David Peat

36 From Certainty to UncertaintyGHFAKBCDLEPythagorean theorem. This theorem states that, for the right-angled triangle ABC,the area of the square BCED (“the square on the hypotenuse”) is equal to the sum ofthe areas ABFG and AHKC (“the sum of the squares on the other two sides”).as the famous theorem of Pythagoras—the square on the hypotenuseof a right angle triangle equals the sum of the squares of the lengths ofthe other two sides (see figure).At the heart of Euclid’s approach lies the notion of mathematicalproof. In his proofs, Euclid starts from one of the axioms, and assumingnothing else, constructs a chain of statements, each following logicallyonto the next. In this way it is possible to arrive at the truth ofeach theorem using a small number of steps and employing logic to gofrom one step to the next. Euclid’s proofs do not involve assumptionsand guesses, neither do they rely on an appeal to “common sense.”Rather they are all constructed with rigorous logic.Newton used the same approach in his great Principles of NaturalPhilosophy, first defining basic terms about space, time, and so on, andthen adopting a small number of axioms as his “laws of nature.” Armedwith these, and proving every statement logically step by step, he wasable to establish truths about the natural world.