240 A SHORT HISTORY OF SCIENCE 



first term is made 1 by division and that of the second is made 



by the substitution x = y . The new equation having the form 



3 a 



e # 



y 3 + ey +/ = we now put y = z , whence z 3 + / = 0, 



3 z 27 2r 



a quadratic equation in z 3 . The solution of the original equation of 

 degree three is thus made to depend on that of an equation of degree 

 one less. 



Similarly if the given equation of the fourth degree is in our 

 notation ax 4 + bx 3 + ex 2 + dx + e = the coefficient of the 

 first term is made 1 by division and that of the second is made by 



the substitution x = y 



4a 



The new equation having the form 

 y*+fy z + gy + h -0. 



We put 7/ 4 + // + gy + h = (y 2 - ay + 0) (/ + ay + 7) 

 whence / = /3 -+- 7 a 2 



9 = (0 - 7) a 



h = /3 7 . 



We obtain a, /3, 7 from these three equations by eliminating two 

 and solving the cubic equation obtained for the other ; that is, the solu- 

 tion of the original equation of degree four is made to depend on that 

 of a new equation of degree one less. 



One of Cardan's scientific inventions was an improved suspen- 

 sion of the compass needle. He was also eminent as an astrologer. 



SYMBOLIC ALGEBRA: VIETA. --Of still greater importance in 

 the history of algebra is F. Vieta (1540-1603) a lawyer of the 

 French court. He won the interest of Henry IV by solving a com- 

 plicated problem proposed by an eminent mathematician, as was 

 the custom of the time, as a challenge to the learned world. This 

 involved an equation of the 45th degree which he succeeded in 

 solving by a trigonometric device. Later he was employed to inter- 

 pret the cipher despatches of the hostile Spaniards. His In Artem 

 Analyticam Isagoge is the earliest work on symbolic algebra. 

 In it known quantities are denoted by consonants, unknown by 

 vowels, the use of homogeneous equations is recommended, the 



