Lecture 1
Introduction and Overview
 

The above problems remained open for almost 2200 years. The final solution employed techniques from abstract algebra and analysis. We will show that it is impossible to construct side of a cube whose volume is 2 by ruler and compass. The word Delian is derived from Delos which was a city in ancient Greece. It is said that almost a quarter of population of Delos died of plague in 428 B.C. A delegation was sent to the oracle of Apollo at Delos to enquire how the plague could be arrested. The oracle replied that the cubical altar to the Sun God Apollo should be doubled. Instead of doubling the volume the faithfuls doubled the sides of the cube thereby increasing the volume eightfold. The second and the third problems also circulated among Greek geometers around the same time. It is not known who solved the Delian problem first. The angle trisection problem was solved by Gauss as a special case of his remarkable solution of the fourth problem. Gauss, barely 19, provided a construction of the 17-sided regular polygon. He also characterized $ n$ for which regular $ n-$gons are constructible by ruler and compass. Recall that a prime of the form $ 2^{2^{m}} + 1$ is called a Fermat prime. Gauss proved that a regular $ n-$gon is constructible if and only if $ n = 2^{r} p_{1} p_{2} \ldots p_{g}$ where $ n \geq 0$ and $ p_{1}, p_{2}, \ldots, p_{g}$ are distinct Fermat primes. Gauss's Theorem solves the angle trisection problem. If $ 20^{o}$ was constructible, then we can construct a regular $ 18$-gon. Since $ 3^{2} \vert 18$, we have a contradiction.

The values of $ n$ for which regular $ n-$gons were known to be constructible up to the time of Gauss were $ n = 2^{m}, 2^{m}.3, 2^{m}.5$ and $ 2^{m}.15$. No one was able to construct a heptagon or a regular $ 17-$gon.

In March, 1796 Gauss made his first mathematical discovery : construction of a 17-sided regular polygon by ruler and compass. He began noting down his mathematical discoveries in a diary which he maintained for the next 19 years. Gauss published % latex2html id marker 11343 $ \lq\lq Disquisitiones \;Arithmaticae''$ in 1801 which has become a classic in mathematical literature. The last result of this is his solution to the fourth problem. Gauss was very proud of this discovery. He desired that a regular polygon of $ 17$ sides be engraved on his tombstone. This wish was not fulfilled. It was fulfilled when a monument to Gauss was built in his birth place Braunschwig. Explicit construction of $ 17-$sided regular polygon was given by Erchinger in 1800. In 1892 Richelot and Schwendenwein constructed a regular $ 257-$gon. Around 1900 Hermes constructed a regular 65537-gon. The manuscript fills a box and it is found in Góttingen. The construction has now been computerized. See an article by Bishop in American Math. Monthly (1978).

Lindemann proved in 1882 that $ \pi$ is not a root of any polynomial with rational coefficients. This proved the impossibility of squaring a circle.