By special request,
we're showing the work.
we're showing the work.
This is all you get, just a triangle and the lengths of its sides.
What's the area of the triangle? Compute the area accurate to 2 decimal places.
1 slice, extra cheese.
You had this once. It's lying around the house somewhere. You just haven't needed it for a long time.
The ancient Egyptians probably could have won the pizza, but the general method, and the proof of it, dates to the late classical Greeks of Alexandria around 75 AD, well after the amazing discoveries of Archimedes (died 212 BC).
It's both elegant and eminently practical. Greek mathematicians often tended to scorn the practical. By the time they cooked this up, the all-practical Romans had conquered them and the whole Mediterranean. The Romans could do what the Greeks never could: Unify, keep the peace (though violence was the first tool the Romans reached for to keep the peace).
Historically, you're looking at something of a farewell from the brilliant and electrifying wizardry of Greek mathematics. Mathematics and science in Europe were preparing to go into a very long and very deep sleep. During that sleep, Arabs and Muslims from Spain to Persia took good care of the legacy of Greek mathematics, and melded it with the ethereal mathematical insights of the Hindus before they re-inspired Europe with math and science again around 1300 AD.
Only the ancient Greeks seem to have been obsessed with proof. For the ancient Egyptians and ancient Babylonians -- from whose knowledge the Greek merchant travellers plagiarized shamelessly -- mathematics was a powerful but disorderly hodge-podge of practical problem-solving tools, a How-To Guide for architecture, surveying, calendar calculations and astronomy.
Only the Greeks perceived an inherent truth and significance in mathematics beyond its mere usefulness in solving practical real-world problems. Only the Greeks perceived and emphasized a deep internal design and structure to the entire body of mathematics.
