In the second edition the work has been revised and some additions have been made both to the text and to the examples; the hints for the solution of the examples have also been considerably increased. March, 1858. In the third edition some articles which experience proved to be difficult for students have been simplified and improved, and a few additional illustrations have been introduced. In consequence of the demand for the work proving much greater than had been originally anticipated, a large number of copies has been printed, and a considerable reduction effected in the price. January, 1862. PLANE CO-ORDINATE GEOMETRY. CHAPTER I. CO-ORDINATES OF A POINT. 1. IN Plane Co-ordinate Geometry we investigate the properties of straight lines and curves lying in one plane by means of co-ordinates; we commence by explaining what we mean by the co-ordinates of a point. Let O be a fixed point in a plane through which the lines X'OX, Y'OY, are drawn at right angles. Let P be any other point in the plane; draw PM parallel to OY meeting OX in M, and PN parallel to OX meeting OY in N. The position of P is evidently known if OM and ON are known; for if through N and M lines be drawn parallel to OX and OY respectively, they will intersect in P. The point O is called the origin; the lines OX and OY are called axes; OM is called the abscissa of the point P; and. T. C. S. 1 ON, or its equal MP, is called the ordinate of P. Also OM and MP are together called co-ordinates of P. 2. Let OM=a, and ON=b, then according to our definitions we may say that the point P has its abscissa equal to a, and its ordinate equal to b; or, more briefly, the co-ordinates of the point P are a and b. We shall often speak of the point which has a for its abscissa and b for its ordinate, as the point (a, b). 3. A distance measured along the axis OX is however most frequently denoted by the symbol x, and a distance measured along the axis OY by the symbol y. Hence OX is called the axis of x, and OY the axis of y. Thus x and y are symbols to which we may ascribe different numerical values corresponding to the different points we consider, and we may express the statement that the co-ordinates of P are a and b, thus; for the point P, x=a and y = b. 4. The lines X'OX, Y'OY, being indefinitely produced divide the plane in which they lie into four compartments. It becomes therefore necessary to distinguish points in one compartment from points in the others. For this purpose the following convention is adopted, which the reader has already seen in works on Trigonometry; lines measured along OX are considered positive and along OX' negative; lines measured along Or are considered positive, and along OY' negative. (See Trigonometry, Chap. IV.) If then we produce PN to a point Q such that NQNP, we have for the point Q, xa, y=b. If we produce PM to R so that MR MP, we have for the point R, xa, y=-b. Finally if we produce PO to S so that OS OP, we have for the point S, x=-a, y=-b. = = 5. In the figure in Art. 1 we have taken the angle YOX a right angle; the axes are then called rectangular. If the angle YOX be not a right angle, the axes are called oblique. All that has been hitherto said applies whether the axes are rectangular or oblique. We shall always suppose the axes rectangular unless the contrary be stated; this remark applies both to our investigations and to the examples which are given for the exercise of the student. |