Page:Popular Science Monthly Volume 67.djvu/649

What are the fundamental principles of geometry? What is its origin; its nature; its scope?

These are questions which have at all times engaged the attention of mathematicians and thinkers, but which took on an entirely new aspect, thanks to the ideas of Lobachevski and of Bolyai.

For a long time we attempted to demonstrate the proposition known as the postulate of Euclid; we constantly failed; we know now the reason for these failures.

Lobachevski succeeded in building a logical edifice as coherent as the geometry of Euclid, but in which the famous postulate is assumed false, and in which the sum of the angles of a triangle is always less than two right angles. Riemann devised another logical system, equally free from contradiction, in which the sum is, on the other hand, always greater than two right angles. These two geometries, that of Lobachevski and that of Riemann, are what are called the non-Euclidean geometries. The postulate of Euclid then can not be demonstrated; and this impossibility is as absolutely certain as any mathematical truth whatsoever.

It was the attainment of this very perception which in fact led to the creation of the non-Euclidean geometry. Says Lobachevski in the introduction to his 'New Elements of Geometry':

The futility of the efforts which have been made since Euclid's time during the lapse of two thousand years awoke in me the suspicion that the ideas employed might not contain the truth sought to be demonstrated. When finally I had convinced myself of the correctness of my supposition I wrote a paper on it [assuming the infinity of the straight].

It is easy to show that two straights making equal angles with a third never meet.

Euclid assumed inversely, that two straights unequally inclined to a third always meet.

To demonstrate this latter assumption, recourse has been had to many different procedures.

All these demonstrations, some ingenious, are without exception false, defective in their foundations and without the necessary rigor of deduction.

John Bolyai calls his immortal two dozen pages (the most extraordinary two dozen pages in the whole history of thought), 'The Science Absolute of Space, independent of the truth or falsity of Euclid's Axiom XI. (which can never be decided a priori).'

Later we read on the title page of W. Bolyai's 'Kurzer Grundriss': 'the question, whether two straights cut by a third, if the sum of the interior angles does not equal two right angles, intersect or not? no one on the earth can answer without assuming an axiom (as Euclid the eleventh)' [the parallel postulate].

With the ordinary continuity assumptions or the Archimedes postulate, it suffices to know the angle-sum in a single rectilineal triangle in order to determine whether space be Euclidean or non-Euclidean.

How peculiarly prophetic or mystic then that the clairvoyant inspiration of the genius of Dante, the voice of ten silent centuries, should have connected with the wisdom of Solomon and the special opportunity vouchsafed him by God a question whose answer would have established the case of Euclidean geometry seven hundred years before it was born, or that of non-Euclidean geometry three thousand years before its creation.

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