Popular Science Monthly/Volume 85/November 1914/Recent Mathematical Activities


By Professor G. A. MILLER


MATHEMATICAL research generally thrives best in seclusion. The results are often embodied in a language which but few understand, and are then stored with a quietude secured and maintained by their own attributes. Now and then there are instances when unsolved mathematical questions get involved with enough external matter to attract general attention. This external matter often consists of an array of names of noted mathematicians who have been unsuccessful in their efforts to solve these questions.

When the solutions of such questions become possible, through special ingenuity or through the gradual development of the necessary elements, there is usually a. stir in which mathematicians join the more heartily on account of its novelty. This fact may be illustrated by the famous memoir on the problem of three bodies by a Finnish mathematical astronomer named Karl F. Sundman, which the president of the Paris Academy of Sciences mentioned during the annual public session held on December 13, 1913.

This academy had previously appointed a committee to examine the work of Sundman, and the committee reported, through the noted French mathematician Émile Picard, that the memoir was epoch making for analysis and for mathematical astronomy. In accord with the recommendation of this committee, the Paris Academy awarded to Sundman the Pontécoulant prize, doubling its usual value. The report of the committee directed attention to the fact that Sundman achieved his results by means of classic mathematical methods.

In the April, 1914, number of Popular Astronomy Professor F. E. Moulton, of the University of Chicago, gave a very interesting popular account of the problem of three bodies and of the actual contribution made by Sundman towards its complete solution. From this account it is easy to see that a long list of eminent names are connected with this problem, including those of Newton, Euler, Lagrange and Poincaré, as well as that of one of the most illustrious American mathematical astronomers—the late G. W. Hill.

About two years ago a certain geometric question relating to the problem of three bodies came suddenly into prominence through an article by H. Poincaré, written shortly before his death, in which he called attention to the fact that he had not succeeded in finding a general solution of this question. It is interesting to note that a young American mathematician, Professor G. D. Birkhoff, of Harvard University, had been thinking along similar lines and soon succeeded in finding a general solution. An article containing this solution was published in the Transactions of the American Mathematical Society, 1913, and a French translation of this article appeared recently in the Bulletin of the French Mathematical Society.

A number of important unsolved mathematical questions are constantly kept before the mathematical public by means of prizes offered by various foreign academies for definite contributions. The prominence of the Paris Academy of Sciences along this line is well known, and it would be difficult to determine the extent of the good influence exerted by these prizes. Moreover, special prizes are not infrequently instituted. The king of Sweden has recently authorized such a prize to be awarded, for important developments in the theory of analytic functions, during the meeting of the sixth international mathematical congress, which is to be held at Stockholm in 1916.

The monetary value of these prizes varies very much, but it generally does not exceed a thousand dollars. For instance, the prize offered by King Gustav V, of Sweden, to which we have just referred, consists of a gold medal and three thousand crowns (about eight hundred dollars) in money. The main value attached to these prizes is the recognition of the importance of the work of the authors honored in this manner, and this is especially valued by the younger investigators.

While prizes have greatly stimulated research activity in mathematics they have not furnished the main stimulus. The opportunities offered by the various journals to make useful development and interesting discoveries promptly known have doubtless furnished a stronger and more permanent stimulus, especially in those cases where the standing of the editors of the journals inspired great confidence. In America we have two journals which have rendered, and are now rendering, preeminent service along this line; viz., the American Journal of Mathematics, founded in 1878 with J. J. Sylvester as editor-in-chief, and the Transactions of the American Mathematical Society, founded in 1900 with E. H. Moore as editor-in-chief.

A current mathematical undertaking whose bigness would seem to entitle it to general interest is the publication of a large encyclopedia devoted to pure and applied mathematics. This work is being published, in parts, in the German and French languages. The first part of the German edition appeared in July, 1898. Since this time a large number of other parts have appeared at irregular intervals, aggregating at present about ten thousand large pages. Several additional parts are now in press, and it seems too early to predict when the entire work will be completed or how large it will become. The published parts would now make twenty large volumes, each containing about five hundred pages.

While this work may appear extensive for an encyclopedia which aims to give in a concise form the fully established mathematical results, yet the French edition promises to become still more extensive. The first part of this edition appeared in August, 1904, and the aggregate of the published parts is at present only about one half as large as that of the German edition. On the other hand, most of the subjects which have been treated in both editions are treated with much more completeness in the French than in the German edition, as might be expected from the fact that the former edition is, in the main, based on the latter. According to the latest announcements some parts of the German edition are to be based on parts which have already appeared in the French edition.

The magnitude of these undertakings is certainly not the main element of interest to the educated man. In fact, the question has been raised whether these encyclopedias are not becoming too large to fulfil one of the main objects in view; viz., to provide a work by means of which the student can determine quickly what has been done along various lines in the mathematical sciences. A keen observer recently made the following significant remark, "the whole encyclopedia, whether German or French edition, seems of late to have run riotously and fruitlessly to leaves."[1] In this connection it may be observed that a big and vigorous tree has normally more leaves than a little one.

One of the main elements of interest in such big educational undertakings is the cooperation which it implies. This is especially true as regards a subject like mathematics, where the certainty and the permanence of conclusions tend to inspire unusual self-reliance and independence. The fact that the directors of these encyclopedias have secured the cooperation of nearly three hundred mathematicians of various nationalities implies that in this field also there is substantial evidence of organized effort on a large scale. It is of interest to observe that American mathematicians are fairly well represented among these collaborators.

A big current mathematical undertaking which affects directly a much larger number of people than the encyclopedias mentioned above is the work which is being done under the direction of the "International Commission on the Teaching of Mathematics." This commission was created during the sessions of the fourth International Congress of Mathematicians, which was held at Rome in April, 1908, and has for its main object a study of the methods and plans of mathematical instruction in different nations. At first it was intended that the commission should confine its work to secondary mathematics, but it soon appeared desirable to include all mathematical instruction in the scope of its investigation.

Sub-commissions were appointed in various countries. The American sub-commission is composed of D. E. Smith, Columbia University; W. F. Osgood, Harvard University, and J. W. A. Young, Chicago University. Under the general direction of these sub-commissions a vast amount of material relating to the mathematical instruction has been collected and published. In our own country this material was published by the U. S. Bureau of Education in the form of thirteen reports. Some of the other countries have not yet completed their work, but about one hundred and sixty such reports have already been published in the twenty-six countries which have joined in this vast undertaking.

In addition to securing these valuable reports the central commission has arranged international meetings for the discussion of some of the fundamental questions relating to mathematical instruction. Such a meeting was held in Paris, France, in April of the present year, and the two subjects under consideration were: (1) The results obtained by the introduction of differential calculus in the advanced classes of the secondary schools, and (2) the place and the role of mathematics in higher technical education.

Some of the leading French mathematicians (including Appell,Darboux, Borel and d'Ocogne) took an active part in the deliberations. Professor Borel emphasized the fact that mathematics is not composed of a linear sequence of theorems such that each depends upon the preceding one. If this were the case, the only possible changes in methods of instruction would relate to what theorems could be omitted in this sequence or what theorems could be substituted for others. On the contrary, the number of different routes leading from first principles to an advanced mathematical proposition is often exceedingly large, and hence arises the possibility of employing widely different methods to achieve the same general results.

In other words, mathematics is a network formed by intersecting thought roads and the chief aim of the International Commission on the Teaching of Mathematics is to secure extensive information as regards the choice of roads in various nations. The Italian member of the central committee, G. Castelnouovo of Rome, stated explicitly in his address during the recent conference at Paris, that the commission did not aim to bring about any great reforms, but aimed to gather facts as regards existing conditions in order that the various nations might be enabled to profit by the experiences of other nations in instituting their own reforms.

In describing mathematics as a network of a certain type of thought-roads, it is not implied that thought is conveyed along these roads as the products of a country are conveyed on a railroad train. On the contrary, thought is developed along these mathematical roads, and the traveler finds continually new difficulties whose solution depends largely upon those encountered earlier. In constructing these roads mathematics is not seeking an intellectual monopoly in order to collect toll from the rest of the intellectual world in succeeding ages. In fact, in most of the newer regions the travelers are too few to encourage such thoughts even if they were not intrinsically repugnant.

There is, however, a considerable number of mathematicians who are interested in constructing unusually attractive toll roads, especially in those regions where travelers are most abundant. Whether the prospects of tolls derived from small royalties constitute the best means to secure improvements in our elementary text-books and whether this system is apt to continue to be efficient are questions which present many difficulties. There appears to be an enormous waste along this line at present resulting from unfruitful duplication.

The financial questions involved in mathematical publications have doubtless much in common with those relating to the publication in other subjects. The journals depend largely upon the universities and the mathematical societies for financial assistance. Lately the American Mathematical Monthly, a journal of collegiate grade, has received financial assistance from more than a dozen colleges and universities, and it has thus been enabled to make many improvements. The Annals of Mathematics, which is a journal of a somewhat more advanced grade, is being published since 1911 under the auspices of Princeton University.

The large mathematical encyclopedias, mentioned above, are being published under the auspices of the Academies of Göttingen, Leipzig, Munich and Vienna, while various governments have been asked to assume the expense of the publication of at least some of the reports prepared under the general direction of the International Commission on the Teaching of Mathematics. The Japanese reports are published both in the Japanese and in the English languages; and all these reports, aggregating already more than ten thousand pages, are for sale by Georg & Co., Geneva, Switzerland.

While mathematical societies generally support publication of advanced grade, they usually have other functions. In many instances membership implies attainments of comparatively high order and hence is attractive in view of the honor and exclusive privileges which it involves. Recently an international mathematical society has been organized with the sole purpose of supporting the publication of the complete works of the most prolific mathematical writer, Leonhard Euler, who died in 1783. Each member of this society is expected to pay at least ten francs annually until this publication is completed, which is expected to require about fifteen years.

Several years ago it was estimated that the complete works of Euler would fill from forty to forty-five volumes, and that the expense would be about half a million francs. As funds had been provided to cover this expense, the publication was begun, but it soon appeared that the estimates were entirely too low and that the expense would be almost twice as large as the original estimate, in view of the additional material found at various places.

The great permanent value of the works of Euler has encouraged the "Schweizerische Naturforschende Gesellschaft zu Lausanne" to make an appeal to all mathematicians, and others interested, to join hands by means of the society mentioned above in securing the completion of this monumental publication. This society seems to be unique in the history of mathematics, but it bespeaks forcibly the spirit of cooperation which has led in recent years to much bigger mathematical undertakings than were possible in former years. The reflex action of these big undertakings on the mathematicians themselves is an element of considerable interest.

The mathematical activities to which we have directed attention in the present article were selected, in the main, on account of their special interest at the present time. The most important activities, however, are those whose permanency has secured for them a place among the fundamental elements which enter unnoticed into our intellectual life, and whose effectiveness is increased by the fact that they are not impeded by effusion. As mathematics is such an old science, the educator naturally looks to its activities with a view to predicting in some measure the future activities of the younger sciences. Hence it is especially interesting to note those activities which imply vigor, and promise for still greater achievements in the mathematical sciences.

  1. E. B. Wilson, Bulletin of the American Mathematical Society, Vol. 18 (1911-12), p. 465.