f^ , PHILOSOVHICAL TRAKSAC'rtONS. [aNNO ] 695-6. 



circles intersect each other, are in all cases equal to the spherical angles they 

 represent ; which is perhaps as valuable a property of this projection as that of all 

 the circles of the sphere on it appearing circles ; but this, not being commonly 

 known, must not be assumed without a demonstration. 



Let EPBL be any great circle of the sphere, (fig. 11, pi. 1 ; ) e the eye placed 

 in its circumference, c its centre, p any point in it, and let fco be supposed a 

 plane erected at right angles to the circle epbl, on which fco we design the 

 sphere to be projected. Draw ep crossing the plane fco in p, and p shall be 

 the point p projected. To the point p draw the tangent apg, and on any point 

 of it, as a, erect a perpendicular ad, at right angles to the plane epbl, and draw 

 the lines pd, ac, dc ; then the angle apd will be equal to the spherical angle 

 contained between the planes apc, dpc. Draw also ae, de, intersecting the 

 plane fco in the points a and d ; and join ad,pd: I say the triangle adp is simi- 

 lar to the triangle adp, and the an^eapd equal to the angle apd. Draw pl, 

 ak parallel to FO ; then by reason of the parallels , ap will be to ad as \k to ad ; 

 but, by Eucl. 3, 32, in the triangle akp, the angle akp = lpe is also equal to 

 APK = EPG, therefore the sides ak, ap are equal, and it will be, as ap to ad 

 so ap to AD. Whence the angles dap, dap being right, the angle apd will be 

 equal to the angle afd, that is, the spherical angle is equal to that on the pro- 

 jection, and that in all cases. Which was to be proved. 



This lemma I lately received from Mr. Ab. de Moivre, though I since under- 

 stand from Dr. Hook that he long ago produced the same thing before the 

 society. However the demonstration and the rest of the discourse is my own. 



Lemma III. — On the globe, the rhumb lines make equal angles with every 

 meridian, and by the foregoing lemma, they must likewise make equal angles 

 with the meridians in the stereographic projection on the plane of the equator : 

 they are therefore, in that projection, proportional spirals about the pole point. 

 Lemma IF. — In the proportional spiral, it is a known property that the angles 

 BPC, or the arches bd, are exponents of the rationes of bp to pc, (fig. 12, pl. ],) 

 for if the arch bd be divided into innumerable equal parts, right lines drawn 

 from them to the centre p, will divide the curve bccc into an infinity of propor- 

 tionals ; and all the lines vc will be an infinity of proportionals between pb and 

 PC, whose number is equal to all the points d, d, in the arch bd ; whence, and 

 by what I have delivered in N°2l6, it follows, that as bd to Bd, or as the 

 angle bpc to the angle bpc, so is the logarithm of the ratio of pb to PC, to the 

 logarithm of the ratio of pb to pc. 



From these lemmata our proposition is very clearly demonstrated ; for, by the 

 first, PB, PC, PC are the tangents of half the eonipiements of the latitudes in 

 the stereographic projection ; and, by the last of them, the differences of Ion- 



