ABSTRACT
It was assumed in ch. I that there can be no measurement, just as there can be no true representation of change of position, unless the space in which it takes place is structured by a system of references or coordinate-system. This is because, in essence, measurement is a form of movement, one which consists in placing a measuring rule against whatever is being measured and transferring it as many times as the part chosen as unit will go into the whole. This assumption may well seem exaggerated, if, to take an example, we have a mental picture of a foot rule with feet and inches already marked on it, standing upright against a wall, with no thought as to how it came to be there; and a fortiori if our thought is of an elementary mensural operation, such as that involved in moving a measuring rule which is taken to be constant, that is, when it is assumed that the rod does not alter in length. But a foot rule as we know it is the end-product of operations carried out in the past, which means that there is no point in studying how children use ready-made foot rules until we have investigated the way they make up their own foot rules or mensural units, even if the latter are crude to a degree and serve only a momentary purpose.
