ALL degrees of arithmetical aptitude may be found among the human races, from the genius of a Newton and a Laplace to an absolute inability to conceive the abstract notion of number aside from concrete facts furnished by direct perception. The savage is not deficient in the perception of the multiple. He never confounds one tree with two, or two with three, or four; and is well aware of the difference between two, three, or ten men, when he is going out to fight them. The thing that he can not do is to abstract the idea of a number from the things to which it is attached, and generalize it, without reference to the concrete objects with which he has seen it associated. He may comprehend two, because it is associated with his two hands and his two feet; three, by the aid of the triads with which he is acquainted, and of the triangle with which those objects may be arranged; four, from the four limbs of animals and the four corners of a square. But his ability to form such conceptions is very limited. The first steps in learning in this direction, in savages and children, are to distinguish the abstract notions of unity and plurality, and in plurality, of two and three from larger pluralities. The difficulty in the way of their reaching a concept of abstract numbers is their inability to form a mental representation: four trees not being identical in the savage's thought with four stones, he can not imagine that there is anything common between them. Nevertheless, he can distinguish clearly enough between four trees and three others, and the two groups will leave quite different impressions in his mind. Four trees in a row will also make a different impression from four trees in a square. He is most struck with differences of distribution in space, and derives from them his notions of differences in plurality. While he is a poor arithmetician he is a good geometrician.
It is by the exercise of this faculty that he finds his way so readily where he has once gone. He recognizes a wood he has been in by the relative distances apart of the trees, their heights, sizes, the inclination of their trunks to one another, the profile of their masses, and their kinds. He learns the landscape by the relief and accidents of the ground, the wave-lines of the horizon, and a thousand details which he fixes upon his memory by a single keen observation so clearly as to give imagination no chance to play tricks with him. He estimates distances by the weakening of tones and the convergence of the lines and planes of the