XOY , but in both cases parallel (or rectilinear).
The co-ordinate y is often called the ordinate, a term repro ducing the Latin adverb ordin ate or ordinatim (arrangedly), long used with the participle app/icata, in translating Apollonius, for the Greek 7- ercuyOvos, especially of parallel chords (in a conic) bisected by their con jugate diameter. A pp/icata itself furnished an equivalent for "ordinate," as in Fermat's appliquee. Abscissa (abscisse), "off cut" (Gr. Lroro,ui) explains itself, as segment cut off from an axis, or diameter (reckoned either from vertex or centre) by a conjugate chord (or ordinate). Both ordinate and co-ordinate, as nouns, were slow in being recognized, being first used in their present sense by Leibniz (1692).
Equations. The whole system of points forming the plane may now be indicated by a symbol (x, y), both x and y "real" and arbitrary, ranging from zero (o) both ways to infinity (from —oo to +oo), each point corresponding to one and only one pair of values (x, y), and conversely. But if x and y be subjected to some condition, as x—y =o, the points (x, y) cannot be just anywhere in the plane, but only in a certain region of .value, viz., on the line bisecting the angle XOY (co); hence x—y=o (fig.
1) or x =y is called the equation of the bisector. To each of its points corresponds a value-pair (x, y) satisfying its equation, and conversely. The equation symbolizes a system of value pairs (all that satisfy it), all de picted together by all the points on the line and, conversely, a o n e-t o-o n e correspondence of point and value-pair. Likewise the system of value-pairs x2-Fy2= 25 symbolizes an as semblage of points, a circle about 0, of radius 5; conversely, such a circle depicts such a sys tem of value-pairs. So for any curve and its symbolizing equation, and for any equation and its depicting curve.
Degrees. All equations of the first degree in point co-ordinates (as lx-1-my+n=o) are depicted by lines, and hence are called linear, and conversely; all of the second degree (as ax2+ 2hxy+by2+ 2jx-F 2fy-Fc=o) by conics (see CONIC SECTION), of which both circles and line-pairs are limiting cases, and con versely; equations of a higher degree, by higher curves, and con versely; the numerical fact of degree corresponding to the geo metric fact of intersection by a line. A curve of the nth order (its equation being of the nth degree in x and y) meets a line in exactly n points, some perhaps coincident, or even nul, non-exist ent in the plane of real x and y. Thus 3x+4y = 25 meets x2.-Fy2= 25 in the double tangent-point (3, 4), while x-Fy = io meets it in the imaginary points x = 5± — 2, y= — 2 (fig. 2). Concen 2 2 tric circles, as meet only in imaginary points at oo .
Transformation. There is often occasion to change the axes of reference. In the simplest change of origin only, OX, OY are pushed, without turning, into new positions 0'X', O' Y' : if 0' be (a', b'), plainly x = x'+a', y = y'-{-b' (fig. 3). If OX, OY be not pushed, but only turned round 0, say counter-clockwise through the angle a, then clearly x = x' cos a -y' sin a, y= x' sin a+y'cos a (fig. 4) . For both changes at once, the formulae are combined. This is the case for rectangu lar axes; for oblique axes we have similar processes with more complex results.
Then for all points of a plane upright on OX, parallel to YOZ, the distance x to