How did mathematical analysis develop after the 16th century?
I
deas about mathematical analysis took a long hiatus after the Greeks. It didn’t
begin to grow again until the 16th century, when the need to examine
mechanics problems became important. For example, German astronomer and
mathematician Johannes Kepler (1571–1630) needed to calculate the area of
sectors in an ellipse in order to understand planetary motion. (Interestingly,
Kepler thought of areas as the sums of lines—a kind of crude form of integra-
tion; even though he made two errors in his work, they canceled each other out
and he was still able to determine the correct numbers.)
By the 17th century, many mathematicians had begun to contribute to the
field of mathematical analysis. For example, French mathematician Pierre de Fer-
mat (1601–1665) made contributions that eventually led to differential calculus.
Bonaventura Cavalieri presented his method of indivisibles,one he developed after
examining Kepler’s integration work. English mathematician Isaac Barrow
(1630–1677) worked on tangents that formed the foundation for Newton’s work on
calculus. Italian mathematician Evangelista Torricelli (1608–1647) added to differ-
ential calculus and many other facets of mathematical analysis. (In fact, collec-
tions of paradoxes that arose through the inappropriate use of the new calculus
were found in his manuscripts. Unfortunately for mathematics, Torricelli died
young of typhoid.) And, of course, the one name most associated with calculus—
Isaac Newton—developed some of his most brilliant work during the 17th century.most important contribution was the method of exhaustion (expanding the measure-
ments of an area to take in more and more of the required area).
For example, Zeno of Elea (c. 490–c. 425 BCE) based many problems on the infi-
nite; Leucippus of Miletus (fl. c. 435–c. 420 BCE), Democritus of Abdera (460–370 BCE;
a student of Leucippus who also proposed an early theory about how the universe was
formed), and Antiphon (c. 479–411 BCE; who some historians believe tried to square
the circle) would all contribute to the method of exhaustion. Eudoxus of Cnidus
(c. 400–347 BCE) would be the first to use the method on a scientific basis. Archimedes
(c. 287–212 BCE; Hellenic)—considered one of the greatest Greek mathematicians—
took mathematical analysis one step further: He more fully developed the theory pre-
sented by Eudoxus that would eventually lead to integral calculus.What is considered one of Archimedes’smost significant contributions
to mathematics?
Archimedes made many significant contributions to mathematics, though not all
210 mathematicians would agree with the label “most significant.” But one of his contri-