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Newton - The Method of Fluxions

There is much confusion around the subject of calculus, what it is and to what extent it played a part in Principia generally and universal gravitation in particular. The word itself has three meanings (OED), the first of which is medical. The remaining two are related, denoting simply a calculation, on the one hand, or a system or method of calculation, on the other. The first of these is obsolete but was in use by Andrew Motte in his 1729 translation of Principia where, for example, in Proposition IV, Book III we find "This we gather by a calculus ...." (Newton 1729). Examples of the second are differential and integral calculus, although today the word is usually taken to mean these two without further qualification, in contrast with the calculus of variations, for example, where the whole phrase is used. In discussions on the use of calculus in Principia, it is this modern sense that is implied and thus it makes sense to establish what Newton had in his particular toolkit, whether or not he actually deployed it in Principia.

Newton's work on integral and differential calculus is contained in the document The Method of Fluxions and Infinite Series and its Application to the Geometry of Curve-Lines (Newton 1736), first published in English translation in 1736 and generally thought to have been written, and given limited distribution, about 70 years earlier. Principia was published, in Latin, in 1687.

John Colson, Newton's translator, says, in his preface, that the main principle upon which the method is founded is "... taken from rational mechanics; which is that mathematical quantities [...] may be conceived as generated by continued local motion". As an example, if a body is in motion the coordinates describing its position, known as fluents, say x and y, will continuously change and the rates at which they do so are variously known as velocities or celerities or fluxions.

The second principle "supposes that quantity is infinitely divisible, or that it may (mentally at least) so far continually diminish, as at last, before it is totally extinguished, to arrive at quantities that may be called vanishing quantities, or which are infinitely little, and less than any assignable quantity". Unsurprisingly perhaps, these infinitesimal quantities had a rough ride philosophically over the centuries. They were, however, remarkably durable and useful throughout and have now been restored to complete respectability.

After an introduction which describes the "method of resolving complex quantities into infinite series of simple terms", there follow 12 problems, all but the first two of which describe the application of the method of fluxions to problems concerning curves. Problem 1 is stated as follows:

The relation of the flowing quantities to one another being given, to determine the relation of their fluxions

Let's say, using a slightly simpler example than Newton does, that the flowing quantities (fluents) are x and y and they are related by the equation . If is the rate of change of x with time (fluxion) and o is a very small increment of time, then in this time x will become or in more familiar notation something like . Substituting and for x and y in the equation we obtain:

Since is given, we can remove these terms. Then dividing throughout by o:

As o approaches zero so will the terms involving o leaving:

In familiar notation, these terms have the form:

where is called the moment of xn. The relation between the fluxions of x and y can now be written:

This can also be written as which is nothing else but in more familiar terms.

Problem 2 deals with the inverse of this process - finding fluents from fluxions. Although Newton proceeds quite logically, it is not entirely clear whether the method of calculation of moments and their inverse that arises is, indeed, of general applicability. However, in Lemma II of Book 2 of Principia, Newton does derive the moment of the product AB, using a rectangle, but his approach drew some criticism from contemporaries and John Colson, in his preface to The Method of Fluxions, gives a lengthy account of it in rebuttal of its critics. A shorter version follows.

A and B are considered to be in flux and in a given time increase by small quantities a and b respectively. The progress of the changes may be represented by three rectangles, as follows.

The difference between the products represented by the outer and inner rectangles can be calculated as:

When B is equal to A, this comes out to 2Aa and when B is equal to A2 to 3A2a which are the moments of A2 and A3 respectively. Continuing the process the moment of An is nAn-1a or:

Problem 3 explains how "to determine the maxima and minima of quantities" and problems 4 to 12 apply the method to various properties of curves. Problem 4 is "to draw tangents to curves".

AB and BD are the abscissa and ordinate of the point D on the curve which passes through E. The line bd, intersecting the curve at d, is parallel to BD and separated from it by the "indefinitely small space" Bb. Dc is equal and parallel to Bb. The line Dd is produced to T. The triangles dcD and DBT are similar so

Newton says:

Since the relation of BD to AB is exhibited by the equation by which the nature of the curve is determined; seek for the relationship of the fluxions, by Problem I. Then take TB to BD in the ratio of the fluxion of AB to the fluxion of BD and TD will touch the curve at the point D.

As an example, take AB and BD to be orthogonal and represented by x and y. With A as the origin, the equation of a right opening parabola with its vertex at E is:

Where h is equal to AE and a is the distance of the focus of the parabola from the vertex. By Problem 1,

Problem 9 is "to determine the area of any curve proposed". This problem demonstrates that the area under a curve can be calculated from the equation of the curve by what is now called integration, as described in Problem 2. The area AFDB under the curve AFD is z. If the line AB is equal to x and BE equal to 1 then the area of the rectangle ABEC is also x.

The two areas are conceived of as generated by lines BE and BD as they move to the right together, perpendicular to AB. Then the increments, or fluxions, of the areas z and x will be in the same ratio as BD and BE. That is

Problem 2 was to find fluents from fluxions and thus the relationship between z and x can be found. In familiar terms, taking BD as y we have

 

Bibliography

Newton,Isaac. The Method of Fluxions, 1671. Translated by John Colson. London: Henry Woodall, 1736.

Newton, Isaac. Philosophiae Naturalis Principia Mathematica (The Mathematical Principles of Natural Philosophy),1687. Translated by Andrew Motte. London, 1729.