NOTE. The book belonging to a ship, in which are entered the courses, distances, winds, &c. from which the daily computation of the ship's place at sea is made, is called the Log Book; the account or register itself, the log s and the computations from the log, the dead reckoning, or account. Since this reckoning is liable to many errors, arising in the measures of distance, determinations of course, effects of tides and currents, and the estimation of lee-way, &c. it is very important to procure corrections frequently, by making astronomical observations suitable for determining the latitude and longitude. Meridian altitudes of the sun, or, when an observation of this kind cannot be obtained, two other altitudes of the same luminary with the intermediate time, for the latitude; and distances of the moon from the sun and certain fixed stars, for the longitude, may be considered as observations best adapted to the general practice of mariners.

END OF NAVIGATION.

CONIC SECTIONS.

CONIC

DEFINITIONS.

SECTIONS are the figures, made by the mutual intersection of a cone and a plane.

2. According to the different positions of the cutting plane, there arise five different figures or sections; namely, a triangle, a circle, an ellipse, a parabola, and a hyperbola only the three last of which are peculiarly called conic

sections.

3. If the cutting plane pass through the vertex of the cone, and any part of the base, the section will evidently be a triangle; as VAB.

B

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.

B

A

5. The section DAB is an ellipse, IMA

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

cone is.

6. The section is a parabola, when the cone is cut by a plane parallel to the side; or when the cutting plane and the side of the cone make equal angles with the base; as ADE.

7. The section is a hyperbola, when the cutting plane makes a greater angle with. the base than the side of the cone makes ; as ADE.

8. And if all the sides of the cone be continued through the vertex, forming an opposite equal cone, and the plane 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 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.