1000000030

will

bt gi,·.o,

t·j: .

rqU.tl

IO hJlflbe rUnI ofllte

lo~,¡rilhms.

IOC

. Ihe hrll IW:l

"i11

exhibil Ihe 10g.Iitl,m10 II!

r. .

+--1

+ 181

: ,

plac's of 6gures;

~nd

fo lruS feries is of greal ufe

ow :

n:s

JX:¡:

~4-;; ~o:

+\ -

lillin up Ihe chiliads omimd by Mr Briggs. For el(-

6r.

Ihall be equa! 10 !he Io¡::arilhm of Ihe ralio, whicb ¡mpie, il is r uired 10 6nd Ibe logarilhmof

2GOOI ;

Ihe eomelrical me.n belweentbe

nUOlbe~

:-1 and =+1, Ihe logarilhmof

0000

is Ihe fame as lhe

logarit.hm

has 10 Ihe orilhmetiCtI meJn,

nito

10 I;,e nUQlber : . If

~,lIilb

Ihe index

preoxed 10 il; and lbe differeocc of

lbe oumber

1000, Ibe 6r1lterm of Ihe feries,

i: .

Ihe logarilbms of 20000 aod 20001, is lhe fame as lhe

di!f<rtn~

of lbe logarilhms of tbe oumbm 1000l) aod

,¡l'

is fuJlicieol for produciog !he logarilhm10 13 or

10001,

vi:.

0.0000434'

Aod

tbis differ-

p!ac?S of 6gure$, and tbe recood

IW~

will gi<e Ihe looa.

eo~

be di,; ded by :, or 8000 , Ihe uOlienl

- /hall

ntbm 10 20 places of figures. BUI If :; be ¡¡remr Ihin.

0.000Q00000542813

and if Ihe logaritbm of Ihe

geomelri~l

mean,

vi:.

jOl05l 70930141 he added 10 tbe quotieot, Ihe rum

will

4.3°1°51 °9845230= Ihe logarilbm of _0001

Wbmfore il is macifell Ihar 10 ha"c tbe logarilbm 10 iD Btlggs's form, aod thecontrary. And, ar,er ,be famo:

14 plaro pf 6gum , tbert is 00 necellily of conlinuing manoer, ,he m ioof nalurallogarilbmsto tb'l ofBriggs's

OUI Ihe quotient be) ood 6 pl.ces of 6gures. BUI if you

will

be foood::86 53 963806.

hnc a mind 10 ha"e Ibe logmlhm 10 10 plaoes of figures

oolf, !he IWO 6rll figures are eoougb. Aod if Ibe loga.

rilbms of !he oumbm abore 20000 are 10 be fouod by

lbis

\\"y,

!he hbour of doiog Ihem \

,11

molllycooúll io

fettiog dowo Ihe oumbers. This feries is eaúly deduced

from Ihe cooúderanoo

Ihe h)lperboli fpaces

aforef~id.

The Grfl 6gure of erery logarithm lowards !he lefl hand,

whi h is fepamed from ¡he retl by a poiol, is aIled Ihe

iode.'C or Ihal loganlbm; beCtufc il poiols OUI Ihe high.

•11

or rem'otcfi' place or Ib¡1 oomber from the place

unily io Ihe i06nile fole of proportiooals 10wmls lbe left

blOd :

Ihus, if Ihe index of Ibe !ogarilhm be l. il fhews

mar

irs highell place lowards Ihe lerl hand is Ibe leolb

place from nnily; aod Iherdoreallloga.ilbms wbich hare

1 ror tbeir index, will be (euod helween tbe lenlb .20d

huodrtdlh plact. iD !he order of oumbers. And for tbe

fame rearoo alllogarilhms \lhich hm

for tbeir iodex,

wiU

be fouod belwem Ibe huodredlh aod tboufaodlh

place, io

lb~

order ofoumbers,

6,.

Whence uoiw f.lly

Ihe iode:c or chm8erillic of ao logarilhm is a1""ys IJ!

by ooe Ihao Ibe oumber of 6gum iD

hole oumhers,

wbich aofwer 10 tbe giren logarilhm

.nd, iD

decimili,

lbe

iode.~

is negaure.

As aU ryllems of logarilhms whm ver, are compofed

f fimilar quantilies, il

'11

be e;.fy 10 formo from.any

fyncmor

10ga~lbms,

aoolher fyllcm iDaoy

gire~

ratio

20d coofequeotly co reduce ooe rabie of logarilhms iOlo ao·

olber of

~ny

given forro. Forasan ootlogaritbm iDIhe

giveo (orm. il 1.0 ilS comfpoodeol logaritbmio aoother

form; fo is aoy olher logarilhm iDIhe

i.eo

formo10 ils

corrJpoodeol logarilhmin Ihercquired fono

¡nd heocc

\\le Ola reduce me logarilhms of lord .pier iOlo me

form of Briggs's, 20d coolrarill'lfe. Fora! .;01 5S509"

(. lord apier's logarilhm of 10, is 10

1.0000000000,

Brig~s't

logarilhm of 10

fo is ¡oy olha lo

~rilhm

iD lord '.pier" formo10Ihe corrcfpcodenl rabulor loga–

ritllm io

r Briggs's forro

ar,d b:.:. ufe Ihe IWO firtl

pumae's C<'nlbntl)' remain Ihe fame

ir lord

apier'. 10-

garilhm of

lO}

one number be divide<! hy

' 02

-S

j ,

Or muhiplied

by 4

'429H,

t .

Ihe m io of ;

.0000,

2.;02;

, be.

as is found by di, io,"!!

1.0<l000.

6,.

3°258,

r. Ihequo"rnl io Ih f,.rOlrr , anolh. pro·

in th: I.u r, \

iII

ive lb orr fpoJ¿tSI logarilbm

Té:

Uf(

Qlftl

.¡p!h:ali4n'¡

LOC AlJ TH~Jl.

is e.ideol, from whal has

beco

faid of tbe coollruc–

tion of logarilbms. Ibal addition of logarilbms mull be

Ihe fame Ibiog as mulliplicalioo io tommoo arilhmetick;

¡nd fubllraaioo iD logarilbms Ihe fame as divifioo : Ihere–

fore, iD molliplicanoo by' logarithms, add tbe logarilbm.

of Ihe mulcipliCtod aod multiplier logetber, Ibeir fum

Ihe 10gaJitbm of Ihe produél•

Dum. logarilhms.

EKample. MultipliC2Dd

8.S

0.1294 189

Muhiplicr

1.00000c0

produa

85 1'92941 8\1

Aod

diriúoo. fublnél lbe logaritbm of tbe dirifor

from Ibe logarilhm of Ibe dirid nd, tbe reml.inder

tbe

logarithm of!be quouent.

oum. 10glritbms.

EKlmple. Dirideod 9712

8 3-98 4H

Diñfor 456

2.6589648

'J.

filJd

'PIt

:nl of a

LOCH IT HN.

Begin

Ihe lef, haod, and ",rile down

whareach.Jj

gure

~'iOIS

of 9. oolYlI'h¡1 the la

fignifi~nl

figurt

wants.

of 10

fo Ihe cumplemeor of' me lo

~bm

of 456.

viz.

2.6

9648,

10352.

Ibe rule of Ihree. Add Ibe logarithms of lbe re–

cond and Ihird lerms logelher, aod fror.t Ihe fumfob–

mé! Ihelogarilhmof Ibe lirll, Ibe remaind.r is Ihe lo.

garilhmof Ihe founh. Or, infi.,d of fublraé!ing a 10-

garilbm, add iu complemenl, aod Ihe ,Juh \ di be tbe

faree.

rolifl

I«<'m

Ó)'

LOCH ITIINS.

!uhipl)' Ibe logarilhm of Ihe oumb.r giren. by Ihe

ind, x o( ,he rouer r<<tulrrd. Ihe produa will be Ihe lo–

garilhmof ,he pOll'er loughl.

Examrlc. Lel Ihe cube of 31 be r(quire<! by laga–

rilhOls. The logarilhmof •2=I .S051 5C('

hich mul,

liplioo hy ;.

H I54500, Ihe log. nlhm of 31 6S. Ihe

cub of

llul iD

uifioS

pOló'm,

t'/Z.

fqumog,

tU-

bio¡¡.