The fourth dimension

THE USE OF FOUR DIMENSIONS IN THOUGHT 87

To take an instance chosen on account of its ready

availability. Let us take 5 two right-angled triangles of ——— a given hypothenuse, but

5

having sides of different

ig. 46. lengths (fig. 46). These

triangles are shapes which have a certain relation to each other. Let us exhibit their relation as a figure.

Draw two straight lines at right angles to each other, the one HL a horizontal level, the other vL a vertical level (fig. 47). By means of these two co-ordinating lines we can represent a double set of magnitudes; one set as distances to the right of the vertical level, the other as distances above the horizontal level, a suitable unit being chosen.

Thus the line marked 7 will pick out the assemblage of points whose distance from the vertical level is 7, and the line marked 1 will pick out the points whose distance above the horizontal level is 1. The meeting point of these two lines, 7 and 1, will define a point which with regard to the one set of magnitudes is 7, with regard to the other is 1. Let us take the sides of our triangles as the two sets of magnitudes in question.

Then the point 7, 1, will represent the triangle whose sides are 7 and 1. Similarly the point 5, 5—5, that is, to the right of the vertical level and 5 above the 5.5 horizontal level—will represent the triangle whose sides are 5 and 5 (fig. 48).

Thus we have obtained a figure consisting of the two points 7, 1,

Fig. 48. and 5, 5, representative of our two triangles, But we can go further, and, drawing an arc

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