(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. 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. t 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 develop a rudimentary science. When the mind came to classify, to define space relations, to summarize the products of human efforts, a second level in human experience was attained. On such a plane worked the minds of the Ancient Egyptians, the Babylonians, and the Chinese. The building of the pyramids of Egypt gave a stimulus to an architecture employing principles of proportion and the simpler facts of plane geometry. The placing of these tombs due north and south led to finding an east and west line. The effects from the overflow of the Nile demanded some knowledge of plane surveying. But the Egyptians continued on the one intellectual level. They knew enough practical geometry for their needs; they formulated rules in mensuration, thus showing an ability to classify their knowledge; but beyond this they did not go. A third and higher level was reached by the Greeks, who, basing their work on the practical geometry of the Egyptians, developed a system of logic which culminated in the great work of Euclid. The growth of man has been in a sense like the growth of the child. First, there was the consciousness of the materials of a science. Second, the use of these materials in a practical, but not necessarily exact way. Later came the stage when the mind began to cultivate its logical powers. It was then that Greece arose with the mind of the full-grown man. There has also been a fourth stage, that in which the theory is put to practical uses. Thus the cycle is made complete and -that which arose from practical needs returns again in the form of theory to be tested and again expanded. THE EGYPTIANS It is well to consider, somewhat in detail, the practical geometry of the Egyptians, not only because it had a bearing on the development of the logical geometry of the Greeks but also because our story would not be complete without some statement of man's achievements in what has above been called the secondary plane of geometric development.. That the Egyptians had great mechanical skill is shown in their building arts. That a nation with so meager a knowledge of geometry built such a structure as the Great Pyramid causes astonishment. Though no great knowledge of geometry was required for this work, we naturally query why a people capable of such mechanical achievements did not pursue further the science that was the basis for their work. 3 A study of the pyramids gives evidence of a knowledge both of astronomy and geometry. The Great Pyramid of Cheops, according to Charles Piazzi Smyth, gives some interesting results: (a) The base is a square. (b) Its height is to twice one base edge as the diameter of a circle is to its circumference. For the truth of this statement, one can refer only to Smyth's measurements.2 (c) The pyramid is orientated to within 4' of the true north and south line. It is reasonable to suppose that the pyramid builders first located the north and south line by means of the polar star. But how did they get a perpendicular to this line? According to Cantor, the philosopher Democritus (cir. 420 B.C.) is quoted as saying, "In the construction of plane figures with proof no one has yet surpassed me, not even the so-called Harpedonaptæ of Egypt." Cantor points out that the name Harpedonaptæ is made up of two Greek words meaning, when taken together, "rope-fasteners." A right angle was formed by stretching a rope around three pegs so placed that the three sides of the triangle formed were in the ratio of 3:45. This statement of Democritus tells us that about 420 B.C. the rope-stretchers had fame as practical geometers in Egypt. The pyramids were built during the fourth dynasty, antedating 3000 B. C. One has a right to judge that the above method of laying off perpendiculars was in use at that time, for at a little later date during the reign of Amenemhat I of the twelfth dynasty, the rope-stretchers plied their art.1 This method of erecting a perpendicular was recognized by other 11-26, 47-66. If we are to accept 1 Smyth, Our Inheritance in the Great Pyramid, pp. 2 In other words here is a ratio that equals 3.14159. these figures and conclusions, we are to think that the early time knew this close approximation to the value of to think, however, that Smyth had extremely good luck in getting his results. That the Egyptians did not know the value so accurately at a later time will be shown from the records. . One is inclined 3 Cantor, Vorlesungen über Geschichte der Mathematik, Vol. I, p. 62. Hereafter referred to as Cantor. See Gow, History of Greek Mathematics, p. 129. Hereafter referred to as Gow. Dümichen, Denderatempel, p. 33. nations. The Hindus made use of it, and the Chinese are credited with knowing that a triangle is right angled when the sides are in the ratio 3:45.1 It is hardly necessary to add that the above method of erecting perpendiculars is in common use to-day. Piazzi Smyth gives us further information regarding the Great Pyramid at Cheops: (d) The angle of the entrance passage on the north is a little over 26°. (e) The angle of the northern air passage is 33° 42'. According to Colonel HowardVyse's statement3, in 2400 B.C. the lower culmination of the polar star was 26°. So Piazzi Smyth draws the conclusion that these passage ways pointed to the lower and upper culminations of the polar star in the year 2400 B.C. The average of the above figures gives an approximation of 30° for the latitude of the Great Pyramid. The latitude as obtained by the French1 in 1799 was found to be 29° 59′ 6′′. We thus see that the Egyptians had knowledge of mathematical astronomy as well as of geometry in the building of this pyramid. The Egyptians were also stimulated to use a form of geometry due to æsthetic influences. In their mural decorations during the period of the fifth dynasty immediately following the building of the Gizeh pyramids, we find evidences of geometric designs embodying principles of symmetry. This is found in particular in the square and its diagonals, the rhombus, and the isosceles trapezoid. One figure shown by Professor Cantor represents two squares one over-lapping the other, so placed as to give the effect of an eight-pointed star. There is also represented the division of the circle into 4, 8, 6, 12, parts by the requisite number of diameters. All of this was accomplished before Greece became interested in geometry. The probable effect of such work on Greece is pointed out by Gow, who says: "To a Greek, therefore, who had once acquired a taste for geometry, a visit to Egypt or Babylon would reveal a hundred 'Gow, p. 130. 2 These angles are referred to the horizon. 3 Smyth, op. cit., p. 57. ▲ Ibid, p. 65. 'Cantor, I, pp. 66-67. The Babylonians divided the circumference into 360° and also made use of various geometric designs in their mural decorations. Cantor, I, p. 98. |