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Jekuthiel Ginsburg's books
Jekuthiel Ginsburg wrote two books, both jointly authored with D E Smith. We give below some information about these two books.
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A history of mathematics in America before 1900 (1934)
Numbers and Numerals. A Story Book for Young and Old (1937)
A history of mathematics in America before 1900 (1934)
Numbers and Numerals. A Story Book for Young and Old (1937)
1. A history of mathematics in America before 1900 (1934), by David Eugene Smith and Jekuthiel Ginsburg.
1.1. 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. Little advance was made upon the mathematics of England as taught in the preceding century, except that the sextant replaced the quadrant and astrolabe for purposes of navigation. In general the tendency was toward the application of mathematics to astronomy and natural philosophy. This union of these subjects with pure mathematics was a healthy one for all three branches, since none of them was sufficiently developed by itself to make a separate course practicable even if desirable.
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 described 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 naturally many exceptions a subject like quaternions, for example, having necessarily to carry with it a considerable knowledge of mechanics; and differential equations having a wide range of applications. Nevertheless the tendency was strongly toward pure analysis and geometry.
The question then arose as to whether this tendency was a healthy one, either for mathematics or the natural sciences. The answer lies with the generation following ours. All we can now say is that the pendulum was motionless in the sixteenth century; that it began to swing toward the applications in the seventeenth and eighteenth centuries; that it reached the limit of oscillation in that direction in the first half of the nineteenth century; that about 1875 it had definitely begun to swing to the side of pure mathematics; and that in the period ending with 1900 it seems to have reached the limit of the movement in that direction. Has the tendency now changed? Does mathematics reach out once more to a closer union with physics, celestial mechanics, the quantum theory, and the material ranges of the sciences in general? It is for the historian of the year 1950 to survey the first quarter of this century and then to answer this question and to venture, if he feels it safe, to prognosticate for the same length of time in the future.
In connection with the survey which can then be made, the historian will have a greater freedom than a contemporary has to pay the tribute due to men now living who, during the closing years of the nineteenth century contributed so greatly to the remarkable development of mathematics in our present generation. The names of a number of these scholars have been mentioned under the topics of their special interests, but no critical evaluation of their work has been attempted and no biographical notes have been given for them. Most of these men will then have passed away and the historian, as he undertakes to analyse the mathematics of America as a whole, will reveal in proper perspective the roles which men now living played in the half century beginning about 1880 and reaching to approximately the present time.
1.2. Contents.
Introduction
Chapter 1. The Sixteenth and Seventeenth Centuries
1. Needs of the Early Settlers
2. Causes of the Low Degree of General Intellectual Effort
3. Early Conditions in the Seventeenth Century
4. New England
5. Early Astronomy
6. The Astrologers
Chapter II. The Eighteenth Century
1. General Survey
2. The Colleges
3. Private Instruction
4. Equipment for Study
5. Textbooks
6. Astronomy, Navigation, and Geodesy
7. Learned Societies and Scientific Periodicals
8. Prominent Names
9. Summary of Conditions in the Eighteenth Century ...
Chapter III. The Nineteenth Century. General Survey
1. The Colleges and Universities
2. European Influences
3. Scientific Societies and Periodicals ....
4. Prominent Names, 1800-1875
Chapter IV. The Period 1875-1900
1. Interest in Mathematical Research
2. The American Mathematical Society
3. European Influence
4. Periodicals
5. Prominent Names and Special Interests
6. American Dissertations
7. General Trend of Mathematics in America, 1875-1900 ...
8. Trend of Important Branches
9. Retrospect
Index
1.3. Summary by D E Smith and J Ginsburg.
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 described 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.
1.4 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. In this volume Doctors Smith and Ginsburg have, in addition, prepared the way for the future scholar who may wish to write an analytical study of the history of mathematics. ...
The subject matter in this small volume is dealt with in four convenient and concise chapters - but virtually grouped into two larger historical aspects, namely, the Colonial period and the Modern or Contemporary period. ...
Finally, the last chapter offers much food for thought. The manner in which the authors have condensed this most important period from 1875 to 1900 reveals their comprehensive grasp of the subject matter. The short life sketches throughout the book could not have been better, as the length of each is according to the importance of the man or woman.
1.5. 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 organised, 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.
1.6. 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".
...
In closing, the authors very fittingly raise the question as to the future: In the fourth period mathematics was "pure" mathematics, except for such subjects as Quaternions and Differential Equations, but earlier there was a strong tendency toward applications. Which tendency is better for mathematics, or for the natural sciences? Has the tendency again turned toward applications?
In conclusion, 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.
1.7. 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.
1.8. 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.
2. Numbers and Numerals. A Story Book for Young and Old (1937), by David Eugene Smith and Jekuthiel Ginsburg. .
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. Little advance was made upon the mathematics of England as taught in the preceding century, except that the sextant replaced the quadrant and astrolabe for purposes of navigation. In general the tendency was toward the application of mathematics to astronomy and natural philosophy. This union of these subjects with pure mathematics was a healthy one for all three branches, since none of them was sufficiently developed by itself to make a separate course practicable even if desirable.
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 described 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 naturally many exceptions a subject like quaternions, for example, having necessarily to carry with it a considerable knowledge of mechanics; and differential equations having a wide range of applications. Nevertheless the tendency was strongly toward pure analysis and geometry.
The question then arose as to whether this tendency was a healthy one, either for mathematics or the natural sciences. The answer lies with the generation following ours. All we can now say is that the pendulum was motionless in the sixteenth century; that it began to swing toward the applications in the seventeenth and eighteenth centuries; that it reached the limit of oscillation in that direction in the first half of the nineteenth century; that about 1875 it had definitely begun to swing to the side of pure mathematics; and that in the period ending with 1900 it seems to have reached the limit of the movement in that direction. Has the tendency now changed? Does mathematics reach out once more to a closer union with physics, celestial mechanics, the quantum theory, and the material ranges of the sciences in general? It is for the historian of the year 1950 to survey the first quarter of this century and then to answer this question and to venture, if he feels it safe, to prognosticate for the same length of time in the future.
In connection with the survey which can then be made, the historian will have a greater freedom than a contemporary has to pay the tribute due to men now living who, during the closing years of the nineteenth century contributed so greatly to the remarkable development of mathematics in our present generation. The names of a number of these scholars have been mentioned under the topics of their special interests, but no critical evaluation of their work has been attempted and no biographical notes have been given for them. Most of these men will then have passed away and the historian, as he undertakes to analyse the mathematics of America as a whole, will reveal in proper perspective the roles which men now living played in the half century beginning about 1880 and reaching to approximately the present time.
1.2. Contents.
Introduction
Chapter 1. The Sixteenth and Seventeenth Centuries
1. Needs of the Early Settlers
2. Causes of the Low Degree of General Intellectual Effort
3. Early Conditions in the Seventeenth Century
4. New England
5. Early Astronomy
6. The Astrologers
Chapter II. The Eighteenth Century
1. General Survey
2. The Colleges
3. Private Instruction
4. Equipment for Study
5. Textbooks
6. Astronomy, Navigation, and Geodesy
7. Learned Societies and Scientific Periodicals
8. Prominent Names
9. Summary of Conditions in the Eighteenth Century ...
Chapter III. The Nineteenth Century. General Survey
1. The Colleges and Universities
2. European Influences
3. Scientific Societies and Periodicals ....
4. Prominent Names, 1800-1875
Chapter IV. The Period 1875-1900
1. Interest in Mathematical Research
2. The American Mathematical Society
3. European Influence
4. Periodicals
5. Prominent Names and Special Interests
6. American Dissertations
7. General Trend of Mathematics in America, 1875-1900 ...
8. Trend of Important Branches
9. Retrospect
Index
1.3. Summary by D E Smith and J Ginsburg.
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 described 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.
1.4 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. In this volume Doctors Smith and Ginsburg have, in addition, prepared the way for the future scholar who may wish to write an analytical study of the history of mathematics. ...
The subject matter in this small volume is dealt with in four convenient and concise chapters - but virtually grouped into two larger historical aspects, namely, the Colonial period and the Modern or Contemporary period. ...
Finally, the last chapter offers much food for thought. The manner in which the authors have condensed this most important period from 1875 to 1900 reveals their comprehensive grasp of the subject matter. The short life sketches throughout the book could not have been better, as the length of each is according to the importance of the man or woman.
1.5. 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 organised, 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.
1.6. 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".
...
In closing, the authors very fittingly raise the question as to the future: In the fourth period mathematics was "pure" mathematics, except for such subjects as Quaternions and Differential Equations, but earlier there was a strong tendency toward applications. Which tendency is better for mathematics, or for the natural sciences? Has the tendency again turned toward applications?
In conclusion, 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.
1.7. 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.
1.8. 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.
2.1. 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. It tells where our modern system of numbers came from, how these numbers came to be used by us, and why they are not used everywhere in the world. It tells us why we have ten "figures," whereas one of the greatest nations of the world at one time used only five figures for numbers below five hundred, and another nation used only three.
The story will take us to other countries and will tell us about their numbers and their numerals. The principal lands we shall visit arc shown on the map facing page 1. When you read about India or Egypt or Iran (Persia) or Iraq, you can look up that particular country on the map.
The story will tell you how some of the numerals of today came to have their present shape, and how the counters in our shops are related to the numerals which you use, and why some numbers are called "whole" and others "fractional," and why we say "three" cheers instead of "two" or "four" cheers. You will find, too, how people wrote numbers before paper was invented, and how such long words as "multiplication" came into use in Great Britain and America, and how it happened that all of us have to stop and think which number in subtraction is the "minuend" and which the "subtrahend," as if it made any difference to you or to us or to anyone else.
We shall find something interesting about the theory which is nowadays so widely advertised under the name of "numerology" really one of the last absurdities that has come down to us in relation to the numbers of today. We shall see that it arose from one of the Greek and Hebrew ways of writing numbers long before these peoples knew the numerals which we all use today. Our story will show how the idea of number separated itself from the objects counted and thus became an abstract idea. We shall also see how this abstract idea became more and more real as it came in contact with the needs of everyday life and with the superstitions of the people.
The story tells many interesting things about measures, a few things about such matters as adding and subtracting, and somethings about the curious and hopelessly absurd number superstitions.
If you come across a few strange words from strange languages, do not think that you must pronounce them or keep them in mind. You will find that they are explained, that they are part of the story, and that they help to tell you something interesting. They are no worse than such relics of the past as "subtrahend" and "quotient."
In other words, here is a story of the ages. We hope that you will like it and will tell it to others.
2.2. 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.
2.3. 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.
2.4. Review by: F A Y.
The Mathematical Gazette 21 (244) (1937), 246.
The first five chapters of Numbers and Numerals are historical, and deal with counting scales, number names, numerals, computation with apparatus and with numerals and, very briefly, with fractions. The two following chapters are more composite. They contain curious obsolete customs and ideas about numbers and more modern number knowledge, introduced wherever possible as number games and puzzles. The last chapter gives the derivations of arithmetical terms. There is a map of the Old World showing the countries mentioned, an index, and the titles of Professor D E Smith's fuller works and other books for further reading.
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. The illustrations are plentiful and fascinating, whether they range from the numerals and suan-pan of the Chinese to the quipu and counting-board of the Peruvians, or trace the development of the Hindu Arabic numerals from their origin to the printed figures of to-day. The reader is given the opportunity of trying his skill in early modes of reckoning.
The book is interesting from cover to cover, and can be read by pupils of thirteen years and older.
2.5. 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.
2.6. 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.
...
Elementary mathematics is under condemnation in many a school today because of the formal and deadening way in which it is being taught there. One way of contributing to its salvation is to lead students to an insight into and understanding of mathematics through natural (even if frequently vicarious) experiences suggested by the study of the evolution of mathematics. As a missionary tract to carry this kind of gospel to many a class-room where it has never been proclaimed before, but where it is greatly needed, this little monograph should have the blessing of all friends of mathematical instruction.
2.7. Review by: Vera Sanford.
The Mathematics Teacher 30 (6) (1937), 300-301.
Louis Agassiz is said to have characterised 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 booklet treats number names, number symbols, computation and number recreations. The last chapter gives the history of certain words in the terminology of arithmetic. On first reading, one is puzzled to know how to appraise this booklet properly for its fine points are many. Yet a review which suggested that the booklet were perfect would by its fulsomeness defeat its purpose. Accordingly, I shall strain at the gnats.
The preface promises a story of numbers, how they came into use, and how our number forms came to have their present shape. It speaks of "the first crude numerals, or number symbols." The preface has the statement, often novel to teachers, that "the idea of number separated itself from the objects to be counted and thus became an abstract idea. We shall also see how this abstract idea became more and more real as it came into contact with the needs of everyday life and with the superstitions of the people."
A gnat is found in the preface when the readers are told that they need not pronounce the foreign words in the text. It would have been difficult to work out a set of diacritical marks, yet they would have helped greatly. How else can you discuss the material - how else can you give an oral report of your readings? Not many suggestions about pronunciation would have been needed.
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. It tells where our modern system of numbers came from, how these numbers came to be used by us, and why they are not used everywhere in the world. It tells us why we have ten "figures," whereas one of the greatest nations of the world at one time used only five figures for numbers below five hundred, and another nation used only three.
The story will take us to other countries and will tell us about their numbers and their numerals. The principal lands we shall visit arc shown on the map facing page 1. When you read about India or Egypt or Iran (Persia) or Iraq, you can look up that particular country on the map.
The story will tell you how some of the numerals of today came to have their present shape, and how the counters in our shops are related to the numerals which you use, and why some numbers are called "whole" and others "fractional," and why we say "three" cheers instead of "two" or "four" cheers. You will find, too, how people wrote numbers before paper was invented, and how such long words as "multiplication" came into use in Great Britain and America, and how it happened that all of us have to stop and think which number in subtraction is the "minuend" and which the "subtrahend," as if it made any difference to you or to us or to anyone else.
We shall find something interesting about the theory which is nowadays so widely advertised under the name of "numerology" really one of the last absurdities that has come down to us in relation to the numbers of today. We shall see that it arose from one of the Greek and Hebrew ways of writing numbers long before these peoples knew the numerals which we all use today. Our story will show how the idea of number separated itself from the objects counted and thus became an abstract idea. We shall also see how this abstract idea became more and more real as it came in contact with the needs of everyday life and with the superstitions of the people.
The story tells many interesting things about measures, a few things about such matters as adding and subtracting, and somethings about the curious and hopelessly absurd number superstitions.
If you come across a few strange words from strange languages, do not think that you must pronounce them or keep them in mind. You will find that they are explained, that they are part of the story, and that they help to tell you something interesting. They are no worse than such relics of the past as "subtrahend" and "quotient."
In other words, here is a story of the ages. We hope that you will like it and will tell it to others.
2.2. 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.
2.3. 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.
2.4. Review by: F A Y.
The Mathematical Gazette 21 (244) (1937), 246.
The first five chapters of Numbers and Numerals are historical, and deal with counting scales, number names, numerals, computation with apparatus and with numerals and, very briefly, with fractions. The two following chapters are more composite. They contain curious obsolete customs and ideas about numbers and more modern number knowledge, introduced wherever possible as number games and puzzles. The last chapter gives the derivations of arithmetical terms. There is a map of the Old World showing the countries mentioned, an index, and the titles of Professor D E Smith's fuller works and other books for further reading.
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. The illustrations are plentiful and fascinating, whether they range from the numerals and suan-pan of the Chinese to the quipu and counting-board of the Peruvians, or trace the development of the Hindu Arabic numerals from their origin to the printed figures of to-day. The reader is given the opportunity of trying his skill in early modes of reckoning.
The book is interesting from cover to cover, and can be read by pupils of thirteen years and older.
2.5. 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.
2.6. 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.
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Elementary mathematics is under condemnation in many a school today because of the formal and deadening way in which it is being taught there. One way of contributing to its salvation is to lead students to an insight into and understanding of mathematics through natural (even if frequently vicarious) experiences suggested by the study of the evolution of mathematics. As a missionary tract to carry this kind of gospel to many a class-room where it has never been proclaimed before, but where it is greatly needed, this little monograph should have the blessing of all friends of mathematical instruction.
2.7. Review by: Vera Sanford.
The Mathematics Teacher 30 (6) (1937), 300-301.
Louis Agassiz is said to have characterised 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.
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The booklet treats number names, number symbols, computation and number recreations. The last chapter gives the history of certain words in the terminology of arithmetic. On first reading, one is puzzled to know how to appraise this booklet properly for its fine points are many. Yet a review which suggested that the booklet were perfect would by its fulsomeness defeat its purpose. Accordingly, I shall strain at the gnats.
The preface promises a story of numbers, how they came into use, and how our number forms came to have their present shape. It speaks of "the first crude numerals, or number symbols." The preface has the statement, often novel to teachers, that "the idea of number separated itself from the objects to be counted and thus became an abstract idea. We shall also see how this abstract idea became more and more real as it came into contact with the needs of everyday life and with the superstitions of the people."
A gnat is found in the preface when the readers are told that they need not pronounce the foreign words in the text. It would have been difficult to work out a set of diacritical marks, yet they would have helped greatly. How else can you discuss the material - how else can you give an oral report of your readings? Not many suggestions about pronunciation would have been needed.
Last Updated December 2023