SOPH. Ant. 822 [322] et seq.
"Many things are wonderful," says the Greek poet, "but nought more wonderful than man, all-inventive man!" And surely, among many wonders wrought out by human endeavor, there are few of higher interest than that splendid system of mathematical science, the growth of so many slow-revolving ages and toiling hands, still incomplete, destined to remain so forever perhaps, but to-day embracing within its wide circuit many marvellous trophies wrung from Nature in closest contest. There are strange depths, doubtless, in the human soul,—recesses where the universal sunlight of reason fails us altogether; into which if we would enter, it must be humbly and trustfully, laying our right hands reverentially in God's, that he may lead us. There are faculties reaching farther than all reason, and utterances of higher import than hers, problems, too, in the solution of which we shall derive very little aid from any mere mathematical considerations. Those who think differently should read once more, and more attentively, the sad history of frantic folly and limitless license, written down forever under the date, September, 1792, boastfully proclaimed to the world as the New Era, the year 1 of the Age of Reason. Perhaps the number of those who would to-day follow Momoro's pretty wife with loud adulation and Bacchanalian rejoicings to the insulted Church of Nôtre Dame, thus publicly disowning the God of the Universe and discarding the sweetest of all hopes, the hope of immortality and eternal youth after the weariness of age, would be found to be very small. This was indeed a new version of the old story of Godiva, wherein implacable, inhuman hate sadly enough took the place of the sweet Christian charity of that dear lady. Let us recognize its deep significance, and acknowledge that many things of very great importance lie beyond the utmost limits of human reason.
But let us not forget, meanwhile, that within its own sphere this same Human Reason is an apt conjuror, marshalling and deftly controlling the powers of the earth and air to a degree wonderful and full of interest. And nowhere have all its possibilities so fully found expression in vast attainment as in those studies preëminently called the mathematics, as embracing all [Greek: mathaesis], all sound learning. Casting about for some sure anchorage, drifting hither and thither over changeful seas of phenomena, a large body of men, deep, clear thinkers withal, some twenty-four centuries since, fancied that they had found all truth in the fixed, eternal relations of number and quantity. Hence that wide-spread Pythagorean philosophy, with its spheral harmonics and esoteric mysteries, uniting in one brotherhood for many years men of thought and action,—dare we say, our inferiors? Why allude to the old fable of the dwarf upon the giant's shoulders? Let us have a tender care for the sensitive nature of this ultimate Nineteenth Century, and refrain. They were not so far wrong either, those old philosophers; they saw clearly a part of the boundless expanse of Truth,—and somewhat prematurely, as we believe, pronounced it the true Land's End, stoutly asserting that beyond lay only barren seas of uncertain conjecture.
But mark what followed! Presently, under their hands, fair and clear of outline as a Grecian temple, grew up the science of Geometry. Perfect for all time, and as incapable of change or improvement as the Parthenon, appear the Elements of Euclid, whose voice comes floating down through the ages, in that one significant rejoinder,—"Non est regia ad mathematicam via." It is the reply of the mathematician, quiet-eyed and thoughtful, to the first Ptolemy, inquiring if there were not some less difficult path to the mysteries. But the Greek Geometry was in no wise confined to the elements. Before Euclid, Plato is said to have written over the entrance to his garden,—"Let no one enter, who is unacquainted with geometry,"—and had himself unveiled the geometrical analysis, exhibiting the whole strength and weakness of the instrument, and applying it successfully in the discussion of the properties of the Conic Sections. Various were the discoveries, and various the discoverers also, all now at rest, like Archimedes, the greatest of them all, in his Sicilian tomb, overgrown with brambles and forgotten, found only by careful research of that liberal-minded Cicero, and recognized only by the sphere and circumscribed cylinder thereon engraved by the dead mathematician's direction.
Meanwhile, let us turn elsewhere, to that singular people whose name alone is suggestive of all the passion, all the deep repose of the East. Very unlike the Greeks we shall find these Arabs, a nation intellectually, as physically, characterized by adroitness rather than endurance, by free, careless grace rather than perfect, well-ordered symmetry. Called forth from centuries of proud repose, not unadorned by noble studies and by poesy, they swept like wildfire, under Mohammed and his successors, over Palestine, Syria, Persia, Egypt, and before the expiration of the Seventh Century occupied Sicily and the North of Africa. Spain soon fell into their hands;—only that seven-days' battle of Tours, resplendent with many brilliant feats of arms, resonant with shoutings, and weightier with fate than those dusty combatants knew, saved France. Then until the last year of the Eleventh Century, almost four hundred years, the Caliphs ruled the Spanish Peninsula. Architecture, music, astrology, chemistry, medicine,—all these arts, were theirs; the grace of the Alhambra endures; deep and permanent are the traces left by these Saracens upon European civilization. During all this time they were never idle. Continually they seized upon the thoughts of others, gathering them in from every quarter, translating the Greek mathematical works, borrowing the Indian arithmetic and system of notation, which we in turn call Arabic, filling the world with wild astrological fantasies. Nay, the "good Haroun Al Raschid," familiar to us all as the genial-hearted sovereign of the World of Faëry, is said to have sent from Bagdad, in the year 807 or thereabout, a royal present to Charlemagne, a very singular clock, which marked the hours by the sonorous fall of heavy balls into an iron vase. At noon, appeared simultaneously, at twelve open doors, twelve knights in armor, retiring one after another, as the hour struck. The time-piece then had superseded the sun-dial and hour-glass: the mechanical arts had attained no slight degree of perfection. But passing over all ingenious mechanism, making no mention here of astronomical discoveries, some of them surprising enough, it is especially for the Algebraic analysis that we must thank the Moors. A strange fascination, doubtless, these crafty men found in the cabalistic characters and hidden processes of reasoning peculiar to this science. So they established it on a firm basis, solving equations of no inconsiderable difficulty, (of the fourth degree, it is said,) and enriched our arithmetic with various rules derived from this source, Single and Double Position among others. Trigonometry became a distinct branch of study with them; and then, as suddenly as they had appeared, they passed away. The Moorish cavalier had no longer a place in the history of the coming days; the sage had done his duty and departed, leaving among his mysterious manuscripts, bristling with uncouth and, as the many believed, unholy signs, the elements of truth mingled with much error,—error which in the advancing centuries fell off as easily as the husk from ripe corn. Whether the present civilization of Spain is an advance upon that of the Moors might in many respects become a matter of much doubt.
Long lethargy and intellectual inanition brooded over Christian Europe. The darkness of the Middle Ages reached its midnight, and slowly the dawn arose,—musical with the chirping of innumerable trouvères and minnesingers. As early as the Tenth Century, Gerbert, afterwards Pope Sylvester II., had passed into Spain and brought thence arithmetic, astronomy, and geometry; and five hundred years after, led by the old tradition of Moorish skill, Camille Leonard of Pisa sailed away over the sea into the distant East, and brought back the forgotten algebra and trigonometry,—a rich lading, better than gold-dust or many negroes. Then, in that Fifteenth Century, and in the Sixteenth, followed much that is of interest, not to be mentioned here. Copernicus, Galileo, Kepler,—we must pass on, only indicating these names of men whose lives have something of romance in them, so much are they tinged with the characteristics of an age just passing away forever, played out and ended. The invention of printing, the restoration of classical learning, the discovery of America, the Reformation, followed each other in splendid succession, and the Seventeenth Century dawned upon the world.
The Seventeenth Century!—forever remarkable alike for intellectual and physical activity, the age of Louis XIV. in France, the revolutionary period of English history, say, rather, the Cromwellian period, indelibly written down in German remembrance by that Thirty-Years' War,—these are only the external manifestations of that prodigious activity which prevailed in every direction. Meanwhile the two sciences of algebra and geometry, thus far single, each depending on its own resources, neither in consequence fully developed, as nothing of human or divine origin can be alone, were united, in the very beginning of this epoch, by Descartes. This philosopher first applied the algebraic analysis to the solution of geometrical problems; and in this brilliant discovery lay the germ of a sudden growth of interest in the pure mathematics. The breadth and facility of these solutions added a new charm to the investigation of curves; and passing lightly by the Conic Sections, the mathematicians of that day busied themselves in finding the areas, solids of revolution, tangents, etc., of all imaginable curves,—some of them remarkable enough. Such is the cycloid, first conceived by Galileo, and a stumbling-block and cause of contention among geometers long after he had left it, together with his system of the universe, undetermined. Descartes, Roberval, Pascal, became successively challengers or challenged respecting some new property of this curve. Thereupon followed the epicycloids, curves which—as the cycloid is generated by a point upon the circumference of a circle rolled along a straight line—are generated by a similar point when the path of the circle becomes any curve whatever. Caustic curves, spirals without number, succeeded, of which but one shall claim our notice,—the logarithmic spiral, first fully discussed by James Bernouilli. This curve possesses the property of reproducing itself in a variety of curious and interesting ways; for which reason Bernouilli wished it inscribed upon his tomb, with the motto,—Eadem mutata resurgo. Shall we wisely shake our heads at all this, as unavailing? Can we not see the hand of Providence, all through history, leading men wiselier than they knew? If not, may it not be possible that we have read the wrong book,—the Universal Gazetteer, perhaps, instead of the true History? When Plato and Plato's followers wrought out the theory of those Conic Sections, do we imagine that they saw the great truth, now evident, that every whirling planet in the silent spaces, yes, and every falling body on this earth, describes one of these same curves which furnished to those Athenian philosophers what you, my practical friend, stigmatize as idle amusement? Comfort yourself, my friend: there was many a Callicles then who believed that he could better bestow his time upon the politics of the state, neglecting these vain speculations, which to-day are found to be not quite unprofitable, after all, you perceive.
And so in the instance which suggested these reflections, all this eager study of unmeaning curves (if there be anything in the starry universe quite unmeaning) was leading gradually, but directly, to the discovery of the most wonderful of all mathematical instruments, the Calculus preëminently. In the quadrature of curves, the method of exhaustions was most ancient,—whereby similar circumscribed and inscribed polygons, by continually increasing the number of their sides, were made to approach the curve until the space contained between them was exhausted, or reduced to an inappreciable quantity. The sides of the polygons, it was evident, must then be infinitely small. Yet the polygons and curves were always regarded as distinct lines, differing inappreciably, but different. The careful study of the period to which we refer led to a new discovery, that every curve may be considered as composed of infinitely small straight lines. For, by the definition which assigns to a point position without extension, there can be no tangency of points without coincidence. In the circumference of the circle, then, no two of the points equidistant from the centre can touch each other; and the circumference must be made up of infinite all rectilineal sides joining these points.
A clear conception of this fact led almost immediately to the Method of Tangents of Fermat and Barrow; and this again is the stepping-stone to the Differential Calculus,—itself a particular application of that instrument. Dr. Barrow regarded the tangent as merely the prolongation of any one of these infinitely small sides, and demonstrated the relations of these sides to the curve and its ordinates. His work, entitled "Lectiones Geometricae," appeared in 1669. To his high abilities was united a simplicity of character almost sublime. "Tu, autem, Domine, quantus es geometra!" was written on the title-page of his Apollonius; and in the last hour he expressed his joy, that now, in the bosom of God, he should arrive at the solution of many problems of the highest interest, without pain or weariness. The comment of the French historian conveys a sly sarcasm on the Encyclopedists:—"On voit au reste, par-là, que Barrow étoit un pauvre philosophe; car il croiroit en l'immortalité de l'âme, et une Divinité, autre que la nature universelle."[A]
[Footnote A: MONTUCLA. Hist. des Math. Part iv. liv. 1.]