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. 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 A Y Р 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 a'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 x = a, y = b, when considered together characterize a point situated at a distance a from the axis of y's, and at a distance b 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 x's, at a distance AN = 6. 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 XV. 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. Y p' P X' M' IY P Z K A M X N P In order to complete our discussion on the equations to a point, let it be required To determine the analytical expression for the distance between two given points which are situated in the same plane. Let the co-ordinates of the first point P, be x', y', and of the second point P, be x', y', so that the equations to these points, whose positions we suppose known, are Of P, { x = x2 sx = x" y = y' ) .....(1) And of l2 { y = y" } .....(2) It is required to express the distance P¡ P, of these points in terms of the given co-ordinates x', y', x", y". Substituting these values of P,Q and P,Q, in (A) we have This formula is quite general, and will apply equally well to the case in which the two points are situated on different sides of the axes. It will only be necessary, in this case, to introduce the changes in the signs which correspond to changes in position; thus, for example, to obtain the distance of two points, one of which is situated in the angle YAX, and the other P, in the angle YAX', we must change the sign of x", which gives us R = √(x2 + x'')2 + (y' — y'')2 Y P2 X M2 A Mi X In fact, if we perform the calculations as in the former case, we find |