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of the axis and the diameter at the point of contact. (a) The normal varies as the square root of the parameter, to the diameter at the point of contact. (b) The normal bisects the angle made by the diameter at the point of contact, with the line drawn from that point to the focus. A. The parts of all diameters intercepted by lines parallel to each other, whether within or without the parabola, are as the rectangles of the corresponding segments of the lines. B. The rectangles of the corresponding segments, into which parallel lines in a parabola divide each other, have to each other a constant ratio. C. If any two straight lines, which meet the curve in two points intersect each other, the rectangle of their corresponding segments, will be as the parameters of the diameters, to which those lines are double ordinates.

Properties of the ellipse. 1. The distance from either focus to the extremity of the axis minor is equal to the semiaxis major. 2. The rectangle of the focal distances from the vertices is equal to the square of the semi-axis minor. 3. The latus rectum is a third proportional to the major and minor axis. 4. If from the foci two straight lines be drawn to any point in the curve, the straight line bisecting the angle adjacent to that contained by these lines is a tangent. COR. Lines drawn from the foci to any point in the curve, make equal angles with the tangent to that point. 5. If tangents be drawn at the extremities of any two diameters of an ellipse, they will form a parallelogram. 6. If from the extremity of any diameter, a line be drawn to the focus, meeting the conjugate diameter, the part intersected by the conjugate will be equal to the semi-axis major. 7. As the square of the axis major is to the square of the axis minor, so are the rectangles of the abscissas of the former, to the squares of their ordinates. A. If from any point in the ellipse, a line be drawn to the minor axis, equal to the semimajor, the part intercepted between that point and the major, is equal to the semi-axis minor. 8. As the square of the axis minor, is the square of the axis major, so are the rectangles of the abscissas of the former, to the squares of their ordinates. 9. If a circle be described on either axis, then any ordinate in the circle, is to the corresponding ordinate in the ellipse, as the axis of that ordinate, is to the other axis. 10. As the square of any diameter, is to the square

of its conjugate, so are the rectangles of its abscissas, to the squares of their ordinates. COR. Every diameter in the ellipse bisects its double ordinate, or lines drawn in the ellipse, parallel to the tangent at its center. B. If straight lines in the ellipse parallel to the conjugate diameter, intersect each other either within or without the ellipse, the rectangles of their corresponding segments are to each other, as the squares of the diameters to which they are parallel. 11. If a tangent and an ordinate to either of the axis be drawn to a point in the ellipse, meeting that axis and axis produced, then the semi-axis is a mean proportional between the distances of the two intersections from the center. C. The distance from the focus to any point of the curve is equal to the ordinate to that point, produced to meet the focal tangent. 13. If ordinates to either axis be drawn from the extremities of any two conjugate diameters, the sum of their squares will be equal to the square of half the other axis. 14. The sum of the squares of any two semi-conjugate diameters is equal to the sum of the squares of the semi-axes. 15. If from the extremity of any diameter a perpendicular be drawn to its conjugate, the rectangle of that perpendicular and the semi-conjugate, is equal to the rectangle of the semi-axes. 16. If perpendiculars be dropped from the foci upon any tangent to the ellipse, the intersection of those perpendiculars with the tangent, will be in the circumference of a circle described upon the axis major. 17. The rectangle of the perpendicular from the foci upon any tangent, is equal to the square of the semi-axis major. D. If tangents be drawn from the vertices to meet any other tangent, the rectangles of the vertical tangents will be equal to the square of the semi-axis minor, and the intercepted part of the other tangent will be the diameter of a circle passing through the foci. 18. The rectangle contained by the straight lines, drawn from the foci to the extremity of any diameter, is equal to the square of half the conjugate to that diameter. E. The squares of any two diameters are to each other, as the rectangles of the segments of one of them, are to the rectangles of the corresponding segments of lines parallel to the other; whether the point of intersection be within or without the ellipse. F. If straight lines in the ellipse intersect each other, either within or without the curve, the rectangles of their corres

ponding segments are to each other as the square of those diameters to which they are parallel.

Properties of the palabola analagous to those of the ellipse. 1. The rectangle of the focal distances from the vertices, is equal to the square of the semi-axis minor. 2. The latus rectum is a third proportional to the major and minor axis. 3. If from the foci, two straight lines be drawn to any point in the curve, the straight line bisecting the angle contained by the line is a tangent. 4. If tangents be drawn at the extremities of any diameter of an hyperbola, they will be parallel to each other. 5. If through the extremity of any diameter, a line be drawn from the focus, to meet the conjugate diameter produced, the part intercepted by the conjugate will be equal to the semi-axis major. 6. As the square of the major axis is to the square of the minor, so are the rectangles of the abscissas of the former, to the squares of their ordinates. Cor. The square of any ordinate to either axis, is less than the square of the same ordinate produced to the conjugate hyperbola, by twice the square of the same axis, to which it is parallel. 7. As the square of the minor axis is to the square of the ma jor, so is the sum of the squares of the semi-minor, and of the distance from the center to any ordinate upon the minor, to the square of the ordinate. 8. If a tangent and an ordinate be drawn from any point of the curve to either of the curves, half that axis will be a mean proportional between the distances of the two intersections from the center. A. The dis. tance from the focus to any point in the curve, is equal to the co-ordinate to that point, produced until it meets the focal tangent. 9. If from the extremity of any diameter, a perpendicular be drawn to its conjugate; then the rectan gle of that perpendicular and the part of it intercepted by the axis major, will be equal to the square of the semi-axis minor. 10. If ordinates to either axis be drawn from the extremities of any two conjugate diameters, the difference of their squares will be equal to the square of half the other axis. 11. The difference of the squares of any two semi-conjugate diameters, is equal to the difference of the squares of the semi-axis. 12. If from the extremity of any diameter a perpendicular be drawn to its conjugate, the rec. tangle of that perpendicular and the semi-conjugate is equal to the rectangle of the semi-axes. 13. If perpendiculars

be dropped from the foci upon any tangent to the hyperbo. la, the intersections of those perpendiculars with the tangent will be in the circumference of a circle described upon the axis major. 14. The rectangle of the perpendiculars from the foci upon any tangent is equal to the square of the semi-axis major. B. If tangents be drawn from the verti ces, to meet any other tangent, the rectangle of the vertical tangents will be equal to the square of the semi-axis minor; and the intercepted part of the other tangent will be the diameter of a circle passing through the foci. 15. The rectangle contained by the straight lines drawn from the foci to the extremity of any diameter, is equal to the square of half the conjugate to that diameter.

SPHERICAL TRIGONOMETRY.

DEFINITIONS. 1. Any circle, which is a section of a sphere by a plane through its center, is called a great circle of the sphere. COR. All great circles of a sphere are equal; and any two of them bisect one another. They are all equal, having all the same radii; and any two of them bisect one another, for as they have the same center, their common section is a diameter of both, and therefore bisects both. 2. The pole of a great circle of a sphere is a point in the superficies of the sphere, from which all straight lines drawn to the circumference of the circle are equal. 3. A spherical angle is an angle on the superfices of a sphere, contained by the arches of two great circles which intercept one another; and is the same with the inclination of the planes of these two great circles. 4. A spherical triangle is a figure upon the superficies of a sphere, comprehended by three arches of three great circles, each of which is less than a semicircle.

PROPOSITION. I. If a sphere be cut by a plane through the center, the section is a circle, having the same center with the sphere, and equal to the circle by the revolution of which the sphere was described. II. The arch of a great circle, between the pole and the circumference of another great circle, is a quadrant. COR. Therefore the straight

line drawn from the pole of any great circle to the center of the sphere is at right angles to the plane of that circle; and, conversely, a straight line drawn from the center of the sphere perpendicular to the plane of any great circle, meets the superficies of the sphere in the pole of that circle. III. If the pole of a great circle be the same with the intersection of the other two great circles; the arch of the first mentioned circle intercepted between the other two, is the measure of the spherical angle which the same two circles make with one another. Cor. If two arches of two great circles which intersect one another in any point, be each of them quadrants, that point will be the pole of the great circle which passes through the extremities of those arches. IV. If the planes of two great circles of a sphere be at right angles to one another, the circumference of each of the circles passes through the poles of the other; and if the circumference of one great circle pass through the poles of another, the planes of these circles are at right angles. COR. 1. If of two great circles, the first passes through the poles of the second, the second also passes through the poles of the first. 2. All great circles that have a common diameter have their poles in the circumference of a circle, the plane of which is perpendicular to that diameter. V. In isosceles spherical triangles the angles at the base are equal. VI. If the angles at the base of a sperical triangle be equal, the trianlge is isosceles. VII. Any two sides of a spherical triangle are greater than the third. VIII. The three sides of a spherical triangle are less than the circumference of a great circle. IX. In a spherical triangle the greater angle is opposite to the greater side; and conversely. X. Ac. cording as the sum of two of the sides of a spherical triangle is greater than a semicircle, equal to it, or less, each of the interior angles at the base is greater than the exterior and opposite angle at the base, equal to it, or less; and also the sum of the two interior angles at the base greater than two right angles, equal to two right angles, or less than two right angles. XI. If the angular points of a spherical triangle be made the poles of three great circles, these three circles by their intersections will form a triangle, which is said to be supplemental to the former; and the two triangles are such, that the sides of the one are the supplements of the arches which measure the angles of the other. XII. The threa

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