OF NEGATIVE AND OF IMAGINARY QUANTITIES. 
97 
this assumption leaves the ideal quantities in their actual state without any 
change, and when the sign of an imaginary operation has been removed in the 
usual way, they also become truly equal on each side of the equation, and from 
this double equality, series readily present themselves expressing the sine and 
cosine of the multiple arc required. 
On the whole, it appears to me that mystery and paradox are entirely ba- 
nished from the science most powerful in eliciting truth, and where they ought 
least to be found, — by considering all quantity as affirmative per se, and admit- 
ting plus and minus only as correlative terms ; consequently, by extending the 
law of the signs, so as to make the multiplication of like signs give that of the 
scale in which the universal antecedent has been assumed, and the multiplica- 
tion of unlike signs the contrary ; and, finally, by excluding all actual exist- 
ence, and thereby all inherent properties, from the symbols of quantities veiled 
by the mark indicative of an operation incapable of being performed, but ar- 
bitrarily endowing them with the properties of real quantities in all expansions 
and formulae, and with the ultimate quality of regaining their actual existence 
whenever the veil is removed, by an operation the inverse of that which had 
originally induced it. 
MDCCCXXXI. 
O 
