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in other cases, that the object may possibly exist still without the subject, or may exist in a new relation to a new subject ; for the object exists only in and through that particular modification of the subject, and, on any other supposition, is annihilated altogether. Thus it is impossible to suppose that a triangle can, in relation to any intelligence whatever, have its angles greater or less than two right angles, or that two and two should not be equal to four ; though it is quite possible to suppose the existence of intelligent beings destitute of the idea of a triangle or of the number two. This is a necessary matter in the strict sense of the term ; a relation which our own minds are incapable of reversing, not merely positively, in our own acts of thought, but also negatively, by supposing others who can do so." (Mansel, Metaphysics, pp. 226-31, ed. 1871.)
206. “ The main branches of mathematical science were formerly stated to be arithmetic and geometry, springing out of the simple notions of number and space.
This is too limited a description. Unquestionably the science of numbers, strictly and demonstratively treated, and that of geometry, or the deduction of the elementary properties of figure from definitions which are entirely exclusive of numerical considerations, must be considered as the elementary foundations, but not as the ultimate divisions, of mathematics. To them we must add the science of operation, or algebra in its widest sense,-the method of deducing from symbols which imply
operations on magnitude, and which are to be used in a given manner, the consequences of the fundamental definitions. The leading idea of this science is operation or process, just as number is that of arithmetic, and space and figure of geometry : it is of a more abstract and refined character than the latter two, only because it does not immediately address itself to the notions which are formed in the common routine of life. It is the most exact of the exact sciences, according to the idea of their exactness which frequently entertained, being more nearly based upon definition than either arithmetic or geometry. It is true that the definitions must be such as to present results which admit of application to number, space, force, time, &c., or the science would be useless in mathematics, commonly so
but it is not the less true that a system of methods of operation, based upon general definitions, and conducted by strict logic, may be made to apply either to arithmetic or geometry, according to the manner in which the generalities of the definition are afterwards made specific.” (English Cyclopædia, Mathematics.)
207. “ The methods of observation of quantity in general are, Numeration, which is precise by the nature of number ; the Measurement of Space and Time, which are easily made precise; the Conversion of Space and Time, by which each aids the measurement of the other ; the Method of Repetition ; the Method of Coincidence or Interferences. (Whewell, Nov. Org. Ren. p. 145.)
208. The branches known as the Natural Sciences can be taught by Object Teaching in so far as facts are needed and can be observed, for the things shown or observed are the immediate objects to be learned. History cannot, from its nature, be taught objectively, except it happen that the learner can be an eyewitness of the events narrated. Language and Literature can be taught by Object Teaching, they being ends unto themselves, and subjects of inspection. Applied mathematics, being themselves illustrations, can hardly be taught illustratively.
209. As soon as perception and determination have given to the mind knowledge of individual facts, the power of retention holds them for ture use they are reproduced and represented in consciousness where Thought seizes them, and constructs science from them. When knowledge of individual objects is gained, the usefulness of the objects ceases. Als beyond is a work of the power of Thought. Hence the stages, for subject matter that will admit of it, are :-(1) Knowledge of facts obtained by Perception and Discrimination, which is the prov
ce of Object Teaching ; (2) The activity of Thought upon this knowledge, aided as it may be by Imagination. In this second stage, Objects are rather a hindrance if present than an aid, because they are so much useless material that should be put aside-Perception has done its work, Discrimination has separated, and now the senses may slumber while Thought is rearing science out of similarities and identities—any energy of attention which is diverted, at this time, towards the objects themselves is so much abstracted from Thought, which is thereby weakened, and Science so much endangered, for Science is not possible without Thought.
“A real experiment is a very valuable product of the mind, requiring great knowledge to invent it and great ingenuity to carry it out. It may be said that a boy takes more interest in the matter by seeing for himself, or by performing for himself, that is by working the handle of the air-pump: this we admit, while we continue to doubt the educational value of the transaction.
The function of experiment, properly so called, in the investigation of the laws and processes of nature can hardly be unduly exalted; but it may be said of the experimenter, as of the poet, that he is born and not manufactured." (I. Todhunter, Conflict of Studies, pp. 16-19.)
210. Ignorance of the province of Object Teaching leads to disaster, in practice, in mental discipline. This teaching addresses the attention of Perception and Discrimination. Then the objects have served their purpose. If the knowledge of facts which the learner has obtained be not wrought up by Thought into Concepts, which are general in their character and form the data for Reasoning, his mind is left far short of discipline. For true intellectual power comes only by constant exercise of Thought, and Thought busies itself only with mental products. Neglect of demanding maximum amounts of work for the powers of Thought of pupils habituates them to superficial scholarship—they may be apt at observing, when objects are placed before them, but they will develop little power of independent research, of vigorous application to thinking, or of the power to generalize. The mere acquisition and memorizing of a number of isolated, heterogeneous, or unrelated facts, is neither learning nor discipline, whether the facts are obtained from personal observation and examination, from oral statements of teachers, or gleaned from books.
(c) ON DISCOVERING METHODS OF TEACHING
211. Having outlined the powers of knowing and the nature of subject matter, it yet remains to investigate the Methods of Teaching, when they are to be applied in the teaching of any given subject. “In every Treatise upon any Science two Points are indispensably required ; the First, that the science which is the subject of it be fully explained ; the second,
that plain Directions be given, how and by what method such science may be attained.” (Longinus, On The Sublime, pp. 1-2, Tr. by Wm. Smith, 1739, London.) It is not the purpose of this discussion to develop a complete Method and Mode in any subject_simply to present the magnitude, importance, and direction in general, of Methods in special studies. To discover a Method to teach any branch is no