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INFINITESIMAL CALCULUS

of gravity and volumes and surfaces of solids, but, as

the project with regard to fluxions. In 1680 Collins sought the assistance of the Royal Society for the publication of the tract, and this was granted in 1682. Yet it remained unpublished. The reason is unknown; but it is known that about 1679, 1680, Newton took up again the studies in natural philosophy which he had intermitted for several years, and that in 1684 he wrote the tract De molu which was in some sense a first draft of the Principia, and it may be conjectured that the fluxions were held over until the Principia should be finished. There is also reason to think that Newton had become dissatisfied with the arguments about infinitesimals on which his calculus was based. In the preface to the De quadralura curvarum (1764), in which he describes this tract as something which he once wrote (“ olim scripsi ”) he says that there is no necessity to introduce into the method of fluxions any argument about infinitely small quantities; and in the Principia (1687) he adopted instead of the method of fluxions a new method, that of “ Prime and Ultimate Ratios.” By the aid of this method it is possible, as Newton. knew, and as was afterwards seen by others, to found the calculus of fluxions on an irreproachable method of limits. For the purpose of explaining his discoveries in dynamics and astronomy Newton used the method of limits only, without the notation of fluxions, and he presented all his results and demonstrations in a geometrical form. There is no doubt that he arrived at most of his theorems in the first instance by using the method of fluxions. Further evidence of Newton's dissatisfaction with arguments about infinitely small quantities is furnished by his tract M ethodns rlijerentialis, published in 1711 by William Jones, in which he laid the foundations of the “ Calculus of Finite Differences.”

24. Leibnitz, unlike Newton, was practically a self-taught mathematician. He seems to have been first attracted to mathematics as a means of symbolical ex ression and P,

i1f=1'°i+é-'i+ - ~ln

1074 he sent an account of his method, called “ transmutation, ” along with this result to Huygens, and early in 1675 he sent it to Henry Oldenburg, secretary of the Royal Society, with inquiries as to Newton's discoveries in regard to quadratures. In October of 1675 he had begun to devise a symbolical notation for quadratures, starting from Cavalieri's indivisible, At first he proposed to use the word oninirz as an abbreviation for Cavalieri's “sum of all the lines, ” thus writing omnia y for that 54-I

which we write “fy¢l.r, ” but within a day or two he wrote “ fy.” He regarded the symbol “ f” as representing an operation which raises the dimensions of the subject of operation line becoming an area by the operation-and he devised his symbol “ d ” to represent the inverse operation, by which the dimensions are diminished. He observed that, whereas “ f ” represents “ sum, ” “ d ” represents “ difference.” His notation appears to have been practically settled before the end of 1675, for in November he wrote fydy=%y2, just as we do now. 25. In July of 1676 Leibnitz received an answer to his inqui-ry in regard to Newton's methods in a letter written by Newton to Oldenburg. In this letter Newton ave a eneral 3 g Corresstatement