Page:Popular Science Monthly Volume 78.djvu/569

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IS EUCLID'S GEOMETRY MERELY A THEORY?

the larger the triangle, the less the sum of its internal angles. Logically pursued, the largest triangle possible would have all its sides parallel and all its angles zero.

This last statement has touched the verge of infinity and that is no doubt treacherous territory. Coming back to our real universe—How can we prove that Euclid is right about it? The real universe is large. If, like the adversary in the Book of Job, we could go to and fro in the great world and walk up and down in it, then we might decide the controversy. But the limit of man's present astronomical measurements is only about 30 light-years—176 millions of millions of miles. Within this compass he has observed no drift or change in the direction of rays of light. If in our real space parallels are not exactly and everywhere equidistant, Euclid's geometry is incorrect. The slightest deviation in parallels would give the victory to Lobachevski or else to the third competitor, Riemann. The three justly claim equal consideration

PSM V78 D569 Euclid hypothesis of triangles.png
Fig. 3.

in the light of present knowledge. Could such drift, if real, escape human observation? Yes; first, because our instruments are not absolutely accurate; and secondly, because eyesight is no infallible test.

If the human eye could survey a sufficiently tremendous expanse, then parallels running through it might present the appearance of the hyperbolic curves limiting the black and white areas in Fig. 4. These curves may represent, and in certain respects they do simulate, the parallels of Lobachevski. They are, to be sure, not parallels, for parallels are by definition straight; however, by placing the eye an inch and a half above the center of this figure, these lines can be made to look straight—a fact that confirms the statement that eyesight is not an infallible test of straightness.