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THE

NEW ENGLANDER.

No. CXXXVI.

JULY, 1876.

ARTICLE I.-THE EDUCATIONAL FORCE OF MATHEMATICS.

MATHEMATICS, as part of a course of academical study, do not encounter the same objection which has been urged against the classical languages. Therefore they have no need of the vindication which has been so abundantly given to them. They have so direct and obvious practical uses; do so evidently subserve material interests, that they command a good price in the very markets of utilitarianism. On this very account, it may happen that their proper educational force and value are less considered.

Classical studies, having no so obvious applicability to material uses, could not have held their place in our systems of education, if scholars had not vindicated them by demonstration of their admirable effects upon the mind.

The people have been reminded, that the men who speak the purest, clearest, strongest English, are not usually those who have studied English only. When Webster poured luminous floods of thought upon the people's minds, or "shook their hearts" with the deep tones of their mighty vernacular,— though every word, and phrase, and idiom were intensely Eng

lish, they knew that those grand powers had derived no small part of their culture from earnest study of Roman and Greek writers. When Everett gathered crowds of the people to listen to his discourse of Washington, or other themes of patriotism; when he led them abroad over wide fields of thought; when in his sonorous diction, his faultless periods, and the magnificence of his various imagery, he exemplified "the infinite loveliness of nature," his hearers knew how long, and with what delight, he had communed with those ancient bards, and orators, and historians whose writings cannot die, whose languages (as he rightly insisted) are improperly called "dead," for they have gloriously outlived the nations that used to speak them.

Made to understand how much classical study has to do with the formation of such minds, the people are willing that much money and labor, and the priceless years of their children should be expended upon them. A Yankee utilitarian sees that to produce a Webster, a Seward, or a Sumner, is quite as practical an achievement as to produce a reaper, a plough, or a sewing machine.

On the other hand, the more obvious connection of mathematical studies with material utilities-with accounts, and landsurveys, and navigation, and civil and military engineeringso easily win the consent of all to their occupying a prominent place in the curriculum, that we may not be giving due consideration to their effects upon the mind itself.

The effect of mathematical study upon the mind may be considered with reference to

I. The cultivation of the powers of reasoning.-These are the powers which are employed in every search for truth, and in every effort to convey the knowledge of truth into other minds. They are directly concerned in all acquisition of knowledge, and all communication of it; in all learning, and in all teaching.

The tendency of mathematical studies to promote accuracy and precision of thinking, and of statement, is obvious. All mathematical processes demand absolute accuracy. They do not tolerate the slightest lack of precision. We conduct all these processes under the conviction that the slightest error will utterly spoil them. We understand that unless we be exactly right, there is no knowing how far we may be wrong;—that

unless our reckoning can be relied upon absolutely, it is not fit to be relied upon at all. Therefore we conduct mathematical processes, not as we carry wood, but as we carry porcelain; not as farmers drive oxen, but as pilots steer vessels. A single heedless step may shatter the costly vase; a moment of inattention may wreck the ship. Scrupulous accuracy, conscientious care to be right, is desirable in every class of studies. They are the successful investigators, the safe intellectual pilots, whom it most decidedly characterizes. For the formation of this habit and character, mathematical study has advantage over every other. In no other is the pupil so easily made sensible of the necessity of precision in every step-of accuracy in every process. In no other does the pupil's mind so readily see that there is no medium between complete success and total failure;—that to make a single mistake is to break one link in a chain, or to loosen one stone in an arch.

The view which the mind has of each step in a mathematical process must also be clear. All mathematical reasoning is demonstration. It all leads to certainty. There are no mathematical opinions; there is only knowledge. When the mind has once apprehended the proper evidence of a mathematical proposition, it cannot possibly have any doubt about it.

Mathematical expressions are eminently free from ambiguity. They convey the same idea to every mind, and always a definite idea. Algebraic symbols and geometrical figures are not like grammatical phrases, liable to different interpretations.

This clearness and definiteness of thought and expression are desirable in every department. The nearest approximation to the impossibility of being misunderstood, is the highest excellence of rhetorical expression. But what writer or speaker ever makes more than an approximation to it. No other study helps to this so much as mathematics. The mind which in youth patiently submits to the severe discipline of mathematical study; which holds itself to the careful processes, requires of itself the distinct conceptions, trains itself to the rigid accuracy of mathematical investigation, and knows the rapture of mathematical discovery,-will not in its maturity, be satisfied with vague conceptions and careless reasoning in any department of thought. Such a mind will not be easily misled by glittering

generalities, dazzled by brilliant declamation, bewildered by cunning sophistry, or satisfied with lazy guessing. Those who have no definite opinions; to whose minds no subject presents itself in clear outline; who are consciously incapable of thorough investigation, and must always be feebly credulous, or as feebly incredulous, or dreamily bewildered;-are not generally persons who in youth loved mathematical studies, and faithfully pursued them. Their minds have not felt any such bracing and balancing influence. More probably they have been specially fond of fictitious reading, and have indulged much in dreamy reveries, the present deliciousness of which is made more fascinating by the fond fancy that, while they thus doze and dream, the wings of their genius are growing.

II. The Cultivation of the powers of imagination.-These are the powers by which the mind represents to itself ideal objects, and scenes, and characters. They are of high importance, not only for purposes of enjoyment, but for purposes of improvement.

It is a wonderful fact, that the human mind is able not only to perceive an object which is presented to its senses-to view and comprehend a scene upon which it looks--to know a character in real life, with which it is conversant-but to present to its own contemplation objects and scenes and characters, different from any real ones which it has known.

The sculptor sees in the rough block of stone, the lovely Venus, the graceful and agile Mercury, the majestic Apollo; and the light of that fair vision guides him in every stroke of his chisel, through all the patient labor which realizes his idea, and presents it to the view and admiration of subsequent generations. The eye of genius beholds landscapes such as the sun nowhere shines upon; and, by means of colors spread upon canvas, or by words written upon paper, in a painting or in a poem-can present them to the delighted contemplation of other minds. The pen of genius has given to our acquaintance characters, not a few, which were never embodied in flesh and blood, but which are vividly present to our imagination, and have no small influence over us. The powers of imagination which such gifted minds have exemplified, exist, in various degrees, in human minds generally ;else works of art might as well be exhibited to brutes as to men.

In education we are concerned for the right culture of these