*Rara arithmetica*(1908),

*History of mathematics*(2 Vols.) (1923, 1925) and

*A source book in mathematics*(1929). The second and third of these books were reprinted around 30 years after the original publication and, since these are reprints and not new additions, we simply include the reviews of these reprints after the reviews of the originals. The "source book" involved contributions from many historians who translated the extracts but Smith was the editor-in-chief and headed the small team choosing which extracts to include. The Preface which we quote from below was written by Smith.

Rara arithmetica: a catalogue of the arithmetics written before the year MDCI with a description of those in the library of George Arthur Plimpton of New York 1.1.

**From the Preface.**One of the first and most important questions for the student of mathematical history is that relating to the available sources of information. In the fields of higher mathematics scholars have been more or less successful in bringing together these sources, and in listing them in bibliographies; but in that humbler field in which primitive mathematics first found root, only a few bibliophiles have sought to preserve the original material, and no one has seriously attempted to catalogue it. Libri, it is true, brought together two large libraries of Rara Arithmetica, but he was neither a true book-lover nor a true scholar, for he gathered his treasures purposely to see them dispersed, his commercial spirit scattering at random what should have been kept intact for the use of scholars. Prince Boncompagni, the most learned of all collectors in this domain, lived to see an unappreciative city ignore his offer to make his magnificent library permanent, and at his death it was scattered abroad, as had been the lesser ones of Kloss and De Morgan. The third great collection of early textbooks which has been made in recent years is that of Mr Plimpton. Of the libraries of arithmetics printed before the opening of the seventeenth century his is the largest that has ever been brought together, not excepting Boncompagni's, and it may well be doubted if another so large will again be collected by one man. De Morgan was able to examine, in the British Museum and elsewhere, less than a hundred arithmetics written before 1601, including all editions; but Mr Plimpton has more than three hundred, a number somewhat in excess of that reached by Boncompagni. Indeed there are few arithmetics of much importance that are not found, in one edition or another, in his library. The writers of these early printed books, not all themselves of the centuries under consideration, were by no means obscure men. Among them was Boethius, whom Gibbon called "the last of the Romans whom Cato or Tully could have acknowledged for their countryman." In the list are the names of Cassiodorus and Capella, who at least represented what there was of culture in their day, and Isidorus, the learned Bishop of Seville. There are also the names of Archimedes, who deemed it a worthy labour to improve the number system of the Greeks; of Euclid, whose contributions were by no means confined to geometry; of Nicomachus and lamblichus, who represent the declining Hellenic civilization, and of Psellus, who was a witness of its final decay. There, too, are the names of the Venerable Bede, of Sacrobosco, and of Bradwardine, all of whom testify to the culture of mediaeval England; of that great Renaissance compiler, Paciuolo; of Tartaglia and Cardan, who helped to make the modern algebra, and of such scholars as Ramus, Melanchthon, and Bishop Tonstall. Worthy as such a list may be, it is rendered none the less so by the names of Widman, Kobel, Borghi, Riese, and Gemma Frisius, mere arithmeticians though they were, for few who read their works can fail to recognize that they powerfully influenced education, not only in their own time but for generations after they had passed away.

1.2.

**Review by: Anon.**

*The School Review***19**(4) (1911), 281.The subtitle states that this book is a catalogue of the arithmetics written before the year 1600 with a description of those in the library of George A Plimpton of New York. This collection of early arithmetics, more than three hundred in number, is the largest that has been brought together; and there are few important arithmetics published before the year 1600 that are not found in it.

1.3.

**Review by: Florian Cajori.**

*Science, New Series***32**(812) (1910), 114-115.As a rule, bibliographical works, though valuable, are uninteresting. The publication which we are reviewing is an exception to the general rule; it is interesting as well as valuable. Every college and school library ought to possess a copy of it. The author aims not only to catalogue the arithmetics in Mr Plimpton's library that were published before 1601 and give a brief statement of their contents, but to supplement this by the titles of other arithmetics known to have been printed during that period. Altogether not less than 500 publications are given, a number which swells to 1,200, if the various editions of each publication are counted. In addition to this a large number of manuscripts, some belonging to the thirteenth century, are catalogued and described.

1.4.

**Review by: Anon.**

*The Journal of Education***71**(20) (1780) (1910), 553.There is no other book to compare with this, and there will be no other. This is a startling statement, but that it is justified will appear when it is understood that in all the world there is no other such collection of arithmetics as this of George Arthur Plimpton, and that there is today no one who could make such admirable use of the material as can David Eugene Smith, who has done this work so well that there is not likely to be occasion for anyone to write upon this world renowned collection, the third ever attempted, the first by an English or American scholar, the largest of rare arithmetics before the seventeenth century ever gathered, and the only one without a commercial spirit and by one with adequate scholarship.

1.5.

**Review by: Lambert L Jackson.**

*Bull. Amer. Math. Soc.***16**(1910), 312-314The title page of this splendid volume modestly states that "the work is a catalogue of the arithmetics written before the year 1601 with a description of those in the library of George Arthur Plimpton of New York." Another appropriate title might be, A brief history, on the bibliographical plan, of the genesis and content of sixteenth century arithmetic. As a bibliography this work is more extensive than any of its predecessors, and is nearly complete for the formative period between 1472 and 1601. There are mentioned over five hundred and fifty different works, which number swells nearly to twelve hundred by the inclusion of the various editions. About four hundred and fifty of the different books are genuine arithmetics, while the others deal partially with algebra, astrology, or the calendar. ... this work is more than a scholarly, well edited digest of all the earlier bibliographies ; it is the result of the examination of the original works in so far as they are extant and are to be found in the libraries of Europe or America. In particular, it is an extensively illustrated catalogue of the arithmetical collection of Mr Plimpton. This collection is already well known to many scholars, and it will now become known to other students of mathematics wherever Professor Smith's work may circulate. ... Professor Smith's work is more than a catalogue, it is a condensed history of arithmetic during its formative period. The numerous notes not only point out the significant features of the specimens under discussion, but they form a comparative study of the subject from many points of view, for example, the comparison of the arithmetic of the Latin schools with that of the trade schools, the comparison of the arithmetics of different nations, and the relation of abacus reckoning to figure reckoning.

(1923).History of mathematics. Volume I. General survey of the history of elementary mathematics 2.1.

**From the Preface.**This work has been written for the purpose of supplying teachers and students with a usable text book on the history of elementary mathematics, that is, of mathematics through the first steps in the calculus. ... The general plan adopted in the preparation of this work is that of presenting the subject from two distinct standpoints, the first, as in Volume I, leading to a survey of the growth of mathematics by chronological periods, with due consideration to racial achievements; and the second, as in Volume II, leading to a discussion of the evolution of certain important topics. ... A general historical presentation is desirable for the purpose of relating the development of mathematics to the development of the race, of revealing the science as a great stream rather than a static mass, and of emphasizing the human element, but ... this ought to lead to a topical presentation by which the student may understand something of the life history of the special subject which he may be studying.

2.2.

**Review by: L E Dickson.**

*Amer. Math. Monthly***32**(10) (1925), 511-512.The excellence of this history of elementary mathematics is due primarily to the many years which Professor Smith has devoted to a first hand study of original sources. He had long collected early and rare books not only for himself, but more extensively for Mr George A Plimpton, also of New York, whose remarkable library contains the most important and extensive collection of early arithmetics in the world, besides many rare books and manuscripts on other fields of elementary mathematics. The constant enthusiastic study of these and other sources during so many years has made Smith an outstanding authority on the history of elementary mathematics. Fortunately he writes in a very clear and pleasing style.

2.3.

**Review by: Anon.**

*The Mathematical Gazette***12**(170) (1924), 126-127.In this preliminary volume, Professor D E Smith, in carrying out his survey by periods, has kept the geographical and historical backgrounds well in view, and has tried to assign to various races of mankind their contributions to the general advance. We had recently come to the conclusion that a work written from this standpoint was to be desired for the use of students and others who, without intending to be professed mathematicians, have sufficient interest in the science as a branch of culture to wish to have it correlated with other branches. We had not, however, ventured to hope for the appearance of such a volume so soon or written by such an expert as Professor Smith, qualified alike by his experience in teaching the subject and by extensive first-hand acquaintance with original sources. We feel sure that it will be welcomed by many readers of the 'Gazette', in whose columns the claims of history on the attention of the teachers of mathematics have been repeatedly urged. It will be further recommended to the class of readers for whom it is chiefly intended by the profusion with which it is illustrated by facsimiles of portions of old books and manuscripts, autograph letters by celebrities, reproductions of portraits, etc., drawn from the treasures in Mr Plimpton's extensive library, and in the collection made by the author in his travels.

2.4.

**Review by: Carl B Boyer.**(of the 1958 reprint)

*Isis***50**(3) (1959), 268-269.Research in the history of mathematics, especially for the pre-Hellenic and medieval periods, has advanced considerably since Smith's two volumes first appeared in 1923-1925; and yet in a sense they have not been outmoded. No one of the subsequent works in the field has had quite the same purpose and appeal, for David Eugene Smith, out of his wide and rich experience, wrote not so much for the technical mathematician or historian of his time as for the never ending circle of readers for whom mathematics and history provide an avocation. The "General Survey," found in volume I, fails to instil historical-mindedness, for it becomes, for the modern period, a catalogue of mathematicians grouped by nationalities; and the "Topical Survey," comprising Volume II, does not go much beyond the elementary aspects, despite the rudimentary treatments of analytic geometry and the calculus. And yet there is throughout the work a soundness of judgment and an aura of enthusiasm which make the latter day unaltered and unabridged republication quite appropriate. The wealth of accurate biographical and bibliographical citations in the footnotes are alone worth far more than the price of the work.

2.5.

**Review by: R Taton.**(of the 1958 reprint)

*Revue d'histoire des sciences et de leurs applications***12**(1) (1959), 77.All those from near and far who are interested in the history of mathematics will appreciate the qualities of seriousness, of precision and of clarity of the two volumes of 'History of Mathematics' by the eminent American historian of mathematics David Eugene Smith (1860-1944). The first volume deals, as we know, with the history of mathematics in general, presented chronologically, while the second contains the historical study of the main areas of elementary mathematics.

2.6.

**Review by: Asger Aaboe.**(of the 1958 reprint)

*Publications of the Astronomical Society of the Pacific***71**(420) (1959), 249-250.This two-volume work appeared originally in 1923-25, the first volume being called "General Survey of the History of Mathematics" and the second, "Special Topics of Elementary Mathematics." The first, in my estimation, is the better of the two volumes. It is, however, a history of mathematicians rather than of mathematics, being mainly devoted to brief biographical sketches arranged essentially in chronological order. There are many amusing anecdotes to be found, but unfortunately little mathematical information for the general reader.

(1925).History of mathematics. Volume II. Special topics of elementary mathematics 3.1.

**Review by: George Sarton.**

*Isis***8**(1) (1926), 221-225.... good as the first volume was, the second is better still. The first volume contained many details which were rather irritating to the historian and especially to the medievalist. The second volume, being divided by topics, contains far more mathematical information than the first and will thus prove more interesting to the reader who is, first of all a mathematician and, only secondarily, a historian. The reading of the second volume has completed my conversion to the author's method of teaching the history of mathematics. I call it the authors' method for, as far as I know, he has followed it consistently ever since he began teaching the history of mathematics at Teachers College, Columbia University, New York, that is, a good many years ago. ... Though I still prefer in a general way that the history of science be told as strictly as possible in chronological order, I admit that the division by topics is far superior from the pedagogical point of view. Now Smith's history is meant primarily for teaching purposes. It offers to the students and especially to the prospective teachers of mathematics a well-balanced diet neither too light nor too heavy - and by means of the learned footnotes, it gives them at the same time abundant opportunities to obtain whatever additional nourishment they may desire. The main advantage of the separate historical treatment of each topic is that it obliges the author to be at once far more concrete and specific. This would become the more obvious that the topics themselves would be more special. For example, if one starts telling the history of trigonometric tables, one is almost driven to explain how those tables were computed and by what technical means they were gradually improved. That is, the history itself becomes more technical and thus more instructive from the technician's point of view.

3.2.

**Review by: Anon.**

*The Elementary School Journal***26**(9) (1926), 716.The second volume of a series of books by David Eugene Smith is an outstanding contribution to the history of mathematics. ... [It] is concerned with special topics of elementary mathematics. While the second volume is primarily for mathematicians rather than for students of education or elementary school teachers, there are certain parts of the book which are of genuine interest and value to the latter groups.

3.3.

**Review by: E M Langley.**

*The Mathematical Gazette***12**(177) (1925), 446-447.We congratulate Dr Smith on the completion of his onerous labours of research and selection, and the teaching world on the valuable addition made to its resources by the publication of the results. He has certainly succeeded well in his object - that of "supplying teachers and students with a usable text-book on the history of elementary mathematics," the term "elementary" expressly including the first steps of the calculus, and actually extended to determinants, geodesy, and non-euclidean geometry.

3.4.

**Review by: Vera Sanford.**

*The Mathematics Teacher***18**(5) (1925), 305-308.Of the general histories of mathematics in English, none fits the requirements of teacher and student so adequately as does this new publication by Dr Smith. Three factors obviously contribute to this: greater length and therefore greater detail than we find in either Ball or Cajori, a dual method of presentation, and the great number of illustrations. None of these, however, suggests the most important thing about the two volumes. They could never have been written except by one who knows and loves books, one who has travelled extensively and observantly, one whose interests are not narrow. Even the casual reader will be impressed by the scope of this work, and the earnest inquirer will appreciate its uniqueness. These volumes are at once more scholarly and more useful than the other general histories of mathematics.

(1929).A source book in mathematics 4.1.

**From the Preface.**The purpose of a source book is to supply teachers and students with a selection of excerpts from the works of the makers of the subject considered. The purpose of supplying such excerpts is to stimulate the study of the various branches of this subject - the present case, the subject of mathematics. By knowing the beginnings of these branches, the reader is encouraged to follow the growth of the science, to see how it has developed, to appreciate more clearly its present status, and thus to see its future possibilities. It need hardly be said that the preparation of a source book has many difficulties. In this particular case, one of these lies in the fact that the general plan allows for no sources before the advent of printing or after the close of the nineteenth century. On the one hand, this eliminates most of mathematics before the invention of the calculus and modern geometry; while on the other hand, it excludes all recent activities in this field. The latter fact is not of great consequence for the large majority of readers, but the former is more serious for all who seek the sources of elementary mathematics. It is to be hoped that the success of the series will permit of a volume devoted to this important phase of the development of the science. In the selection of material in the four and a half centuries closing with the year 1900, it is desirable to touch upon a wide range of interests. In no other way can any source book be made to meet the needs, the interests, and the tastes of a wide range of readers. To make selections from the field, however, is to neglect many more sources than can possibly be selected. It would be an easy thing for anyone to name a hundred excerpts that he would wish to see, and to eliminate selections in which he has no special interest. Some may naturally seek for more light on our symbols, but Professor Cajori's recent work furnishes this with a satisfactory approach to completeness. Others may wish for a worthy treatment of algebraic equations, but Matthiessen's 'Grundzüge' contains such a wealth of material as to render the undertaking unnecessary. The extensive field of number theory will appeal to many readers, but the monumental work of Professor Dickson, while not a source book in the ordinary sense of the term, satisfies most of the needs in this respect. Consideration must always be given to the demands of readers, and naturally these demands change as the literature of the history of mathematics becomes more extensive. Furthermore, the possibility of finding source material that is stated succinctly enough for purposes of quotation has to be considered, and also that of finding material that is not so ultra-technical as to serve no useful purpose for any considerable number of readers. Such are a few of the many difficulties which will naturally occur to everyone and which will explain some of the reasons which compel all source books to be matters of legitimate compromise.

4.2.

**Review by: George Sarton.**

*Isis***14**(1) (1930), 268-270.The extracts are preceded by short bio-bibliographical introductions which are generally well done, but very unequal. This inequality is obviously due to the diversity of collaborators; the editor might have been a little more vigilant in this respect. Reading a few of these biographies, at random, I cannot help grumbling a little. ... The fact is that this source book will be of little value except to mathematicians, whereas one giving some space to the earlier history would have appealed to a much larger audience. After all for which kind of readers are there source books compiled? I would have assumed they were compiled for the elite of the educated public, but not for the very experts. For example, a mathematician curious of the past reads a history of mathematics and then tries to obtain the old books which interest him most. He will hardly depend on extracts for the chances are that these do not contain the very things he is looking for. On the other hand the philosopher, who is not a mathematician, will not be able to do much with this volume, while one devoted to the earlier period would have proved very stimulating.

4.3.

**Review by: L P Copeland.**

*Amer. Math. Monthly***37**(6) (1930), 310-312.The material in the "Source Book in Mathematics" covers such an extensive range that the book will be of use to teachers and students in preparatory schools as well as in colleges. The selections illustrate in a most illuminating way the theory and the processes that lie at the foundation of many of our most important mathematical concepts. ... A volume as attractive as this in form and in content is certainly destined to arouse interest in mathematics and its history and hence deserves an honoured place in every mathematical library.

4.4.

**Review by: Anon.**

*The Monist***41**(2) (1931), 313-315.The purpose of the volume, as stated by the compiler, is to supply teachers and students with selected excerpts from the works of creative mathematicians, and thus to stimulate the study of various branches of the general subject. By knowing the beginnings of these branches, the reader is encouraged to follow the growth of the science, to appreciate more fully its present status, and thus to see its future possibilities. The author has brought together ninety-six excerpts, each translated or edited by an expert. ... The selection of material for such a work is a very difficult task. In this task, the compiler had the assistance of an advisory committee, consisting of Professors Archibald, Cajori, and Dickson. Professor Smith, and all others who have in any way been connected with the publication of this book, are deserving of high compliments on the result.

4.5.

**Review by: Robert H Oehmke.**(of the 1959 reprint)

*Journal of the American Statistical Association***55**(292) (1960), 773-774.It was the intention of the pro-gram that led to this work "to present the most significant passages from the works of the most important contributors" to mathematics during the last three or four centuries. The natural choice for this task, at the time, was one of the outstanding men in the activity of the history of mathematics, David E Smith. ... the work still is a most valuable aid to the student of mathematics or the history of science and such a readable collection of articles deserves a place in every library.

4.6.

**Review by: J M Hammersley.**(of the 1959 reprint)

*Journal of the Royal Statistical Society. Series A (General)***123**(3) (1960), 340-341.A distinguished mathematician once said: "Mathematics is different from other fields of study in that at the end of a mathematical proof you really can write Q.E.D." This interesting source book teaches, amongst other valuable lessons, that there is no such thing as mathematical proof in the above sense. Proof, in short, is our name for an argument that conforms to the conventions of the moment, or which is enough to satisfy a contemporary referee or a contemporary examiner, or is the fallacious outcome of muddled thought.

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