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(5th.) If a Cone be cut by a Plane, as I K, any where parallel to its Axis V L, the Section so made will be an Hyperbola, as PIqK. The Line drawn down through the Middle of the Hyperbola, as I K, is called its Axis, or intercepted Diameter. The Part produced above the Section till it meet the other Side of the Cone continued, as S I, is called the Tranfverfe Diameter; and the Line drawn from the Vertex of the Cone V, perpendicular to the Tranfverse Diameter SI, to C, is called the Semi-conjugate Diameter of the Hyperbola.

All Lines drawn at Right Angles across the Axis are called Ordinates; and that which passes through its Focus (for every Hyperbola, as well as Parabola, hath one Focus or Burning Point near the Top of the Plane, as at f) is called the Latus Rectum, as in the Parabola, Ellipsis, and Circle.

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That Part of the Axis which is contained between the Vertex I, and the Ordinate, wherever drawn, is called the Alfcissa, as in the Parabola.

The Middle of the Tranfverse Axis or Diameter SI is called the Center, as at C, of the Hyberbola. From which Point may be drawn two Right Lines out of the Section, called Afjmtotes, because they will always approach nearer to the sides of the Hyberbola, yet would never touch them, though both they and the Sides of the Hyperbola were infinitely extended.

These are all the Sections that can be cut from a Cone; and from a due Observation of them, it is evidently seen how one Section degenerates into another. As the Circle into an Ellipsis; the Ellipsis into a Parabola; the Parabola into a Hyperbola; and the Hyperbola into a plain Triangle. And the Center of the Circle, which is its Focus, divides itself into two Foci so soon as the Circle begins to degenerate into an Ellipsis; but when the Ellipfis changes into a Parabola, one End of it flies open, one of its Foci vanishes, and the remaining Focus goes along with the Parabola when it degenerates into a Hyperbola. And when the Hyperbola degenerates into a plain Triangle, this Focus becomes the vertical Point of the Triangle, that is, the Vertex of the Cone. So that the Center of the Cone's Base may be truly faid to pass gradually through all the Sections until it arrive at the Vertex of the Cone, still carrying its Latus Rectum along with it, where both, the Focus and Latus Rectum, become coincident, and terminate in the Vertex of the Cone.

THE

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From the Square of the Transverse Diameter, fubtract the Square of the Conjugate; the Square Root of the Remainder will be the Distance of each Focus from the Center, or Middle of the Ellipfis.

Example.

Suppose the Transverse Diameter AB of an Ellipfis be 24 Inches, and the Conjugate CD be 18 Inches, what is the Distance of each Focus F from the Center at S?

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Operation. 144-81-63, whose Root is 7.938 Inches,

the Distance of each Focus from the Center S.

1

. Each of the dotted Lines drawn through the two Foci parallel to the Conjugate is called the Latus Rectum; and the Length may be found by this Proportion:-As the Transverse Diameter A B, is to the Conjugate CD; so is the Conjugate CD, to the Latus Rectum bd, or e f.

Problem 2.

To delineate an Ellipfis, having the Tranverse and Conjugate Diameter given.

Let A B be the longer Diameter, and CD the shorter, as in the following Figure.

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Two Right Lines drawn from any Point in the Curve of the Elliplis to the two Foci, are together always equal to the Transverse Diameter, viz. fc + C F = AB, or fG + GF = AB; and this holds good for every Point in the Circumference of the Ellipfis. Whence is derived the

following Method of delineating that Figure.

Construction. First set the two Diameters across each other at Right Angles, and exactly in the Middle at M. Next, take Half the longer Diameter AM or MB in the Compaffes, and setting one Foot in C or D, make a Dash each Way across the longer Diameter AB, at the Points F and f, which will be the two Foci or Centers on which the Ellipfis must be described. Then in the Points Fand f stick up two Pins, and round them put a Thread fo long, that it being doubled may reach from F to B, or from f to A, and tie it fast. Lastly, inserting a Black Lead Pencil between the Threads, draw them strait, and move the Pencil round, keeping the Threads moderately stretched, so will the Point of the Pencil describe the Periphery of the Ellipfis required.

Note. The Orbits of all the Planets are Elliptical, and the Sun is placed in or near to one of the Foci of each of them; and that in which he is placed, is called the lower Focus.

Problem 3.

The two Diameters of an Ellipsis being given, to find its Circumference.

Rule.

Add the Squares of the two Diameters together, and xtract the Square Root of their Sum; then to the double of that Root add of the shorter Diameter; this last Sum will be the Circumference very near.

Crample.

Suppose the Tranverse Diameter A B be 24 Inches, and the Conjugate CD 18 Inches, what is the Circumference of the Ellipsis?

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Operation. AB 576+CD 324 = 900, whose √30; then 30 × 2 = 60; and 60 + 6 = 66 Inches, the Circumference required.

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