ing a foreign language, one is as interested in obtaining case forms and tense forms as in declining or conjugating a given word. This is particularly true of such languages as have different declensions for the noun and the adjective. If a foreigner would use an adjective of such a language as an attribute, he must combine two different systems of forms and, if the gender is also subject to change, he must let whole paradigms pass muster in his mind. Current grammar, however, does not meet the needs of the beginner in Latin either with regard to the comprehension or with regard to the application. It fixes all his attention on the inflection of words; and the categories, although they govern the inflections, are imparted, so to say, in an underhand way, for they must be acquired along with the forms. For example, while learning to decline mensa and to conjugate amo, the boy must at the same time learn the principles of syntax which are responsible for the declension and conjugation, though he is apparently learning forms only. Instead of being equally divided, the whole burden is thus laid on the beginning of etymology, while what is most important and of the greatest educational value, is treated as something incidental.

7. The demand that grammar be correlated with the classics, is made primarily for those grades in which both are subjects of instruction. But it should govern also that elementary instruction in language which precedes the reading of the classics. The development of Latin instruction has influenced the methodology of elementary language instruction. In the Middle Ages there was only one didactic instrument employed in elementary language instruction-the grammar; but, being written in Latin, it served also as a reader. And besides, the student heard Latin on all sides, for Latin was, in some measure, a living language still. The Humanists compiled easy texts, which, though they accompanied the first grammatical instruction, were nevertheless not correlated with it throughout. The language books of the 18th century were the first to correlate grammatical instruction with the reading of the classics: they illustrated by sentences, in a sort of running commentary, the rules of gramThe most successful of these books was Meierotto's Latin Grammar in Examples, 1785, where we find, instead of paradigms, series of sentences, mostly proverbs, which embody the respective rules, and thus form an admirable method for teaching rules by examples. Mager employed the genetic method in

mar.

his French language book and recommends it for Latin instruction; he proceeds from the simple sentence and, by introducing into it different modifiers, illustrates successively the different forms. Mager's and Meierotto's are the most important experiments made in this line, but they both fall into the same error of occupying the pupil too long with unconnected materials and of thus making him learn too many words that are of no help to him in reading the classics, while it is precisely for this that the elementary language instruction should prepare. Yet practically all exercise books in use labor under the same defects and lack, moreover, the redeeming features of Mager's and Meierotto's books.

After what has been said, it is clear that the first exercise book should cover the work of one year; it should prepare for the first Latin reading whether this be the Epitome historia sacræ or Lhomond's Viri illustres Græcia or Roma or anything similar so long as it is a connected whole-and should also smooth the way for an organically arranged grammar. For the reading it should prepare by taking its vocables from the text to be read; and for the study of grammar it should smooth the way by exemplifying in sentences the most important forms. Meierotto's work may serve as a model for such an exercise book, for the latter should contain suitable proverbs to illustrate the rules of grammar.

To pass from such an exercise book to the grammar, would be but a step, because the grammar would differ from the exercise book only in that it derives the rule from the example and places the rule before the example.

CHAPTER XLIV.

On the Organico-Genetic Treatment of Mathematics.

1. The method of our school mathematics is, like that of our school grammar, handed down from the ancients, but is opposed to the organico-genetic principle. We have shown above (ch. XLI, 5) that this Euclidean method combines synthesis and analysis in such a way that the presentation proceeds synthetically, but in detached sections, namely, in theorems and problems, whose proofs and solutions respectively are given analytically. Euclid's method is strictly exact and skeptically careful

to accept nothing that is not proved, and is truly artistic in its structure, for no theorem is introduced before the means for its analysis are at hand, and each theorem contributes its share of such means for analyzing the succeeding theorems. These high merits have gained for the method an exceptional authority and have led to its being applied to other materials, especially to philosophical truths, which, when proved more geometrico, seemed to be unassailable.

But the method is, in reality, less intelligible and less perspicuous than the simple synthesis which presents the theorems in a continuous development. Hence modern thinkers have also criticized it severely as being partly the very reverse of the genetic method. Herbart, for one, finds fault with it for using arbitrary auxiliary lines and computations: "The mind that wanted to sink into the depths of the subject itself, is deflected and hurried hither and thither through a number of narrow, crooked by paths. . . . To be sure, we must believe the proof, for considered step by step there has been no logical slip. But not seeing through the whole, quite contrariwise, every separate step making a separate thinking act, one would almost as willingly take the word of a clever teacher for it." He demands that the analysis let itself drift through the given concepts to those which embody the internal necessity of the theorem. Hegel, on the other hand, takes exception to the fact that in Euclid "the various concepts that are needed for the argumentation, are adduced from anywhere and made to serve as preparatory material. The proof is not the genesis of the relationship that constitutes the content of the theorem. . . ; it is but an external reflection that enters from without into the inner nature of the subject. In other words, it concludes from external circumstances to the inner property of the relationship.

3

Trendelenburg's criticism is most thorough as well as most pertinent. "In Euclid," he says, "the most important theorems are proved only from the external relationship and by incidental views, but not in accordance with the elements necessarily given with the concept of the thing. If the full and complete theorem comes first to be followed by the proof, the

1 Herbart's A B C of Sense-Perception and Introductory Works, transl. by W. J. Eckoff, New York, 1896, p. 164.

2 Subjektive Logik, Werke V, pp. 302 ff. and pp. 274 ff.

3 Logische Untersuchungen, II, pp. 370 ff.; for other references see Mager, Die genetische Methode, pp. 23 ff.; cf. T. Wittstein, Die Methode des mathematischen Unterrichts, Hannover, 1879.

whole resembles a series of rigid statements, which obtain a footing and then entrench themselves. There is everywhere an artistic concatenation, but nowhere a growth and development. The method that we recommend (the genetic) leads farther, for it directs us to find what lies in the nature of the thing and not to confirm, by a discovered relationship, that which has been obtained from outside sources. The theorem is gained anew and not supported merely externally."

2. The Euclidean method, as well as the ancient mathematical method in general, is connected with the middle position which mathematics held among the ancients. Its function was to lead beyond the empirical and prepare for the cognition of ideas, but by way of practice. What Plato said of arithmetic is equally true of mathematics in general: it is the arena of study and practice such as can nowhere else be found.' Thus we find in mathematics a technical element-technical, not in the sense of practical application after the manner of surveyors, a view expressly excluded-but in the sense in which we contrasted the technical with the logical. Hence the form of the problem, the construction, the combinative method, the careful providing of all the parts needed.

3

This viewpoint is not at all foreign to mathematics. We have elsewhere characterized mathematics as the science of problems, and have found one factor of its educative content in its many-sided experimenting and trying, its manipulations and operations, and its use of questions and answers. The two elements of mathematics, the logico-speculative and the technico-combinative, are combined in different ways in the single departments of this subject. They are more easily separated in arithmetic, so that a purely synthetical course of study can be paralleled with a course in exercises. In geometry, however, the combination of the elements can not be exhausted by a continuous presentation. Here the threads cross at so many points that it does not suffice to follow them up singly: their knots also must be examined and opened. Geometry consequently can not be presented in a purely genetic way, or in the

1

Rep., VII, p. 526: ἅ γε μείζω πόνον παρέχει μανθάνοντι και μελετῶντι, οὐκ ἂν ῥᾳδίως οὐδὲ πολλὰ ἂν εὔροις ὡς τοῦτο.

2 Supra, ch. XL, 3. Both elements may be recognized in the combination of theorems and problems; the former are proved and the final formula 15: ὅπερ ἔδει δείξει, the latter are solved and concluded with the words: ὅπερ ἔδει ποιῆσαι.

3 Supra, ch. XXIII, 2 and 3.

form, perhaps, of a genealogical tree of cognitions. Instead, the latter grow from their principle, the nature of space, simultaneously in different directions, so that they can not be assembled without repetition and combination, and this excludes all arbitrariness. Therefore we can not agree with Herbart who says (1. c.), that the internal necessity of a theorem is not discovered as long as there are two or more proofs that make the thing equally clear. On the contrary, if we have but one proof, we can not be certain of having discovered the true position of a theorem, because only a plurality of proofs corresponds to the various relationships that are possible between this and other theorems. Herbart is likewise unfair in missing "philosophical transparency" in the presentation of mathematics. Trendelenburg also assigns to mathematics what belongs properly to philosophy, for he says (.c.), "No science is so well adapted as mathematics to develop the content of a thing from its concept.' For such a development mathematics has, admittedly, the best major premises, but among the minor premises there are so many of equal value that one must needs select, combine, and

construct.

We have taken the position that mathematics should be reinstated as a middle member between the empirical disciplines and philosophy,' and for that reason we had to defend, though not without some qualifications, the Euclidean method. The fault of Euclid does not lie in his connecting the logical and the technical elements-this is unavoidable and even desirable in instruction-but in his interweaving the two elements, even where it would be advantageous to the logical element to keep them separate. We shall make no mistake in following the principle laid down before: synthesis, if possible; analysis, if necessary. Applied to the present subject, this means: adopt the purely synthetic method, identical here with the genetic, as long as cognitions can be obtained from the essence of the magnitudes and the nature of number and space; but analyze, combine, construct, wherever it may be necessary either because of the complexity of the subject or because of the need of exer

cises.

In order to illustrate what has been said and at the same. time to give a view of the many problems made available by the genetic arrangement of school mathematics, we shall add a

1 Supra, ch. XXXVII, 2.

2 Supra, p. 230.