JSiifcettanea Curio fa. 239 



-t-ACxBH=) ABxBH-|-ACxBD+ ACxDH* 

 Wherefore FRx AC— BDx AC (that is, BRx 

 AC) = ABxBH+ACxDH. But (by Propo- 

 und* IV.) ACx DH=Space AGF. Therefore 

 BRx AC— (ABHL-f AGF= by Cor. I. Propo- 

 fition V.) 2BAH. 



PROP. VII. THEOREM, 



(Fig. 37.) If in the Logarithmical Curve 

 LAG (xvhofe Subtangent HS, given, ts 

 equal to the Line a, determined, as at 

 Cor. II. Prop. II.) be taken the pint 

 A, xvhofe diftance AC from the AJymp- 

 tote HP, is equal to the Subtangent 

 HS ; and from the points H, and P 

 (taken at Liberty in the Affymptote, 

 and equally dijl ant from the point C) 

 he eretted the Lines HL, PG, Ordi- 

 nate* to the Logarithmical Curve, the 

 half Sum of which is equal to HD or 

 PF : Then the points D and F, jhall be 

 pofited in the Curve of the Catenaria ? 

 correfponding to the right Line AC. 



L et AB be put — x, and confequently CB 

 or QH the half Sum of the Ordinates HL, 

 PG, will = a-\-x. Let the half difference of 

 them be put ~y\ whence HL-^-i-j,. 

 and PG =2 a~\-x—y. And lince from the Na- 

 ture of the Logarithmical Curve, CA is a 

 mean Proportional between them, aa fhall = 



