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4. If the plane cut the cone parallel
to the base, or make no angle with it, the section will be a circle ; as ABD.
5. The section DAB is an ellipse, IVAA when the cone is cut obliquely through both sides, or when the plane, thus cutting it, is inclined to the base in a less angle than the side of the
6. The section is a parabola, when the cone is cut by a plane parallel to the
or when the cutting plane and the side of the cone make equal angles with the base; as ADE.
7. The section is a byperbola, when the cutting plane makes a greater angle with the base than the side of the cone makes ; as ADE.
3. And if all the sides of the cone be continued through the vertex, forming an opposite equal cone, and the plarre be also continued to cut the opposite cone, this latter section will be the opposite byperbola to the former ; as DBe.
9. The vertices of any section are the points, where the cutting plane meets the opposite sides of the cone, or the sides of the vertical triangular section ; as A and B.
Hence the ellipse and the opposite hyperbolas have each two vertices; but the parabola only one, unless we consider the other as at an infinite distance.
10. The Axis, or Transverse Diameter, of a conic section
a is the line or distance AB between the vertices.
Hence the axis of a parabola is infinite in length, Ab being only a part of it.
11. The centre C is the middle of the axis.
Hence the centre of a parabola is infinitely distant from the vertex.
And of an ellipse the axis and centre lie within the curve ; but of a hyperbola without.
12. A Diameter is any right line, as AB or DE, drawn through the centre, and terminated on each side by the curve ; and the extremities of the diameter, or its intersections with the curve, are its vertices.
Hence all the diameters of a parabola are parallel to the axis, and infinite in length ; and therefore Ab and De are only parts of two diameters. And hence also every diameter of the ellipse and hyperbola have two vertices; but of the parabola only one ; unless we consider the other as at an infinite distance.
13. The Conjugate to any diameter is the line drawn through the centre, and parallel to the tangent of the curve at the vertex of the diameter. So FG, parallel to the tangent at D, is the conjugate to DE ; and HI, parak lel to the tangent at A, is the conjugate to AB.
Hence the conjugate HI of the axis AB is perpendicular
And hence there is no conjugate to a diameter of the parabola, unless it be considered as at an infinite distance from the vertex.
14. An Ordinate to any diameter is a line parallel to its conjugate, or to the tangent at its vertex, and terminated by the diameter and curve. So DK, EL, are ordinates to the axis AB; and MN, NO, ordinates to the diameter DE.
Hence the ordinates to the axis are perpendicular to it.
15. An Absciss is a part of any diameter contained between its vertex and an ordinate to it ; as AK or BK, or DN or EN.
Hence in the ellipse and hyperbola, every ordinate has two abscisses ; but in the parabola, only one ; the other vertex of the diameter being infinitely distant.
16. The Parameter of any diameter is a third tional to that diameter and its conjugate.
17. The Focus is the point in the axis, where the ordihate is equal to half the parameter ; ás K and L, where DK or EL is equal to the semipárameter.
Hence, the ellipse and hyperbola have each two foci ; but the parabola only one.
18. If DAE, FBG be two opposite hyperbolas, having AB for their first or transverse axis, and ab for their second or conjugate axis ; and if dae, fbg be two other opposite hyperbolas, having the same axes, but in a contrary order, namely, ab their first axis, and AB their second; then these two latter curves dae, fbg, are called the conjuo gate hyperbolas to the two former DAE, FBG ; and each pair of opposite curves mutually conjugate to the other:
19. And if tangents be drawn to the four vertices of the curves, or extremities of the axes, forming the inscribed rectangle HIKL ; the diagonals HCK, ICL of this rectangle are called the asymptotes of the curves. And if these asymptotes intersect at right angles, or the inscribed rectangle be a square, or the two axes AB and ab be equal, then the hyperbolas are said to be right-angled, or equilateral
SCHOLIUM. VOL. II.
By the Traverse Table.
Seek difference of latitude 96, and departure 210, in the table ; the nearest found ars 976 and 2094, which correspond with 65° course, and 231 distance. Again, with comiddle latitude 54° as course, and 210 departure, is found 260 difference of longitude in the column of distance.
The bearing and distance are found by plane sailing, as before.