Locating a Point in Two or Three Dimensional Space 1

H_AVING observed how children locate a point on a straight line by measuring the distance from its origin to the point in question, we will now study their responses when asked to locate a point in two- or three-dimensional space. We shall be led to two important conclusions. (1) Sooner or later, children measure in two or three dimensions by means of paired measurements along the two axes of a right angle. Therefore, later in part three, we study angular measurement as such. (2) Children cannot locate a point in two or three dimensional space without first evolving a coordinate system. The importance of rectangular coordinates was made abundantly clear in our study of the construction of Euclidean space as a whole (C.C.S. chs. XIII, XIV), and again in the present enquiries in conjunction with the grouping of changes of position (ch. I), relations of distance and length (chs. III and IV), and finally measurement itself (ch. II). Here the problem of rectangular coordinates re־appears in metrical terms.