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Section 1. Every Ellipfis is proportion'd, and all such Lines as relate to it are regulated by the Help of one general Theorem.

As the Rectangle of any two Absciffæ : is to the Square

of Half the Ordinate which divides them :: fo is the Theorem.

Rectangle of any other two Abscissä: to the Square of Half that Ordinate which divides them.

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B

n

Demonftration.
Let the annexed Figure represent a Right Cone, cut thro' both
Sides by the Right Line TS; then

V
will the Plain of that Section be an
Ellipfis (by Sect. 3. Ghap. 1.) TS
will be the Tranverse Diameter,
NC N and b a b will be Ordinates
rightly apply'd; as before Again,

DA
if the Lines D dand K k be parallel
to the Cone's Base, they will be

KA
Diameters of Circles (by Set. 2.
Chap. 1.) Then will CK and
Ta D be alike. Also, A Sad and
ASC k will be alike.

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I 2

Ergo 118 a:ad: :SC: Ck2
And 2 TC: CK:: 1a: aD per Theorem 13.

13 Sax Ck=adxSC

.-14Tax CK=TCXaD
2 X 315 SaxCkXTaxCK=adX SCXTCXaD. Per Axiom 3.

But 6 C KXGk=ONC
And 7 a DXadro ba{ per Lemma Seet. 2.
Then for CKxCk, and a D x ad, take o N Cand o ba
5,6,78 Sax Taxa NC=TC X SC Xaba Per Axiom 5.
Hence 9! Sax Taiba :: TC X SC: NC. See Page 1946

Q: E. D.

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Or, the Truth of these Proportions may be otherwise prov'd by a Circle, without the Help of the Cone, thus : Let any Ellipfis be circumscrib'd and inscrib’d with Circles, as in the following Figure ; then from any Point in the circumferib'd Circle's Periphery, as at B, draw the Right Line B a, parallel to the femiconjugate Diameter Nc, then will ba be a Semi-ordinate righıly apply'd to the transverse Diameter TS; as before. Again, from the Point b (in the Ellipfis's Periphery) draw the Right Line bd parallel to the Transverse TS; and draw the Radius BC. Then will A B C a and Cfd be alike.

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Therefore 1

BC:Ba:: Cf:dc
per Theorem 13.

N
TC=BC, NC=Cf

d.
But 2

and ba=dC
Confeq. 3

TC:Ba:: NC:ba T
Or 4.TC:NC :: Barba
4 in O's 5 ATC: NC:: 0 Ba: Oba

S Tax Saro Ba

=
But 6

?
per

Lem. Sect. 2. Ch. I.
Therefore 7

S

Tax Sa: (ba:: TC
RXSC= TG:CNC, as before,

a

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1

And so for any other Abscisa and their Semi ordinates, These Proportions being found to be the true and common Properties of every Ellipsis, all that is farther requir'd in (or about) that Section may be eally deduced from them. Sect, 2. To find the Latus Kedum, or Kight Parameter

of any Elli, fis. There are several Ways of finding the Latus Rectum, but I think none so easy, and thews it so plainly to be the Third principal Line in the Ellipsis, as the following.

As the Transverse Diameter: is in Proportion to . the Conjugate :: fo is the Conjugate : to the Latus

Rectum.
Viz. (in the following Fig.) TS : Nn:: Nn: LR the Latus ReElum.

Demontration.
From the last Proportions take either of the Antecedents, and,
its Consequent, viz, either TCXSC:ONC; or Tax Sa: Oba,

B bb

and

Theorem. {

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But 2

f

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and make TS the third Term, to which find a fourth Proportio-
nal, and it will be = LR:
Thus i TGXSC:O NC::TS:LR

N
STC=SC

L and NC=Cn Therefore

3

IC XSC=0 TS
And 410NC= Nn

T 업 1, 3, 4, 5/10 TSIO Nn::TS:LR

R 5:16: TSXLR=aNnXTS

n 6X4171 OTSXLR=NnXTS

TSXLRONn
7;TS 8{ which gives the following Analogy
viz. 9TS:Nn:: Nn: LR

STCXSC:ONC :: Ta X Sa: Oba
Again, 10

12 by common Properties. I,

10 u TS: LR:: TaxSa : Oba. From hence'tis evident that LR, thus found, is that Ordinate by which the other Ordinates may be regulated and found. Therefore (according to its Definition Sect 3. Chap. 1.) it is the true Latus Rellum. 0. E.D.

Confe&tary. Hence it follows, that if the transverse and conjugate Diameters of any Ellipsis are given (either in Lines or Numbers) the Latus Reclum may be easily found ; and then any Ordinate, whose Distance from the Conjugate is given, may be found; as above.

Seet. 3. To find the Focus of any Ellipsis. The Focus is the Distance of the Latus Reetum from the Conjugate or Middle of the Ellipsis (vide Definition 4, Page 364.) and that Distance is always a Mcan Proportional between the half Sum and half Difference of the transverse and conjugate Diameters, which gives this Theorem.

From the Square of half the Transverse subiralt beorem.

the Square of Half the Conjugate, the square Rost of their Difference will be the Distance of each Focus

from the Middle or common Center of the Ellipsis, That is, supposing the points f and F to be the two Foci, viz. fC=CF, and TC= IS.NC= Nn. Then, TC +NG:fC:: FC: TC--NC, Ergo FC=OTG ONG. Consequently, FC=VQT-ONC.

Demon

Demonttration.

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f

F

2

First, ITS XLR=Nn, by 8 Step of the last Proces

And 2 TSOLR::TFXSFLF, common Properties. That is, 3 TS: LR::TC+CFXTG-CF:: LR=LF SADLRX TS =

N
4
2 TC-CFXLR

[ 4;LR SELRXTS=aTC-CF 146 TSxL R=NnO=NG T 5, 617 0 NC=OTC_OCF 7 +180CF=TC-ONC

R 8 w? 9{CF=V OTC_ONG

n Now from hence is deduced that notable Propofition, upon which is grounded the usual Method of describing an Ellipsis, and drawing of Tangents, &c.

If from the two Foci of any Ellipfis there be Propofition.

drawn two Right Lines, so as to meet each other in any Point of the Ellipfis's Periphery, the Sum of

those' Lines will be equal to the Transverse. Viz. f NXNF=7$.fL+LF= is. Or fB+BF=TS,&c.

Demonftration.
First,
O CFX ONC=OTC

N
{ by 8th of the last.

L But, 2

SOCFTONC=ONF
by Theorem II.

TI
SONF=OTC

S I, 2 3) {

by Axiom 5. 3 we ? 4 NF ETC

R Hence 2 NF-2 TC=TS

B Again, 5 TS: LR:: TFXFS:0 LF, by common Properties. Conseq. 6 TS : į LR::TF X FS :Ó LF

But, ITS TC. And LR=LF

Ergo 7 TC: LF::TC + CF X TC CF: 0 LF 2

8 TC X LF = 0 TC But, 9 ofF+OLF = OFL, by Theorem 11. That is, 1014 Ó CF+OLF=of L, for 2 CF=fF 8 X 411 4 OTC-40CF= 4TC XLF 10+11 12 4 OTC+0 LF = 4TCXLFtOfL 12 134 O TC-4TCXLF+OLF=ofL

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OCF

B b b 2

13 ww2

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13 w?
W 14

2 TC-LF=fL
14+LF| 15 | 2 TC=fL+LF. But 2 TC=TS
Ergo TS=fL+LF.

Q. E. D.

And this Proposition must needs hold true to every Point in the Ellipsis's Periphery, viz. at B, &c. As will evidently appear to any one who rightly considers, That, as a Thread just the Length of the Diameter of any Circle having its two Ends tyd together, and then mov'd about a Poiot in the Center (viz. by making it a double Redius) will, by drawing another Point in its Extremity, describe the Periphery of a Circle ; (vide Definition Page 280) even so, if a Thread just the Length of the transverse Diameter (TS) having its two Ends so fix'd upon the two Foci (f and F) that it may be mov'd about them, by drawing a Point in its Extremity (viz. at its full Stretch) it will describe the true Periphery of an Ellipfis.

Now, altho' this easy Way of describing, or, as usually phras'd, drawing an Ellipsis, be mechanical, and known even to moft Joiners, Carpenters, &c. yet it gives as compleat and clear an Idea of that Figure as any other way whatsoever; and by describing it thus about its two Foci, as a Circle is about its Center, doth plainly sew that those two Points are not improperly call'd particular Centers in Definition 4, Sect. 3, Chap. 1, for each of them bears much the same Respect to the Ellipsis's Periphery, as the Circle's Center doch to its Periphery.

Sect. 4. To describe or delineate an Ellipsis several Ways. There are several (other) Ways of describing an Ellipsis, both Geometrically and Numerically, according to peculiar Occasions, but I shall only mention two or three of them, leaving the rest to the Learner's Genius. Now, in order to that Work, it will be convenient to consider what Lines are requifite to limit or bound its Form, which I take to be chiefly these following.

1. If the Transverse and Conjugate are given, the Ellipfis is perfectly limited ; (vide Conseetary Page 363 ) for if T S and Nn be fet at Right Angles in their Middle at C, and TC or CS be set off from N, orn, both Ways upon the Transverse to f and F, (viz. make f N = TC= NF) chen will those Points f and F be the two Foci (by 4th Step of the last Process) and then the Ellipsis may be describ'd as above.

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