theory of
DEDEKIND CUT
s on this approach developed
by Eudoxus.
In geometry, Eudoxus was the first to establish that
the volume of a cone is one-third the volume of the
cylinder that surrounds it, and also that the volume of
a pyramid is one third the volume of a prism of the
same base and height. A
RCHIMEDES OF
S
YRACUSE
(ca.
287–212
B
.
C
.
E
.) made use of these results in his famous
treatise On the Sphere and Cylinder, citing Eudoxus as
the person who first proved them.
Eudoxus maintained an active interest in astron-
omy throughout his life. He built an observatory in the
city of Cnidus and made careful note of the motion of
the planets and stars across the night skies. Outside of
mathematics, Eudoxus is best known for his ingenious
theory of planetary motion based on a system of 27
nested spheres. Using advanced techniques in three-
dimensional geometry, Eudoxus was able to use this
model to explain the puzzling retrograde motion of the
heavenly bodies.
Euler, Leonhard (1707–1783) Swiss Analysis, Geom-
etry, Number theory, Graph theory, Mechanics, Physics
Born on April 15, 1707, in Basel, Switzerland, genius
Leonhard Euler was, beyond comparison, the most pro-
lific mathematician of all time. With over 850 books
and papers to his name, Euler made fundamental con-
tributions to virtually every branch of mathematics of
his day. He formalized the notion of a
FUNCTION
(and
introduced the notation f(x) for it), and thereby
changed the focus of mathematics from a study of fixed
curves and lines to a more powerful study of transfor-
mation and change. (He was the first, for instance, to
regard the special functions from
TRIGONOMETRY
as
functions.) Euler published works on
ANALYSIS
,
DIFFER
-
ENTIAL CALCULUS
,
INTEGRAL CALCULUS
,
DIFFERENTIAL
EQUATION
s,
NUMBER THEORY
,
GEOMETRY
,
LOGIC
,
COM
-
BINATORICS
, approximations for π, planetary motion
and astronomy, navigation, cartography, mechanics,
and more. He introduced and made popular many of
the standard symbols we use today (such as ifor √
–
–1, π
for
PI
,
E
for his famous number, and Σfor
SUMMATION
).
It is simply not possible in a short piece to give proper
justice to the phenomenal quantity of contributions
Euler made to the study of mathematics.
Euler obtained a master’s degree in philosophy
from the University of Basel in 1723, following the
path his father set for him to study theology. Euler’s
interests, however, lay with mathematics, and Euler
remained at the university another three years to pur-
sue a course of study in the subject. In 1727 he sub-
mitted an entry for the Grand Prize of the Paris
Academy of Sciences on the best arrangement of
masts on a ship. Euler won second prize, which gar-
nered the attention of the scientific community as an
outstanding young graduate. Euler accepted a position
at the St. Petersburg Academy of Science that year and
was promoted to a full professor of the academy just
three years later.
In 1736 Euler published Mechanica (Mechanics), a
landmark piece that introduced rigorous mathematical
techniques as a means for studying the subject. He won
the Grand Prize of the Paris Academy in 1738, and
again in 1740, and by this time was a highly regarded
scholar. At the invitation of Frederick the Great, Euler
Euler, Leonhard 173
Leonhard Euler, the most prolific of famous mathematicians, made
significant contributions to almost every field of pure and applied
mathematics studied at his time while also establishing new
courses of research. Much of the mathematical notation used
today was either introduced or made standard by Euler. (Photo
courtesy of Topham/The Image Works)