But even if the ultimate geometrical truth is hidden behind the veils of Maya,[6] we are yet left with the consolation that the method of approximation, even when applied to a relatively modest degree, produces remarkable results in the realm of numbers. Let us consider for a moment in the simple figure of a circle the ratio between the circumference and the radius.
[6]Maya = appearance.
As we know, this ratio is constant, and is called in honour of the man who first gave a trustworthy value for it, Ludolf's number, namely, π (pi). Thus it makes no difference whether we consider a circle as small as a wedding-ring, or as large as a circus arena, or even one the radius of which is as great as the distance of Sirius. And it makes just as little difference what happens to the circle whilst it is being measured; the above ratio must remain constant.
But here, too, a contradiction makes itself heard, issuing from one section of modern science. It calls to mind the saying of Dove that when professors are not quite sure about a thing they always preface their remarks with the phrase: "it is well known that" ... We should be well advised in avoiding this method of expression altogether, for even when we feel quite sure, the ghost of the unknown lurks behind what we fain would call well known.
The theorem that all circles without exception are subject to the same measure-relation belongs a priori to the synthetic judgments. But fields of thought have been discovered in which the a priori has lost its power. Mathematics—once a quintessence of synthetic statements a priori—is now regarded as being dependent on physical conditions. Physical conditions, however, are empirical and subject to change. Therefore, since the a priori is not subject to change, we encounter a discrepancy. It leads to the question: Is the Euclidean geometry with which we are familiar the only possible geometry? Or, in particular: Is π the only possible measure-relation?
Einstein replies in the negative. He not only shows how another geometry is possible, but he also discloses what once seemed inconceivable, namely, that if we wish to describe the course of the phenomena of Nature exactly by means of the simplest laws, it is not only impossible to do so with the help of Euclidean geometry alone, but that we have to use a different geometry at every point of the world, dependent on the physical condition at that point.
From the comparatively simple example of two systems rotating relatively to one another, Einstein shows that the peripheral measurement of a rotating circle, as viewed from the other system, exhibits a peculiarity which does not accompany the radial measurement. For, according to the theory of relativity, the length of a measuring rod is to be regarded as being dependent on its orientation. In the case quoted, the rod undergoes a relative contraction only when applied along the circumference, so that we count more steps than when we measure the circumference of the same circle at rest, that is, in non-rotation. Since the radius remains constant in each case, we get a relatively greater value for π, which shows that we are no longer using Euclidean geometry.
Yet, formerly, before such considerations could even be conceived in dreams, this π was regarded as absolutely established and immutable; and observers used every possible means of determining its value as accurately as possible.
In Byzantium there lived during the eleventh and twelfth centuries a learned scholar, Michael Psellus, whose fame as the "Foremost of Philosophers" stretched far and wide, and whose mathematical researches were regarded as worthy of great admiration. This grand master had discovered by analytical and synthetical means that a circle is to be regarded as the geometric mean between the circumscribed and the inscribed square, which gives to the above quantity, as may easily be calculated, the value √8, that is, 2.8284271.... In other words, the length of the circumference is not even three times that of the radius.
We have the choice of regarding the result of Psellus as an approximation, or as mere nonsense. Every schoolboy who, in a spirit of fun, measures a circular object, say a top, with a piece of string, arrives at a better result, but the contemporaries of Psellus accepted this entirely wrong figure with credulous reverence, and continued to burn incense at the feet of the famous master. It is all very well for us of the present to call him a donkey. We have just as much right in saying that mathematicians differ, not in their natures, but only in the order of their brain functions. If a man like Psellus missed the mark by so much, it is possible that men like Fermat or Lagrange may also have erred occasionally or even consistently.