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expense of the Royal Society, in 1687. Meanwhile, about the year 1669, he had made his other great discovery of the non-homogeneity of light, and the differing refrangibility of the rays of which it is composed; by these fundamental facts revolutionising the whole science of optics. His Treatise on Optics, in which these discoveries and their consequences were developed, was first published in 1704; and along with it a Latin tract, entitled ‘De Quadratura Curvarum,” containing an exposition of the method of fluxions; of which, however, the Principia had already 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 may 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 honour of the prior invention belongs to Newton.

JAMES GREGoRY, AND oth ER contempora RIEs of

We must dismiss some other distinguished names with * Very brief mention. James Gregory, who died in 1673 at the age of only thirty-six, after having been successively Professor of Mathematics at St. Andrews and 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,

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published in 1712, was made up. “Many of the disco-
veries 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 attend-
ing the inquiry; at other times explaining their advan-
tages, and keeping up a spirit and energy for improve-
ment. 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 Mer-
senne.” “ Roger Cotes died in 1716, at the age of
thirty-four, after having, in the estimation of his contem-
poraries, 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 pub-
lications 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
* Abridg, of Phil. Trans., i. 338.

“gave the rules for finding the fluents of fractional expressions, whether rational or irrational, greatly generalised and highly improved by means of a property of the circle discovered by himself, and justly reckoned among the most remarkable propositions in geometry.” 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.”f Cotes superintended 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 1715, is characterised by Playfair as having “added a new branch to the analysis of variable quantity.” “A single analytical formula,” Playfair adds, “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 diffi

* Dissertation on Progress of Math. and Phys. Science, p. 531. # Article on Cotes, in Penny Cyclopædia, viii. 87.

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cult 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.” 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 formulae.”f


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

* Dissertation, p. 532.

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

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