( "4 )

i, z6lZl6y6n669v6n.y45OOyi72.6698O5

105-7) 434.19 Lcc. (- - - -41087461810146814347315886368

2 i» 1117x49) 41087 &c. 11x587115-44181819460074.

5-'» 1117x49) 11158 8cc. (.6583x35184376175

f in 1 117149) 6583X &C. . - - 41088x9765

fini 1 17149) 41088 &C. ( .1932

Summa 1, 36171783601759187886777711115117

Which is the Logarithm os 13 to Thirty two Places, and obtained by 5Divisions with very small Divisors, all which is much leis Work than simply-Multiplying the Series into the sitid Multiplicator o, 43419 &c.

From the Logarithm given to find what Ratio it expresses, is a Probiern thathas not been so much considered as the former, but which is solved with thelike caso, and ,demonstrated by a like Proceß, front the sinne general Theoremof Mr. Newton. For as the Logarithm of the Ratio of 1 to 1 - 4 - q was prov-

edtobe 1 -+ q\ m — 1, and that of the Ratio of 1 to 1 — q to be 1 —

1.— q\ m - so the Logarithm, which we will from heneeforth call L, bang

I

given, I -+- L will be equal to t ■+ 5I”* in the one Laie; and 1 — L will1,

be equal to x — j| m in the otber : Consoquently 1 - 4 - L|’ n will be equal toj - 4 - q, and 1 — L| ” to 1 — 5 ; that is, according to Mr. Newtons seid Rule,I -4-77 jL-+ 2 L 2 - 4 - J m 3 IJ - 4 - ^w 4 L 4 + L*,&c, will

be = to 1 -+■ <7, and 1.— m L ~h — m z L z — —;»! L ’ -4 - 4 L 4 —

~ m f L *, &c. will be equal to 1 — 5, m beiug any infinite Index what-

foever; which is a füll and general Proposition from the Logarithm given tofind the Number, be the Species of Lqgarithms what it will. But if Na-pierh Logarithm be given, the Multiplication by m is siived, (which Multipli-pation is indeed no pther than the reaucing the other Species to bis) and the

Series will be- more simple, -1-/5. i 4 L + - L 2 - 4 - - 4 - — L>-

- • • • * • *•" ‘ ‘ * 1 2 6 24

-4- — L 5 , 8cc. or 1 — L - 4 - L 2 — — L3 -i- L4 — — L f,&c.

120 2 6 24 I2Q

This Series, especially in great Numbers, Converges so flowly, that it, wer«tobe wilhed it could be coritracted.

If one term of the Ratio, whereof L is the Logarithm, be given, the o-ther term will be had easily by the fame Rule,- Foris L were Napier s Loga-rithm--