68 A SHORT HISTORY OF SCIENCE 



ing with an isosceles right triangle, he describes a semicircle on 

 each of the three sides. By the theorem just quoted the semi- 

 circle on the hypotenuse is equal in 

 area to the sum of the other two. If 

 the larger semicircle is taken away 

 from the entire figure, two equal lunes 

 remain; if the two smaller semi- 

 circles are taken away, the triangle remains. Therefore the 

 two lunes are together equivalent to the triangle, and the 

 area of each may be determined. The gulf between rectilin- 

 ear and curvilinear figures has at last been successfully crossed. 

 A second attempt employs three equal chords instead of two, and 

 incidentally the theorem that the square on the side of a triangle 

 is greater than the sum of the squares on the other two sides when 

 the angle opposite the first side is greater than a right angle. 

 Other interesting and still more complicated attempts are pre- 

 served. 



A third classical problem was that of the so-called " duplication 

 of the cube." One of the older Greek tragedians attributed to 

 King Minos the words referring to a tomb erected at his order : 



Too small thou hast designed me the royal tomb, 

 Double it, yet fail not of the cube. 



At a somewhat later period it is related that the Delians, suf- 

 fering from a disease, were bidden by the oracle to double the size 

 of one of their altars, and invoked the aid of the Athenian geom- 

 eters. Hippocrates transformed the problem of solid geometry 

 into one in two dimensions by observing that it is equivalent to 

 that of inserting two geometrical means between given extremes. 

 In our modern algebraic notation, the continued proportion 

 x:y = y:z = z:a leads to the equations y 2 = xz, z 2 = ya, whence, 



1 2 



eliminating z, t/ 3 = ax 2 , y = a 3 X s ; y and z are the desired means 

 between x and a, and by putting a = 2x the problem is solved. 

 No such algebraic notation existed at this time, however, 

 and the geometrical methods invented by later Greek mathema- 

 ticians were necessarily very complicated, as will appear below. 



