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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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