has considerable weight; nevertheless it would be a mistake to discard these ballots. The teacher of mathematics, more than any one else, has the opportunity to observe pupils pursuing this subject. Moreover, he, more than others, is apt to indulge in introspection regarding his own experiences as a pupil. Those who insist on eliminating the votes cast by mathematicians, will find the count to be 413 to 123, still a substantial majority of 77 per cent favoring mathematics. Our statistical results are exhibited in the table on page 303, which shows also the votes of the different groups of thinkers. For ancient and modern times taken together, the vote on the value of mathematics in liberal education is 71 to 26 among philosophers, psychologists, and theologians; 58 to 21 among educators; 186 to 5 among mathematicians; 18 to 4 among scientists who were not mathematicians; 47 to 24 among literary men and statesmen; 219 to 48 among business men.

We have based our statistics upon a "fair sample" of witnesses. We believe that this record of human experience teaches mankind truths which it should not be necessary to acquire all over again by trial and error.

And yet, the validity of our statistics favoring mathematics may be questioned on the ground that the ideas of the theoretical educators are new ideas which of necessity are ideas of minorities. It is true that advanced thinkers at first stand alone. When Pasteur first advanced his germtheory of fermentation, he represented a minority. But does it follow that majorities are always in the wrong and minorities always in the right? Is there a sanctity in minorities? On the contrary, is it not a fact that minorities are usually in the wrong, that most of the hypotheses advanced in science are not true? It is indeed well that new ideas should be obliged to fight for existence, and it is a fact that most of them fall in combat. The history of science is largely a graveyard of fallen theories. Educators denying the great value of mathematics in liberal education are a minority; their arguments have been considered by some of our best thinkers, but found to be unconvincing.

The bulk of their arguments refer to defective teaching of mathematics and therefore call for the better training of teachers, but not for the virtual elimination of algebra and geometry from the required courses in secondary schools.

Another question arises, whether recent measurements and statistics in education, in spite of their defects, should not be accepted as superior to the observations of even the best thinkers and seers of the past. When in physics, Fraunhofer invented the spectroscope, he obtained results on spectral lines which completely displaced the previous crude determinations. When the elder Herschel, with his great reflecting telescope, discovered Uranus, this was at once added to the old planetary group. Why do the measurements in education not meet with equally immediate and hearty acceptance? To us there seem to be two principal reasons. One we have already noted, namely, that measurements of intelligence are incomparably more difficult than measurements of material phenomena. There are very many variable and uncertain factors in mind study. The mind is like an ocean whose depths have not been sounded. In physics one careful experiment suffices to establish a point. In education, it is admitted by all investigators that it is not safe to accept as generally valid, any statistical observations made upon only a few individuals. Again, the methods that have been devised for mental tests contain no radically new modes of procedure. They are labor-saving devices-short questions, frequently of the nature of little puzzles, requiring immediate answer. They aim to ascertain in an hour or two what is ordinarily obtained through long and tedious examinations, or from observations of an individual, extending over long stretches of time. For purposes of immediate and approximate classifications these tests are very serviceable. But for delicate measurement, it is not evident that these instantaneous tests are more reliable, or indeed as reliable as the older observations, in which the time for both observer and observed extended, perhaps, over a quarter or half a century.

Still another possible objection to our historical mode of attack is that, in the sifting of truth, ideas rather than votes should be considered. The importance of ideas we heartily concede. We respect the illumination cast by privileged intellects. We have studied our problem from that angle and find the arguments in favor of mathematics to surpass those against it, even more decisively than does the mere count of witnesses. Lack of time prevented a full presentation of views. On the one side, we have the wellnigh unanimous arguments of mathematicians and the arguments of leading philosophers, educators, and business men. The arguments against mathematics come almost exclusively from writers who possessed only a meager knowledge of this science, or who were unfortunate in the kind of mathematical instruction they themselves received, or who had no idea of the aims of mathematical instruction as held by the more progressive and far-sighted teachers of this science. Some of the arguments of the opponents relate to the ill effects of a study of mathematics when pursued for a long time, to the exclusion of all other subjects. Such arguments are beside the point, for no one advocates such an exclusion.

Certain other possible objections should be considered. The philosophers, statesmen, educators, and mathematicians, whose judgments were consulted are, of necessity, drawn from the distinguished men of the ages. The views of the inconspicuous men, as a rule, are not preserved. The claim has been pressed with insistence that, to attribute the greatness of men to their inclusion of mathematics in their school studies, is to reason post hoc, ergo propter hoc, where one runs the risk of attributing the effect to the wrong cause; perhaps mathematics did not contribute at all to their greatness. With equal insistence we state that our argument has not been of that character. We have not taken a census of successful men who are known to have studied mathematics and attributed their success to mathematics. What we have done is to list the men who from their own personal experience in life, or from their observa

tions of others, themselves arrived at the conclusion that mathematics does possess or does not possess great educational value. We have taken a census of "experiences" and "observations." These experiences and observations are the more valuable because of the prominence of the men. Their very ability is somewhat of a guaranty against the error of attributing the effects to the wrong causes. Moreover, if among our 727 witnesses, such errors did occur, they are not likely to have fallen systematically on one side of the question. A comprehensive view of our situation is this: Those who wholly reject the voice of the past in these matters are drifting helplessly from one fad to another, for the reason that present-day measurements of educational values have not yet attained the precision necessary to serve as safe guides.

The conclusions of our historic study are reinforced by the results of our modern neglect of elementary mathematics in the United States. While in Europe there have been fluctuations in the time-allowance for mathematics in schools, this subject has been taken much more seriously than it is taken now in America. With us, there has been for some years a propaganda against this study in our schools. Both parents and pupils have been assured by educational theorists that algebra and geometry are comparatively unimportant and useless. Even teachers of mathematics have been seized by that idea. In such an atmosphere it is rare for a pupil in mathematics to acquire ideals of procedure, appreciation of logical methods, a recognition, for example, of the wide reach in modern affairs of the concept of functional relationship. This lack of ideals has caused instruction to languish. No longer can teachers say with Tyndall: "I have seen the boy's eye brighten . . . . with a pleasure of which the ecstacy of Archimedes was but a simple expansion." There is a total absence of the idealism. of Plato, who insisted on arithmetic and geometry, not only for practical needs of life, but for the training of the mind, "for the soul itself." The effects are felt in our higher institutions. Recent reports indicate an increased number