129 
troversy respecting the first solution of biquadratic equations; 
and it is not unreasonable to infer that the first solution of 
a biquadratic known to English students, through the 
excellent treatise on the theory of equations by Todhunter, 
as Des Cartes, may justly be claimed by Bombelli, Oughtred, 
and Harriot. 
In Wallis’s Algebra, published in 1685, there is a copious 
account of the works of Vieta’s specious arithmetic, Oughtred 
and his Clavis, Harriot’s Algebra, published in 1631, Hr. 
Pell’s Algebra, and several other works of European interest. 
By multiplying out the form. (A) there results 
— (a + /3 + V + + (a/3 + a?/ + + /3v + /3^ + vh)x^ 
— (a(3y + aj3B + + ayS)x 
+ a/3r^ == 0 (3) 
If this biquadratic is to coincide with (2) it can do so only 
by determining the indefinite coefficients a, /3, r, S so as to 
satisfy the following equations : 
a+/3 + p + S = 0 (4) 
a/3 + a?^ + a^ + (3^ + j3h + = a (5) 
al3v + avh + + af3d = ( 6 ) 
a[3vh — c (7) 
These equations, though admirable in discovering the 
various relations amongst the roots themselves, are but of 
little use in finding each root in functions of a, b, c. For 
instance square (4) and compare the result with (5), then 
a^ + /32 + + -2a... 
a^ + /3« + v® + a3= T35.... 
a^ + /3^ + v^ + a^ = 2a2-4c 
a^ -t- /3® + V® + ± 5a6. . 
..( 8 ) 
..(9) 
ao) 
(11) 
The value of a’* + /3'‘ + + a” can be obtained by Newton’s 
process in terms of a, 6, c. 
3. The form (B) which is said to be due to Hes Cartes, 
but according to Wallis is equally due to Bombelli, Ouo-htred, 
and Harriot : 
Multiply out, then, 
iJ3 ■¥ V — a^)x^ + a(i^ — /3)o; + /3»/ = 0 (12) 
