various objects which people her beautiful domains may stand or move in unbroken silence; but "In reason's ear they all rejoice, And utter forth a glorious voice." I say no more in explanation of this preliminary point. I wish to show that the applicability of it does not cease when we enter the formidable precincts of the exact sciences; but that the Great Creator has so implicated together law and enjoyment, has so wedded order and beauty, that even the severest processes of mathematics are not divested of all poetical charms, while their results are connected with the highest and most admirable poetry. The illustrations of this subject I arrange under three classes, viz. I. Those drawn from the Doctrines of Mathematics. II. Those drawn from its Operations and Results. III. Those drawn from its History. I. As to the Doctrines of the Mathematics, there is obviously some difficulty in pointing out their poetical character. In order to realize and perceive it, one must both be familiar with them, and must hold a liberal mind in respect to what is a poetical characteristic. The question, What is a poetical characteristic? though a necessary preliminary, it is not possible here to discuss. It is sufficient to say that that object which, in itself, or in its essential relations, excites the emotions which pertain to the sublime or the beautiful, causes a glow both of the imagimay lawfully be regarded as pos nation and the feelings, sessed of a poetical characteristic. Whatever it may be, — whether a scene in nature, or a trait of humanity, or an abstract truth, if its presentation to the mind excites the imagination and the feelings with that glow of pleasure or interest which springs from the beautiful and the sublime in any of their forms, it is so far poetical. Agreeably to what a distinguished writer has said, "All objects and passions which lift our thoughts from the dust, and stir the soul strongly, almost every thing which has in it a strong principle of impulse or elevation, has a claim to be considered poetical." * What I have to say, therefore, is, that many of the doctrines of the mathematics precisely answer this description. They excite and elevate the mind which contemplates them. They call out, with strong impulse, the imagination, and stir the feelings with emotions of the sublime and the beautiful. This is true even of much that may be found in the severest abstractions of the science. The most abstruse demonstrations are sometimes felt, by reason of their order, precision, and neatness, to have exquisite beauty. These, however, being unsuited to satisfy expectation in a public discourse, cannot be largely dwelt upon on the present occasion; but our illustrations must be drawn principally from the "mixed mathematics." "Pure mathematics" comprise arithmetic, algebra, geometry, fluxions, etc. Mixed mathematics is the application of these to natural philosophy, comprising astronomy, optics, mechanics, etc. Yet I do not despair of showing, intelligibly, that even in the pure mathematics there are doctrines whose sublimity brings them within the province of poetry. For instance, there is the doctrine of the asymptotes, which teaches that a right line may approach forever to a certain curve, and yet, though infinitely extended, will not touch it; and, although thus forever approaching it, never comes nearer by any appreciable quantity. What an idea for the imagination to dwell upon is this! Lines forever approaching towards each other, yet never able to meet! And so with the whole doctrine of infinite natures and quantities. Fontenelle has a work called the "Germ of the Infinite." There is poetry in the very title. In this book we find demonstrated such principles as these: If upon an infinite plane be drawn two infinite and parallel lines at a given distance, the area intercepted between them will likewise be infinite; but yet it will be infinitely less than the whole plane, and even infinitely less than the angular or sectoral space intercepted between two infinite lines that are inclined, though at never so small an angle. Because, in the one case, the given finite distance of the parallel lines diminishes the infinity in one of the dimensions; whereas, in the sector there is infinity in both dimensions. And thus there are two species of infinity in surfaces -the one infinitely greater than the other.* Is there any poetry which more severely taxes the imagination than demonstrated propositions like this? Poetry is said to dwell in the region of the imagination. Milton, and Dante, and Shakspeare, are so great because of the extraordinary power and efforts of their imagination. But no men have made so severe requisitions of this faculty as the mathematicians. They astonish us with their imaginary quantities, just as the poets with their imaginary beings. Many of their most interesting processes are connected with these imaginary quantities, and their grandest discoveries accomplished by their aid. They take the letter x, or n, which means they know not what, (it may be less than a thousandth part of one, or it may be several millions,) and pursuing this shapeless image through the dark, wind ing passages of an intricate investigation, leading they know not whither, they, by and by, come out into the broad daylight of clear and certain knowledge. Nay, remarkable enough, as I am told, this unmeaning formula is oftentimes not only an unknown, but an impossible quantity, and yet by means of it they arrive at important results. very The doctrine of fluxions presents an illustration of what we are saying, in a somewhat different point of view. This powerful instrument, by which such great intellectual enterprises have been achieved, - a branch of science most subtile and abstruse, embraces in its idea the very soul of poetry, and uses in its nomenclature the most beautiful forms of metaphor; its common expressions being borrowed from the vocabulary of the fancy. The fundamental idea of fluxions is derived from that of growth, or increase. It looks on all things not according to the vulgar, prosaic idea, as made up of accumulated particles and atoms, but as formed by motion. It says that a point flows on and becomes a line; that a line flows on and becomes a surface; a surface flows on and becomes a solid thus investing the most difficult and profound calculations with a drapery of poetical imagery, and proceeding to solve some of the hardest mysteries of calculation by an effort of fancy, which, in Milton, would have been called sublime. And thus the remark of Madame de Stael holds good, that "Imagination, far from being an enemy to Truth, brings it forward more than any other faculty of the mind." It is fair to say, then, that in their doctrines concerning infinity, concerning imaginary quantities, and flowing quantities, the mathematicians have something of the poets. II. Let us look next at some of the Operations and Results of Mathematics. Perhaps none can be found better to answer our purpose than conic sections; pertaining, as this operation does, primarily, to the order of pure mathematics, and dealing in the driest and most intricate investigations, and thence passing on to a series of applications which copiously minister to the sentiments of the beautiful and the sublime. Here, then, is a cone — a regular figure, but to the common eye exhibiting no very remarkable properties - a form which the uninitiated might look at and pass by suggesting nothing of very special beauty or grandeur. But the mathematician sees in it the secret of the universe. To him it is emblematic of the sublimest discoveries of science, and brilliant with the records of mind's most wonderful achievements. Cast a glance on his operations with it. First, he cleaves the cone from its apex to its base, and remarks that the outlines of that section describe a triangle. He divides it horizontally, (as one says society is divided in the Old World,) and discovers a circle. He cuts it obliquely, and finds in that section an ellipse. He cuts it again, in a line parallel to the side, and presents to view a parabola. He cuts it still once more, making a greater angle with the base than the side of the cone makes, and thus obtains the hyperbola. Having by these five sections obtained five figures, (the five conic sections,) he is curious to know their properties, and relations, and uses. He, accordingly, applies to them the scrutinizing power of his geometry; measures their curves, their angles, their dimensions, their axes, their conjugates, their ordinates, abscisses, parameters, and asymptotes; and, while the busy world, peeping into his study, smiles contemptuously on the man who can find entertainment in poring for day and night over these strange diagrams, hunting after their abstract relations, and calculating |