96 
MR. DAVIES GILBERT ON THE NATURE 
Let ^ the original arc, v the sine, and y the cosine, x the cosine of the arc n z, 
then as 2 : — y : : 1 : ^ 1 — y l v z = therefore 
x y 
" ' " ~ 'll # ^ 
V i — x~ " V 1 — y 2 
No integration can however be effected of these quantities in their actual form ; 
but changing the signs of the terms in both denominators. 
■ n, 
y 
and 
\/ x 2 — 1 ' \/ y 2 — 1 
the nat. log. of (x + V x 2 — l) = n X the nat. log. of (y + */ -if _ 1) 
and x + \/x 2 — 1 = (3/ + Vy 2 - 1 ) n =y nJ r n y 
n — 1 
n — 1 n — 2 — . n — 1 n — 2 
+ n ~2 ~y -y 2 - 1 + '—ar-y 
. ^/^/2 — J 
n — 3 
+/V - 1 
But since y is taken as the cosine of the original arc, and x is the cosine of 
the multiple arc, and consequently each less than unity, it is obvious that 
the second term on the left of the equation, and that every even term of the 
expansion on the right, can exist only in the potential form of an ideal quantity ; 
and a conclusion has thence been drawn (but as it seems to me on no solid 
principle whatever), that since real and imaginary quantities occur on each 
side of the equation, and they are of a nature completely heterogeneous one 
to the other, each must be respectively equal ; but this mode of reasoning 
clearly imputes qualities to mere symbols beyond those originally imparted to 
them. The double equality, on my principles, depends entirely on its being as- 
sumed ; as in the solution of cubic equations. 
When a -f b have been substituted for x in the equation x? — q x r — 0, 
and it becomes changed into a 3 + & + (3 a b — q) x a b r ■=■ 0, two un- 
known quantities exist with but one relation ; another may therefore be 
assumed ; and that which obviously reduces the expression to the most simple 
form is obtained by making 3 ab — q = 0. 
In the same manner x and y, the cosines of two arcs, having but one rela- 
tion, admit of another being assumed; any relation might be taken, but the 
one clearly indicated is that which makes the real terms on both sides equal : 
