1000000030

when CG is = 2. Alfo,

fince~]

= 3, the fum

).°986122, &c.

of the mas belonging to

:~82,

and

will be the area of AFGH, when CG=3. Again, fince

2X2

o.s=S,

and 2XS=IO; by adding

Ad=.223I,

tic.

AD=0.1823,

tic.

and

Ad=.

t053,

tic.

together, their

fum is 0.51°8,

&c.

and Ihis added tO 1.0986,

tic.

Ihc area

oí

AFGH, when CG=3. You will have

! .6093379124341004=AFGH, when CG is

and

adding that of

to this, gives

2.302S8S0929940457

=AFGH, when CG is equal to 10: and fince 10XIO:

J(lO;

and 10X 100=1000; and.v

SXloXo.98=7,

and

1000XI.091

d 1000Xo·998

JOXI.I=lI, an ----=13, an ----=

7XII

499; it is plain that the area ,AFGH may be found by

tbe compofition of thoreas found before, when CG=I"O,

000, or any other of the numbers above mentioned;

and aH thefe mas are Ihe hyperbolic logarithms of thofe

feveral oumbers.

Having thus obtaioed the hyperbolic logarithms of the

numbers 10, 0,98,°'99, 1.01, 1.02; if the logarithms

of Ihe four lalI of Ibem be divided by the hyperbolic lo,

garitbm

2.302S850, &c.

of 10, and the index

beadd·

cd; or, whicb is Ihe fame thing, if it be multiplied by

reciprocal 0.H42944819032SI8, tbe value of the

fubtangent of the logaritbmic curve, to which Briggs's lo,

g1rithms are adapted, we {ball have the true tabular loga.

rithms of98, 99, 100, 101, 102. Thefe are to be inlerpo·

laled bylen iotervals, and Ihen we fhall have Ihe logarilhms

ofa11 the numben between 980 and 1020; and all betweeo

;80 alid 1000, being agaio inlerpolated by len inlervals,

the table will be as il were confiruéled. Then from Ihefe

we are

&tt Ihe logarithms of all the prime numbm,

aod tbeir muhiples lefs Ihan

lOO,

which may be done by

addirion'

and '

fubuaélioo ooly: for

84X1020_2 ;

994S

8X9963_ .

.v'9L

99_ •

~,_

•

-----3, -S,

---7,--11, X

-13,

984

102_ 17 ,

88 -19' 993 6 -23' 9 86 -29' 992

· '4XI3-

'~x2¡-

'2Xq-

=3 t • 999=0.7 • 9 8 4=4 1 • 9 8 9=43 • 9 8 7=47'

, 27

'24 '23

':t '

J1.

I!...=n' 997 1 =59' 9 88 '=61' .9949=67'

l/X17

'13XI3

'2X81

'3X49

W=7

1 '

=73' ..mJ=79' 99

=83'

'8XI 7

'7Xt8

'12 ' 7X16

=89; 9

94 = 97; and thos having tbe logarithms

6XJ7

of all the numbers IcCs than 100, you have nOlhing 10

do but

interpol~le

the fevcral times, Ibrough ten inter.

vals.

Now the void places may be

(¡ll~d

up by the following

theorem. Let

be a number, whofe 10garithOl is waOl'

td; let

be the difference be,llVeen that and the tillO

nemn numbm. cqually diOant on each fide, whofe lo·

;¡qritbOls are already found; and lel

be half the diffcr.

L O G

eoce of th:ir logarithms: then the req\tired log.,ilhm

,dx dxl

o t le oum er

wdl be had by addlog

d+:-

+-_,

2r.

2IJ

&c.

to tbe logarithm of the lel\'tr nu",ber ; for if the

numbers

ar~

reprefenled by

Cp,

~G,

CP,

(íbid.

2.)

and the ordlnates

PI,

be ralfed; if

be wrote for

X¡

CG, and

for GP, or

Gp,

the area

p/()P

or -

n 2n·

3n" tic.

will be to tbe area

p/HG,

as the differeoce

belweeo the logarithms of lhe extreme numbers or

2Ó

to the

di~erence

between the logarithms of the ¡ell'er,

~nd

of the mlddle one; which, therefore, will be

dx óx'

d,-l

-;; +;;

+t;,

tI~/!. +~,

tic.

J2nl

-;; 3

~,tlr.

.S"

The IWO 6rfi terms

of this feries, being fuf!i.

•

cieot for the confiruélion ,of a canon of logarithms, eveo

pl~ces

of figures, proyided the number, whofe 10-.

garnhm

to be 'found, be lefs tban 1000; which caonot

very troublefome, becaufe

is eilher 1 or 2: yel it ii

not necelfary to

iot~rpolate

all the places by help of this

rule, fince the logarithms of

~umbers,

which are produ.

ced by tbe multiplicanoo or divilion of the

numb~r

Iafr

found, may be obtaioed by tbe oumbers whofe logarithms

were had before, by tbe addition or fubtraélion of

tb~ir

I?garithms. Moreover, by Ihe difference of their loga.

ruhms, and by tbeir fec'ond aod thírd differeoces; if

cell'ary, the void ,place! may,

fupplird more

exp~dili.

ouay; tbe rule afore.golDg being to be applied ooly

wh~rC'

the continuatioo of

fo~

fuU places is \IIaoted, 'in

ord~

to obtaio thefe differences.

By the fame method rules may be found for tbe iQler·,

calation of logarithms, when of three oumbers Ihe loga·

rilbm of the lelJ'er and of tbe middle oumber are giyen,

or of the middle number and the' greater; aod tbis

al·

though the oumbm fhouJd nOI be in arithmetical pro·

greffioo. Alfo by purfuing the fieps of this method,

rules may be eafily- difcoyere<! for the confiruélion of aro

ti6cial fines and tangen", without ¡he help of the natural

tables. Thus far the great Newton. who fays. in one.

of his letters to Mr Leibnilz, thal he was fo mucb de·

lighted with the conflruélion of logarithms, at his 6r(1

fetting OUt in tbofe fiudies, that he IVas afhamed to tell

10 how maoy places of figures

had carried them althat

time: and this was

b~fore

the ym t666; becaufe, he

fays, the plague made bim lay afide tbofe fiudies, aod

tbink of other things,

Dr. Keil, io his

Tr~atife

of Log.,ithms, at the end

of his Commandine's Euclid, givcs a

f~ries.

by mean' of

which may be found eafily and expcditiounJ the loga·

rithms of

hrg~

numbers. Thus, let z be an odd numbtr,

wh.ofe logarithm is fought: then

nl~ll

thc numbers Z-t

aud

z+t

be even, and accordinglytheir log3rnhms, and

thedifference of

th~

10g.Hithms will be had. ",hich let be

caBed

Thereforc, alfo tbe logarithm of a nUOlher,

which is a

s~ometric.1

mean bctween

Z- I

and

will