Here, then, is the result of several years of labor by a somewhat radical organization, fostered by excellent mathematicians, and carried on in a country where elementary geometry is held in highest esteem, and where Euclid was thought unsuited to the needs of the beginner. The number of propositions remains substantially the same as in Euclid, and the introduction of some unusable logic tends to counterbalance the improvement in sequence of the propositions. The report provoked thought; it shook the Euclid stronghold; it was probably instrumental in bringing about the present upheaval in geometry in England, but as a working syllabus it has not appealed to the world as the great improvement upon Euclid's "Elements" that was hoped by many of its early advocates.

The same association published later, and republished in 1905, a "Report on the Teaching of Geometry," in which it returned to Euclid, modifying the "Elements" by omitting certain propositions, changing the order and proof of others, and introducing a few new theorems. It seems to reduce the propositions to be proved in plane geometry to about one hundred fifteen, and it recommends the omission of the incommensurable case. This number is, however, somewhat misleading, for Euclid frequently puts in one proposition what we in America, for educational reasons, find it better to treat in two, or even three, propositions. This report, therefore, reaches about the same conclusion as to the geometric facts to be mastered as is reached by our later textbook writers in America. It is not extreme, and it stands for good mathematics.

In the United States the influence of our early wars with England, and the sympathy of France at that time, turned the attention of our scholars of a century ago from Cambridge to Paris as a mathematical center. The influx of French mathematics brought with it such works as Legendre's geometry (1794) and Bourdon's algebra, and made known the texts of Lacroix, Bertrand, and Bezout. Legendre's geometry was the result of the efforts of a great mathematician at syllabus-making, a natural thing in a country that had early broken away from Euclid. Legendre changed the Greek sequence, sought to select only propositions that are necessary to a good understanding of the subject, and added a good course in solid geometry. His arrangement, with the number of propositions as given in the Davies translation, is as follows:

Book I.Rectilinear figures31
Book II.Ratio and proportion14
Book III.The circle48
Book IV.Proportions of figures and areas51
Book V.Polygons and circles17
——
Total for plane geometry161

Legendre made, therefore, practically no reduction in the number of Euclid's propositions, and his improvement on Euclid consisted chiefly in his separation of problems and theorems, and in a less rigorous treatment of proportion which boys and girls could comprehend. D'Alembert had demanded that the sequence of propositions should be determined by the order in which they had been discovered, but Legendre wisely ignored such an extreme and gave the world a very usable book.

The principal effect of Legendre's geometry in America was to make every textbook writer his own syllabus-maker, and to put solid geometry on a more satisfactory footing. The minute we depart from a standard text like Euclid's, and have no recognized examining body, every one is free to set up his own standard, always within the somewhat uncertain boundary prescribed by public opinion and by the colleges. The efforts of the past few years at syllabus-making have been merely attempts to define this boundary more clearly.

Of these attempts two are especially worthy of consideration as having been very carefully planned and having brought forth such definite results as to appeal to a large number of teachers. Other syllabi have been made and are familiar to many teachers, but in point of clearness of purpose, conciseness of expression, and form of publication they have not been such as to compare with the two in question.

The first of these is the Harvard syllabus, which is placed in the hands of students for reference when trying the entrance examinations of that university, a plan not followed elsewhere. It sets forth the basal propositions that should form the essential part of the student's preparation, and that are necessary and sufficient for proving any "original proposition" (to take the common expression) that may be set on the examination. The propositions are arranged by books as follows:

Book I.Angles, triangles, parallels25
Book II.The circle, angle measure18
Book III.Similar polygons10
Book IV.Area of polygons8
Book V.Polygons and circle measure11
Constructions21
Ratio and proportion6
——
Total for plane geometry99

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.