■*■«■ 



APPENDIX, CojJtainmg the Inveftigation of a For- 

 mula for the ReEiificatmt of any Arch of an Ullipfe, 



I. TT is now generally underftood, that by the redlification of 

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

 ftraight line exadlly equal to it, but alfo the method of expreC- 

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

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

 is evidently in the latter fenfethat we muft underftandthe term 

 reBification, when applied to the arches of conic fedlions, feeing 

 that it has hitherto been found impoffible, either to exhibit 

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

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

 ber of terms. 



With refpedl to the recflification of the circle and parabola, 

 there feems to remain but little farther to be defired. 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 formula:, involving cir- 

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

 plicity, as if they involved only algebraic funftions of ftraight 

 lines ; and as to the latter curve, it is well knovi^n that the for- 

 mula exprcfling 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 affign the numerical value of any portion of the curve. 



Mm2 2. The 



