To our primitive ancestors measuring probably meant much the same thing as counting. The content of a flock of sheep was estimated merely by counting heads, and the length of a journey was recorded as being so many days' marches, the method being perhaps that of cutting a notch in a stick at the close of each day. It must soon have emerged that a measurement by integral numbers, while adequate for a flock of sheep, was quite unsuitable for the length of a journey, the reason, stated in modern scientific phraseology, being that sheep are “atomic” while a journey is not. At a quite early stage of civilisation, men must have appreciated the need for two distinct kinds of measurement—measurement by integers and measurement by continuously changing quantities. For a primitive people, all measurement of substance could be most readily expressed in terms of the space or area occupied, so that the two types of measurement reduce to counting and to the measuring of areas and volumes. Hence arise the two fundamental sciences of arithmetic and geometry; the shepherd was the primitive arithmetician, the land-surveyor and builder the primitive geometer.

Starting from such a basis it was natural that the earliest of thinkers should suppose that space was continuous, and that the objects that filled it were atomic. For space had always been the special province of the geometer, who had treated it as suited for continuous measurement, while objects in space had always provided occupation for the arithmetician, whose professional knowledge was concerned primarily with integers.