His first original work, in 1665, aged 23, concerned infinite (power) series. In particular, he proved the binomial theorem

(his notation was quite different). This had long been known for integral r, but Newton proved it for rational, positive or negative r, for which it is a power series; for example, he found power series expansions of 1/root(1-x²), 1/(1+x²) , etc and their derivatives and antiderivatives by termwise differentiation. He simply regarded power series as polynomials of infinite degree, and did not consider convergence. His intuition guided him in avoiding divergent series. Thus he was able to find power series for sin, cos, tan, arcsin, arccos, arctan and ln (1+x).

The method of fluxions Over time, Newton produced three different foundations for his calculus, but there is no doubt that the one he used for his discoveries, and his most popular presentation, was to look on a curve as the path of a moving particle, so the first and second derivatives always exist and represent velocity and acceleration. Both x and y = f(x) are fluents or flowing quantities, and dot x and dot y are their fluxions or rates of change with respect to time. So the slope of a curve would be dot y/ dot x , what we would call the parametric representation of the derivative. Similarly he has dot dot y for the fluxion of dot y and y' for the fluent whose fluxion is y , ie the antiderivative of y . The existence of these functions is justified by the existence of instantaneous velocities.

The second method, which Newton used to actually compute the derivatives of various functions was the " little o " notation, exactly as in Fermat, so o is an infinitesimal or infinitely small quantity.

The third is the " method of first and last ratios " which is similar to our current ideas of limits. Newton intended this method the replace Exhaustion as a logical foundation for his calculus.

Newton was slow to publish. He was persuaded by Edmund Halley, his friend, promoter and Professor at Oxford, later Astronomer Royal, to publish his first and most important publication, Principia Mathematica , the Mathematical Principles of Natural Philosophy. This contains a complete development of calculus as well as dynamics and their application to astronomy. In particular it contains Newton's three laws of motion, the Law of Gravitation (particles of mass m and M at distance d attract each other with force mM/d²)), and a mathematical justification based on this for Kepler's three laws of planetary motion:

1. planets move in elliptical orbits with the sun at a focus,
2. the line joining the planet to the focus sweeps out equal areas in equal times, and
3. if the major axis is a and the period t, then t² is proportional to a³ .

The Principia is regarded as the most revolutionary scientific treatise of all time. It presents the foundations of physics and astronomy in geometric form.

The extract on first and last ratios is not intended to be a method of calculating derivatives, but of logically justifying them. Lemma 1 is really an attempt to define limits and a proof that they are unique. The other lemmas in Section 1 form a list of ways of using limits, and the properties of Leibniz 'characteristic triangle'. They discuss the ratio between change in its sides and change in the curve, and change in velocity of a moving particle. On p. 373 we have an attempt to justify Newton's methods of dealing with infinitesimals by reference to the method of Exhaustion. The actual algorithms of calculus appear in Book 2.

Newton's third and clearest account of the calculus is Tractatus de Quadratura curvorum , written in 1676 but published as an appendix to his Opticks in 1704. Instead of infinitesimals and fluents, he tries to justify "first and last ", " prime and ultimate" and " nascent and evanescent " ratios as follows:

suppose you want the ratio of changes to x and xⁿ, n rational. Let o be the increment in x , so (x + o)ⁿ - xⁿ is the increment in xⁿ. Then the ratio of the increments, for any o , is 1:(nx^n-1 + n(n-1)/2 ox^n-2 + ...) The ultimate ratio is this quantity when o = 0 , namely 1:nx^n-1.

The Tractatus also contains definitions of fluents and fluxions, first and last ratios, integrals for volumes and surface ares of solids of revolution, related rate problems, and algorithms for power series and intrinsic differentiation.

Book I, Section I: Of The Methods Of First And Last Ratios ...

OF THE MOTION OF BODIES.

SECTION I.

Of the method of first and last ratios of quantities, by the help whereof we demonstrate the propositions that follow.

LEMMA I.

Quantities, and the ratios of quantities, which in any finite time converge continually to equality, and before the end of that time approach nearer the one to the other than by any given difference, become ultimately equal.

If you deny it, suppose them to be ultimately unequal, and let D be their ultimate difference. Therefore they cannot approach nearer to equality than by that given difference D; which is against the supposition.

Perhaps it may be objected, that there is no ultimate proportion, of evanescent qualities; because the proportion, before the quantities have vanished, is not the ultimate, and when they are vanished, is none. But by the same argument, it may be alledged, that a body arriving at a certain place, and there stopping, has no ultimate velocity: because the velocity, before the body comes to the place, is not its ultimate, velocity; when it has arrived, is none. But the answer is easy; for by the ultimate velocity is meant that with which the body is moved, neither before it arrives at its last place and the motion ceases, nor after, but at the very instant it arrives; that is, that velocity with which the body arrives at its last place, and with which the motion ceases. And in like manner, by the ultimate ratio of evanescent quantities is to be understood the ratio of the quantities not before they vanish, nor afterwards, but with which they vanish. In like manner the first ratio of quantities is that with which they begin to be. And the first or last sum is that with which they begin and cease to be (or to be augmented or diminished). There is a limit which the velocity at the end of the motion may attain, but not exceed. This is the ultimate velocity. And there is the like limit in all quantities and proportions that begin and cease to be. And since such limits are certain and definite, to determine the same is a problem strictly geometrical. But whatever is geometrical we may be allowed to use in determining and demonstrating any other thing that is likewise geometrical.

It may also be objected, that if the ultimate ratios of evanescent quantities are given, their ultimate magnitudes will be also given: and so all quantities will consist of indivisibles, which is contrary to what Euclid has demonstrated concerning incommensurables, in the 10th Book of his Elements. But this objection is founded on a false supposition. For those ultimate ratios with which quantities vanish are not truly the ratios of ultimate quantities, but limits towards which the ratios of quantities decreasing without limit do always converge; and to which they approach nearer than by any given difference, but never go beyond, nor in effect attain to, till the quantities are diminished in infinitum. This thing will appear more evident in quantities infinitely great. If two quantities, those difference is given, be augmented in the ultimate ratio of these quantities will be given, to wit, the ratio of equality; but it does not from thence follow, that the ultimate or greatest quantities themselves, whose ratio that is, will be given. Therefore if in what follows, for the sake of being more easily understood, I should happen to mention quantities as least, or evanescent, or ultimate, you are not to suppose that quantities of any determinate magnitude are meant, but such as are conceived to be always diminished without end.