A History of the Teaching of Elementary Geometry

THE BEGINNING OF GEOMETRY AMONG PRIMITIVE PEOPLE In this attempt to trace the historic development of the teaching of elementary geometry, the word teaching will be used both in its widest sense and as restricted to methods of the school room. In the more general use of the word, we shall be concerned with the manner in which man began to formulate the science, with the additions to the subject-matter in the various epochs, and with the books written to spread this knowledge. In studying the development of the teaching of geometry1 in any epoch, four factors will be considered: 1. The contributions to the subject-matter. 2. The text-books and books read by the learned. 3. The methods of teaching geometry. 4. The place of the subject in the curriculum. The history of geometry as such will be considered only so far as is necessary for a foundation for the present study. Naturally this will form a large part of our information about the early teaching of the subject. Keeping in mind these four aspects of the matter, our subject will be treated first chronologically, bringing it down to the present time. Certain modern problems will then be considered in the light of the foregoing historic material.

Geometry as studied in the schools to-day has two values: (a) It is a study which has practical applications in mensuration and in the related fields of science. (b) It is a means of logical discipline. With respect to the general historic development of the subject-matter of geometry, we shall find: (a) That the practical side alone was recognized in pre-grecian geometry.

1 Where contradictions do not arise the term geometry will generally be employed for elementary geometry.

(b) That geometry as a pure science (apart from its applications) was developed by the Greeks and reduced to a coherent logical system by Euclid. In accordance with these two lines of development, we shall show that two chief aims have characterized the teaching of geometry: (a) The practical aim, under which geometric principles have been applied in the general field of mathematics or in the related fields of science. This aim was dominant in the later Greek period, was not without influence in the Middle Ages, and has been recognized in geometry instruction in most of the countries here considered; (b) The disciplinary or logical aim, under which the instruction has been directly or indirectly from Euclid. Directly, where the text of Euclid has been closely adhered to; indirectly, where books based on Euclid have been followed. The opposition of the logical and practical points of view is fundamental in presentday teaching, and will occupy us constantly in this essay.

The pupil to-day is not ready for his logical geometry unless he has some practical experience on which to base his logic. It was so with man's first logical geometry. Greece based her geometry on the practical work of the Egyptians. It will therefore be necessary for us to consider this pre-grecian geometry.

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Geometry arose, like all science, out of man's contact with nature. We may postulate that man's first efforts to interpret and adjust himself to nature were intuitive. We are familiar with this intuition in the habits of animals. All animals make use of what we term a geometric principle that a straight line is the shortest path between two points. Boys cross lots without first learning that one side of a triangle is less than the sum of the other two. These facts of nature are used because they are serviceable. The Indian fastened his pony to a stake, enabling him to graze in a circle. He knew that the longer the rope, the greater the area covered, but no exact relation between area and radius occurred to him. Though this early stage of intellectual development, that of intuition, does not necessarily lead up to the domain of abstract principles, still all science has its genesis in man's efforts to seek an adjustment with nature. In this instinctive stage we find man employing principles common to a higher plane. of civilization. The Indian chose the cone-shaped tee-pee for economic reasons. The mound builders1 1 Carr, The Mounds of the Mississippi Valley, p. 64.

in their ground plans employed the square and the circle, besides shapes of irregular form. Some of their mounds illustrated the truncated cone. But we cannot draw the conclusion that the mound builders had any conception of geometric principles. If they had used the principle of orientation in the building of their structures, then some real geometric notions could be traced to them. But, as far as known, such orientation was lacking.

The employment of what we call geometric design is not common to man alone. The beaver builds his dome-like structure as skillfully as the Esquimau does his. The white ants of Africa build hills twenty-five feet high, ingeniously honeycombed with galleries. While the honey bee builds his cell according to what we call geometric design, he is instead moulding nature to meet his immediate needs. In many of our so-called geometric constructions we merely copy what nature has already revealed to us. After ages of contact with nature, man was finally led to an understanding of the laws existing in and between her various forms. So nature has been the first source from which man has drawn his geometric inspirations and out of this contact with nature ideas have been developed.

Primitive man, and like him the child, does not consider per se the forms which nature has built around him, but observes them as necessary constituents of his well-being. He observes and thinks of the plane surface in terms of its serviceability. The same is true of the use of space forms. That there is a science in which these forms play a part is a world not yet disclosed. The same is true of the concept of number. While birds recognize a difference between a few and many grains of corn, the exact notion of how many, in all probability, never occurs to them. Even man did not need exact knowledge of number until the economic conditions of his life demanded an exact measure of discrete things. He frequently expresses his quantitative powers with respect to nature in terms of his own physical capacities, as when we speak of stone's-throw, fingerbreadth, span, hand, ell, cubit, fathom, day's journey, and the like.

When the relations between space and its measure meant more to man's spiritual and physical well-being, he began to

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