In mathematical studies, the æsthetic interest of form, or the active interest of actual performance of problems, is not the sole or even the chief interest that should be appealed to. But the pupil should feel that he is making a progressive mastery of the principles of number. It is a pleasure to apply a rule, to solve a problem neatly; but it is a still greater pleasure to comprehend thoroughly the meaning of the rule, to grasp and to feel its universality, so that although it is not worth while, as Herbart suggests, to urge the ultimate function of mathematics in the life of the world, it is quite worth while to set up those immediate ends of interest such as appear in the activity of solving problems, in the æsthetic appearance of the work upon paper or board or slate, and in the comprehension of mathematical principles. These ends are near at hand; they can be made to appeal to the pupil through the quality of the work that the teacher demands of him. The same is true in the natural sciences. Even though the ultimate function of biology is an idea too remote or too complex for the child to grasp with enthusiasm, the immediate mastery of a principle in physics, or the discovery of a law of plant life, or of a fact in chemistry, may be an end in which the pupil’s most intense interest can be excited. 102. Geometry has other advantages of association, advantages we have begun only recently to turn to account in earnest. Figures made of wood or pasteboard, drawings, pegs, bars, flexible wires, strings, the use of the ruler, of compasses, of the square, counted coins arranged in long or short, in parallel or diverging series,--all these may be offered to the eye _ad libitum_ and connected with other concrete objects. They may be made the basis of systematic employment and exercises, and this will be done more and more when the fact is once grasped that concrete ideas possessing the _proper degree of strength_ constitute the surest foundation of a branch of instruction whose success depends on the manner in which the pupil forms in his mind the ideas of spatial relations. This is not grasped, of course, by those who regard space once for all as a form of sense-perception common to all minds alike. A careful study of the data of experience will convince the practical educator that the opposite is true; for in this respect individual differences are very marked. Pupils rarely hit upon geometrical constructions unaided; the aptitude for drawing, that is, for imitating the objects seen, is met with more often. It is easy by abstraction to form arithmetical concepts out of the apprehension of geometrical relations. To do so should not be regarded as superfluous, not even when the pupil has already fully entered upon his work in arithmetic.