OF NEGATIVE QUANTITIES. 
249 
of them can be so expanded without the intervention of any geometric consi- 
derations, and to point out that the series obtained will involve c the circum- 
ference of the circle, and thus to prove that there is a connexion between 
algebra and geometry ; if the object had been to obtain all the values of l 1 in 
a series, it would have been more convenient to have introduced geometrical 
considerations as in my treatise. 
To the second part of this objection, viz. that the geometric representation 
can only be analogical and not a true algebraic representation of the roots ; it 
may be replied, that the geometric representation of the square roots of nega- 
tive quantities rests on the same foundation as the geometric representation, 
or any other representation of the negative quantities themselves. A nega- 
tive quantity arithmetically considered is a mere absurdity, being the dif- 
ference which arises from subtracting a greater quantity from a less ; but 
algebraists having found that operations might more easily be performed 
by considering negative quantities in the abstract, endeavoured to establish 
their real existence, and with this view they made the following hypothe- 
ses ; that, if a line drawn in one direction be represented by a positive quan- 
tity, a line drawn in the opposite direction will be represented by a nega- 
tive quantity ; that the sum of a positive and a negative quantity is to be 
found by subtracting the less from the greater, and prefixing the sign of the 
greater ; that subtraction is to be performed by changing the sign of the quan- 
tity to be subtracted, and proceeding as in addition ; that the product of a po- 
sitive and a negative quantity is negative, and the product of two negative 
quantities, positive ; and having made these hypotheses, they proved, by ex- 
amining into the nature of algebraic operations, that the results arrived at by 
means of these hypotheses must be correct ; therefore they concluded that these 
were true hypotheses ; and their truth being established, they were admitted 
as fundamental principles of algebra : and in the same way other true hypo- 
theses were established relative to the representation of negative quantities : 
such as ; if time to come be represented by a positive quantity, time past will 
be represented by a negative quantity, & c. I call these algebraic principles, 
hypotheses ; for though most algebraists have considered them as propositions, 
and have endeavoured to establish their truth by direct demonstration, yet 
their reasoning is unsatisfactory, for they always treat of negative quantities 
2 K 
MDCCCXXIX. 
