Mifcettanea Curiofa. a 3 



ing Circles: But this not being vulgarly known, 

 muft not be afliimed without a Demonftr avion. 



Let £jBP£ in Fig. u Tab. 2. be any great 

 circle of the Sphere, E the Eye placed in its 

 Circumference, C its Center, P any point 

 thereof, and let FC O be fuppofed a plain 

 erefted at right Angles to the ."Circle E BPL, 

 on which F CO we defign the Sphere to be 

 projeded. Draw E P crofling the Plain F 

 CO in p, and p fhail be the Point P projeded. 

 To the point P draw the Tangent A P G 

 and on any point thereof, as A, ereft a per- 

 pendicular AD, at right angles to the plane 

 EBP L, and draw the lines PD, AC, DC: 

 and the AngleAPD fhall be equal to the Sphe- 

 rical Angle contained between the plains A P 

 C, DPC. Draw alfo AE, D E, interfering 

 the plain FCO in the points a and A% and 

 joyn ad, p d: I fay the Triangle a d p is fimu- 

 lar to the triangle AD P. And the Angle apd 

 equal to the Augle APD. Draw PL, A K, pa- 

 rallel to FO, and by reafon of the parallels, a 

 p will be to a d as AK to At) : But f by Evcl. 

 3. 32O in the triangle AKP, the angle AKP bt 

 LPE is alfb equal to APK^ EPG, wherefore 

 the fides AK, AP, are equal, and 'twill be as 

 a p to ad fo AP to AD. Whence the angles 

 DAP, dap being right, the angle APD will 

 be equal to the angle apd; that is, the Sphe- 

 rical Angle is equal to that on the Projection, 

 and that in all Cafes. Which was to be proved. 



This Lemma I lately received from Mr. Ah. 

 de Molvre, though I fince underftand from Dr. 

 Hookj, that he long ago produced the fame 

 thing before the Society. However the demon- 

 ftration and the reft of the Difcourfe, is my 

 own. 



C 4 Lemma 



