8 6 Mr. Woodhouse on the Independence of the 
and that the relations between the symbols or general terms 
were to be established by giving the true meaning to the con- 
necting signs, which indicate not so much the arithmetical 
computation of quantities, as certain algebraical operations. It 
was further observed, that, from certain established formulas, 
abridged symbols or general terms might be formed, which 
consequently must have their signification dependent on such 
formulas ; and that, although the parts of certain abridged ex- 
pressions could not separately be arithmetically computed, yet 
the expressions themselves might be legitimately employed in 
all algebraic operations. 
The chief object of my paper was to shew, that operations 
with imaginary quantities, as they are called, were strictly and 
logically conducted, that is, conducted after the same manner 
as operations with quantities that can be arithmetically com- 
puted : the question, whether calculation with imaginary sym- 
bols is commodious or not, was then slightly discussed. I have 
since attentively considered it, and, what usually happens in 
such cases, my inquiries have been extended beyond their origi- 
nal object ; for, actual research has convinced me of what there 
were antecedent reasons for suspecting, that not only in the 
theory of angular functions, demonstration is most easy and 
direct by giving to quantities their true and natural* represen- 
tation ; but, that the introduction of expressions and formulas 
not analytical, into analytical investigation, has caused much 
ambiguity, confused notion, and paradox; that it has made 
/ 
. + -*s— -^=T} 
&c. I call the natural representations of the cosines, sines, &c. of an arc x; because, 
admitting the algebraical notation, they, by strict inference, adequately, unambigu* 
ously, and solely, represent the cosines, sines, &c. 
