CHAPTER III
A BRIEF HISTORY OF GEOMETRY
The geometry of very ancient peoples was largely the mensuration of simple areas and solids, such as is taught to children in elementary arithmetic to-day. They early learned how to find the area of a rectangle, and in the oldest mathematical records that have come down to us there is some discussion of the area of triangles and the volume of solids.
The earliest documents that we have relating to geometry come to us from Babylon and Egypt. Those from Babylon are written on small clay tablets, some of them about the size of the hand, these tablets afterwards having been baked in the sun. They show that the Babylonians of that period knew something of land measures, and perhaps had advanced far enough to compute the area of a trapezoid. For the mensuration of the circle they later used, as did the early Hebrews, the value π = 3. A tablet in the British Museum shows that they also used such geometric forms as triangles and circular segments in astrology or as talismans.
The Egyptians must have had a fair knowledge of practical geometry long before the date of any mathematical treatise that has come down to us, for the building of the pyramids, between 3000 and 2400 B.C., required the application of several geometric principles. Some knowledge of surveying must also have been necessary to carry out the extensive plans for irrigation that were executed under Amenemhat III, about 2200 B.C.
The first definite knowledge that we have of Egyptian mathematics comes to us from a manuscript copied on papyrus, a kind of paper used about the Mediterranean in early times. This copy was made by one Aah-mesu (The Moon-born), commonly called Ahmes, who probably flourished about 1700 B.C. The original from which he copied, written about 2300 B.C., has been lost, but the papyrus of Ahmes, written nearly four thousand years ago, is still preserved, and is now in the British Museum. In this manuscript, which is devoted chiefly to fractions and to a crude algebra, is found some work on mensuration. Among the curious rules are the incorrect ones that the area of an isosceles triangle equals half the product of the base and one of the equal sides; and that the area of a trapezoid having bases b, b', and the nonparallel sides each equal to a, is ½a(b + b'). One noteworthy advance appears, however. Ahmes gives a rule for finding the area of a circle, substantially as follows: Multiply the square on the radius by (16/9)2, which is equivalent to taking for π the value 3.1605. This papyrus also contains some treatment of the mensuration of solids, particularly with reference to the capacity of granaries. There is also some slight mention of similar figures, and an extensive treatment of unit fractions,—fractions that were quite universal among the ancients. In the line of algebra it contains a brief treatment of the equation of the first degree with one unknown, and of progressions.[16]
Herodotus tells us that Sesostris, king of Egypt,[17] divided the land among his people and marked out the boundaries after the overflow of the Nile, so that surveying must have been well known in his day. Indeed, the harpedonaptæ, or rope stretchers, acquired their name because they stretched cords, in which were knots, so as to make the right triangle 3, 4, 5, when they wished to erect a perpendicular. This is a plan occasionally used by surveyors to-day, and it shows that the practical application of the Pythagorean Theorem was known long before Pythagoras gave what seems to have been the first general proof of the proposition.
From Egypt, and possibly from Babylon, geometry passed to the shores of Asia Minor and Greece. The scientific study of the subject begins with Thales, one of the Seven Wise Men of the Grecian civilization. Born at Miletus, not far from Smyrna and Ephesus, about 640 B.C., he died at Athens in 548 B.C. He spent his early manhood as a merchant, accumulating the wealth that enabled him to spend his later years in study. He visited Egypt, and is said to have learned such elements of geometry as were known there. He founded a school of mathematics and philosophy at Miletus, known from the country as the Ionic School. How elementary the knowledge of geometry then was may be understood from the fact that tradition attributes only about four propositions to Thales,—(1) that vertical angles are equal, (2) that equal angles lie opposite the equal sides of an isosceles triangle, (3) that a triangle is determined by two angles and the included side, (4) that a diameter bisects the circle, and possibly the propositions about the