ABSCISS
, Abscisse, or Abscissa, is a part or segment cut off a line, terminated at some certain point, by an ordinate to a curve; as AP or BP.
The absciss may either commence at the vertex of the curve, or at any other fixed point. And it may be taken either upon the axis or diameter of the curve, or upon any other line drawn in a given position.
Hence there are an infinite number of variable abscisses, terminated at the same fixed point at one end, the other end of them being at any point of the given line or diameter.
In the common parabola, each ordinate PQ has but one absciss AP; in the ellipse or circle, the ordinate has two abscisses AP, BP lying on the opposite sides of it; and in the hyperbola the ordinate PQ has also two abscisses, but they lie both on the same side of it. That is, in general, a line of the second kind, or a curve of the first kind, may have two abscisses to each ordinate. But a line of the third order may have three abscisses to each ordinate; a line of the fourth order may have four; and so on.
The use of the abscisses is, in conjunction with the ordinates, to express the nature of the curves, either by some proportion or equation including the abfcifs and its ordinate, with some other fixed invariable line or lines. Every different curve has its own peculiar equation or property by which it is expressed, and different from all others: and that equation or expression is the same for every ordinate and its abscisses, whatever point of the curve be taken. So, in the circle, the square of any ordinate is equal to the rectangle of its two abscisses, or AP.PB = PQ2; in the parabola, the square of the ordinate is equal to the rectangle of the absciss and a certain given line called the parameter; in the ellipse and hyperbola, the square of the ordinate is always in a certain constant proportion to the rectangle of the two abscisses, namely, as the square of the conjugate to the square of the transverse, or as the parameter is to the transverse axis; and so other properties in other curves.
When the natures or properties of curves are expressed by algebraic equations, any general absciss, as AP, is commonly denoted by the letter x, and the ordinate PQ by the letter y; the other or constant lines being represented by other letters. Then the equations expressing the nature of these curves are as follow; namely, for the circle , where d is the diameter AB; parabola - px = y2 , where p is the parameter; ellipse - t2 : c2 :: tx - x2 : y2, hyperbola t2 : c2 :: tx + x2 : y2, where t is the transverse, & c the conjugate axis.
