der Sprache. We shall do well in following Ziller's advice to study the native and foreign literatures as parallel with one another. Both modern and ancient authors should therefore be read in pairs. Homer and Vergil, Shakespeare and Schiller, Sophocles and Schiller, Longfellow and Tennyson, Tennyson and Vergil-all these can be read simultaneously. The details of prosody and rhetoric should also be taught more from the viewpoint of being common to all languages. The matter, as now taught by each language teacher, is torn apart and its organic wholeness is thus not realized.

3

Every textbook should have, beside the historical appendix, also a terminological appendix to explain all the technical terms used in the body of the book. The termini stick longest in the memory; and the explanations, if properly connected with them, will also be remembered more easily, and the full understanding of the technical terms will finally be no small aid to the understanding of the whole subject. Again, the list of technical terms will bring home the intimate connection between the word and the thing expressed. The pupil with a talent for philology will feel at home among the words, but he will also be taught never to separate the word from the thing that it signifies. Another pupil with a talent for the realia will be taught the scope and value of language studies.

Such historical and terminological appendices to our textbooks in history, geography, mathematics, physics, natural history, etc., would bring together ancient and modern elements and would attain what von Wilamowitz-Mollendorff attempted in his Greek Reader," which has but this one defect that is it far beyond the class of students for whom it is intended.

4. Of all the school subjects, mathematics would seem to be least adapted to be correlated with the other studies; and, as a matter of fact, it is isolated in modern education. Though its peculiar position in the family of the sciences may partly ac1 Cf. Hornemann, Gedanken und Vorschläge zu einer Parallelgrammatik, Hannover, 1888.

2 Grundlegung zur Lehre vom erziehenden Unterricht, XIX, p. 465, etc.; cf. supra, ch. XXII, 1.

3 Cf. Warren, Vergil and Tennyson in Essays of Poets and Poetry (New York, 1909) and Mustard, Tennyson and Vergil in Classical Essays in Tennyson (London, 1904).

4 Supra, ch. XI, 4.

5 Greek Reader, selected and adapted with English notes from Prof. von Wilamowitz-Mollendorff's Griechisches Lesebuch, by E. C. Marchant, London,

count for this isolation, yet the principal causes for it are the scant attention, given to the threads connecting mathematics with the other subjects and the wilfulness with which the schools abandoned the courses of study in which this branch was properly correlated. In the system of the liberal arts, arithmetic and geometry were continued in the theory of music and astronomy, and the latter was the capstone of the entire course. The restoration of this relationship would mean a great gain for our school mathematics: a course of mathematical instruction that led up to the elements of spherical astronomy would be better rounded off and more compact than anything we have at present, and by aiming at something definite and tangible it would bring out the connections between mathematics and other interests and studies. The starting-point of mathematical instruction should, like its aim, be of the sensuous world, just as the discovery of the limitations of the latter is one of the functions of mathematics. Figuring connects arithmetic with the circle of experience, and the attempt has been made to establish similar connections for geometry by means of geometrical object lessons. The science of geometrical sense-perception should be considered, as Fresenius puts it, as "a grammar of nature." By co-ordinating the object lessons with drawing and natural history we shall obtain what is called form study, a subject whose content connects it with various other branches, but whose chief aim is to fit the mind for systematic geometry.'

A further means for correlating mathematics-and one that will at the same time correlate also physics, a related subjectconsists in the proper use of the history of mathematics and physics. Both these sciences are traceable to the ancients, though they did not teach physics in the schools. The teacher should therefore not rest satisfied with the mere names of the Pythagorean theorem, the Archimedean principle, the Hippocratic moons, but should use the occasion for presenting the presuppositions and methods of the ancient scholars. Our school geometry is at present following too closely the methods of Euclid, without, however, avowing its indebtedness to the Alexandrian mathematician. The contrary course would be the right thing: to acknowledge gratefully the debt we owe the ancient teacher, but to exchange his rigid demonstrative form for the more pliant genetic method."

1 Cf. supra, ch. XXXI, 2 and ch. XXXII, 3. 2 For the details see infra, ch. XLIV.

The historical connection of mathematics with music may recall that mathematics as well as physics are internally connected with the fine arts. But on account of the subordinate position that the fine arts occupy at present in the school curricula, we lose sight of these internal relations. Acoustics, for example, is basic for the theory of music; optics, for the theory of drawing; and mathematics, for the science of the beautiful. Mathematics and æsthetics are, indeed, so closely interconnected with each other that they have been said to coincide. "Esthetics," says Zeising, "coincides, in a certain sense, with mathematics, the only difference being that mathematics is concerned with nothing else than the rationality of the sense-perception of space and time, whereas æsthetics inquires into the effect of this rationality upon the feelings." The science of the golden section, which abounds in educational elements, occupies the borderland between mathematics and æsthetics.

5. When arranging a course of study, the educator should remember that the units currently established in the schedule of recitation periods, viz., Latin, history, physics, etc., are not the only ones that deserve attention. Other important units are: language studies, literature studies, classical antiquities, knowledge of the home environment, knowledge of one's native country, etc. These domains are apportioned among different branches and teachers, which is unavoidable. But still, it remains a deficiency that can be remedied only by a careful attention to the interconnections. Consequently the school should, in every possible way, do justice to the actual interrelation of the materials. The unavoidable framework should not conceal the thing itself, and the lines which we draw for facilitating the study of the picture should not cut up into disjointed parts what is to be appreciated in its entirety.

The teacher of every branch must ever be alive to the points of contact between his and the related subjects. He must draw the attention of his pupils to these points, must let them realize the importance of being familiar with the borderlands, and correct any defects he might note in this regard. This presupposes, of course, that the teacher be interested not only in the subject he is teaching, but in all the branches taught in the school, and that he may never be out of touch with any of them. These are no exaggerated demands, for we can not expect more of the Neue Lehre von den Proportionen des menschlichen Körpers, Leipzig, 1854,

p. 122.

pupils than of the teacher, and the teacher must in all things be a model to his charges.'

CHAPTER XXXVII.

The Psychological Principle of Grading.

1. In establishing the moral aims as the principle of the organization of the educational content and in recognizing the need of correlating the school subjects, we might follow both Plato and Herbart. But Plato alone must be our guide in grading the studies, as Herbart does not accept the views expressed on this subject in the Republic. Both thinkers agree, indeed, on the one hand, in making poetry the beginning and philosophy the end of the liberal arts course. But, on the other hand, Plato recognizes mathematics as the middle member that connects the two extreme members, while Herbart knows no middle term.

This divergence, though apparently insignificant, is grounded on basic differences. Plato considers the sensuous world and the ideal world to be essentially different, and consequently regards the faculty that perceives the former as essentially different from that which perceives the latter. As mathematics is the rational cognition of the sensuous, it appeared to his mind as the intermediary between sense and understanding. Christian thinkers adopted Plato's views, and made mathematics, under the form of the Quadrivium, the middle step between language studies on the one hand and philosophy and theology on the other. This position is the true reason why mathematics was, in the Middle Ages, not confined to surveyors and merchants. Mathematics has thus retained its position not only because of tradition, but also because of the agreement between the Christian and the Platonic idealism." But Nominalism, which obtained the ascendancy toward the end of the Middle Ages, denied that universal terms have any objective, real existence corresponding to them, and maintained that they are mere words, or names, mere vocal utterances, flatus vocis. Thus the

1Cf. Correlation in Monroe, Cyclopedia of Education, and Burns, Correlation and the Teaching of Religion in the Catholic Educational Association Bulletin, Vol. XI, No. 1, pp. 37-49.

2 Cf. supra, ch. XXIII, 5, and Vol. I, ch XIX, 1.

fundamental difference between the sense and the spirit, between sense-perception and mind-perception was denied, and there was no more need of a connecting link between sensuous and rational cognition. Therefore the system of the seven liberal arts, which is traceable to Plato, was in principle as much opposed by Nominalism as by its more outspoken enemies, the Humanists, who ultimately destroyed it. Similarly, the modern curricula do not in principle allow mathematics to function as the connecting link between the empirical and the rational elements of instruction; but actually mathematics still occupies a middle position, at least in those schools where philosophy continues to be taught. We contend that what is actually the case should again be recognized also in principle, and then the structure reared by the thinkers in the olden times would reappear. This structure is, if we may continue the simile, built in the basilican style. It is ancient in origin, yet Christian in character. But the Renaissance, misled by the Gothic annex, failed to recognize the beauty of the whole, and so began to build over it, and the modern age has still further hid it by several characterless additions.

2. Language studies, mathematics, philosophy form a series that is based on psychological grounds, and that is of prime import for arranging the sequence of studies. The elementary character of language studies consists in their being empirical: inductively they reduce the language phenomena to rules, and the latter admit of immediate and varied application. Language studies are, at the same time, the key to varied material knowledge, since poetry and prose deal with the material of senseperception. Mathematics, however, is a rational science. Its content is obtained chiefly by deductive reasoning, and it starts the mental processes of discovering causes and of drawing conclusions.1 But it is elementary in that it deals with sensuous quantities, and thus it clings, one may say, to the balustrade of the sensuous world. Furthermore, mathematics admits, as the "science of problems,"" of immediate application of its content, and thus it also leads the pupil back to the sensuous world. Philosophy employs, like mathematics, rational and deductive methods; logic is related to language studies, and admits of varied application; but it and psychology and ethics, still more, demand such exercises in abstract thinking as exclude almost all concrete illustrations.