Page:Popular Science Monthly Volume 67.djvu/651

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In vain would one seek to treat analytically propositions of this species, even as all the theory of parallels. One would never succeed, just as one would not be able to do without synthesis for measuring plane rectilineal figures, or solids terminated by plane surfaces. It is incontestable that in the beginnings of geometry or mechanics, analysis can not serve as sole method.

One may compress the circle of synthesis; but it is impossible completely to suppress it.

From this, however, it follows that all investigations, such as those of Sophus Lie which start with the idea of number-manifold, involve a petitio principii, if interpreted directly as researches on the foundations of geometry. In the same way, the non-Euclidean geometry stops the old wrangle as to whether the axioms of geometry are a priori or empirical by showing that they are neither, but are conventions, disguised definitions, or unprovable assumptions pre-created by auto-active animal and human minds.

As Lambert insisted, for the space problem the mathematical treatment is in essence the treatment by logic. The start is from a system of axioms, assumptions. We postulate that between the elements of a system of entities certain relations shall hold, e. g., two points determine a straight, three a plane. There is to be shown that these axioms are independent and not contradictory, presupposing pure logic and the applicability to the entities of an arithmetic founded by and made of pure logic. That the assumptions considered should be axioms of geometry, they must satisfy a further condition, which Hilbert formulates thus:

A system of assumptions is called a system of axioms of geometry if it gives the necessary and sufficient and independent conditions to which a system of things must be subjected in order that every property of these things should correspond to a geometric fact, and inversely; so that therefore in Hertz's sense these things should be a complete and simple picture of geometric reality.

The physiologic-psychologic investigation of the space problem must give the meaning of the words geometric fact, geometric reality.

It is the set of assumptions which makes the geometry what it is, which determines it. Thus, in my 'Rational Geometry,' one system of assumptions about the elements, points and straights on a plane, makes Euclidean planimetry. Another set makes Riemannean planimetry, in which when we picture it as in Euclidean space, we may call the straights straightests (great circles), and the plane sphere.

In the light of all this we see how the importance of non-Euclidean geometry for the teacher is still emphasized by the text-books of France, which have never recovered from Clairaut and Legendre. Even the latest and best French geometry, that of Hadamard, published under the editorship of Gaston Darboux, never presents nor consciously considers the question of its own foundations. It seems childishly unconscious of what is now requisite for any geometry pretending to be scientific or rigorous. This lack of foundation is allowable in a preliminary