Hinton - The Fourth Dimension.pdf

5
THE USE OF FOUR DIMENSIONS IN THOUGHT 87
To take an instance chosen on account of its ready
availability. Let us take
two right-angled triangles of
5
1
Fig. 46.
7
a given hypothenuse, but
having sides of different
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,
VL
7 the one HL a horizontal level, the
other VL a vertical level (fig. 47).
By means of these two co-ordin-
1
HL
ating lines we can represent a
double set of magnitudes; one set
as distances to the right of the ver-
Fig. 47. tical 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 point whose
distance from 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
7,1 (fig. 48).
O
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