We proceed to treat first of Trigonometry; then of Coördinates, and their immediate applications; and lastly of the Conic Sections. § 1. Plane Trigonometry, has for its object the solution of problems concerning plane triangles; the sides of which are always straight lines. It is subdivided, in reference to the different kinds of triangles, into Right Angled and Oblique. In any right angled triangle, ABC, Plate VII. Fig. 13, if, from the vertex, A, of one of the acute angles, as a centre, and with the hypothenuse, AC, for a radius, we describe an arc of a circle, the side, BC, opposite to the vertex used, becomes the sine, and the adjacent side, AB, becomes the cosine, of the angle, A, in question. The cosine, prolonged, becomes another radius of the same are; and the prolonged part, BD, beyond the triangle, is called the versed sine of the arc or angle in question. If now we apply a scale, on which the hypothenuse or radius shall be equal to unity or 1, the sine and cosine will be expressed by decimals, which are called the natural sine, and cosine, of the angle in question. But if we take the radius equal to 10,000,000,000, (whose logarithm is 10), and then find the logarithms of the corresponding lengths of the sine and cosine, we shall have the logarithmic sine, and cosine, of the same angle. Thus, angles may be designated by their sines, or cosines. Again, if from the same vertex, A, as a centre, and with the base, AB, as a radius, we describe an arc, then the other leg, BC, is called the tangent, and the hypothenuse, AC, is called the secant of the same angle. The tangent and secant of the complement of an angle, or what it wants of 90°, are called the co-tangent and co-secant of the angle itself. It is chiefly by means of Tables of the sines and co-sines, tangents and co-tangents of angles, that all problems of Trigonometry are resolved. In every plane triangle, we must have given at least three parts, sides and angles, one of which at least must be a side, in order to find the other parts. Thus, in a right angled triangle, if we have given the base, and angle at the base, the right angle being of course known, then, the base is to the perpendicular, or other leg, as the cosine of the angle at the base, is to the sine of the same angle; and the base is to the hypothenuse, as the cosine of the angle at the base, is to radius, or the sine of 90°. In an oblique angled triangle, the sides are proportional to the sines of the opposite angles: also, the sum of any two sides is to their difference, as the tangent of the half sum of the two opposite angles, is to the tangent of their half difference: and finally, the sine of half of either angle, is equal to radius multiplied by the half sum of the three sides minus one of the adjacent sides, this multiplied by the same half sum minus the other adjacent side, and the whole divided by four times the product of the two adjacent sides, adjacent to the angle sought. §2. Spherical Trigonometry, has for its object the resolution of spherical triangles, formed by arcs of great circles on the surface of a sphere. The angles of such a triangle, (Plate VII. Fig. 14), are those formed by the planes of its sides, with each other; and its sides are measured as arcs, by the number of degrees which they subtend, at the centre of the sphere. In spherical triangles, the sines of the sides are proportional to the sines of the opposite angles. In a right angled spherical triangle, if we omit the right angle, we have five parts left, sides and angles; one of which being called the middle part, two of the others become adjacent parts, and the other two, the opposite parts; taking however not the oblique angles and hypothenuse themselves, but their complements in their stead. Then, radius into the sine of the middle part, will be equal to the product of the tangents of the adjacent parts, and also equal to the product of the cosines of the opposite parts. These rules, called Napier's Analogies, may be applied to oblique angled spherical triangles; by dividing them each into two right angled triangles, by means of an arc drawn from one vertex, perpendicularly, to the opposite side. § 3. The invention of Coördinates, has furnished the means of representing geometrical curves, by the medium of algebraic equations. For this purpose, we imagine two straight lines, XX' and YY', Pl. VII., Fig. 15, to be drawn in the plane which contains the given curve; and these lines are called the axes of coördinates: the origin of coördinates being their point of meeting. Generally, the axis which extends across the figure, from right to left, is called the axis of abscissas; and the other, the axis of ordinates. If, then, from any given point, m, we draw a vertical line, m n, until it meets the axis of abscissas, this line is called the ordinate of that point; and the distance, An, from the foot of this ordinate, on the axis of abscissas, to the origin of coördinates, is called the abscissa of the same point. Thus, the position of the point, in the plane under consideration, is fixed by means of its abscissa and ordinate; which, being parallel to the axes, are generally perpendicular to each other, and together are called the coördinates of that point. = Suppose, now, that we imagine a series of points, at different distances from the origin of coördinates, but so situated that the ordinate of each point shall be equal to its abscissa. Then will all these points lie in one and the same straight line, Ab, Fig. 15, passing through the origin, and making an angle of 45° with each of the axes, when they are rectangular: and the equation of this straight line. would be y x; calling x the abscissa, and y the ordinate, in general. By giving any particular value to 2, it determines the corresponding value of y, and defines some particular point of this straight line. For the origin itself, we have x = 0, and y = 0. and in general the abscissa will be 0 for any point situated on the axis of ordinates, and the ordinate will be zero for the axis of abscissas. In like manner as above, the equation y x, is that of a line, AC, Fig. 15, passing through the origin, and having the ordinate for each point, the double of its abscissa. A line whose equation is y 1 x + 10, would be parallel to this last, but would cut the axis of ordinates at a distance from the origin expressed by the number 10. The equation y=x+5, might represent the line de, Fig. 5; the coefficient, determining its oblique direction. = = § 4. We must pass on to the Conic Sections. If we suppose a cone, (Plate VII., Fig. 16), to be bounded by an infinite number of consecutive straight lines, all passing through its vertex, V, and together composing its convex surface, these lines are called its |