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The diffusion of calculus

|    the Analyse des infinimentes petites    |    integration of differential equations    |    the problem of the brachistocrona    |    the birth of the concept of function    |   

The Analyse des infinimentes petites.

An essential contribution to the development and the spreading of the Leibniz calculus came from several members of the Bernoulli family. In 1691 Johann spent a period in Paris and attended the meetings of the cultural circle that had formed around father Malebranche. Among others he met there, was the marquis Guillaume Françoise de l'Hospital (1661-1704) to whom he gave lessons on the new method of calculus. From the handwritten notes related to those lessons came the treatise Analyse des infiniments petits which de l'Hospital published anonymously in Paris in 1696. This first systematic exposition of differential calculus which was completed fifty years later by Jacob with the Lectiones mathematicae de methodo integralium for the calculus related issue defined by Leibniz as "summatorius" and "integralis" became a great success, and the text book which formed generations of mathematicians. The Analyse begins by setting a few definitions, among which we find the differential, and two postulates or "requests or assumptions":

The infinitesimal part by which a variable quantity is continually increased or decreased is called the difference of that quantity [...]
It is evident that the difference of a constant quantity is null or zero, or (and it is the same) that constant quantities have no difference [...].
I. Request or assumption
It is required that two quantities with an infinitesimal difference can be used indifferently one instead of the other [...]
II. Request or assumption
It is required that a curve can be considered as the collection of an infinity of straight lines, each being infinitesimal, or (which is the same thing) as a polygon of an infinite number of sides, each one infinitesimal, which from the angles formed between them, determine the curvature of the line [...]

The rules of differentiation for ordinary calculations are then expounded and the following sections of the volume are dedicated to the application of the calculus to geometry problems such as finding tangents, determining the maxima, minima and flexes of the curve, the study of the curvature, evolutes, caustics and envelopes.
The so called "rule of de l'Hospital" can be found in section IX.

* Exhibit IV.1

Guillaume Francois de L'Hospital
Analyse des infiniments petits

Let AMD (AP=x, PM=y, AB=a) be a curve such that the value of the ordinate y is expressed by a fraction in which the numerator and the denominator both become zero when x=a, that is when the point P falls on a given point B. The question then is, to what should the value of the ordinate BD.

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Integration of differential equations

The problem of the integration of differential equations emerged at the beginning of calculus as "the inverse problem of tangents ". In brief, the issue is to find a curve when a relation between the tangent and the coordinates is given, which in the simpler cases can be written analytically in the form $dy/dx=f(x,y)$. The first problem of this kind was posed by Florimond De Beaune and solved by Descartes without using differential calculus. The issue was to find a curve whose subtangent had a constant value.
In the Nova methodus, Leibniz himself proposed a solution to this as an immediate application of the calculus. After this, many mathematicians tried out their solutions with different formulations and variants.
In modern terms one could say that, recalling the fact that the subtangent of a curve $y=y(x)$ is the quantity $y/y'$, the problem leads to an equation of the type $y'=ay$, the solution of which is $y=Ae^{ax}$.
A variant to the problem that leads to an equation of the type $y'=a(x-y)$ is taken into consideration by Johann Bernoulli who gives the construction of the curve using a logarithmic curve.

* Exhibit IV. 2

Johann Bernoulli Solutio problematis Cartesio Propositi in Opera [...] The problem is therefore the following: find the curve AI such that its ordinate KI is to the subtangent KM as a given segment N is to the segment IL of the ordinate determined by the curve AI and by the line AL which forms together with the axis AK an angle equal to half a right angle. Solution: [...] back to top

The problem of the brachistochrone

Given two points on a vertical plane, find among all the segments of the curve that link them, the trajectory that a heavy particle needs to cover to go from one point to the other such that the time taken is the lowest possible: the question is proposed in 1696 by Johann Bernoulli in the Acta eruditorum.
This is one of the first problems of the calculus of variations. In modern terms, it means minimising the integral which expresses the time of descent in function of the curve. On the different nature of this problem in relation to the questions of maximum and minimum that had been solved up to then, Jacob Bernoulli observes:

Up to now, geometers have used the method of maxima and minima for those problems in which among infinite parts or functions of a single given curve one looks for the maximum and the minimum; only that they have not thought of applying it there where among the infinite ungiven curves, one looks for one to which one can ascribe a certain maximum or minimum.

Newton, Leibniz, de l'Hospital, Johann Bernoulli and Jacob Bernoulli, who published their solutions in Acta in the following year, all determined, independently and with different techniques, the "brachistocrone" (from the Greek $\beta\rho \acute{\alpha} \chi \upsilon \sigma \tau o \varsigma$, the shortest and $\chi\rho\tilde{\omega}\nu o \varsigma$, time), also called "oligochrone" or "curva celerrimi descensus".
Surprisingly, the new curve is the already known cycloid, for which Huygens had not long before that demonstrated its isochronic property, that is that the oscillations of a particle around a point of equilibrium on a cycloid trajectory are completed in equal times independently from their amplitude.
The solution proposed by Johann brings the problem of the fastest trajectory down to the issue of the path followed by a ray of light through a medium with a suitably chosen variable index of refraction. Using the law of Snell to determine the path of a ray of light through this means, he considers a certain number of strata and then considers their number as tending to infinite.
The solution of his brother Jacob is more geometrical and also quite elaborate, but contains a more general idea that can be used also in other cases. To determine the points of the curve, Jacob considers infinitesimal increments of abscissas, such as EC, and ordinates, such as GE, and also increments obtained from variations which are infinitesimal compared to these, such as GL compared to GE. With geometrical considerations and simple relations between velocity and spaces travelled, he operates on these infinitesimal quantities and infinitesimal quantities of infinitesimal quantities, such that in the end the abscissas and the ordinates have to satisfy the relation $\frac{EG}{\sqrt{HC}} : \frac{GI}{\sqrt{HE}} = CG : GD$, which defines the cycloid.

* Exhibit IV. 3

Jacob Bernoulli De curva celerrimi descensus In the development, we propose to the famous Nieuwentijt the use of differential-differentials (which he unjustly refuses), there where we are compelled to assume that the bit GL is even smaller than EG and GI which are already infinitely small. Without this, I do not see how the solution to the problem can emerge. EG and GI are in fact elements of the abscissa AH, just like CG, GD are elements of the curve itself, and HC, HE elements of its ordinate, and CE, EF its elements; so that having brought the problem down to pure geometry, one ends up looking for the curve the elements of which are directly proportional to the elements of the abscissa and inversely proportional to the elements of the ordinates. In truth what I observe is that this property belongs to the isochrone of Huygens, from now on also said oligochrone, commonly known to geometers as cycloid [...]

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The birth of the concept of function

The word "function" appears for the first time in a manuscript written by Leibniz in 1673 called Methodus tangentium inversa seu de functionibus . It then reappears in articles published subsequently and repeatedly in the correspondence with Johann Bernoulli. The differential calculus, both of Leibniz and Newton, is centered on the subject of curves. Typical problems are those of finding tangents and the quadrature of given curves or of finding curves starting from their properties.
Starting from the idea of the curve as a location, the points of which satisfy a certain equation of the type $P(x,y)=0$, where P is in general a polynomial of the variables x and y, gradually the focus tends to fall on the fact that the "applied", or the ordinates, depend on operations carried out on the abscissas.
In 1718 Johann Bernoulli, gives the following definition in an article which appeared in the archives of the Académie de Paris:

I shall call the function of a variable quantity, a quantity that is composed in any fashion by this variable quantity and by constants.

Nevertheless it is with Euler that the concept of function becomes preponderant on the less flexible concept of relation. The definition given at the beginning of the Introductio in analysin infinitorum is that of an analytical expression constructed by taking the variable $x$ obtaining the result through a set of operations.
The idea of Euler is that every function of a variable can be represented as a set of powers of that variable of the type $A+Bx+Cx2+Dx3+...$. He in fact finds these developments for all the usual functions, including trigonometric ones $\sin x$, $\cos x$, $\arctan x$, and the exponentials and logarithmics $e^x$ and $\log(1+x)$. He then declares "If anyone were to doubt, the doubt will be vanquished by the development of every function", without obviously being able to bring the demonstration. Going ahead, he produces a generalisation of the first definition, including variables at any kind of power, and not only with natural numbers:

In order for this explanation to have an even wider scope, any power should be included and not only the powers of z that have positive exponents. At that point there will be no more doubt that every function z can be transformed into an infinite expression of this type:

$$Az^{\alpha}+Bz^{\beta}+Cz^{\gamma}+Dz^{\delta}+...$$

where $\alpha$, $\beta$, $\gamma$, $\delta$...denote any kind of number.

The idea of function is still distant from the modern one. The fact that the function can be described within its interval of definition by means of a single analytical expression becomes a matter of utmost importance. Only later, following various discussions regarding in particular vibrating chords, Euler will accept the possibility of there being the so called "discontinuous" functions, described by one expression within a certain interval, and by a different one in another.

* Exhibit IV. 4

Leonard Euler Introductio in analysin infinitorum A constant quantity is a determinate quantity which maintains the same value. Quantities of this type are numbers of any kind, therefore they are that which always keeps the value it has taken; and if constant quantities of this type need to be represented with a symbol, we use the first letters of the alphabet a, b, c, etc. In ordinary analysis where the only quantities taken into consideration are the determinate quantities, these first letters of the alphabet usually indicate the known quantities and the last ones the unknown quantities; but in sublime analysis, this distinction is no longer valid, since here one is looking to another difference between quantities and that is that some are taken as constant, and others instead as variables [...] . A function of variable quantities is an analytical expression composed in some way by this variable quantity and by numbers or constant quantities. back to top

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