LOGARITHMS. 
of figures; but, by reafon of the cipher which ftands in 
the fecond place, it is eafy to fee that it muft contain 
only feven ; which therefore gives feven for the logarithm 
of foqr. The logarithm of 16 is then exprefi'ed by the 
number of places of figures in the product of the 20th 
power of 2 into itfeif; and is therefore denominated by 13. 
That of 256 is denoted by the 80th power of 2, contain¬ 
ing 25 places of figures. The logarithm of 2, therefore, 
having been already exprefi'ed by the 10th power of 2, 
will be again exprefi'ed by the 100th power. Adding, 
therefore, the number of places contained in the 80th 
poyver, viz. 25, to 7, the number of places contained in 
the 20th, we have 32 for the next exprefiion of that loga¬ 
rithm. On account of the cipher which ftands in the fe- 
cond place of one of the factors, however, we mult deduct 
one from the number; and thus we have 31 for the loga¬ 
rithm of 2, which is a confiderable approximation. Pro¬ 
ceeding in this manner, at the 1000th power of 2, we 
have 302 for the logarithm of 2; at the 10,oooth 
power we have 3011 ; at the 100,000th power, 300103; 
at the 1,000,oooth, we have 301030 ; and at the 
10,000,000th power, we obtain 3010300; 'which is as ex- 
a£t as is commonly exprefl'ed in the tables>of logarithms ; 
but, by proceeding in the fame manner, we may have it 
to any degree of exaftnefs we pleafe. Thus, at the 
100,000,000th power, we have 30103000; and at the 
1,000,000,000th, the logarithm is 301029996, true to eight 
places of figures. 
Mr. James, Gregory, in his Vera Circuli Hyperbola: Qua- 
dratura, printed at Padua in 1667, having approximated to 
the hyperbolic afymptotic fpaces by means of a feries of 
infcribed and circumfcribed polygons, from thence fliows 
hoWto compute the logarithms, which are analogous to 
the areas of thole fpaces; and thus the quadrature of the 
hyperbolic fpaces became the lame thing as the computa¬ 
tion of the logarithms. He here alfo lays down various 
methods to abridge the computation, with the afliftance 
of Come properties of numbers themfelves, by which the 
logarithms of all prime numbers under 1000 may be com¬ 
puted, each by one multiplication, two divifions, and the 
extraction of the fquare root. And the fame fubjeCt is 
farther purfued in his Exercitationes Geometricae. In 
this latter place, he firlt finds an algebraic exprefiion, in 
an infinite feries, for the logarithm of —-j— -j an ^ t * ien 
thelike for the logarithm of—-—; and, as the one feries 
1 — a 
has all its terms pofitive, while thofe of the other are al¬ 
ternately politive and negative, by adding the two toge¬ 
ther, every fecond term is cancelled, and the double o. 
the other terms gives the logarithm ol the product of 
691 
1 4" a 
and 
or the logarithm of—-—■> that is of the 
ratio of 1 —a to 1 +« : thus, he fiat's, 
firlt, a — la 2 -f -§a 3 — %a 4 See. =iog. of -- --- ; 
and, a -j- |a 2 -}- la 4 Sec. = log. of — 
therefore, %a%a 3 + ^a s -J- § -a v Sec. = log. of 
1 -j- a . 
which may be accounted Mr. James Gregory’s method of 
making logarithms. 
In 1668, Nicholas Mercator published his “ Logarith- 
motechnia, live Methodus conftruendi Logarithmos, no¬ 
va, accurata, & facilis;” in which he delivers a new and 
ingenious method for computing the logarithms upon 
principles purely arithmetical; and here, in his modes of 
thinking and exprefiion, he clofely follows the celebrated 
Kepler in his writings on the fame fubjeCt; accounting 
logarithms as the meajures of ratios, or as the number of 
ratiunculse contained in the ratio which any number bears 
to unity. Purely from thefe principles, then, the num¬ 
ber of the equal ratiunculse contained in fome one ratio, 
as of io to i, being fuppofed given, our author fhows 
how the logarithm, or meafure, of any other ratio may 
be found. . But this, however, only by the bve, as not be¬ 
ing the principal method he intends to teach, as his lad 
and belt. Having fhown, then, that thefe logarithms, or 
numbers of fmall ratios, or meafures of ratios, may be all 
properly reprefented by numbers, and that of 1, or the 
latio 01 equality, the logarithm or meafure being always 
o, the logarithm of 10, or the meafure of the ratio of Jo 
to 1, is moft conveniently reprefented by 1 with any num- 
bei of ciphers; lie then proceeds to fiiows how the mea- 
fures of all other ratios may be found from this laft fup- 
pofition; and he explains thefe principles by fome exam¬ 
ples in numbers. 
In the latter part of the work, Mercator treats of his 
otnei method, given by an infinite feries of algebraic terms, 
which are colle&ed in numbers by common addition onlv’ 
He here fquares the hyperbola, and finally finds that the 
hyperbolic logarithm of 1 -f a is equal to the infinite fe¬ 
ries a—\a~-\-la 3 — \a 4 , See. which may be confidered as 
Mercator’s quadrature of the hyperbola, or his general ex¬ 
prefiion of an hyperbolic logarithm in an infinite feries. 
This method vyas farther improved by Dr. Wallis, in 
the Phil. Tianl. for the year 1668. The celebrated New¬ 
ton invented alio the fame feries for the quadrature of the 
hyperbola, and the conftruriion of logarithms; and that 
before the fame were given by Gregory and Mercator, 
though unknown to one another, as appears by his letter 
to Mr. Oldenburg, dated October 24, 1676. The expla¬ 
nation and conftruction of the logarithms are alfo farther 
purfued in his Fluxions,publifhed in 1736 by Mr. Colfon. 
Di. Ilalley , in the Phil. Eranl. for the year 1695, gave a 
very ingenious efl'ay on the conltruriion of logarithms, in- 
titled, “A moft compendious and facile Method for Con- 
ftrudmg the Logarithms, and exemplified and demonftrated 
fiom the Nature ot Numbers, without any regard to the 
hyperbola, with a fpeedy method for finding the Number 
from the given Logarithm.” Inftead of the more ordinary 
definition of logarithms, viz. numeroriun proporticnalium aqui- 
diferentcs comtes, the learned author adopts this other nu- 
men rationum exponentes, as better adapted to the principle 
on which logarithms are here conftruried, confidering 
them as the number of ratiunculae contained in the given 
ratios whole logarithms arein queftion. Tn this way he 
ni ft arnves at the logarithmic feries before given by New- 
ton and others ; and afterwards, by various combinations- 
and lections of the ratios, lie derives others, converging 
lliil fafter than the former. Thus he found the logarithms 
of feveral iatios, as below, viz. when multiplied by the 
modulus peculiar to the fcaJe of logarithms : 
7 \f + '3? 3 — If See. the log. of 1 to 1 -f. q, 
7 + if + if + if Sec. the log. of 1 to 1 — 
X X ^ X ^ X ^ 
+ the log- of « to b, or 
X X^ X^ 
T + 7 F + ^3 + Jff &c * the farae lo g' of « to b , or 
b 
2X 
2X 
+ 73 + 771 + — Sec. the fame log. of a to b. 
rS 
2 x 6 
+ 4^ + 6l^ + 8l« &C * the l0S * of Vab t0 * z ’ 0v 
y J 
+ ^4 &c - the fame lo S- of fab to | * ; 
where a, b, q, are any quantities, and the values of x, y, z, 
are thus, viz. xzzb — a, z zz b a, y — ab -f- \z 2 . 
Dr. Halley alfo, firft of any, performed the reverfe of 
the problem, by afiigning the number to a given loga¬ 
rithm ; viz. 
— 2= 1 4- / -j- \l 2 -J-/ ! -J- -- l 4 Sc c. or 
o. 2.3 2-. 3 *4- 
Z-czci—l+lL 2 - 1 -/ 3 -}-—L—^c. 
b a. 3. a.. 3 A- 
. wlier^ 
