(60 
1 — ^[ ^ in the other : Confequently i U"^ will be equal 
to I 4" f 5 I ■ — to I ' — q ; that is^ according to Mr. 
Newton $ faid Rule, i -[-/^L+ i m^V-]- i n.^ U + \j 
+ -A^ D &c. will be = 1 4- and i — »^ L + ^ L* 
— im^V + i^w'^L'^ — p!^??^* LJ &:c. will beequ.al to i — 
f» being any infinite Index whatfoever, which is a tiill and 
general Propofition from the Logarithn; given to find the 
Number^ be the Species of Logarithm what it will. But if 
Napeir's Logarithm be given, the Multiplication by m is faved^ 
(which Multiplication is indeed no ocher than the reducing 
the other Specks to his) and the Series will be more fimple, 
VIZ.. I + L 4- { L L + 1 L^ 4- L^ + tI ^ L» &c or I — L 
+ i L L — i L' 4- 4 L^ — V &c. This Series, efpeci- 
ally in great Numbers, converges fo flowly, that it were to 
be wifiied it could be contraded. 
If one term of the ratio, whereof L is the Logarithm, be 
given,the other term will be had eafily by theXame Rule : For 
if L were Napeirs Logarithm of the ratio of a the lefler to b 
the greater term, h would be the Produd of a into i 4" L 
4-|LL4-^LLL&c. =^ + ^L+ |tf LL4-J^L' &c. 
But if h were given.^z would be = b — b L4- k b LL-— J b L' &c. 
Whence by the help of the Cbtliads, the Number appertain- 
ing to any Logarithm vvill be exadly had to the utmoft ex- 
tent of the Tables. If you feek the neareft next Logarithm, 
whether greater or lefler, and call its Number a if lefler, or 
b if greater than the given L,and the difference thereof from 
the faid neareft Logarithm you call /; it will follow that the 
Number anfwering to the Logarithm L will be either a into 
i4./_{-i//+j///4--^/4 41_i-/^&c. orelfe b into i — / 
4-1 //—I /// 4-tJ y^—^lsM &c. wherein as / is lefs, the 
Senes will converge the fwifter. And if the firfl: 20000 Lo- 
garithms be given to fourteen places, there is rarely occafion 
for the three firft fl:eps of this Series to find the Number to as 
many places. But for U/^^^^j great Canon of 100000 Loga- 
rithms, which is made but to ten places, there is fcarce ey,er 
need for more than the firft ftep a^A^a lov a^^malm one 
cafe, or d{Q b — blovb — mblm the other, to have the 
Number true to as many Figures as thofe Logarithms cpnfift 
of. 
If 
