BO 
L O G 
L O G 
LOG 
LM, which let be continually proportional, 
and represent, numbers, viz. Ail, 1 ; CD, 10; 
EE, 10'.), &c. then shall we have two pro- 
gressions of lines, arithmetical and geometri- 
cal: for the lines AC, AE, AG, &c. are in 
arithmetical progression, or as 1, 2, 3, 4, 5, 
&c. an 1 so represent the logarithms to which 
the geometrical lines AB, CD, EE, &c. do 
correspond. For since AG is triple at the 
right line AC, the number Gil shall be in 
the third place from unity, if CD is in the 
first; so likewise shall LM be in the fifth 
place, since AL = 5 AC. If the extremi- 
ties of the proportionals S, cl, B, D, F, & c. 
are joined by right lines, the figure SBML 
will become a polygon, consisting of more or 
less sides, according as there are. more or less 
terms in the progression. 
If the parts AC, C E, EG, &c. are bisected 
in the points c, e, g, i, /, and there are again 
raised the perpendiculars cxl, ef, git, ik , lm, 
which are mean proportionals between AB, 
CD; Cl), EE, i kc . then there will arise a 
new series of proportionals, whose terms be- 
ginirng from that which immediately follows 
unity, are double of those in the first series, 
and the difference of the terms is become 
less, and approaches nearer to a ratio of equa- 
lity, than before. Likewise, in this new se- 
ries, the right lines AL, Ac, express the dis- 
tances of the terms LM, cd, from unity, viz. 
since AL is ten times greater than Ac, LM 
shall be the tenth term of the series from 
unity; and because Ac is three times greater 
-than Ac, ef ‘will be the third term of the se- 
ries if cd! is the first, and there shall be two 
mean proportionals between AB and ef ; and 
between AB and LM there will be nine mean 
proportionals. • And if the extremities of the 
tines lid, Df, Yl), Ac. are joined by right 
lines, there will be a new polygon made, 
consisting of more but shorter sides than the 
last. 
If, in this manner, mean proportionals are 
continually placed between every two terms, 
the number of terms at last will be made so 
great, as also the number of the sides of the 
polygon, as to be greater than any given 
number', or to be infinite; and every side of 
the polygon so lessened, as to become less 
than any given right line; and consequently 
the polygon will be changed into a curve- 
lined figure: for any curve-lined figure may 
be conceived as a polygon whose sides are 
infinitely .small and infinite in number. A 
curve described after tins manner, is called 
Jogarithmical. 
It i ; manifest, from this description of the 
logarithms; curve, that all numbers at equal 
distances are continually proportional. It 
is also plain, that if there are tour numbers, 
A B, C i), I K, L M, such that the dis- 
tance between the first and second, is equal 
to the distance between the third and fourth, 
let the distance from the second to the third 
be what it will, these numbers will be pro- 
portional. For because the distances AC, 
I L, are equal, A I> shall be to the increment 
D ■?, as 1 K is to the increment M T. Where- 
fore, by composition, A B : DC : : IK: 
M L. And contrariwise, if four numbers 
are proportional, the distance between the 
first and second shall be equal to the distance 
between the third and fourth. 
I he distance between any two numbers, 
is called the logarithm of the ratio of those 
numbers ; and, indeed, does not measure the 
ratio itself, but the number of terms in a I 
given series of geometrical proportionals, j 
proceeding from one number to another, and 
defines the number of equal ratios by the 
composition whereof the ratios of numbers 
are known. 
LOGARITHMS are numbers so contrived | 
and adapted to other numbers, that the sums 
and differences of the former shall correspond 
to, and shew, the products and quotients of the ! 
latterE 
Or, more generally, logarithms are the nume- 
rical exponents of ratios; or a series of numbers 
in arithmetical progression, answering to another 
series of numbers in geometrical progression. ; 
Thus, 
0, 1,2, 3, 4, 5, Indices, or logarithms. 
1, 2, 4, 8, 16, 82, Geometric progression. 
Or, 
0, 1, 2, 3, 4, 5, Indices, or logarithms. 
1, 3, 9, 27, 81,243, Geometric progression. 
Or, 
0. 1, 2, 3, 4, 5, Ind. or log. 
1, 10, 100, 1000, 10000, 100000, Geo. prog. 
Where it is evident that the same indices serve 
equally for any geometric series ; aud conse- 
quently there may be an endless variety of sys- 
tems of logarithms to the same common num- 
bers, by only changing the second term, 2, S 1 or 
10, Ac. of the geometrical series. 
It is also apparent, from the nature of these 
series, that if any two indices be added together, 
their sum will be the index of that number 
which is equal to the product of the two terms, 
in the geometric progression, to which those in- 
dices belong. 
Thus, the indices 2 and 3, being added to- 
gether, are — 5 ; and the numbers 4 and 8, or 
the terms corresponding with those indices, be- 
ing multiplied together, are ~ 32, which is the 
number answering to the index 5. 
And, in like manner, if any one index be sub- 
tracted from another, the difference will be the 
index of that number, which is equal to the 
quotient of the two terms to which those in- 
dices belong. 
Thus the index 6, minus tire index 4, is — 2; 
and the terms corresponding to those indices are 
64 and 16, whose quotient is — 4 ; which is the 
number answering to the index 2. 
For the same reason, if the logarithm of any 
number are multiplied by theindex of its power, 
the product will be equal to the logarithm of 
that power. 
Thus, the index or logarithm of 4, in the 
above series, is 2 ; and if this number is multi- 
plied by 3, the product will he = 6 ; which is 
the logarithm of 64, or the third power of 4. 
And, if the logarithm of any number is di- 
vided by the index of its root, the quotient will 
be equal to the logarithm of that root. 
Thus, the index or logarithm of 64 is 6 ; and 
if this number is divided by 2, the quotient will 
be — 3 ; which is the logarithm of 8, or the 
square root "of 64. 
The logarithms most convenient for practice 
are such as are adapted to a geometric series 
increasing in a tenfold proportion, as in the last 
of the above forms ; and are those which arc to 
be found, at present, in most of the common 
tables upon this subject. 
The distinguishing mark of this system of lo- 
garithms is, that the index, or logarithm, of 1 
is 0; that o' 10, 1 ; that of 100, 2; that of 
1000 3, Ac. And in decimals the logarithm of 
.1 is — 1 ; that of .01, — 2 ; that of .001, — 3, Ac. 
From whence it follows that the logarithm of 
any number between 1 and 10 must be 0 and 
some fractional parts, and that of a number be- 
tween 10 and 100 will be 1 and some fractional 
parts ; arid so on for arty other number whatever. 
And since the integral part of a logarithm is 
always thus readily found, it is usually called the 
4 
index, or characteristic; and is commonly omit- 
ted in the tables ; being left to be supplied by 
the operator himself, as occasion requires. 
Of the M, iking of Logarithms. Whatever arith- 
metical progression we apply to a geometrical 
one, the terms of it are logarithms only to that 
series to which we apply them, and answer the 
end proposed only for those particular num- 
bers ; so that if we had logarithms adapted only 
to particular geometrical series, they would be 
but of little use. The great end and design of 
these numbers is the ease and expedition which 
they afford in long calculations, by saving the 
laborious work of multiplication, division, and 
the extraction of roots : but this end would ne- 
ver be completely answered, unless logarithms 
could he adapted to the whole system of num- 
bers, 1, 2, 3, 4, Ac. And as here lie the chief 
excellence and merit of the contrivance, so also 
the difficulty. For the natural system of num- 
bers, 1, 2, 3, 4, Ac. being an arithmetical, and 
not a geometrical series, seems rather fit to be 
made logarithms of. than to have logarithms 
applied to it. But this difficulty may he easily 
removed, by considering, 
i That though the whole system of natural 
| numbers, 1, 2, 3, 4, Ac. is not in geometrical 
! progression, and cannot, by any means, be made 
I to agree with such a series, yet it may be 
j brought so near it, as to be within any assign- 
j able degree of approximation ; which may he 
I conceived, in general, thus: suppose a fraction 
: indefinitely small to be represented by x, and a 
geometrical series arising front 1, in the ratio of 
1 to 1 4-.V, to he 1, (1 -j-A-) l ,(l +xf, (1 4-.v)k 
(! — }— a.-) ', Ac. Then some of these terms mu: t 
come indefinitely near to all the natural numbers, 
1, 2, 3, 4, Ac. ; because, amongst numbers which 
1 arise by extremely small increments, some of 
them must exceed, or fall short of, any deter- 
minate mumber, by an indefinitely little exeess 
i or defect. 
If, therefore, in the places of the terms of 
this series, which approach indefinitely near to 
any of the natural numbers, we suppose these 
natural numbers themselves to be substituted, 
j then will this serjes be in geometrical progres- 
sion, to an exactness which may be called inde- 
| finite ; because the approximation of its terms 
| to the natural numbers can never end but goes 
| on in infinitum. 
j And since this imagined geometric series com- 
j prebends, indefinitely near, the whole system of 
; natural numbers, 1, 2, 3, 4, Ac. so the indices 
! of its terms comprehend a whole system of lo- 
! garithms, which are adapted to this system of 
numbers, and may be extended to any length 
j we please. For though the natural system of 
I numbers make not, by themselves, a complete 
! geometrical series, yet they are conceived as a 
i part of such a series, and their logarithms are 
j the indices of their distances from unity in that 
1 series ; or, more generally, they are the cor- 
j responding terms of an arithmetical series ap- 
i plied to that geometrical one. 
: But, again, it must be observed, that an inde- 
■ finitely small fraction cannot be assigned : and, 
therefore, in the actual construction of loga- 
rithms, we must be content with a determinate 
degree of approximation. Whence, according as 
we take x, in the series 1, (1 -j-.v) 1 , (1 -j- x) 2 , 
(l-f-*)’, ( 1 -f- x) l , Ac. the approximation of 
iis terms to the natural numbers will be in dif- 
ferent degrees of exactness : for the less x is, the 
nearer will be the approximation ; but then the 
more are the number of involutions of 1 -j- v, 
necessary to come within any determinate de- 
gree of nearness to the natural number assigned. 
Thus then we may conceive the possibility of 
making logarithms to all the natural numbers, 
1, 2, 3, 4, Ac. to any determinate degree of 
exactness ; viz. by assigning a very small frac- 
tion for v, and actually raising a series, in the 
ratio of I to 1 -j- x, and taking for the natural 
