History of the Problems

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Three Famous Problems

### Source of the contents for this section

Most of the contents of this section come from "History of Greek Mathematics Vol. I From Thales to Euclid" (ref. 1) by
T.L.Heath.

A very unique Origami solution by H.Abe, and another Origami approximation by Haga are added.
### History of the problem

#### Problem Definition:

To construct the edge of a cube that is double the volume of a given cube

************** Delian_problem.dwg** *************

#### Origin of the Problem :

The origin of this problem is not clear even among the experts on this subject.
The book "A History of Greek Mathematics" by Sir Thomas Heath, quotes one of the

prominent mathematician, Eratosthenes's comment.

"when the god proclaimed to the Delians by the oracle that, if they would get rid of a plague

they should construct an altar double of the existing one, their craftmen fell into great

perplexity in their efforts to discover how solid could be made double of a (similar) solid;

they therefore went to ask Plato about it, and he replied that the oracle meant, not that the god

wanted an altar of double the size, but that he wished, in setting the task, to shame the Greeks

for their neglect of mathematics and their contempt for geometry"

#### Rule of the Game ( Platonian Rule ) :

##### Tools allowed

the compasses and unmarked straight-edge
##### Permitted use of these 2 tools

1. The drawing of a straight line of indefionite length through two given points.
2. The construction of a circle with center at a given point and passing through a second given point.

3. The procedure must be a finite number of operations

#### Summary of Early Attempts:

#### Proof of impossiblility:

Carl Friedrich Gauss (1777 - 1855) had stated without proof that the problems of trisecting an angle and doubling a cube cannot be solved
with ruler and compasses. But the first published proof was given by Frenchman, Pierre Wantzel (1814 - 1848), in 1837.

#### References:

- Heath,Thomas L. :"History of Greek Mathematics Vol. I From Thales to Euclid" Dover 1981

- Heath,Thomas L. :"A Manual of Greek Mathematics" Dover 1963 original 1931

- Heath,Thomas L. :"A History of Greek Mathematics Vol. II" Dover 1981 original in 1921

- Heath,Thomas L. :"The Works of Archimedes Dover" 2002 original in 1912

- Knorr,Wilber Richard :"The Ancient Tradition of Geometric Problems" Dover 1993

- Knorr,Wilber Richard :"The Textual Studies in Ancient and Medieval Geometry",Birkhauser, 1989

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