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4. The two Points, which I call particular Centers of an Ellip. Jis,'(for a Reason which Mall be shew'd farther on) are two Points in the Transverse Diameter, at an equal Distance each Way from the Conjugate Diameter, and are usually callid Rodes, Foci, or burning Points.
5. All Right Lines within the Ellipsis that are parallel to one another, and can be divided into two equal Parts, are callid Didi, nates with Respect to that Diameter which divides them: And if they are parallel to the Conjugate, viz. at Right Angles with the Transverse Diameter, then they are call’d Ordinates rightly apply'd. And those two that pass through the Foci are remarkable above the rest, which, being equal and situated alike, are callid both by one Name, viz. Latus Kedum, 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 B A B bS) is call'd a Parabola.
1. A Right Line being drawn thro? the Middle of any Parabola (as S A) is call'd its Axis, or intercepted Diameter.
2. All Right Lines that interfect or cut the Axis at Right-Angles fas B B and b b are supposed to cut or cross S A) are callid Ordinates rightly apply'd (as in the Ellipsis) and the greatest Ordinate, as B B, which limits the Length of the Parabola's Axis (SA) is ufually call’d the Base of the Parabola.
3. That 3. That Ordinate which passes thro? the Focus, or burning Point of the Parabola, is call'd the Latus Reétum, or Right Parameter (as in the Ellipfis) because hy it aļl the other Ordinates are proportioned, and may be found.
4. The Nodę, Focus, or burning Point of the Parabola, is a Point in its Axis, (but not a Center, as in the Ellipfis) distant from the Vertex, or Top of the Section, (viz. from S) juft i part H of the Latus Rectum; as shall be thewn farther on.
B 5. All Right Lines drawn within a Parabola parallel to its Axis are callid Diameters; and every Right Line, that any of those Dia. meters doth bisect or cut into two equal Parts, is laid to be an Or. dinate to the Diameter which bisects it.
the Mithe sections or intet and the
If a Cone be any where cut by a Right Line, either parallel to its Axis, (as S A, or otherwise as x N) fo as the cutting Line being 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 B bS) is calls an Dyperbola.
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 7x, &c. is calld the Transverse Diameter, or Transverse Axis of the Hyperbola.
2. All Right Lines that are drawn within an Hyperbola, at Right Angles to its Axis, are callid Ordinates rightly apply'd; as in the Ellipsis and Parabola,
3. That Ordinate which passes thro' the Focus of ihe Hyperbola is calls Latus Re&tum, or Right Parameter, for the same Reason as in the other Sections.
4. The middle Point of the Tranfuerfe Diameter is call'd the Center of the Hyperbola ; from whence may be drawn two Right Lines (out of the Section) calld asymptotes, 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, Parabela, and Hyperbola, are all the Plains 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.
Besides the 'foregoing Definitions, it may not be amiss to add, by Way of Observation, how one Section may (or rather doth) change or degenerace into anocher,
An Ellipsis being that Plain of any Section of the Cone which is between the Circle and Parabola, 'cwill be easy to conceive that there may be great Variety of Ellipfis produced from the same Cone; and when the Section comes to be exactly parallel to one Side of the Cone, then doth the Ellipsis 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 Ellipsis may ; for so soon as ever it begins to move out of that Pofition, (viz. from being parallel to the Cone's Side) it degenerates either into an Ellipsis, or into an Hyperbola : That is, if the Section inclines towards the Plain of the Cone's Base, 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 Ellipses.
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 Tosceles 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 so soon as ever the Circle begins to degenerate into an Ellipsis; but when the Ellipfis changes into a Parabola, one End of it Aies open, and one of its Foci va
nishes, and the remaining Focus goes along with the Parabola when it degenerates into an Hyperbola : And when the Hyperbola degenesates into a plain Tosceles Triangle, this Focus becomes the vertical Point of the Triangle (vix, the Vertex of the Cone); so 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, still carrying its Latus Rectum along with it: For the Diameter of a Circle being that Right Line which passes 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 Cir. cle's Latus Rectum : And altho' it loses the Name of Diameter when the Circle degenerates into an Ellipfis, yet it retains the Name of Latus Rectum, with its first Properties, in all the Sections, gradually shortening as the Focus carries it along from one Section to another, until at last it and the Focus become co-inci. dent, and terminate in the Vertex of the Cone.
I have been more particular and fuller in these Definitions than is usual 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 seem a little strange, and at first hard to be understood, yet, when they are well consider'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.
СНА Р. ІІ.
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 Abscissæ: is to the Square
Jof Half the Ordinate which divides them :: fo is the Theorem.
3 Rectangle of any other two Abscissæ : to the Square Lof Half that Ordinate which divides them.
Demonftration. Let the annexed Figure represent a Right Coné, cut thro' both Sides by the Right Line TS; then will the Plain of that Section be an Ellipsis (by Sect. 3. Chap. 1.) IS will be the Tranverse Diameter, NC N and b a b will be Ordinates rightly apply'd; as before Again, if the Lines D dand K k be parallel to the Cone's Base, they will be Diameters of Circles (by Set. 2. Chap. 1.) Then will s TC K and Ta D be alike. Allo, A Sa d and ASC k will be alike.