described by P, in those times, with its uni- 
form motion. 
Hence the chief properties of logarithms 
are deduced. They are the measures of 
ratios. The excess of the logarithm of the 
antecedent above the logarithm of the con- 
sequent, measures the ratio of those terms. 
The measure of the ratio of a greater quan- 
tity to a le.'ser is positive ; as this ratio, 
compounded with any other ratio, increases 
it. The ratio of equality, compounded 
W'ith any other ratio, neither increases nor 
diminishes it ; and its measure is nothing. 
The. measure of the ratio of a lesser quan- 
tity to a greater is negative ; as this ratio, 
compounded with any other ratio, diminishes 
it. The ratio of any quantity A to unity, 
compounded with the ratio of unity to A, 
produces the ratio of A to A, or the ratio of 
equality ; and the measures of those two 
ratios destroy each other when added toge- 
ther ; so that when the one is considered as 
positive, the other is to be considered as ne- 
gative. By supposing the logarithms of 
quantities greater than o a (which is sup- 
posed to represent unity) to be positive, 
and the logarithms of quantities less than it 
to be negative, the same rules serve for the 
operations by logarithms, whether the 
q\iantities be greater or less than o a. When 
op increases proportionally, the motion ofp 
is perpetually accelerated ; for the spaces 
ac,cd,de, &c. that are described by it in 
any equal times that continually succeed 
after each other, perpetually increase in the 
same pr oportion as the lines oa, oc,od, &c. 
AVhen the point p moves from a towards o, 
and op decreases proportionally, the mo- 
tion of p is perpetually retarded ; for the 
spaces described by it in any equal times 
that continually succeed after each other, 
decrease in this case in the same proportion 
as op decreases. 
If the velocity of the point p be always 
as the distance op, then will this line in- 
crease or decrease in the manner supposed 
by Lord Neper ; and the velocity of the 
point p being the fluxion of the line op, will 
always vary in the same ratio as this quan- 
tity itself. This, we presume will give 
a clear idea of the genesis, or nature of lo- 
garithms ; but for more of this doctrine, see 
Maclaurin’s Fluxions. 
Logarithms, construction of. The first 
makers of logarithms, had in this a very la- 
borious and difiicult task to perform ; they 
first made choice of their scale or system 
of logarithms, that is, what set of arithmeti- 
cal progressionals should answer to suen a 
LOGARITHMS. 
set of geometrical ones, for this is entirely 
arbitrary ; and they chose the decuple geo- 
metrical progressionals, 1, 10, 100, 1000, 
10000, &c. and the arithmetical one, 0, 1, 
2, 3, 4, &c. or, 0.000000 ; 1.000000 ; 
2.000000 ; 3.000000 ; 4.000000, &c. as the 
most convenient. After this they were to 
get the logarithms of all the intermediate 
numbers between 1 and 10, 10 and 100, 
100 and 1000, 1000 and 10000, &c. But 
first of all they were to get the logarithms 
of the prime numbers 3, 5,7, 11, 13,17,19, 
23, &c. and when these were once had, it 
was easy to get those of the compound 
numbers made up of the prime ones, by 
the addition or subtraction of their loga- 
rithms. 
In order to this, they found a mean 
proportion between 1 and 1 0, and ils loga- 
rithm will be one half that of 10 ; and so 
given, then they found a mean proportional 
between the number first found and unity, 
which mean w'ill be nearer to 1 than that 
before, and its logarithm will be one half of 
the former logarithm, of one-fourth of that 
of 10 ; and having in this manner continu- 
ally found a mean proportional between 1 
and the last mean, and bisected the loga- 
rithms, they at length, after finding 64 such 
means, came to a number 
1.0000000000000001278191493200323442, 
so near to 1 as nof to differ from it so much 
part, and found its loga- 
‘V' ICOOOO'OOOOOOOOOOO 
ritinn to be 
0.00000000000000003351115123125782702 
and 
00000000000000012781914932003235 to be 
the difference whereby 1 exceeds the num- 
ber of roots or mean proportionals found by 
extraction ; and then, by means of these 
numbers, they found the logarithms of any 
other numbers whatsoever ; and that after 
the following manner; between a given 
number, whose logarithm is wanted, and 1, 
they found a mean proportional, as above, 
until at length a number (mixed) be found, 
such a small matter above 1, as to have 1 
and 15 cyphers after it, which are followed 
by the same number of. significant figures ; 
then they said, as the last number men- 
tioned above, is to the mean proportional 
thus found, so is the logarithm above, viz. 
0.00000000000000005551115123125782702 
to the logarithm of the mean proportional 
number, such a small matter exceeding 1, 
as but now mentioned ; and this logarithm 
being as often doubled as the number of 
mean proportionals, (formed to get that 
number) will be the logarithm of the given 
