The main thrust of much of Cauchy’s work was to
make mathematics rigorous and precise. He insisted on
providing clarity, precision, and rigor in all the courses
he taught and in his published work. His famous 1821
text was in fact a course book for students developed
with the intention of “doing calculus the correct way.” It
is fair to say that Cauchy influenced the entire course of
mathematical research by pointing out, and demanding,
the need for absolute clarity and care in the development
of new (and even previously established) ideas. It is said
that the great mathematician P
IERRE
-S
IMON
L
APLACE
(1749–1827), after attending a lecture given by Cauchy
on the importance of the convergence of an infinite
series, quickly ran home to check the convergence of all
the infinite series he had used in his already published
popular text Mécanique céleste on celestial mechanics.
In 1826 Cauchy published seminal works in the
field of number theory, and in 1829 he defined, for the
first time, the notion of a complex function of a com-
plex variable.
Cauchy left Paris and the brewing politics of the
royal regime in 1830. Upon his return a year later, he
refused to swear an oath of allegiance to the new
regime and consequently lost his academic position. It
was not until the overthrow of Louis Philippe in 1848
that Cauchy regained his university position. Even
though Cauchy’s publication rate slowed considerably
during this trying time, he did accomplish important
work on the theory of differential equations and appli-
cations to mathematical physics during this period.
Cauchy left a standing mark on the development of
calculus with his work on refining the logical basis of
the subject and greatly influenced the study of complex
functions. A number of fundamental concepts in the
field of analysis are named in his honor, including a pair
of equations known as the Cauchy-Riemann equations
that determine whether or not a complex function is
differentiable. Cauchy died in Sceaux, near Paris,
France, on May 22, 1857. His collected works, Oeuvres
completes d’Augustin Cauchy (The complete works of
Augustin Cauchy), collated under the auspices of the
Académie des Sciences, were published throughout the
years 1882 to 1970 in a total of 27 volumes.
Cavalieri, Bonaventura Francesco (1598–1647) Ital-
ian Geometry Born in Milan, Italy, in 1598 (his exact
birth date is not known), mathematician, and disciple of
G
ALILEO
, Bonaventura Cavalieri is best remembered for
his 1635 work Geometria indivisibilibus continuorum
(A new geometry of continuous indivisibles) in which he
introduced his famous “method of indivisibles” for
determining the areas and volumes of curved figures.
This work is considered a forerunner to the entire theory
of
INTEGRAL CALCULUS
.
While still a boy, Cavalieri joined the religious
order Jesuati in Milan. In 1616, he transferred to the
monastery in Pisa, where he met Galileo and developed
an interest in mathematics. Even though Cavalieri
taught theology for many years and became a deacon
in the order, he actively pursued employment as a
mathematician. In 1629 he received a position as a
chair of mathematics at Bologna.
By this time Cavalieri had developed his method
of indivisibles. Based on A
RCHIMEDES
’ method of
exhaustion and J
OHANNES
K
EPLER
’s theory of the
infinitely small, Cavalieri’s technique provided a
means to rapidly compute the area and volumes of
certain geometric figures previously deemed too diffi-
cult for analysis. Cavalieri’s famous 1635 work
describing these methods, however, was not well
received and was widely attacked for its lack of rigor.
In response, Cavalieri published a revised piece,
Exercitationes geometricae sex, which successfully
settled all concerns. This second piece was acknowl-
edged as a masterpiece and deemed a necessary text
of study for all 17th-century scholars.
Cavalieri also studied and wrote extensively on the
topics of
LOGARITHM
s,
CONIC SECTION
s,
TRIGONOME
-
TRY
, optics, and astronomy. He developed a general rule
for computing the focal length of lenses, and described
the principles and design of a reflecting telescope.
Cavalieri died on November 30, 1647, in Bologna,
Italy. His name appears in all high-school geometry
textbooks of today for the principle he devised.
See also C
AVALIERI
’
S PRINCIPLE
.
Cavalieri’s principle Italian mathematician B
ONAVEN
-
TURA
C
AVALIERI
(1598–1647) identified a general princi-
ple today known as Cavalieri’s principle:
Solids of equal height have equal volumes if
cross-sections made by planes parallel to the
bases at the same distances from these bases
have equal areas.
Cavalieri’s principle 67