In the 18th century, it was certainly believed that all determinant values of a variable could be expressed as numbers. Contrary to the modern view, the particularization of the variable x in the function f(x) was problematic in the 18th century. However, there is a very important consequence of the concept of a variable as a quantity in general.
In the second case, the determination of one variable in no way limits the meanings of the others.
AN ALTERNATIVE DEFINITION OF FUNCTIONS
Therefore, in the 18th century, the real novelty of the concept of functions was not the appearance of functionality in mathematics, but the fact that functionality was subjected to calculations. As a result, mathematicians emphasized a particular aspect of a function depending on the circumstances: the relation was emphasized in applications or when the context allowed an intuitive discussion; form in analytical manipulation. He pointed out that the calculus cannot be defined in everyday terms and that even the part of the final value analysis from which the differential calculus is developed is not sufficient for this purpose.
Therefore, he had to introduce the basic concepts of calculus (variables, functions, infinitesimal numbers and differential ratios) in an intuitive way. The definitions of the 1755 preface thus differ from the definitions that Euler gave elsewhere in a formal or analytical way (in [16], for variables and functions, and in chapters III, IV and V of the first part of Institutiones calculi differentalis - that is, in the treatise in the strict sense of the word- for infinitesimal numbers and differential ratios. In the preface of 1755, Euler initially defined a variable simply as a continuously increasing or decreasing quantity (see footnote no. 7).
For example, if the amount of gunpowder is fixed and one changes the angle of fire, then the range and time of the trajectory have also changed. It is precisely a dependence of this kind that characterizes a function: "Quantities that depend on others in this way (in which, when the latter changes, the former is also changed), are referred to as functions of the latter. The intuitive meaning of the word "function" (in my terminology, the functional relation) was sufficient for the scope of the preface of [19] (but not for analytical investigation).
CONDITIONS FOR THE REPRESENTIBILITY OF FUNCTIONAL RELATIONS AS FUNCTIONS
Since they could not exhibit a form of the type f(x), they simply used the equation F(x,y)=0 as form. He gave the example of an equation of the fifth degree, which is the same example mentioned in [16]. Initially, this class of peculiar transcendental functions consisted exclusively of the exponential and logarithmic functions.
In the second half of the 18th century, there were many attempts to invent new functions. There is another very important aspect of the representativeness of a functional relation as a function, which I referred to several times above. To clarify this last point, consider Euler's construction of the exponential function i.
In the final analysis, the construction of an exponential function refers to the construction of a curved line passing through the point (n, an) and this guarantees the existence of the function. This brings us to a key aspect of 18th-century analysis: the intuitive image of a function was a segment line, or a piece of a curved line described by other lines. Although the table of values of a given function was one of the tools mathematicians had to have in order to understand that function, the table of values was not the image of the function.
LOCAL AND GLOBAL POINTVIEWS
Not only did the numerical not logically precede the continuous, but on the contrary the discrete could arise from the continuous and be seen as an interruption of the continuous. P was therefore a multivalued function because it takes two real values, while 3 P had to be considered a single-valued function because it takes one real value and two complex values. Real functions really mattered; complex functions were not an autonomous object of study: rather they were useful tools for the theory of real functions and their use appears to be limited to exceptional circumstances.
In arithmetic, geometry and mechanics, functions and variables have a natural scope and therefore mathematicians were obliged to take into account the limitations imposed by the nature of the particular problem under investigation. When the results obtained from the use of generality were applied to other sciences, they had to be subjected to appropriate reinterpretations that adapted them to concrete circumstances. This approach is an aspect of the mathematical method for studying natural science in the 18th century, which Dhombres [11] referred to as the "functional method".
Solving a problem mathematically, appropriate symbols replaced concrete quantities and their relationships are conceived as forms and equations. The solutions of these equations would be interpreted in relation to the specific problem and eliminating anything that was meaningless to this particular problem. The results are obtained without any restrictions regarding the convergence of the series; only at the time of application was the numerical sense of the series (and therefore convergence) significant.
THE LAW OF CONTINUITY
In [20] Euler said that curves or functions were discontinuous if they were the union of more than one equation: the formal aspect, the analytical expression, was indicated by the term equation, while the functional relation was indicated by the term curve. and function, the one often used in place of the other. Since the purpose of [20] was the application and interpretation of certain results of the calculus, Euler now emphasized the intuitive aspect of functional relation by the word 'function' (as in the preface of [19]), and resorted to to 'comparison' to indicate the formal aspect. According to my terminology, only functional relations were G-discontinuous and could be regarded as arbitrary or as lacking a definite law of formation (eg the relation between the Cartesian coordinates of a curve formed by a free stroke of the be manually traced).
In the controversy over the vibrating string, d'Alembert believed that the solution to the problem could only be interpreted by means of G-continuous functional relations, because calculus was founded in functions derived from a functional relation (see [43] ). In [20; 21], Euler added the new E-discontinuous functions to old continuous functions without changing the concept of the latter. Similarly, if one integrates a function Z(x,y) of the variables x and y with respect to x, one obtains.
The character of the quantity f(y) is determined by the nature of the problem and it can even be a quantity that is not expressed by a form, but can be considered as the ordinate of a curve whose abscissa is y (that is, a G - functional link broken). Since integration naturally contains an element of arbitrariness, Euler believed that the integral calculus of functions of more than one variable could directly provide a functional relation, without the intermediate step of form. Despite the fact that Euler was forced to admit that the use of a geometric notion immediately in an analytical context caused a "slight defect", the Eulerian solution of the vibrating string problem was substantially accepted in the 18th century.
INCOMPLETE FORMALISM
In retrospect, the controversy over the vibrating string has raised the question of the lack of analytical tools to describe certain more complicated phenomena: it in fact demonstrated the limited nature of the 18th analysis and its overall inadequacy for more sophisticated investigations instead of its local inadequacy. This will become clearer if you remember that the formal rules of calculus were understood in the 18th century as a generalization of the arithmetic of rational numbers; in that sense calculus was part of universal arithmetic. In modern terminology you would say that rational numbers and quantities (after Descartes' interpretation of the multiplication of line segments) had the same algebraic structure and that they formed the algebraic structure of the calculus.
The topological structure was instead provided by the topological properties of variable quantities, which played the role of the modern numerical continuum (see Section 6). Crucially, in my view, 18th-century mathematicians never perceived the possibility of constructing symbol systems that possessed properties distinct from those of the set. Euler's and Lagrange's calculus therefore lacked the essential characteristics of modern formalism.
However, it is not wrong to call it formal because it studied the 'form' of the relationships between quantities. Similarly, Lagrange said: “I hope that the solutions I will give will interest geometers both in terms of the methods and the results. Analysis was appreciated for its greater generality (e.g. the symbols f(x), g(x) do not refer to a specific function, but to the object function in general; while the diagram of a curve always has its own specificity has);
CONCLUSION
In the second half of the eighteenth century, such usage spread so far that the adjective 'analytic' was added to the noun 'function' to designate function in a technical sense (that is, all functions used in the mathematical analysis are considered). 19. This terminological evolution did not affect the substance of the matter.20 A functional relation in itself was never intended as an object of study in analysis: it was considered an object of study in analysis only insofar as it was embodied in a form endowed with by a special rule. Consequently, it is very difficult to undermine the 18th calculus by means of counterexamples derived from assigning a certain value to a variable, for the simple reason that a theorem of type (T) was a theorem concerning abstract quantities (variables) and not on their values. .
20 It seems to have left its mark only in the proliferation of definitions similar to the preface of [1755]. Bos, Differentials, Higher Order Differentials, and Derivatives in Leibnizian Calculus, Archive for the History of the Exact Sciences. Calculus as algebraic analysis: some observations on mathematical analysis in the 18th century, Archives for the History of the Exact Sciences.
Ivor Grattan-Guinness, The Development of the Foundations of Mathematical Analysis from Euler to Riemann, Cambridge (Mass.) and London: M.I.T Press, 1970. Giovanni Ferraro, Differentials and differential coefficients in the Eulerian Foundations of the calculus, Historia. Giovanni Ferraro, Convergence and formal manipulation of series in the first decades of the eighteenth century, Annals of Science.