From Archimedes to statistics: the area of the sphere

Archimedes (287-212) has shown that when you put a sphere in a cylinder in the tightest manner, the lateral area of the cylinder is the area of the sphere itself. This is not an easy result, and Archimedes was so proud of it that he asked that the corresponding picture should be engraved on his tomb in Agrigente in Sicily : 150 years after his murder during the siege of Syracuse by an ignorant Roman soldier (imagine Werner von Braun killed by a GI in 1945), this detail of the sphere inside the cylinder enabled Cicero to discover the grave and to restore it in the year 75. Actually, the Archimedes’ result is even more precise. Indeed if you cut off the whole by a plane perpendicular to the axis of the cylinder, the remainders of the cylinder and of the sphere have still the same area. I learnt of this result when I was a I5 years old schoolboy (with the original proof that I shall sketch in a few seconds). In these far away times, the proof was actually belonging to the syllabus of the ”classe de Première” of the lycées, the 11th grade of the French system. Although these syllabus have changed, I have observed recently that the statement is still fascinating for my grand children. The principle of the proof given by Archimedes can be easily rediscovered by somebody who likes mathematics, even elementary ones. However, it uses a clever trick which simplifies the calculation : you are stalled if you do not have it. Finally, it still has two minor defects : slight lack of rigor and complication. There are some other proofs, which use classical calculus learnt during first years at the university. I will also give you a completely different proof which uses a weapon borrowed to statistics : the normal distribution.