| GEOMETRY: ANCIENT AND MODERN. |
By Professor EDWIN S. CRAWLEY,
UNIVERSITY OF PENNSYLVANIA.
AMONGST the records of the most remote antiquity we find little to lead to the conclusion that geometry was known or studied as a branch of mathematics. The Babylonians had a remarkably well-developed number system and were expert astronomers; but, so far as we know, their knowledge of geometry did not go beyond the construction of certain more or less regular figures for necromantic purposes. The Egyptians did better than this, and Egypt is commonly acknowledged to be the birthplace of geometry. It was a poor kind of geometry, however, from our point of view, and should rather be designated as a system of mensuration. Nevertheless it served as a beginning, and probably was the means of setting the Greek mind, at work upon this subject. Our knowledge of Egyptian geometry is obtained from a papyrus in the British Museum known as the Ahmes Mathematical Papyrus. It dates from about the eighteenth century B. C, and purports to be a copy of a document some four or five centuries older. It is the counterpart of what to-day is called an engineer's hand-book. It contains arithmetical tables, examples in the solution of simple equations, and rules for determining the areas of figures and the capacity of certain solids. There is no hint of anything in the nature of demonstrational geometry, nor any evidence of how the rules were derived. In fact, they could not have been obtained as the result of demonstration, for they are generally wrong. For example, the area of an isosceles triangle is given as the product of the base and half the side, and that of a trapezoid as the product of the half-sums of the opposite sides. These rules give results which are approximately correct so long as they are applied to triangles whose altitude is large compared with the base, and to trapezoids which do not depart very far from a rectangular shape. Whether the Egyptians ever came to realize that these rules were erroneous we cannot say, but it is known that long after the Greeks had discovered the correct ones they were still in use. Thus Cajori, 'History of Mathematics,' page 12, says: "On the walls of the celebrated temple of Horus at Edfu have been found hieroglyphics written about 100 B. C, which enumerate the pieces of land owned by the priesthood and give their areas. The area of any quadrilateral, however irregular, is there found by the formula a+b2 x c+d2." [a and b for one pair of opposite sides and c and d for the others.] It is plausibly argued that a superstitious tra-
