Encyclopaedia Britannica; or, A dictionary of arts and sciences, compiled upon a new plan … > Volume 2, C-L

LOG (9
.generated. Hence logarithms, formed after this model,
are called Naper’s Logarithms, and fometimes Natural
Logarithms.
When a ratio is given, the point p defcribes the dif¬
ference of the terms of the ratio in the fame time. When
a ratio is duplicate of another ratio, the point /> defcribes
the difference of the terms in a double time. When a
ratio is triplicate of another, it defcribes the difference
of the terms in a triple time ; and fo on. Alfo, when a
ratio is compounded of two or more ratios, the point/»
defcribes the difference of the terms of that ratio in a
time equal to the fum of the times in which it defcribes
the differences of the terms of the Ample ratios of which
it is compounded. And what is here faid of the times of
the motion of p when op increafes proportionally, is to be
applied to the fpaces defcribed by P, in thofe times, with
its uniform motion.
Hence the chief properties of logarithms are deduced.
They are the meafures of ratios. The excefs of the lo¬
garithm of the antecedent above the logarithm of the con¬
sequent, meafures the ratio of thofe terms. The meafure of
the ratio of a greater quantity to a leffer is pofitive; as this
ratio, compounded with any other ratio, increafes it. The
ratio of equality, compounded with any other ratio, nei-
-ther increafes nor diminifhes it; and its meafure is nothing.
The meafure of the ratio of a leffer quantity to a greater
is negative ; as this ratio, compounded with any other
ratio, diminifhes it. The ratio of any quantity A to u-
nity, compounded with the ratio of unity to A, produces
the ratio of A to A, or the ratio of equality; and the
meafures of thofe two ratios deflroy each other when ad¬
ded together ; fo that when the one is confidered as po¬
fitive, the other is to be confidered as negative. J3y fup-
pofing the logarithms of'quantities greater than oa (which
is fuppofed to reprefent unity) to be pofitive, and the lo¬
garithms of quantities lefs than it to be negative, the
fame rules ferve for the operations by logarithms, whe¬
ther the quantities be greater or lefs than oa. When
0/increafes proportionally, the motion of p is perpetually
accelerated ; for thefpaces ac, cd, de, fee. that are de-
feribed by it in any equal times that continually fucceed
after each other, perpetually increafe in the fame propor¬
tion as the lines oa, oc, od, he. When the point p
moves from a towards o, and o/>decreafes proportionally,
the motion of p is perpetually retarded ; for the fpaces
deferibed by-it in any equal times that continually fuc¬
ceed after each other, decreafe in this cafe in the fame
proportion as op decreafes.
If the velocity of the point p be always as the diftance
op, then will this line increafe or decreafe in the manner
fuppofed by lord Naper ; and the velocity of the point p
being the fluxion of the line op, will always vary in the
fame ratio as this quantity itfelf. This, we prefume, will
give a clear idea of the genefis, or nature of logarithms;
but for more of this doftrine, fee Maclaurin’s Fluxions.
Ccnjlruftion ^Logarithms.
The fiifl: makers of logarithms had in this a very
laborious and difficult talk to perform ; they firft made
choice of their fcale orfyftem of logarithms, that is, what
isl of arithmetical progrdlionals Ihould anfwer to fuch a
So ) LOG
fet of geometrical ones, for this is entirely arbitrary; and
they claufe the decuple geometrical progreflionals, i, 10,
100, 1000, loooo, ha and the arithmetical one, o, i, 2,
3,4. or0,000000; x.oooooo; 2,000000; 3,000000;
4,000000, he. as the mofl; convenient. After this they
were to get the logarithms of all the intermediate num¬
bers betveen 1 and 10, 10 and 100, 100 and 1000,
1000 and 10000, he. But firft of all they \vere to get
the logarithms of the prime numbers 3, 5, 7, n, 13,
17. 19, 23, he. and when thefe were once had, it w. 5
eafy to get thofe of the compound numbers made up of
the prime ones, by the addition cr fubtradlion of their
logarithms.
In order to this, they found a mean proportion be¬
tween 1 and 10, and its logarithm will be ~ that of 10;
and fo given, then they found a mean proportional be¬
tween the number firft found and unity, which mean will
be nearer to 1 than that before, and its logarithm wdl! be
4 of the former logarithm, or 4 °f that of 10 ; and ha¬
ving in this manner continually found a mean proportional
between 1 and the laft mean, and bifefted the logarithms,
they at length, after finding 54 fuch means, came to a
number 1 ©000000000000001278191493200323442,6)
near to 1 as not to differ from it fo much as
togtre-ffaoVogoco~o-o~oc6h Part> an^ found its logarithm
to be0.00000000000000005551115123125782702, and
00000000000000012781914932003235 to be the dif¬
ference whereby 1 exceeds the number of roots or mean
proportionals found by extraflion ; and then, by means
of thefe numbers, they found the logarithms of any other
numbers whatfoever ; and that after the following man¬
ner : between a given number, whofe logarithm is wanted,
and 1, they found a mean proportional, as above, until
at length a number (mixed) be found, fuch a finall matter
above 1, as to have 1 and 15 cyphers after it, which are
followed by the fame number of fignificant figures ; then
.they faid, as the laft number mentioned above is to the
mean proportional thus found, fo is the logarithm above,
viz. 0.00000000000000005551115173125782702, to
the logarithm of the mean proportional number, fuch afmall
matter exceeding 1 as but now mentioned ; and this lo¬
garithm being as often doubled as the number of mean
proportionals (formed to get that number) will be the
logarithm of the given number. And this was the me¬
thod Mr. Briggs took to make the logarithms. But if
they are to be made to only feven places of figures, which
are enough for- common ufe, they had only occafion to
find 25 mean proportionals, or, which is the fame thing,
to extrad the T-rrp3TT-rth root of 10. Now having the
logarithms of 3, 5, and 7, they eafily got thofe of 2, 4,
6, 8 and 9; for fince xf-=2, the logarithm of 2 will be
the difference of the logarithms of 10 and 5, the loga¬
rithm of 4 will be two times the logarithm of 2, the lo¬
garithm of 6 will be the fum of the logarithm of 2 and 3,
and the logarithm of 9 double the logarithm of 3. So,
alfo having found the logarithms of 13, 17 and 19, and
alfo of 23 and 29, they did eafily get thofe of all the
numbers between 10 and 30, by addition andffubtradioa
only; and fo having found the logarithms of other prime
numbers, they got thofe of ether numbers compounded of
them. But