The Renaissance. There is a much-studied problem
whether and in what sense there indeed was a Renais-
sance—in the fifteenth and sixteenth centuries, say—as
institutionalized by Jacob Burckhardt; that is, whether
there is a marked-off era which interposes itself be-
tween medieval and modern times. (For a history of
the problems see W. K. Ferguson, K. H. Dannenfeldt,
T. Helton, also E. Panofsky.) George Sarton, the leading
historian of science, once made the statement (which
he later greatly modified) that “from the scientific point
of view, the Renaissance was not a renaissance” (quoted
in Dannenfeldt, p. 115). Of course, nobody would deny
that there was a Copernicus in the sixteenth century
or that the introduction of printing created a great
spurt in many compartments of knowledge, scientific
or other; but the question, which Sarton's statement
was intended to answer, is whether during the era of
the Renaissance, disciplines like physics, chemistry,
biology, geology, economics, history, philosophy, etc.,
were developed in a manner which sets these two
centuries off from both the Middle Ages and the seven-
teenth century.
Now, with regard to mathematics no such doubts
need, or even can arise. There was indeed a mathe-
matics of the Renaissance that was original and dis-
tinctive in its drives and characteristics. Firstly, there
was a school, mainly, but not exclusively, represented
by Germans (Peurbach, Regiomontanus, and others),
that sharply advanced the use of symbols and notations
in arithmetic and algebra. Secondly, and strikingly, an
Italian school sharply advanced the cause of the alge-
bra of polynomial equations when it solved equations
of the third and fourth degree in terms of radicals
(Scipione del Ferro, Ludovico Ferrari, Nicolo Tartag
lia, Geronimo Cardano). Thirdly, a French school,
mainly represented by François Viète, achieved a syn-
thesis of these two developments. And finally, an “in-
ternational” school laid the foundation for the eventual
rise of analysis by introducing and studying two special
classes of functions, trigonometric functions and loga-
rithms. Regiomontanus, Rheticus, Johann Werner, and
later Viète gradually made trigonometry an inde-
pendent part of mathematics, and Henry Briggs and
John Napier (and perhaps also Jost Bürger) created the
logarithmic (and hence also exponential) functions.
After that, for over three centuries trigonometric and
logarithmic functions were the stepping-stones leading
to the realm of analysis, for students of mathematics
on all levels.
There is a near-consensus among social historians
that the rise of arithmetic and algebra during the
Renaissance was motivated “preponderantly by the rise
of commerce in the later Middle Ages; so that the
challenges to which the rise of algebra was the response
were predominantly the very unlofty and utilitarian
demands of counting houses of bankers and merchants
in Lombardy, Northern Europe and the Levant”
(Bochner, Role..., p. 38). On this explanation, the
socioeconomic needs that were thus satisfied were not
those of the “industrialist” but those of the merchant,
in keeping with the fact that in the late Middle Ages,
and soon after, the general economy was dominated
not by the producer but by the trader (ibid.).
