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The master's next aim must be to cultivate the power of abstract mathematical reasoning. With a view to this end, he may advantageously avail himself of the knowledge, obtained by the pupils from the solids, in the manner above described. Here, then, he will lead them to deduce the necessary consequences from the facts which they know to be true, and then invite them to examine the object and see whether their reasoning has led to a correct result. Thus, if a child has ascertained and knows that two sides of different planes are requisite to form an edge, and that a certain solid (an octahedron) is bounded by eight triangular planes, he will be required to determine from these data the number of edges which that solid has. He will reason thus:-Eight triangular faces have twenty-four sides; two sides form one edge: therefore, as many times as there are two sides in these twenty-four sides, so many edges that body must have,—that is, twelve edges. This result being obtained, the object is presented to him for examination, and he perceives by actual observation the truth of that conclusion at which he had arrived by abstract reasoning.

These lessons form the basis of the Introduction to

Geometry, and their results are, correct ideas of the subject matter of the subsequent lessons, adequate expressions for these ideas, and sound knowledge of the definitions, which form the connecting link between physical and abstract truths.

In the former part of this work, a mode of accomplishing these points is set forth in the second part, the further development of the power of abstract reasoning is connected with a direct preparation for the study of Euclid's Elements. That work exhibits a series of mathematical reasonings and deductions, arranged in the most perfect logical order, so that the truths demonstrated rest, in necessary sequence, on the smallest possible number of axioms and postulates. But, admirable as it may be in itself, viewed simply in relation to the science, it is not, viewed pædagogically, an elementary work. It is fitted for the matured, and not for the opening mind. The judicious teacher will desire to present to his pupils the subject matter of Euclid in such a mode and in such order as that in studying it the higher faculties of their minds may be most effectually exercised and improved. For thus only can the intellectual food be assimilated to the

intellect itself be received, as it were, into its substance, and nourish, and strengthen, and expand its powers.

These Lessons on Form present a mode in which these principles are applied: other modes, perhaps better ones, may be arranged; we but say with


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Si quid novisti rectius istis
Candidus imperti; si non, his utere mecum.

It has been found in the actual use of these Lessons for a considerable period, that a larger average number of pupils are brought to study the Mathematics with decided success, and that all pursue them in a superior manner. There is much less of mere mechanical committing to memory, of mere otiose admission and comprehension of demonstrations readymade, and proportionably more of independent judgment and original reasoning. They not only learn Mathematics, but they become Mathematicians.

Hence, when Euclid's Elements and the higher branches of Mathematics are to be read, the pupils are found competent to demonstrate for themselves the greater part of the propositions, and have recourse to books only for occasional correction or im

provement of their processes, and for fixing more firmly in their memory the results.

These advantages arise from the application of a principle generally neglected in early education, but deserving of attentive consideration and universal adoption; namely, that "Every course of scientific instruction should be preceded by a preparatory course, arranged on psychological principles." FIRST


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