no Mr. Woodhouse on the Truth of Conclusions 
a + b v/ — i for x, in the series x — 1 — — See. 
1 i .2.3 “ 1 .2. 3.4.5 
proved for the sine of an arc x, is the same as what results from 
expanding e b , e~ b , cos. a, sin. a, & c. and combining the terms 
after the manner directed by the signs -f , x, & c. A like expla- 
nation is to be given of the second expression. 
If m be an integer, c the semi-circumference, and a — c, 
then cos. a= o, and the first expression becomes sin. [a -f- b\/ — 1) 
= \[e b -}- e~ b ) sin. a. According to the explanation I have given, 
this expression is very perspicuous and intelligible ; but Euler, 
inattentive to its true meaning, gives it an air of mystery and 
paradox, when he says that an impossible arc may have a real 
sine. 
The symbol \ Euler proved equal to 0.20787 957, 
&c. To understand its meaning, we must recur to the form from 
which it was derived : now, according to the definition that has 
been given of equality between imaginary expressions, it may 
be shewn that 
(a-f&v/ — 1 ) m+Kv/ ~i —f l e nx [cos.(mx^ny.l.r)^K/TZ^sm.{mx-\-ny.l. 
r being = y/ a* + b\ sin. x = cos. x — -^. 
Now, if a be put = 0, m = o, b=i,n — 1, the expression 
[a -j- b V — i) m + " ^ “ 1 becomes [y/ — 1 ~~ \ and the ex- 
pression to which it is equal becomes e ~~ '* (c circumference). 
Or the meaning of the symbol may be thus explained, x x is the 
same as e x log * x .*. if\/ — lbeputfor*, [y/ — i^~ l — e x/ ~‘ llos ‘ x/ ~ I ; 
but it has appeared that log. v/ — 1 is an abridged symbol for 
