obtained by Means of imaginary Quantities. 107 
sentatives of real things ; and demonstration would be defined 
to be, a method of shewing the agreement of remote ideas by a 
train of intermediate ideas, each agreeing with that next it ; or, 
in other words, a method of tracing the connection between 
certain principles and a conclusion, by a series of intermediate 
and identical propositions, each proposition being converted into 
its next, by changing the combination of signs that represent 
it, into another shewn to be equivalent to it. 
Exactly according to this plan have the foregoing proposi- 
tions been demonstrated: the symbol for the sine of x is 
, for the cosine of y is 
and 
the connection was traced between 
+ e~ 
and \ x 
J x+y) V - 1 _ -r (x+y)V: 
+ * 
-( X ~y)V 
V^~l 
zV — I 
by a series of transformations, each 
of which was shewn to be lawful, by referring to what e x ^ ~~ 1 
&c. was made to represent, and to the nature of the operations 
directed to be performed by the signs x, +, &c. thus, the trans- 
formation of e x ^ ~ 1 x (e y ^ ~~ 1 + e~ y ^ ~ 1 ) into e x ^ ~ 1 x e? ^ ~~ 1 
e x V 1 x e —y d — 1 j s lawful because the same series results, 
whether ey ^ — 1 and e~ y ^ ~~ 1 be first expanded, and then each 
term of their sum be combined with e x ^ ~ *, or whether e x ^ — 1 
be separately combined with each term of the developements 
for ey ^ — 1 and e~~ y ^ — *, and then the resulting terms added 
together : again, the transformation of e x ^ ~ 1 x ey ^ ~~ 1 into 
ei x + y) v/ — 1 is lawful, because the same series results, whether 
Pa 
