physical and mathematical sciences. 295

divisibles, would assign the small area corresponding toeach term, and the sum of these would give the wholearea required.

But such methods as those of indivisibles were operose,tedious, and unconnected by any very general principle.The fertility of Newton ’s genius soon supplied this partof the process also, and devised a method which, by theapplication of certain easy but highly general rules, gavethe ready solution of all problems of this nature. Aletter of Dr. Barrow, in the collection above referredto, in 1669 , mentions, that some years before Newton had shown him a MS. treatise, but would not publishit, entitled “ Analysis per sequationes numero termin-orum infinitas,” in which not only were such seriesesemployed, but the general connecting principle andmethod clearly pointed out, though certainly not givenin precise formal rules, with a peculiar notation.

Newton first gave to the public an account, though avery brief one, of his method, in the second lemma ofthe second book of his Principia, in 1687 ; and theprecise rules and notations were made known by Dr.VPallis, in the second volume of his works, in 1 6Q3, wherehe extracts an account of them from two letters ofNewton , written in 1692 . This method, then, whichits inventor had called fluxions, was known for yearsonly to his friends; but those friends soon included, bymeans of their correspondence, some of the first ma-thematicians of Europe . In the earlier stage of theinvestigation, however, the different serieses employedfor quadratures were the principal objects of attention,and these were soon communicated to some of the mosteminent geometers.

Leibnitz visited England in 1673; and, having formedan acquaintance with Oldenburgh, secretary to theRoyal Society , continued a correspondence with himon his return to the Continent. At the time of hisvisit, he was but slightly conversant with mathematics ;but his powerful mind grasped every subject: and,after his return, he was soon able to enter into thoseu 4