( ^4) 
And this IS the Refult of the DG<5lrine of Mercatcr^ as im- 
proved by the Learned Dr. Wattis. But by the fecond Theo- 
rem, 'viz,. — 4- — i + — ? the fame Logarithms are ob- 
tained by fewer Iteps. To wit, 
4r*^73m"^ 922640625' '"^^261^8617187^^' ^ 
3_ , 2 2 J 2 
47 3 1 1469 "I" 114672503^"^^ 3546561843241 
which was invented and demonftraced in the Hyperbolick 
Spaces Analogous to the Logarithms, by the Excelknt Mr. 
James Gregory ^ in his Exercitationes Geometrica^ and fince fur- 
ther profecuted by the aforefaid Mr. Speidall^ in a late Trejtife 
in Engli[h by him. publillied on this Subjed. But the Demon- 
ilration as I conceive was never till now perfeded without 
the confideration of the Hyperbola, which in a matter purely 
Arithmetical as this is, cannot (b properly be applyed. But 
what follows I think I may more jalily claim as my owiiyviz,. 
That the Logarithm of the ratio of the Geometrical Mean to 
the Arithmetical between 22 and 24, or of '✓528 to 23 will 
be found to be either 
I ■ I . I , I 
10^8 + 1119364+888215:334"'^ 626487882248^^-®'* 
105-7 3 5'4279^5'79 + 6y96765'y 8485285- 
' All thefe Series being to be mukiplyed into 0,4342944819 &c. 
if you defign to make the Logarithm of Briggs, But with 
great Advantage in refped: of the Work, the faid 43-42944819 
ice. is divided by 105-7, and the Quotient thereof again di- 
vided by three times the Square of 1057, and that Quotient 
again by | of that Square, and that Quotient by | thereof, 
and fo forth, till you have as many Figures of your Logarithm 
as you defire. As for Example, The Logarithm of the Geo- 
nietrical Mean between 22 and 24 is found by the Logarithms 
of 2, 3 and II to be 
1.36131696126690612945009172669805' 
105:7)43429 &:c,( 41087462810146814347315886368 
^ in 1117249)41087 &c.( i225'852i5'44i8i82946oo74 
f in 1117249)12258 &c.( ^5'83235i84376i75 
I in 1117249)65832 &C.C ^ 4208829765 
I in 1117249)42088 6cc.( 2 9^0 
Summa 1.36172783601759287886777711225117^ 
