CO-ORDINATES. What is called the method of C. is an invention of Descartes, whereby algebra and the calculus may be employed in geometrical investigations. The method is sometimes called algebraical geometry—sometimes, and more properly, analytical geometry; and it is commonly treated under the heads "geometry of two dimensions," and "geometry of three dimensions," according as it is applied to investi pue the properties of figures all in one plane, or of curved surfaces. The method is capable of popular explanation. C. are lines so measured off from a fixed point, called the origin of C., along fixed lines passing through it. called the axis of C., as to deter mine by their quantities the position of any other point relative to the origin. The first step is to find how to determine the position of a point in a plane. 'lake any fixed .point in it for the origin of •C., and through it draw two fixed lines—the co-ordinate axes—at right angles to one another. Then, if the perpendicular distance of the point from each of these axes be given, its position will be determined. Referring to Fig. 1, if P be the point, and 0 be taken for the origin of C., OX, OY for the axes, then if we know NP or OM, the perpendicular distance of P (rain OY, and measure off from 0, 031 on the axis OX, and through M raise a line perpendicular to OX, P must lie in this line, for it con tains all the points in the plane which are at the perpendicular distance 031 from the axis OY. Similarly, if ON or PN, the perpendicular distance of from the axis OX, be known, and we measure that dis tance off from 0 along OY, and throug,h N draw a perpendicular to OY, the point must be in that perpendicular. It is therefore at the intersection of the perpendiculars through 31 and N respectively. When, as in the figure, the fixed lines are at right angles to one another, the C. 031, ON are called the rectangular C. of the point. Let us now see what use can be made of this mode of determining the position of the point, for the discovery of the properties of lines and surfaces. As the values of the C. change for the different points in the plane, they are denoted by the variables x and y. Now, if we suppose the point P to begin
to move according to a determinate law, and the C. to change their • magnitudes so as always to be its C., knowing the law of P's motion, we are able to express in algebraical language the law of the corresponding changes in its co-ordinates. For instance, if P moves so as to be always at the same distance from 0, OP is constant, and (47th Prop. Euclid, Book I.) the square on OP is equal to the sum of the squares on 031 and PM. Putting this into algebraical language, we have the equation, 2.2 y' = 11', or y = ± tr Ili— where R = OP. This is called the equation of the circle referred to its center as origin, and to rectangular C.; and it expresses the law according to which the changes of the C. must take place: and from this equation, combined with that to a straight line, etc., every property of the circle may be determined. If P move so that the sum of the distances from two fixed points shall be always the same, and we express the relation between x and y in that case, we should have the equation of an ellipse. This suffices to show in a general way the nature of the method. Equations between x and y are called the equations of the lines, whether straight or curved, traced out by the point P; and by means of them, though they but express relations between quantities, the qualifies of the lines to which they refer may, by artifices explained in every treatise on the subject, be detected. Nay, by assuming equations between x and y, and examin ing the lines which points represented by them would trace, many singular curves have been discovered. There arc a variety of conditions to be attended to in the interpretation of such equations, depending on the assumptions set out with, in choos ing the origin and axis. The axis of x or OX being taken to time right of the origin, and the axis of y or OY being perpendicular to it and above it, x and y are counted positive when they are measured along their axes to the right of and above the origin respectively, and negative when they are measured to the left and downwards respectively. Suppose x = 031 = ON, and y = 3IP