Abstract
In the history of mathematics there are a number of problems which have agitated the imagination of the greatest minds for centuries. Three problems proved elusive: the problem of squaring the circle, the doubling the cube, and the trisecting the angle. The problems are to be solved purely by using a compass and an unmarked ruler, a straightedge. Ferdinand von Lindemann proved in 1882 that the first problem has no solution while Pierre Wantzel showed in 1837 that the solutions of the latter two problems are also impossible.
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Notes
- 1.
Squaring of the circle refers to finding the area of a circle of a given radius. In modern terms, it is the question of finding the exact value of π.
- 2.
The problem of doubling the cube, also called the Delian problem, is to find the length of the side of a cube which makes its volume twice as big as the original cubic volume. In modern terms, it amounts to the determination of \( \sqrt[3]{2} \).
- 3.
Dividing a given angle in three equal parts using only a compass and a ruler.
- 4.
According to the legend, the emperor Chung Kang relied on his astronomers to track and interpret heavenly motions. It was a serious job. Eclipses were believed to be caused by a dragon eating the Sun, and were bad omens for the emperor. The monster had to be frightened away with drums, gongs and arrows fired into the sky. When two state astronomers, He and Xi got drunk and failed to predict an eclipse, the emperor had no time to prepare a response. Although the Sun apparently survived the dragon’s attack, the pair were beheaded.
- 5.
Thales may have travelled to Babylonia in his youth and gained access to the extensive records of astronomical observations which dated from the time of the ruler Nabonassar (747 BC). By that time the Babylonians, just as Chinese in their own quarters, and many others, had been recording celestial events for several 1000 years. These records formed the basis for predicting lunar eclipses, and to some extent, solar eclipses. The methods may have been already known before 585 BC, even though written evidence for this knowledge has survived only from later centuries.
After centuries of continuous monitoring of celestial events, a period of 18 years and 10–11 days (called the Saros cycle) was discovered in lunar eclipses, after which similar eclipses start to repeat themselves. Another shorter cycle is 47 months long. Thales may have witnessed, or at least heard of, a nearly total solar eclipse in Babylonia on May 18th, 603 BC. If he suspected that also solar eclipses follow the Saros cycle, he could have predicted a solar eclipse on May 28, 585 BC. Alternatively, he may have known that 23.5 months after a lunar eclipse a solar eclipse has a high probability. This period is exactly one half of the 47 month lunar eclipse cycle, and he must have understood that the opposite alignment of the Earth-Moon-Sun happens half-way through this cycle. He most likely observed the July 4, 587 BC lunar eclipse which would have lead to the same predicted date. Perhaps he knew of both methods which gave him confidence. Anyway, he and the warriors were lucky in that the Halys river battle happened to be on the narrow strip, about 270 km wide, where the eclipse was total. A more common occurrence of a partial eclipse where the Sun is only partly covered by the Moon, is seen over a wider region, but it is not such an eerie and chilling experience as the total eclipse.
- 6.
\( \mathrm{F}={\mathrm{Gm}}_1{\mathrm{m}}_2/{\mathrm{r}}^2 \) where F is the gravitational force of attraction between two bodies of masses m1 and m2, separated by distance r from each other. G is the universal gravitational constant.
- 7.
Two of the authors (JA and VO) came close to the discovery of the figure “8” stable orbit a decade before Moore. They would have had to pursue the orbit longer to prove the case which was not yet possible.
- 8.
In more recent times, similar opinions have been expressed by Newton (“Nature is pleased with simplicity”) and Einstein (“nature is the realization of the simplest conceivable ideas”). The reader will find more about this in Mario Livio’s book “Is God a Mathematician?” (Simon & Schuster, New York, 2009).
- 9.
The Almanac based on Copernicus’ work was called Prutenic Tables. They soon replaced the older Alfonsine Tables, based on Ptolemy’s model. Over the years the tables of Ptolemy were developed by Arab scholars, and in the process of transmitting the Greek and Arabic knowledge to the west, the new Alfonsine tables were created in the thirteenth century Spain.
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Valtonen, M., Anosova, J., Kholshevnikov, K., Mylläri, A., Orlov, V., Tanikawa, K. (2016). Classical Problems. In: The Three-body Problem from Pythagoras to Hawking. Springer, Cham. https://doi.org/10.1007/978-3-319-22726-9_1
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DOI: https://doi.org/10.1007/978-3-319-22726-9_1
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