§ 23. 3. “Our lessons ought to start from the concrete, and end in the abstract.” What Mr. Spencer says under this head well deserves the attention of all teachers. “General formulas which men have devised to express groups of details, and which have severally simplified their conceptions by uniting many facts into one fact, they have supposed must simplify the conceptions of a child also. They have forgotten that a generalisation is simple only in comparison with the whole mass of particular truths it comprehends; that it is more complex than any one of these truths taken simply; that only, after many of these single truths have been acquired, does the generalisation ease the memory and help the reason; and that, to a mind not possessing these single truths, it is necessarily a mystery. Thus, confounding two kinds of simplification, teachers have constantly erred by setting out with “first principles,” a proceeding essentially, though not apparently, at variance with the primary rule [of proceeding from the simple to the complex], which implies that the mind should be introduced to principles through the medium of examples, and so should be led from the particular to the general, from the concrete to the abstract.” In conformity with this principle, Pestalozzi made the actual counting of things precede the teaching of abstract rules in arithmetic. Basedow introduced weights and measures into the school, and Mr. Spencer describes some exercise in cutting out geometrical figures in cardboard, as a preparation for geometry. The difficulty about such instruction is that it requires apparatus, and apparatus is apt to get lost or out of order. But if apparatus is good for anything at all, it is worth a little trouble. There is a tendency in the minds of many teachers to depreciate “mechanical appliances.” Even a decent black-board is not always to be found in our higher schools. But, though such appliances will not enable a bad master to teach well, nevertheless, other things being equal, the master will teach better with them than without them. There is little credit due to him for managing to dispense with apparatus. An author might as well pride himself on being saving in pens and paper. § 24. 4. “The genesis of knowledge in the individual must follow the same course as the genesis of knowledge in the race.” This is the thesis on which I have no opinion to offer. § 25. 5. From the above principle Mr. Spencer infers that every study should have a purely experimental introduction, thus proceeding through an empirical stage to a rational.