230 Carriot on the Theory of 



rily at any diftance from the point M, draw the line RS pa- 

 rallel to MP, and, through the points R and M, draw the 

 fecant RT' ; then we fhall evidently have T'P : MP : : MZ 



M7 

 : RZ, and therefore T'P, or TP + T'T = MP %. Now, 



RZ 



if we imagine RS. moving parallel to itfclf, to approach con- 

 tinually to MP, it is evident that the point T' will, at the 

 fame time, more and more approach to the point T; and con- 

 sequently, that we may render the line T T as final! as we 

 pleafe, without affecting the proportion above eftablifhed. If, 

 then, wc neglect the quantity T' T in the equation juft found, 

 there will indeed be an error in the refulting equation TP = 



JHP___, to which the former will now be reduced: but 



this error may be dim'mifhed as much as we pleafe, by making 

 RS approach to MP as nearly as is ne.ceflary ; that is, the 

 proportion of the two members of this Iaft equation will differ 

 as little as we pleafe, from the proportion of equality. 



T ... , MZ iy + RZ , c 



In like manner we have - ■■- = ■ -, . (See 



RZ 2a — ix — MZ y 



Article 3.) And this equation is perfectly accurate, what- 

 ever be the poiiikm ot the point R ; that' is, whatever be the 

 values of MZ and RZ. But, the nearer RS approaches to 

 MP, the frhaller will be the lines MZ and RZ; and hence, 

 it' they be neglected in the fecond member of that equation, 



the error which will refult in the equation -_ - = — =- — , 



HZ a — .v 



will, as in the former cafe, be thereby rendered as fmall as 



we think proper. 



This being fo, without paying any regard to errors, which 



I have it always in my power to diminifh as much as I pleafe, 



MZ M7 

 I treat the two equations TP = MP ~-jj r y, and = 



_: — -, as if both of them were perfectly accurate ; fubfli- 

 « — x 



tilting, therefore, in the former the value of — - rj expreffed 



in the latter, I have TP = MP -2L_ ; that is, TP =-^— t 



a — .v a — x 



as before. 



This 



