Equally self-evident it will appear that, instead of representing only an additional exercise, separated from the rest of the instruction, perfect correlation with the same has to be established, if anything like serious results or benefits are to be expected. In other words, manual or industrial training can be summed up as the experimental adjunct of the abstract studies, verifying the correctness of the conclusions arrived at, in the shape of laws, theories, or principles, and demonstrating their practical adaptability. A pupil having some idea about the actual use of the things he learns is, without comparison, the superior of one who only hopes to find it out some time at college—if ever. As, nevertheless, the curriculum of a general popular, say, public-school or artisan education, varies to some extent, at least, from a preparation for a future profession, the special course in the experimental departments of the two will have to differ respectively.
Objective teaching, or the practical acquaintance with one's surroundings and Nature's chief subdivisions, will remain common to both—the value of such instruction being enhanced through the so called Socratic method of cross-questioning, but at a point of abstract concentration certain parts of the said objective instruction, say, in form and number, may in their further development form a line of demarkation. Form might lead to practical working draughts in the manual training of the first case, and number, entering here as the necessary accessory, would serve only for short immediate calculations. A more mathematical handling of the subject, subjecting facts to more minute calculations, and early introduction of the mechanical equations of cause and effect, will form the central pivot in the second higher grade of schools. Certain generally lightly treated truisms may be added in shape of suggestions to enable any worthy pedagogue to start logically in the progressive line of our educational innovation.
1. Mathematics has its origin in the concrete and not in the abstract, and therefore is more easily approached and more successfully taught on this basis. One has to start with actual things—dimensions, forms—especially when dealing with pupils of the elementary grades.
2. Space, notwithstanding Hamilton's arguments, viz., Stewart's, is conceivable to us only conjointly with the actual experience of muscular exertion; its notion originates with the turning of the eye of the new-born child and our pedimetric or other dynamic measurements.
3. Language is by no means our only agency for making ourselves understood; a few lines, if properly drawn, will tell a better story about many things in technics than a long-worded lawyer's version. The short-hand expression, sketching, is therefore an in-
