Book V. Polygons and circle measure 11

Constructions 21 Ratio and proportion 6 ---- Total for plane geometry 99

The total for solid geometry is 79 propositions, or 178 for both plane and solid geometry. This is perhaps the most successful attempt that has been made at reaching a minimum number of propositions. It might well be further reduced, since it includes the proposition about two adjacent angles formed by one line meeting another, and the one about the circle as the limit of the inscribed and circumscribed regular polygons. The first of these leads a beginner to doubt the value of geometry, and the second is beyond the powers of the majority of students. As compared with the syllabus reported by a Wisconsin committee in 1904, for example, here are 99 propositions against 132. On the other hand, a committee appointed by the Central Association of Science and Mathematics Teachers reported in 1909 a syllabus with what seems at first sight to be a list of only 59 propositions in plane geometry. This number is fictitious, however, for the reason that numerous converses are indicated with the propositions, and are not included in the count, and directions are given to include "related theorems" and "problems dealing with the length and area of a circle," so that in some cases one proposition is evidently intended to cover several others. This syllabus is therefore lacking in definiteness, so that the Harvard list stands out as perhaps the best of its type.

The second noteworthy recent attempt in America is that made by a committee of the Association of Mathematical Teachers in New England. This committee was organized in 1904. It held sixteen meetings and carried on a great deal of correspondence. As a result, it prepared a syllabus arranged by topics, the propositions of solid geometry being grouped immediately after the corresponding ones of plane geometry. For example, the nine propositions on congruence in a plane are followed by nine on congruence in space. As a result, the following summarizes the work in plane geometry:

Congruence in a plane 9 Equivalence 3 Parallels and perpendiculars 9 Symmetry 20 Angles 15 Tangents 4 Similar figures 18 Inequalities 8 Lengths and areas 17 Loci 2 Concurrent lines 5 ---- Total for plane geometry 110

Not so conventional in arrangement as the Harvard syllabus, and with a few propositions that are evidently not basal to the same extent as the rest, the list is nevertheless a very satisfactory one, and the parallelism shown between plane and solid geometry is suggestive to both student and teacher.

On the whole, however, the Harvard selection of basal propositions is perhaps as satisfactory as any that has been made, even though it appears to lack a "factor of safety," and it is probable that any further reduction would be unwise.

What, now, has been the effect of all these efforts? What teacher or school would be content to follow any one of these syllabi exactly? What textbook writer would feel it safe to limit his regular propositions to those in any one syllabus? These questions suggest their own answers, and the effect of all this effort seems at first thought to have been so slight as to be entirely out of proportion to the end in view. This depends, however, on what this end is conceived to be. If the purpose has been to cut out a very large number of the propositions that are found in Euclid's plane geometry, the effort has not been successful. We may reduce this number to about one hundred thirty, but in general, whatever a syllabus may give as a minimum, teachers will favor a larger number than is suggested by the Harvard list, for the purpose of exercise in the reading of mathematics if for no other reason. The French geometer, Lacroix, who wrote more than a century ago, proposed to limit the propositions to those needed to prove other important ones, and those needed in practical mathematics. If to this we should add those that are used in treating a considerable range of exercises, we should have a list of about one hundred thirty.

But this is not the real purpose of these syllabi, or at most it seems like a relatively unimportant one. The purpose that has been attained is to stop the indefinite increase in the number of propositions that would follow from the recent developments in the geometry of the triangle and circle, and of similar modern topics, if some such counter-movement as this did not take place. If the result is, as it probably will be, to let the basal propositions of Euclid remain about as they always have been, as the standards for beginners, the syllabi will have accomplished a worthy achievement. If, in addition, they furnish an irreducible minimum of propositions to which a student may have access if he desires it, on an examination, as was intended in the case of the Harvard and the New England Association syllabi, the achievement may possibly be still more worthy.

In preparing a syllabus, therefore, no one should hope to bring the teaching world at once to agree to any great reduction in the number of basal propositions, nor to agree to any radical change of terminology, symbolism, or sequence. Rather should it be the purpose to show that we have enough topics in geometry at present, and that the number of propositions is really greater than is absolutely necessary, so that teachers shall not be led to introduce any considerable number of propositions out of the large amount of new material that has recently been accumulating. Such a syllabus will always accomplish a good purpose, for at least it will provoke thought and arouse interest, but any other kind is bound to be ephemeral.[32]

Besides the evolutionary attempts at rearranging and reducing in number the propositions of Euclid, there have been very many revolutionary efforts to change his treatment of geometry entirely. The great French mathematician, D'Alembert, for example, in the eighteenth century, wished to divide geometry into three branches: (1) that dealing with straight lines and circles, apparently not limited to a plane; (2) that dealing with surfaces; and (3) that dealing with solids. So Méray in France and De Paolis[33] in Italy have attempted to fuse plane and solid geometry, but have not produced a system that has been particularly successful. More recently Bourlet, Grévy, Borel, and others in France have produced several works on the elements of mathematics that may lead to something of value. They place intuition to the front, favor as much applied mathematics as is reasonable, to all of which American teachers would generally agree, but they claim that the basis of elementary geometry in the future must be the "investigation of the group of motions." It is, of course, possible that certain of the notions of the higher mathematical thought of the nineteenth century may be so simplified as to be within the comprehension of the tyro in geometry, and we should be ready to receive all efforts of this kind with open mind. These writers have not however produced the ideal work, and it may seriously be questioned whether a work based upon their ideas will prove to be educationally any more sound and usable than the labors of such excellent writers as Henrici and Treutlein, and H. Müller, and Schlegel a few years ago in Germany, and of Veronese in Italy. All such efforts, however, should be welcomed and tried out, although so far as at present appears there is nothing in sight to replace a well-arranged, vitalized, simplified textbook based upon the labors of Euclid and Legendre.

The most broad-minded of the great mathematicians who have recently given attention to secondary problems is Professor Klein of Göttingen. He has had the good sense to look at something besides the mere question of good mathematics.[34] Thus he insists upon the psychologic point of view, to the end that the geometry shall be adapted to the mental development of the pupil,--a thing that is apparently ignored by Méray (at least for the average pupil), and, it is to be feared, by the other recent French writers. He then demands a careful selection of the subject matter, which in our American schools would mean the elimination of propositions that are not basal, that is, that are not used for most of the exercises that one naturally meets in elementary geometry and in applied work. He further insists upon a reasonable correlation with practical work to which every teacher will agree so long as the work is really or even potentially practical. And finally he asks that we look with favor upon the union of plane and solid geometry, and of algebra and geometry. He does not make any plea for extreme fusion, but presumably he asks that to which every one of open mind would agree, namely, that whenever the opportunity offers in teaching plane geometry, to open the vision to a generalization in space, or to the measurement of well-known solids, or to the use of the algebra that the pupil has learned, the opportunity should be seized.

FOOTNOTES:

[32] The author is a member of a committee that has for more than a year been considering a syllabus in geometry. This committee will probably report sometime during the year 1911. At the present writing it seems disposed to recommend about the usual list of basal propositions.

[33] "Elementi di Geometria," Milan, 1884.

[34] See his "Elementarmathematik vom höheren Standpunkt aus," Part II, Leipzig, 1909.