io6 Mr . Woodhouse on the Truth of Conclusions 
.*. adding the quantities that are vertical to each other, the series 
Now, each horizontal row consists of a series of cosines of 
may immediately be obtained from the expression deduced in 
proposition 3d. 
I think it superfluous to give more examples, since the object 
of this memoir is rather to shew the logical justness of a method, 
than its commodiousness or extent : all other propositions rela- 
tive to lines drawn in a circle, when expressed by aid of the 
symbol s/ — 1 , the same principle of explanation regulates ; 
the principle once understood, the operations become mecha- 
nical, require attention, but are attended with no real mental 
difficulty. 
e* v '~" x 4- e~ x x/ ~~ 1 
It is inaccurate to call - — - — ^ an imaginary value 
of the cosine of an arc : the expression expanded is a real one. 
By use of the symbol s/ — 1, and of the forms proved to ob- 
tain in the combination of real quantities, a mode of notation 
is obtained, by which we may express sines and cosines, &c. re- 
latively to their arc. 
If the process by which the foregoing propositions have been 
established require illustration, I would ask what demonstration 
is, when the characters employed are signs of ideas, or repre- 
-f &c. 
arcs in arithmetical progression ; and the sum of each series 
