THE RISE OF FUNCTIONS
rightly insists that with Oresme’s anticipation of functions something very
new was added to the classical Greek mathematics of Euclid, Archimedes,
and Apollonius.
Since the second half of the 19th century, Oresme’s standing as a harbinger
of analysis in general and of the concept of function in particular has been
steadily on the rise. But, on hard tangible evidence, it seems impossible to
assay what Oresme’s effect on subsequent developments of analysis actually
was. His works were known in the 15th and 16th centuries [6, p. 88]. But he,
or his mathematical works, are apparently never mentioned by name in the
decisive 17th or 18th centuries [6, p. 165], and may have been unfamiliar
to them. It can only be recorded that in the 19th century the Great Reha-
bilitation of the Middle Ages, which had become a state of mind, somehow
remembered Oresme and began to restore his achievements one by one.
Renaissance. It may be said that in the 16th and 17th centuries almost
anything mathematics achieved stimulated the eventual emergence of
functions. Thus, the rise of formulaic algebraic expressions undoubtedly
contributed to the rise of functions which can be given by such expressions.
Also, the intensive preoccupation with logarithms could not but lead to
the introduction of the pair of functions {log x, ex} and to the realization
that these functions are inverses of each other. Finally, Irigonometrists may
have sensed that the addition theorem
sin (x + y) = sin x cos y + cos x sin y
is a “functional equation” by which to define sin x and cos x for angles
greater than 360o; this suggested itself to me when reading the great work
of von Braunmiihl [7], although I would not be able to adduce a specific
reference.
It can be said even more affirmatively that the analysts of the 17th century,
certainly beginning with Descartes and Fermat, always had functions in
their thinking, even though they spoke of “curves,” as they had to. (The
term “function,” a dictionary word since the 16th century, began to be
used as a mathematical term only in 1694, in a publication of Leibniz.)
Thus, Fermat certainly dealt with functions in his famous paper on maxima
and minima, which he composed sometime between 1629 and 1638 (and in
which the word “analyst” occurs several times). He considers a general
“parabola”
У = P{×},
in which P(x) is a polynomial of any degree, and he asserts that its maxima
and minima occur among those points for which