Greek Geometry

The greeks viewed the number line as consisting of positive rational numbers. Legend has it that Pythagoras drowning Hamas in a pique after proving that the square root of two is irrational.

Greek geometry is also planar and they used geometric constructions to solve what we consider algebraic manipulations. The constructions used compass and straightedge only.

Their number system used a decimal system, the septadecimal system (base 60) of the babylonians, and the accounting based system of the Romans. IOW, they used what was available.

As for written material, they did their calculations (constructions) in the dust and when completed maintained a permanent copies on terra cotta plates and rock baa relief (talk about a hard copy).

Rap

I was thinking about geometric counting while on a bike ride,, and recalled that the Greeks used a counting system based on geometric shapes. For instance triangular numbers.

A Triangular Number the elements of the sequence 1,3,6,10, 15, 21,.....,n(n+1)/2.

Similarly, the greeks (and Euclid) based other sets of numbers on platonic shapes, squares, pentagons, hexagons, heptagons, octagons...well you can see the pattern. And would analyze numbers based on these shapes in terms of triangular numbers.

If you think in terms of these numbers, you can see that it would only include nonzero counting numbers.

Rap

Rap^2 - the Greeks never used Babylonian hexadeximal systems, their mathematics were developed one full millenium before the Romans got around to that line of business, and Pythagoras had true insight into what came to be Riemannian (non-planar) geometry when he posted with reservations as an axiom his #4, about parallel lines not meeting at infinity.

Wishing you good cycling <G>

P.S. before I forget, irrational numbers were well known to the Greeks even before Zeno and Anaximander. That was the basis for most of Zeno's paradoxes - though differential calculus was to come later.

To see why your statements are flat-out wrong, consider the Greeks calculating pi to 20 decimal places

I agree that Zeno conceived the concept of an infinite limit well before differential calculus and that they discovered irrational numbers, but if I remember my math history pi as irrational came much later.

As for the accuracy of pi, the greeks were able to determine tis from their knowledge of and their knowledge of construction of regular polygons. (the Egyptian mathematicians used a similar determination of pi).

As for the babylonian system (base 60) somehow I find it hard to believe a Greek trader and beancounter wasn't aware of the utility of this system.

The Roman Numerals granted were later but there was a Greek number codes that were roughly decimal until they reached the thousands, then they resorted to something akin to Roman Numerals for 1000, 2000, 3000, 10,000, 20,000, 100,000. Again I opine this was for beancounting and commerce. However, the Greek engineers and scientists (as well as Roman ones) used a number system that was roughly decimal when preforming calculations that involved multiplication and division.

I find the geometric numbers fascinating, particularly as their development resulted in the discovery of several series formula.

As for the 4th postulate and the Riemannian consequence, the Greeks realized that the world was round and that plane geometry wasn't a good model for global navigation.

As for answering pepito's initial question I'd like to reference Euclid's Elements.

Rap

Thanks Rap - Euclid's 4th was meant, not Pythagoras' in my post, btw, but you understood what I meant!

Geometric series fascinate me also.

As to the original poster - yes, the Greeks didn't have zero, but I think that was introduced by Arab mathematicians, not Indians, in later centuries.

P.S. Perseus (English translations) and this site (with English and French translations) >
http://philoctetes.free.fr/index2.htm
> are excellent sources for the actual original Greek texts.

I am eternally thankful to that Tunisian that kept Euclids Elements out of the Roman fires at the Library of Alexander. I am constantly amazed when I look at the Elements sophistication and application knowing that the latest book is over 2500 years old.

Besides the roots of plane and solid geometry the Greeks laid the foundation of the theory of numbers. And the Euclidian method of finding the greatest common denominator of two whole numbers is nothing short of inspirational.

Unfortunately I'm unable to read the original Aramaic translation and have to suffer along with a more modern translation (The King James Elements?).

Yes zero was invented by the arabs, who were led by that Imam Al-Ja-Bra in order to unlease a "weapon of Math destruction".

Rap

Rap - another reason to admire the early Greeks is that all their philosophers were trained in mathematics; the sign over Plato's academy in Athens read: "Let no-one ignorant of geometry enter here!"