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On the past of a famous theorem: the predecessors of a theorem of Pythagoras
255-267Views:7The well-known Theorem of Pythagoras asserts a relation among the sides of any right-angled triangle. It can be found any secondary school textbook. An interesting question whether this result due to the Pythagoreans from the VIth century BC, or it was known in earlier civilizations. The first answer is a vague yes. According to the legends the Egyptian rope-stretchers used a triangle with sides 3,4,5 units to create right angle. But are there real evidences that this result was known earlier? We will argue that in almost all river-valley civilizations it was known and used. -
On two long lasting delusions in the history of equations
147-158Views:10Almost everybody was thought, that the 9th century Moshlem mathematician al-Khwarismi was the inventor of two powerful methods – called by him as al-jabr and al-muqabala – in solving quadratic equations. The second belief is that between Leonardo's Liber abaci and Luca Pacioli's Summa... happened nothing interesting in algebra. We will show that both beliefs are false by giving examples from the antiquity and analyzing Mediaeval Italian manuscripits. -
The first clear distinction between the heuristic conjecture and the deductive proof in the ancient mathematics
397-406Views:7The mathematics of the ancient river-valley cultures was purely empirical, while the classical Greek mathematics was entirely deductive without any written sign of the heuristic arguments. In the forthcoming Hellenistic period there were significant changes. One of them is that in spite of the rigorous (deductive) proofs some heuristic arguments appeared in separate treatises. We show a nice example due to Archimedes.
"We have learned from the very pioneers of this science not to have regard to mere plausible imaginings when it is a question of the reasonings to be included in our geometrical doctrine." – Proclus
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