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An Introduction To Non-aristotelian Systems And General Semantics.

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ON GEOMETRY
623
Let us take the surveyor's network and label the curves by consecutive numbers in each family. The essential point is that these numbers, (let us call them the X and Y numbers) do not represent either lengths, or angles or other measurable quantities, but are simply labels for the curves, much as when we label streets by numbers.
But such numbering does not lead us far. We must introduce some measure relations. We have at our disposal a measuring chain and the arbitrary meshes of the network which we have introduced. The next step is to measure the ■mall meshes one after another and plot them on our map. When this is done we have a complete map similar in structure to our region. Because of the smallness of the meshes we can consider them as small parallelograms, and such parallelograms can be denned by the lengths of two adjacent sides and one angle.
We may, however, proceed differently, as shown on Fig. 10.
Let us select one mesh, for instance the one bounded by the curves, 3 and 4, and by the curves 7 and 8. Let us consider a point P within the mesh, and let us denote its distance from the point by s. This distance
could be directly measured. Let us draw from the
point P parallels to our mesh lines and label the intersections with mesh lines by A and B, respectively. Let us also draw PC perpendicular to the xaxis.
The points A and B then also have numbers or labels or gaussian co-ordinates in our network. The co-ordinate of A may be determined by measuring the side of the parallelogram on which A lies and the distance of A from O. We can
regard the relation called the ratio of these two
Fig. 10
lengths as the increase of the x co-ordinate of
A towards O. We shall denote this increase itself by x, choosing 0 as the origin of the gaussian co-ordinates. Similarly, we determine the gaussian co-ordinate of y of B as the ratio in which B cuts the corresponding side. We see that these two ratios, which for brevity we call x and y, are the co-ordinates of our point P.