Trego's Mountain Ear

We take arabic numbers for granted, and rarely think how difficult multiplication and division would be with anything other than our positional system.  If you glance at Roman numbers like IV and XXVI, and think how difficult division would be, then realize that Roman numerals were an improvement on what the old Greeks had to work with, the true brilliance of A squared plus B squared equals C squared comes through.

Still, the illustration suggests that the Pythagorean theorem was probably developed by working stiffs who knew that a triangle with sides in a 3:4:5 ratio would always have a right angle.  The Sumerians built with right angles and a more unwieldy numbering system than even the ancient Greeks.  Picture, if you will, doing multiplication and division with written words and not the numbers we regard as normal.  Those early builders may not have had the theory – but they did know that the 3:4:5 ratio gave them right angles.  Pythagoras and Euclid stood on the shoulders of those Sumerian craftsmen.

It has taken me most of a lifetime to realize that the great thinkers of classical Greece stood on the shoulders of those working stiffs who had developed particular aspects of thought but did not have a culture of science to integrate those discoveries.  Pythagorus and Euclid harvested the low-hanging fruit that craftsmen and laborers discovered and left available over the preceding two millennia.  The Sumerian math system was based on 12 and 60 – counting knuckles on one hand, and using individual thumb and fingers on the other to keep track of the dozens.  Still, even that system was easier to use than the system Pythagorus had on hand.  And the Sumerian system is still with us for measuring seconds, minutes and hours.

Pythagoras inheriting the unwieldy Greek system, invented a new system of numbering (from TRIANGULAR NUMBERS AND PYTHAGOREAN TRIPLES – A SURPRISING RELATIONSHIP which is worth clicking and reading for a great explanation)

While we think of Pythagoras as Greek, he lived and taught in southern Italy.