## David Eugene Smith - More History Books

We have looked at some extracts of reviews and the Preface of Louis Charles Karpinski and David Eugene Smith's

*The Hindu-Arabic numerals*(1911) on a separate page. Again on another page we have given some extracts of Prefaces and reviews of Smith's*Rara arithmetica*(1908),*History of mathematics*(2 Vols.) (1923, 1925) and*A source book in mathematics*(1929). Here we look at extracts from reviews of further historical works by D E Smith. Although all these texts are on historical topics, some are at a research level while others are intended for school pupils or their teachers.- (with Yoshio Mikami)
**A history of Japanese mathematics**(1914).

1.1.**From the Preface by D E Smith.**

Although for nearly a century the greatest mathematical classics of India have been known to western scholars, and several of the more important works of the Arabs for even longer, the mathematics of China and Japan has been closed to all European and American students until very recently. Even now we have not a single translation of a Chinese treatise upon the subject, and it is only within the last dozen years that the contributions of the native Japanese school have become known in the West even by name. At the second International Congress of Mathematicians, held at Paris in 1900, Professor Fujisawa of the Imperial University of Tokyo gave a brief address upon 'Mathematics of the old Japanese School', and this may be taken as the first contribution to the history of mathematics made by a native of that country in a European language. The next effort of this kind showed itself in occasional articles by Baron Kikuchi, as in the*Nieuw Archief voor Wiskunde*, some of which were based upon his contributions in Japanese to one of the scientific journals of Tokyo. But the only serious attempt made up to the present time to present a well-ordered history of the subject in a European language is to be found in the very commendable papers by T Hayashi, of the Imperial University at Sendai. The most important of these have appeared in the Nieuw Archief voor Wiskunde, and to them the authors are much indebted. Having made an extensive collection of mathematical manuscripts, early printed works, and early instruments, and having brought together most of the European literature upon the subject and embodied it in a series of lectures for my classes in the history of mathematics, I welcomed the suggestion of Dr Carus that I join with Mr Mikami in the preparation of the present work. Mr Mikami has already made for himself an enviable reputation as an authority upon the wasan or native Japanese mathematics, and his contributions to the Bibliotheca Mathematica have attracted the attention of western scholars. He has also published, as a volume of the Abhandlungen zur Geschichte der Mathematik, a work entitled Mathematical Papers from the Far East. Moreover his labours with the learned T End M, the greatest of the historians of Japanese mathematics, and his consequent familiarity with the classics of his country, eminently fit him for a work of this nature. Our labours have been divided in the manner that the circumstances would suggest. For the European literature, the general planning of the work, and the final writing of the text, the responsibility has naturally fallen to a considerable extent upon me. For the furnishing of the Japanese material, the initial translations, the scholarly search through the excellent library of the Academy of Sciences of Tokyo, where Mr End M is librarian, and the further examination of the large amount of native secondary material, the responsibility has been Mr Mikami's. To his scholarship and indefatigable labours I am indebted for more material than could be used in this work, and whatever praise our efforts may merit should be awarded in large measure to him. The aim in writing this work has been to give a brief survey of the leading features in the development of the wasan. It has not seemed best to enter very fully into the details of demonstration or into the methods of solution employed by the great writers whose works are described. This would not be done in a general history of European mathematics, and there is no reason why it should be done here, save in cases where some peculiar feature is under discussion. Undoubtedly several names of importance have been omitted, and at least a score of names that might properly have had mention have been the subject of correspondence between the authors for the past year. But on the whole it may be said that most of those writers in whose works European scholars are likely to have much interest have been mentioned.

1.2.**Review by: Arnold Reymond.**

*Isis*2 (2) (1919), 410-413.

As the preface of this book suggests, the history of Japanese mathematics is little known. [Previous books] despite their importance, were far from exhausting a matter which by nature is vast and complex. With this book Smith and Mikami publish an original and in large part unpublished work for which we can only congratulate them. The enterprise was not easy because the documents relating to the early history of Japan are few, since most of which have been destroyed by fire. In the presence of this difficulty, Smith and Mikami have been obliged to resort to more or less plausible conjectures; but they do honour to their erudition and scientific probity.

1.3.**Review by: Louis C Karpinski.**

*Science, New Series***40**(1036) (1914), 675-676.

This interesting story of Japanese mathematics is presented in most attractive garb. The paper, the type and the illustrations make of it a work which it is a delight to handle. ... This work should appeal to a wide circle of readers, to the students of the history of science, to all interested in Japanese civilisation and even to the general reader, for much of the work is non-technical. Certainly this book will contribute to a juster and broader appreciation of the Japanese genius.

1.4.**Review by: Louis C Karpinski.**

*Amer. Math. Monthly***21**(5) (1914), 152-153.

This book is a beautiful specimen of the printer's art. The paper, the type, and the illustrations make it a work which it is a delight to read almost aside from the text. I hasten to add that text and content are in harmony with the dress. The sympathetic portrayal of the development of Japanese mathematics, largely indigenous and as the authors well state, "like her art, exquisite rather than grand," will appeal to a wide circle of readers and will contribute to a juster and broader appreciation of the Japanese genius. The history of Japanese mathematics is divided into six periods. The first extends to 552 A. D. and was almost entirely a native development. The second period, from 552 to 1600, was characterized by the predominance of Chinese mathematics. The third period was a kind of Renaissance culminating in the appearance of Seki, the most famous Japanese mathematician. The fourth period, 1675 to 1775, and the fifth, 1775 to 1868, are marked by the complete development of the native mathematics, the wasan, in the second of these two periods somewhat influenced by European mathematics. The sixth is the period of the present, the mathematics of the world which knows nothing of political and racial boundaries.

1.5.**Review by: Anon.**

*The Mathematical Gazette***7**(111) (1914), 339-341.

Until very recently (from 1900 on) the mathematics of China and Japan has been closed to all European and American students. The aim followed by Prof Smith and Mr Mikami has been to give a brief survey of the leading features in the development of the "wasan" or native Japanese mathematics as distinguished from the "yosan" or European mathematics. The European literature, the general planning of the work, and the final writing of the text is chiefly the work of Professor Smith; while Mr Mikami is responsible for furnishing the Japanese material, the initial translations, search through the library of the Academy of Sciences of Tokyo (where T Endo, the greatest of the historians of Japanese mathematics, is librarian), and a further examination of the large amount of native secondary material. ... all will agree that Prof Smith and Mr Mikami have produced a book which is really of absorbing interest, whether we are more interested in the Japanese or in what they discovered and how, often independently of Western influences, they discovered it.

1.6.**Review by: Anon.**

*The Mathematics Teacher***6**(3) (1914), 183-184.

It is only in recent years that Japanese mathematics and mathematicians have attracted any attention in the western world. It is therefore very appropriate that at this time two men well qualified for the task should give us a history of Japanese mathematics. It starts back with the earliest period and comes down to quite recent times and makes very interesting reading. **Number stories of long ago**(1919).

2.1.**Review by: Anon.**

*The Journal of Education***89**(24) (2234) (1919), 672.

Now and here, for the first time, the wonder tale is told of how the young world laboriously learned to count and how, centuries later, it learned to add, to subtract, to multiply, and to divide. The story of numbers is the story of Ching and Chang and Wu from the Land of the Yellow Dragon; of Anam and Lugal from the Tigris and Euphrates; of Menes and Ahmes from the banks of the Nile; of Hippias from Greece and Titus from the Seven Hills of Rome; of Gupta, Mohammed, Gerbert, and the rest. It begins with the story of Ching, a barbaric little figure in leopard skins, who played with his turtles at the foot of Mount Yu. His computations were confined to One and Two. Menes, who lived on the banks of the Nile, could count to four. Then, as the centuries passed, folk learned to count on the fingers of one hand, until the great discovery was made that all ten fingers could be used to still better advantage. And so it is to this very day that most of the world counts by tens. But all this is only the beginning of the story of numbers. ... How the world learned to multiply is the story of Leonard, and Cuthbert, and Johann. ... There is the story of how in the course of no less than twenty centuries our own numbers slowly assumed their present form ...

2.2.**Review by: W F Bushell.**(of the 1948 edition)

*The Mathematical Gazette***33**(305) (1949), 231-232.

This is the first English edition of the well-known book published in America in 1919. There are some slight differences. The cover is more attractive. The print and pages are a little larger, and, although the coloured pictures are omitted, there are a few extra diagrams in the text to take their place. The author quotes Plato: "Do not then train boys to learning by force and harshness; but direct them to it by what amuses their minds." So he provides a story-teller, who is represented as talking to children about the gradual invention of numbers, in accordance with man's needs, up to the Hindu-Arabic numerals of today.**Our Debt to Greece and Rome: Mathematics**(1923).

3.1.**Review by: Louis C Karpinski.**

*Classical Philology***18**(4) (1923), 358-360.

Mathematics is a fundamental development of human intelligence, analogous to literature, the drama, art, and philosophy. To no one nation can be ascribed literature or art or mathematics. What is desired is to properly estimate the contribution of Greece and Rome in the development. In mathematics the successive stages are largely inevitable and must be recognized as discoveries rather than inventions. To the reviewer t he failure of the book under review consists in ascribing to Greek and Roman influence developments and details only remotely connected with classical ideas and further in depreciating the contributions of other people. In the wealth of superfluous details the fundamental contributions of Greece are in truth minimized rather than given their just appreciation.

3.2.**Review by: D'Arcy W Thompson.**

*The Classical Review***38**(7/8) (1924), 207-208.

Greek learning, as scholars know it, has few defenders nowadays, and fewer devotees; but as practical men we are curious to know the precise extent of our classical inheritance, the value in dollars of the wisdom of the Greeks. Professor David Eugene Smith finds it easy to show, in the little book before us, that Greece left us a legacy of Mathematics which has proved worth its weight in gold. Some hundred and fifty small pages are not much to tell the story in; but there are bigger books for those who want them, and those who want a little one may be well content with what Professor Eugene Smith supplies, for his book is excellent of its kind.

3.3.**Review by: George Abram Miller.**

*Science, New Series***58**(1502) (1923), 288-290.

The volume gives a very appreciative popular account of the mathematical contributions by the Greeks and the Romans, and brings out a number of historical facts which are not usually found in a history of mathematics. Hence, it will doubtless be read with profit by many mathematicians as well as by others to whom its popular style and very meagre use of technical mathematics should appeal strongly. Mathematics has been called a Greek science, not only by those who find it difficult but also by those who are in position to understand its nature and who are familiar with the fundamental contributions of the Greeks along this line. It should, however, not be assumed that the Greeks developed the greater part of the mathematics of our times. They merely made a good start along certain important lines.

3.4.**Review by: Milton W Humphreys.**

*The Classical Weekly***18**(2) (1924), 13-16.

It is needless to point out the difficulties that beset one who undertakes to write on the subject treated in this work (one of the series entitled Our Debt to Greece and Rome). ... The author, who is Professor of Mathematics in Teachers College, Columbia University, has done his work as successfully as could be expected of anyone. ... One useful and interesting feature is the explanation of the names given to mathematical operations and geometrical figures. Occasionally other matters of a linguistic nature are discussed. ... It is to be hoped that the pupils in our High Schools will be encouraged to read this excellent book, especially the General Survey and the Conclusion.

3.5.**Review by: S M Barton.**

*The Sewanee Review***32**(2) (1924), 253-254.

The editors of this series were fortunate in securing as the author of this book David Eugene Smith, whose History of Mathematics is the most complete and the most scientific work of its kind in our language. ... To those who know something of the scholarly attainments of Professor Smith and his habits of research, it goes without saying that the work is as accurate as such a work could well be. It is a difficult task to prepare a brief account of any period of history, and Professor Smith is to be congratulated on his success.

3.6.**Review by: Vera Sanford.**

*The Mathematics Teacher***17**(2) (1924), 122-123.

If we were asked to itemize our mathematical debt to Greece and Rome, I fancy we would all write down demonstrative geometry at once, but that then we would hesitate, wondering what other specific things we might add. After reading this little volume, we are in the opposite predicament: is there any thing that we study in mathematics today that does not have its root in the mathematics of one or other of those two countries? ... to those of us who have had the good fortune to be in Dr Smith's classes in the History of Mathematics, this book will bring many recollections of those days. We will supplement it with the obiter dicta that were of necessity crowded from these few pages. For others who have been less fortunate as yet, the book cannot replace these experiences , but it certainly offers more than the proverbial "half a loaf."**The progress of arithmetic in the last quarter of a century**(1923).

4.1.**Review by: G T Buswell.**

*The Elementary School Journal***24**(5) (1924), 394-395.

David Eugene Smith gives an excellent summary of the developments of the last twenty-five years in the subject of arithmetic. The content of the book is organized into three major divisions. The first division traces the development in the aims and subject-matter of arithmetic. By specific reference to topics in typical books of the present time and in those of a generation ago the author vividly portrays the progress made. ... The second division, which is the largest of the three, treats of the progress in the teaching of arithmetic. ... The last division deals with textbooks in arithmetic. ... The general tone of the book is optimistic. Both the textbooks and the teaching of arithmetic show clear progress during the past generation. The book should be widely read by teachers of arithmetic, to whom it is addressed, and by principals of elementary schools.

4.2.**Review by: H A W.**

*Peabody Journal of Education***1**(5) (1924), 292.

Arithmetic grows just like a tree! And it is still growing! It sends out branches of various kinds; some become strong and of great length, others wither. Some bear good fruit, others poor. Movements, wise and foolish, that have made arithmetic what it is today; what the psychologist knows about arithmetic; the big experiments in the teaching of arithmetic - these furnish many pages of interesting discussion in this little volume. Then there is a most practical section on writing and printing a textbook in arithmetic. Read it before starting yours!

4.3.**Review by: E C Hinkle.**

*The Mathematics Teacher***16**(8) (1923), 507-508.

All teachers interested in arithmetic will welcome this little book. It is really an essay of seventy-eight pages, besides the introduction and the publishers' supplement. "It exploits no special method and no narrow curriculum, but it registers the progress of the past quarter of a century and it records the present status of the oldest and most important of all the sciences - the science of elementary arithmetic." The author discusses all the tendencies and movements of the last quarter of a century in arithmetic as sub-topics under the three main captions: progress in purpose and in topics, progress in the teaching art, and the textbook. In his characteristic style he does more than merely "register" and "record;" he evaluates in terms of his own experience and thinking and often in terms of his own prejudices.**The progress of algebra in the last quarter of a century**(1925).

5.1.**Review by: Anon.**

*The Journal of Education***99**(3) (2463) (1924), 79.

It is well for a man like David Eugene Smith to tell the story of the change as he has done in "The Progress of Arithmetic." The best feature of this book is the heroic way in which he ignores the senseless flights of psychology into the arithmetic class. Take this paragraph for instance: - "The customs of business are observed instead of the opinions of a limited number of teachers. It should be very rare for a textbook to find itself out of harmony with commercial usage. This is one reason why it is better in short division to write the quotient below the dividend, in spite of the well-known but futile argument of some teachers to the contrary." "The Progress of Arithmetic" places the emphasis upon*progress*rather than upon*newness*. All newness is not progress and Dr Smith is most specific in classifying genuine advance, warning against needless detour.**Le comput manuel de Magistera Nianus**(1928).

6.1.**Review by: George Sarton.**

*Isis***11**(2) (1928), 385-387.

Dr Smith, to whom we owe already so many studies on mediaeval arithmetic, has increased our debt materially by this very thorough investigation. He chose to write the results of it in French, - and very well, too - and the book is charmingly published.- (with Jekuthiel Ginsburg)
**A history of mathematics in America before 1900**(1934).

7.1.**From the authors summary:**

From 1500 to 1600 the aims and achievements [in mathematics] were hardly commensurate with those of a mediocre elementary school of our time. From 1600 to 1700, they were not even equal to those of our high schools of low grade, but the purpose was more definite than in the preceding century. The objective was now to give to those seeking it enough work in astronomy to predict an eclipse and to find the latitude of a ship at sea, and enough mensuration to undertake the ordinary survey of land. From 1700 to 1800 the general nature of the work in the colleges was that found in the two great universities of England, but it was far from being of the same quality. The courses then began to include algebra, Euclid, trigonometry, calculus, conic sections (generally by the Greek method), astronomy, and "natural philosophy" (physics). The prime objective was still astronomy. ... From 1800 to 1875 America began to show a desire to make some advance in both pure and applied mathematics, independently of European leadership. The union of mathematics, astronomy, and natural philosophy was still strong, and the pursuit of mathematics for its own sake was still somewhat exceptional. From 1875 to 1900, however, a change took place that may well be de scribed as little less than revolutionary. Mathematics tended to become a subject per se; it became "pure" mathematics instead of a minor topic taught with astronomy and physics as its prime objective. American scholars returning from Europe brought with them a taste for abstract mathematics rather than its applications. There were many exceptions. ... Nevertheless, the tendency was strongly toward pure analysis and geometry.

7.2.**Review by: Frederick E Brasch.**

*Isis***22**(2) (1935), 553-556.

Interest in the history of scientific thought in American colonies is notable, particularly in medicine, chemistry, astronomy and mathematics. This small volume before us reminds us of the words of Sir William Dampier, "That local history is of extreme importance towards a better understanding of a larger history of a subject." However, in the case of this book, we find it transcends more than sectional or local history, as it treats of one country and almost a complete period; nevertheless, it does aid in forming a part of the larger aspect of the history of mathematics. ... It affords a great deal of satisfaction to review a book written so ably by mathematical scholars who have themselves the prerequisite historical background for colonial studies, such as the joint authors of "A History of Mathematics in America before 1900." At the outset we must congratulate the authors on having placed before mathematicians and other scholars a small volume giving the synopsis of the mathematical progress in the United States from early colonial period to almost contemporary generations.

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

*Amer. Math. Monthly***42**(3) (1935), 166-168.

The authors of this slim volume have contributed an admirable, though necessarily brief sketch of mathematical developments in the United States down to the close of the 19th century. The presentation is well organized, systematic, and furnishes, with considerable clarity, an excellent overview of the subject in proper historical perspective. The style, unfortunately, is somewhat matter-of-fact and impersonal, in contrast to the more intimate and at times dramatic account given by the late Professor Cajori in 'The Teaching and History of Mathematics in the United States' (1890). ... it would seem only honest to state that while the book leaves a little to be desired in the matter of style and details, nevertheless, considering its brevity (necessitated by conforming with companion volumes of the series) it unquestionably represents a welcome addition to the literature of historical and expository mathematics.

7.4.**Review by: Irby C Nichols.**

*National Mathematics Magazine***9**(2) (1934), 59-61.

The subject is a difficult one to treat satisfactorily and the Carus Monograph Committee are certainly to be congratulated upon their selection of persons to do the work. All American lovers of mathematics will be pleased with the results. The word "America" as used in the title of the book includes "territory north of the Rio Grande River and the Caribbean Sea". The time covered is treated in four chapters: "The Sixteenth and Seventeenth Century", "The Eighteenth Century", "The Nineteenth Century", and "The Period 1875-1900". ... the present reviewer wishes to express the genuine inspiration he has derived from reviewing this little volume; he is certain that its influence will fully justify the wisdom of the committee who conceived the piece of work and the authors who have so well executed it.

7.5.**Review by: Vera Sanford.**

*The Mathematics Teacher***27**(7) (1934), 353-354.

The authors have confined them selves to the territory now included in the United States and the Dominion of Canada, but to all intents and purposes the work is virtually limited to the former. ... The sixteenth and seventeenth centuries are treated briefly in a single chapter. The eighteenth century is given greater space. The nineteenth century is considered first in a general survey and then in a section devoted to the period from 1875 to 1900 which comprises half of the volume.

7.6.**Review by: Raymond C Archibald.**

*Bull. Amer. Math. Soc.***41**(1935), 603-606.

The Committee on the Carus Monographs had a happy inspiration when it was led to induce Professor Smith to prepare this history. He was in every way qualified for the task - through his unique knowledge of the subject, through his attractive literary style, and through the excellence of his judgment in dealing with a great mass of material and in presenting its essence in well-balanced and compact form. All of these qualities are very much in evidence in the little volume under review. Only one who has had considerable experience in such matters can truly appreciate the great amount of research which went into the preparation of the manuscript. In this research Professor Smith had the valuable assistance of Professor Ginsburg of Yeshiva College, the editor-in-chief of 'Scripta Mathematica'. ... On the whole the work is exceedingly valuable and suggestive, and American mathematicians must be highly grateful to the authors for thus notably contributing to their enlightenment and edification. **Portraits of Eminent Mathematicians, with Brief Biographical Sketches Portfolio I**(1936).

8.1.**Review by: R B McClenon.**

*Amer. Math. Monthly***45**(1) (1938), 39.

This portfolio of portraits of men distinguished for their contributions to mathematics is a splendid collection which aside from its intrinsic interest provides a good example of modern methods of photogravure. The men included in the collection are Archimedes, Copernicus, Vieta, Galileo, Napier, Descartes, Newton, Leibniz, Lagrange, Gauss, Lobachevsky, Sylvester.

8.2.**Review by: Anon.**

*The Mathematics Teacher***30**(1) (1937), 39-40.

We heartily recommend this portfolio to teachers who wish not only to make their class room more colourful, but who wish also to create interest in their pupils for the work in mathematics. Most classrooms in mathematics are drab enough at best. By using some of these portraits ... teachers of mathematics can help to create a better atmosphere for the pupils.- (with Jekuthiel Ginsburg)
**Numbers and Numerals. A Story Book for Young and Old**(1937).

9.1.**From the Preface.**

This is a story of numbers, telling how numbers came into use, and what the first crude numerals, or number symbols, meant in the days when the world was young.

9.2.**Review by: J A Drushel.**

*The Clearing House***12**(7) (1938), 443-444.

The scope of this little book is told best in the chapter titles: Learning to Count, Naming the Numbers, From Numbers to Numerals, From Numerals to Computation, Fractions, Mystery of Numbers, Number Pleasantries, Story of a Few Arithmetic Words.

9.3.**Review by: H F M.**

*The High School Journal***20**(5) (1937), 201.

There is much material in this monograph that would be of great interest to boys and girls of the upper elementary grades and the high school. It is written so that much if not all of the thought may be easily understood by such pupils. Because it is so brief it should serve as a splendid introduction to a more comprehensive history of mathematics and should stimulate the interest of pupils in such history so that they will read some of the more complete works on this subject.

9.4.**Review by: F A Y.**

*The Mathematical Gazette***21**(244) (1937), 246.

The pupils who use this book will be fortunate. They will read what others have not read - because history of mathematics is so rarely included in a school curriculum - the story of one of the great factors in civilisation. At successive stages the authors show a new need arising and describe man's efforts to increase mathematics and mould it to the purpose, until they have given all the essential facts about numbers and computation and the development, form and use of the numerals of all times. To cover the ground of whole numbers so completely in thirty-four pages is a masterpiece of condensation, more noticeable because the matter is given in simple words and explained as to a beginner.

9.5.**Review by: Anon.**

*Journal of Educational Sociology***11**(6, The Challenge of Youth) (1938), 383-384.

'Numbers and Numerals' is good enough to find a welcome place in many arithmetic classes and in many teachers-college libraries.

9.6.**Review by: U G Mitchell.**

*Amer. Math. Monthly***45**(1) (1938), 41-42.

During the last twenty-five years there has been a growing recognition that familiarity with the history of mathematics increases the power of a teacher to give students insight into the nature and meaning of mathematics. But thousands who are now teaching elementary mathematics are not likely to have opportunity to study the history of mathematics at any college or university where there are facilities for teaching it well. In some way the history of mathematics must be brought to these teachers if they are to have it at all. It is probably with some such idea in mind that the fifty-page booklet under review has been issued by*The Mathematics Teacher*free to all members of the National Council of Teachers of Mathematics and for sale at an almost negligible price to others.

9.7.**Review by: Vera Sanford.**

*The Mathematics Teacher***30**(6) (1937), 300-301.

Louis Agassiz is said to have characterized a scholar as a man who can discuss his subject in technical terms with experts, who can express it in language that can be understood by the man in the street, or who can tell it as a fairy tale for children. People familiar with Professor Smith's works will classify his 'History of Mathematics' and his 'Rara Arithmetica' under the first heading and his 'Number Stories of Long Ago' under the third. 'Numbers and Numerals', written in collaboration with Professor Ginsburg, belongs to the second class although the editor's note suggests that it may be used as supplementary reading material in school classes in mathematics and in the social studies. The more able pupil in the junior high school and in the senior high school presents much the same problem as does the average man of Agassiz's statement. It is from the point of view of using this booklet for supplementary material in schools that the following digest and comment has been written. **The Wonderful Wonders of One-Two-Three**(1937).

10.1.**Review by: Lenore John.**

*The Elementary School Journal***38**(6) (1938), 477.

It is frequently recommended that the mathematics courses of the intermediate grades and of the junior high school include informational material relating to the history and the characteristics of our number system. Lack of suitable reading materials for pupils has perhaps been the chief reason why this recommendation has not been followed more generally. A recent book by a well-known historian in the field of mathematics will, therefore, be welcomed by many teachers. ... Children of the intermediate grades will enjoy reading the book, but it will also appeal to junior high school pupils whose previous mathematical diet has not included such material.

10.2.**Review by: William D Reeve.**

*The Mathematics Teacher***30**(7) (1937), 348.

This is a new book by the author of "Number Stories of Long Ago" who himself enjoys the various and interesting things that can be done with numbers. He gives us here the story of how numbers grew from notches on a stick to figures in a row. To a child who can read and understand this book, numbers become some thing much more important and real to him than is often the case in a dreary classroom with some humdrum teacher. One finds here the story of numbers, Roman numerals and place value, the magic square and other magic, and other interesting topics. Surely here is fun and useful information for many young readers.**Portraits of Eminent Mathematicians, with Brief Biographical Sketches Portfolio II**(1938).

11.1.**Review by: Tomlinson Fort.**

*Amer. Math. Monthly***45**(7) (1938), 467.

Portraits of the following men make up Portfolio Number Two: Euclid, Cardan, Kepler, Fermat, Pascal, Euler, Laplace, Cauchy, Jacobi, Hamilton, Cayley, Chebyshev, Poincaré. I have only praise for the present work, as for its predecessor published in 1936. Portraits published by Professor Smith some years ago are now a classic in American mathematics.

11.2.**Review by: Henry Schroeder.**

*National Mathematics Magazine***13**(1) (1938), 55-56.

The following are a few quotations taken from some of the biographies:

"Giorolamo Tirabosci, historian of Italian literature, refers to Cardan as to a man who was more credulous of dreams than any frivolous women, ... while at the same time he felt that he was one of the most profound and fertile geniuses that Italy has ever produced, and that in mathematics and in medicine he had made rare and valuable discoveries."

"At an age when boys in our country would be in a high school class, he was a companion of world-renowned professors, opening for them new paths leading through the vast maze of mathematics - still a puzzle to all who study his brief career and marvel at his achievements. Such a boy was Blaise Pascal."

"Laplace lived in a period of one of the world's greatest wars - the French Revolution. ... He was not always a careful writer, however, and often when he was himself puzzled he would write 'It is easy to see that ...,' when it was not at all easy for even first-rate astronomers to see it at all."

"Life in the country village was, however, by no means easy, and the Cauchy family felt the pinch of poverty as did most of France. ... He had shown himself an indefatigable worker in all branches of mathematics - higher algebra, differential equations, the theory of probability, mathematics as applied to physics and astronomy, the calculus of variations, determinants, the foundations of mathematics, and the theory of functions. In all he published nearly eight hundred important memoirs distributed among these various branches."

"In the year 1900, however, mathematics had branched out so widely that no one could grasp the whole subject. It was in this period that Poincaré was in his prime, and of him it could have been said, without too much exaggeration, that he knew the greatest branches of the subject, particularly as applied to astronomy and physics, more completely than any of his contemporaries. As a matter of interest to teachers who place implicit confidence in the various types of tests it may be worth while to know that, when Poincaré had reached maturity, the Binet tests were tried upon him. It is said, on good authority, that the results showed him to be a man of extremely low intelligence-this man who in all the world ranked as one of the world's greatest scholars."

11.3.**Review by: J A Drushel.**

*The Clearing House***13**(3) (1938), 186-187.

This portfolio consists of a beautiful, full-page, photo-engraved cover design by Rutherford Boyd, of a two-page introduction, and of thirteen portraits of eminent mathematicians from Euclid to Henri Poincaré. Each portrait is accompanied by a well-written, two-page biographical sketch in each of which the early education of the subject gets proper emphasis.**Addenda to Rara Arithmetica**(1939).

12.1.**Review by: Lambert L Jackson.**

*Amer. Math. Monthly***46**(8) (1939), 504-506.

As indicated by the title, this pamphlet is a supplement to the 'Rara Arithmetica' by the same author and publisher, first issued in 1908. ... the Addenda, taking the same form of presentation as the 'Rara', contains a large amount of technical information; in fact, it is an exhaustive summary of the findings of research during the last thirty years in the history of sixteenth-century arithmetic. It will serve all scholars in this field, particularly lovers and collectors of rare books, cataloguers, bibliographers, and dealers interested in early publications.

12.2.**Review by: F A Yeldham.**

*The Mathematical Gazette***23**(256) (1939), 409.

Students who use 'Rara Arithmetica, 1908', know that it owes much of its value as a history of Renaissance arithmetic to the descriptions it contains of the European arithmetical books of the fifteenth and sixteenth centuries in the library of Mr G A Plimpton. Mr Plimpton was adding to his books and manuscripts when 'Rara Arithmetica' was being prepared, and continued to do so until his death in 1936, after which the library was presented to Columbia University. 'Addenda to Rara Arithmetica' brings the catalogue of the early arithmetics in the library up to this date.

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