Equality, once having come to be definitely conceived, was readily applicable to other phenomena than those of magnitude.  Being predicable of all things producing indistinguishable impressions, there naturally grew up ideas of equality in weights, sounds, colours, etc.; and indeed it can scarcely be doubted that the occasional experience of equal weights, sounds, and colours, had a share in developing the abstract conception of equality—­that the ideas of equality in size, relations, forces, resistances, and sensible properties in general, were evolved during the same period.  But however this may be, it is clear that as fast as the notion of equality gained definiteness, so fast did that lowest kind of quantitative prevision which is achieved without any instrumental aid, become possible.

The ability to estimate, however roughly, the amount of a foreseen result, implies the conception that it will be equal to a certain imagined quantity; and the correctness of the estimate will manifestly depend upon the accuracy at which the perceptions of sensible equality have arrived.  A savage with a piece of stone in his hand, and another piece lying before him of greater bulk of the same kind (a fact which he infers from the equality of the two in colour and texture) knows about what effort he must put forth to raise this other piece; and he judges accurately in proportion to the accuracy with which he perceives that the one is twice, three times, four times, etc., as large as the other; that is—­in proportion to the precision of his ideas of equality and number.  And here let us not omit to notice that even in these vaguest of quantitative previsions, the conception of equality of relations is also involved.  For it is only in virtue of an undefined perception that the relation between bulk and weight in the one stone is equal to the relation between bulk and weight in the other, that even the roughest approximation can be made.

But how came the transition from those uncertain perceptions of equality which the unaided senses give, to the certain ones with which science deals?  It came by placing the things compared in juxtaposition.  Equality being predicated of things which give us indistinguishable impressions, and no accurate comparison of impressions being possible unless they occur in immediate succession, it results that exactness of equality is ascertainable in proportion to the closeness of the compared things.  Hence the fact that when we wish to judge of two shades of colour whether they are alike or not, we place them side by side; hence the fact that we cannot, with any precision, say which of two allied sounds is the louder, or the higher in pitch, unless we hear the one immediately after the other; hence the fact that to estimate the ratio of weights, we take one in each hand, that we may compare their pressures by rapidly alternating in thought from the one to the other; hence the fact, that in