accurate geodesic measurements of triangulation in the Hartz Valley.

But all these observations proved negative: space presented itself as Euclidian. Nevertheless there was an idea amongst men of science, that more accurate observations and the development of the mechanical consequences of non-Euclidian geometry with regard to astronomical problems, would certainly favour the legitimacy of non-Euclidian postulates as physical hypotheses. This was done by Einstein, when he built up his theory of Relativity by means of Riemann's geometrical conceptions. As a matter of fact, without the wonderful development of non-Euclidian geometry, Einstein's achievement would have been impossible. But are we to conclude, with him, that the geometry of the universe is not Euclidian? This affirmation is simply too bold and premature, for it has no real import. And we must confess that Einstein's cosmological considerations on this topic are the least convincing portion of his work.

Euclidian geometry, says Einstein, must be abandoned, because in a system of reference rotating relatively to a system at rest, the geometrical behaviour of the bodies, which are affected by the Fitzgerald-Lorentz contraction, does not correspond to Euclidian geometry. In simpler words, the path of moving bodies in nature is never a Euclidian straight line, because of the action of gravitational fields, which is always effective; the universe therefore, is not Euclidian.

The fallacy of this argument is easily seen, if we point out that because the path of a moving body is influenced by the presence of gravitational fields, it does not follow at all that the Euclidian straight line path is not there. As a matter of fact, it is there, although not actually followed by the moving body. Let us consider for a moment a photograph of the solar eclipse of May, 1919. We see plainly the magnificent curve described by the rays of light of remote stars passing near the sun; but at the same time, we see that there is a Euclidian

straight line path between these very stars and the point where their rays of light touch the earth. It is not followed by these rays of light, because, according to the Principle of Least Action, the shortest way between two points in a spatiotemporal continuum is influenced by the presence of gravitational fields. And this is so far true that if the sun, which causes the gravitational field in this case, were taken further and further away, the path of these same rays of light would have a decreasing curvature, thus tending asymptotically towards a Euclidian straight line. That path would be adequately Euclidian, if per impossibile the sun and all other gravitational fields were removed from the universe.

Another argument familiar to relativists is that the geometry of the universe cannot be Euclidian, because Einstein's theory is based on Riemannian geometry, where space is curved and the straight line illimited but finite. This argument, however, looked through its adequate proportion, means only that Riemann's geometry is more convenient than any other for the description of the universe. In fact, there is a principle in the theory of groups of transformations, called the Principle of Equivalence, which enabled Poincaré, Klein and others to transpose any system of metrical geometry into any other. By means of a biunivocal correspondence, illustrated with an appropriate vocabulary, between two geometries, any Riemannian concept, for instance, is shown to be equivalent to a Euclidian concept. It follows then, that the group of natural phenomena explained by the theory of Relativity can be interpreted by means of Euclidian geometry: as axiomatic geometry alone makes no affirmations on the reality accessible to experience, but only axiomatic geometry completed with physical propositions, it is possible, whatever be the nature of reality, to keep Euclidian geometry. Geometry [G] does not enunciate anything on the behaviour of real objects, but geometry together with the sum [P] of physical laws; it is the sum [GP] which can be checked by experience. It is then

always possible to take [G] as Euclidian and make appropriate assumptions with reference to some parts of [P]; it is only necessary to take the remainder of [P] such as the sum [GP] is in agreement with experience. In the case of Relativity, however, it is more convenient and less complicated to describe the universe as Riemannian. Convenience is then the condition of the choice of the world's geometry; and we must draw the attention to the word "description" in physical science, which has the profoundest significance for Epistemology. Because a description is always by means of the accidents: the essence of the thing described is left untouched by this operation.

Now we come to the ontological aspect of our argument. With all the essential difference between the object and method of geometry and of physics, there must be a close connexion between these two disciplines. Because, on the one hand, physical sciences cannot reach their actual degree of certitude without the help of mathematics, and on the other hand, mathematics would be useless if it had no practical value, considering also that its origin is empirical. In fact, when the thinking person stops to reflect upon the fact that the existence of Neptune was pointed out to the astronomer before his telescope had noticed this wanderer in the remote heavens; when he learns that the mathematician by a theory related to the solution of the problem of finding the roots of an algebraic equation, is able to say that there are not more than thirty-two distinct types of crystals; when he remembers that the existence of wireless telegraphy is due to deductions of Maxwell by means of theorems that depend upon imaginary quantities; when, to give a last instance, he considers that the abstruse non-Euclidian geometry of Riemann and the tortuous theory of absolute

* D. Wrinch, “On certain Methodological Aspects of the Theory of Relativity," in Mind, April, 1922; and B. Russell, Introduction to Mathematical Philosophy.

differential calculus of Ricci and Levi-Civita enabled Einstein to work out his momentous law of gravitation, which is, as says Sir J. J. Thomson, one of the highest achievements of human thought; he will undoubtedly endeavour to penetrate this mysterious riddle which has perplexed all the great seekers of the unknown: How is it possible that geometry and mathematics in general, which are constructions of the human mind, independent, in their structure and development, of all experience, adapt themselves so wonderfully to the objects of reality? Is Reason able to discover by its sole activity, the very properties of the existing universe?

By denying the reality of matter for the benefit of extension, Descartes was led to the conclusion that geometry and mathematics in general (because of his invention of analytical geometry) are the science of Reality, the science which could penetrate the ultimate essence of its object; and Nature would be completely known when the edifice of mathematics would be completed. And the modern style logicist, with all the restrictions he makes in the Cartesian doctrine, still holds that Reason is the ruler of things as well as the ruler of thought. For him, mathematics, mechanics, physics and every science which uses mathematical expressions are a developed aspect of logic; so that there is no incompatibility between the laws of chemistry, for instance, and the laws of thought, as Leibniz said, Dum Deus calculat fit mundus.

But on the other hand, we must remember that when the mathematician tries his creative power of imagination on ideal constructions, he does not think of any practical utilization of the results he obtains. While to the physicist mathematical systems are operators enabling him to act more successfully on matter, to the mathematician the construction of an abstract theory is an end in itself independently of its applications. As Professor Bouasse says, the creator of new mathematical forms does not care whether his inventions correspond to some reality. The forms in themselves interest

him more than anything else, for they enlarge the readyreckoner of mathematical forms. It is another question whether in a near or remote future, physical phenomena will consent to lodge in those structures. The algebraist prepares in advance sets of moulds which will be utilized by the physicist according to his convenience. But he does not think of that convenience when he makes them; although history shows us that many times the solution of a physical problem has led physicists to invent new mathematical forms, as for example, when the ideas of Faraday led Maxwell to the mathematical exposition of the electromagnetic theory.

The independence of mathematicians towards reality is shown by the fact that a great number of mathematical and specially of geometrical constructions do not find or rather cannot have a corresponding reality. We have mentioned the theory of hyperpolyhedra which has no application in nature. In the same way it is impossible to give an adequate geometrical description of a flower. Nature is far more complicated than geometry and mathematics; so that if we are to describe the external world, mathematics must be supplemented with qualitative principles. It is certain then that mathematics are not at all co-extensive with reality: On the one hand, reality outruns them by its imprevisibility; and on the other hand, it is overstepped by them by all the distance between existence and possibility.

However, it is always possible to reconcile the real with that overwhelming creation of virtual relations. For Reason, however disinterested one may think it, has a utilitarian function. With the same activity, reason deduces a proposition from other propositions, and relations between natural phenomena. If quantitative relations, which are the object of mathematics, agree with the laws of nature as well as with the laws of thought, it is because of the conformity of the order in nature and the order in thought. We are adapted to our environment, to the world in which we live, in such a way as to make possible not only our material living, but also our