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e o
N
e
S E
e
T
o
N S.
eHipfe
an~
llyperboh ¡ it bifeéls the angle eontained by
the lines dra\Yo tOthe foei.
In
,11
the fcélions , lióes pmllcl to the taogent are
ordioms to the diameter pafling through the point
of
contaél j and
in
the ellipfe and hyperbola, the diameten
parallel to the tangent, and thofe paffing through the
points of eontáGl, are mmually eoojugate to eaeh other.
If
ao ordinate be dra\Yo from a point to a diameter and
a
langentfrom
lb..
fame point \Yhieh meetsthediameter pro·
dueeJ. j in the Jl3rabola tbe pan of thediamcler bel\Vixl
lhe ordinale aod langenl will be bifeéled in lhe vertex j
and in the ellipfc ano hyperbola, lhe femi·diameter \ViII
be a mean proponion belwixt the fegments of the dia·
meler bel\Yixl the eenter aad ordinate, and bellVixt the
centre anol angent.
The paralltl()gramformed
by
tangenls draIVn Ihrough
the venices of any eonjuga!e diamelers, in Ihe fame el·
lipfe or hyperbala., will be equal 10 eaeh olher.
Propertiu peC/lJiar
lo
tbe Hyperbok
As Ihe hyperbola has fome eurious propenies ariíing
from ils atrymplotes, whieh appear at firfl viewalmo!!
inctedible, \Ve lIiall bricfly demonnrote Ihem.
1 .
The hyperbola and its atrymplOles never meet : if
not, 'Iet lhemmeel in S, fig,
6.
j thenby Ihe properly of
lhe curve Ihe reélangle KXE is 10 SX q as GEQ 10
GM~
or EPQ ; Ihal is, as Gxq tOSX q j wherefore, KXE \ViII
be equal 10 Ihe fquare of GX ; bU! the reélangle
KXE, logelher with Ihe fquare of GE, is alfo equal to
tbe fquarc of GX j \Vhieh is abfurd.,
2:
If a line be draIVn lhrough a hyperbola pmllel lO
its Cecond axis, Ihe reélangle, by Ihe fegments of that
line, betwixI Ihe poinl in lhe hyperbola and lbe atrymp.
lotes, IVi ll be equal
10
the fquare of the feeond axis .
. For, if SZ, fig.
6,
be drawn perpendicular 10theCecond
axis,
by
lhe propertyof the curve, thefquare of MG, that
is, Ihe Cquare of PE, is 10 lhe fquare of GE, as lhe
fq uares of ZG and the fquare of MG logelher, 10 lhe
fquare of SZ or GX: and the fquares of RX and GX
are in Ihe falOe proporlion, beeaufe Ihe triangles RXG,
PEG are equiangular j Iherefore the fquares ZG and
MG are equal 10 Ihe fquare of RX j from \Vhich taking
lhe equal fq uam of SX and ZG, Ihere remains the
reélangle RSV, equal lO the fquare of MG .
3.
Henee, if righl lines be dra\Yn parallel 10Ihefeeond
axis, eutting an hyperbola and ils atrymptotes, Ihe reélan.
gles eontained helwixt the hyperbola and points where
Ihe lines CUt Ihe atrymptotes will be equal lO each o·
Iher j for Ihey are feverally equal 10 Ihe fquare of the
fecond axis.
4'
If
from any poinls,
d
and S, in ahyperbola, there
be dra\Vn lines pmllel tO Ihe atrymplotes
d.
SQ.andSb
de,
Ihe reélangle uoder
d.
and
de
\ViII be equal tO Ihe rcél.
angle under
~
and S
h
j alfo the parallelograms
da,
G
e,
and
S~ h,
\Vhich are equiangular, and confequenl'
Iy proponion.1 to Ihe reélanRles, are equal.
For draw YW KV paraJlel 10 the fccond axis, the
reélanglc Y
dW
is equal to the reélangle RSV j II'here·
fore,
\VD
is 10 SV as RS is to
d
Y.
BUl becaufe
the lriangles
R02,
AYO, and GSV
f
dW,
are
e–
quiangular, W
d
is to SV as
ed
10
S
h,
and RS
is to O
Y
as S<Z!o
d.
j \Vherefore,
de is
to
S
h
as
SQ..to
da :
and Ihe reélangle
J
e,
da,
is equal tO Ihe
rcélangle
Q2,
S
h.
5.
The aflymploles always approach nearer Ibehyper.
bola.
For, becaufe the reélangle under SQ..and SÓ, or
Q9,
is
e~ual
10 the reélangle under
da
and
de,
or
AG,
and
Qg
is greater than
a
G
j therefore
a
d
is grealer
Ihan~.
6.
The atrymplOtes come oearer Ihe hyperbola than
any affignable diflance.
Let X be any fmaJlline. Take anypoint, as
d,
in Ihe
hyperbola, and dra\V
da,
de,
pm lJel 10 Ihe alfymp–
loteS j and as X is to
d.,
fo lel
a
G be 10
GQ.
Draw
~
paralld tO
a
d,
meeting the hyperbola in
S,
Ihen
Q2
\ViII be equal 10 X. For the reélangle
S~
\ViII be
equal to Ihe reélangle
da G
j and CIlofequelitly
SQ.
is to
da
as
AG
to
G<Z:.
Ir any poinl be taken in Iht atrymptole be!ow
<t
it
can eafily be lhown Ihat its diflance is lefs Ihan lhe
Jine
X.
ArenJ eonlnincd by Coníe SeBioll!.
T HE area of a parabola is equal 10 T,lhe area of
a
circumfcribed paralellogram.
The area of an ellipfe is equal to Ihe area of a cirele
\Vhofe diameter is a mean proportional betwixI its great·
er and letrer axes.
If
t\Vo lines,
a
d
and
02,
be dralVn paraJle! to oneof
theatrymplotes of an
hyperbol~,
Ihe fpaee.
Qll
d,
bound·
ed by Ihefe paraJlel Iines, the atrymplOteS and the hy·
perbola \ViII he equal to the logarithm of
a
Q,
whofe
module is
Q
d,
fuppofing
a
G equal to unity.
CUY'Ualure
o[
wmie SemollJ.
THEcurvature of any cooie feélion, al Ihe Tenices of
its a>cis, is equal to the curvalure of a cirele whofedia·
meter is equal
10
lhe parameler of its axis.
If a tangent be dra\Yn from any other point of a conie
feélion, the curvature of the fcélion in that point will
be
equal 10 Ihe curvature of a cirele 10 "hieh Ihe fame line
is a
langr.nt, and \Vhieh CUlS off from the
~iameler
of
the (eélion. dmvn lhrough lhe poin!, a pan equal tO
ils parameler.
U[eJ
o[
Conie SefliollJ.
A~v
body, projeéled from Ihe furfaee
oC
Ihe
e~rt~,
defcnbes a parabola, 10 \Vhich the di reélion wherelll tt
~s
projeélcd is a tangent; and the difiance of lhe direéllix
IS
:qual to the heighl from \Vhich a body mul! fall
10
ac–
qutre the velocilY \Vherewith it is proje<'led: hen.e the
propenies of Ihc.parabola are Ihe foundation of gunnery.
AII bodies aélcd on by a central force, "hich decrca·
fts as ¡he fquare of the dinances inereafes, and impref.
fed \Vith any projcélile mOl ion, making any angle wit.h
Ihe direuion of the central force, mul! dcfcrihe COl1le
fcéli ons, having Ihe ecotral force in one of the foei,
and
will