This system should be written on the blackboard, and the pupils should then copy and explain it. They should also give the proper headings for the horizontal and the vertical rows. They should, in brief, learn to know how much is condensed into the thirteen words; and one member of the class should discuss the scheme connectedly. Hitherto the method applied was analytical, since it ascended from the individual and specific to the generic; and now that the latter has been found, it will be well to proceed synthetically, i. e., group the particular under the categories obtained. In the first place, we should supplement what is suggested by the system. Active and medial forms should be placed beside the passive, and the reduplicated forms should be given; ἔφηνα should be placed beside ἔλυσα; φανῶ, beside λύσω; specimens of contract verbs, beside λύω; and specimens of the other classes of verbs ending in w, beside TUTTO. The athematic formations on the pure stem may be grouped, as in grammar, according to the final letters of the stem, etc. What has been grouped together in the system, should be compared with what is said in the grammar, as this brings out the relation between the two principal conjugations: the verbs in μι are represented by φημί, δείκνυμι, and ἔστην, and these forms may be separated from the others by a broken line. The system is applied by letting the pupils find forms that will fit into it, and to this end they should propose questions to one another. A further application of the insight into the formation of the verb would be to compare the corresponding Latin forms. It is not difficult to demonstrate the analogy between onμí and fers, fert, etc.; between Aúw and lego; between τύπτω and fecto; ἔλιπον and legi; ἔλυσα and scripsi; and λύσω and amabo. Another method of applying what has been found, is to reflect upon the way which has led to the discoveries; and this will prove a preparation for the study of logic. In the present case the divisions have been applied most. The four divisions are based on the crossing of the following concepts: athematic, thematic, formation on the pure stem, formation on the enlarged stem. And these concepts themselves fall under the concept of formation-media. In the system itself the two concepts, kind of formation and tense, are crossed. 2. To illustrate the method of applying analytical development in the instruction in logic, we shall develop the concepts analysis and synthesis. We have, in ch. XLI, 2, developed these concepts synthetically, but we shall now, for the purpose of practical instruction, show their analytical derivation. This will, at the same time, give us an opportunity to illustrate how logic can be correlated with the other branches of the curriculum.1 The ascending thought-process of analysis and the descending one of synthesis are understood most easily in their application to concepts, in the form, consequently, of abstraction and determination. In the first place, the teacher should illustrate how new concepts are obtained from the old, namely, on the one hand, by abstraction, and, on the other, by determination. Examples taken from grammar are particularly appropriate. The discovery that Greek, Latin, German, English, and other languages are cognate, gave rise to the new and higher concept of the Indo-Germanic languages. On the other hand, the study of the history of the English language and of its dialects gave rise to a number of lower and narrower concepts: Old English, Middle English, Yorkshire dialect, etc., so that from the one concept, English language, there has been an ascending and a descending process, in the direction of the genus as well as of the species. Natural history, too, on the one hand, groups the species under the genus and, on the other, divides the species into families, and thus offers a large number of illustrations. Witness, on the one hand, the concepts of classes: canis, felis, mammalia, ruminantia, etc.-which had in many cases to be expressed by new terms-and, on the other hand, the concepts of the families within the species. A similar process of ascending and descending can be observed in judgments, theorems, and cognitions. The theorem of Pythagoras, for instance, is the generalization of what has been observed in certain right triangles with commensurable sides. These angles are known as the Pythagorean triangles, whose sides are in the proportion of 3:45, of 68: 10, of 5:12:13, etc. The theorem of Pythagoras, however, can be amplified in a double way. First, the relation which it expresses between squares, is true of all similar figures, so that all similar polygons constructed upon the legs are equal to the similar polygon constructed upon the hypotenuse. Secondly, its formula: ca2+b2 in the changed form: ca2 + b2 – 2ab cos y can be applied to all triangles. The theorem of Pythagoras will then appear as only a special case of more general theorems. Every application of the theorem-for example, to apply it to = 1 Supra, ch. XXIV, 3 and ch. XXXVI, 2. find a side of the triangle—is a determination; therefore, also a specialization. The latter is continued still further by substituting numbers for the letters. The process of generalization is called induction, if the ascending process of the reasoning is based on the bringing in (émáyev, inducere) and comparing of a plurality of facts, chiefly of observed facts. The reverse is deduction, which is the process of gaining new knowledge by determining more closely general truth. The element common to abstraction, generalization, and induction is the ascending process; or, to use another illustration, the backward movement to the generic as to the conditioning. The element common to determination, specialization, and deduction is the descending process or the advance toward the specific. The ascending process is regressive and the descending process is progressive. The ancients, however, termed the regressive process analysis, for they had in mind the resolving and the separating of the special traits; and the progressive process they called synthesis, for here they had in mind the adding and joining together of such traits. The analysis is, as is apparent from its etymology, a resolving (ává up and λveiv to loose) of a thing into its essential elements, a separating, therefore, of the essential traits; while the synthesis implies a πрóσleσis, the adding of more specific determination. The words of Alexander of Aphrodisias, quoted in ch. XLI, 2, may assist in fixing these points in the mind. The following "system" is a summary of the results of this development: We may now consider our task done, but we might also trace the concepts analysis and synthesis beyond the domain of logic, for they signify solution and composition in general. In grammar the objects of these activities are words, i. e., language elements; in metrics, the time durations; in drawing, the lines; in chemistry, the elements, etc. But in logic this is different, because here we have no resolution into parts, but a resolution of the thought contents-viewed as elements of knowledge -into more simple or more general elements, and a corresponding composition. 3. For illustrating developmental instruction in mathematics. we select a group of algebraic problems, which, though found in text-books, still do not receive sufficient attention in the schools. And this is to be regretted, particularly since it is just such problems as we have in mind that afford an opportunity for correlating mathematics with other disciplines. We mean the problems dealing with the meeting of moving bodies, which belong to elementary mechanics, and are, therefore, based only upon the concepts of magnitude, space, and motion. The teacher should, above all, try to let the problems issue, as far as possible, from each other, and he should, therefore, begin with the simplest case and then follow it up to see how it can be varied and complicated. The simplest case we have in the following question: where will two bodies meet that start out at the same time from the points A and B respectively, and that move toward each other with the same velocity? It is obvious that the centre of the course will be the meeting-point. But we must even here show that in answering the question where? we answer also the question when? and that for the following reasons: the bodies meet at the time that they arrive at the centre of the course, or in other words, in half the time it would take them to pass over the whole course. The simplest modification would be to let one body start out before the other. The distance covered by the first body must be deducted before the course is halved. Here is a second. mcdification of the original problem: let the velocity of the two bodies be unequal, the one making m and the other n units of distance in one unit of time. Because AB as well as m and n are distances, the constructive solution naturally comes first. The distances are brought into connection by being placed along the sides of an angle, on the one side AB, on the other m along AC, and n along CD, a prolongation of AC. Then draw the line DB and a parallel to it at C, which will intersect AB at E. According to the theorem of proportional lines, E is, then, the meeting-point of the two bodies. The pupils are to give the algebraic solution also. It will be most simple with the meeting-point known and the velocity required. The velocity of the two moving bodies will be equal, if the meeting-point lies halfway between A and B. If the meeting-point divides AB into % and 3, the proportion of the velocity will obviously be 1:2; if it divides it into % and %, the proportion will be 2:3; if into 10 and 10, the proportion will be 3 :7; and in general, if it divides it into and m + n the proportion will be n : m. n m m + n', This solution is analytic and inductive. If, however, the meeting-point be required, the solution must be synthetic and deductive. We must remember that two things are known of the two distances over which the bodies pass until they meet, namely the sum AB and the proportion m:n. The two distances can be determined without an equation by considering the following: If m and n be equal, then AE as well as EB were the half of AB, therefore 1⁄2AB; but AB requires another factor to yield AE, and again another factor to yield EB. But these factors must, like the two halves, result in I. But this will be the case only in two magnitudes which have m and n as factors and m + n as divisor, therefore in m m m + n hence the two distances are AB and m + n and n m + n n m + n AB. and In the third place, the pupils should make the technical, algebraic solution, in which either one of the partial distances. or both are unknown and where the equations containing what is unknown must be formed. A further variation would be to consider not only the partial distances as unknown, as hitherto, but also the whole distance or one number of the proportion as the thing required. The proportion found should now be applied to special cases. Instead of the two moving bodies, the teacher may give problems dealing with two travellers, two railroad trains, armies, etc., and particular attention should be given to the actual velocity of the given motions. And it will be labor profitably spent to have this velocity even memorized, for such examples are well adapted, not only to illustrate abstract relations of magnitude, but also to develop and widen the sensuous horizon by imparting knowledge of life and nature. Only at this point should problems be given which deal with objects moving in the same direction. Here we need con |