ABSTRACT
The previous chapter studied one way in which the simplest curve, the circle, can be constructed. The present chapter sets out to study how far children can learn to understand another way of constructing not only circles, but also more complex curves, such as spirals and cycloids (the so-called mechanical curves). This other way is by following paths of movement. As applied to spirals, however, it involves two reference systems instead of one and introduces problems which—like that of the sum of the angles of a triangle— cannot be finally solved before the level of formal operations. If a cylinder is revolved while an ant moves along its length, keeping to whatever portion is the uppermost, the total movement so described will be a spiral. Similarly if a red disc or lantern is attached to the rim of a cartwheel in motion the disc or lantern would describe a cycloid curve. In order to predict either of these the horizontal and rotational movements must both be taken into account. This is more difficult than the loci problems as it requires the synthesis of two separate paths.
