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14 THE NATURE OF MATHEMATICS cided ; the point thus indicated was marked by a peg R. The result was to form a triangle PQR whose angle at P was a right angle, and the line PR would give an east and west line. A similar method is constantly used at the present time by practical engineers, and by gardeners in marking tennis courts, for measuring a right angle. This method seems also to have been known to the Chinese nearly three thousand years ago, but the Chinese made no serious attempt to classify or extend the few rules of arithmetic or geometry with which they were acquainted, or to explain the causes of the phenomena which they observed. The geometrical theorem of which a particular case is in- volved in the method just described is well known to readers of the first book of Euclid's Elements. The Egyptians must probably have known that this theorem is true for a right- angled triangle when the sides containing the right angle are equal, for this is obvious if a floor be paved with tiles of that shape. But these facts cannot be said to show that geometry was then studied as a science. Our real knowledge of the nature of Egyptian geometry depends mainly on the Rhind papyrus. The ancient Egyptian papyrus of Rhind, which was written by an Egyptian priest named Ahmes considerably more than a thousand years before Christ, and which is now in the British Museum, contains a fairly complete applied mathe- matics, in which the measurement of figures and solids plays the principal part ; there are no theorems properly so called ; everything is stated in the form of problems, not in general terms but in distinct numbers. For example : to measure a rectangle the sides of which contain two and ten units of length ; to find the surface of a circular area whose diameter is six units. We find also in it indications for the measurement of solids, particularly of pyramids, whole and truncated. The arithmetical problems dealt with in this papyrus which, by the way, is headed " Directions for knowing all dark things "- contain some very interesting things. In modern language, we should say that the first part deals with the reduction of fractions whose numerators are 2 to a sum of fractions each of whose numerators is 1. Thus ^ is stated to be the sum of -Jf, -^g-, yfj-, and -j^-. Probably Ahmes had no rule for forming the component fractions, and the answers given
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