The Differential and Integral Calculus

Пολλά τμ̀ δεLυὰ, κοὐδὲυ ὰυ-

ϑρώπου δειυότερου πέλει·...

περιϕραδὴς αυήρ!

SOPH. Ant. 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 Notre 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 μάϑησις, 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 harmonies 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 flouting 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 timepiece 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 trouveres 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 ThirtyYears’ 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 infinitely small 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 Geometricæ," 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 universeller."1

The Italian Cavalleri had, before this, published his "Geometry of Indivisibles,” and fully established his theory in the “ Exercitationes Mathematicæ,” which appeared in 1647. Led to these considerations by various problems of unusual difficulty proposed by the great Kepler, who appears to have introduced infinitely great and infinitely small quantities into mathematical calculations for the first time, in a tract on the measure of solids, Cavalleri enounced the principle, that all lines are composed of an infinite number of points, all surfaces of an infinite number of lines, and all solids of an infinite number of surfaces. What this statement lacks in strict accuracy is abundantly made up in its conciseness; and when some discussion arose thereupon, it appeared that the absurdity was only seeming, and that the author himself clearly enough understood by these apparently harsh terms, infinitely small sides, areas, and sections. Establishing the relation between these elements and their primitives, the way lay open to the Integral Calculus. The greatest geometers of the day, Pascal, Roberval, and others, unhesitatingly adopted this method, and employed it in the abstruse researches which engaged their attention.

And now, when but the magic touch of genius was wanting to unite and harmonize these scattered elements, came Newton. Early recognized by Dr. Barrow, that truly great and good man resigned the Mathematical Chair at Cambridge in his favor. Twenty-seven years of age, he entered upon his duties, having been in possession of the Calculus of Fluxions since 1666, three years previously.

Why speak of all his other discoveries, known to the whole world ? Animi vi prope planetarum, motus, figuras, cometarum semitas, Oceanique œstus, suâ Mathesi lucem prœferente, primus demonstravit. Radiorum lucis dissimilitudines, colorumque inde nascentium proprietates, quas nemo suspicatus est, pervestigavit. So stands the record in Westminster Abbey; and in many a dusty alcove stands the “ Prineipia,” a prouder monument perhaps, more enduring than brass or crumbling stone. And yet, with rare modesty, such as might be considered again and again with singular advantage by many another, this great man hesitated to publish to the world his rich discoveries, wishing rather to wait for maturity and perfection. The solicitation of Dr. Barrow, however, prevailed upon him to send forth, about this time, the “ Analysis of Equations containing an Infinite Number of Terms,”-a work which proves, incontestably, that he was in possession of the Calculus, though nowhere explaining its principles.

This delay occasioned the bitter quarrel between Newton and Leibnitz,— a quarrel exaggerated by narrow-minded partisans, and in truth not very creditable, in all its ramifications, to either party. Newton, in the course of a scientific correspondence with Leibnitz, published in 1712, by the Royal Society, under the title, “ Commercium Epistolicum de Analysi promotâ,” not only communicated very many remarkable discoveries, but added, that he was in possession of the inverse problem of the tangents, and that he employed two methods which he did not choose to make public, for which reason he concealed them by anagram-

matieal transposition, so effectual as completely to extinguish the faint glimmer of light which shone through his scanty explanation.2 The reference is obviously to what was afterwards known as the Method of Fluxions and Fluents. This method he derived from the consideration of the laws of motion uniformly varied, like the motion of the extreme point of the ordinate of any curve whatever. The name which he gave to his method is derived from the idea of motion connected with its origin.

Leibnitz, reflecting upon these statements on the part of Newton, arrived by a somewhat different path at the Differential and Integral Calculus, reasoning, however, concerning infinitely great and infinitely small quantities in general, viewing the problem algebraically instead of geometrically,-and immediately imparted the result of his studies to the English mathematician. In the Preface to the first edition of the “ Principia,” Newton says, “ It is ten years since, being in correspondence with M. Leibnitz, and having instructed him that I was in possession of a method of determining tangents and solving questions involving maxima and minima, a method which included irrational expressions, and having concealed it by transposing the letters, he replied to me that he had discovered a similar method, which he communicated, differing from mine only in the terms and signs, as well as in the generation of the quantities.” This would seem to be sufficient to set at rest any conceivable controversy, establishing an equal claim to originality, conceding priority of discovery to Newton. Thus far all had been open and honorable. The petty complaint, that, while Leibnitz freely imparted his discoveries to Newton, the latter churlishly concealed his own, would deserve to be considered, if it were obligatory upon every man of genius to unfold immediately to the world the results of his labor. As there may be many reasons for a different course, which we can never know, perhaps could never hope to appreciate, if we did know them, let us pass on, merely recalling the example of Galileo. When the first faint glimpses of the rings of Saturn floated hazily in the field of his imperfect telescope, he was misled into the belief that three large bodies composed the then most distant light of the system,-a conclusion which, in 1610, he communicated to Kepler in the following logograph:-


It is not strange that the riddle was unread. The old problem, Given the Greek alphabet, to find an Iliad, differs from this rather in degree than in kind. The sentence disentangled runs thus :-


And yet we have never heard that Kepler, or, in fact, Leibnitz himself, felt aggrieved by such a course.

But Leibnitz made his discovery public, neglecting to give Newton any credit whatever; and so it happened that various patriotic Englishmen raised the cry of plagiarism. Keil, in the “ Philosophical Transactions” for 1708, declared that he had published the Method of Fluxions, only changing the name and notation. Much debate and angry discussion followed ; and, alas for human weakness ! Newton himself, in a later edition of the “ Principia,” struck out the generous recognition of genius recorded above, and joined in terming Leibnitz an impostor, -while the latter maintained that Newton had not fathomed the more abstruse depths of the new Calculus. The “ Commercium Epistolicum ” was published, giving rise to new contentions ; and only death, which ends all things, ended the dispute. Leibnitz died in 1716.

The Calculus at first found its chief supporters on the Continent. James and John Bernouilli, Varignon, author of the “ Theory of Variations,” and the Marquis de l’Hôpital, were the first to appreciate it ; but soon it attracted the attention of the scientific world to such a degree that the frivolous populace of Paris had even a Well-known song with the burden, “ Des infiniment petits.” Neither were opponents wanting. Wrong-headed men and thick-headed men are unfortunately too numerous in all times and places. One Nieuwentiit, a dweller in intellectual fogbanks, who had distinguished himself byproving the existence of the Deity in one of his works, made about this time what he doubtless considered a second discovery. He found a flaw in the reasoning of Leibnitz, namely, that he (Nieuwentiit) could not conceive of quantities infinitely small! A certain Chever also performed sundry singular mathematical feats, such as squaring the circle, a problem which he reduced to the single question, Construere mundum divinœ menti analagum, and showing that the parabola, the only conic section squared by ancient or modern geometers, could never be quadrated, to the eternal discomfiture and discredit of the shade of Archimedes. Leibnitz used every means in his power to engage these worthy adversaries in a contest concerning his Calculus, but unfortunately failed. Bishop Berkeley, too, author of the “ Essay on Tar-Water,” devout disbeliever in the material universe, could not resist the Quixotic inclination to run a tilt against a science which promised so much aid in unveiling those starry splendors which he with strenuous asseveration denied. He published, in 1754, “The Minute Philosopher," and soon after, “ The Analyst, or the Discourse of a Mathematician,” showing that the Mathematics are opposed to religion, and cultivate an incredulous spirit,- such as would never for a moment listen, let us hope, to any theory which proclaims this green earth and all the universe “such stuff as dreams are made of,” even though the doctrine be ecclesiastically sustained and backed with abundant wealth of learning. Numerous were the defenders, called out rather by the acknowledged metaphysical ability of Bishop Berkeley than by any transcendent merit in these two tracts; and among others came Maclaurin.

Taylor’s Theorem, based upon that first published by Maclaurin, is the foundation of the Calculus by La Grange, differing from the methods of Leibnitz and Newton in the manner of deriving the auxiliaries employed, proceeding upon analytical considerations throughout. Of his “ Théorie des Fonctions,” and that noblest achievement of the pure reason, the “Mécanique Analytique,” we do not propose to speak, nor of the later developments of the Calculus, so largely due to his genius and labors. These are mysteries, known only to the initiated, yet capable of raising their thoughts in as sublime emotion as arose from the view of the elder, forgotten mysteries, which Cicero deemed the very source and beginning of true life.

We have seen how, and through whose toil, this mightiest instrument of human thought has reached its present perfection. Now, its vast powers fully recognized, it has become interwoven with all Natural Philosophy. On its sure basis rests that majestic structure, the “Mécanique Céleste” of La Place. Its demonstration supports with undoubted proof many doctrines of the great Newton. Discovery has succeeded discovery ; but its powers have never yet been fully tested. “ It is that field of mathematical investigation,” says Davies, “ where genius may exert its highest powers and find its surest rewards.” Looking back through the long course of events leading to such a magnificent result, looking up to that choral dance of wandering planets, all whose courses and seasons are marked down for us in the yearly almanac, can we not find in these manifestations something on the whole quite wonderful, worthy of very deep thankfulness, heartfelt humility withal, and far-reaching hope ?

In an age of many-colored absurdity, when extremes meet and contradictions harmonize,-when men of gross, material aims give implicit confidence to the wildest ravings of the supernatural, and pure-minded men embrace French theories of social organization,-when crowds of dullards all aflame with unexpected imagination assemble in ascension-robes to await the apocalyptic trump, and Asiatic polygamy spreads unmolested along our Western rivers, - when the prediction is accomplished, “ Old men dream dreams and young men see visions,” and the most practical of the ages bids fair to glide ghostly into history as the most superstitious,- it is well, it can but be well, to contemplate reverently that Reason, which Coleridge, after Leighton, calls “an influence from the Glory of the Almighty.” In the contemplation of the spirit of man (not your animula, by any means !) there is earnest of immortality which needs not that one rise from the dead to confirm it. In view of the Foresight which guides men, we may trust that all this tumultuous sense of inadequacy in present institutions, this blind notion of wrong, far enough from intelligent correction, is, after all, better than sluggish inaction.

  1. MONTUCLA. Hist, des Math. Part. iv. liv. 1.
  2. This logograph Newton afterwards rendered as follows: “Una methodus consist it in extractione fluentis quantitatis ex æquatione simul involvente; altera tantùm in assumptione seriei pro quantitate incognitâ ex quâ ceteræ commodè derivari possunt, et in collatione terminorum homologorum asquationis resultantis ad eruendos terminos seriei assuraptæ.”