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by Geeky Nigeria

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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