A measure system is a group of congruent transformations of space into itself. Consider a rigid body occupying all space. Let this body be moved in any way so that the particles of the body which occupied points P_{1}, P_{2}, P_{3}, etc., now occupy points Q_{1}, Q_{2}, Q_{3}, etc. Then any point P_{1} in space is uniquely related to the corresponding point Q_{1} in space by a one-to-one transformation with certain characteristics. By the aid of these transformations we can achieve the definition of distance in a way which definitely determines the distance between any two points, provided that we can define what we mean by a congruent transformation without introducing the idea of distance. If we introduced the idea of distance, we should simply say that a congruent transformation is one which leaves all distances unchanged, _i. e._, if P_{1}, P_{2} are transformed into Q_{1}, Q_{2} then the distance P_{1}P_{2} is equal to the distance Q_{1}Q_{2}. But mathematicians have succeeded in defining congruent transformations without any reference to distance. There are alternative groups of such congruent transformations, and each group gives a different measure system for space. The distance P_{1}P_{2} may equal the distance Q_{1}Q_{2} for one measure system, and will not equal it for another measure system. All these different measure systems are on the same level, equally applicable. A being with a strong enough head could think of them all at once as applying to space. The result so far as it interests us in respect to the theory of relativity is explained on pp. 197-200, ending with "The most extraordinary fact ... same metrical system." This final sentence bears on Poincaré's assertion that the measure system adopted is purely "conventional." I presume that by "conventional" a certain arbitrariness of choice is meant; and in that case, I must express entire dissent. It is true that within the circle of geometrical ideas there is no means of giving any preference to any one measure system, and any one is as good as any other. But it is not true that if we look at a normal carriage wheel, and at an oval curve one foot broad and ten feet long, we experience any arbitrariness of judgment in deciding which has approximately the form of a circle. Accordingly to Poincaré the choice between them, as representing a circle, is entirely conventional. Again, we equally form immediate judgments as to whether a body is approximately rigid. We know that a paving stone is rigid, and that a concertina is not rigid. This again necessitates a determinate measure system, selected from among the others. Accordingly we conclude that (i) each being does, in fact, employ a determinate measure system, which remains the same, except possibly for very small variations, and (ii) the measure systems of different human beings agree, to within the limits of our observations. These conclusions are not the less extraordinary because no plain man has ever doubted them.