Infinitesimals are fictions, useful instruments for dealing with finite quantities, so Euler was able to argue that the calculus algorithm is not about differentials, but about functions. The same criticism was leveled at the method of the first and last reasons: indeed, in the expression ((x+h)n-xn)/h was first treated as finite. The point of the question was to know what meaning to attach to the equation a+dx=a.
In section 114 of the Introductio in analysin infinitorum (first section of chapter V) Euler stated that if a is a number greater than one and and are infinitesimal (or rather fractions so small that they are almost equal to nothing ), then it follows that. He suggested the idea that the difference between az1 and az2 of the exponential function az can be made equal to a finite small quantity. Euler's use of evident quantity corresponds to Euclid's use of the "common notion" in the Elements: a part is less than the whole.
According to Euler, an infinite quantity was not an actual quantity in the proper sense of the word, but a short way of indicating how "true" quantities vanish. Euler justifies the need to introduce the use of infinity for numbers by observing that the sum of the numbers 1+2+3+4+. 34 Euler refers to the fiction of the infinitesimal part of a line as a reference to the representation of infinitesimals in a diagram;.
Further aspects of infinite numbers and infinitesimals
The usual notion of a sum of a series, of the limit of a series and function, of an improper integral is not regarded as intrinsic to the real nature of mathematical objects (and thus the only possible one); on the contrary, it is assumed that different and arbitrary definitions of sum, limit and improper integral can be given. Moreover, nowadays every object in mathematics is introduced through an appropriate definition, even if it is basically arbitrary. Given this context, it is not surprising that Euler wanted to use the symbol (-1) without defining it: if he had behaved differently, it would have been anomalous compared to standard practice.
Such a justification in the case of (-1), rather than being linked to the rather confused idea of infinite parity41, was based on the conscious acceptance of calculus as an algebraic formalism and of the principle of generality in algebra. However, it is not my intention in this article to examine this point in detail (although I will return to it briefly in the final section). I merely intend to emphasize the point that Euler's use of infinity also includes (7), and that a translation of the vague Eulerian notions into the precise terminology of non-standard analysis eliminates aspects that were thought to be uniform.
A further notable aspect of the Eulerian notion of infinite quantity and infinite size is the significance that could be attributed to the equation "0=0" and. This idea is the basis of the distinction between arithmetic and geometric equations, with which Euler aims to provide an improved mathematical explanation of the equation between infinitesimals (0=0) and between infinite numbers. He notes that andx=a (n is any number) is true, being verified not only in the arithmetical sense ((andx)-a=ndx that is ndx=0) but also in a geometric sense.
Euler notes that "the infinitesimal quantity dx2 vanishes before dx", since the quantities dx+dx2 and dx (both vanishing: dx+dx2=dx=0) go to zero in the same way (dx+dx2):dx=1 + dx=1. In modern terms, A and B are asymptotically equal43 or have the same asymptotic behavior.44 Nevertheless, while it is quite clear what is meant by the relation between infinitesimal and infinite quantities and , it is. 44 In particular, the idea of the approach of two quantities, which we can see here in function, rather than the usual definition of limit, makes us think of the following generalized definition:….
Of course, there is no intention to attribute to Euler modern asymptotic concepts46 or the definition of the Peano derivative: this would be as difficult as saying that Euler uses infinitesimals in the sense of non-standard analysis. I merely noted that Eulerian notions possessed many facets which, in the hindsight of modern knowledge, we can appreciate, but which were obscure to Euler: therefore the idea of vanishing quantity contained subtleties that made us think of the standard concept of limit or modern infinitesimals or of asymptotic processes or generalized limit concepts.
The calculus as the calculus of functions
In the 3rd and 4th chapters of [1755], where the rules of calculus are formulated,47 the introduction of differential coefficients was much more laborious. In Chapter 1, Euler defined finite differences of any order49 and established rules for the sum and product of finite differences. Based on the study of these functions (which made up the universe of the most frequently studied functions), he stated that the difference y for any function y can be expressed in the form
Euler, however, preferred to give a direct definition of the differential dy of the function y(x) (using y=P+Q+R+S+.. ) and only an indirect definition the differential coefficient.50 For example, to determine the differential of y=xn, he considered. Although the calculus dealt with finite quantities dy/dx, in the actual construction of the calculus it regarded dy/dx as the actual ratio of the differences dy and dx. Euler used differentials on the basis that the differential dy of the function y(x) could be regarded as the increment of y(x) when x has an infinitely small increment dx [namely Euler based on the formula dy=y (x+dx)-y(x).
Thus, in the calculation of the differential xn, there is no reference to the marginal process and the short-lived quantity, contrary to the case presented in the preface of the Institutiones in De usu. Consequently, dV is obtained by calculating the differentials of V, each time assuming that two of the variables are constant [Euler. These were considered particularly problematic from the beginning of calculus and were, for example, the subject of Nieuwentijdt's attack on Leibniz.
In §.126 of the first part of Institutions, Euler stated that the second differential was simply the differential of the first differential (since the second difference was simply the difference of the first difference) [Euler. Instead, in chapter IV of Institutions, Euler used the complex construction described above to introduce higher-order differential coefficients: he seemed to be concerned about dealing with it. The issue of the choice of progression of variables and the indeterminacy of higher-order differentials in eighteenth-century calculus is treated in detail by Bos (see, in particular [Bos.
Differentiation is not a pointwise defined operation;58 in other words, the differential is not defined at a specific point x0 of the domain of the function. He said that the sum y of z=y is not unique but of the type y=x+C, where C is an arbitrary constant. 58 Today, a derivative is defined locally in the sense that, given a point x0 of the domain of a function f(x), the derivative is defined at the point x0.
No adequate development of this idea took place: Euler never actually dealt with such functions, merely stating that it was possible to define the differential coefficient and that they were part of the solution of the equations of partial derivatives.61.
Conclusion
The only, and in any case partial, exception to the approach to the calculus as an algorithm that transforms analytic expressions into analytic expressions, concerns the treatment of equations of partial derivatives. For example, Lagrange rejected the use of vanishing quantities and infinitely large quantities; however, he accepted the attempt to give a systematic translation of the calculus of differentials into a calculus of finite quantities as well as the use of rules that are applied globally to analytical expressions and not locally to functions in the modern sense. Soon after, Cauchy rejected this latter aspect of Eulerian theory, but accepted the fundamental idea that the algorithm of the calculus consisted in determining the differential coefficient.
Cauchy's use of the idea of limit was explicit, but the use of infinitesimal numbers and infinite numbers persisted, provided they could be effectively reduced to limits (accordingly, a case like (-1) was excluded). 62 In this article I have not gone into detail about the interpretations given to the calculation of Eulerian zeros in the eighteenth century. I shall simply cite the fact that Lagrange regarded the fundamentals of the calculus of Euler and d'Alembert as analogous and contrasted both with the Leibnizians.
De progressionibus transcendentalibus, vel quarum termini generales algebraice tradi non possunt, Opera a) Methodus universalis summas seriei convergentium quam proxime fieri potest, Opera. Ratio ortus 20th century mathematica, Archivio pro Historia Exacta Scientiarum, apparet. Lagrange Mutationem Accede ad Fundationes Calculi Variationum Archive de Historia Exactae Scientiae.
Calculus as Algebraic Analysis: Some Observations on Mathematical Analysis in the 18th Century, Archives for the History of Exact Sciences. Aspect of Euler's attempt to transform calculus into algebraic calculus, Quaderns d'història de l'ingenieria. Giovanni Ferraro, Differentials and differential coefficients in Euler's foundations of calculus, Historia Mathematica.
Giovanni Ferraro, Convergence and formal manipulation of series in the first decades of the 18th century, Annals of Science. Giovanni Ferraro, Pure and Mixed Mathematics in the Work of Leonhard Euler in Computational Mathematics: Theory, Methods and Applications, c.
