160 On tie 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 conftruction 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 faidto 

 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 fubjeft, involved a contradiction. Thus, 

 when it was required to divide a given line, fo that the rectan- 

 gle under its fegments, fhould be equal to a given fpace, it 

 was evident, that if this fpace 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 proportions concerning the maxima and the minima 

 of quantities, or the limits of the poflible relations which quan- 

 tities may ftand in to one another. 



8. Such 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 directly contrary to that which has now been affigned. In- 

 ftances would be found where the lines that, by their interfection, 

 were to determine the thing fought, inftead of interfering one 

 another, as they did in general, or of not meeting at all, as in 

 the above mentioued cafe of impoflibility, would coincide 



with 



