we are to seek any causes for Yale's action other than the obvious ones mentioned at the beginning of this article, we are to seek them in this quarter. The measure is radical only on its surface; in its character it is essentially conservative. Radical perhaps it may be in putting women's colleges and their graduates on absolutely the same level as men's; but conservative it certainly is in maintaining the old distinction between college and university: between the college life of the undergraduate students, women as well as men, on the one hand, and on the other hand the true university work of special education, with the radically different problems which it presents for solution.

YALE UNIVERSITY,

NEW HAVEN, CONN.

ARTHUR T. HADLEY.

VIII.

DISCUSSIONS.

CERTAIN VIEWS OF HERBART ON MATHEMATICS AND NATURAL SCIENCE.

Herbart always writes as a mathematician. His psychology and his philosophy are intwined with the calculus. He becomes at times a special pleader for mathematics, as Herbert Spencer does for science: it should be found in the "beginning, middle, and end" of every course of study for "six hours weekly." Yet when we learn that Herbart includes in the term Mathematik all that is not covered by the term history (Geschichte), his thought becomes easier to understand as well as more worthy of heed.

Herbart's writings upon pedagogy abound in suggestions for the co-ordination of studies, and many of these hints are pertinent to questions now in debate. "As the numerous studies that are included in ancient literature form a complete whole, whose central idea is interest in man, so also must the natural sciences be developed (unter sich geordnet) into a similar whole, in order to establish interest in nature, with which also the interest in mathematics stands in close relation."1 As applied to the education of youth, this means that a manysided interest is always to be the source of appeal, of apperception; and that arithmetic, algebra, geometry, physics, botany, etc., are not to be taught as isolated subjects, but as mutually helpful manifestations of nature, as the humanities are of

man.

The history of education, also, repeats itself. In 1802, while Docent at Göttingen, Herbart began to advocate the introduction of geometry, trigonometry, physics, and astronomy among the studies of the younger lads of Germany. He argued that as man grows in knowledge of nature and in dominion over it, chiefly along mathematical lines, so also he should study the sciences that deal with nature with reference to their mathe

Ideen zu einem pädagogischen Lehrplan (1801), Pädagogische Schriften, ed. Willmann, Bd. I, 81.

matical interpretation.

Conversely, as artist, artisan, and scholar all use their knowledge of mathematics chiefly in its application to natural objects in the processes of manufacture or of investigation, it would be wise to learn mathematics principally by studying these applications. If this should be done, there would be no more need of sundering mathematics and nature-study, than of separating arithmetic from geometry. In his monograph on Pestalozzi's Idee eines ABC der Anschauung, Herbart says: "The true place and rank of natural science is not yet sufficiently definite. When it once secures this, its inseparable companion, mathematics, will also be given possession of its rights."

" 2

The case is put even more strongly in the Umriss Pädagogischer Vorlesungen, which presents Herbart's most matured thought: "Mathematical studies, from simple arithmetic to the higher mathematics, must be connected with the study of nature and with practical experiments, in order to gain entrance to the inner thought of the pupil. For even the most elementary mathematical instruction shows itself unpedagogic as soon as it forms for itself alone an isolated mass of concepts, while it either exerts little influence on character (der persönlichen Werth), or still more often falls to the care of a speedy forgetfulness." "For the real scholar a knowledge of mathematics is indispensable, because without it a thorough study of natural science is utterly impossible. . . . . Of course, one may, in the flowery diction of popular lectures, as in a camera obscura, view a series of interesting pictures; striking apparatus heightens the strangeness of the impression; something is remembered, the rest is forgotten: little becomes sap and blood."*

We in this country can hope for little progress of the kind so much desired, until a wider assent is given to thoughts like these: "Mathematical drill must not by any means be allowed to detain the learner too long in a narrow circle, but he should always be advancing in knowledge while gaining skill in applying that already known. If the object were simply to arouse self-activity, then the rudiments would easily be enough to afford any number of problems in which the pupil would enjoy his growing skill, would even find delight in his own little discoveries, without ever suspecting the extent of the science. Pädagogische Schriften, ed. Willmann, Bd. I, S. 124. 3 Ibid., Bd. II, S. 523.

Ibid., Bd. II, S. 263.

Many problems are to be likened to sallies of wit, which are enjoyable at an appropriate time, but must not be allowed to intrude upon the time for work. One should not stop merely to use skillful devices to explain things that become selfevident with further progress. Problems based on natural science are incomparably more serviceable than mere practice problems, as the sciences are so much more readily explained by the use of mathematics when these stand in relation to technical knowledge."

For those who are to pursue more advanced courses, Herbart's objective point is the differential and integral calculus, as the key to his philosophy; but for the elementary school, trigonometry is proposed as the goal. The means by which this is to be reached are very ingeniously worked out, and are given below. Little or no attention is to be paid to demonstrations until plane trigonometry and the empirical use of logarithms have become familiar. Instead of seeking, like the English-speaking peoples, to develop logical sequence of thought by means of demonstrations, reliance is placed upon training the perception. Doubtless Pestalozzi's influence upon Herbart is seen here. "There is no question as to whether there is a close relation between clearness of perception and soundness of judgment-between keen vision and clear-cut thought; as to whether by the study of natural science the clear head is prepared to deal with abstract ideas."

In introducing the discussion of method in mathematics, Herbart says with truth: "That aptitude for mathematics is more rare than for other studies, is merely apparent, and due to the late period and negligent manner in which the subject is begun. That mathematicians are seldom disposed to adapt themselves to children is natural. In arithmetic they have neglected to begin with simple combinations and geometrical forms, and have sought to teach demonstrations to children in whom the mathematical imagination had not been awakened." In this connection Herbart advocates the use of lines, angles, and geometric figures in teaching number-all devices which have long since found favor. Plane surfaces of various forms, and angles, are to be measured and made the source of simple problems. The measurement of angles in degrees, is the first step in preparation for geometry and for trigonometry. Alge6 Ibid., Bd. II, S. 223.

5 Ibid., Bd. II, S. 627.

Ibid., Bd. II, S. 621.

bra is introduced early, beginning with the use of letters to denote magnitudes, continuing under the form of literal arithmetic in connection with number work, and soon taking on the form of general symbolic statement. "The essential thing is the training of the eye in estimating distances and angles, and the connecting of this training with easy arithmetical problems. The object is not simply to sharpen the observation of visible objects, but especially to arouse the geometric imagination and to connect the arithmetical thought with it. In this lies the usually slighted, yet necessary preparation for mathematics. The means used must be concrete objects. . . . . The best to begin with are triangles cut from thin pieces of hard wood. There are needed seventeen pairs of these, all rightangled and having one side of the same length in common. In order to find these triangles, let a circle be drawn with a radius of four inches, then draw tangents and secants for 5°, 10°, 15°, etc. to 85°. The various uses that may be made of these triangles are easily conceived. The tangents and secants must be actually measured by the pupils, and the length correctly recorded--at first only in whole numbers and tenths. On these are based simple problems, the immediate aim of which is that the pupil may acquire the habit of heeding so simple objects. It goes without saying that the exercise of the senses does not take the place of geometry, much less of trigonometry, but prepares the way for these sciences. When the pupil comes to study the measurement of plane surfaces, the wooden triangles are laid aside, and the geometric construction takes the place of actual perception of a concrete object. At the same time arithmetic and algebra begin to treat of simple proportion, and later of powers, roots, and logarithms."

With learners Herbart never makes the mistake of studying mathematics for its own sake alone, but always introduces its immediate application to natural objects and phenomena. "Pure mathematics" is reserved for a gymnastic of the reason after it has developed. Even then analytic demonstration is subordinated to the conception of mathematics as interpreter of the rational universe.

The idea is advanced that the immature mind gains its first

Note the early objective use of decimals. Reference to any plane trigonometry will make clear the advantage of first approaching the trigonometric functions in this way.

9 Ibid., Bd. II, S. 623.