ABSTRACT
In the last chapter we saw how the simplest curve is constructed, that is, the circle. However, the method we used is not the only method available to the child in elaborating such shapes, for the notion of‘locus’ can be appreciated by him only in its most elementary applications. In order to study the circle further and to advance to the construction of more complex curves, we need to use a more direct intuition of movement than was necessary for the study of equidistances. Especially appropriate are those curves known as ‘mechanical curves’. These were well-known to the Greeks but paradoxically omitted from Euclid’s geometry because they cannot be drawn with ruler and compasses alone. In this chapter, the last in Part III, we shall study how far children learn to understand how, by following paths of movement, we can construct circles, spirals and cycloids. A study of the last two curves should prove all the more rewarding in that it provides an opportunity of carrying a stage further the enquiry which began with the circle, since they introduce problems which cannot be solved before the level of formal operations. The present study in the field of geometrical generalization therefore corresponds with the research into the sum of the angles of a triangle (ch. VIII). Questions concerned with spirals and cycloids require the level of formal operations because they involve two reference systems instead of one. If a cylinder is slowly rolled about its axis while an ant is made to walk down its length, keeping throughout to whatever portion is uppermost, the total movement so described will be a spiral. Similarly, if we were to attach a lantern or a red disc to the rim of a cartwheel in motion, the lantern or disc would describe a cycloid curve (i.e. a series of hoops). For a child to predict what curves will result, he must mentally combine the rectilinear movement of the ant with the circular movement of the cylinder, or the circular motion of the red disc with the forward motion of the wheel. In either case, we have two simultaneous movements each of which must be referred to its own coordinate system. In order to reconstruct the total motion the observer must determine a few points and then generalize from them, as in the case of the loci in ch. IX, but the problem is more difficult because it requires the synthesis of two separate paths.
