obtained by Means of imaginary Quantities . 109 
sitions relative to lines belonging to a circle, given by M. La- 
grave, by Euler, (Introductio in Analysin Infinitorum, p. 
198.) and by Bossux. (Mem. de l’Acad. 1769, p. 453.) I am, 
however, of opinion, that the method of representing sines, co- 
sines, &c. by their abridged algebraical symbols, (such as is 
given in this Paper,) is the most easy and extensive in its ap- 
plication. * 
It will be consistent with the purpose of the present memoir, 
to consider some of the expressions which I imagine are alluded 
to, by those who complain of the abuses, paradoxes, &c. intro- 
duced by negative and impossible quantities. 
The quantity — - — ~?===, which John Bernouilli proved to 
be the circumference of a circle, is merely an abridged symbol, 
founded on a form proved for real quantities : the sense in which 
it is to be understood is this, that if in the series for log. x, viz. 
(x — x"' 1 ) — \ (a : 1 — X ~ 2 ) + i (x 3 — x ~ 3 ) — &c. s/ -~T is 
substituted for x , and the terms multiplied by 4 and divided by 
%/ — 1, the resulting series expresses the circumference of a 
circle. 
The expressions 
(1) sin. (a-f b v / —i) = e~ b ) sin. a -f- [e b -± - e~ b ) co s.a, 
(2) cos. [a -j- b \/ — 1) + cos - [ a — b s/ — 1).= [e b -{- e~~ b ) cos .a, 
are due to Euler : the sense in which alone they are to be un- 
derstood is this, that the series which results from substituting 
• M.Bossut does not sum any series beyond that of the fourth power of sines 
and cosines of arcs in arithmetical progression : he contents himself with saying, that 
the general law for cos. q 4. cos. 2 q &c. may easily be discovered. 
