Diocles's Solution of the Delian Problem

​
move E
The Demonstration constructs a cissoid and and uses it to show Diocles's solution of the problem of doubling the cube, also known as the Delian problem.
Suppose that a cube of side length
L
is given; it has volume
3
L
. To double the cube means to construct another cube with twice the volume as the original,
2
3
L
, so the side of the new cube would be
3
2
L
. Using an unmarked ruler and compass, it is impossible to construct a line segment
3
2
as long as a given line segment. However, Diocles solved the problem with the aid of a cissoid.
Let
c
be a circle of radius
a
and center
O
. Let
E
and
F
be points on
c
equidistant to the diameter
AB
and on opposite sides of
AB
. Let
DC
be the diameter perpendicular to
AB
and let
G
and
H
be the perpendicular projections of
E
and
F
onto the diameter
DC
. Then
P
, the point of intersection of the lines
EC
and
HF
, lies on a cissoid.
Since
HF
is a mean proportional between
DH
and
CH
,
DH:HF=HF:CH
. By similarity,
CG:GE=CH:HP
. It follows
DH:HF=HF:CH=CH:HP
, since
CG=DH
and
GE=HF
.
Let
K
be the intersection of
DP
and
OB
. Move
E
so that
K
is the midpoint of
O
and
B
(the cyan point). It follows that
DO:OK=DH:HP=2
. Then
2
CH
=HF·HP
and
HP=DH·CH
. So
3
HF
:
3
CH
=2
and
HF:CH=
3
2
.

References

[1] T. Heath, A History of Greek Mathematics, Vol. 1, Oxford: Clarendon Press, 1921 pp. 264–266.

External Links

Cube Duplication (Wolfram MathWorld)
Linkage for the Cissoid
The Eratosthenes Machine for Finding the Cube Root of Two
Geometric Problems of Antiquity
Constructing the Cube Root of Two

Permanent Citation

Izidor Hafner
​
​"Diocles's Solution of the Delian Problem"​
​http://demonstrations.wolfram.com/DioclessSolutionOfTheDelianProblem/​
​Wolfram Demonstrations Project​
​Published: September 28, 2012