Part IV. 3. That Ordinate which paffes thro' the Focus of the Hyperbola is call'd Latus Rectum, or Right Parameter, for the fame Reason as in the other Sections.

4. The middle Point of the Tranfverfe Diameter is call'd the Center of the Hyperbola; from whence may be drawn two Right Lines (out of the Section) call'd Afymptotes, because they will always incline (that is, come nearer and nearer) to both Sides of the Hyperbola, but never meet with (or touch) them, altho' both they and the Sides of the Hyperbola were infinitely extended; as will plainly appear in its proper Place.

These five Sections, viz. the Triangle, Circle, Ellipfis, Parabola, and Hyperbola, are all the Planes that can poffibly be produced from a Cone; but of them, the three last are only called Conick Sections, both by the ancient and modern Geometers.

Scholium.

Befides the 'foregoing Definitions, it may not be amifs to add, by way of Obfervation, how one Section may (or rather doth) change or degenerate into another.

An Ellipfis being that Plane of any Section of the Cone which is between the Circle and Parabola, 'twill be eafy to conceive that there may be great Variety of Ellipfes produced from the fame Cone; and when the Section comes to be exactly parallel to one Side of the Cone, then doth the Ellipfis change or degenerate into a Parabola. Now a Parabola, being that Section whose Plane is always exactly parallel to the Side of the Cone, cannot vary, as the Ellipfis may; for fo foon as ever it begins to move out of that Pofition, (viz. from being parallel to the Cone's Side) it degenerates either into an Ellipfis, or into an Hyperbola: That is, if the Section inclines towards the Plane of the Cone's Bafe,. it becomes an Ellipfis; but if it incline towards the Cone's Vertex, it becomes an Hyperbola, which is the Plane of any Section that falls between the Parabola and the Triangle. And therefore there may be as many Varieties of Hyperbola's produced from one and the fame Cone, as there may be Ellipfes.

To be brief, a Circle may change into an Ellipfis, the Ellipfis into a Parabola, the Parabola into an Hyperbola, and the Hyperbola into a plane fifceles Triangle: And the Center of the Circle, which is its Focus or burning Point, doth, as it were, part or divide itself into two Foci fo foon as ever the Circle begins to degenerate into an Ellipfis; but when the Ellipfis changes into a Parabola, one End of it flies open, and one of its Foci vanishes,

nishes, and the remaining Focus goes along with the Parabola when it degenerates into an Hyperbola: And when the Hyperbola degenerates into a plane Ifofceles Triangle, this Focus becomes the vertical Point of the Triangle (viz. the Vertex of the Cone); so that the Center of the Cone's Base may be truly faid to pafs gradually thro' all the Sections, until it arrives at the Vertex of the Cone, ftill carrying its Latus Rectum along with it: For the Diameter of a Circle being that Right Line which paffes thro' its Center or Focus, and by which all other Right Lines drawn within the Circle are regulated and valued, may (I prefume) be properly called the Circle's Latus Rectum: And altho' it lofes the Name of Diameter when the Circle degenerates into an Ellipfis, yet it retains the Name of Latus Rectum, with its firft Properties, in all the Sections, gradually fhortening as the Focus carries it along from one Section to another, until at laft it and the Focus become co-incident, and terminate in the Vertex of the Cone..

I have been more particular and fuller in thefe Definitions than is ufual in Books of this Subject, which I hope is no Fault, but will prove of Use, especially to a Learner: And altho' they may perhaps feem a little strange, and at first hard to be understood, yet, when they are well confider'd, and compar'd with a Cone cut into fuch Sections as have been defined, they will not only be found true, but will also help to form a true and clear Idea of each Section.

CHA P. II.

Concerning the Chief Properties of an Ellipfis.

NOTE, If the tranfverfe Diameter of an Ellipfis, as TS in the following Figure, be interfected or divided into any two Parts by an Ordinate rightly apply'd, as at the Points A, C, a, &c. then are thofe Parts TA, TC, Ta, and SA, SC, Sa, &c. ufually called Abfciffe (which fignifics Lines or Parts cut off) and by the Rectan gle of any two Abfciffe is meant the Rectangle of fuch two Parts as, being added together, will be equal to the Tranfverfe Diameter.

As TA+SA=TS. And TC + SC=TS.

Or TA+SA=TS, &c.

Section

Section 1.

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

Theozem.

As the Rectangle of any two Abfciffe: is to the Square of Half the Ordinate which divides them :: fo is the Rectangle of any other two Abfciffe: to the Square of Half that Ordinate which divides them.

V

Ꮴ

Let the annexed Figure represent a Right Cone, cut thro' both Sides by the Right Line TS; then will the Plane of that Section be an Ellipfis (by Sect. 3. Chap. 1.) TS will be the Tranfverfe Diameter, NCN and bab will be Ordinates rightly apply'd; as before. Again, if the Lines Dd and Kk be parallel to the Cone's Bafe, they will be Diameters of Circles (by Sect. 2. Chap. 1.) Then will TaD be alike. Alfo,

ASCk will be alike.

TCK and
Sad and

Ergo Sa ad SC C kl
And 2 TC: CK:: Ta: aDS

K

S

D

aDS per Theorem 13.

I3 S ax Cka dx SC 2.4 Tax CK=TCxaD

2 x 35 Sax Ckx Tax CK adxSCx TCxa D. Per Axiom 3. But 6 C K x C k = □ NC |

And

7a Dxadba per Lemma Sect. 2. Then for CK x Ck, and a D x ad, take NC and

ba

5, 6, 78 Sax Tax NC TC SC x □ ba Per Axiom 5. SC×□ Hence 9 Sax Taba :: TC x SC: □ NC. See Page 194.

Q. E. D.

Or

Or, the Truth of thefe Proportions may be otherwife prov'd by a Circle, without the Help of the Cone, thus: Let any Ellipfis be circumfcrib'd and infcrib'd with Circles, as in the following Figure; then from any Point in the circumfcrib'd Circle's Periphery, as at B, draw the Right Line Ba, parallel to the femiconjugate Diameter NC, then will ba be a Semi-ordinate rightly apply'd to the tranfverfe Diameter TS, as before. Again, from the Point b (in the Ellipfis's Periphery) draw the Right Line bd parallel to the Tranfverfe TS; and draw the Radius BC. Then will ABC a and ▲ Cfd be alike.

And fo for any other Abfciffa and their Semi-ordinates.

a

Thefe Proportions being found to be the true and common Properties of every Ellipfis, all that is farther requir'd in (or about) that Section may be eafily deduced from them.

Sect. 2. To find the Latus Rectum, or Right Parameter of any Ellipfis.

There are several Ways of finding the Latus Rectum, but I think none fo eafy, and fhews it fo plainly to be the Third principal Line in the Ellipfis, as the following.

Theozein. {

As the Tranfverfe Diameter: is in Proportion to the Conjugate::fo is the Conjugate: to the Latus

Rectum.

Viz. (in the following Fig.) TSNn:: Nn: LR the Ratus Rectum.

Demonstration.

From the laft Proportions take either of the Antecedents, and its Confequent, viz. either TCx SC: NC, or Tax Sa: Bbb

ba,

and

and make TS the third Term, to which find a fourth Proportional, and it will be = LR:

Thus ITCXSC:□NC::TS: LR
STC=SC

But 2 and NC=Cn

Therefore 3 TCX SCOTS
And 4ONC÷□ N n
1, 3, 4 50TS:□Nn::TS: LR
560TSXLR=NnxTS

6 x 4 7 TSXLR=NnxTS
STS XLR=□ Nn

7TS 8

N

L

T

F

f

R

n

which gives the following Analogy

viz. 9TS: Nn:: Nn: LR

Again, 10 TCx SC:

NC:: Tax Sa: □ ba

by common Properties.

1, 1011TS: LR:: Tax Sa: Saba.

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 Rectum. Q. E. D.

Confectary.

Hence it follows, that if the tranfverfe and conjugate Diameters of any Ellipfis are given (either in Lines or Numbers) the Latus Rectum may be eafily found; and then any Ordinate, whose Distance from the Conjugate is given, may be found; as above.

Sect. 3. To find the Focus of any Ellipfis.

The Focus is the Distance of the Latus Rectum from the Conjugate or Middle of the Ellipfis (vide Definition 4, Page 364.) and that Distance is always a Mean Proportional between the half Sum and half Difference of the tranfverfe and conjugate Diameters, which gives this Theorem.

From the Square of half the Transverse fubtract the Square of half the Conjugate, the fquare Root of Theorem. their Difference will be the Distance of each Focus -from the Middle or common Center of the Ellipfis.

That is, fuppofing the Points f and F to be the two Feci, viz. fC CF, and TCTŠ. NC = ÷ Nn. Then, TC + NC:ƒC:: FC: TC-NC. Ergo FC = □TCNC. Confequently, FC=DTC-ONG.

Demon