1000000030

980 )

L O G

~!nerat(J .

ll~~re lor.~ridllns,

formeJ

,I:¡~r

\:'is muJel, fet of

~com,"ical

Gr.cs

, for Iloi! is entirc!y arbi(nry; lnr!

are cllJeo N'per's Logadlilm¡, ano fometÍmes

;tUral theyc¡,,,r( Ihcuceuplc gcolUwical pror,reffion,ds, 1, 10,

Logaruhms.

100, lOCO, 100('0,

.t, ..

,ndlhealilhn:elical une, O, 1,2 ,

When a miQ is given,

th~

point

d:faibes t!:e dif· 3..1'

(;·c

or o,OOCCOO; 1,000000;

~,OOOOOO;

),000000;

ference of the tcrms 01

Ih~

ralio in the f. me lime When 4,000000,

6c.

Ih( moa convenien!. Ilfter IIIIS Ih¡'Y

a r.!tio is ouplicm of anuth<r ratio, thc roint

oefcribcs \Vere lOget Ihe 10gJrithms 01'

311

the int"mcdlatC nulO.

th~

difl"c;enct uf the \erms in 3 double time. \'1hcn

bers bct.lcen I ano 10, 10 and

I (,(l,

100 ,nd 1000,

ratio is trll'licm of anotha, il d,fcrib!s the d¡!Icrence 1000 .od 10000,

&c.

But fi rll of

,11

they \Ver< 10 gel

01 the m ms in a triple time ; and fo oo. Alfo, when a Ihe logalilhllis of rhe prime oumberr. 3, 5, 7, 11 , 13,

n .io is compounded of twO or more ratios , thc peint

17. 19, 23,

&c.

aod \I'heo thele wereonle had, it \V

defcribes Ihe difTerence of the terms of that ratio in a eafy to

~et

thufe of tbe compouod numbcrs made up uf

time eq"al to the f"OI of the times in which it defcribes Ihe prime ones, by the aduitioo or fubm/1iun of tildr

the oiff,rcnecs of the terms of the fi mple ratios of \Vhidl 10gMi hllls.

it is compounded. And wbat is her: (air! of the limes of

In order to this, thry

(ou~d

a mean proponion be.

lhe mo:ionof

when

DI'

increafes

pruponion~lIy,

is to be t\Vcen 1 aod tO, and its loganthm..·ill be

{th~t

01 10 ;

Rpplid to the fpacesder"ibed by

iD thofetimes, with aod (o given, Ihen they fuund

mean proponional be–

its ,,"iform motion.

tween ,he number

(Idl

fOllnd and unity, which OleanwiU

H~n·e

the ehief pro?erties of

log~rithm\

are dedueed. be nemr to 1 !lun tlm before, aod in logarithmwiJl be

T hcy

3rt

the meafurcs of ratios. The excefs of the lo·

of Ihe forme r logarithm, or

of that of 10 ; and h3·

garithmof the antecedent abo.,e thelogarithmof the con· viog inthis manner continually fou nd

meanproponional

Lqucnl. meafum theratio ofthore lerms. Themeafureof bet\Veen I Rod the

I.a

mean, and bifeéled the logarithms,

theratioof a

gre~lerquantity loaldrtris pofilive ;

asthis lhey al length, afler fi nding

fu eh meaol, came 10 a

.ratio, compouodedwith anyolher ralio, inereafes it, The number 1000000000000000 1278191493200323442, fo

ratio of equalily, compouoded with any other ratio, nei· near to 1 as nOl 10 dill'er from it fo mueh as

ther ioereafes noruiminilhes it; and irs meafure is nothiog.

.~'"~dth

parl' aod found ilS logarilhm

The meafure of the ratio of a le{fer quantity to agreater tobe0.000000000000000055

5123125782702,aod

is negative; as this ratio, eompounded with any other 00000000000000012 78 191493 200323) tO be the dif–

ratio, diminilhes it. The ratio of any quantity

to u· ference whereby 1 exeeeds the number of roots or meao

nilY, eompounded wilh the ratio of unity10

produces proportionals found by extra/1ion; and Iheo, by means

the ratio of

01 the ratio of equalily; and the of thefe numbm, they found thelogarithms of any other

mearures of thofe twOu :ios dearoyeaeh olher \Vhen ad· numbm whatfoever ; and that after the foJlowiog mao–

ded together; fo that when lhe one is eonfidered as po· Der : betweena givennumber, whofe logarithm is wanled,

fitive, theother is tobe coofidered as oegalive. Byfup. and 1, they found a mean proponional, as abofe, until

pofing the logarithms of

q~aotities

greater than

(which at length anllmber (mixed) be found, (ueh afmall mmer

is (uppofed to reprerent uoi,y) to be pofitive, and the lo· above 1, as 10have 1 aod 15 cyphm after it, \Vhieh are

llarithms of quantities lefs than it to be negative, the followed by the fame Aumber of Ggoificant figures ; then

{ame rules ferve for the operations by logarithms, whe- Ihey

f~id,

as the lafl oumber mentioned abofe is 10 the

ther Ihe quantilies be greater or le(s thao

Da.

Wheo mean proportional thus found, fo is the logarithm above,

increafes proponionally, themotionof

is perpetually

viz.

0.0000000000000000555t1151 23 125782702, to

aeeelermd ; for thefpaces

Qe.

eJ,

dl,

&c.

Ihat are de· thelogarithmofthemean proponiooalnumber,fuehafmall

{cnoed by it in an)' equal times that continually fueeeed malter exeeediog 1 as but nowmentioned; and Ihis,lo–

after eaeh olher, perpetually increafe in the fame propor· garilhmbeiog

ofteo doubled as the number of meao

tioo

the lines

oa"oe, od, 6 e.

When the poiot

proponionals (formed tOget that number) wllI be lhe

moves from

to\Vords

and opdeereafes proponionally, logarithmof Ihe given number.· And this was the me–

lhe motion of

is perpttually retarded ; for Ihe fpaees thod Mr. Briggs took

make the logarithms. But if

drfcribtd by it in any equal times that contioually fue- they are tobe made toooly fevenplaces of figures, whieh

"td afler eaeh olher, decreafe io lhis caCe io the fame are enough for commno ufe, they had ooly occafion

proportioo as

deereafes.

fiod 25 meanproponionals, or, whieh is the fame Ihiog,

If the.•e1ceity of tbe point

be always as the dinance to eXlra/1lhe

~Ib

root of 10. Nowhaving the

.p,

Ihtn

\Viii

this lioe increafe or deereafe in the maooer logaritbms of

¡,

and

they eafily got thofe of 2,

fJPFofed

lord Nzper ; and the velocity of the POiOI

8 and 9; for fioee J/'-=2, the logarithm of 2 will be

beiog Ibe Buxion of Ihe lioe

op,

will alwa)s vary in the the difTereDee of Ihe logarilhms of 10 aod

Ihe loga–

f~m<

ratioas this quantityit(eir. This, we pre(ume, will rilhm of 4 will be two limes Ibe

lo~arithm

of 2, the lo·

give a

c1car

idea of Ihe genefis, or DalUreof log.rithms; -garithmof

\ViII be the fum of the logarithm of 2 and

but for more of this dol1rioe, fee Maclaurio's Fluxioos. aod the logarithm of 9 double the logarilhmof 3. So,

Conjlrufli<n

1 LoGARtTHMS .

The fi,a

m.~ers

of logarithms had in this a . ory

laborillus ano d,lIieuh talk tO perform; they Grn made

elloic, of their

fe~le

or fyflem of logari!llm!, thal is, \Vhat

:!:!

Df uithmtti

progreOiop&ls lhou!J

~.ofw(f

tOfueh a

.Ifo having found Ihe logarirhms of 13, 17 aod 19, aod

al(o of 23 and 29, they did e.fily get rhofe of all Ihe

numbers belween 10 and 30, by aeditioo

aroe

fubll.ilion

ooly; aod (o

havin~

founu rhe logarilhms 01 other p,ime

numbers, they gOl tho(e of o\her

num~ lf!

eomrounded of

U,(ID,

CI/t