the same definite integral, a number, and so xand tare
dummy variables. (The variable x, however, is not a
dummy variable in the
INDEFINITE INTEGRAL
∫x2dx.
This expression is a function of the specific variable x.)
duplicating the cube (Delian altar problem, doubling
the cube) One of the problems of antiquity (like
SQUARING THE CIRCLE
and
TRISECTING AN ANGLE
) of
considerable interest to the classical Greek scholars is
the task of constructing a cube whose volume is twice
that of a given cube. Legend has it that this problem,
known as duplicating the cube, arose during the Greek
plague of 428
B
.
C
.
E
. It is said that the oracle of Delos
instructed the people of Athens to double the size of the
cubic altar to Apollo as an attempt to appease the god.
They were unable to accomplish this feat.
A
POLLONIUS OF
P
ERGA
(ca. 260–190
B
.
C
.
E
.) solved
the problem with the use of
CONIC SECTIONS
, but schol-
ars later decided to add the restriction that only the
primitive tools of a straightedge (that is, a ruler with no
markings) and a compass be used in its solution. The
difficulty of the problem increased significantly.
If we assume that the side-length of the original
cube is aunits long, then one is required to construct a
new length bso that b3= 2a3. Consequently, b=
3
√
—
2a,
and so the problem essentially reduces to the challenge
of constructing a length
3
√
—
2 units long using only a
straightedge and compass.
The theory of
CONSTRUCTIBLE
numbers shows that,
in this setting, any quantity of rational length can be
constructed, and that if two lengths l1and l2can be
produced, then so too can their sum, difference, prod-
uct and quotient, along with the square root of each
quantity. It seems unlikely that a length of
3
√
—
2, being
neither rational, nor the square root of a rational num-
ber, could be produced. Indeed, in 1837, French mathe-
matician Pierre Laurent Wantzel (1814–48) proved that
the number
3
√
—
2 is not constructible and, consequently,
that the problem of duplicating the cube is unsolvable.
(To see that
3
√
—
2 is not rational, assume to the contrary
that it can be written as a ratio of two integers:
3
√
—
2 = .
Then 2q3= p3. If the number phas mfactors of 2, then
the quantity p3has 3mfactors of 2. Consequently, so
too must 2q3. But this is impossible, as the number of
factors of 2 in 2q3must be 1 more than a multiple of 3.
This absurdity shows that
3
√
—
2 cannot be a ratio of two
integers. A similar argument shows that
3
√
–
2 does not
equal the square root of a rational quantity either.)
Dürer, Albrecht (1471–1528) German Geometry Born
on May 21, 1471, in Nürnberg, Germany, artist
Albrecht Dürer is remembered in mathematics for his
significant accomplishments in the development of
descriptive
GEOMETRY
and its applications to the theory
of art. In four famous texts, Dürer explained the theory
of proportions and described ruler-and-compass tech-
niques for the construction of regular polygons. He
explored the art of placing figures in a manner that is
pleasing to the eye, thereby beginning a developing the-
ory of
PERSPECTIVE
, and began a study of the shadows
cast by three-dimensional objects. Dürer is noted as the
first scholar to publish a mathematics book in German,
and also as the first Western scholar to give an example
of a
MAGIC SQUARE
.
Dürer studied painting and woodcut design as a
young man. He apprenticed with the leading producer
of altarpieces of his time, Michael Wolgemut, until the
age of 20 and learned to appreciate the role mathemat-
ics could play in the design of artistic works. After read-
ing the works of E
UCLID
(ca. 300–260
B
.
C
.
E
.), as well as
a number of famous texts on the theory of architecture,
Dürer traveled to Italy, the site of the Renaissance
revival of mathematics, to study the mathematics of
shape, motion, and perspective. Around 1508 Dürer
began collating and processing all the material he had
studied with the aim of producing one definitive text on
the mathematics of the visual arts. This work was never
completed, but he did later publish his four volumes on
the theory of proportions Underweysung der Messung
mit Zirckel und Richtscheyt in Linien, Ebnen, und
gantzen Corporen (Treatise on mensuration with the
compass and ruler in lines, planes, and whole bodies)
in 1525.
Dürer is noted for his inclusion of the following
array of numbers in the background of his 1514
engraving Melancholia:
16 3 2 13
51011 8
96712
41514 1
p
–
q
Dürer, Albrecht 149