loo On the ORIGIN and 



31. Another fpecies of impoffibility may frequently arife 

 from the porifmatic cafe of a problem, which will very much 

 affect the application of geometry to aftronomy, or any of the 

 fciences of experiment, or obfervation. For when a problem is 

 to be refolved by help of data furnifhed by experiment or ob- 

 fervation, the fir ft thing to be considered is, whether the data fo 

 obtained, be fufficient for determining the thing fought , and 

 in this a very erroneous judgment may be formed, if we reft 

 fatisfied with a general view of the fubject : For though the 

 problem may in general be refolved from the data that we are 

 provided with, yet thefe data may be fo related to one another 

 in the cafe before us, that the problem will become indeter- 

 minate, and inftead of one folution, will admit of an infinite 

 number. 



Suppose, for inftance, that it were required to determine 

 the pofition of a point F, (fig. 4.) from knowing that it was fi- 

 tuated in the circumference of a given circle ABC, and alio 

 from knowing the ratio of its distances from two given points 

 E and D ', it is certain, that in general thefe data would be fuf- 

 ficient for determining the fituation of F : But neverthelels, 

 if E and D mould be lb lituated, that they were in the fame 

 Straight line with the centre of the given circle ; and if the 

 rectangle under their diftances from that centre, were alfo e- 

 qual to the fquare of the radius of the circle, then, as was 

 {hewn above, (§ 12.) the pofition of F could not be deter- 

 mined. 



This particular inftance may not indeed occur in any of the 

 practical applications of geometry ; but there is one of the fame 

 kind which has actually occurred in aftronomy : And as the 

 hiftory of it is not a little lingular, affording befides an 

 excellent illuftration of the nature of Porifms, I hope to be ex- 

 cufed for entering into the following detail concerning it. 



32. Sir Isaac Newton having demonstrated, that the tra- 

 jectory of a comet is a parabola, reduced the actual determina- 

 tion 



