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and safeguard his freedom with the zeal of his European colleague. It is too commonly assumed that loyalty implies lying.

The investigators in pure mathematics form a small army of about 2,000 men and a few women.' The question naturally arises, What is this little army trying to accomplish? A direct answer is that they are trying to find and to construct paths and roads of thought which connect with or belong to a network of thought roads commonly known as mathematics. Some are engaged in constructing trails through what appears an almost impassable region, while others are widening and smoothing roads which have been traveled for centuries. There are others who are engaged in driving piles in the hope of securing a solid foundation through regions where quicksand and mire have combined to obstruct progress.

A characteristic property of mathematics is that by means of certain postulates its thought roads have been proved to be safe and they always lead to some prominent objective points. Hence they primarily serve to economize thought. The number of objects of mathematical thought is infinite, and these roads enable a finite mind to secure an intellectual penetration into some parts of this infinitude of objects. It should also be observed that mathematics consists of a connected network of thought roads, and mathematical progress means that other such connected or connecting roads are being established which either lead to new objective points of interest or exhibit new connections between known roads.

The network of thought roads called mathematics furnishes a very interesting chapter in the intellectual history of the world, and in recent years an increasing number of investigators have entered the field of mathematical history. The results are very encouraging. In fact, there are very few other parts of mathematics where the progress during the last 20 years has been as great as in this history. This progress is partly reflected by special courses in this subject in the leading universities of the world. While the earliest such course seems to have been given only about 40 years ago, a considerable number of universities are now offering regular courses in this subject, and these courses have the great advantage that they establish another point of helpful contact between mathematics and other fields.

Mathematical thought roads may be distinguished by the facts that by means of certain assumptions they have been proved to lead safely to certain objective points of interest, and each of them connects, at least in one point, with a network of other such roads which

1 Between 5 and 10 per cent of the members of the American Mathematical Society are women, but the per cent of women in the leading foreign mathematical societies is much smaller. Less than 2 per cent of the members of the national mathematical societies of France, Germany, and Spain are women, according to recent lists of members. The per cent of important mathematical contributions by women does not appear to be larger, as a rule, than that of their representation in the leading societies. The list of about 300 collaborators on the great new German and French mathematical encyclopedias does not seem to include any woman. Possibly women do not prize sufficiently intellectual freedom to become good mathematical investigators. Some of them exhibit excellent ability as mathematical students.

were called mathematics, uabhuara, by the ancient Greeks. The mathematical investigator of the present day is pushing these thought roads into domains which were totally unknown to the older mathematicians. Whether it will ever be possible to penetrate all scientific knowledge in this way and thus to unify all the advanced scientific subjects of study under the general term of mathematics, as was the case with the ancient Greeks,' is a question of deep interest.

The scientific world has devoted much attention to the collection and the classification of facts relative to material things and has secured already an immensely valuable store of such knowledge. As the number of these facts increases, stronger and stronger means of intellectual penetration are needed. In many cases mathematics has already provided such means in a large measure; and, judging from the past, one may reasonably expect that the demand for such means will continue to increase as long as scientific knowledge continues to grow. On the other hand, the domain of logic has been widely extended through the work of Russell, Poincaré, and others; and Russell's conclusion that any false proposition implies all other propositions whether true or false is of great general interest.

During the last two or three centuries there has been a most remarkable increase in facilities for publication. Not only have academies and societies started journals for the use of their members, but numerous journals, inviting suitable contributions from the public have arisen. The oldest of the latter type is the Journal des Sçavans, which was started at Paris in 1665, while the Transactions of the Royal Society of London, started in the same year, should probably be regarded as the oldest of the former type. These journals have done an inestimable amount of good for the growth of knowledge and the spread of the spirit of investigation. At the present time more than 2,000 articles which are supposed to be contributions to knowledge in pure mathematics appear annually in such periodicals. In addition to these there is a growing annual list of books.

The great extent of the fields of mathematics and the rapid growth of this literature have made it very desirable to secure means of judging more easily the relative merit of various publications. Along this line our facilities are still very meager and many serious difficulties present themselves. In America we have the book reviews and the indirect means provided by the meetings of various societies and by such publications as the "American Men of Science."

The most important aid to judge contemporaneous work is furnished by a German publication known as the Jahrbuch über die Fortschritte der Mathematik. In this work there appear annually about 1,000 pages of reviews of books and articles published two or three years earlier. These reviews are prepared by about 60 different

1 The term mathematics was first used with its present restricted meaning by the Peripatetic School. Cantor, "Vorlesungen über Geschichte der Mathematik," vol. 1 (1907), p. 216.

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mathematicians who are supposed to be well prepared to pass judgment on the particular books and articles which they undertake to review. While these reviews are of very unequal merit, they are rendering a service of the greatest value.

The main object of such reviews is to enable the true student to learn easily what progress others are making, especially in his own field and in those closely related thereto. They serve, however, another very laudable purpose in the case that they are reliable. We have the pretender and the unscrupulous always with us, and it is almost as important to limit their field of operation as to encourage the true investigator. "Companions in zealous research" should be fearless in the pursuit of truth and in the disclosure of falsehood, since these qualities are essential to the atmosphere which is favorable to research.

While the mathematical investigator is generally so engrossed by the immediate objects in view that he seldom finds time to think of his services to humanity as a whole, yet such thoughts naturally come to him more or less frequently, especially since his direct objects of research seldom are well suited for subjects of general conversation. If these thoughts do come to him they should bring with them great inspiration. Who can estimate the amount of good mathematics has done and is doing now? If all knowledge of mathematics could suddenly be taken away from us there would be a state of chaos, and if all those things whose development depended upon mathematical principles could be removed, our lives and thoughts would be pauperized immeasurably. This removal would sweep away not only our modern houses and bridges, our commerce and landmarks, but also most of our concepts of the physical universe.

Some may be tempted to say that the useful parts of mathematics are very elementary and have little contact with modern research. In answer, we may observe that it is very questionable whether the ratio of the developed mathematics to that which is finding direct application to things which relate to material advantages is greater now than it was at the time of the ancient Greeks. The last two centuries have witnessed a wonderful advance in the pure mathematics which is commonly used. While the advance in the extent of the developed fields has also been rapid, it has probably not been relatively more rapid. Hence, the mathematical investigator of to-day can pursue his work with the greatest confidence as regards his services to the general uplift both in thought and in material betterment of the human race. All of his real advances may reasonably be expected to be enduring elements of a structure whose permanence is even more assured than that of granite pillars.

1 In 1726, arithmetic and geometry were studied during the senior year in Harvard College. Natural philosophy and physics were still taught before arithmetic and geometry. Cajori, "The Teaching and History of Mathematics in the United States," 1890, p. 22.

THE CONNECTION BETWEEN THE ETHER AND MATTER.1

By M. HENRI POINCARÉ.

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When M. Abraham came to me and asked that I close this series of sessions of the Société française de Physique, I was at first inclined to refuse. It seemed as if each subject had been completely discussed and that I could have nothing to add to that which had already been so well said. I could only try to put in words the impression which seemed to emerge as a summary of all the discussions, and that impression was so definite that each of you must have felt it. I did not see how I could make it any clearer by forcing myself to put it into words. But M. Abraham insisted with such good grace that I resigned myself to the inevitable difficulties of which the greatest is to repeat what each one of you has long since felt, and the least is to run through a maze of diverse subjects without the time to dwell on any one of them.

One thought must at once have struck all those present. The old mechanical and atomic hypotheses have, during recent years, become so plausible that they have ceased to seem like hypotheses; atoms are no longer just a convenient fiction. It seems almost as if we could see them, now that we know how to count them. A theory assumes reality and gains in probability when it accounts for new facts. Yet this may result in different ways. Generally it has to be enlarged to include the new data. Sometimes it loses in precision as it becomes broader. Sometimes it becomes necessary to engraft upon it an accessory hypothesis which plausibly fits in with it, but which nevertheless is somewhat foreign to it, and contrived expressly to fit a certain case. Then it can scarcely be said that the new facts confirm the original hypothesis, only that they are not inconsistent with it. Or, again, there may be between the new facts and the old, for which the hypothesis was originally conceived, such an intimate connection that whatever theory renders account of one must, because of that connection, render account of the other as well. Then the new data which fall in with the old are really only apparently new.

1 An address delivered before the Société française de Physique, April 11, 1912. Reprinted by permission from Journal de Physique, Paris, 5th series, vol. 2, May, 1912.

It is quite different when we discover a coincidence which could have been predicted, and is thus not the result of chance, and especially when that coincidence is a numerical value. Now, there are coincidences of this last nature which have recently brought confirmation to our atomic views.

The kinetic theory of gases has thus received unexpected corroboration. New theories have been very closely patterned after the kinetic theory, for instance, the theory of solutions as well as the electronic theory of metals. The molecules of a dissolved substance, as well as the free electrons to which metals owe their electrical conductivity, behave just as do the molecules of a gas within its inclosure. The parallelism is perfect and can be followed even to numerical values. Thus what seemed doubtful becomes probable. Each one of these three theories, if it had to stand by itself, would seem only an ingenious hypothesis for which we might substitute other explanations equally probable. But when, as in each of the three cases, a different explanation would be necessary, the coincidences found would be inadmissible as the result of chance, whereas the kinetic theories make the coincidences necessary. Further, the theory of solutions quite naturally leads us to that of the Brownian movements, where it is impossible to consider the thermal agitation as a theoretical fiction, since it is actually seen under the microscope. The remarkable counting of the number of atoms by Perrin completed the triumph of the atomic theory. What carries our conviction are the multiple concordances among the results obtained by completely different procedures. But a short time ago we would have thought ourselves fortunate if the numbers found had the same number of digits; we would have asked only that the first significant figure should be the same. That first figure we know to-day. What is more remarkable, we are now discussing even the most diverse properties of the atoms. In the processes used with the Brownian phenomenon, or in those used for the law of radiation, we do not deal directly with the number of atoms, but with their degrees of freedom of movement. In that process where we consider the blue of the sky, the mechanical properties of the atoms come into play; the atoms are looked upon as producing an optical discontinuity. Finally, when we take in hand radium, what we observe is the emission of projectiles. Here, were there discordances, no embarrassment would have been felt, but happily there were none.

The atom of the chemist is now a reality. But that does not mean that we have reached the ultimate limit of the divisibility of matter. When Democritus invented the atom he considered it as the absolutely indivisible element within which there would be nothing further to distinguish. That is what the word meant in Greek. It was for that reason that it was coined. Beyond the atom he wished

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