DECLINE OF ALEXANDRIAN SCIENCE 125 



A triangle with sides 13, 14, 15 is selected as an illustration. Its 

 area is 



V21 X 6 X 7 X 8 = 84. 



This work seems to have become a standard authority for 

 generations of surveyors, and thus in course of time to have lost 

 much of its identity by successive changes. The whole spirit of 

 the work is rather Egyptian than Greek, that of the practical 

 engineer as distinguished from that of the mathematician, thus in 

 a measure a reversion to the aims of the Ahmes manuscript. " Let 

 there be a circle with circumference 22, diameter 7. To find its area. 

 Do as follows. 7 X 22 = 154 and $ = 38|. That is the area." 



Some of Hero's methods indicate knowledge of the new trigo- 

 nometry of Hipparchus and of the principle of coordinates. Thus 

 he finds areas of irregular boundary by counting inscribed rec- 

 tangles, a process corresponding to the use of coordinate paper. 



From Hero date such time-honored problems as that of the 

 pipes. A vessel is filled by one pipe in time ti, by another in 

 time ti. How r long will it take to fill it when both pipes are used ? 



He defines spherical triangles and proves simple theorems about 

 them : for example, that the angle-sum lies between 180 and 

 540. He determines the volume of irregular solids by measuring 

 the water they displace. Having by a blunder introduced V 63 

 he confuses it with V63. 



INDUCTIVE ARITHMETIC. NICOMACHUS. As in the case of 

 astronomy, progress in geometry now lags and finally ceases alto- 

 gether. About 100 A. D. a final era of Greek mathematical science, 

 predominantly arithmetical in character, begins with Nicomachus 

 of Judea, whose work remained the basis of European arithmetic 

 until the introduction of the Arabic arithmetic a thousand years 

 later. He enunciates curious theorems about squares and cubes, 

 for example : In the series of odd numbers from 1, the first term 

 is the first cube, the sum of the next two is the second, of the 

 next three the third, etc., doubtless simple observation and 

 induction. He refers to proportion as very necessary to " natural 

 science, music, spherical trigonometry and planimetry," and 

 discusses various cases in great detail. 



