392 
LOGARITHMS. 
where / is the logarithm of the ratio of a the lefs to b 
the greater of any two terms. 
We mult very briefly mention a few others who have 
treated of this fubjeft.—Mr Abraham Sharp, of York- 
fhire, made many calculations and improvements in loga¬ 
rithms, Sec. The molt remarkable of thefe were, his qua¬ 
drature of the circle to 72 places of figures, and his com¬ 
putation of logarithms to 61 figures, viz. for all numbers 
to 100, and for all prime numbers to 1100. 
The celebrated Mr. Roger Cotes gave to the world a 
learned traift on the nature ’"and conftrudtion of loga¬ 
rithms-. this was fir'd printed in the Phil. T'ranf. N° 338. 
and afterwards with his Harmonia Menfurarum in 1722, 
under the title Logometria. This traift has jultly been 
complained of, as very obfeure and intricate; and the 
principle is fomething between that of Kepler and the 
method of fluxions. He invented the terms modulus and 
modular ratio ; this being the ratio 
_ II 3 3 3 , „ 
of i 4 - 1 -i-1-!-— Sec. to 1, or 
1 2 2.3 2.3-4. a- 3 - 4-5 
x 
—+ - 
1 
&c. 
of I to I 
2.3 2.3.4 2.3.4-5 
that is, the ratio of 2-718281828459 Sec. to 1, 
or the ratio of 1 to 0-367879441171 &c. 
the modulus of any fyftem being the meafure or logarithm 
of that ratio, which in the hyp. logarithms is 1, and in 
Briggs’s or the common logarithms is 0-434294481903 &c. 
The learned Dr. Brook Taylor gave another method of 
computing logarithms in the Phil. Tranf. N° 352. which 
is founded on thefe three principles, viz. iff, That the 
i’um of the logarithms of any two numbers is the loga¬ 
rithm of the produdt of thofe numbers. 2d, That the lo¬ 
garithm of 1 is o ; and confequently, that, the nearer any 
number is to 1, the nearer will its logarithm be to o. 3d, 
That the produdf of two numbers or fadtors, of which the 
one is greater and the other lefs than 1, is nearer to 1 than 
that fadfor is which is on the lame fide of 1 with itfelf; 
fo of the two numbers § and ^, the produdf § is lefs than 
1, but yet nearer to it than is, which is alfo lefs than 1. 
And on thefe principles he founds an ingenious, though 
not very obvious, approximation to the logarithms of 
given numbers. 
Conjlru&ion of Logarithms by Fluxions. 
From the definition and defeription of logarithms given 
by Napier, and of which we have already taken notice, it 
appears that the fluxion of his, or the hyperbolic loga¬ 
rithm of any number, is a fourth proportional to that 
number, its logarithm, and unity; or, which is the 
fame, that it is equal to the fluxion of the number 
divided by the number: for the defeription Ihows, that 
zi : za or 1 :: £1 the fluxion of zi : za, which therefore 
is =2 — ; but za is alfo equal to the fluxion of the loga¬ 
rithm A, &c. by the defeription; therefore the fluxion of 
the logarithm is equal to — > the fluxion of the quantity 
divided by the quantity itfelf. The fame thing appears 
again at art. 2. of that little piece in the appendix to his 
CnjlruBio Logarithmonm, intirled “ Habitudines Logarith- 
morum Se fuorum Naturalium Numerorum invicem;” 
where he obferves, that, as any greater quantity is to a 
lefs, fo is the velocity of the increment or decrement of 
the logarithms at the place of the lefs quantity to that at 
the greater. Now this velocity of the increment or de¬ 
crement of the logarithms being the fame thing as their 
fluxions, that proportion is this; x : a :: flux. log. a : 
flux. log. x. Hence, if a be == 1, as at the beginning of 
the table of numbers, where the fluxion of the logs, is the 
index or charadferiftic c, which is.alfo one in Napier’s or 
She hyperbolic logarithms, and 43429, Si c. in Briggs’s, 
the fame proportion becomes x : 1 c : flux. log. at; but 
-She conftant fluxion of the numbers is alfo i, and there¬ 
fore that proportion is alfo this; x : fc :: c : — = the fluxion 
X 
of the logarithm of x ; and in the hyperbolic logarithms, 
x • 
where c is = 1, it becomes - = the fluxion of Napier’s or 
the hyperbolic logarithm of *. This fame property has 
alfo been noticed by many other authors fince Napier’s 
time. And the fame or a flmilar property is evidently 
true in all the fyftems of logarithms whatever, namely, 
“ that the modulus cf the fyftem is to any number as the 
fluxion of its logarithm is to the fluxion of the number.” 
Now from this property, by means of the doftrine of 
fluxions, are derived other ways for making logarithms, 
which have been illuftrated by many writers on this 
branch; as Craig, Jo. Bernoulli, and alrnoft all the writers 
on fluxions. And this method chiefly confifts in ex¬ 
panding the reciprocal of the given quantity in an infinite 
feries, then multiplying each term by the fluxion of the 
faid quantity, and laftly taking the fluents of the terms; 
by which there arifes an infinite feries of terms for the 
logarithm fought. So, to find the logarithm of any num¬ 
ber N, put any compound quantity for N, as fuppofe 
——5 then the flux, of the log, or — being — _ _ 
f 6 N 
XX X x^x 
— Tr &c ’ the fluents give log. of N, or log. of 
n + x 
n 
n 
n H 
x 2 
272 ^ 
3 « d 
472 
-» & c - And writing —* for 
x n—x x 
x gives log.___ 
n n 
n n + x 
-zz: i-f- 
n±x n 
n 
becaufe 
, or log. 
X 3 
372 3 
71 
See. Alfo, 
, 71 Ztx 
;= o—log- ; we 
71 
have log.- 
n -f- a: 
71 X 
— -J- — -f- 
n—x n 
x x z 
71 272 2 
X 3 X 4 
-J- 77-j-) See. anil log. 
3 n 
4 n H 
-7 -&c. 
3« 3 4?* 4 
zn 2 1 3 n 3 
And by adding and fubtra(fling any of thefe feries to or 
from one another, and multiplying or dividing their cor- 
refponding numbers, various otlier feries for logarithms 
may be found, converging much quicker than thefe do. 
In like manner, by afluming quantities otherwife coml 
pounded for the value of N, various other forms of loo-a- 
rithmic feries may be found by the fame means. 
Ex. Given a Logarithm, to find the Number.- 
-Let 1 H-jv be 
any number, and y its logarithm ; then j = 
m be¬ 
ing the modulus ; hence, y xy — mx—o. AH"ume*=: 
ay -J- by 2 -J- cy 3 -j- Sec. then x — ay zbyy -j- ^cy 2 y -f See. 
Subititute thefe values of x and x into j -{■ xy — mx — o, 
and we have, 
y -J- ayy -J- by 2 y -f- &c. \ _ 
— may —zmbyy ■ —3 mcy 2 y — Sec.J °> hence, 
1 — ma = o, a — 2 mb — o, b — 3 me = o, Sec. Therefore, 
_ 1 1 __ a 3 b i 
a — ; b — — — ; c — — --— ; Sec. hence. 
- y _1 Z_ 
A m 2 in 2 
+ 
2 m 
y 3 
.3 m 
3 »i 2-3»2 3 
— -J- Sec. Therefore, i-f-x—i-f- 
}" 
y y 
-"1-7 4- &c. the number whofe logarithm 
m 2 m 2 2.3 m 3 1 0 
is 
y 2 * y 3 
If m — 1, then x 4 -x = x 4 -y 4 -[- -- \- Sec. is the 
2 2.3 
number whofe hyp. log is y. 
To find the Modular Ratio -.—Every logarithm is mca- 
fure of the ratio of its correfponding number to 2 ; iv nee, 
y y 2 y 3. 
*~ + 
Sec. 
y is the meafure of the ratio of i -f- 
v 2 
-f -— 
m 2 m 2 J a.3 
