LOG 
B, D, F, &c. be joined by right lines, the 
figure S B M L will become a polygon, con- 
sisting of more or less sides, according as 
there is more or less terras in the progres- 
sion. 
If .the parts AC, C E, EG, &c. be 
bisected in the points c, e, g, i, I, and there 
be again raised the perpendiculars cd, e f, 
g h, ik, I m, which are mean proportionals 
between A B, C D ; C D, E F, &c. then 
there will arise anew series of proportionals, 
whose terras beginning from that which im- 
mediately follows unity, are double of those 
in the first series, and the ^fference of the 
terms are become less, and approach 
nearer to a ratio of equality than before. 
Likewise, in this new series, the right lines 
A L, A c, express the distances of the terms 
L M, c d, from unity ; viz. since A L is ten 
times greater than A c, L M shall be the 
tenth term of the series from unity ; and, 
, because Ae is three times greater than 
A c, e/ will be the third term of the series 
if c d be the first, and there shall be two 
mean proportionals between A B and ef ; 
and between A B and LM there will be 
nine mean proportionals. And if the ex- 
tremities of the lines B d, T)/, FA, &c. be 
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 be 
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 this manner, is called logaritbmical. 
It is manifest from this description of the 
logarithmic curve, that all numbers at equal 
distances are continually proportional. It 
is also plain, that if there be four numbers, 
A B, CD, I K, LM, such that the distance 
between the first and second be equal to the 
distance between the third and the fourth ; let 
the distance from the second to the third be 
what it will, these numbers will be propor- 
tional. For because the distances A C, 
I L, are equal, A B shall be to the incre- 
ment D s, as I K is to the increment M T. 
Wherefore, by composition, A B : D C : : 
I K : M L. And, contrarywise, if four 
numbers be proportional, the distance be- 
LOG 
tween the first and second shall be equal 
to the distance between the third and 
fourth. 
The distance between any two numbers, 
is called the logarithm of the ratio of those 
numbers ; and, indeed, doth not measure 
the ratio itself, but the number of terms in 
a given series of geometrical proportionals, 
proceeding from one number to another, 
and defines the number of equal ratios by 
the composition whereof the ratio of num- 
bers are known. 
LOGARITHMS, are the indexes or ex- 
ponents (mostly whole numbers and deci- 
mal fractions, consisting of seven places 
of figures at least) of the powers or roots 
(chiefly broken) of a given number; yet 
such indexes or exponeirts, that the several 
powers or roots they express, are the natu- 
ral numbers 1, 2, 3, 4, 5, &c. to 10 or 
100000, &c. (as if the.giveii number be 10, 
and its index be assumed 1.0000000, then 
the 0.0000000 root of 10, which is .1, will be 
the logarithm of 1 ; tlie 0.301036 root of 10, 
which is 2, will be the logarithm of 2 ; the 
0.477121 root of 10 which is 3, will be the 
logarithm of 3 ; the 0.612060 root of 10, 
the logarithm of 4; the 1.041393 power 
of 10 the logarithm of 11 ; the 1,P79181 
power of 10 the logarithm of 12, &c.) 
being chiefly contrived for ease and ex- 
pedition in performing of arithmetical 
operations in large numbers, and in tri- 
gonometrical calculations; but they have 
likewise been found of extensive service 
in the higher geometry, particularly in 
thus. 
the method of fluxions. They are gene- 
rally founded on this consideration, that if 
there he any row of geometrical propor- 
tional numbers, as 1, 2, 4, 8, 16, 32, 64, 
128, 256, &c. or 1, 10, 100, 1000, 10000, 
&c. And as many arithmetical progres- 
sional numbers adapted to them, or set 
over them, beginning with 0. 
5 0, 1,- 2, 3, 4, 5, 6, 7, &c. ? 
’> I 1, 2, 4, 8, 16, 32, 64-, 128, &c. ( 
S 0) 2, 3, 4, &c. I 
) 1 , 10 , 100, 1000, 10000, &c. s 
Then will the sum of any two of these 
arithmetical progressionals, added together, 
be that arithmetical progressional which 
answers to, or stands over the geometrical 
progressional, which is the product of the 
two geometrical progressionals over which 
the two assumed arithmetical progressionals 
stand : again, if those arithmetical pro- 
gressionals be subtracted from each other, 
the remainder will be the arithmetical pro- 
gressional standing over that geometrical 
progressional which is the quotient of the 
