382.Those who are familiar with elementary calculus and want to make sure that they have the "correct" view of the foundations, which have been frequently confused in the past, should put this book on the "must" list. The treatment is historical, the emphasis is on concepts.

**1.3. Review by: William L Schaaf.**

*The Mathematics Teacher***32**(6) (1939), 284.On rare occasions the literature of scientific thought becomes notably enriched by an out standing work such as Dantzig's 'Number: The Language of Science' or E T Bell's 'Men of Mathematic's. Another such welcome occasion is provided by the appearance of Dr Boyer's un deniably scholarly and carefully documented study. It is a thoroughly adequate examination of the origins and evolution of the fundamental notions underlying modern analysis. It is far more, however, than a mere chronological ac count or a compelling historical narrative; it is a searching analysis and critical evaluation of the well-sources of certain crucial ideas, and of their subsequent transformations, reappearances and ultimate refinements. As such, it must be regarded as a genuine contribution, for which both laymen and scholars owe the author a debt of gratitude. The book is doubly welcome: in addition to an agreeable lucidity of style and singular clarity of thought, the interpretation is born of that rare combination of competent mathematical insight and sympathetic historical under standing. Moreover, an adequate history of the calculus needed to be written, despite the wealth of material scattered here and there.

**1.4. Review by: William E Byrne.**

*National Mathematics Magazine***14**(7) (1940), 427.In this book the author traces the development of calculus concepts from the time of the early Greeks up to the end of the nineteenth century. ... Both students and instructors alike should find this volume of interest, since many of the conceptual difficulties of early mathematicians still trouble students in classrooms every day. A more detailed account, along the same line, of the growth of calculus from the Cauchy-Riemann-Weierstrass period to the present would prove of value to all of us.

**1.5. Review by: I Bernard Cohen.**

*Isis***32**(1) (1940), 205-210.The author states that "this is not a history of the calculus in all its aspects, but a suggestive outline of the development of the main concepts." As such, it is really a history of mathematics from the focal point of the derivative and the integral and Mr Boyer traces the main ideas of the subject from antiquity to modern times. The book represents a prodigious amount of labour, and the richness of material will make it a source book for all mathematicians. ... By far the most interesting chapter is that on the "medieval contributions," especially in the fine account of the mathematical work of the "Calculator," Richard Suiseth, who lived in the first half of the fourteenth century. ... It is much to be regretted that the other chapters are not written as carefully and as clearly as this one on the medievals. ... Unfortunately, at times Mr Boyer makes very loose statements such as that Pascal "was not so much a creative genius as a mathematician, scientist, and philosopher, with a remarkable flair for clarifying ideas which had been somewhat vaguely set forth by others, and for supplying these with a reasonable basis". This in face of his highly original treatment of conics, and the latter part of the statement in face of some of the religious and mystical ravings concerning mathematics! ... But for the few occasional misstatements of the kind instanced above, the book is an admirable one, covering a much needed field of investigation. It is to be hoped that in the near future, Mr Boyer will revise the book to bring it all up to a high standard of excellence. For Mr Boyer's contributions are highly original and very valuable.

**1.6. Review by: John Oulton Wisdom.**

*Mind, New Series***49**(194) (1940), 248-253.We are fortunate in having a mathematical historian who is a worthy successor to the late Professor Florian Cajori. We have at last a wholly adequate account of the growth of the calculus. The scope of the present work is clearly indicated by its sub-title [A Critical and Historical Discussion of the Derivative and the Integral]. Mr Boyer has done more than give a clearly written historical record: he always keeps his matter alive by judicious criticism and by keeping before his readers those aspects of present-day mathematics towards which the historical conceptions of the calculus have evolved. Even though he refers to large numbers of minor mathematical figures of the past he always succeeds in distilling the point for which he brings them before us. This balance of judgment throughout makes history illuminating instead of, as is often the case, misleading.

**1.7. From the Preface of the 1949 edition.**It is gratifying to find sufficient demand for a work on the history of the calculus to warrant a second printing. This appears to be a token of the increasing awareness in academic circles of the need for a broad outlook with respect to science and mathematics. Amazing achievements in technology notwithstanding, there is a keener appreciation of the fact that science is a habit of mind as well as a way of life, and that mathematics is an aspect of culture as well as a collection of algorithms. The history of these subjects can never be a substitute for work in the laboratory or for training in techniques, but it can serve effectively to alleviate the lack of mutual understanding too often existing between the humanities and the sciences. Perhaps even more important is the role that the history of mathematics and science can play in the cultivation among professional workers in the fields of a sense of proportion with respect to their subjects. No scholar familiar with the historical background of his specialty is likely to succumb to that specious sense of finality which the novitiate all too frequently experiences. For this reason, if for no other, it would be wise for every prospective teacher to know not only the material of his field but also the story of its development.

**Reprinted as:**The History of the Calculus and its Conceptual Development (1959), by Carl B Boyer. **2.1. Review by: Morris Kline.**

*Mathematical Reviews*, MR0124178**(23 #A1495)**.This book is a reprint of the work which originally appeared under the title

*The concepts of the calculus*[1939 and 1949]. It is a rounded history of the development of the concepts of the subject as opposed to the theorems and applications of the calculus.**2.2. Comments by: Charles Gillispie.**

*Isis***67**(4) (1976), 610-614.'The Concepts of the Calculus' owes a certain debt to Ernst Mach. Its subtitle is "A Critical and Historical Discussion of the Derivative and the Integral," and Boyer does not hesitate to instruct mathematicians as well as historians and philosophers on matters of taste, rigour, and the nature of mathematics itself, as well as its historiography. It is a history, not of problems or technicalities, but of the foundations of mathematical operations and reasoning, beginning with Zeno's paradoxes and ending with Cauchy's theory of limits and with Dedekind, Weierstrass, and Cantor on continuity, number, and infinity. The book is no mere survey, however. Re-reading it, I have been struck with the many points of detail, about Descartes, say, or Newton or Leibniz, which I had not known or remembered or which I thought somehow I had learned from more recent works of scholarship. Gradually, it was borne in on me that Boyer's books are based on a far wider and more intimate command of all the sources than I had ever appreciated ...

A History of Analytic Geometry (1956), by Carl B Boyer. **3.1. From the Preface.**The history of analytic geometry is by no means an uncharted sea. Every history of mathematics touches upon it to some extent; and numerous scholarly papers have been devoted to special aspects of the subject. What is chiefly wanting is an integrated survey of the historical development of analytical geometry as a whole. ... there seemed to be room for an historical volume of modest size devoted solely to coordinate geometry. It soon became evident that, in view of the amount of material available, some limitations would have to be imposed if the work were to remain within reasonable proportions. Not all relevant details could be included ... The present history covers only such parts of analytic geometry as might reasonably be included in an elementary general college course. Consequently developments of the last hundred years or so are largely omitted. The manuscript of this work was completed about half a dozen years ago, and major portions of it have appeared from time to time in 'Scripta Mathematica'.

**3.2. Review by: Howard Eves.**

*The American Mathematical Monthly***64**(7) (1957), 518.Here is an integrated survey of the history of analytic geometry from the earliest glimmerings in ancient Mesopotamia and Egypt up through the death in 1868 of the subject's most powerful and prolific contributor, Julius Plucker. Leaning largely upon the earlier, but either incomplete or inconveniently inaccessible, works of Loria, Tropfke, Wieleitner, and Coolidge, Professor Boyer has in scholarly fashion sketched the fascinating story of the emergence and development of this important branch of mathematics, filling in many lacunae and correcting many errors perpetrated by early writers and perpetuated by subsequent historians of mathematics. There is no unanimity of opinion among historians of mathematics as to who invented analytic geometry, nor even as to what age should be credited with the invention, and this matter certainly cannot be settled without an agreement as to the defining characteristics of the subject. Professor Boyer steers an admirably restrained course here, and though he bespeaks his own views he gives a fair account of existing opinions. ... Teachers and students of analytic geometry, and all interested in the history of mathematics, owe Professor Boyer a very real debt of gratitude for performing the patient, painstaking, and time consuming research that undoubtedly went into the preparation of this fine book.

**3.3. Review by: René Taton.**

*Isis***50**(4) (1959), 489-492.We know the seriousness and interest of numerous articles published by C B Boyer and robust qualities of his work on the origins and development of the calculus, where the accuracy of the documentation is combined with a very laudable concern for clear and relatively elementary presentation. It is to an audience "of mathematics students and researchers in the field of the history of thought" that C B Boyer looks to succeed with his book 'History of Analytic Geometry'. This book treats, therefore, those aspects of analytical geometry that can be included in an elementary course at a college in the US and, leaving aside most of the contributions of the last hundred years, ends with an analysis of the work of Plücker.

**3.4. Review by: Arthur Rosenthal.**

*Science, New Series***125**(3252) (1957), 823-824.The greater part of the History of Analytic Geometry was first published as a series of articles in Scripta Mathematica, volumes 16 through 21 (1950-55). It is gratifying that the complete book, in final form, has now appeared.

The Rainbow: From Myth to Mathematics (1959), by Carl B Boyer. **4.1. Review by: Oystein Ore.**

*Philosophy of Science***27**(2) (1960), 207-208."The rainbow has had hosts of admirers - more, perhaps, than any other natural phenomenon can boast - but it has had few biographers." (Preface) This lacuna has now been filled in an admirable manner, for there is little of importance in regard to mankind's speculations about the causes of the rainbow which cannot be found in Boyer's new book. In fact, its scope is wider, it is more nearly a history of optics from the Greeks until the beginning of the last century. ... At a dinner gathering of English poets in 1817 Lamb and Keats agreed that the poetic charm of the rainbow had suffered through Newton's hard headed mathematical theory and the guests drank a toast: "Newton's health and confusion to mathematics." Boyer's book should remedy some of this regret for, after having read it, it is impossible not to regard the rainbow and its details with greatly intensified interest.

**4.2. Review by: Peter Diamandopoulos.**

*Isis***55**(2) (1964), 218-220.The subject of Mr Boyer's book is attractive; his way of presenting it opaque; the moral insinuated, obscure. He recounts man's attempts to understand the rainbow. He shows that as more satisfactory explanations of the phenomenon are increasingly reached from age to age there is a growing understanding of the complexities of nature. Strangely enough at the end of the book the meaning of these advances is open to radically opposed interpretations of physics. In view of Boyer's avowed purpose to take his topic as truly representative of scientific progress, the circumstances surrounding the scientific explanation of the rainbow ought, I think, to reveal crucial truths about science and its feats. Quite disappointingly, however, they don't. What we get instead is the impression that physics strives after ever-receding objects, and its method thrives on abstractions and weird observations. The development of one of the oldest physical disciplines - optics - is mechanically retraced as though there were no lesson to be imparted. This is sad, especially since the historical data deployed by Boyer are richly suggestive and his way of handling them most scrupulous. ... Philosophers of science unacquainted with the history of physics may get something out of this book. I am not so sure what the benefits will be for historians of optics.

**The 1967 edition.****4.3. Review by: Peter Hamshere.**

*The Mathematical Gazette***72**(461) (1988), 259-260.Carl Boyer is well known to those interested in the history of mathematics for his 'History of mathematics' (1968) and for 'The history of the calculus and its conceptual development' (1939). The story of man's fascination with the rainbow and the attempts to explain its colours and its formation was first published in 1959. ... As a case study in the history of science the story of the rainbow shows that progress in science is not a steady advance with more adequate explanations following each other at regular intervals but is a progress by fits and starts. It shows that the accumulation of observations however accurate is not a substitute for ideas. Neither are instruments a substitute for insight - the idea of a globe of water as a magnified raindrop disappeared between Theodoric and Descartes but the globe of water as a gadget persisted. The book is well produced and is easier to read than the complexity of the subject matter would lead one to believe. It has excursions into poetry and is full of erudition. It should prove of great interest to all who would like to understand how mathematics "explains" and underpins qualitative explanations of phenomena and to those who like intellectual detective stories.

**4.4 Comments by: Charles Gillispie.**

*Isis***67**(4) (1976), 610-614.[Carl Boyer] always wanted mathematics to be united with its history and the history of mathematics with the history of science generally. 'The Rainbow, from Myth to Mathematics' (1959) is history of physics, the narrative culminating in the mathematization of the account of the phenomenon in the modern theory of Airy, Stokes, and Potter. Though Boyer never forgets the perennial appeal the rainbow holds for the imagination, it remains a technical book. Perhaps the title led certain reviewers to expect a romantic treatment. In later years he felt that he might have overextended his vein in stepping even this far outside the boundaries of mathematics itself. I cannot agree. I do not know a more instructive example of the way in which understanding of a phenomenon has evolved in the interplay between the mathematical techniques available and the prevailing preference in physical models. Nowhere does the author lead his reader to conclude that some certain formulation is to be considered the right theory - not even at the end.

A History of Mathematics (1968), by Carl B Boyer. **5.1. Review by: Howard Eves.**

*American Scientist***56**(4) (1968), 468A-469A.The English-reading world has been fortunate in possessing some fine texts dealing with certain special epochs and with certain special areas of the history of mathematics, but for a long time it has lacked a good, up-to-date, broad, and comprehensive history of the subject. Professor Boyer has now remedied this lack and has given us, compressed into about 700 pages, a book that may be destined to remain for some time the best thing of its sort in the English language. Arranged in strictly chronological sequence, we have here an authoritative, scholarly, well-balanced, and eloquent narration of the story of the development of mathematics from its primitive origins to Bourbaki and the "New Mathematics." ... As a truly magnificent reference work, the book should unquestionably be in every college, university, and sizeable public library. Its suitability as a class room text for an undergraduate college course in the history of mathematics will be a matter of opinion. Some instructors will find the text too massive for a comfortable one-semester coverage, and may feel that it lacks certain other pedagogical requirements. But surely it will be in the instructor's personal library and will be one of the chief references of the course. We owe Professor Boyer our sincerest thanks for having written this superb book.

**5.2. Review by: P S Jones.**

*The American Mathematical Monthly***76**(10) (1969), 1163-1165.This book is intended to be a textbook for a course in the history of mathematics for upper division college students. It can, then, be evaluated as a history of mathematics whose content and emphases are conditioned by the textbook function, and as a textbook which has some of the elements of a historical treatise. In a historical treatise one looks for scholarship in marshalling the facts, for insight and a broad view in portraying the changing trends, themes, and interrelationships both within mathematics and between mathematics and other aspects of the culture. One may also be concerned for the coverage and for the quality of the evaluation of forces directing progress. This book scores high in all of these respects. ... As a textbook, 'History of Mathematics' is long and not easy reading. This well-written book is packed with condensed information and interpretation. There are many places where a good and eager student could and should fill in the details of a derivation. The data and the motivation are there. However, this taxes both the students who do it and those who do not. The author's literary style is pleasing to the reviewer, but not easy for his students, even though it may be good for them. ... the book rates an excellent score as both a scholarly product which sets a good example for young scholars, and as a textbook in an undergraduate course. Although it lacks the range and the occasional flamboyance of E T Bell's 'Development of Mathematics', mathematicians too will find it good reading and the best book in English from the viewpoint of historical scholarship.

**5.3. Review by: Judith Victor Grabiner.**

*Science, New Series***163**(3863) (1969), 171.At last there is a history of mathematics that can be recommended without reservation. Making full and critical use of recent scholarship, Boyer has avoided major errors of fact or interpretation. And unlike several currently popular handbooks, the work is neither too concise nor too elementary. The guiding principle of Boyer's book is that continuity in the development of mathematical ideas is the rule rather than the exception. Important ideas of modern mathematics, such as infinity, coordinate geometry, and the striving towards generality and rigour, are discussed in their ancient and medieval settings. The author makes judicious assessments of the influence and importance of individuals and schools, and illustrates his generalisations with well-chosen examples. ... Boyer has produced a work which should be welcome to mathematicians, teachers, and historians alike.

**5.4. Review by: A Prag.**

*The British Journal for the History of Science***5**(1) (1970), 89.To write a comprehensive history of mathematics from its primitive origins to the efforts of Bourbaki requires skill, courage and a reliable fund of expert knowledge - all of which Boyer brings to his task, and so we find here a very useful college textbook, sensibly divided into some 27 chapters of manageable size, supported by an abundantly rich and up-to-date bibliography, both in footnotes and collectively at the end-as one might expect from a scholar whose own careful and detailed studies have ranged wide and have often become standard works in their particular field. One's envy that there are places of higher learning where such textbooks are widely wanted is somewhat modified when one sees the drudgery of a set of exercises at the end of each chapter; they contain, however, not only the usual essay-type revision questions, but also simple sums to test whether the plain mathematical content has been assimilated. They may appear too straightforward, but this is, after all, a book for a rather general public, and a keen student could find here the challenge to probe more deeply and the guidance perhaps to read an original text himself. ... The book frequently shows its origin as a text for a course of lectures, and what might have been lightly spoken cannot so comfortably be read in print ... altogether this admirably documented survey by an expert is a welcome achievement, even if one continues to base proper appreciation of Boyer's merits in the history of mathematics rather on his earlier monographs.

**5.5. Review by: Howard L Prouse.**

*The Mathematics Teacher***62**(7) (1969), 597.The book 'A History of Mathematics' blends historical and mathematical material into a volume that presents the history of mathematics in a clear, concise, and intellectually appealing manner. Historical perspectives are enhanced by a stricter adherence to chronological arrangement than is found in other successful books on this subject. However, 'A History of Mathematics' is not merely a chronology of the development of mathematics from prehistoric times until today. It is a book in which mathematical and historical materials are thoroughly integrated so that the reader can discern the human element present in the growth and development of mathematics.

**5.6. Review by: Kenneth May.**

*Mathematical Reviews*, MR0234791**(38 #3105)**.As one might expect, the exposition is carefully footnoted to primary and secondary sources and is more reliable than E T Bell's

*The development of mathematics*[1940], at the cost of being less amusing. The author has included a tremendous amount of well-indexed detail, without allowing the work to become an unreadable compendium like F Cajori's*History of mathematics*[1931]. ... Although any respectable history course for mathematicians must be based on extensive reading of primary sources and high quality historical studies such as those of van der Waerden and Neugebauer, a comprehensive text-book is valuable for orientation and guidance. This book is the best available. It is also suitable for the student or mathematician who needs a reliable and readable general survey of the history of his discipline.**5.7. Review by: J E Hofmann.**

*Isis***60**(4) (1969), 553.The well-known author has set himself the goal of providing an overview of the history of mathematics, which begins with the earliest ideas and ends with an look at the main points of the current position. He addresses in this well-equipped work, illustrated with numerous portraits, facsimiles and figures, primarily those readers who are looking not for details and peripheral personalities, but who want to understand the big picture. Is references numerous article in the literature and the accompanying exercises give one the opportunity to become acquainted with characteristic thoughts from earlier times.

**Revision by Uta C Merzbach.**A History of Mathematics (1991), by Carl B Boyer and Uta C Merzbach. **6.1. Review by: Mary Kim Prichard.**

*The Mathematics Teacher***85**(5) (1992), 394-395.Uta C Merzbach had the job of revising one of the best-written and most useful texts on the history of mathematics. In his preface to the first edition, Boyer said that his book represented "a stricter adherence to the chronological arrangement and a stronger emphasis on historical elements" than comparable texts. The first edition also had a multicultural emphasis and followed the development of mathematics in Western civilizations as well as Chinese, Indian, and Arabic civilizations. Merzbach revised the first twenty-two chapters of Boyer's work very little. ... The chapters that dealt with nineteenth- and twentieth century mathematics underwent more substantial revision. These changes not only make the book more readable but also expand on recent developments in mathematics. ... I highly recommend this new edition as a reference book and as a textbook. Even though the problems are gone, I will probably adopt it the next time I teach our history-of-mathematics course.

**6.2. Review by: Anon.**

*Biometrics***49**(2) (1993), 674.The first edition of this book was published in 1968 as an American college textbook; it differed from its nearest rivals in its strict chronological presentation and in its emphasis on historical detail as well as mathematical accuracy. It has long been a standard work. Since 1968 there have been much research, undergraduate teaching, and publication on the history of mathematics. Merzbach's revision of Boyer's book accommodates these developments by adding to, rather than reappraising or amending, the original text. ... Boyer wrote in the first edition that the modest scope of his book precluded mention of every productive mathematician. This revision omits many leading twentieth-century mathematicians. It also fails to take due account of many important and rapidly growing twentieth-century areas of mathematical research. For example, less than two pages are devoted to the impact of electronic computers, and there are no entries in the index to statistical theory, control theory, stochastic processes, or catastrophe theory.