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ANALYTICAL GEOMETRY 41
not on this circumference does not have an x and y such that
the constant relation o; 2 + 7/ 2 = c 2 is satisfied for it.
There are two points to be noticed in the above general state-
ment. Firstly, I have said that the curve " may be expressed,"
and so on. By this I mean that it is possible and not neces-
sarily always true that the curve may be so considered. We
can imagine curves that cannot be represented by a finite
algebraical equation. Secondly, about the fundamental lines
of reference the "axes" as they are called. One of these
axes we have called the "#-axis," and the distance measured
by the number x is sometimes called " the abscissa " ; while
the line of length y units which is perpendicular to the end of
the abscissa farthest from the origin, and therefore parallel to
the other axis ("the y-axis ") is called " the ordinate." The
name " ordinate " was used by the ancient Roman surveyors.
The lines measured by the numbers x and y are called the
" co-ordinates " of the point determining and determined by
them. Sometimes the numbers x and y themselves are called
" co-ordinates," and we will adopt that practice here.
Sometimes the axes are not chosen at right angles to one
another, but it is nearly always far simpler to do so, and in
this book we always assume that the axes are rectangular.
The whole plane is divided by the axes into four partitions,
the co-ordinates are measured from the point called " the
origin" where the axes cross. Here the interpretation in
geometry of the " negative quantities " of algebra which so
often seems so puzzling to intelligent beginners gives us a means
of avoiding the ambiguity arising from the fact that there
would be a point with the same co-ordinates in each quadrant
into which the plane is divided.
Consider the cc-axis. Measure lengths on it from the origin,
so that to the origin (0) corresponds the number 0. Let OA,
measured from left to right along the axis, be the unit of
length; then to the point A corresponds the number 1. Then
let lengths AB, BC, and so on, all measured from left to right,
be equal to OA in length ; to the points B, C, and so on,
correspond the numbers 2, 3, and so on. Further to the point
that bisects OA, let the fraction ^ correspond ; and so on for
the other fractions. In this way half of the x-axis is nearly
filled up with points. But there are points, such as the point
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