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If a Right Cone be cut directly thro’ its Axis, the Plain or Superficies of that Section will be a plain Tosceles Triangle
, as HVG Fig 2, viz. the Sides (H V and VG) of the Cone will be the Sides of the Triangle, the Diameter (HG) of the Cone's Bale will be the Base of the Triangle, and (VC) its Axis will be the perpendicular Height of the Triangle.
If a Right Cone be cut (any where) off by a Right-line parallel to its Base, as hg (it will be easy to conceive, that) the Plain of that Section will be a Circle, because the Cone's Base is fuch: wherein one thing ought to be clearly understood, which may be laid down as a Lemma, to demonstrate the Properties of the following Sections.
If any two Right Lines, infcribd within a Circle, do out or cross each other (as big doth bb in the annexed
Figure) the Rectangle made of the Segments of one Lemma.
of the Lines will be equal to the Ričiangle made of the Segments of the other Line. . (See Theorem 15. | Page 315.)
b O B A.
If a Right Cone be (any where) cut off by a Right Line that cuts both its Sides, but not parallel to its Base (as TS in the followirg Figure) the Plain of that Section will be an Ellipfis (ongarly celled an Oval) viz an oblong or imperfect Circle, which hath feveral Diameters, and two particular Centers. That is,
1. Any Right Line that divides an Ellipfis into two equal Parts is call'd a Diameter ; amongst which the longest and the fortes are particularly distinguish'd from the rest, as being of most general Ule; the other are only applicable to particular Cases.
2. The longeft Diameter (as TS) is call'd the Transverse Diameter, or Transverse Axis, being that Right Line which is drawn thro' the Middle of the Ellipsis, and doth shew or ļimit its Length
3. The shortest Diameter, call'd the Conjugate Diameter, is a Right Line that doth intersect or cross the Transverse Diameter at Right Angles, in the Middle or common Center of the Ellipfis (as N n) and doth limit the Ellipfis's Breadth.
4. The two Points, which I call particular Centers of an Ellisfis, (for a Reafon which shall be shew'd farther on) are two Points in the Transverse Diameter, at an equal Distance each Way from the Conjugaté Diameter, and are usually called fodes, Foci, or bura: ing Points.
5. All Right Lines within the Ellipsis that are parallel to one a. nother, and can be divided into two equal Parts, are call's Drdi 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 2. And those two that pass through the Foci are remarkable above the rest, which, being equal and situated alike, are call'd both by one Name, viz. Latus Redum, or Right Parameter, by which all the other Ordinates are regulaced and valued ; as will appear farther on.
If any Cone be cut into two parts by a Right-line parallel to one of its Sides (as S A in the following Scheme) the Plain of that' Section (viz. Sb B A B bS) is call’da 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 (ps B B and bb are supposed to cut or cross S A) are caña Ordinates rightly apply'd (as in the Ellipsis) and the greatest Ordinate, as BB, which limits the Length of the Parabola's Axis (SA) is u: sually call'd the Bake of the Parabola.
3. That Ordinate which passes thro'
V the Focus, or burning Point of the Pa. rabola, is call'd the Latus Rectum, or Right Parameter (as in the Ellipfis) because by ic 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) just part
G of the Latys Rectum; as shall be thewn farther on.
B 5. All Right Lines drawn within a Parabola parallel to its Axis are call's Diameters; and every Right Line, that any of those Diameters doth bisect or cut into two equal Parts, is laid to be an 06dinate to the Diameter which bisects it.
If a Cone be any where cut by a Right Line, either parallel to its Axis, (as S A, or otherwise as * N) so as the cutting Line being continued throʻ one side of the
T Cone" (as at S or *) will meet with the other side of the Cone if it be continued or produced beyond the Vertex
V 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'
S the Middle of any Hyperbola, viz. within the Section, (as S A, or < 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
H the other Side of the Cone continued,
N CA viz. TS or Tx, &c. is call’d 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, arc call d Ordinates rightly apply'd ; as in the Ellipsis and Parabola.
3. That Ordinate which passes thro' the focus of the Hyperbola is callid Latus Rectum, or Right Parameter, for the same Reason as in the other Sections.
4. The middle Point of the Transverse Diameter is call the Center of the Hyperbola ; from whence may be drawn two Right Lines (out of the Section) callid asymptotes, because they will alw .ss 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 Hyperbala were infinitely extended; as will plainly appear in its proper Place.
These five Sections, viz. the Triangle, Circle, Ellipfis, Parabolo, and Flyper bola, are all the Plains that can possibly 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 ratber doth) change or degenerate into another. - An Ellipfis being that Plain of any Section of the Cone which is between he Circle and Parabola, 'twill be easy to conceive that there may be great Variety of Ellipfis produced from the same
and when the Section comes to be exactly parallel to one Side of the Cone, then doth the Ellipsis change or degenerate inco 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 Hiperbola, 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 Ijosceles 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 Ellipsis changes into a Parabola, one End of it flies open, and one of its Foci vâ
nilhes, and the remaining Focus goes along with the Parabola when it degenerates into an Hyperbola : And when the Hyperbola degenerates into a plain loceles 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 Se&tions, until it arrives at the Vertex of the Cone, ftill carrying its Latus Re£tum 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 Circle's Latus Rectum : And altho' it loses the Name of Diameter when the Circle degenerates into an Ellipsis, yet it retains the Name of Latus Reflum, 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-incident, 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 confider'd, and compar'd with a Cone cut into such 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.
Concerning the Chief Properties of an ellipüs.
OTE, if the transverse Diameter of an Ellipfis, as I S in the
following Figure, be interfeited or divided into any two Paris by an Ordinate rightly apply'd, as at the Points A, C, a, '&c. then are those Parts TA TC, Ta, and S A, SC, Sa, &c. usually called Abscissæ (which signifies Lines or Parts cut off) and by the Rettangle of any two Abfcifíe is meant the Rectangle of such iwo Parts as, being added together, will be equal to the Transverse Diameter. As T'A+SA=TS. And TC+SC=TS. Or TA+SA=TS, &c.