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2. The longest Diameter (as TS) is call'd the Tranfverfe Diameter, or Tranfverfe Axis, being that Right Line which is drawn thro' the Middle of the Ellipfis, and doth fhew or limit its Length.

3. The fhorteft Diameter, call'd the Conjugate Diameter, is a Right Line that doth interfect or cross the Tranfverfe Diameter at Right Angles, in the H Middle or common Center of the Ellipfis (as Nn) and doth limit the Ellipfis's Breadth.

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4. The two Points, which I call particular Centers of an Ellipfis, (for a Reafon which shall be fhew'd farther on) are two Points in the Tranfverfe Diameter, at an equal Distance each Way from the Conjugate Diameter, and are ufually call'd Nodes, Foci, or burning Points.

5. All Right Lines within the Ellipfis that are parallel to one another, and can be divided into two equal Parts, are call'd Droi nates with Refpect to that Diameter which divides them: And if they are parallel to the Conjugate, viz. at Right Angles with the Tranfverfe Diameter, then they are call'd Ordinates rightly apply'd. And thofe two that pafs through the Faci are remarkable above the rest, which, being equal and fituated alike, are call'd both by one Name, viz. Latus Rectum, or Right Parameter, by which all the other Ordinates are regulated and valued; as will appear farther on.

Sect. 4.

If any Cone be cut into two Parts by a Right-line parallel to one of its Sides (as SA in the following Scheme) the Plain of that Section (viz. Sb BABbS) is call'd a Parabola.

1. A Right Line being drawn thro' the Middle of any Parabola (as SA) is call'd its Axis, or intercepted Diameter.

2. All Right Lines that interfect or cut the Axis at Right-Angles (as B B and bb are fuppofed to cut or cross S A) are call'd Ordinates rightly apply'd (as in the Ellipfis) and the greatest Ordinate, as BB, which limits the Length of the Parabola's Axis (SA) is ufually call'd the Bafe of the Parabola.

3. That

3. That Ordinate which paffes thro the Focus, or burning Point of the Parabola, is call'd the Latus Rectum, or Right Parameter (as in the Ellipfis) because by it all the other Ordinates are proportioned, and may be found.

4. The Node, Focus, or burning Point of the Parabola, is a Point in its Axis, (but not a Center, as in the Ellipfis) diftant from the Vertex, or Top of the Section, (viz. from S) juft part H of the Latus Rectum; as fhall be fhewn

farther on.

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5- All Right Lines drawn within a Parabola parallel to its Axis are call'd Diameters; and every Right Line, that any of those Diameters doth bisect or cut into two equal Parts, is faid to be an Ordinate to the Diameter which bifects it.

Sect. 5.

If a Cone be any where cut by a Right Line, either parallel to its Axis, (as SA, or otherwife as x N) fo as the cutting Line be

ing continued thro' one Side of the Cone (as at S or x) will meet with the other Side of the Cone if it be continued or produced beyond the Vertex V, as at T; then the Plain of that Section (viz. the Figure Sb B BbS) is call'd an Hyperbola.

1. A Right Line being drawn thro' the Middle of any Hyperbola, viz. within the Section, (as S A, or x N) is call'd the Axis or intercepted Diameter (as in the Parabola) and that Part of it which is continued or produced out of the Section, until it meet with the other Side of the Cone continued, viz. TS or Tx, &c. is call'd the Tranf

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verfe Diameter, or Tranfverfe Axis of the Hyperbola.

2. All Right Lines that are drawn within an Hyperbola, at Right Angles to its Axis, are call'd Ordinates rightly apply'd'; as in the Ellipfis and Parabola,

3. That

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.

Thefe five Sections, viz. the Triangle, Circle, Ellipfis, Parabola, and Hyperbola, are all the Plains that can poffibly be produced from a Cone; but of them, the three laft 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 Plain 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 Ellipfis 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 Plain 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 Plain 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 Plain 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 plain Ifofceles 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 plain Ifofceles Triangle, this Focus becomes the vertical Point of the Triangle (vix. the Vertex of the Cone); fo that the Center of the Cone's Base may be truly said to pass 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 presume) 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 these Definitions than is ufual in Books of this Subject, which I hope is no Fault, but will prove of Ufe, especially to a Learner: And altho' they may perhaps feem a little ftrange, and at firft 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.

N

CHA P. II.

Concerning the Chief Properties of an Ellipfis.

OTE, 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 applv'd, as at the Points A, C, a, &c. then are thofe Parts TA, TC, Ta, and S A, SC, Sa, &c. ufually called Abfciffe (which fignifies 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+SATS. And TC+SC=TS.
Or TASA=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.

Theorem.

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.

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Let the annexed Figure represent a Right Cone, cut thro' both

Sides by the Right Line TS; then will the Plain of that Section be an Ellipfis (by Sect. 3. Chap. 1.) TS will be the Tranverfe Diameter, NC N and b a b will be Ordinates rightly apply'd; as before Again, if the Lines D d and K k be parallel to the Cone's Bafe, they will be Diameters of Circles (by Sect. 2. Chap. 1.) Then will ATCK and Ta D be alike. Alfo, AS a d and ASCk will be alike.

Ergo Saad: SC: Ck2

K

V

S

D

And 2 TC: CK:: Ta: aD per Theorem 13.

aDS

I • 3 SaxCk = a dx SC

2

... 4 Tax CK=TCXaD

2 X 35 SaxCkxTaxCK=ad×SC×TCxaD. Per Axiom 3.

But 6 C KX C k = □ N C

And 7a a DX ad=ba per Lemma Sect. 2.

Then for CKX Ck, and a D X ad, take □ NC and
5,6,7 8 Sax Tax□ NC = TCX SC X
Hence 9 Sax Taba:: TCX SC:

ba

ba Per Axiom 5. NC. See Page 194.

Q. E. D.

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