GEOMETRY: ANCIENT AND MODERN. 257 



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, how- 

 ever, from our point of view, and should rather be designated as a sys- 

 tem of mensuration. Nevertheless it served as a beginning, and prob- 

 ably was the means of setting the Greek mind, at work upon this sub- 

 ject. 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 arithmeti- 

 cal 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 prod- 

 uct 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 Egyp- 

 tians 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 + b c + d „ 



2 — * — 2 * *- a a ^and for one pair of opposite sides and 



c and d for the others.] It is plausibly argued that a superstitious tra- 



VOL. LVIII.— 17 



