When each of our surveyors, both going at the same speed, has finished his task of measuring the round of the earth, will they both have the same result? Evidently not. As the super-observer in the sun will see, the yard of the surveyor who travels eastward is shortened by velocity in virtue of the Fitzgerald-Lorentz contraction. On the other hand, the yard of the surveyor who travels westward does not experience this contraction, as the super-observer on the sun, in reference to whom he remains stationary, would see. Consequently the two surveyors reach different figures for the earth’s circumference, the one who travels westward finding a result a few yards less than that of the other. Yet it is obvious that when they proceed to measure the earth’s diameter, travelling at the same speed, the two observers will reach the same figure for it. Hence the π which expresses the proportion of the earth’s circumference to its diameter on the ground of actual measurement differs according as the measurer travels in the direction of the earth’s rotation or in the opposite direction. Therefore, as the real values of π are different, they cannot be the unique and quite definite figure of classical geometry. Therefore the real universe does not conform to this geometry. These differences, in the illustration we have given, are due to the earth’s rotation. From the standpoint of gravitation the earth’s rotation has centrifugal effects which modify the centripetal influence of weight. We have seen, moreover, that for the surveyor whose speed equals that of the earth’s rotation the value of π is smaller than for the observer whose speed seems to be double that of the rotation. Thus the effects of weight being the reverse of those of rotation, or of centrifugal force, it follows (it would be just as easy to prove this as the preceding) that the effect of weight is to give π something less than its classical value. In a word, in the universe real circumferences traced upon gravitating masses, such as stars, are, in proportion to their diameters, less than they are in the Euclidean geometry. The difference is generally very slight, it is true. But there _is_ a difference. If we put a mass of a thousand kilogrammes in the centre of a circle that is ten metres in diameter, the figure π will differ in reality from its Euclidean value by less than one-thousand-million-billionth. In the neighbourhood of such formidable masses of matter as the stars are, the difference may be far greater, as we shall see. This is the origin of the divergences between Newton’s law of gravitation and that of Einstein: divergences which observation has settled in favour of the latter. But we will not anticipate. * * * * *