quadratic y cubic y quadrato-cuhic, and higher Surds. 33 
— cV ahc' = — cm\ which solved gives W ahc^ = 
involve to the 5th power ; then, abc^ = ^ ^ 
— V c^ — ^cm. 
^ 
Divide by ~c^, and transpose ; then, — c^-\- ^mc‘^ ^ = ± 
— gmc + V c~ — /\gnc : squared, gives 
c*— lomc' + 35 w^^ c* —50^3. -f 25m+. c =.c ^ — iomc^+ 
— ^ab. -^zoahrn. — zoabtn*. 
-f \a^b*. 
— — 4’7iV. 
Transpose, divide by 4c, and arrange ; then, * * — ^abcm^ 
t^abc^m = 0 ; or, =: abc . ^rrd — ab — ac — be ; 
which is an equation of 5 dimensions, free from surds. 
This equation, if, instead of + Wb + %/r = 0, were 
substituted, -|- ^°Vb + c ~ 0, would contain no other 
than quadratic surds ; if, + 'Wb + 'Wc = 0, no higher 
than cubic surds ; wherefore, if the exterminacion of any 
number of surds of the 5th power from an equation be ad- 
mitted, since the number of surds of any lower order which 
may be exterminated is unlimited, an equation consisting of 
any number of surds, whose indices are in any manner com- 
pounded of the factors, 2, 3, and 5, may be totally freed from 
surds. 
If a formula for the solution of any equation of 6 dimen- 
sions were known, any number of surds of the 7th power 
might be taken away from an equation : As such a formula, 
however, is, I suppose, at present altogether unknown, we 
may be contented with the extermination of 3 surds of the 
7th power, which may be accomplished, because, a formula 
Cor the solution of cubic equations is known, 3, the index of 
MDCCCXIV. F 
