APPENDIX, containing the Inveftigation of a For- 

 mula for the Reclifcation of any Arch of an Ullipfe. 



i. ]f T is now generally underftood, that by the rectification of 

 Jl a curve line, is meant, not only the method of finding a 

 ftraight line exactly equal to it, but alfo the method of expret- 

 fing it by certain functions of the other lines, whether ftraight 

 lines or circles, by which the nature of the curve is defined. It 

 is evidently in the latter fenfe that we muft underftand the term 

 reElificatioti, when applied to the arches of conic fections, feeing 

 that it has hitherto been found impomble, either to exhibit 

 ftraight lines equal to them, or to exprefs their relation to their 

 co-ordinates, by algebraic equations, conufting of a finite num- 

 ber of terms. 



With refpect to the rectification of the circle and parabola, 

 there feems to remain but little farther to be d^fired. The deter- 

 mination of any arch of the former of thefe curves is a problem 

 which fo often occurs, and its folution is rendered fo eafy, by 

 the aid of trigonometrical tables, that formulas, involving cir- 

 cular arches, are confldered as nearly of the fame degree of fim- 

 plicity, as if they involved only algebraic functions of ftraight 

 lines ; and as to the latter curve, it is well known that the for- 

 mula exprefling its length is compofed of two parts, the one an 

 algebraic, and the other a logarithmic function of the co-or- 

 dinates, fo that, by means of a table of logarithms, we can 

 quickly afhgn the numerical value of any portion of the curve. 



Mm2 2. The 



