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shown him to be in complete possession twenty years before, and which he had made use of in a paper written, according to his own account, in 1666, and undoubtedly communicated to Dr. Barrow, and by him to Mr. Collins, in 1669. This paper, entitled Analysis per Aequationes numero terminorum Infinitas, was published in 1711. The question of the invention of the fluxionary or differential calculus, as is well known, gave occasion to a warm and protracted dispute between the partisans of Newton and those of his illustrious continental contemporary, Leibnitz; but it is now admitted on all hands, that, whatever claim Leibnitz also


have to be accounted its independent inventor (and there can scarcely be a doubt that he has a good claim to be so accounted), the honor of the prior invention belongs to Newton.



We must dismiss some other distinguished names with a very brief mention. James Gregory, who died in 1675 at the age of only thirty-six, after having been successively Professor of Mathematics at St. Andrews and at Edinburgh, had in his short life accomplished more than any of his contemporaries except Newton. He is popularly remembered chiefly as the inventor of the first reflecting telescope ; but his geometrical and analytical inventions and discoveries were also numerous, and some of them of the highest order of merit. His nephew, David Gregory, Professor of Mathematics at Edinburgh, and afterwards Savilian Professor of Astronomy at Oxford, was also an able mathematician, and published some valuable works on geometry, optics, and astronomy. The Newtonian Theory of universal gravitation is said to have been taught by him at Edinburgh before it was introduced into any other European university. It is remarkable that when this David Gregory died, in 1708, he and two of his brothers held professorships in three British universities, — himself at Oxford, James at Edinburgh, and Charles at St. Andrews. The last-mentioned, too, was succeeded, upon his resignation in 1639, by his son, named David.

John Collins (b. 1624, d. 1683) is the author of several practical works and of a good many papers in the Philosophical Transactions ; but he was most useful in promoting the publication of the works of others; it is said that Wallis's History of Algebra, Barrow's Optical and Geometrical Lectures, and various other publications owed their seeing the light principally to his instigation and encouragement. He also kept up an extensive epistolary intercourse with the other scientific men of the day : it was principally from the letters and papers he left behind him that the Commercium Epistolicum, or volume of correspondence on the invention of fluxions, published in 1712, was made up. “ Many of the discoveries in physical knowledge,” says Dr. Hutton, “ owe their chief improvement to him ; for while he excited some to disclose every new and useful invention, he employed others in improving them. Sometimes he was peculiarly useful by showing where the defect lay in any branch of science, and pointing out the difficulties attending the inquiry; at other times explaining their advantages, and keeping up a spirit and energy for improvement. In short, Mr. Collins was like the register of all the new acquisitions made in the mathematical sciences; the magazine to which the curious had frequent recourse; which acquired him the appellation of the English Mersenne.” 1

Roger Cotes died in 1716, at the age of thirty-four, after having, in the estimation of his contemporaries, given promise of becoming one of the greatest mathematicians that had ever existed: Newton himself is reported to have said, “ If Cotes had lived we should have known something.” Cotes's mathematical papers were published, in 1722, under the title of Harmonia Mensurarum, by his cousin Dr. Robert Smith (author of a work on optics), and his Hydrostatical and Pneumatical Lectures in 1738 by the same editor. Of all the publications that appeared in the early stages of the fluxionary calculus, Professor Playfair conceives that none is more entitled to notice than the Harmonia Mensurarum of Cotes. In this work, he observes, a method of reducing the areas of curves, in cases not admitting of an accurate comparison with rectilinear spaces, to those of the circle and hyperbola, which Newton had exemplified in his Quadratura Curvarum, was extended by Cotes, who also 66

gave the rules for finding the fluents of fractional expressions, whether rational or irrational, greatly gener

1 Abridg. of Phil. Trans. i. 338.

Cotes super

alized and highly improved by means of a property of the circle discovered by himself, and justly reckoned among the most remarkable propositions in geometry."1 Another eminent authority describes the Harmonia as “the earliest work in which decided progress was made in the application of logarithms, and of the properties of the circle, to the calculus of fluents.” 2 intended the printing of the second edition of Newton's Principia, published in 1713, and prefixed to it a preface which immediately acquired for him a wide scientific reputation.

The last of these early English cultivators of the new calculus whom we shall mention is Dr. Brook Taylor, a geometrician and analyst of great profoundness and originality, whose Methodus Incrementorum, published in 1915, is characterized by Playfair as having " added a new branch to the analysis of variable quantity.” “A single analytical formula,” Playfair adds,

66 in the Method of Increments has conferred a celebrity on its author which the most voluminous works have not often been able to bestow. It is known by the name of Taylor's Theorem, and expresses the value of any function of a variable quantity in terms of the successive orders of increments, whether finite or infinitely small. If any one proposition can be said to comprehend in it a whole science, it is this : for from it almost every truth and every method of the new analysis may be deduced. It is difficult to say whether the theorem does most credit to the genius of the author, or the power of the language which is capable of concentrating such a vast body of knowledge in a single expression.” 3 Taylor's Theorem has since its first announcement been, in the language of the late Professor Leslie, “ successively modified, transformed, and extended by Maclaurin, Lagrange, and Laplace, whose names are attached to their respective formulæ.” 4

1 Dissertation on Progress of Math. and Phys. Science, p. 531.
2 Article on Cotes, in Penny Cyclopædia, viii. 87.
3 Dissertation, p. 532.

4 Dissertation on the Progress of the Math. and Phys. Sciences in the Eighteenth Century, in Encyclopædia Britannica, p. 599.


THE example and discoveries of Newton, and especially the publication of the Principia, had, before the end of the seventeenth century, given a new direction and character to scientific speculation, and even to what was generally understood by the term science, in England. The day of little more than mere virtuosoship, in which the Royal Society had taken its rise and commenced its operations, had given place to that of pure

science in its highest forms and most lofty and extensive applications.

Next to the development and application of the fluxionary calculus, the field in which, as might have been expected, the impulse given by Newton produced the most brilliant results, was that of astronomy. The Royal Observatory at Greenwich was founded by Charles II., for the benefit of astronomy and navigation, in 1676 ; and the appointment of Astronomer Royal (or Astronomical Observator, in the official style) bestowed upon John Flamsteed, then about thirty years of age, and already distinguished as a cultivator of astronomical science. Flamsteed held this office till his death in 1719; and during that space of time made and published a voluminous series of observations, from the commencement of which his late biographer Mr. Baily dates the commencement of modern astronomy. “Nor,” observes another writer, to whose masterly contributions to the history of the mathematical sciences we have been repeatedly indebted in the preceding pages, “ can such chronology be disputed, if we consider that we now return to Flamsteed's observations as the earliest with which it is desirable to compare those of our day, and also that Flamsteed's Catalogue is the first which attained a precision comparable to that of later times.” 1 What is here alluded to is a catalogue of above 3300 stars,

“ whose places,” as has been remarked, were more accurate than any determined in the next fifty years, and whose selection and nomenclature have served as basis to every catalogue since that time.”2 A portion of this Catalogue was first published, without Flamsteed's consent, in 1712, by a committee appointed by the government, of which Newton, Wren, and Gregory were members, and under the immediate superintendence of Halley, by

1 Article on Flamsteed, in Penny Cyclopædia, x. 296.
2 Article on Greenwich Observatory, in Penny Cyclopædia, xi. 441.


whose name the work, entitled Historiæ Cælestis Libri Duo, is commonly known. Flamsteed considered himself, and apparently with good reason, to have been very ill used in this transaction ; 1 and, having at last succeeded in recovering from the government all the copies of Halley's book that remained unsold, he committed them to the flames, with the exception of a portion of the sheets, out of which he formed part of the first volume of a new work, with the title of Historia Coelestis Britannica, the printing of which, however (in three volumes, folio), was not completed till 1725, six years after the author's death. It was carried through the press by his widow, with the aid of his assistants Mr. Crosthwait and Mr. Abraham Sharp, the latter of whom had attained great distinction as an accurate observer. This work is characterized by the writer of the article on Flamsteed in the Penny Cyclopædia as occupying the same place in practical astronomy which the Principia of Newton holds in the theoretical part. It was to Flamsteed that Newton (who afterwards quarrelled with his old friend, and abused him in no measured terms, on the misunderstanding that arose about the first publication of his catalogue) was indebted for all the observations of the moon which he made use of in the illustration and verification of his lunar theory. “ The first edition of Newton's Principia,” to quote again the publication just referred to, “ had appeared shortly before Flamsteed had supplied himself with his best instruments; and at Newton's request many of Flamsteed's observations of the moon, reduced as well as was then practicable, were communicated to him to aid in perfecting the theory deduced from the principle of universal gravitation. The time at which these observations were made was in fact a most critical one — when the most accurate observations that had been made were needed for the support of the most extensive philosophical theory that man had invented.” 2

1 See the particulars, as for the first time brought to light by Mr. Francis Baily, in his new edition of The British Catalogue of Stars, corrected and enlarged, with an account of the life of Flamsteed prefixed. Lond. 1835.

2 Article on Greenwich Observatory, in Penny Cyclopædia, xi 141.

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