DEVELOPMENT OF MATHEMATICAL THOUGHT. 645 



infinite series. The solution of an equation being called 

 finding its roots, it was for a long time assumed that 

 every equation has as many roots as are indicated by 

 its degree. A proof of this fundamental theorem of 

 algebra was repeatedly attempted, but was only com- 

 pleted by Gauss in three remarkable memoirs, which 

 prove to us how much importance he attached to rigorous 

 proofs and to solid groundwork of science. The second 

 great doctrine in which the conceptions of the continuous 

 and the infinite presented themselves was the expansion 

 of mathematical expressions into series. In arithmetic, is. 

 decimal fractions 1 taken to any number of terms were of series, 

 quite familiar; the infinite series presented itself as a 

 generalisation of this device. A very general formula 



Gauss. 



'Werke,' vol. iii. pp. 1 and 71). A 

 very good summary of this proof 

 is given by Hankel ( ' Complexe 

 Zahlen-Systeme,' p. 87). A purely 

 algebraical demonstration of the 

 same theorem, not involving con- 

 siderations of continuity and ap- 

 proximations, was also given by 

 Gauss in the year 1816, and re- 

 produced by others, including 

 George Peacock, in his 'Report,' 

 quoted above, p. 297. Hankel 

 (loc. cit., p. 97) shows to what 

 extent Gauss's proof supplemented 

 the similar proofs given by others 

 before and after. 



1 Decimal fractions seem to have 

 been introduced in the sixteenth 

 century. Series of other numbers, 

 formed not according to the decimal 

 but to the dyadic, duodecimal, or 

 other systems, were known to the 

 ancients, and continued in use to 

 the middle ages. The dyadic sys- 

 tem was much favoured by Leibniz. 

 It was also known that every 

 rational fraction could be de- 

 veloped into a periodical decimal 



fraction. Prominent in the re- 

 commendation of the use of deci- 

 mal fractions was the celebrated 

 Simon Stevin, who, in a tract 

 entitled ' La Disme ' attached to 

 his ' Arithme"tique ' (1590, trans- 

 lated into English, 1608), described 

 the decimal system as " enseignant 

 facilement expe"dier par nombres 

 en tiers sans rompus tous comptes 

 se rencontrans aux affaires des 

 hommes." Prof. Cantor ( ' Gesch. 

 der Math.,' vol. ii. p. 616) says, 

 " We know to-day that this pre- 

 diction could really be ventured 

 on that indeed decimal fractions 

 perform what Stevin promised." 

 At the end of his tract he doubts 

 the speedy adoption of this device, 

 connecting with it the suggestion 

 of the universal adoption of the 

 decimal system. The best account 

 of the gradual introduction of deci- 

 mal fractions is still to be found in 

 George Peacock's ' History of Arith- 

 metic' ('Ency. Metrop.,' vol. i. p. 

 439, &c.) 



