Thus, for example, let b, c, A, be given, and let it be required to determine a, independently of the angles B, c. By (α), we have Whence cos. a cos. A sin. b sin. c + cos. b cos. c. From which equation a is determined, but the expression is not in a form adapted to logarithmic computation; we can, however, effect the necessary transformation by the introduction of a subsidiary angle. Add and subtract sin. b sin. c on the right hand side of the equation. Then cos, a cos. A sin. b sin. c + cos. b cos. c + sin. b sin. c sin. b sin, c cos. b cos. c + sin. b sin. c + sin. b sin. è cos. A — sin. b sin. c sin. b sin. c vers. A =cos. (bc) cos. (b — c) + sin. b sin. c vers. A vers. a = vers. (b — c) + sin. b sin. c vers. A sin. b sin. c vers. A vers. (bc) A} = vers. (bc) sec.2 0 from which a may be determined by the tables, being known from the equation. In like manner in case II, where two angles and the included side were given, we first determined the remaining sides, and then we were enabled to find the remaining angle. Now, let us suppose, that A, B, c, are given, and that we are required to find C independently of a and b. cos. cos. A cos. B sin. A sin. B cos. c sin. A sin. B cos. C=1 sin A sin. B (1 =1+cos. (A + B) + cos. A cos. B vers. c) + cos. A cos. B sin. A sin. B vers. c. + sin. A sin. B vers. c {1+ =2 cos. 2 sec. If we assume tan." = sin. A sin. B vers. c In case III, where two sides and the angle opposite to one of them were given we first determined the angle opposite to the other side, and then the remaining angles and the remaining side in succession. Now, let us suppose, that a, b, A, are given, and that we are required to determine the angle C and the side c, independently of the angle B and of each other, under a form adapted for logarithmic computation. .. sin. C cos. + cos. C sin. = cot. a sin. b tan. A cos. ◊ sin. (C + 4)'= cot. a sin. b tan. A cot. a tan. b. sin. sin. cos. b tan. A whence C is known, being previously determined from equation. whence c may be found, ◊ being previously determined from the equation. tan. tan. b cos. A. In like manner, in case IV, when two angles and the side opposite to one of them were given, we first determined the side opposite to the other angle, then the remaining side and the remaining angle in succession. Now, let A, B, α, be given, and let it be required to determine c and C, independently of b and of each other, and under a form adapted to logarithmic computations. If we take the formula (6). whence c may be determined, being previously known from equation whence C may be found, being known from equation tan. tan. B cos. a. In the fifth and sixth cases, any one of the angles or sides required, may be found independently of the rest by the formulæ referred to. EXAMPLES IN SPHERICAL TRIGONOMETRY. (1.) In the right-angled spherical triangle ABC, the hypothenuse AB is 65°5', and the angle A is 48°12'; find the sides AC, CB, and the angle B. (2.) In the oblique-angled spherical triangle ABC, given A B=76° 20′, BC=119° 17', and ▲ B= 52° 5′; to find AC and the angles A and C. (3.) In an oblique spherical triangle the three sides are a=81°17′, b=114°3′, c=59°12′; required the angles A, B, C. ANALYTICAL GEOMETRY OF TWO DIMENSIONS. If we reflect on the nature of Geometrical Problems, we shall perceive that the greater number of them depend ultimately on finding the distance of one or more unknown points, from other points or straight lines, whose position is already known and determined. If, therefore, we have a method which enables us to determine analytically the position of a point, with reference to certain other points or straight lines whose position is known, we shall be in a state to resolve all kinds of geometrical problems. Let there be two straight lines AX, AY, whose position is known and determined, situated in the same plane at right angles to each other, and let P be any point in the same plane whose position we are required to determine. N Y P A M From the point P let fall PM, PN, perpendiculars on AX and AY. Then it is manifest that the point P will be determined, if we know the length of the sides AM, AN, of the rectangle AP. For these sides are the distances of the point P from the two fixed straight lines AX, AY, so that, if we draw from the points M and N two straight lines, respectively parallel to AY and AX, the point where they intersect will be the point required. The two fixed lines AX, AY are called Axes. The distance AM or PN of the point P from the axis AY is called the Abscissa of the point P, and is usually designated algebraically by the letter x. The distance AN or PM of the point P from the axis AX is called the Ordinate of the point P, and is usually designated algebraically by the letter y. The two distances x and y are together denominated the Co-ordinates of the point P. The two axes are distinguished from each other by calling the axis AX, along which the abscissas are reckoned, the Axis of Abscissas, or the Axis of x's; and in like manner the axis AY, along which the ordinates are reckoned, is called the Axis of Ordinates, or the Axis of y's. The point A is called the Origin of Co-ordinates, since it is from this point that the distances are reckoned. EQUATIONS OF A POINT. The characteristics of every point situated on the axis of y's is x = 0, since that equation indicates that the distance of the point in question from that axis is nothing. Similarly the characteristic of every point situated on the axis of 's is y=0 Hence the system of two equations, x = 0, y = 0, characterizes the point A the origin of co-ordinates, since these equations can hold good at the same time for no other point. In general the two equations = a, y = b, when considered together characterize a point situated at a distance a from the axis of y's, and at a distance from the axis of x's. The first of these equations, when considered separately, belongs to all the points of a straight line drawn parallel to the axis of y's, at a distance AM = a, and the second to all the points of a straight line drawn parallel to the axis of a's, at a distance AN = b. Hence the system of two equations together belongs to the point P, in which these lines intersect, and belongs to this point alone. These expressions are thus, as it were, the analytical representations of the point, and for this reason are called the Equations of the point. We must always consider, in the expressions a and b, not only the absolute or numerical values of the distances of the point from the two axes, but likewise the signs by which they may be affected, according to the position of the point in the plane of the axes AX and AY. For, according to the conventions explained in the first chapter of Analytical Plane Trigonometry, if we agree to consider as positive, distances such as AM reckoned along AX to the right of the point A,` we ought to consider as negative, distances such as AM' reckoned to the left of the same point. In like manner, if we consider as positive, distances such as AN reckoned along AY upwards from the point A, we must regard as negative, distances such as AN' reckoned along AY downwards from the point A. If, then, we exhibit the different signs with which a and b may be affected, we shall have four systems of equations to characterize the four different positions of the point P. D' X' M' צו P A M X Y P N P |