-151 



The application of Fagnani's theorem arifes from the following circum- 

 flance. While the feries for the whole quadrantal arc of an excentric el- 

 lipfe, is ufelels from its flow degree of convergency, a part of the 

 quadrantal arc commencing at the extremity of the axis minor, may 

 be found by a feries fulEciently converging, and the remainder of the 

 elliptic quadrant (the feries for which would be diverging) may be 

 obtained from its relation to the firfl arc, by that theorem.* 



Therefore, as connefted with the above fubjeft, I have fubjoined a 

 geometrical demonftration of Fagnani's theorem ; which demonftration is 

 derived from a property of the ellipfe. which I believe to be new. 



Lemma. 



Let AED be a femi-ellipfe on the axis major AD, C the centre 

 and AFD the circumfcribing circle. Let alfo any ordinate GE be pro- 

 duced, to meet the circle in F. Draw FC interfering the ellipfe in 

 O and QOP parallel to FG, then if QC be drawn interfering the 

 tangent EN in N, CN will be perpendicular to the tangent EN, 

 and equal to CO. 



Demonftration, 



Draw the tangents FT and TE which interfeft in a point T of 

 CA produced. The triangles POC and FTG are fimilar, therefore 



TG 



* It may be proper to remark an error in a paflage of the " Excerpta ex eplftolis Newtoni" 

 noticed, I believe, by none of the commentators. Two feries are given for computing the 

 length of an elliptic arc, and in finding the length of the quadrant, the femiaxis is direfled to 

 be bifefled, and the arcs correfponding to the two abfciflas to be found by the two feries 

 (Page 312, art. 7, vol. i. Horfley's edit.) But the direi5lion in this palFage is, as may be rea- 

 dily (hewn, entirely impraflicable, whenever the ratio of the axis major to the axis minor 

 exceeds the fubduplicate ratio of 5 : 3. For then one] of the feries will be diverging. 

 When the excentricity is confiderable, no divifion of the femiaxis major will render the appli- 

 cation of the Newtonian feries ufeful, as the convergency of one or other will not be 

 greater than that of the common theorem for the whole circumference of an ellipfe. 



