1005 / 1042
1.
o
G
nla
f.1I:C
ll:c lI'.y auo\'e "inleJ al. for
fi,)Jin~
",e lo·
g"il!'OlS of Ihe prime nlln,bers is fo intolmblyIJborious
"od
lrollbl~lome.
lhe more O,ilful nmhcOlalicians lhal
CJlIle afler
,h~
r,r!l invenlors. employing Ih,ir lhoughlS
nbulll
a~bre\'ialiog
Ihis method. had a vaflly mere eafy
aad
OlOn way ofi'«ed 10 Ihem from lhe conlemplalion
nnd
m~nfuration
of hype.bolic fpaces conlained belween
the ronions of ao afymp,ole. righl lines perpendicular 10
ir, and Ihe cum of the hypcrbola: for if ECN (Plate
CIlI.
li~.
6. nO1.) be an hyperbola. and AD. AQ..lhe
afymrtores. and AB. AP.
A<z:.
&e.
lakeo upon one of
II,em,
he
reprefcnted by numbm. and lhe ordinales BC.
PM,
QtI.
6e.
be drawn from Ihe
f~veral
points B,
p.
~
&e.
10 Ihe eun'e. Ihen will lhe quadrilinear fpaces
BCMP.
PMN~
&e viz.
their numerical meafures be
the
lo~a rilhms
of the quolienl' of Ihe divifion of AB by
AP.
AP by AQ:.
be.
fince when AB. AP.
A~
&e.
are conlinual proponiooals. the faid fpaces are equal. as
is
d~monnraled
by feveralwriters concerning conie Jtaions.
Se
H
VPERBOLA.
Having faid that thefe hyperbolic fpa·ees. numerieally
exprefred. may be taken for logarithms. we fhall neXI
give a fpeeimen. from the &reat Sir Ifaae Newton. of lhe
melhod how 10 meafure thefe fpaees. and eonfequently
of the eonaruaion of logarithms.
Let CA
(ibid.
0 °
2.) =AF be =1. and
AB=Ab=t
i
theu will _tI be=BD. and
-~=pd;
and pu'lting
I X '
I-X
thefe expreffioos iOlo feriefes, il will be
l~x=l-xtX.'
......xltX·_xl.
be.
and _1_=ltxtx'txltx·txl.
be.
.
l-X
;¡od
2-~x¡+x';_;I;tx';-XI;,
be.
and _.-
I~
1-
==x:¡'x¡tx';txl;tx·~tx¡x·.
&c.
and taking the flu·
X"
xl ,,4
cnt!. we fhall have lhe ma AFDB =x--t--t
•
2
3 4
¡
u
~ ~
0
x
5
•
&c.
and the area
AFdh.
=t,t3'+:¡+S'
2X
l
oc.
and the fum
6dDB=2xt-tTx¡+~x'+fx9,
be.
. 3
Now if AB or
ah
be
~FX,
Cb
being=0.9. and CB=J.l,
by pUlling this value of xin the equalions abo
ve,
\Ve fhall
have lhe area
bd
DB=0.200670695462 151 I for lhe
ternu of lhe feries will lland as you fee in lhis table,
. 0.2000000000000000 = fira
l:;l .
6666666666666 = feeond
I
3
4óooooooooo
= lhird
o
2857 14286:founh
l' ..,
2222222 = fifth
;-
18182 =fixlh
~
,
154 = fcvenlh .
S.
1=eighlh
J
~
0.200('7069\46l 1511
lf
Ihe par15
A.I
and ADof Ihis arca be added feparale–
Iy. and the
Idr<r
DA be t.ken fromthe greater
d.~,
we
x'
x
6
x
8
nlall hav: i\J-A D:x'+ - t-t-.
6,.
=
2
3 4
VOL .
JI.
~¡umb.
67.
2
T.
O G
=0.0r0°5°33 j8 j350 14. for lhe lerms
rcdu~cd
to d:·
cim.lswdJ Jland Ihus :
om00000000000000 = fira
l;;l
500000000coo = fecond
I
3
3333lB3l3 = .lhird
?,
25°00000 = founh
~;.
200000 = fiflh
J
n
1667=fixlh
i
14=fmnth
~
0.010°503)585350 14·
Now if lhis diffcrenee of the arcas be added 10. and
fubrraéted from lheir fum btfore found. half lhe agre·
gale,
.iz.
0.1°53605156578263 \ViII be lhe greater am
Ad,
and half lheremaindcr••iz. 0.0953101798°4;249.
will be Ihe Iefrer area AD.
By lhe fame
labl~s,
lhefe areas AD and
Ad.
will be
oblained airo \Vhen AIl=.'\.b are fuppofed 10 be
, h
or
CB=I .Ol. and
Cb=O'99,
ir lhe ourr.bers are bUl duly
uansferred 10 10IVer plaees. as
r
erm
6666666666 = ,e
7
0nd of lhe
0.0200000000000000 = firn
~T
4°°000 = Ilmd
~ .
28 = founh enes.
Sum=0.D200006667066694 = arca
hD.
0.0001000000000000 = fira
~
Term
50000000 = feeond of lhe
3333 = lhird (eries.
------
0.0001000050003333 = area Ad-AD.
Half lhe agregale 0.0100503358 535014=Ad, Rnd
half lhe remaioder••iz.o.009950B0853168 1=AD.
And fo pUlling AB=Ab=m¡¡. or CB=I .OOI aod
Cb=
0'999. Iherewill be
oblainedAd:o.ooIOOO¡C003335835,
and AD=0.OC0999500t3310835.
Afler [he faOle maoner, if AB=Ab, be=O'2, or 0.02,
or O.OOl. Ihefe areas will arife.
Ad=0.2 ljI435513142097. and
AD=0. 1 8 2321557~939546.
or
Aá=0.0202027073175194. and
AD=o.109801627 2961797. or
Ad=0.o02002.
and AD=o,oo l.
From lhefe arcas thus found, olhm may be eafily had
.
.
fi
1.2 1.2
from addilloo and fubuaaron only. For IOce --8
X -
o. 0.9
=2, lhe fum of Ihe areas
b~longing
10 lhe ratios
!..:!
and
!..:!
(lhal is. infifling upon lhe
patlS
of lhe abo
0.8
0.9
feifs 1.2, 0.8; and 1.2. 0.9)
"izo
6 (;
d
5
1D =o 182)2.
6c.
0'4054 5,
' e.
an
I.
Ad
=0. 10 5Jó, 6,.
addeu lhus,
Sum=0.28768.
&e.
7.
0'4°546.
be.
S
0.28768,
&c.
TOlal=0693 14.
&r.
= l'le arca of AFHG.
t
10
l'
\I'hen