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 analysis developed since the time of the Renaissance in the form of Euclid,—of course only in outward form, for into the spirit of them he was quite unable to penetrate."[16]

The contemporaries and immediate successors of Newton in Great Britain were men of no mean merit. We have reference to Cotes, Taylor, Maclaurin, and De Moivre. We are told that at the death of Roger Cotes (1682–1716), Newton exclaimed, "If Cotes had lived, we might have known something." It was at the request of Dr. Bentley that Cotes undertook the publication of the second edition of Newton's Principia. His mathematical papers were published after his death by Robert Smith, his successor in the Plumbian professorship at Trinity College. The title of the work, Harmonia Mensurarum, was suggested by the following theorem contained in it: If on each radius vector, through a fixed point O, there be taken a point R, such that the reciprocal of OR be the arithmetic mean of the reciprocals of $$\scriptstyle{OR_1,~OR_2,~\cdots~OR_n,}$$ then the locus of R will be a straight line. In this work progress was made in the application of logarithms and the properties of the circle to the calculus of fluents. To Cotes we owe a theorem in trigonometry which depends on the forming of factors of $\scriptstyle{x^n-1}$. Chief among the admirers of Newton were Taylor and Maclaurin. The quarrel between English and Continental mathematicians caused them to work quite independently of their great contemporaries across the Channel.

Brook Taylor (1685–1731) was interested in many branches of learning, and in the latter part of his life engaged mainly in religious and philosophic speculations. His principal work, Methodus incrementorum directa et inversa, London, 1715–1717, added a new branch to mathematics, now called "finite differences." He made many important applications of it, particularly to the study of the form of movement of vibrating