ASTRONOMY OF EIGHTEENTH CENTURY. 265 



notion of Leibnitz lend itself more easily to new developments, and his 

 notation also to be a more manageable and effective instrument than 

 that proposed by the great English geometer. 



The English mathematicians of the period succeeding Newton were 

 men of conspicuous talents, but their labours were directed to extend- 

 ing the bounds of the ancient geometry by its own rigorous methods, 

 rather than to extensions of the new calculus, which was giving men 

 the means of investigating the universe of things around them. The 

 names of several of these English mathematicians will be remembered, 

 indeed, as contributors of some improvements in algebra and in geo- 

 metry, and as those to whom are due the best editions of the works 

 of the ancient geometers. The names of GREGORY (1661 1708), 

 WARING (1736 1798). EMERSON (1701 1782), T. SIMPSON (1710 

 1761), and R. SIMSON (1687 1768), maybe mentioned here as illus- 

 trating these remarks. ROGER COTES (1683 1716), who was pro- 

 fessor of experimental philosophy at Cambridge, gave, however, several 

 improvements to the rules for the inverse or integral calculus, and some 

 ingenious methods for areas of curved figures. 



DR. BROOK TAYLOR (1685 1731) published in 1715 his " Method 

 of Increments/' a work which added greatly to the resources of the new 

 calculus, and exhibited many ingenious applications of it to physical as 

 well as to mathematical questions. Dr. Taylor's name will always be 

 associated with a particular theorem, in which he shows that the value 

 of any function whatever of a variable can be expressed by a series of 

 terms formed according to a certain law. Some writers on the dif- 

 ferential calculus have made this theorem the foundation of the whole 

 science. 



The validity of the reasoning on which the differential calculus re- 

 posed was called in question by Bishop Berkeley, an Irish prelate who 

 is well known for his extreme "idealism "in metaphysics, which he 

 carried so far as to refuse to acknowledge the existence of an "object 

 world'"' at all. In a tract called " The Minute Philosopher," published 

 in 1734, and again in a later one called "The Analyst," he contends 

 that it is unmeaning and absurd to suppose that any finite ratio can exist 

 between two infinitely small quantities, which he humorously designates 

 as "the ghosts of departed quantities." The Bishop considers that 

 geometry is opposed to religion, and that by a perverse contradiction 

 of mind the geometers are intractable, and, unbelieving in religious 

 matters, they believe in the mysteries si fluxions. He declares the new 

 calculus is erroneous and obscure in its principles, and that, if geometri- 

 cal truths are deducible by its principles it is because one error in these 

 compensates for another ; and, finally, he is of opinion that Newton 

 himself did not understand it. Berkeley's attack did good by showing 

 that the language of the calculus laid it open to objections of the kind 

 he urged. Replies to these objections were published by two Cam- 

 bridge professors and by others, notably by MACLAURIN (1698 1746), 



