1000000030

e o

part of a tangent paralle! to ir, and inrereepted berwixt

Ihe alTymptotes, is ealled a feeond diamerer.

Ao ordinate to any feélion is a line

biC~éled

by a dia–

meter and the abCeilTa, the pan of rhe diameter cut off'

by the ordinare.

Coojugare diamertrS in the ellipfe and hyperbola are

fuch as mutually bifeél lines parallel to the other ; and

a third proportional to two eonjugate diameten is ealled

the latus reélum of that diameter, whieh is the firfl in

the proportioo.

In the parabola, the lines drawn from any point to

the focus are equal to perpendieulars to the direélrix;

being both equa! to the par! of the thread Ceparated

from the ruler.

In the ellipfe, the IWO lines drawn from any point in

Ihecurve to the foei are equal to eaehother,

bei~g

equal

Ihe leogth of the thread; they are alCo equal to the

traoCverCe axis. lo the hyperbola me diffcrenee of the

lines drawn from any point to the foei is equal, being e–

qual to the difl'erenee of the lengths of thc ruler and

thread, and is equal to the tranfverfe axis.

From thefe fundamental properties all the

ot~ers

are

derived.

The ellipfe return. ioto itfelr. The parabola and

hyperbola may be extended wilhout Dmit.

Every line perpeodieular to the direélrix of a parabola

meet! it inone point, and faUs afterwards within it; and

every line drawa from the focus meets it ia onc poinr,

and falls afterwards without it. And every line that paf–

fes through aparabola, not perpendicular tO the direélrix,

will meet it again, but oaly once.

. Emy lioe pafling through the eenter of 3n ellipfe i.

blCeéled by it; the tranf,erfe axis is the greatell of all

theCe lioe.; the lelTer axis lhe lea(!; aod thefe nearer

the tranfverfe axis greater than thofe more

remot~.

. I? the hyperbola, everyline pafliog through theceoter

1I blfeéled by the

oppo~te

hyperbola, and the traofverfe

axis i. the leall of

, 11

thefe lines; alfo the feeond axis i.

the lea(!ofall the feeond diameters. Everylinedrawn from

the eenter within the aogle eontain.d by the alTymptotet,

mem it once, and falls aftcrwards within it; and every

lioe drawo through thc eeotcr IVilhoUI that angle never

mm. it; and a line which CUlS one of the alTymptoles,

aod CUll the otherextcnded beyond lhe eenter, will meet

bOlh the oppofite h)'perbolas in ooe POiOL

a lioe G M, lig. 4. be drawll from

~'poinl

in a pa–

rlbola perpendicular

the axis, it IVill be an ordinate

thelKil, and il! fquarc will be

equ~1

to lhe reélangle IIn–

~er

lhe abfeilT. MI and latus reélllm; for, uwufe GMC

1I a right augle,

Mq i, equal tO lhcdlfrm nee of (;cq

¡nd CMq; but GCis equal lo CE, whichis eqllal lo

~lfl;

t~erefore

GMqis equal to BMq-ClvIq; whieh, hwufe

e land lBare"111.1, is

F.II

( .

2.)

egu.11

louf liu"s 1"<

rcllangle under MI antl lB, or equal to lhe rc.'langle

Under MI anJ lhe

/,rtUI

"ffum.

H~nce

it folle,IVs, lh.u if difrerent ordinatr; b.

clr~·.\·n

lh. axis, lheir fquates

bci.:~

c.eh

egu,.1

th; reél·

angle under tht abfell!:1 Jnd 1.lllISreélulIl. "ill be IJ clch

tlher in Ihe proponi"n uf lhe

abfe.rr"

, wJoich is 1 he

f'me prtoptrlY 's

W;¡S

fllelVn ud"r. lu lakc 1".lcc in Ihe

T 1

parabala cut from the cone,

be the fame.

and prom tboCe curves tO

This property i, extended alfo to the ordinates of o–

ther diameterl. whofe fquares are equa! to the reélangle

under the abfcilTas and pm.meters of their refpeélive

diameters.

In the ellipfe, the f'iuare of the ordiaate is to the reél–

angle under the Cegments of the diameter, as the fquare

of the diameter paraUeI to the ordinate tO the Cquare of

the diameter to whieh it is drawn, or as the 6r(! diame–

ter to its latus reaum; that is, LKq(6g. 5.) is to EKF

as EFq to GHq.

In the hyperbola, the Cquare of the ordinate is tOo

the reélangle eontained under me Cegmentl of the dia–

meters betwixt itl vertices,

thefquare of the diameter

parallel tO the ordinate to the fquare of tbe diameter to

which it is dralVn, or as the firll diameler to irs

{alUI

,dluln ;

that is,

q il to EXK as MNq toKEq.

Or if an ordinate be drawn to a feeond diameter, irl

fquare will be to the fum of the fquares of the feeond

diammr, and of the line intereepted betwixt the ordinate

and centre, io the fameproponion ; that is, RZq (fig. 6.)

is to ZGq added to GMq, as KEq to MNq. Thefe are

the mo(! important properties of lhe eonic feaions : ano,

by mean. of thefe, it is demonllrated, that the figures

are the fame deCeribed on a plane as cut from the eone;

whieh we have demonllrated in the eafe of the parabola.

EqllationJ of

th~

ConicSe[fion¡

Au derived from the above propenie.. The equa–

tion of any curve, is an algebraie expreflion, which de–

notes lhe relation betwixt the ordinate and abfeilTa; the

abfeilTa being equal to

.nd the ordinate equal

be lhe parameter of a parabola, theo

px;

which is an equalion for all parabolas.

be the diameter of an ellipfe,

its parameter :

then

ax- xx

:: ; :

and

=.!.. X-;;;=;; · an

equation for all ellipfes.

be a tranCeverfe diameter of a hyperbola, , it:

parameter; Ihen

J '

a x

fu ::

aod

J '

'!" Xa.+xx.

be a feconGdiamcter of no hypcrbola, then

o, f u::

andy' =.t

Xaa f xx ;

whic;, a;c e-

quatioos for all hypabolas.

As all lhefe CIIII.I ions arc

exprelT~d

hy tite

f~(Grc\,

pOIVcrs uf "

~n,t

,dIconie feal,,", ¿re curves of

lh~

fecelnd order; and cOOllddy, lhe lo,os uf erer)' qC'I'

drarlc

equdti(ll1

is a

wl1i~ 1~~~i()O,

ar.rl

is a

p.l1 ;.b· .. ,

I:l,

'd–

l'f re, or hypnb"!3, >.ccording as lhe fo'

Ir.

~~' lhe

eOI;.'–

tlon cOllelponds IVuh rhe .boY< ones, 01 Wilh I:,m.

0:11<1

d,duct<! flIJm

lir.tl

(b~n

ill a Jiffacll l llllnnrl Wlt], le–

fl'~él

tOlhe f,[:illn.

Cm.'ml

C!,U

¡ ..

COIIÍ;

Seti\JnJ.

T ANGENl' 10"

p".lhola bifcéls lhe anR'e cont:.in .

e.1

h)'

lhl' linls J\.I\II¡ 10 lhe fuc\I: ¡¡nd dircClnx ; ii. .:::

~'I;r:,