The Italian Cavalleri had, before this, published his "Geometry of Indivisibles," and fully established his theory in the "Exercitationes Mathematicae," 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 propè divinâ, planetarum motus, figuras, cometarum semitas, Oceanique aestus, suâ Mathesi lucem praeferente, 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 "Principia," 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 anagrammatical transposition, so effectual as completely to extinguish the faint glimmer of light which shone through his scanty explanation.[B] 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.
[Footnote B: This logograph Newton afterwards rendered as follows: "Una methodus consistit in extractione fluentis quantitatis ex aequatione simul involvente; altera tantùm in assumptione seriei pro quantitate incognitâ ex quâ ceterae commodè derivari possunt, et in collatione terminonim homologorum aequationis resultantis ad eruendos terminos seriei assumptae.">[
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:—
SMAISMRMILMEPOETALEVMIBVNENGTTAVIRAVS.
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:—
ALTISSIMVM PLANETAM TERGEMINVM OBSERVAVI.
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.