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, paral lel to the tangent at A, is the conjugate to AB. Hence the conjugate HI of the axis AB is perpendicular to it. 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 16. The Parameter of any diameter is a third proportional to that diameter and its conjugate. 17. The Focus is the point in the axis, where the ordinate is equal to half the parameter; as K and L, where DK or EL is equal to the semiparameter. 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 conju 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. By the Traverse Table. Seek difference of latitude 96, and departure 210, in the table; the nearest found are 976 and 209'4, 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. from E, and parallel to CA. Continue DA till it intersect EB in B. EB will be the difference of longitude. Computation. The bearing and distance are found by plane sailing, as before. The bearing and distance being found, as already shewn; under the same course with 118 as difference of latitude, is found departure or difference of longitude 254. The difference in these results is owing to the odd minutes in the course being rejected. EXAMPLE OF A TRAVERSE. Suppose a ship to sail from latitude 43° 25′ N. on the following courses, viz. SWbS. 63 miles, SSW W. 45 miles, S6E. 54 miles, and SWbW. 74 miles. Required the latitude arrived at, and the difference of longitude made good. Solution by Middle Latitude Sailing. The difference of latitude and difference of longitude, corresponding to each course and distance, are found to be as in the following table. VOL. II. U U Courses, |