User:Boris Tsirelson/Sandbox1
Birth and infancy of the idea
Some tables compiled by ancient Babylonians may be treated now as tables of some functions. Also, some arguments of ancient Greeks may be treated now as integration of some functions. Thus, in ancient times some functions were used implicitly, without being recognized as special cases of a general notion.
Further progress was made in the 11th century by Al-Biruni (Persia), and in 14th century by the "schools of natural philosophy" at Oxford (William Heytesbury, Richard Swineshead) and Paris (Nicole Oresme). The concept of function was born, including a curve as a graph of a function of one variable and a surface - for two variables. However, the new concept was not yet widely exploited either in mathematics or in its applications.
Power series
Further progress appears in the 17th century from the study of motion (Johannes Kepler, Galileo Galilei) and geometry (P. Fermat, R. Descartes). A formulation by Descartes (La Geometrie, 1637) appeals to graphic representation of a functional dependence and does not involve formulas (algebraic expressions):
If then we should take successively an infinite number of different
values for the line y, we should obtain an infinite number of values for the line x, and therefore an infinity of different points, such as C, by means of which the required curve could be
drawn.
The term function is adopted by Leibniz and Jean Bernoulli between 1694 and 1698, and disseminated by Bernoulli in 1718:
One calls here a function of a variable a quantity composed in any manner whatever of this variable and of constants.
This time a formula is required, which restricts the class of functions. However, what is a formula? Surely, y = 2 x2 - 3 is allowed; what about y = sin x? Is it "composed of x"? "In any manner whatever" is now interpreted much more widely than it was possible in 17th century.
... little by little, and often by very subtle detours, various
transcendental operations, the logarithm, the exponential, the trigonometric functions, quadratures, the solution of differential equations, passing to the limit, the summing of series, acquired the
right of being quoted. (Bourbaki, p. 193)
But on the first stage the notion of an algebraic expression is quite restrictive. More general, possibly ill-behaving functions have to wait for the 19th century.