only to the demand on the part of students for the education offered, but to the demand from the field for the product of the school as shown by the relation of output to the growth of the mineral industry. If now there be drawn in this same diagram, a curve whose ordinates represent the value of the mineral product of the country in millions of dollars, the dotted line is obtained. Its correspondence with the curve of graduates is not only probably unexpected, but it is most striking. At first thought the correspondence might appear fortuitous. However it covers too many years to be accidental. If the growth of the value of the mineral product of the country represents fairly the growth of the mineral interest, and we commonly accept such to be the case, the close agreement of the curves would seem to clearly indicate that the output of mining graduates keeps very satisfactory pace with the growth of the industry to which they are destined. In other words, that the number of graduates does "increase at least as fast as the mining interest" in spite of the fact that the latter "grows prodigiously."

But if the schools are really turning out a fairly satisfactory "article of mining engineer," it would appear that work formerly in the hands of unschooled men would to some extent at least pass to those with college training and that, therefore, the number of graduates ought to increase somewhat faster than the value of the mining product. A closer inspection of the figures shows that with the six schools under consideration this is really the case. The numbers for a single year are of little value, but if we divide the twelve years available for comparison into three periods of four years

each, we shall find that these schools graduated for the first of these periods one man to (roughly) 13.6 millions of product; in the second period, one to 10.2 millions, and for the last four-year period, one to 9.4 millions of the total mineral product. The persistence of this gain seems significant.

But if it is true, as supposed, that the mining school product is gradually taking the place of the so-called practical man, this should appear as well in the ratio of graduates turned out to the number of persons engaged in the mining industry. Previous to the census of 1900 it is not easy to get definite figures as to the total number of persons employed in mining pursuits, and, rather than make or take estimates of this total, the decision was made to take the number of wage earners employed in mining as given by the U. S. census for the basis of this comparison. This number at least seems to have been definitely determined and it will perhaps be accepted as fairly representing the condition of the mining industry with reference to numbers employed. It is of course not to be supposed that the number engaged in engineering and superintendence bears a constant ratio to that of the wage earners. The modern plan of consolidation, where possible, tends to reduce this ratio, while the extension of exploration work and the opening of new properties tend to increase it. Yet such variations may be neglected where, as in the case in hand, great accuracy is not required.

The number of wage earners in mining was taken from the tenth, eleventh and twelfth census for the respective years. The number for the current year was estimated by considering the rates of increase for

the two decades between 1880 and 1900. In hundred thousands these are as follows:

Joining the points determined by means of these numbers, there is obtained the line marked "wage earners," which we choose to take as approximately showing the increase in numbers of mining people.

The divergence of the other three curves from this line is striking. If the evidence here presented is to be accepted we may conclude that both the enrollment in our mining schools and their output of graduates shows a steady and marked increase as compared with the number of persons engaged in the mining industry.*

It is fair to conclude then that judged to-day on the same basis and by the same reasoning as that on which in the early nineties it was proposed to condemn them, the American mining schools, to quote again the optimistic Richards, "have amply proved the necessity for their existence."

The most enthusiastic friend of any one of these schools, if at all familiar with its work, will hardly claim that it has succeeded in completely solving the problems that are presented to it and has attained either the ideal curriculum or methods for training a mining engineer. Far less likely will this claim be made by anyone belonging to the school and engaged in the effort

* Attention might be drawn to the increase in value of output per wage earner employed as shown in the diagram. This fact is of course well known from other sources.

to preserve some semblance of relation between its resources and its needs, and between its ideals and its attainments. But whatever may be the defects in individual schools, whether in curricula or the more important methods of instruction, the American mining schools are meeting a real need, and, if we may accept what appears to be the verdict of the mining world, by whose judgment their work must stand or fall, they meet the need in a fairly acceptable way.

SOME HINTS ON TEACHING MATHEMATICS TO

ENGINEERING STUDENTS.

BY FLORIAN CAJORI,

Professor of Mathematics and Dean of the School of Engineering, Colorado College.

The tendency of the present time is to arithmetize mathematics. The earlier explanation of irrational numbers, like that of fractions, involved the idea of measurement. Formerly an irrational number was defined as the expression of the incommensurable ratio of two geometrical quantities-that is, as the ratio between the quantities which have no common measure. But this mode of treatment involves certain logical difficulties which G. Cantor, K. Weierstrass and others have endeavored to remove by treating irrational number in a manner free of ratio and measurement, and of all geometrical considerations. The pupil is warned to take no notice of geometric figures or diagrams. Among them, if geometry is studied, "geometry without diagrams is the order of the day."

The question naturally arises whether in the training of engineers the teacher should endeavor to follow in the footsteps of the modern logicians of mathematics. Should he aim at extreme mathematical rigor? Probably the majority of teachers agree that this should not be the case, that it is well, as a rule, to lead beginners along the same road that the race has taken in the acquirement of knowledge. In the historical development of mathematics the naïve treatment has invariably pre