numbers such terms of that series as are nearest 
to them, and their indices for the logarithms. 
But then, to construct logarithms in this man- 
ner, to such an extent of numbers, and degree 
of exactness, as would be necessary to make 
them of any considerable use, is next to impos- 
sible, because of the almost infinite labour and 
time it would require. This, however, is an in- 
troduction for understanding the method of the 
noble inventor, who undoubtedly first took the 
hint of making logarithms from the considera- 
tion of the indices of a geometrical series ; and 
by means of the principles and known proper- 
ties of these progressions he first formed his 
tables, and adapted them to the practical pur- 
poses intended. 
To find the logarithm of any of the natural numbers-, 
1, ‘2, 3, 4, Itfc. according to the method + NaPIER. — 
1. Take the geometrical series, 1, 10, 100, 1000, 
10,000, &c. and apply to it the arithmetical se- 
ries 1, 2, 3, 4, &c. as logarithms. 2. Find a geo- 
metric mean between 1 and 10, 10 and 100, or 
any other two adjacent terms of the series be- 
twixt which the number proposed lies. 3. Be- 
tween the mean, thus found, and the nearest 
extreme, find another geometrical mean, in the 
same manner ; and so on, till you are arrived 
within the proposed limit of the number whose 
logarithm is sought. 4. Find as many arithme- 
tical means, in the same order as you found the 
geometrical ones, and the last of these will be 
the logarithm answering to the number re- 
quired. 
Examples. I.et it be required to find the loga- 
rithm of 9. 
Mere the numbers between which 9 lies are 
1 and 10. 
First, then, the log. of 10 is 1, and the log. of 
1+0 
1 is 0 ; therefore ■ — h — — .5 is the arithmetical 
mean, and \/(l X 10) = \/10 = 3.1622777 — 
geometric mean : whence the logarithm of 
3.1622777 is .5. 
Secondly, the log. of 10 is 1, and the log. of 
1 J_ # 5 
3.1622777 is .5 ; therefore — — .75 = arith- 
metical mean, and + (10 X 3.1622777) = 
.5.6234132 = geometric mean : whence the log. 
of 5.6234132 is .75. 
Thirdly, the log. of 10 is 1, and the log. of 
1 -L .75 
5.6234132 is .75; therefore — .875 = 
arithmetical mean, and ^/(10. x 5.6234132) = 
7.4989421 — geometric mean : whence the log. 
of 7.4989421 is .875. 
fourthly, the log. of 10 is 1, and the log. of 
- , , 1 + - 8 75 
7.4989421 is .875; therefore — — .9375 
— arithmetical mean, and (10 X 7.4989421) 
— 8.6596431 =r geometric mean : whence the 
log. of 8.6596431 is .9375. 
Fithlv, the log. of 10 is 1, and the log. of 
1 -4- .9375 
8.6596431 is .9375; therefore — 
.'96875 = arithmetical mean, and (10 X 
8.6596431) = 9.3057204 — geometric mean: 
whence the log. of 9.3057204 is .96875. 
Sixthly, the log. of 8.6596431 is .9375, and 
the log. of a 3057204 is .96875 ; therefore 
.9375 + . 90*75 , 
1 — .953125 anth. mean, and 
2 . 
jf (S.6596431 X 9.3057204) = 8.9768713 = 
geometric mean : whence the log. of 8.9768713 
is .95312.5. 
And, proceeding in this manner, after 25 ex- 
tractions, the logarithm of 8.9999998 will be 
found to be .9542425 ; which may be taken for 
the logarithm of 9, because it differs from it 
only by and is therefore sufficiently 
exact for all practical purposes. 
VOL. II. 
LOGARITHMS. 
f i 
And in the same manner the logarithms of 
almost all the prime numbers were found ; a 
w irk so incredibly laborious, that the unre- 
mitted industry of several years was scarcely 
sufficient for its accomplishment. 
To determine the hyperbolic logarithm (I.) of any 
given number (N). The hyperbolic logarithm of 
any number Is the index of that term of the lo- 
garithmic progression, which agrees with the 
proposed number multiplied by the excess of 
the common ratio above unity. 
Let, therefore, (1 -\- >)" be that term of the 
logarithmical progression, 1, (1 + a)’, (1 + ,r) 2 , 
(1 -|- x ) ’, . ( 1 -j- a)*, & c. which is equal to the 
required number (N). 
Then will (1 -}- x) n — N, and 1 + .v =N"; 
and if 1 +y be put = N, and m = — , we 
• 71 
T 
shall have 1 -f- x = N n — (1 +y) m — 1 -{- my 
the next thing to be done is, to shew how the 
logarithms of fractional quantities may be found. 
And, in order to this, it may be observed, that 
as we have hitherto supposed a geometric series 
to increase from an unit on the right hand, so 
we may now suppose it to decrease from an unit 
towards the left ; and the indices, in this case, 
being made negative, will still exhibit the loga- 
rithms of the terms to which they belong. 
Thus Log. — 3 — 2 — 1 0 +1+3 +3, &c. 
Nam ? < 1 10 100 1000, &c 
lOOOlOOlO 
Whence + 1 is the logarithm of 10, and — 1, 
the logarithm of ; + 2 the logarithm of 100, 
and — 2 the logarithm of &c. 
And from hence it appears that all numbers, 
consisting of the same figures, whether they be 
integral, fractional, or mixed, w r iil have the de- 
cimal parts of their logarithms the same. 
Thus, the logarithm of 5874 being 3.7689339, 
the logarithm of +, T +, TtToo’ &c. part of 
it will be as follows : 
. m — 1 m — 1 m — 2 
+ x —— y + m X ~Y~ X — — y -,&c. 
And, consequently, x — my + m X — 'y 1 
m — 2 m — 2 
+ m X — — — X — y ■> See. where m being 
rejected in the factors m — 1, m — 2, m — 3, 
&e. being indefinitely small in comparison of 
1, 2, 3, &c. the equation will become a- = my 
my 1 my 5 my 1 . 
2 3 T’ C ' 
Hence — - (nx — I.) — y — — f- ^ 
m K J 2.3 4 
+ & c . — hyperbolic logarithm of N, as 
was required. 
The hyperbolic logarithm (L) of a number being 
given, to find the number (N) itself, which answers to 
it. Let (1 + x)" be that term of the loga- 
rithmic progression, 1, (1 + a) 1 , (1 + x) 2 , 
0+*)\ (1+*) 4 . & c. which is equal to the 
required number N. 
Then, because (1 + x)" is universally = 1 + 
n — 1 „ , n — 1 n — 2 
+ ti X 
+ n X “ 
2 1 ' 2 
& c. we shall have 1 + nx + n X 
+ 
n — 1 « — 2 
n X — — X — — x\ &C. = N. 
But since n, from the nature of the logarithms, 
is here supposed indefinitely great, it is evident 
that the numbers connected to it by the sign — 
may all be rejected, as far as any assigned num- 
ber of terms. 
For* as 1,2, 3, Sec. are indefinitely small in 
comparison to n, the rejecting of those numbers 
can very little affect the values to which they 
belong. 
If, therefore, 1, 2, 3, &c. be thrown out of 
n — 1 n — 2 n — 3 
the factors — — , — - — , — - — > &c. we shall 
ll 2 X 2 !pX 2 « 4 A - 4 
have l + ++- + ++ + 2.+4’ &C - 
= N. 
But nx (= L) is the hyperbolic logarithm of 
(1 + x) u , or N, by what has been before spe- 
L 3 
cified ; and therefore 1 + L + f- + 
IT 
2X1.4 
, &c. “ N number required. 
N um 
I.ogar 
thms. 
5 8 7 
4 | 
3.7 
6 
8 
9 
3 
3 
9 
5 8 7 
4 ! 
2.7 
6 
8 
9 
3 
3 
9 
5 8.7 
4 1 
1.7 
6 
8 
9 
3 
3 
9 
5.8 7 
4 1 
0 7 
6 
S 
9 
3 
3 
9 
.5 8 7 
4 1 
— 1.7 
6 
8 
9 
3 
3 
9 
.0587 
4 
— 2.7 
6 
8 
9 
3 
3 
9 
005 8 7 
4 
— 3.7 
6 
8 
9 
3 
3 
9 
From this it also appears, that the index, or 
characteristic, of any logarithm, is always one 
less than the number of figures which the na- 
tural number consists of : and this index is con- 
stantly to be placed on the left hand of the de- 
cimal part of the logarithm. 
When there are integers in the given number, 
tire index is always affirmative; but when there 
are no integers, the index is negative, and is to 
be marked by a line drawn before it, like a ne- 
gative quantity in algebra. 
Thus, a number having I, 2, 3, 4, 5, &c. in- 
teger places, the index of its log. is 0, 1, 2, 3, 4, 
Sec. And a fraction having a digit in the place 
of primes, seconds, thirds, fourths, See. the index 
of its logarithm will be — 1, — 2, — 3, — 4, Sec. 
It may also be observed, that though the in- 
dices of fractional quantities are negative, yet 
the decimal parts of their logarithms are ahvavs 
affirmative ; and all operations are to be per- 
formed by them in the same manner as by ne- 
gative and affirmative quantities in algebra. 
In taking out of a table the logarithm of any 
number not exceeding 10000, we have the de- 
cimal part by inspection ; and if to this the pro- 
per characteristic be affixed, it will give the 
complete logarithm required. 
But if the number, whose logarithm is re- 
quired, be above 10000, then find the logarithm 
of the two nearest numbers to it that can be 
found in the table, and say, as their difference * 
the difference of their logarithms * * the differ- 
ence of the nearest number and that whose lo- 
garithm is required * the difference of their 
logarithms, nearly; and this difference being 
added to, or subtracted from, the nearest loga- 
rithm, according as it is greater or less than the 
required one, will give the logarithm required, 
nearly. 
Thus, let it be required to find the logarithm 
of 367182. 
The decimal part of 3671 is by the table 
.5647844 ; and of 3672 is .5649027 ; 
/. The C 367100 is 5.5647844 7 
log. of i 367200 is 5.5649027 £ 
Their diff. 100 .0001183 cliff. 
Nearest No. C 3672C0 
Given No. £ 367182 
Of the Method of using a Table of Logarithms . — 
Having explained the method of making a table 
of the logarithms of numbers greater than unity. 
18 diff. 
Therefore ICO .0001183 H 18 ; .0000212. 
