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LESSONS ON FORM;
AN INTRODUCTION TO GEOMETRY.
BY C. REINER,
TEACHER OF MATHEMATICS IN CHEAM SCHOOL.
PRINTED FOR TAYLOR AND WALTON,
BOOKSELLERS AND PUBLISHERS TO THE UNIVERSITY OF LONDON,
UPPER GOWER STREET.
Bacon has made an observation to this effect- that a man really possesses only that knowledge, which he in some sort creates for himself. To apply to intellectual instruction the principle implied in these words was the aim of Pestalozzi. It is a principle admitting of various degrees, as well as modes of application, in the different branches of human knowledge ; but in no one can it be more extensively applied than in Geometry. That.science is peculiarly the creation of the human mind, in which, independent of external nature, and complete in its own resources, it builds up the solid but airy fabric of its abstractions. It needs no laboratory to test its conclusions, no observatory to obtain data for its calculations; rendering aid to other sciences, it asks none for itself.
Hence, that teacher will act most in conformity with the genuine character of the science, and consequently will render the study of it the most interesting and the most improving, who invites and trains his pupils to create the largest portion of it for themselves. In Geometry, the master must not dogmatise, either in his own person or through the medium of his book; but, he must lead his pupils to observe, to determine, to demonstrate for themselves. In order to accomplish this, he must study the intellectual process in the acquisition of original mathematical knowledge; and having ascertained what are the conditions of successful investigation, he must so arrange his plan of instruction as that these conditions may be perfectly supplied. He cannot fail to perceive that the leading requisites are a clear apprehension of the subject matter, and well-formed habits of mathematical reasoning. To these must of course be added a familiar acquaintance with the science as far as it has been elaborated. The master, led by these considerations, will, in directing the first labours of his pupils, consider it as his especial aim, to enable them to form clear apprehensions of the subject matter of Geometry, and then to develope the power of mathematical reasoning. Aware that clearness of apprehension can take place only when the idea to be formed is proximate to some idea already clearly formed—when the step, which the mind is required to take, is really the next in succession to the step already taken, he will commence his instruction exactly at that point where his pupils already are, and in that manner which best accords with the measure of their development. As his pupils are unaccustomed to pure abstractions, he will not commence with abstract definitions. But supposing them, through the medium of · Lessons on Objects' to have had their attention directed to the forms which matter assumes, he will present in his first lessons a transition from the promiscuous assemblage of forms to a particular group of them, consisting of the sphere, the cone, the pyramids, the prisms, and the five regular bodies. In conformity with the plan pursued in Lessons on Objects,' the pupils will examine these solids, state what they perceive at the first glance, then by more close and attentive examination, directed by the master, discover and supply the deficiencies in their first perception, and afford him an occasion for connecting their new ideas with adequate technical expressions.