minded person that the whole number of women in all nations and all times who may have been said to be so placed as justly to be considered in the comparison, is far less than that of the men so placed in any great nation in a single century. It is only within the last few decades that any considerable number of girls have grown up with any other notion than that serious intellectual work in their sex is a monstrosity; and only in England and America has a different view of the matter been widely entertained even in our time, the "woman movement" having attained an important character in Germany only within the past five or ten years.

In the second place, I have endeavored to emphasize the fact that even this numerical exclusion of all but an extremely small fraction of the sex does not begin to measure the disadvantage of women in the comparison. Every one must recognize that the minute fraction which may properly be considered at all has not been surrounded by the atmosphere, affected by the agencies, impelled by the stimuli, which exercise so incalculable an influence upon human achievement; but there is a not unnatural tendency to think that after all there ought to have been some women who had risen superior to all these things. It is for this reason that I have dwelt on the utter absence of intellectual greatness in periods of national decadence, and on the universally acknowledged influence of general conditions upon the flourishing of literature, art, and science. But surely the ordinary differences in these conditions, which have been uniformly found sufficient wholly to prevent the emergence of genius among men, are insignificant in comparison with the unfavorable difference which has always existed in the conditions surrounding women, in every direction of intellectual effort.

A final word as to the importance or unimportance of the whole discussion. There would be no harm in leaving the question entirely open; what is to be deplored is an erroneous belief that it has been settled. In a matter of keen human interest however unsubstantial or speculative that interest may be any error is to be deplored, simply as error. But in this case there is another and more special reason for regret.

It is that the conclusion which I have been engaged in controverting is sure to be understood by the generality of people as meaning vastly more than in its exact terms it professes to convey. Even those who are not "the generality" slide imperceptibly into this exaggeration of its purport. The most that could be claimed as shown by history, even were the considerations adduced in the present article wholly ignored, would be that women cannot reach the highest heights; yet we see the very able and gifted writer of the article to which this is a reply, belittling achievements of members of her own sex which are of undeniable greatness, a thing which can hardly be ascribed to anything else than the bias due to a preconceived theory. Whether or not any woman can be as great as the greatest men, it is quite certain that some women can be as great as very great men; for some women have been.

The capacity for doing excellent work in the most difficult departments of university study, positive experience has now shown to be no more abnormal among women than among men. Yet we see surviving to our own day — and probably, if the truth were known, still very widely entertained the notion that, leaving out a possible lusus naturæ here and there, women are incapable of doing high university work. In a recent number of a prominent Review, I find a Lecturer on History in the University of Cambridge making the utterly ridiculous statement that he had "never seen a woman's paper equal to a man's"; which, if understood literally, would mean that the ablest of the women whose papers have ever come under his eye was not equal to the most stupid of the men. This doubtless is not what he meant to say, but the expression shows the persistence in his mind of an utterly baseless belief in woman's essential inferiority. Any one whose memory extends back twenty-five years will remember the time when the belief was practically universal that women were incapable of mastering the higher mathematics. Go back a little farther, and we find a schoolmaster in one of the principal towns of Massachusetts set down as a visionary because he proposed to undertake to teach girls fractions. A century ago no less a man than Kant

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declared the unfitness of women for the study of geometry. "It is generally believed in Germany," writes Professor Klein,1 one of the greatest of living mathematicians, "that mathematical studies are beyond the capacity of women"; but he assures us that the women who have attended the mathematical course at Göttingen "have constantly shown themselves from every point of view as able as their male competitors." And it may be remarked that the mathematical work here referred to is as far beyond anything that was taught in America before the opening of the Johns Hopkins University as the work in our best colleges in those days was beyond that of a country school.

It is because the view combated in this article not only is lacking in foundation, but tends to strengthen the hold of beliefs which still cling to the majority of persons, though they have been amply proved to be erroneous, that I feel it to be important that it should be opposed. It is impossible to determine the relative powers of men and women; it will be long before experience can show, even with a moderate degree of probability, what limits there may be to the possibilities of woman in the realm of intellect. Let us not, in the meanwhile, belittle the actual work of women, in pursuance of a baseless dogma of essential inferiority. Let us refrain, for instance, from saying, with Mr. Gosse, that women cannot write poetry requiring art "because they lack the artistic impulse," when we know not only that they have written such poetry, but that paintings like those of Miss Mary Cassatt or Mme. DemontBreton, not to speak of older names, show the possession of an extremely high artistic impulse. Let Americans, at least, not talk glibly of women's power in scientific discovery being essentially inferior to men's, until such time as some American mathematician receives as high recognition as that bestowed by the French Academy on the work of Sonia Kovalewski, the judgment being pronounced without knowledge of the writer's sex. Let us not regard the results of women's attempts in poetry and

1 "Les Femmes dans la Science." By A. Rebiere, Paris, 1897 (page 318).

music as utterly fatal to aspirations however high, when we remember that our country has thus far produced neither a great composer nor, in the high sense of the word, a great poet. Let us not lay too great stress on the fact that "in dramatic literature no woman has ever gained for herself any lasting fame," when it is remembered that America has never produced a drama of even moderate excellence; while, on the other hand, I find Professor Kuno Francke, of Harvard, saying in The Nation a few weeks ago, of a drama recently written by a German woman, Gisela von Arnim, the wife of Hermann Grimm, that its chief scene is "one of the most affecting in dramatic literature," that the personages of the play are "characters of genuine grandeur," and that in it the longings and aspirations of the author "have found a supreme poetic expression." In a word, as to what woman may do in the future, let us frankly acknowledge that the future alone can decide, the experience of the past being far too slight to furnish the materials for a forecast; and as to what women have done in the past, or are doing in the present, let us recognize it as what it is, and not as what, in accordance with an unproved generalization, we imagine it. must of necessity be.

THE MATHEMATICIAN AND THE ENGINEER1

It is not a little remarkable that the pure mathematician still retains a very exaggerated view of the value of the services he is, in the immediate present, capable of rendering to the practicing engineer. What the value of his researches may be to engineers of a future generation is another matter, and judging from past experience, it may very well be high; but as matters stand, there is no justification whatever for such a claim as that recently made in Nature by Mr. D. M. Mair. This writer, in an otherwise not unfair appreciation of the actual state of affairs, makes the following statement: "The engineer has an outfit of mathe

1 From Engineering, May 9, 1913.

matical tools sufficient for his ordinary needs, but at times he meets with problems for which his tools are useless. He may then spend thousands of pounds on the determination of some point which the mathematician, could have settled for a fivepound note." We believe that this belief is not uncommon among mathematicians, but on considering the many advances made in engineering during the past twenty-five years, it is impossible to recall a single instance in which this view could be even partially justified. The fact is, the aid of the mathematician is not wanted until the engineer by hard thinking and by careful experiment has solved in some fashion the problem in view. The mathematician may then, taking the engineer's coefficients and experience as a basis, show that certain methods of calculation may be shortened, and a few hours may thus be saved and labor economized. The earlier of the builders of continuous girders, for example, were, we believe, unacquainted with the theorem of three moments, and had to determine the reactions at the various piers by a more or less tentative process. Direct methods, of which the theorem of three moments is one, are now available, by the use of which a considerable reduction may be made in the amount of arithmetical work necessary.

An analogous case arose some little time ago, when a firm found it necessary to calculate the critical speed of the rotor of a generator on the assumption that the direction of the shaft. was fixed at each bearing. A direct solution of this problem is not, of course, difficult, but actually the solution was found by a system of trial and error. A little more time was needed than by the direct process, which was at least partially compensated for by the fact that each successive approximation served to check the accuracy of the preceding one.

Even in this matter of time-saving, however, the services of the mathematician, pure and simple, are steadily becoming of less and less immediate importance, since the average engineer is himself acquiring a greater and greater mastery of mathematical methods, and he has the great advantage of realizing much more adequately than it would be possible for an "external" mathematician to do, the physical characteristics of the problem