( ^os ) 
feiSting the plan FCO In the points a and d ; and joyn a J^ 
pd: 1 fay the Triangle adph fimular to the triangle j^DF, 
and the angle af d equal to the angle APD» Draw PL, AK 
parallel to FO, and by reafon of the parallels^ a p will be to 
ad 3LS AK to AD : But ( by Eucl. z, ;2. ) in the triangle 
AKP, the angle AKPr=LPE is alfo equal to APKrzrEPG 
wherefore the fides AK, AP are equal, and twill be, as 
ap to ad loAP to AD. Whence the angles DAP, dap 
being right, the angle APD will beequal to the angle ap d 
that is, the Spherical Angle is equal to that on the Proje(5lionj 
and that in all Cafes. Which was to he proved, 
' . This Lemma I lately received from Mr. Ak de Moivre-^ 
though I /ince underftand from Dr. Hook that he long ago 
produced the fame thing before the Society. However the 
demonftraticn and the reft of the difcourle is my own. 
Lemma III. On the Ghhe^ the Rhumb Lines make equal 
angles with every Meridian, and by the aforegoing Lemma^ 
they muft likewife make equal angles with the Meridians in 
the Stereographick TrojeBion on the plain of the Equator : They 
are therefore, in that Projedion, Proportiaml Spirals about 
the Pole Point. 
Lemma IV. In the Vreportional Spiral it is a known proper- 
ty that the angles BPC or the ^ — ^ 
arches BD, are £;cpfj«e«f/ of the LP 
rationes of ^Pto PC ; for if the T^^^\\ C 
arch BD be divided into innu- / \\ ^'""'"^^ / 
merable equal parts , right / \\ I ^^"^""^""^xm 
lines drawn from them to the / \\ / / 
Center P , ftiall divide the / \ \ / / 
Curve BccC into an infinity of / \y\ / 
proportionals^ and all the lines ~/ cv \ 
Pc fhall be an iofiA*iity of pro- / 
portionals between PB and PC Bk:::__3A^^ 
whofenumbeif is equal to all' the points d,d, in the arch 
BD : Whence, and by what I have delivered in Num.216, 
it follows, that as to Bd, or as the angle BPC to the 
angle BPc, fo is the Logarithm of the ratio of PB to PC, to 
the Logarithm of tbe ratio di PB to Pc, 
Froif! 
