quadratic, cubic, quadrate^ cubic, and higher Surds, 31 
u „ + 3 ^’'* + 3 ^’- -{■22ob^c 
• or +1518&C. a' 4- 3 1 686V. a + 9246V 
+3^*. -f 3 1686c*. -f2206c^ 
+ 3c^ -fc‘^ 
+ 3 • ^ 
3 +876. 
+ 339 <^- 
a 
-1536*. 
- 3357 *^- 
+ 1089c*, 
+ 4 ^^ 
+ 1656V 
-f 2646c* 
+ 2 2C3 
+ 3 . 
3 4 - 339 &- 
+ 87C. 
4- 10896*. 
—33S7bc. a 
-153c* 
4226^ 
4 2646 *c 
4 i656tf* 
44^^ 
^y/bc' = 0, 
Compare this with the equation, s d'e v de* = 0 : 
the resulting equation, s "i" v^y'de* = o^stvde^ being 
accordingly computed, will be free from surds. It will be of 
12 dimensions; but may be depressed to one of g. Instead 
of continuing the operation to shew this, I refer to the exter- 
mination of surds of the 5th, and of the 7th power, to be given 
below, for the manner in which 'some equations, resulting on 
involution, are depressed. 
In surds of the 5th power, the quantity or factor, necessarily 
subjected to the radical sign, may be of 4 dimensions, but not 
higher: whence, if the solution of any biquadratic equation be 
admitted, any number of surds of the 5th power may be taken 
away from an equation ; and here it may be observed, that, 
as to the matter in hand, it is of no importance, whether the 
biquadratic equation may be solved in possible terms, or not ; 
for the value, in numbers, of any particular quantity, or factor, 
is not required ; it is only required to obtain the quantity, or 
factor, of a single dimension, in order to deprive it, by invo- 
lution, of its radical sign. 
When an equation consists of 3 surds of the 5th power, the 
biquadratic equation is virtually a quadratic. 
Let + *Vb 4 “ = 0 (2 ;) cz 4- 6 4 “ 5 ^Va*b 4- 10 
W a^h'' 4 " 4 " 5 * == — c ; put cz 4- ^ 4 " = 5 ^'^* 
*.• (3;) Wa^b 4“ 2 Wa^b* 4* 2 4" Wdb'" = ^ m\ this, 
