powers.

We are now to consider the influence of Mathematics

upon them.

Let it not be supposed that mathematical studies have only a restraining or repressive effect upon the powers of imagination. Rightly viewed, and rightly used, they furnish, at the same time, healthful stimulus and safe regulation.

The processes of geometry and trigonometry, and the conic sections, directly familiarize the mind with forms and figures. Using diagrams as representative of all possible figures, in the limitless space, the mind at once possesses itself of the univer sal truths concerning the relations and laws of these figures, and of the lines and angles which define them. Three dots upon a black-board, three signal poles set up on a field, three stars in the sky, are all seen to have identical mathematical relations. By the same mathematical principles and processes the positions and distances of them all may be exactly determined.

Can any human mind become the possessor of such a power and such an instrument; find itself able thus to carry up into the sky the same measurements which it used on the ground; able to speak in the precise terms of mathematics, and in the confident tone of demonstration, of distances reckoned in millions of miles; know itself the possessor of an instrument of computation, which loses none of its power and none of its accuracy, carried how far soever into the infinite space, and not feel its powers of imagination quickened, as well as its powers of reasoning strengthened? Will such a mind behold no forms but those made visible in material bodies; no landscapes save those which nature or art has realized? Will such a mind compute no magnitudes save those of existing bodies; no distances. beyond those within which visible suns are shining? Will such a mind limit its thought to the uppermost arch of the telescopic heavens? Nay, there, where reason reverently folds her wings, the wings of imagination will still be outspread. She will pursue her adventurous flight, through the shining spaces, heaven above heaven, filled with statelier systems, and glowing with clearer radiance, on and on, toward that supreme heaven which is filled with the uncreated and unapproachable light.

From even so daring flight, she evermore returns with no wildness in her eye, no stain upon her plumage. Her sober

sister Reason greets her with a complacent smile, for in all that flight she has been guided by Reason's own maxims, nor has . one stroke of her wing done violence to any of Reason's demonstrations..

A mind that is habituated to mathematical processes, invigorated by mathematical exercises, disciplined to mathematical methods, and emboldened by mathematical triumphs, is not likely to be wild in its adventures of imagination. Its flights may be high, but they will be steady; they may be daring, but they will be sustained.

Such was the mind of the Scottish Chalmers, whose cumulative periods pile themselves in cloud-like magnificence above you, now glowing with sun-lit splendors, anon darkening the landscape with awful shadows, while still you feel the granite solidness of his thought firmly supporting your steady footsteps. And no marvel, for that strong mind had experienced both the invigorating influence, and the wholesome regulation of mathematical study; and the earliest of his intellectual triumphs was in the mathematical class-room at St. Andrews, where, as his biographer tells us, "He was ready to guide his students steadily and consecutively along a strictly scientific course, but as they trod that path, he would have all their bosoms to glow with the same philosophic ardors which inflamed his own; for to him the demonstrations of geometry were not mere abstractions to be curiously and unmovedly gazed at by the cold eye of speculation. A beauty and a glory hung over them, which kindled the most glowing emotions in his breast. . . . And all that his beloved science was to himself, he would have her become to the youths in the class-room around him." "Under his extraordinary management," writes one of his pupils, "the study of mathematics was felt to be hardly less a play of the fancy than a labor of the intellect."

Nor may this rightly be set down as the triumph of an extraordinary genius, clothing with fictitious charms that which, of itself, is dull and dry and uninteresting. It was the work of an appreciative mind, doing simple justice to a noble science, and beautiful as noble, unveiling charms which before faulty or inadequate methods had too effectually concealed.

Those who are familiar with Chalmers's writings, especially his

"Astronomical Discourses," have in them most ample illustration of the efficacy of mathematical study in both nourishing and regulating the powers of imagination. Under such nurture and such regulation, these powers are developed in harmony with the powers of reasoning; these supporting and invigorating those, and those refreshing and adorning these. So grows the tree that is "planted by the rivulets of water," the sturdy trunk upholding and nourishing the leafy top; the ample foliage gathering, from all the air, refreshment and life for the trunk and the root.

Let the student of mathematics know to what appreciation of beauty this science can elevate him, and into what wide fields. of rapturous contemplation she will conduct him. Let him. dutifully submit to her discipline, and make himself master of her methods, remembering always that the first step towards triumphant mastery must be obedient subjection; his reward shall be, not only an enlarged power of abstract reasoning, but a vastly increased capacity for intellectual enjoyment. Let him know that if, in his youth, he will be simply faithful to this science, patient, tractable, diligent, she will do for the eyes of his mind just what one of her daughter sciences does for bodily vision, by her telescopic and microscopic lenses. She will make the boundary of the field of vision indefinitely recede, and she will bring into view ten thousand various forms of beauty and of life, too minute for perception by the unaided organ. She will not only aid his business, and his labor, and his acquisition of solid knowledge, but she will cheer his life by her joyous companionship; she will walk with him over all the fields of nature, and through all the galleries of art, and along all the paths of labor; and her frequent suggestions, and the continual application of her lessons will lighten his labor, will heighten his appreciation of every beauty, and will steadily deepen the tide of his enjoyment.

III. Moral Culture. It is not an accident, nor a blunder, whereby we have transferred the terms of mathematics into the language of ethics. We speak of a right action as intelligibly as of a right line or angle; of an upright man as of an upright column. As often as we speak of moral rectitude, of a line of conduct, of a rule of action, of square business transactions, or of "crooked whiskey;" as often as we call sinning a fall, and tendencies toward it inclinations, and steadfast virtue up

rightness, we illustrate the affinity of mathematical with ethical truths, in virtue of which the same terms are equally expressive of both.

Not to insist too far upon these etymological analogies, it is an essential consideration, that the careful and scrupulous habits of thought which mathematical study requires and cultivates is equally necessary to right moral culture.

The youth who patiently forms his mind to habits of scrupulous accuracy in mathematical studies, who constantly and patiently strives to conduct mathematical processes with perfect accuracy-we will not say that he can thereby gain the essence of virtue, but we do say that he is thereby learning the method of virtue, and is forming habits most helpful to the practice of virtue.

Admirably is this illustrated by the authentic biography of Washington. Few books had he in his youth, few teachers, and scanty school privileges. But his was a youth of diligent, faithful, successful study. The records of his mathematical studies have been preserved, and are among the most interesting memorials of him.

Looking over his copy-books, observing their scrupulous neatness and pains-taking accuracy, seeing how he was training himself to rigid correctness, attentively considering the plots of his surveys, the distinct setting forth of the elements of all his computations, and the full and clear presentation of all his processes, in all their unimpeachable accuracy, who does not perceive a real and beautiful correspondence between that faithful and conscientious self-discipline of the boy and the Aristidean integrity of the man?

A youth who despises such pains-taking accuracy and carefulness, may achieve some sort of success, but the blunders and the blots in his life will be apt to bear a pretty accurate ratio to those in his copy-books. He may become a brilliant man, but not a reliable one. Men may admire him, but they will not trust him.

Now, after all, what human tribute is it so good to receive as, that one's neighbors and his country should show that they have no interests too precious or too sacred to be entrusted to his care? Nay, what divine tribute is more blessed than this "Thou hast been faithful?"

IV. Religious Culture.-No other mental exercises are better fitted than the mathematical to prepare the mind for the most worthy views of God. Who else can so intelligently consider the "heavens the work of God's fingers," as he who can carry his reliable measurements into them; can accurately survey their mighty spaces; can calculate the distance and size and weight of the heavenly bodies; their motions also, and the dimensions of their orbits, and the periods of their revolutions?

"The undevout astronomer is mad." Mathematical science enables us to attain views of God at once the most grand and the most sober, the best fitted to stir the mind to its utmost depths, and to tranquilize it with the deepest solemnity.

Religion evermore looks toward and into eternity. Now, by mathematics we cannot indeed compute infinity of duration any more than by mathematics we can measure infinity of space, or estimate infinity of power. But this science which teaches us how to compute all that is computable, and to measure all that is measurable, does surely best prepare us rightly to regard all that lies beyond its reach. In the very processes of measuring the fields of time, it brings us to the best positions from which to look out upon that ocean whose further shore its best instruments do not enable us to see-which really has no further shore.

We have referred to a Scotchman of the last generation, eminentin both mathematics and theology, even more eminent in practical Christian philanthropy. We recall a scene in the Scottish General Assembly, in which in the ripeness of his powers and of his piety, he found occasion to repudiate a pamphlet which he himself had written in his early days of superficial religious thought, and of secular ambition. Adverting to his ambitious pursuit of mathematics and his low estimate of the work of a pastor, he exclaimed-" What, Sir, is the object of mathematical science? Magnitude, and the proportions of magnitude. But then, Sir, I had forgotten two magnitudes. I thought not of the littleness of time; I recklessly thought not of the greatness of eternity."

Never surely did Chalmers more truly honor mathematical science than when he thus illustrated the vigor and decisiveness of thought which it had given him, in so eloquently rebuking his own foolish preferring of the earthly to the heavenly, the temporal to the eternal.