Abstract
Through his oracle at Delos, Apollo informed the Delians that if they wanted to be rid of the plague they must construct a new cubical altar that exactly doubled the volume of the existing one. The Delian Problem then was to construct a cube having a side \( \sqrt[3]{2} \) times as long as a side of the original cube. This problem also has the somewhat misleading name The Duplication of the Cube. According to another legend, Eratosthenes reported that the problem was sent to the geometers at Plato’s Academy in Athens. Plato is reported to have said that the god had assigned the task to shame the Greeks for their neglect of mathematics and their contempt for geometry. It was not that the Greeks could not construct segments of the required length by various methods but that they could not do so using only the ruler and compass. That was their task. There is little doubt that the Greeks soon suspected the problem had no solution. However, they lacked the algebra to prove this fact. Our task is to prove the ancient Greeks necessarily failed because they were asking for the impossible. To do this, we must formulate our problems in the language of algebra.
The methods of coordinate geometry allow us to translate any geometric statement into the language of algebra, and though this language is less elegant, it has a larger vocabulary.
Hilda Hudson
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
Author information
Authors and Affiliations
Rights and permissions
Copyright information
© 1998 Springer Science+Business Media New York
About this chapter
Cite this chapter
Martin, G.E. (1998). The Ruler and Compass. In: Geometric Constructions. Undergraduate Texts in Mathematics. Springer, New York, NY. https://doi.org/10.1007/978-1-4612-0629-3_2
Download citation
DOI: https://doi.org/10.1007/978-1-4612-0629-3_2
Published:
Publisher Name: Springer, New York, NY
Print ISBN: 978-1-4612-6845-1
Online ISBN: 978-1-4612-0629-3
eBook Packages: Springer Book Archive