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Summary In this paper I discuss some of the historical and philosophical aspects of mathematical education, especially in foundational subjects where many of the principal issues seem to lie. Historical examples are chosen from the calculus, mathematical analysis, set theory and logic, but the philosophical points discussed also apply to other branches of mathematics. The paper falls into four parts, which are alternately mainly historical and philosophical. The first part is motivated partly by problems arising in school mathematics (although the subject matter is treated from a more advanced viewpoint), and the second attempts to develop some general methodological considerations. The third part is relevant mainly to university mathematics, while the final part contains a number of broad conclusions, including the question of the role of the history of mathematics in mathematical education.

Many approximate solutions of the problem of constructing a true sea-chart or Mercator's projection were worked out in the sixteenth and seventeenth centuries. It was not until after the middle of the latter century that it became widely known that the solution was very closely related to logarithmic tangents. However, unpublished manuscripts in private possession in England show that the problem had been completely dealt with much earlier by Thomas Harriot (1560–1621), the contemporary of Napier and Briggs, who is best known in the history of mathematics for his work on algebra. Harriot was successively scientific and navigational adviser to Sir Walter Ralegh and under the patronage of the earl of Northumberland. As a young man he produced an approximate solution of the Mercator problem, which is now lost. He then developed theories of the conformality of stereographic projection, the rectification and quadrature of the equiangular (logarithmic) spiral, used the exponential series, devised interpolation formulae, and applied these results to the calculation of the so-called meridional parts (latitudes croissantes) used in the construction of a Mercator chart, which was most probably completed in 1614. Transcriptions of many of the relevant manuscripts are included in the article.

If we consider the excellent summary of Hindu geometry made long ago by M. Chasles (1), and the various scholarly essays upon the subject that have of late appeared, it may seem an unnecessary labor 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 MahavTracarya, by Professor Rangacarya. In the second place, there is now an opportunity for a comparison of results that was wanting until the publication of Mahavlr'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. Add to these two reasons the further one that Isis is not a journal for the specialist in the history of mathematics, but is intended for the general student of the growth of human culture, and the justification for attempting an article of this nature is complete. The problem in its large aspects is set forth in three questions: (1) Who were the great geometers of India ? (2) How extensive was their knowledge of geometry ? (3) In how far was this original with India as a country, and with each of these scholars individually ? And first, who were the great geometers of India? The earliest one whose name has come down to us is Xryabhata, who was born in 476 A. D. To be sure there were two Aryabhatas, and it is within the range of possibility that it is the younger of the two, perhaps a century or two later, that we owe some or all of the work attributed

(1902). On Ball's History of Mathematics. The American Mathematical Monthly: Vol. 9, No. 12, pp. 280-283.

Many words which are often used have at least two very different meanings. Thus, the word ( mathematics ) is sometimes used to denote the processes which have been and are now used to discover entities or truths of a certain kind, and sometimes to denote the entities or truths discovered by means of these processes. From this verbal confusion has grown a very harmful confusion which has shown itself repeatedly in discussions on the principles of mathematics, when some who rightly perceived the great importance of discovery mistakenly imagined that logic was at fault in ignoring the processes of discovery ((intuition ). We speak of (the history of mathematics ), and this, in the fullest sense, would seem to mean the history of those processes which have been used to attain to knowledge of those truths which mathematicians, qta mathematicians, seek to find out. At least, this would presumably be the ideal history of mathematics: as it is, we can only even approach this ideal in a very few cases. Mathematicians usually consider that the description of their processes of thought is of little importance, and that the chief interest of their work lies in the concepts and the truths and proofs of them that they reach, or think that they reach, or try to reach. Consequently, the history of mathematics is usually made up of only a part of the above description of processes, and the description and comparison of the various notions that men have had of mathematical truths, mathematical entities, and mathematical methods. Again, we speak of ((the principles of mathematics ), when we are not speaking of the principles that may have guided mathematicians in their search for certain truths, but of the logical analysis of these truths.

I Was Recently confronted with the assertion that teachers of mathematics would soon be seeking other fields as that subject was fast approaching a state of obsoleteness in the high school curriculum. This was of course an extreme view but nevertheless in line with the trend of opinion that makes necessary the justifying of the place of mathematics by reorganized and enriched teaching that will be adequate for our present type of high school pupil.

FLORIAN CAJORI, professor emeritus of mathematics at the University of California, died at his home in Berkeley, I4 August 1930, after a brief attack of pneumonia, in the seventy-second year of his age. For forty years his name had been a familiar one throughout the world, recalling a genial personality and a prolific writer on various topics connected with the history of mathematics and physics. He was born at St. Aignan near Thusis, Switzerland, 28 February 1859. His mother was CATHERINE (CAMENISCH), and his father was GEORG CAJ6RI (the family name was spelled thus, or as CAJOERI (2)), an engineer and contractor, well known in his part of Switzerland for important bridges and highways that he built. Professor CAJORI received his early education at Zillis, and in the canton school at Chur. At the age of I6 years he came to America, an elder brother having preceded him, and made his way at once to Whitewater, Wisconsin, where he continued his education at the normal school during the years I876-78. After some teaching in a country school he entered the University of Wisconsin where he received the degrees of Bachelor of Science in I883, and of Master of Science in i886. From January I884 to June I885 he was a graduate student of mathematics at The Johns Hopkins University, and received his first academic appoint

Recenzja

As a result of two recent studies by KARL PEARSON (1) valuable information is at hand for forming a true estimate of the important position which ABRAHAM DE MOIVRE occupies in the history of mathematics. These studies are based on a consideration of : (a) part 4 of JEAN BERNOULLI'S Ars Conjectandi,(2) which contains what is properly called (c Bernoulli's Theorem )); and (b) a pamphlet of MOIVRE dated November 12, 1733. The chief purpose of this paper is to summarize some of PEARSON'S results, to remove misapprehension which one of his paragraphs may cause, and to present a facsimile of the MOIVRE pamphlet, which is entitled: Approximatio ad Summam Terminorum Binomii (a +b)" in Seriem expansi.

More than two thousand years ago, Pythagoras lectured on the power of number, meaning only integers and fractions. He reiterated again and again, “Number rules the universe; the essence of all things is number.” Then one day he evaluated the diagonal of a unit-sided square. Alas, there was no number as he understood number which gave an answer.

The centre of interest now shifts from St Andrews and Edinburgh to Glasgow. The troubles that afflicted Scotland during the 17th Century bore heavily on Glasgow University and more particularly on the position of Mathematics in the University; but in 1691 a distinct Professorship of Mathematics was founded, and from that date the old system of Regents disappeared from Glasgow so far as Mathematics was concerned. The first occupant of the Chair was George Sinclair, who is now chiefly remembered by the controversy in which James Gregory held up Sinclair's Treatise Ars nova et magna to ridicule. It is not fair however to take Gregory's pamphlet as a final estimate of Sinclair's contributions to science; Sinclair laid himself open to attack, but he rendered great service to the mining industry of Scotland and deserves the gratitude of posterity in spite of his many eccentricities. His contributions to mathematics however are of no importance, but during his tenure of the Chair the number of students grew rapidly and the new professorship made a good start.

The following essay is adapted from one with the same title read to the British Society for the History of Science on 20 October 1958—the anniversary, by a striking coincidence, of the birth of W. H. Young (1863–1942). To his memory I dedicated the talk, and now rededicate its publication, not only because I am his daughter and of all that means, but because he invented a method, the method of monotone sequences, which shows the powerfulness of inequalities as a mathematical tool supremely.

In the Gazette (176) for May, 1925, Miss J. M. Brown wrote a delightful article on “ Mathematical Clubs ” in Schools. It is interesting today to recall some of the ideas mentioned.“ The obvious function of a Mathematical Club is to stimulate the interest which may have been aroused in the classroom, or if possible to awaken an interest hitherto dormant.”The History of Mathematics was considered a starting point but was not found to be a very suitable topic for arousing enthusiasm. Mention is made of graph work (there was no Cundy and Rollett, Mathematical Models, in those days)—and of interest shown in Flatland (Dr. Abbott).

The student or teacher who consults several references to obtain information about some historical point in the development of mathematics may, at first, be shocked to find definite discrepancies. Further study may reveal that while in certain cases the reference seems to be definitely in error, in other cases the point may be debatable. Sometimes the error seems clearly to be typographical; in other situations the author has been careless in drawing too-general conclusions; in still others the situation is simply one in which even competent research has not yet produced clear-cut evidence. Unfortunately, in some cases, reading of the references indicates an error caused by repeating an unchecked statement made by some previous author.

1. The Association and Mathematical Education. The inclusion of a survey of mathematical education as a part of the semi-centennial celebration of the Mathematical Association of America is appropriate for many reasons. For example, the by-laws of the Association state that one of its objectives is "to assist in promoting the interests of mathematics in America, especially in the collegiate field . .. by conducting investigations for the purpose of improving the teaching of mathematics.... " In pursuit of its goals the Association has published the Carus Monographs of which the fifth, The History of Mathematics in America before 1900, by David Eugene Smith and Jekuthiel Ginsburg implicitly contains much of significance for the history of mathematical education in the country, especially at the college level. However, the past concern of the Association for mathematical education at the secondary level should neither be ignored nor allowed to lapse. It was particularly significant that the Association was mentioned on page one of the first volume of The Mathematics Teacher to be issued under the aegis of the National Council of Teachers of Mathematics. In 1921 the newly founded Council took over this journal from the Association of Teachers of Mathematics in the Middle States and Maryland. In the first issue of that year C. M. Austin, first president of the National Council, explained, in words which have a familiar ring, the needs and reasons for the founding of this new organization as follows: "During this same period (the previous ten years) high school mathematics courses have been assailed on every hand. So-called educational reformers have tinkered with the courses and they, not knowing the subject and its values, in many cases have thrown out mathematics altogether or made it entirely elective. Individual teachers and local organizations have made a fine defense to be sure, but there could be no concerted action. Finally, the Mathematical Association of America came to the rescue and appointed a committee to study the situation and to make recommendations. Already two valuable reports have been issued and others are in preparation. The pity of it is that this work, wholly in the realm of the secondary schools, should have to be done by an organization of college teachers. True, they have generously called in high school teachers to help, but the fact is that it remained for the college people to initiate the work. They could do it because they possessed a live, vigorous organization [1]. (The final report of the committee of the Association to which Austin referred was the famous "1923 report.") It is, I believe, altogether appropriate that this Association which is still live and vigorous should continue its interest in mathematical education, not only by recognizing it in the program of this, its semi-centennial, but also in the continuing concern showed by the work of its Committee on the Undergraduate Program which has a panel on teacher training. I was, therefore, greatly pleased when the Committee asked me to speak on

IF WHAT he heard is correct, Girolamo Cardano was born much to the dismay of his parents, who were not married. They attempted several unsuccessful abortions before his birth on 24 September 1501. Girolamo's father, Fazio, although he practiced law in Milan, found more time for other interests, such as medicine, mathematics, and occult lore. Fazio was highly esteemed as a scholar by his con temporaries in Milan. He apparently, for example, consulted with Leonardo da Vinci about flying machines. Perhaps the similarities evidenced be tween the father and the son may be at tributed in part to the fact that Cardano was educated by his father until the age of nineteen. At this time, with the moral support of his mother, he was finally per mitted to attend his father's alma mater in Pavia to study medicine. The school closed a year later because of war, and he transferred to the University of Padua. There began the stormy life of Cardano that ended in 1576. Much of this life un folds as we examine the character of this remarkable man. It seems that nowhere else does one find a man's character so thoroughly crucified as in the writings about Cardano in most books on the history of mathematics or science. In general, students of the history of mathematics consider him to have been

It was a hard struggle for mankind to free himself from the notion that geometry had anything to do with the real world.

I want to begin with an explanation as to my position in coming here to address you. I am not an expert upon the history of mathematics. I have come in order to tell you something of my experiences in trying to teach a little of the history of mathematics to sundry university students. This had its origin in the determination of Professor Milne, even while he was teaching at Clifton, to propagate a knowledge of the history of mathematics amongst students who, as a rule, know absolutely nothing of it. He continued on that crusade in a true missionary spirit at the University of Leeds.

Many conclusions useful for the future of mathematics might result from a sustained investigation into the relationship between the history of mathematics and the history of the societies in which mathematics has developed. For example, is it posible to reach some conclusion about the nature of the social, economic, and political milieu which is optimal for the development of science in general and of mathematics in particular?

By a curious chronological coincidence, the tercentenary of Newton’s birth in 1642 is succeeded by the centenary of another outstanding event in the history of mathematics; the discovery of Quaternions in 1843 by Sir W. R. Hamilton. It seems a suitable occasion for drawing attention to the extent to which conceptions due to Irish mathematicians of that epoch are interwoven into the technique employed in recent developments in mathematical physics.

Recently K. Rychlik1 has published for the first time a manuscript by B. Bolzano containing a theory of real numbers. This publication is of great importance for the study of the history of mathematics in the nineteenth century, since it shows that as early as 1 83 0-1 83 5 an attempt was made at an arithmetical foundation of the theory of real numbers. Such a foundation was afterwards realised by K. Th. Weierstrass, H. Ch. R. Meray, G. Cantor and J. W. R. Dedekind (1858) in the second half of the nineteenth century. Though A. L. Cauchy2 had observed in 1827 that irrational numbers may be considered as limits of sequences of rational numbers, this is something quite different from Bolzano's attempt to develop a theory of real numbers based on a method of approximation . Rychlik's edition contains besides the text of the manuscript a collection of fifteen mathematical footnotes, a Vorwort and a Nachwort in which he tries to evaluate and justify Bolzano's theory. There is also a foreword by L. Rieger containing valuable observations concerning Bolzano's theory. In order to make clear the reason for this study I describe Rychlik's position concerning Bolzano's theory as it appears from the Vorwort and the Nachwort. I refer to Rychlik's edition by R, followed by the number of the page. It should be stressed that both Rychlik and Rieger are aware that Bol

The main subject of this essay is to describe and to analyse the polemics on the concept of function in the years around I900, especially the discussions between Baire, Borel, and Lebesgue. The roots of these discussions lie much earlier in the history of mathematics, extending back to the eighteenth century. For this reason we shall give some information about this earlier period and also quote from the works of Euler, Lagrange, Bolzano, Cauchy, Dirichlet, and Riemann. They serve merely as an introduction to the main subject, the years around I9OO. Completeness for this earlier period is not attempted, the more so because further research for the initial developments might be necessary. For this historical background, the following points should be kept in mind.

SINCE the time Pythagoras reputedly discovered a relationship between the length of a vibrating string and the musical harmonic series, scholars have known a connection between mathe matics and music. Through the course of history, the alliance has been recognized in varying degrees as musicians either looked toward a more scientific view of art or zealously guarded their domain as a field based on intuition and inspira tion. The history of mathematics abounds with accounts of mathematicians who

I saw David Brewster the other day, who received me kindly and spread out his bank draft for fifteen guineas like a man. He told me further that a translation was for certain to be set about and that I as certainly should have the first offer of it. The work is a thing I can work at if the “gea of life” be in me at all, and for that cause alone I propose to accept.

(1915). History of Mathematics. The American Mathematical Monthly: Vol. 22, No. 9, pp. 299-304.

The Rule of False Position, Single and Double, is a method of solving equations of the First, degree in one unknown by guessing the answer and then adjusting the guess in terms of the conditions of the problem. As the name suggests, there might be one such guess or there might be two. The word position in this ease is used in the sense of a “thing on which one takes a stand”—in its meaning here it is allied to the idea of letting or allowing, forming a tenet or proposition on which one expects to act. Supposition would be closer to our thinking.

Before reviews devoted to particular branches of science had come into existence, results obtained by individual research were diffused by means of letters to those who were likely to be interested. At a later date they were published in encyclopaedic journals, in literary periodicals, and in the Proceedings of the various academies. This makes it extremely difficult to obtain accounts both accurate and complete of much that has been written in this or that particular subject. To assist in removing this difficulty, and to bring within the reach of the student the results of historical research, I have suggested, for instance, an examination of the periodicals, etc., which, while not exclusively devoted to science, may nevertheless contain material of the greatest value to us in the history of science. The suggestion had the honour of receiving the approval of one eminent bibliographer, and, in his turn, he did for three rare German periodicals what I had done for two of the most celebrated Italian literary journals.

1. There can be no doubt as to the fact that the natural numbers of the ancients were numbers having neither a positive nor a negative sign. We find it necessary to insist upon the recognition of this truth, because articles and books on the history of elementary mathematics yield much information on how the notion of negative numbers gradually arose, but it is impossible to find any indication as to how the notion of positive numbers came about, and also as to the role that was played by numbers without signs in the history of mathematics.