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in dispute among the learned, and upon some of them there is not a perfect agreement even to this day. The foundation of all geometry is in our idea of space, and even that is a subject of irreconcilable differences of opinion. The followers of the immortal Kant (and even many of those who in most things else break away from his paths, follow him here), declare that space has no existence other than in the mind of the observer, and some of them further affirm that it is an idea never attained by those born blind. Space, according to these philosophers (and I think they comprise a majority of those who have shown a decided turn for metaphysical speculation), is only a way in which we view things; it is a law of thought, and the proof that it is so, lies in the fact that we necessarily conceive of things as occupying space. Since we cannot even mentally divest ourselves of the idea of space, it must be, say these men, a part of ourselves and non-existent out of ourselves.

On the contrary, many geometricians of the highest ability believe, as all ordinary men believe, in the reality of space external to the mind.

external to the mind. To them the doctrine of “the laws of thought,” the doctrine that space and time are forms put upon things by the mind, appears the very quixotism of philosophy. It amounts to saying that we know that space does not exist because we cannot help believing that it does exist.

But, while the fundamental conceptions of geometry, while the very existence of its subject-matter, space, is thus in dispute, and the learned world so equally divided on the point, is geometry to be excluded from the course of public instruction, or shall , any believer in space hesitate in his public teaching to proclaim his faith, and to utter his protest against the transcendental delusions and errors which have accompanied the growth of transcendental philosophy? Certainly no man would ask it.

Again, there are those who deny that space is the subject matter of geometry. The French school of positive philosophy say that geometry is the science of measuring extended bodies. Our own countryman, H. C. Carey, in the introductory chapters of his splendid work on Social Science, denies to geometry the name of science. Science, he says, is the knowledge of external nature, and the mathematics is only a peculiar language to aid in investigating nature and recording results. With still grosser ideas, another American writer maintains that a line is an infinitesimal thread, and a point an atom, thus reducing geometry to the science of extended bodies considered as extended.

Does, therefore, any teacher think that we should not carefully guard the student of geometry from supposing that he is studying either material shapes or mere abstract rules of measuring material things?

The like conclusion, that the mere fact of the ex

unjust in compelling them to pay taxes to sustain such schools?

The writer of this article is a Protestant of the most radical tyl does not stop with Martin Luther, but who protests against all obst:

istence of differences of opinion concerning a science, does not unfit it to take its place in a course of public instruction, might be drawn from all the other sciences. We frequently find men, through a deficiency of mathematical clearness, opposing the Newtonian doctrine of gravity, and stating new theories of mechanics ; — or putting out new ideas concerning optics. Even in the Reports of the Proceedings of the American Association for the Advancement of Science, you will find a paper explaining the zodiacal light, the whole paper being based on the astounding assumption that both ends of a straight line may be far below the horizon, and yet a large part of the line remain above the horizon.

I might go into chemistry and find still more striking instances of differences of opinion. I need not allude to bygones, such as the contest concerning the nature of chlorine, but may appeal to the present state of the science. Who shall set bounds to the doctrines of allotropism, and assure us that the whole list of elemental substances is not after all but a series of allotropic forms of one and the same substance ? Yet if this be so, who shall be able to define the science of chemistry? Nevertheless the science is pursued and taught and justly considered an essential element in a course of public instruction. It is even demanded as a practical science for the use of agriculturists, although the most vital points concerning the value of mineral and organic manures are still in

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dispute. We may therefore set it down as certain, that differences of opinion concerning the truth or falsehood of a doctrine do not constitute, in the judgment of wise men, any reason for omitting the discussion of those doctrines from the course of public instruction. Nor, on the other hand, is perfect unanimity of opinion upon a point, or perfect certainty of a doctrine, any reason why such points should be included in our general course of study. No mathematician doubts the truth, or the utility, of that little book on the division of superficies, published by John Dee and Frederic Commandine, and thought by Dee to be from the hand of the great Euclid, but no one has ever proposed to incorporate it, or any part of it, into a course of academical study.

It is, therefore, manifest that the questions of certainty or uncertainty, and difference or unity of opinion, are only secondary considerations in deciding whether a certain study should or should not enter into the course of instruction in this or that form.

The general principles which I have already enunciated concerning the relation of each part to the whole, and to the whole hierarchy of sciences, to the length of the curriculum, and to the capacities of the individuals, these general principles are sufficient to decide the minor as well as the greater questions. Those parts of geometry are to be introduced which bear most directly upon the progress of the student in higher mathematical studies, and to the acquire

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unjust in compelling them to pay taxes to sustain such schools ? The writer of this

article is a Protestant of the most radical ty does not stop with Martin Luther, but who protests against all obs ment of mechanical, chemical, botanical, or zoological knowledge. Those parts are to be excluded which are not thus connected. And inasmuch as no knowledge is wholly disconnected with other branches (the work to which I just alluded as resuscitated by Dee, is capable of some useful applications in surveying, etc.), we must omit those things which have fewest or least important connections, and retain those which are most directly in the highway of truth.

So in religion. It makes no essential difference whether there is, or is not, an agreement of opinion on a point; it does not necessarily follow that it is important if all are agreed, or that it is unimportant if men are disagreed upon it. The question rather is, Is this doctrine important in its connection with other matters? Does it throw light on the course of history? Does it have a connection with the physical sciences ? Does it bear directly on the moral character of the pupil? Does it bear upon his religious character, his habits of piety?

We have tested, on such general principles, the question whether religion ought, or ought not, to enter into the course of public instruction, and have decided that it should. But shall we teach Theism or Atheism, Pantheism or Exotheism ? By the last term, I designate that steadfast ignoring of religious questions to which an exclusive attachment to physical researches may lead one.

lead one. But this and blank Atheism we have already condemned as unfit to be taught

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