126 A SHORT HISTORY OF SCIENCE 



Mathematics had passed from the study of the philosopher to 

 the lecture-room of the undergraduate. We have no more the grave 

 and orderly proposition, with its deductive proof. Nicomachus 

 writes a continuous narrative, with some attempt at rhetoric, with 

 many interspersed allusions to philosophy and history. But more im- 

 portant than any other change is this, that the arithmetic of Nico- 

 machus is inductive, not deductive. It retains from the old geometrical 

 style only its nomenclature. Its sole business is classification, and 

 all its classes are derived from, and are exhibited in, actual numbers. 

 But since arithmetical inductions are necessarily incomplete, a general 

 proposition, though prima facie true, cannot be strictly proved save 

 by means of an universal symbolism. Now though geometry was 

 competent to provide this to a certain extent, yet it was useless for 

 precisely those propositions in which Nicomachus takes most interest. 

 The Euclidean symbolism would not show, for instance, that all the 

 powers of 5 end in 5 or that the square numbers are the sums of the 

 series of odd numbers. What was wanted, was a symbolism similar 

 to the ordinary numerical kind, and thus inductive arithmetic led 

 the way to algebra. Gow. 



PTOLEMY AND THE PTOLEMAIC SYSTEM. - - With Claudius 

 Ptolemy, in the second century of our era, Greek astronomy 

 reaches its definitive formulation. In the 260 years which had 

 elapsed since Hipparchus no progress of consequence had been 

 made. 



Of Hipparchus, from whom he inherited so much, Ptolemy 

 writes : 



It was, I believe, for these reasons and especially because he had 

 not received from his predecessors as many accurate observations as 

 he has left to us, that Hipparchus, who loved truth above everything, 

 only investigated the hypotheses of the sun and moon, proving that 

 it was possible to account perfectly for their revolutions by combi- 

 nations of circular and uniform motions, while for the five planets, 

 at least in the writings which he has left, he has not even com- 

 menced the theory, and has contented himself with collecting sys- 

 tematically the observations, and showing that they did not agree 

 with the hypotheses of the mathematicians of his tune. He explained 

 in fact not only that each planet has two kinds of inequalities but also 



