Fermat's Last Theorem by Simon Singh Page A

was predicting the phases of the moon far into the future with high accuracy – information which could be used to draw up vital navigation tables. Newton had already shown that it is relatively easy to predict the orbit of one body around another, but in the case of the moon the situation is not so simple. The moon orbits the earth, but there is a third body, the sun, which complicates matters enormously. While the earth and moon attract each other, the sun perturbs the position of the earth and has a knock-on effect on the orbit of the moon. Equations could be used to pin down the effect of any two of the bodies, but eighteenth-century mathematicians could not incorporate the third body into their calculations. Even today it is impossible to predict the exact solution to the so-called ‘three-body problem’.
    Euler realised that mariners did not need to know the phase ofthe moon with absolute accuracy, only with enough precision to locate their own position to within a few nautical miles. Consequently Euler developed a recipe for generating an imperfect but sufficiently accurate solution. The recipe, known as an algorithm, worked by first obtaining a rough-and-ready result, which could then be fed back into the algorithm to generate a more refined result. This refined result could then be fed back into the algorithm to generate an even more accurate result, and so on. A hundred or so iterations later Euler was able to provide a position for the moon which was accurate enough for the purposes of the navy. He gave his algorithm to the British Admiralty and in return they rewarded him with a prize of £300.
    Euler earned a reputation for being able to solve any problem which was posed, a talent which seemed to extend even beyond the realm of science. During his stint at the court of Catherine the Great he encountered the great French philosopher Denis Diderot. Diderot was a committed atheist and would spend his days converting the Russians to atheism. This infuriated Catherine, who asked Euler to put a stop to the efforts of the godless Frenchman.
    Euler gave the matter some thought and claimed that he had an algebraic proof for the existence of God. Catherine the Great invited Euler and Diderot to the palace and gathered together her courtiers to listen to the theological debate. Euler stood before the audience and announced:

    With no great understanding of algebra, Diderot was unable to argue against the greatest mathematician in Europe and was left speechless. Humiliated, he left St Petersburg and returned to Paris. In his absence, Euler continued to enjoy his return to theological study and published several other mock proofs concerning the nature of God and the human spirit.

    Figure 5. The River Pregel divides the town of Königsberg into four seperate parts,
A
,
B
,
C
and
D
. Seven bridges connect the various parts of the town, and a local riddle asked if it was possible to make a journey such that each bridge is crossed once and only once.
    A more valid problem which also appealed to Euler’s whimsical nature concerned the Prussian city of Königsberg, now known as the Russian city of Kaliningrad. The city is built on the banks of the river Pregel and consists of four separate quarters connected by seven bridges. Figure 5 shows the layout of the city. Some of the more curious residents of Königsberg wondered if it was possible to plot a journey across all seven bridges without having to stroll across any bridge more than once. The citizens of Königsberg tried various routes but each one ended in failure. Euler also failed to find a successful route, but he was successful in explaining why making such a journey was impossible.

    Figure 6. A simplified representation of the bridges of Königsberg
    Euler began with a plan of the city, and from it he generated a simplified representation in which the sections of land were reduced to points and bridges were replaced by lines, as shown in Figure