History of the infinitesimal calculus

The history of the Infinitesimal calculus is related to two Mathématicien S: Isaac Newton and Gottfried Wilhelm von Leibniz.

Context

At the 17th century, two problems impassion the mathematicians: that of the tangent and that of the squaring S. the first consists in finding, starting from a unspecified Courbe, the various tangents with the curve. The second lies in the calculation of the surface generated by a curve. Many are those which are interested in these problems and various solutions give some: Descartes, Wallis, and others. However, two scientists, Isaac Newton and Leibniz, each one on their side will make research and will put at the day a general and simple solution with these problems. By doing this, they will introduce into the world of mathematics a new concept which is the base of the today analyzes: infinitesimal calculus.

Isaac Newton

Large mathematician and Physicist English of the 17th century, Newton is regarded as one of the founders of the infinitesimal calculus. Taking as a starting point Descartes and Wallis of which it had read the writings, it indeed poses the problem of the tangents which it quickly connects to that of squaring. However, he writes rather little on this subject (only three writings) and will be published very late by fear of criticisms. As of 1669, Newton, taking as a starting point Wallis and Barrow, connects the problem of squaring to that of the tangents: the Dérivée is the opposite procedure of the integration. It is interested in the infinitesimal variations of the mathematical quantities and the surface generated by these movements . Its method most famous remainder that of the Fluxion S. Très influenced by its work of physicist, it regards the mathematical quantities as generated “by a continual increase” and compares them with the space generated by the “bodies moving”. Into the same spirit, it introduces time as a universal variable and defines the fluxions and the flowing ones. The flowing ones (X, there, Z,…) are quantities “increased gradually and indefinitely”, and the fluxions ( $\ dowry X, \ dowry there, \ dowry z$ ) “speeds whose flowing ones are increased”. It poses the problem “being given the relations between the flowing quantities, to find the relation between their fluxions. ”.

Here for example the solution which it gives for $y=x^n$ :

Is O, an time interval infinitely small.

\ dowry X o

and

\ dowry o

will be increases infinitely smalls in X there and Y.

y= x^n \,

By replacing X and there by

x+ \ dowry X o

and

y+ \ dowry there o

y+ \ dowry there O = (x+ \ dowry X O) ^n

Then while developing by the formula of the binomial which it showed:

y+ \ dowry there O = x^n + nox^ {n-1} \ dowry X + \ frac {N (n-1)}{2} o^2 \ dowry x^2 x^ {N2} + \ ldots

Then, it cuts off

y= x^n \,

and divided by O.

\ dowry there = nx^ {n-1} \ dowry X + \ frac {N (n-1)}{2} o^2 \ dowry x^2 x^ {N2} + \ ldots

Lastly, it neglects all the terms containing O, and obtains:

\ dowry there = nx^ {n-1} \ dowry X

, which points out the well-known formula

(x^n) '= nx^ {n-1} \,

The intuition is well there, but Newton misses conviction. He would like to get rid of the infinitesimal quantities that it does not manage to base on rigorous principles. In its method “of the first and last reasons”, it will be satisfied with the relationship between fluxions, which will enable him to avoid “neglecting” terms, letting O “disappear” in the report/ratio. It approaches then our current concept of limit, comparing that with the idea “speed instantaneous” of a body. Not that which it has before arriving, nor that which it has afterwards, but that which it has at the time when it arrives. In Principia , it expresses its opinion thus: “The ultimate reports/ratios in which the quantities disappear are not really report/ratio of ultimate quantities, but the limits towards which the report/ratio of quantities, decreasing unbounded, always approach some: and towards which they can also approach some close that one wants. ” It is incredible to see at which point this design approaches the definition even of the limit used today: F (X) tends towards has if being given ε given positive unspecified, there exists α such as: |x-a|<α => |F (X) - F (a)|<ε. However, Newton does not generalize this definition and its concept of limit remains held for the reports/ratios of fluxions, with what approaches our calculation of derivative. And even thus, it is in the incapacity to base its differential calculus on rigorous bases. The concept of infinitesimal value is still too new and is highly criticized, being for certain only one “phantom of disappeared quantities”.

Gottfried Wilhelm von Leibniz

Whereas Newton hesitated to publish its discoveries, another mathematician: Wilheilm Leibniz was interested in this same problem and made similar discoveries. Its approach is however very different. Indeed, Leibniz is at the beginning a Philosophe and discovers mathematics only in 1672 when it meets Christian Huygens at the time of a voyage to Paris. It is then inspired by works by Descartes, Pascal, Wallis and others. Very quickly, it establishes the link between the problem of the tangents and that of squaring by noticing that the problem of the tangent depends on the report/ratio of the “differences” of the ordinates and the X-coordinates and that of squaring, of the “nap” of the ordinates. During its work on the combinative ones, it observes this indeed:

1,4,9,16 being the continuation of the squares

1,3,5,7 the continuation of the differences of the squares:

1+3+5+7=16

Its work in philosophy pushes it to consider the infinitely small differences and it draws soon the conclusion: ∫dy = there, ∫ being a sum of infinitely small values and Dy an infinitesimal difference.

Indeed, Leibniz puts forth at the same time the philosophical assumption of the existence of components infinitely smalls of the universe. All that we perceive being only the sum of these elements. The relationship with this mathematical research is direct. He explains sometimes also these infinitesimal elements by making an analogy with the geometry: the dx is with X, which the point is with the right-hand side. What pushes it on the assumption of impossibility of comparing differential values with “true” values. Just like Newton, it will privilege the comparisons between report/ratio. The clear and practical notation which it sets up (that which we let us use today) allows of fast and simple calculations. Being interested in the dy/dx report/ratio, it identifies it with the directing coefficient of the tangent, being justified by the study of the triangle formed by an infinitely small portion of the tangent and two infinitely small portions of the parallels to the x-axis and that of the ordinates. Thus, it expresses for example the directing coefficient of the tangent to the curve representative of y=x ²:

$\ mathrm Dy = (X + \ mathrm dx) ^2 - x^2 = 2x \ mathrm dx + (dx) ^2 \,$

\ frac {\ mathrm Dy} {\ mathrm dx} = 2x + \ mathrm dx

And finally, by neglecting D X :

\ frac {\ mathrm Dy} {\ mathrm dx} = 2x

It solves also the problems D (X + there), D (x.y), D (x/y), D (xⁿ) in optics to create a true algebra of the infinitely smalls. But it undergoes many criticisms, similar to that which one did in Newton: For which reason does neglect one the infinitesimal ones in the end result? And if they are equal to 0, how can one submit their report/ratio? Itself has evil to base its theory on solid concepts and tends to regard the infinitesimal values as tools, as well as the imaginary numbers, which “would not exist” not really. But even thus, its detractors remain numerous.

At the 19th century

It is only at the 19th century that the concept of limit will be truly clarified. And it is only as well as differential calculus will be able to really develop. Because, indeed, it is not on reports/ratios that Newton and Leibniz work, but of course of the limits of report/ratio, and it is this concept which is the base of all the remainder. It is at that time that the real number as we know it truly is introduced in the mathematicians. As much at Newton that at Leibniz, it is this design which misses and which prevents them from basing the limit on rigorous bases. The number as they conceive it still is very inspired of the vision of Euclide. And without the characterization of the density and the intrinsically infinite character of realities, the concept of limit cannot be born.

It is what that their detractors, with the number of which George Berkeley, will be very numerous. One will show also Leibniz to have copied the work of Newton. Will follow of many arguments and the personal attacks between the two men. We can however note that their approaches are very different and it is this difference which brings all its richness and all its inventiveness to this new concept. They is also these two visions of the things, in their common points and their dissimilarities, which can allow us, still today, to better include/understand this concept of infinitesimal value. Indeed, it is his relationship with physics, philosophy or the geometry, that themselves had to make to conceptualize it, which gives us the possibility of seizing it as a whole.

In spite of criticisms, the simple methods and clear developments by Newton and Leibniz, in particular the new formalism introduced by Leibniz, make it possible to solve the problems of the tangents and the squaring, which worried much the mathematicians of the time. And this is why they were gradually made accept to their foundation on strong foundations at the 19th century. We must still much, ourselves, with these concepts, and still besides we use the formalism of Leibniz in a current way and even that of Newton in physics. And this is why, still today, Newton and Leibniz, beyond their quarrels, are regarded as the founders of the infinitesimal calculus.

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