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algebra to one year's work, but students entering with this minimum preparation are sometimes not allowed to proceed with the regular mathematical classes in the university. Freshmen courses in mathematics differ widely, but the Variety of most common subjects are advanced algebra, plane trigo- courses in nometry, and solid geometry. The most common subjects mathematics of a somewhat more advanced type are plane analytic geometry, differential and integral calculus, and spherical trigonometry. Beyond these courses there is much less uniformity, especially in those institutions which aim to complete a well-rounded undergraduate mathematical course rather than to prepare for graduate work. Among the most common subjects beyond those already named are differential equations, theory of equations, solid analytic geometry, and mechanics.

A very important element affecting the mathematical courses in recent years is the rapid improvement in the training of our teachers in the secondary schools. This has led to the rapid introduction of courses which aim to lead up to broad views in regard to the fundamental subjects. In particular, courses relating to the historical development of concepts involved therein are receiving more and more attention. Indirect historical sources have become much more plentiful in recent years through the publication of various translations of ancient works and through the publication of extensive historical notes in the Encyclopédie des Sciences Mathématiques and in other less extensive works of reference.

The problem presented by those who are preparing to teach mathematics may at first appear to differ widely from that presented by those who expect to become engineers. The latter are mostly interested in obtaining from their mathematical courses a powerful equipment for doing things, while the former take more interest in those developments which illumine and clarify the elements of their subject. Hence the prospective teacher and the prospective engineer might appear to have conflicting mathematical interests. As a matter of fact, these interests are not con

flicting. The prospective teacher is greatly benefited by the emphasis on the serviceableness of mathematics, and the prospective engineer finds that the generality and clarity of view sought by the prospective teacher is equally helpful to him in dealing with new applications. Hence these two classes of students can well afford to pursue many of the early mathematical courses together, while the finishing courses should usually be different.

The rapidly growing interest in statistical methods and in insurance, pensions, and investments has naturally directed special attention to the underlying mathematical theories, especially to the theory of probability. Some institutions have organized special mathematical courses relating to these subjects and have thus extended still further the range of undergraduate subjects covered by the mathematical departments. The rapidly growing emphasis on a college education specially adapted to the needs of the prospective business man has recently led to a greater emphasis on some of these subjects in several institutions.

The range of mathematical subjects suited for graduate students is unlimited, but it is commonly assumed to be desirable that the graduate student should pursue at least one general course in each one of broader subjects such as the theory of numbers, higher algebra, theory of functions, and projective geometry, before he begins to specialize along a particular line. It is usually taken for granted that the undergraduate courses in mathematics should not presuppose a knowledge of any language besides English, but graduate work in this subject cannot be successfully pursued in many cases without a reading knowledge of the three other great mathematical languages; viz., French, German, and Italian. Hence the study of graduate mathematics necessarily presupposes some linguistic training in addition to an acquaintance with the elements of fundamental mathematical subjects.

Historical studies make especially large linguistic demands in case these studies are not largely restricted to predigested material. This is particularly true as regards the

older historical material. In the study of contemporary mathematical history the linguistic prerequisites are about the same as those relating to the study of other modern mathematical subjects. With the rapid spread of mathematical research activity during recent years there has come a growing need of more extensive linguistic attainments on the part of those mathematicians who strive to keep in touch with progress along various lines. For instance, a thriving Spanish national mathematical society was organized in 1911 at Madrid, Spain, and in March, 1916, a new mathematical journal entitled Revista de Matematicas was started at Buenos Aires, Argentine Republic. Hence a knowledge of Spanish is becoming more useful to the mathematical student. Similar activities have recently been inaugurated in other countries.

college mathematics

Until about the beginning of the nineteenth century the History of courses in college mathematics did not usually presuppose a mathematical foundation carefully prepared for a superstructure. According to M. Gebhardt, the function of teaching elementary mathematics in Germany was assumed by the gymnasiums during the years from 1810 to 1830.1 Before this time the German universities usually gave instruction in the most elementary mathematical subjects. In our own country, Yale University instituted a mathematical entrance requirement under the title of arithmetic as early as 1745, but at Harvard University no mathematics was required for admission before 1803.

On the other hand, L'Ecole Polytechnique of Paris, which occupies a prominent place in the history of college mathematics, had very high admission requirements in mathematics from the start. According to a law enacted in 1795, the candidates for admission were required to pass an examination in arithmetic; in algebra, including the solution of equations of the first four degrees and the theory of series; and in geometry, including trigonometry, the applications of algebra to geometry, and conic sec1 Internationale Mathematische Unterrichtskomission, Vol. 3, No. 6 (1912), page 2.

tions. It should be noted that these requirements are more extensive than the usual present mathematical requirements of our leading universities and technical schools, but L'Ecole Polytechnique laid special emphasis on mathematics and physics and became the world's prototype of strong technical institutions.

The influence of L'Ecole Polytechnique was greatly augmented by the publication of a regular periodical entitled Journal de l'Ecole Polytechnique, which was started in 1795 and is still being published. A number of the courses of lectures delivered at L'Ecole Polytechnique and at L'Ecole Normale appeared in the early volumes of this journal. The fact that some of these courses were given by such eminent mathematicians as J. L. Lagrange, G. Monge, and P. S. Laplace is sufficient guarantee of their great value and of their good influence on the later textbooks along similar lines. In particular, it may be noted that G. Monge gave the first course in descriptive geometry at L'Ecole Normale in 1795, and he was also for a number of years one of the most influential teachers at L'Ecole Polytechnique.

A most fundamental element in the history of college mathematics is the broadening of the scope of the college work. As long as college students were composed almost entirely of prospective preachers, lawyers, and physicians, there was comparatively little interest taken in mathematics. It is true that the mental disciplinary value of mathematics was emphasized by many, but this supposed value did not put any real life into mathematical work. The dead abstract reasonings of Euclid's Elements, or even the number speculations of the ancient Pythagoreans, were enough to satisfy most of those who were looking to mathematics as a subject suitable for mental gymnastics.

On the other hand, when the colleges began to train men for other lines of work, when the applications of steam led to big enterprises, like the building of rail

1 Journal de l'Ecole Polytechnique, Vol. 1 (1896), part 4, page lx.

roads and large ocean steamers, mathematics became a living subject whose great direct usefulness in practical affairs began to be commonly recognized. Moreover, it became apparent that there was great need of mathematical growth, since mathematics was no longer to be used merely as mental Indian clubs or dumb-bells, where a limited assortment would answer all practical needs, but as an implement of mental penetration into the infinitude of barriers which have checked progress along various lines and seem to require an infinite variety of methods of penetration.

The American colleges were naturally somewhat slower than some of those of Europe in adapting themselves to the changed conditions, but the rapidity of the changes in our country may be inferred from the fact that in the first half of the nineteenth century Harvard placed in comparatively short succession three mathematical subjects on its list of entrance requirements; viz., arithmetic in 1802, algebra in 1820, and geometry in 1844. Although Harvard had not established any mathematical admission requirements for more than a century and a half after its opening, she initiated three such requirements within half a century. It is interesting to note that for at least ninety years from the opening of Harvard, arithmetic was taught during the senior year as one of the finishing subjects of a college education.1

The passage of some of the subjects of elementary mathematics from the colleges to the secondary schools raised two very fundamental questions. The first of these concerned mostly the secondary schools, since it involved an adaptation to the needs of younger students of the more or less crystallized textbook material which came to them from the colleges. The second of these questions affected the colleges only, since it involved the selection of proper material to base upon the foundations laid by the secondary schools. It is natural that the influence of the colleges 1 F. Cajori, Teaching and History of Mathematics in the United States, 1890, page 22.