54 A SHORT HISTORY OF SCIENCE 



a denoting any positive integer. How this method was discovered 

 remains a matter of conjecture. 



We may recognize here the characteristic elements of the in- 

 ductive method, first, observation of the particular fact that in a 

 certain right triangle, with sides 3, 4, and 5, the sum of the squares 

 on the two sides is equal to that on the hypotenuse ; second, the 

 formation of the hypothesis that this may be true also for right 

 triangles in general; third, the verification of the hypothesis in 

 other particular cases. Then follows the deductive confirmation 

 of the hypothesis as a law for all right triangles. 



PYTHAGOREAN PHYSICAL SCIENCE. - - It has been already noted 

 that one of the most fundamental principles of the Pythagorean 

 school was the significance attached to number in connection 

 with all sorts of phenomena, the regular motions of the heavenly 

 bodies, the musical tones, etc. There is a tradition that Pythag- 

 oras, walking one day, meditating on the means of measuring 

 musical notes, happened to pass near a blacksmith's shop, and 

 had his attention arrested by hearing the hammers as they struck 

 the anvil produce sounds which had a musical relation to each 

 other. It was found that vibrating cords emitted tones de- 

 pendent in a simple way on their length ; for example, cords of 

 lengths 2, 3, and 4 giving a tone, its fifth and its octave re- 

 spectively. The monochord used in studying these numerical 

 relations is said to have been the first apparatus of experimental 

 physics. It was even supposed that each of the various 

 heavenly bodies and the sphere of the fixed stars had a char- 

 acteristic tone, these all uniting to produce the so-called " music 

 of the spheres." 



TERRESTRIAL MOTION ; PHILOLAUS, HICETAS. The universe 

 was believed to consist of the four elements, earth, air, fire, water, 

 - to be a sphere with a spherical earth at its centre, and to have 

 life. Pythagoras identified the morning and evening stars, and 

 attributed the moon's light to reflection. It is of peculiar interest 

 that later Pythagoreans, in particular Philolaus, about 400 B.C., 

 attributed the apparent daily motion of the heavenly bodies from 

 east to west not to their own actual motion but to a motion of 



