. 
160 On the ORIGIN and 
that could occur, carefully remarking whatever change might 
arife in the conftruction, from any change that was fuppofed to 
take place among the magnitudes which were given. 
Now, as this cautious method of proceeding was not better cal- 
culated to avoid error, than to lay hold of every truth that was 
connected with the main object of enquiry, thefe geometers foon 
obferved, that there were many problems which, in certain circum- 
ftances, would admit of no folution whatever, and that the gene- 
ral con{truction by which they were refolved would fail, in con- 
fequence of a particular relation being fuppofed among the 
quantities which were given. Such problems were then faid to 
become impoffible ; and it was readily perceived, that this al- 
ways happened, when one of the conditions prefcribed was in- 
confiftent with the reft, fo that the fuppofition of their being 
united in the fame /udyjec?, involved a contradiction. Thus, 
when it was required to divide a given line, fo that the reftan- 
gle under its fegments, fhould be equal to a given {fpace, it: 
was evident, that if this {pace was greater than the fquare of 
half the given line, the thing required could not poflibly be 
done; the two conditions, the one defining the magnitude of 
the line, and the other that of the rectangle under its fegments, 
being then inconfiftent with one another. Hence an infinity of. 
beautiful propofitions concerning the maxima and the minima 
of quantities, or the limits of the poffible relations which quan- 
tities. may ftand in to one another. 
8..Sucu cafes as thefe would occur even in the folution of 
the fimpleft problems ; but when geometers proceeded to the 
analyfis of fuch as were more complicated, they muft have re- 
marked, that their conftructions would fometimes fail, for a rea- 
fon dire@ly contrary to that which has now been affigned. In- 
{tances would be found where the lines that, by their interfection, 
were to determine the thing fought, inftead of interfeCting one 
another, as they did in general, or of not meeting at all, as im 
the above mentioned cafe of impoflibility, would coincide 
with. 
