Message #27796

[Rebecca]

Fluxions

Up to the time of the publication of the Principia in 1687 the method of fluxions which had been invented by Newton, and had been of great assistance to him in his mathematical investigations, was still, except to Newton and his friends, a secret. One of the most important rules of the method forms the second lemma of the second book of the Principia. Though this new and powerful method was of great help to Newton in his work, he did not exhibit it in the results. He was aware that the well known geometrical methods of the ancients would make his new creations seem less strange and uncouth to those not familiar with the new method. The Principia gives no information on the subject of the notation adopted in the new calculus, and it was not until 1693 that it was communicated to the scientific world in the second volume of John Wallis's works.

Newton's admirers in the Netherlands informed Wallis that Newton's method of fluxions passed there under the name of Gottfried Leibniz's Calculus Differentials. It was therefore thought necessary that an early opportunity should be taken of asserting Newton's claim to be the inventor of the method of fluxions, and this was why this method first appeared in Wallis's works. A further account was given in the first edition of Newton's Optics (1704).

To this work were added two treatises, entitled Accedunt tractatus duo ejusdem authoris de speciebus & magnitudine figurarum curvilinearum, the one bearing the title Introductio ad Quadratura Curvarum, and the other Enumeratio linearum tertii ordinis. The first contains an explanation of the doctrine of fluxions, and of its application to the quadrature of curves; the second, a classification of 72 curves of the third order, with an account of their properties.

The reason for publishing these two tracts in his Optics, from the subsequent editions of which they were omitted, is thus stated in the advertisement: "In a letter written to Leibniz in the year 1679, and published by Dr Wallis, I mentioned a method by which I had found some general theorems about squaring curvilinear figures on comparing them with the conic sections, or other the simplest figures with which they might be compared. And some years ago I lent out a manuscript containing such theorems; and having since met with some things copied out of it, I have on this occasion made it public, prefixing to it an introduction, and joining a Scholium concerning that method. And I have joined with it another small tract concerning the curvilinear figures of the second kind, which was also written many years ago, and made known to some friends, who have solicited the, making it public."

In 1707 William Whiston published the algebra lectures which Newton had delivered at Cambridge, under the title of Arithmetica Universalis, sive de Compositione et Resolutione Arithmetica Liber. We are not accurately informed how Whiston obtained possession of this work, but it is stated by one of the editors of the English edition "that Mr Whiston, thinking it a pity that so noble and useful a work should be doomed to a college confinement, obtained leave to make it public." It was soon afterwards translated into English by Raphson; and a second edition of it, with improvements by the author, was published at London in 1712, by Dr Machin, secretary to the Royal Society. With the view of stimulating mathematicians to write annotations on this admirable work, 's Gravesande published a tract, entitled Specimen Commentarii in Arithmetican Universalem; and Maclaurin's Algebra seems to have been drawn up in consequence of this appeal.