192 ON THE MAGNITUDE OF 



little about the early history of the subject, but it is clear that the 

 primitive astronomers must have found the quantities to be mea- 

 sured too small for detection with their instruments, and even in 

 modern times the problem has proved to be an extremely difficult 

 one. Aristarchus of Samos, who flourished about 270 B.C., seems 

 to have been the first to attack it in a scientific manner. Stated 

 in modern language, his reasoning was that when the moon is 

 exactly half full, the earth and sun as seen from its centre 

 must make a right angle with each other, and by measuring the 

 angle between the sun and moon, as seen from the earth at that 

 instant, all the angles of the triangles joining the earth, sun, and 

 moon would become known, and thus the ratio of the distance of 

 the sun to the distance of the moon would be determined. 

 Although perfectly correct in theory, the difficulty of deciding 

 visually upon the exact instant when the moon is half full is so 

 great that it cannot be accurately done even with the most power- 

 ful telescopes. Of course, Aristarchus had no telescope, and he 

 does not explain how he effected the observation, but his conclu- 

 sion was that at the instant in question the distance between the 

 centres of the sun and moon, as seen from the earth, is less than 

 a right angle by i/3oth part of the same. We should now express 

 this by saying that the angle is 87 degrees, but Aristarchus knew 

 nothing of trigonometry, and in order to solve his triangle he had 

 recourse to an ingenious, but long and cumbersome geometrical 

 process which has come down to us, and afi"ords conclusive proof 

 of the condition of Greek mathematics at that time. His conclu- 

 sion was that the sun is nineteen times further from the earth than 

 the moon, and if we combine that result with the modern value of 

 the moon's parallax, viz., 3, 422*38 seconds, we obtain for the solar 

 parallax 180 seconds, which is more than twenty times too great. 



The only other method of determining the solar parallax 

 known to the ancients was that devised by Hipparchus about 150 

 B.C. It was based on measuring the rate of decrease of the dia- 

 meter of the earth's shadow cone by noting the duration of lunar 

 eclipses, and as the result deduced from it happened to be nearly 

 the same as that found by Aristarchus, substantially his value of 

 the parallax remained in vogue for nearly two thousand years, and 

 the discovery of the telescope was required to reveal its erroneous 



