396 THE POPULAR SCIENCE MONTHLY. 



Equally self-evident it will appear that, instead of representing 

 only an additional exercise, sej^arated from the rest of the instruc- 

 tion, 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 adapta- 

 bility. A pui)il 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, never- 

 theless, the curriculum of a general popular, say, public-school or 

 artisan education, varies to some extent, at least, from a prepara- 

 tion 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 com- 

 mon 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 j)arts of the said object- 

 ive 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 mathemat- 

 ical handling of the subject, subjecting facts to more minute cal- 

 culations, 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 inno- 

 vation. 



1. Mathematics has its origin in the concrete and not in the ab- 

 stract, and therefore is more easily approached and more success- 

 fully 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., Stew- 

 art's, is conceivable to us only conjointly with the actual experi- 

 ence 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 our- 

 selves 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- 



