[ 5° ] 



Properties of the Equilateral Hyperbola. 



VIII. Theorem. Let a, a, a reprefent abfcifTas meafured 

 from the centre on the axis of an equilateral hyperbola, and 

 0, o\ 6' correfponding ordi nates : let alfo the hyperbolic area 

 contained by the femi axis (- unity), diflance from the centre 

 to the extremity of the ar % and the arc, the abfciffa of which is a 

 and ordinate o", be equal to the fum of the areas contained in the 

 fame manner 'by the femi axis, dift-. and arcs the abfciffas and 

 ordinites of which are a, a' and o, o' : then will a -^^ a a -\- o d 

 and = a d -\- a' 0. 



Dem. Let the area ACV (fee fig.) = EC V + BC V, let the 

 double ordinates F E ^, /^ G B, ^ H A be produced to meet the 

 affymptote C w' /y N Y X »« « W/, and let fall the perp'. aw\ bx\ ey\ 

 VN, EY, BX, AW. Becaufe ACV = EVC + BCV and 

 becaufe (by prop, hyperb.) CVN = ECY = BCX = ACW 

 •.•VNEY + VNBX = VNAWorVNEY = BAWX:andit 

 has been proved by many writers on conies that when thefe 

 areas are equal 



CN: CY:: CX: CW 

 orVN: EY:: BX: AW 

 Whence it follows that 



CV: Era:.- B«: hp 

 or I : a — o : : a —a . a — o 



in 



