466 
THE SOLUTION OF EQUATIONS 
He also gave two or three other solutions, but the one here 
shown is particularly clear and simple. In his work in equa 
tions he was greatly aided by his new symbolism (p. 430). 
Hudde’s Contribution. Although Descartes contributed to the 
solution of the cubic equation by his convenient symbolism and 
by his work on equations in general, he made no specific con 
tribution of importance. The next writer to materially simplify 
the work of Vieta was Hudde (c. 1658). Taking advantage of 
Descartes’s symbolism, he brought the theory of the cubic equa 
tion to substantially its present status. He is also the first 
algebraist who unquestionably recognized that a letter might 
stand for either a positive or a negative number. 1 
His method of solving the cubic equation is to begin with 
and let 
x = qx + r 
x=y + z, 
so that y 8 4- 3/*+ 3 T- 2 + z% ~ <l x + r - 
He then lets 
and 
which gives 
Hence 
and so 
and 
Hence 
J 3 * + 2 Z = V 
3 ¿y + 3 -3’ 2 y = qx, 
y = i q! z ' 
y = r — z 3 = 2 y q 8 !y 
* 3 = I r ± 
y/1 1 
V ~r r 7r*T 
y = JfTvd r 2 - Jyq a = B - 
x = yA+ y B, 
which satisfies both his assumptions. 2 
Equation of the Fourth Degree. After the cubic equation had 
occupied the attention of Arab scholars, with not very signifi 
cant results, the biquadratic equation was taken up. Abu’l- 
Faradsh 3 completed the Fihrist c. 987, and in this he refers to 
1 Eneström, in Bibi. Math., IV (3), pp. 208, 216. 
2 The problem, as worked out by Hudde, is given in Matthiessen, Grundzüge, 
P- 374- 
3 Abü’l-Faradsh (Faraj) Mohammed ibn Ishaq, known as Ibn Abi Ya'qub al- 
Nadim. The title is Kitdb al-Fihrist (Book of Lists). See the Abhandlungen, VI, 1.