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NEWTON S FIRST EXPOSITION OF THE CALCULUS . First edition of Newton s first exposition of his fluxional calculus. Originally written in 1671, in Latin, this was Newton s first comprehensive presentation of his method of fluxions which, according to Hall might have effected a mathematical revolution in its own day (Philosophers at War, pp. 65-6). It should properly be placed first in the great trilogy of Newton s major works: Fluxions, Principia (1687) and Opticks (1704). Newton s Methodus fluxionum was originally prepared in 1671, but remained unpublished until this English translation by John Colson. In it he presents a method of determining the magnitudes of finite quantities by the velocities of their generating motions. At its time of preparation, it was Newton s fullest exposition of the fundamental problem of the calculus, in which he presented his successful general method. Newton prepared this treatise just before his death and entrusted the Latin manuscript to Henry Pemberton, who never published it. In the preface, Colson writes "I thought it highly injurious to the memory and reputation of our own nation, that so curious and useful a piece should be any longer suppressed." The engraved plate demonstrates the concept of fluxions in the shooting of two birds at once. The method of fluxions was not published in its original Latin until 1779, in Samuel Horsley s Opera omnia. Newton wrote three accounts of the calculus. The composition of the first, a tract entitled De analysi per aequationes numero terminorum infinitas, resulted from Newton s reception from Isaac Barrow, in the early months of 1669, of a copy of Mercator s Logarithmotechnia, a work which contained the series for log(1 + x). The work, in which Newton demonstrated his much more general methods of infinite series, was not published until 1711, when William Jones included it, along with a number of other tracts, in his Analysis per quantitatum series. In De analysi, however, Newton "did not explicitly make use of the fluxionary notation or idea. Instead, he used the infinitely small, both geometrically and analytically, in a manner similar to that found in Barrow and Fermat, and extended its applicability by the use of the binomial theorem. … It will be noticed that although the work of Newton contains the essential procedures of the calculus, the justification of these is not clear from the explanation he gave. Newton did not point out by what right the terms involving powers of o were to be dropped out of the calculation, any more than Fermat or Barrow … His contribution was that of facilitating the operations, rather than of clarifying the conceptions. As Newton himself admitted in this work, his method is shortly explained rather than accurately demonstrated " (Boyer, The Concept of Calculus, p.191). It was first in Methodus fluxionum that "Newton introduced his characteristic notation and conceptions. Here he regarded his variable quantities as generated by the continuous motion of points, lines, and planes, rather than as aggregates of infinitesimal elements, the view which had appeared in De analysi . … In the Methodus fluxionum Newton stated clearly the fundamental problem of the calculus: the relation of quantities being given, to find the relation of the fluxions of these; and conversely" (ibid., pp. 192-3). In Newton s third exposition, De quadratura, which was composed some twenty years after Methodus fluxionum and published as an appendix to the Opticks, "Newton sought to remove all traces of the infinitely small" (ibid.). "It was often lamented that the world had had to wait so many years to see Newton s masterpiece on fluxions. It is astonishing to realize that publication sixty years beforehand would have changed the history of the calculus and would have avoided for Newton any controversy over priority. In 1736 all the results contained in Newton's treatise were well known to mathematicians. However, it was too concise for a b.
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