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bols? Nay, who, having seen them, is able to obtain from the symbol, any just idea of their distinctive features, without a close and vigorous exercise of the constructive imagination?

Or take, if you will, the diagrams in solid geometry. Suppose that you wish to represent a cube. Draw it in perspective, and how exactly will the figure, in the relative direction and length of its receding lines, or the comparative magnitude of their included angles, correspond to the actual facts of the solid itself? Draw it according to the isometrical projection, and, while you secure a technical conformity in the lines and angles, to the form of the body represented, the figure fails to give you the receding or remote elements, in their proportions or relations, as the mind apprehends them. Your exact geometrical diagram must, then, be, of necessity, either false to the objective form, or false to the subjective image.

THE ALLEGED DEFECT GROWS OUT OF THE VERY NATURE OF LANGUAGE, AND THE EXACT ACCORDANCE OF THE DIAGRAMS WITH IT.

But the case is even stronger than this. Let the objector look nearer home. He complains that the diagrams do not, and cannot, represent all the relations and offices of the various elements in the sentence, with equal accuracy. But how is it in the case of the language itself? Speech, with all its wonderful subtlety and fullness, does not, and does not even pretend to, present all of our ideas in their real relations, with any greater accuracy. For example, you employ several adjectives as alike relating to and modifying the same noun; you use the same adjective as relating alike to two or more

nouns, and modifying them in common; you give two or more subjects a similar and common reference to the same predicate; you present several objects as similarly dependent on one verb, or different subsequents as holding a common relation to the same preposition; or you give one pronoun a diverse, and even widely separate dependence on two altogether dissimilar parts of speech; and yet, by no artifice whatever, either in oral or written expression, can you give the elements having this common relation or dependence, a grammatical position or connection exactly similar. In the necessary order of the words, only one of the terms thus applied in common, can hold the exact place indicative of its relation or dependence : the relation or dependence of the others must, by a. purely logical exercise, be carried on, either through or over the intervening terms, to the desired point. If, then, the diagrams fail, it is just where, and precisely as, the language which they represent fails;-in other words, the two coincide exactly in the point raised by the objector. One of two conclusions is, then, inevitable, either the language itself is first, and equally, in fault, or the diagrams as a means of representation, are intrinsically more consistent and accurate because of the very feature which the objector condemns. It will be seen, then, that this objection, like the former one, recoils upon the objector himself it shows him to be as ignorant of the language, as he is of the diagrams.

4th Objection.-They are unnecessary.

As a final objection, it is sometimes urged, that the diagrams, if not a positive hindrance, are, at least, unnecessary. The ground taken by the objector is, that, granting all that may be claimed as to the capacity of the dia

grams to perform the work allotted them, it is not only possible for all the facts involved to be clearly apprehended in abstract, or without sensible illustration, but it is better discipline for the mind to be compelled to seek out and master them, through this abstract contemplation alone.

THE SAME HOLDS WITH EQUAL FORCE AGAINST GRAPHIC ILLUSTRATION IN OTHER BRANCHES OF SCIENCE.

To this objection, the answer is at once, that the same thing may be said of graphic illustration in other branches of science. The objection, if valid, holds, for example, equally good against the use of diagrams in geometry. In considering any proposition, it is, doubtless, possible for the process of demonstration to be carried on without the aid of a sensible figure, or diagram, and the discipline of such an exercise will be, beyond question, far more severe than that of the ordinary method. Nay, more, the use of the diagram in the study of geometry may be said to be even more inexcusable; since it is possible for the mind to construct the requisite figure in pure thought, and to proceed with the demonstration in accordance with its conceived elements and facts alone.

THE PROPOSED DISCIPLINE IS NOT ACCORDANT WITH NATURE, OR ACCESSIBLE THROUGH MORE PROFITABLE CHANNELS.

And yet, while this idea has been broached as practically of great importance, scholars have neither been foolish nor rash enough to press its adoption. And this, for the simple reason that so long as the spiritual activity is not only leagued with, but is somewhat subject to the

physical nature, so long the outward sense may and should be made helpful to the inward understanding, by affording it a sensible means of fixing attained facts, and bringing the whole truth, as it were, bodily before the apprehension. As to the matter of discipline, it is quite as clear that so long as the field of science is boundless and inexhaustible, instead of seeking development and discipline through the agency of needless detention and difficulty, the mind may far more wisely look for it in the direction of the more extended and inspiring advances rendered possible through these graphic aids to the readier and clearer comprehension of truth.

THIS ABSTRACT DISCIPLINE IMPRACTICABLE IN THE CASE OF THE YOUNG MIND.

But beyond this, however possible this abstract comprehension and retention may be in the case of the mature and somewhat disciplined mind, it can not but be altogether beyond the capacity of intellect both immature and untrained. Such intellect imperatively demands all the aid which can be afforded by graphic illustrations or diagrams. They are indispensable auxiliaries in its efforts to climb from the sensible, in the midst of which it has its being, and on which it is so dependent for its earliest ideas, to the region of the abstract or the ideal, in which it may possibly, at the last, become able to move unaided and alone.

THE PREVAILING TENDENCY OF THE LATER METHODS OF INSTRUCTION IS TOWARDS CONSTANT OBJECTIVE, OR SENSIBLE ILLUSTRATION.

And this cardinal principle is practically sustained by the entire tendency of modern instruction. Setting like

a flood-tide as it does (perhaps with a somewhat excessive impetuosity) towards objective means and methods in instruction, it gives evidence of a deep-seated conviction that the natural out-workings of the child's intellect are properly suggestive of the natural avenues to his intellect. Hence, everywhere, the endeavor of advancing science, and enlightened education for the young, is, to bring truth within the reach of objective presentation, and to perfect the art of sensible, or graphic illustration. But if this effort is possessed of any validity and excellence, there can be no just reason why the truths of grammatical science may not be thus symbolized in diagrams, as well as those of any other branch. The objection urged is then simply illogical and absurd.

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