DEVELOPMENT OF MATHEMATICAL THOUGHT. G4 5 



infinite series. The solution of an equation being called 

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

 every equal icui has as many roots as are indicated by 

 its degree. A proof of this fundamental theorem of 

 algebra was repeatedly attempted, Ijut 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 mathenuitical expressions into series. In arithmetic, 15. 



. Doctrine 



decimal fractions taken to any number of terms were of series. 



"^ Gauss. 



quite familiar ; the infinite series presented itself as a 

 generalisation of this device. A very general formula 



' Werke,' vol. iii. pj). 1 aiul 71). .\ 

 very good suiiiiiuiry of this proof 

 is given liy Hankel ( ' Couiplexe 

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

 algebraical deuion.stration of the 

 same theorem, not involving con- 

 sidt-rations of continuity and ap- 

 pruximatifjn.s, was also given by 

 Gauss in the year 1816, and re- 

 produced by others, including 

 George Peacock, in his ' Report,' 

 (juoted above, p. 297. Hankel 

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

 extent Gauss's proof supplemented 

 the siniiiar proofs given by othei-s 

 before and after. 



^ 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 sj'stems, were known to the 

 ancients, and continued in use to 

 the middle ages. The dyadic sys- 

 tem wa.s much favoured by Leibniz. 

 It was also known that every 

 rational fraction could be de- 

 veloped into a periodical decimal 



fraction. l*i-ominent 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 ' Arithmetique ' (1590, trans- 

 lated into English, 1608), described 

 the decimal system as " enseignant 

 facilement expodier par nombres 

 entiei's sans rompus tons 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 sj)eedy adoption of this device, 

 connecting with it the suggestion 

 of the universal adoption of the 

 decimal system. The best account 

 of the grail ual introduction of deci- 

 mal fractions is still to be found iii 

 George Peacock's ' History of Aiilh- 

 metic' ( ' Ency. ^tetrop.,' vol. i. p. 

 439, &c.) 



