1003 / 1042
L
o
G
979
L O G
for<, by compofition,
A
n:
De: : 1K: ML.
And,
contranwife, if four numhers be proponi,m,l, thedi·
(lJnce bwvecn the firfl,nd fccond OIJIJ
b~
cqual to the
dillance bctween lhe third and founh.
The dlllance betwcen any tlVO numbm , is ealled
the logarithm of the ratio of lhofe numhtrs; and, in·
deed, doth nOl meafure the ratio i,felf, but the numo
ber
01
terms in a given felies of geometl'lc¡1 propor·
tionals, proceeding fromone number to another, and
defines Ihe number of equdl ratios by lhe compufitioD
whereof the ratio of numbers is known
LOGARITHMS, are the indexes or exponenlS (moflly
whole numbm and decimal fraétions, confifling of f, .
ven places of ligures al leafl) of Ihe powers or roots
(chiefly broken) of a given number. ; yet (ueh indexes
Or exponeDts, thal the (everal powers or rOOIS they
cxprefs, are Ihe nalural numbm,
l, 2,
3,
4,
S,
&&.
10
loor 100000,
&&.
(as, ir lhe given number be
lO,
and in index be a/Tumcd 1 0'000000. lhen the
0.0000000 root of lO, which is 1, will be the lo·
garilhm of l ; Ihe 0.3°1°36 root of 10, whieh is
2,
will be lhe logarilhm of
2;
the 0,477 121 root of 10,
whieh is 3, will be the logarithm of 3; Ihe 1.6t2060
tOOI of lO, the logarilhm of
4;
the 1.04' 393 power
of tOthe logarilhm of 11; Ihe 1.079 181 power of
10 the logarilhm of
11,
&&.)
being chiefly contrived
for ea(e and expediliort in performing of arilhmelical
operations in large numbers, and in trigonometrical cal·
culations; bUl they havclikewi(e becn(ound ofoaenfive
fmice in lhe higher geometry, panicularly in theme·
Ihod o( fluxions. They are generally fou nd-.d on this
confideralion, Ihal if lhere be any rolV of geollletrical
proponional oumbers, as t, 2,
4,
8, t6, 32, 64,
128,256,
&,
or 1, tO, 100, tOOO, 10000,
&&.
and
as many arithmcticdl progrellional numbers adapled tO
Ibem, or (el over Ihem, beginning \Vilh
o,
Ihu 5°, 1, 2, 3,
4,
S,
6,
7,
&,.'2
s,
~
1',
2, 4,8, t6, 32. 64, 128, 0,.5
5
o,
1, 2,
3,
4,
&,.
'2
or,
(l. 10, l OO, 1000, 10000,
&c.
S
lheo \ViII the fum of any t\Vo of thefe arithruetical
progrellionals, added logether, be Ihal arilhmetical pro·
grellional which an(IVers to or flands over thegcome·
tric.1progrefliondl, which is lhe produét of the tlVO geo·
metricalprogrellionals over whichthe tWO a/Tumed arith·
m~tical
pr"grellionals lIand: again , if thofe arithme·
ticalprogrellionals be fubtr aéted(rom each olher, the re·
mainder will be the arithmetical progrellional llanding
over that geometrical progrcllional which is the quo·
tient of the divifion o( the tlVO geomelrieal progref
fionals belonging tO the tlVO firO
a/Tum~d
arilhm!liral
progrellionals; and the doulJle, triple,
&r..
of . ny
one of the arithm tical
progrellion.ls. lI'ill : e the a
rithmeticJI progrelfion. lllanding overthe fquarc, eobe,
O,.
of th'l
g~ometrical
progreOiQnal
~llIch
rhe alTu·
m~d
arilhmetic,1 progrcllion.llll,nds over, as \Vell ' s the
t,
T'
6,.
o( thal arilhmetic,1 progllflional lI'ill be
the geol1letl
ic~1 pro~rellional
a"(,,,erins
10
th~
fquare
rOOt, cube ront,
Óc.
01'
thc ., lIhmctie.1protireftional
ovcr it; and f",mhrnce "i(e; the foll,,\\
in~
enmmon,
though lame and impcI
ítél
ddiniuun uf
IU~¡rJlhnll
;
1i/~,
j .
That lhcy are (omany arithmelieal progrtllionals,
M'
(",ering IOlhe famcnllmGer of geoOJetrieal ones." Where.
.s, Ir any one louks ioto Ihe tables o( logarilhms, he will
fi nd. that the(e do oot all run
00
in an alÍthmetic.1 pro·
grellion, nor the numbers they anfwer
10
in a geometrr.
cal Dne; thefe lall being Ihem(elves arithmctical progre(.
fion,ls. Dr W.Jlis, in his hiflory of algebra, calls lo.
garilhms Ihe indexes of the ratios of numbers
10
one
anolher. Or H.lley, in Ihe philofophical tranfaétioos,
nO
2
t6, fays, theyare the exponemenl9 of the ratios
o( unity
10
numbm. So al(o Mr COles, in his Har·
monia Men(urarum, fays, Ihey are the numerical mea·
fures of ratios. But all thefe definilions convey but avery
con(u(ed oOlion o( logarilhms. Mr Maclaurin, in his
Trealife of Fluxioos, has explained the nalural aod ge·
nefis of logarilhms agreeably to lhe nOtioDof ·their fir(l
inventor lord Naper. Logarilhlflllhen, and Ihe quao.
tities to which they corre(pond, may be fuppofed tO
be
generated by the motion o( a poinI; and ir Ihis poinl
moves over equal fpaces in equal times, the line defcri–
bed by it increa(es equaJly
Ag.ina hne deereafes proponiooably, when thepoint
lhal moves over it def:ribes fueh pam in equal limes as
are alw,ys in Ihe fame conllan( ratio to lhe lioes frolll
whieh they are fubduéled, or
10
Ihe diflaDces o( lha!
poiot, al the beginning o( Ihofe lines, fromagiveo terlll
io Ihat line. In like manner, a lioe may
in~reafe
pro–
ponionably, if in eqaal tiines the moving poinI defcribel
fpaces proponional to in difiances (rom
a
cenaio lerm at
Ihe beginning o( each lime. Thus, io the 5r!! cafe, let
a,
(Plate CIV.
hg.
3')
be
10
ao,
,dIO
co, de
10
do,
cito e
.,fg
lO
l o,
allVays in Ihe fame ratio o(
<L
R
to
<U>;
aod fuppofe lhe poiot
P
fm
Out
from
a,
defcribing.
a"
,d,
de,
ff,
Ig,
in equal pans of Ihe lime; and
lel the fpace defcribed by
P
in any gil'en time be al.
ways in the (ame ratio
10
the difl.nee of
P
from
o
al
the
beglOning of lhat lime; then will the righl lioe
a'
de.·
creafe proponionably.
Jn hke m,nner, the line.
a, (ibid.
nO 3') inerea(C1
proponiooally, ifthe poiotp, inequa! times, defcribes the
(paces
a&,
,d, de,
fg,
ó,.
fo Ihal
a&
isto
a
0,
,d
tOo
c,,
de
to
d.,
(;,
io a conflanl ralio. If we oow fup.
pofe a point
P
defcribing (he line AG
(ibid.
nO 4.) with
ao uniformmotion. while the point
p
de(cl ibes
a
line io·
creafing or decreafing proportionally, Ihe line A
P,
d~·
fcrihed by
P,
lVith thlS uni(orm motion, in the fame time
thdt •
0,
by increafing or demafing rroponionally, be.
comes equal tO •
p.
is the logarithm of
o
p.
Thus
A
C,
A
D, AE,
él,.
ar~
the logarithms of •
e, o
d,
oc,
&'.
refpetllvcly ; and
oa
is the quanlity
whof~
logarithm is
fuprofcd equal tOnOlhing.
\Ve have hcre abOra8ed fromnumbers, thal the doc·
trine rnay be Ihe more generol; bm it is plain, that i(
AC,
AD,
A
E, (;,.
be fuppofed,
! ,
1,
3,
Ó'.
in a·
ritllRletic progrelfion;
o,, e
d.
or,
t e.
will
b~
iD
g~o·
metric
pru~rdlion
and lhat the logarithm 'of
oa,
which.
may be takl n ¡'or unity, is nothing.
LOld N. per, in his fidl fdltOle of logarithms, furpofes,
t"al wlllle
o
p
InHe.fes or decreab proponionally, Ihe
"niform mOllon o( the p,'iO!
P,
by whieh the
log~ri!llm
oi' •
p IS
r,ner."J, is e'I,,?llo Ihe l'dO';I)' o(f at
a;
!1m
is, al lhe term
uf
lime
~
heDIhe log.trttltms bcgin lO be
geoerat
J..