A Commentary on the Heiberg Manuscript of Archimedes.

*The Monist***19**(2) (1909), 225-230.If there ever was a case of appropriateness in discovery, the finding of this manuscript in the summer of 1906 was one. In the first place it was appropriate that the discovery should be made in Constantinople, since it was here that the West received its first manuscripts of the other extant works, nine in number, of the great Syracusan. It was furthermore appropriate that the discovery should be made by Professor Heiberg, facilis princeps among all workers in the field of editing the classics of Greek mathematics, and an indefatigable searcher of the libraries of Europe for manuscripts to aid him in perfecting his labours. And finally it was most appropriate that this work should appear at a time when the affiliation of pure and applied mathematics is becoming so generally recognized all over the world.

- (with Clara C Eaton)
Rithmomachia, the Great Medieval Number Game.

*Amer. Math. Monthly**18*(4) (1911), 73-80.When the subject of number games shall be adequately treated, and the long and interesting story is told of how the world has learned to handle the smaller numbers quite as much through play as through commerce, the climax will probably be found in the chapter relating to the Battle of Numbers, the Rithmomachia of the Middle Ages. For here was a tournament worthy of intellectual foes, a play that outranked chess as much as chess surpasses mere dicing, and a game that was by its very nature closed to all save selected minds that had been trained in the Boethian arithmetic, the Latinized Nicomachus, the last great effort in the Pythagorean philosophy of numbers.

The Geometry of the Hindus.

*Isis***1**(2) (1913), 197-204.If we consider the excellent summary of Hindu geometry made long ago by M Chasles (in 1875), and the various scholarly essays upon the subject that have of late appeared, it may seem an unnecessary labour if not indeed a presumption to attempt to do more with our present fund of knowledge. Nevertheless there are two reasons for assuming to write upon a topic that has been so well considered in the past. In the first place, we have very recently come into possession of a quantity of new material through the completion of the translation of the Ganita-Sara-Sangraha of Mahaviracarya, by Professor Rangacarya. In the second place, there is now an opportunity for a comparison of results that was wanting until the publication of Mahavir's work. Therefore, while not pretending to set forth any great discovery, a writer may properly feel justified in seeking to present some of the salient features of the Hindu geometry more clearly than has heretofore been done.

The Journal "Isis".

*Science, New Series***41**(1047) (1915),132-133.I beg to call your attention to one of the incidents of the war which is likely to be overlooked in the midst of all the excitement of daily battles and the destruction of life and property. I refer to the devotion to scholarship, to duty, and to educational ideals shown by Dr G Sarton, of Wondelgem-lez-Gand, editor of 'Isis', in continuing the publication of this important journal in spite of the invasion of his city and country, and under circumstances that must be most trying. 'Isis' was founded in 1913, its purpose being to consider the historical development of all the various human disciplines, a field not covered by any other publication.

Notes on De Morgan's Budget of Paradoxes.

*The Monist***27**(3) (1917), 474-477.In a work requiring the large amount of reading involved in editing a book like the 'Budget of Paradoxes', and particularly in the condensing of the results to the proper proportions for footnotes to aid the reader, it was, of course, inevitable that a certain number of inaccuracies would occur. It is also evident that many more notes might profitably have been added to elucidate the meaning of the text, or to correct the original where this would be warranted. De Morgan was a careless writer and many of his errors are mentioned in the footnotes; but numerous others exist, some of which are patent to any reader and others of which might profitably have been set forth by the editor.

On the Origin of Certain Typical Problems.

*Amer. Math. Monthly***24**(2) (1917), 64-71.One thing which impresses the student of mathematical problems is that several which he would naturally classify as purely fictitious and of the nature of pleasing puzzles apparently had their origin in genuine applications of mathematics to questions of real life. Of these I shall mention only four, although the list could be greatly extended.

- (with Jekuthial Ginsburg)
Rabbi Ben Ezra and the Hindu-Arabic Problem.

*Amer. Math. Monthly***25**(3) (1918), 99-108.When Browning put in the mouth of Rabbi ben Ezra the words "The Future I may face now I have proved the Past," he wrote better than he knew, for no scholar of the twelfth century had proved the Past more thoroughly than he, and few could face the Future with greater confidence.

The First Work on Mathematics Printed in the New World.

*Amer. Math. Monthly***28**(1) (1921), 10-15.If the student of the history of education were asked to name the earliest work on mathematics published by an American press, he might, after a little investigation, mention the anonymous arithmetic that was printed in Boston in the year 1729. It is now known that this was the work of Isaac Greenwood who held for some years the chair of mathematics in what was then Harvard College. If he should search the records still further back, he might come upon the American reprint of Hodder's well-known English arithmetic, the first textbook on the subject, so far as known, to appear in our language on this side of the Atlantic.

Two Mathematical Shrines of Paris.

*Amer. Math. Monthly***28**(2) (1921), 62-63.In a city like Paris, that has for centuries been one of the intellectual foci of the world, there are many spots where the devotee to mathematics may well stand with bared head. The houses in which great savants were born, or lived, or died are often known or can be ascertained by the searcher after historical spots, and their tombs may be found in Père la Chaise or in the few churches that escaped in the period of vandalism which swept away so many Gothic temples in the making of modern Paris.

An Interesting Fourteenth Century Table.

*Amer. Math. Monthly***29**(2) (1922), 62-63.There has recently come into the possession of the library of Columbia University an interesting mathematical roll written apparently in the south of England about the close of the fourteenth century. ... The roll consists of a table showing the widths corresponding to various lengths of a rectangular piece of land containing an acre.

Historical-Mathematical Paris.

*Amer. Math. Monthly***30**(3) (1923), 107-113.The World War has naturally turned the steps of many of our advanced students to the paths their intellectual ancestors trod soon after the American and French revolutions, namely, to Paris. There will still be large numbers who go to Germany and England, and many who go to Italy, but for some years to come it is probable that Paris will attract American students more than it ever has in the past and more than any other single city of Europe. For these students, and for the more casual visitor of mathematical tastes as well, this article has been prepared in the hope that new interest may be added to their sojourn in what is, all things considered, the most attractive city of the world. Having spent much time there during repeated visits spread over a period of more than forty years, I have naturally come to know a considerable number of the places of mathematical interest, and my collection of autograph letters of those who have made the science what it has become in the last three or four centuries has supplied considerable information as to where the various writers and their correspondents lived and laboured and died.

Historical-Mathematical Paris.

*Amer. Math. Monthly***30**(4) (1923), 166-174.Few of the older mathematicians had abiding places on the right bank of the Seine, most of them, as already seen, preferring the Quartier Latin. Guillaume Budé, the prime mover in the founding of the College de France, however, died at No. 203 of Rue Saint-Martin, as an inscription shows. This is a long street running parallel to the Boulevard de Strasbourg from near the Conservatoire des Arts et Metiers to the Seine. Budé was one of the leading scholars of his day and wrote a work

*De asse et partibus ejus libri V*(1516) which was very well known. Even the Louvre, which we ordinarily look upon as the ancient abode of royalty and the modern home of the graphic and plastic arts, has a certain connection with mathematics. In the years immediately preceding the Revolution it was the seat of the academies and the lodging place of certain savants.The First Printed Arithmetic (Treviso, 1478).

*Isis***6**(3) (1924), 311-331.Printing was introduced into Italy in 1464 by Juan Turrecremata, abbot of the monastery of Subiaco, near Rome. The first book from an Italian press appeared in the following year, and thirteen years later the first printed arithmetic was published at Treviso, then a day's journey north of Venice. At present the traveller going from Venice to Feltre, where Vittorino was born, or on to Belluno to begin his automobile journey to the Val d'Ampezzo, sees the five tall towers of Treviso an hour after taking his train. ... The various arithmetics of the earlier days make mention of it in connection with Padua, Venice, and other centres of trade, so that it is natural to expect that any textbook published there would be purely mercantile in character. That this arithmetic is of that nature will be seen from the translation of certain of the most interesting and characteristic portions ...

In the Surnamed Chosen Chest.

*Amer. Math. Monthly***32**(6) (1925), 287-294.This cryptic title, from "A Death in the Desert," shows that Browning was a genuine bibliophile, one who loved to speak of the ancient "parchment, of my rolls the fifth," which "lies second in the surnamed Chosen Chest." The title is manifestly selected for this paper merely for the purpose of arousing at least a slight degree of curiosity, of securing the attention of those who love old books, and of tempting some friend, or friends, to taste the pleasures of knowing of other chosen chests than those which he, or they, may guard with jealous care.

The Surnamed Chosen Chest.

*Amer. Math. Monthly***32**(8) (1925), 393-397.In the "Surnamed Chosen Chest" there are many oriental manuscripts and early printed books, - some hundreds in all. It is proposed in this article to mention a few of the most interesting manuscripts of the mathematical classics of the East. There are not many copies of these works in this country, and it is probable that a brief list of some of the more important ones will be of service to scholars, not merely in the history of mathematics but in the field of oriental languages as well.

In the Surnamed Chosen Chest.

*Amer. Math. Monthly***32**(9) (1925), 444-450.It might naturally be supposed that few mathematicians would ever have been honoured by portrait medals struck in recognition of their services to the world. The issuing of such medals was common in the case of kings and of those who fought their battles, and the church has recognized her prelates and saints in the same manner. In recent years the stage and the societies of the fine arts in general have followed the older custom, and certain scientists of great repute have been similarly honoured. These tokens of esteem have usually, however, been bestowed upon men who have had some influence with those in power, either the power of office or that of wealth. Only in the nineteenth century did governments, through learned societies, make a real beginning in honouring the deceased scholars of their countries, and in this way many of the medals of mathematical interest came to be struck. In this collection there are between 125 and 150 medals of this nature, not counting ancient coins selected to represent the development of the numerals, and counters (jetons) that were used in computation in medieval times.

The Poetry of Mathematics .

*The Mathematics Teacher***19**(5) (1926), 291-296.Weierstrass, than whom few men of his calling were better able to speak with authority, once remarked that "a mathematician who is not somewhat of a poet will never be a perfect mathematician"; and Thoreau, who is not often suspected of possessing the mathematical tastes that really were his, went even farther when he wrote: "We have heard much about the poetry of mathematics, but very little of it has yet been sung. The ancients had a juster notion of their poetic value than we. The most distinct and beautiful statements ... Just as mathematics, to the mathematician and to one who teaches the science, is filled with poetry, so poetry welcomes mathematics to herself, arranging her message in meter and her sonnets with mathematical precision. In like manner she reveals, as the mathematician does, the beauties of symmetry, and she commonly designates her rhythmic lines as "numbers" ...

The First Great Commercial Arithmetic.

*Isis***8**(1) (1926), 41-49.In speaking of the first printed arithmetic (Treviso, 1478), in an article which recently appeared in Isis (vol. VI, pages 311-331), the present writer mentioned the fact that in the decade 1480 to 1490 there were at least sixty-three mathematical works printed in Italy. Among these was the 'Nobel opera de arithmetica', as the name appears in the first edition, or the 'Libro dabacho' (or 'Libro de Abacho') as it appears later. It is now proposed to set forth some of the details relating to this work which, although the second commercial arithmetic in point of time, may properly be called the first one of considerable importance and influence in this field.

Was Paul Guldin a Plagiarist?

*Science, New Series***64**(1652) (1926), 204-206.[G A Miller's] criticism, expressed in some eight hundred words, may be set forth more clearly as follows: "Guldin's Law was original with him because Tropfke says so." The added statement that "other well-known mathematical historians have recently expressed the same view" illustrates Professor Miller's habit of making general assertions without adducing any proof. It is an unfortunate habit and is probably one reason why his book reviews command so little attention. Doubtless another reason lies in the fact that he always seeks to show that every book is bad, that no author is a scholar, that no one is a master of style and that no historian goes to the sources. He illustrates the type of those who, as a French critic has expressed it, failing to succeed themselves, rail at those who "arrive." ... Speaking of Professor Miller's reviews in general, the present writer does not believe that his method and purpose of book reviewing are worthy. He feels that a reviewer should set forth clearly and succinctly a statement of the purpose of a book, of the way in which this purpose has been developed, of the scholarship shown by the writer, of the style in which the work is written and of the kind of help that is given and the kind of readers who will find it of value. On the other hand, he should point out evidences of poor scholarship, of poor arrangement of material and of poor style, provided these are significant.

The Twentieth Anniversary of "Scientia".

*Amer. Math. Monthly***34**(6) (1927), 317-318.It seems appropriate that

*Amer. Math. Monthly*should call attention at this time to the fact that twenty years have elapsed since the first appearance of the international publication 'Scientia', and to extend to its sponsors congratulations upon the success which it has achieved. Founded in 1907 by Signori Bruni, Dionisi, Enriques, Giardina, and Rignano, it was published under their direction until 1925, since which time it has been under the editorship of Dr Rignano alone- (with Salih Mourad)
The Dust Numerals Among the Ancient Arabs.

*Amer. Math. Monthly***34**(5) (1927), 258-260.Among the books composed by Eastern Moslem scholars in the tenth century, several have special interest with respect to elementary computation and the gobar (ghobar) numerals. One was written by 'Ali ibn Ahmed, Abu'l-Qasim, al-Antaki (that is, from Antioch), al-Mujtaba (the chosen), who lived at Baghdad and died in 987. It was called Kitab al-takht al-kabir fi'l kisab al-Hindi ("The great book of the board on Hindu arithmetic").

A Source Book in the History of Mathematics.

*Amer. Math. Monthly***35**(6) (1928), 280-282.Professor Gregory D Walcott of Hamline University, St Paul, Minnesota, has for some time been planning a series of source books in the history of the sciences, covering the period 1500-1900. To carry out this project the American Philosophical Society has recently secured from the Carnegie Foundation a grant sufficiently large to permit of a beginning in the publication of the series. It is proposed to use this sum to underwrite the publication of the first one or two volumes, and to use the income from the sales to allow for the publication of others. It is expected that the second volume will relate to mathematics. It will be edited by a committee consisting of Professors R Archibald, Florian Cajori, and David Eugene Smith (chairman). The general tentative plan for this volume is to publish articles of about 6-20 pages each, giving

(1) The title, author and source of the material selected.

(2) The name and position of the editor of the article.

(3) A brief statement, in a few lines, of the reason why the material has been considered important enough to have place in the publication.

(4) The original text, transcribed and checked with great care.

(5) A translation into English, in case the original is in a foreign language.

(6) A brief explanation, by notes or commentary, of any obscure statements needing elucidation. This feature must necessarily be very limited, the space at the Committee's disposal being devoted principally to the source material itself.Unsettled Questions Concerning the Mathematics of China.

*The Scientific Monthly***33**(3) (1931), 244-250.... we need the help of expert sinologues upon the following points:

(1) The authenticity of ancient texts, determined by a new and scientific study of the most (apparently) reliable manuscripts and block books extant.

(2) The approximate dates of these texts and of any interpolations and commentaries, determined in the same way.

(3) The evidence of foreign influence in such matters as the calendar, problems, and methods of treatment of equations, including those of the indeterminate class and the approach to determinants.

In all this work the historian of western mathematics can assist in stating probabilities as to dates, these being based upon knowledge of European, Indian, or Arabian achievements; but the other questions involved are for the experts in the study of ancient Chinese texts.Note on a Greek Papyrus in Vienna.

*Amer. Math. Monthly***39**(7) (1932), 425.There has recently been published by the Nationalbibliothek at Vienna, in its 'Mitteilungen aus der Papyrussammlung', "Eine stereometrische Aufgabensammlung im Papyrus Graeccus Vindobonensis 19996," an event of considerable interest in the history of mathematics. ... The manuscript consists of a number of fragments of a papyrus roll containing thirty-eight problems relating to the metrical side of solid geometry, twenty-three being accompanied by drawings. Although the manuscript is not unique as to content, since it is quite like two others that have been described, one in the Field Museum in Chicago and the other in Berlin, it loses nothing of its value because of this fact.

The Ganesh Prasad Prize.

*Science, New Series***81**(2107) (1935), 487.The great interest shown by Hindu scholars during the last few years in the history of mathematics in India is well known. During nearly a century the subject had been so neglected ... In view of the present activities shown by Hindu scholars it is interesting to know that the Calcutta Mathematical Society has recently announced for the subject of the competition for the "Krishna Kumari-Ganesh Prasad Prize and Medal" the following: "Lives and works of the ten Famous Hindu Mathematicians: Aryabhatta, Varamihir, Bhaskara I, Lalla, Bramhagupta, Sridhar, Mahavir, Sripati, Bhaskara II, Naryana." ... In view of the excellence of various recent articles by Hindu scholars, it may be expected that the winner in the contest will offer to the English-speaking world a work of outstanding importance.

Possible Boundaries in the Early History of Mathematics.

*Amer. Math. Monthly***45**(8) (1938), 511-515.Having been invited by the editors of this Memorial Volume to contribute an article upon some appropriate topic in which Professor Slaught would be interested if he were with us at this time, a brief historical topic has been selected. Having known Dr Slaught for many years, often discussing with him the teaching of mathematics and the types of literature which were best adapted to our work, we were naturally led to considering at various times the story of the origin and development of the subject of our major interest. I well recall that in one of our visits we dwelt upon the possibility of the opening of new regions in the early history of the subject, necessarily in the fields of numbers, and then of geometry, and finally of some crude form of algebra. We were both familiar with the current literature of the subject, but we visualized a much older era to be revealed by earlier material which scholars in the field of archaeology might discover. He was greatly interested in the work being done at that time in his own university circle, particularly in the excavations then being carried on in Iraq. It was natural, therefore, that we should be led to discuss the possibility of discoveries in other fields of archaeology and even of medieval documents.

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