Archimedes was the earliest thinker to develop the apparatus of an infinite series with a finite limit ...starting on the conceptual path toward calculus. Of the giants on whose shoulders Isaac Newton would eventually perch, Archimedes was the first.
The fundamental definitions of the calculus, those of the derivative and integral, are now so clearly stated in textbooks on the subject... that it is easy to forget the difficulty with which these basic concepts have been developed.
The precision of statement and the facility of application which the rules of the calculus early afforded were in a measure responsible for the fact that mathematicians were insensible to the delicate subtleties required in the logical development... They sought to establish calculus in terms of the conceptions found in traditional geometry and algebra which had been developed from spatial intuition.
The Calculus of Variations owed its origin to the attempt to solve a very interesting and rather narrow class of problems in Maxima and Minima, in which it is required to find the form of a function such that the definite integral of an expression involving that function and its derivative shall be a maximum or a minimum.
If a cone is cut by surfaces parallel to the base, then how are the sections, equal or unequal? If they are unequal then the cone would have the shape of a staircase; but if they were equal, then all sections will be equal, and the cone will look like a cylinder, made up of equal circles; but this is entirely nonsensical.
Pascal's aritmentic triangle and Leibniz' harmonic triangle enjoy a certain inverse relationship... These considerations implanted in Leibniz' mind a vivid conception that was to play a dominant role in his development of the calculus—the notion of an inverse relationship between the operation of taking differences and that of forming sums of the elements of a sequence.
Fermat applied his method of tangents to many difficult problems. The method has the form of the now-standard method of differential calculus, though it begs entirely the difficult theory of limits.
That method [of infinitesimals] has the great inconvenience of considering quantities in the state in which they cease, so to speak, to be quantities; for though we can always well conceive the ratio of two quantities, as long as they remain finite, that ratio offers the to mind no clear and precise idea, as soon as its terms become, the one and the other, nothing at the same time.
In the famous dispute regarding the invention of the infinitesimal calculus, while not denying... the priority of Newton... some... regard Leibnitz's introduction of the integral symbol as alone a sufficient substantiation of his claims to originality and independence, so far as the power of the new science was concerned.
In connection with the study of curves Fermat proceeded to apply the idea of infinitesimals to the questions of quadrature and of maxima and minima as well as to the drawing of tangents. In this he seems to have anticipated the work of Cavalieri, but the date of his discovery is unknown.
The calculus was the first achievement of modern mathematics, and it is difficult to overestimate its importance. ...it defines more unequivocally than anything else the inception of modern mathematics, and the system of mathematical analysis, which is its logical development, still constitutes the greatest technical advance in exact thinking.
Two unequal magnitudes being set out, if from the greater there be subtracted a magnitude greater than its half, and from that which is left a magnitude greater than its half, and if this process be repeated continually, there will be left some magnitude which will be less than the lesser magnitude set out.
You may find this work (if I judge rightly) quite new. For I see no reason why I should not proclaim it; nor do I believe that others will take it wrongly. ...it teaches all by a new method, introduced by me for the first time into geometry, and with such clarity that in these more abstruse problems no-one (as far as I know) has used...
This method of mine takes its beginnings where Cavalieri ends his Method of indivisibles. ...for as his was the Geometry of indivisibles, so I have chosen to call my method the Arithmetic of infinitesimals.
Around 1650 I came across the mathematical writings of Torricelli, where among other things, he expounds the geometry of indivisibles of Cavalieri. ...His method, as taught by Torricelli... was indeed all the more welcome to me because I do not know that anything of that kind was observed in the thinking of almost any mathematician I had previously met.
It is an extremely useful thing to have knowledge of the true origins of memorable discoveries, especially those that have been found not by accident but by dint of meditation. ...the art of making discoveries should be extended by considering noteworthy examples of it.
Among the most renowned discoveries of the times must be considered that of a new kind of mathematical analysis, known by the name of the differential calculus; and of this... the origin and the method of the discovery are not yet known to the world at large.
Its author invented it nearly forty years ago, and nine years later (nearly thirty years ago) published it in a concise form; and from that time it has... been a method of general employment; while many splendid discoveries have been made by its assistance... so that it would seem that a new aspect has been given to mathematical knowledge arising out of its discovery.
Now there never existed any uncertainty as to the name of the true inventor, until recently, in 1712, certain upstarts... acted with considerable shrewdness, in that they put off starting the dispute until those who knew the circumstances, Huygens, Wallis, Tschirnhaus, and others, on whose testimony they could have been refuted, were all dead.
And it seems still more difficult, to conceive the abstracted Velocities of such nascent imperfect Entities. But the Velocities of the Velocities, the second, third, fourth and fifth Velocities, &c. exceed, if I mistake not, all Humane Understanding. The further the Mind analyseth and pursueth these fugitive Ideas, the more it is lost and bewildered; the Objects, at first fleeting and minute, soon vanishing out of sight.
[T]o conceive a Part of such infinitely small Quantity, that shall be still infinitely less than it, and consequently though multiply'd infinitely shall never equal the minutest finite Quantity, is, I suspect, an infinite Difficulty to any Man whatsoever; and will be allowed such by those who candidly say what they think; provided they really think and reflect, and do not take things upon trust.
Insomuch that we are to admit an infinite succession of Infinitesimals... in an infinite Progression towards nothing, which you still approach and never arrive at.
All these Points, I fay, are supposed and believed by... Men who pretend to believe no further than they can see. ...But he who can digest a second or third Fluxion, a second or third Difference, need not, methinks, be squeamish about any Point in Divinity. ...{W]ith what appearance of Reason shall any Man presume to say, that Mysteries may not be Objects of Faith, at the fame time that he himself admits such obscure Mysteries to be the Object of Science?
[T]he modern Mathematicians... scruple not to say, that by the help of these new Analytics they can penetrate into Infinity itself: That they can even extend their Views beyond Infinity: that their Art comprehends not only Infinite, but Infinite of Infinite (as they express it) or an Infinity of Infinites.
But, notwithstanding all these Assertions and Pretensions, it may be justly questioned whether, as other Men in other Inquiries are often deceived by Words or Terms, so they likewise are not wonderfully deceived and deluded by their own peculiar Signs, Symbols, or Species.
But if we remove the Veil and look underneath, if laying aside the Expressions we set ourselves attentively to consider the things themselves... we shall discover much Emptiness, Darkness, and Confusion; nay, if I mistake not, direct Impossibilities and Contradictions.
Two illustrious men, who adopted his method with such ardour, rendered it so completely their own, and made so many elegant applications of it that Leibnitz several times published in the journals, with a disinterestedness worthy of so great a man, that it was as much indebted to them as to himself. ...I am speaking of the two brothers James and John Bernoulli.
Every branch of the new geometry proceeded with rapidity. Problems issued from all quarters; and the periodical publications became a kind of learned amphitheatre, in which the greatest geometricians of the time, Huygens, Leibnitz, the Bernoullis, and the marquis de l'Hopital combated with bloodless weapons; the honour of France being ably supported by the marquis for several years.
John Bernoulli who... learned the analysis of infinites from the writings of Leibnitz, ... advances not only that the method of fluxions did not precede the differential calculus, but that it might have originated from it; and that Newton had not reduced it to general analytical operations in form of an algorithm, till the differential calculus was already disseminated through all the journals of Holland and Germany.
Barrow's lectures failed to attract any considerable audiences, and on that account he felt conscientious scruples about retaining his chair. Accordingly in 1669 he resigned it to his pupil Newton, whose abilities he had been one of the earliest to detect and encourage.
[Isaac Newton's] subsequent mathematical reading as an undergraduate was founded on Kepler's Optics, the works of Vieta, Schooten's Miscellanies, Descartes's Geometry, and Wallis's Arithmetica infinitorum: he also attended Barrow's lectures.
[Isaac Newton] took his BA degree in 1664. There is a manuscript of his written in the following year, and dated May 28, 1665, which is the earliest documentary proof of his discovery of fluxions. It was about the same time that he discovered the .
In 1704 published his Optics, containing an account of his emission theory of light. To this book two appendices were added; one on cubic curves, and the other on the quadrature of curves and his theory of fluxions. Both of these were old manuscripts which had long been known to his friends at Cambridge, but had been previously unpublished.
The Greeks developed the about the 5th century B.C.Zeno of Elea (c. 450 B.C.) was one of the first to introduce problems that led to a consideration of magnitudes. He argued that motion was impossible, for this reason:
Antiphon (c. 430) is one of the earliest writers whose use of the is fairly well known to us. In a fragment of Eudemus (c. 335 B.C.)... we have the following description:
It is known that Hippocrates of Chios (c. 460 B.C.) proved that circles are to one another as the squares of their diameters, and it seems probable that he also used the ... Archimedes tells us that the "earlier geometers" had proved that spheres have to one another the triplicate ratio of their diameters, so that the method was probably used by others as well.
The first great advance, after the ancients, came in the beginning of the seventeenth century. Galileo (1564-1642) would appear to have led the way, by the introduction of the theory of composition of motions into mechanics; he also was one of the first to use s in geometry, and from the fact that he uses what is equivalent to "virtual velocities" it is to be inferred that the idea of time as the independent variable is due to him.