252. Aptitude for mathematics is not rarer than aptitude for other studies. That the contrary seems true, is owing to a belated and slighted beginning. But that mathematicians are seldom inclined to give as much time to children as they ought is only natural. The elementary lessons in combination and geometry are neglected in favor of arithmetic, and demonstration is attempted where no mathematical imagination has been awakened. The first essential is attention to magnitudes, and their changes, where they occur. Hence, counting, measuring, weighing, where possible; where impossible, at least the estimating of magnitudes to determine, however vaguely at first, the more and the less, the larger and the smaller, the nearer and the farther. Special consideration should be given, on the one hand, to the number of permutations, variations, and combinations; and, on the other hand, to the quadratic and cubic relations, where similar planes and bodies are determined by analogous lines. =Note.=--This is not the place for saying much that might be said concerning that which renders early instruction in mathematics unnecessarily difficult. But it may be remarked in brief that some of these difficulties arise from the terminology, some from the teacher’s accustomed point of view, and some from the multiplication of varying requirements. (1) The phraseology used forms an obstacle, even to the easiest steps in fractions. The fraction ⅔, for example, is read two-thirds, and, accordingly, ⅔ × ⅘, two-thirds times four-fifths, instead of, multiplication by two and by four, and division by three and by five. The fact is overlooked that the third part of a whole includes the concept of this whole, which cannot be a multiplier, but only a multiplicand. This difficulty the pupils stumble over. The same applies to the mysterious word _square root_, employed instead of the expression: one of the two equal factors of a product. Matters grow even worse later on when they hear of roots of equations. (2) Still more might be said in criticism of the erroneous view according to which numbers are recorded as sums of units. This is true as little as that sums are products; two does not mean two things, but doubling, no matter whether that which is doubled is one or many. The concept of a dozen chairs is not made up of 12 percepts of single chairs; it comprises only two mental products,--the general concept chair and the undivided multiplication by 12. The concept one hundred men likewise contains only two concepts,--the general concept man and the undivided number 100. So, also, in such expression as six foot, seven pound, in which language assists correct apprehension by the use of the singular. Number concepts remain imperfect so long as they are identified with series of numbers and recourse is had to successive counting.