We've seem elsewhere that the Pythagoreans developed a "Figurate Geometry", wherein structures such as line segments, triangles, squares, pentagons, cubes, etc., can be built from dots in sand or stones or pegs in pegboard, etc. But the "incommensurability of the diagonal of a square to its sides" means that THE DIAGONAL cannot be explained FIGURATIVELY (by dots or whatever). This failure gave support to the philosophical enemies of the Pythagoreans.

Chief among these were the Eleatics. Their leader, Parmenides (x-y), claimed that MOTION IS AN ILLUSION -- that there is no CHANGE IN THIS WORLD.

His chief pupil was Zeno (490-430 BC) who developed three "Paradoxes" to confound the Pythagoreans and others. These paradoxes have been resolved only in modern times. Their resolution has helped to build up our CIVILIZATION.

I'll concentrate upon only one of these, "The Paradox of Achilles and The Tortoise".

Achilles (hero of The Trojan War and Homer's ILIAD) was alleged to be the fastest of all humans. Zeno imagines Achilles in a race with the Tortoise, one of the slowest of land animals.

However, Achilles gives the Tortoise a head start -- say, one UNIT OF LENGTH. (In present terms, it could be a yard or a meter).

But, before Achilles can run that UNIT (to try to catch up with the Tortoise), he must first run HALF THAT UNIT DISTANCE. And before he can run that HALF-DISTANCE, he must run HALF-OF-THE-HALF-OF-THE-DISTANCE. And HALF OF THAT. HALF OF THAT. Etc.

Numerically, we have 1/2 + 1/4 + 1/8 + 1/16 + 1/32 + 1/64 + .... ARITHMETICALLY, there's NO END TO THE HALVING. And, reasoned Zeno, ARITHMETIC CANNOT ADD UP AN INFINITE NUMBER OF NUMBERS. Hence, either ACHILLES cannot catch up with the Tortoise -- OR (!!!!!) ARITHMETIC CANNOT DESCRIBE MOTION!!!!!

This argument FROZE ALL ATTEMPTS TO DESCRIBE MOTION ARITHMETICALLY until Westerners in the Renaissance picked up on results of Islamic scholars and broke this deadlock in thinking.

Historians fail to tell you that The Roman Catholic Church had Galileo (x-y) put under house arrest for the rest of his life not merely for agreeing with Copernicus that the Earth moved around the Sun. The even greater complaint against Galileo was that he was a "Pythagorean" for using arithmetic to describe "A Law of Falling Bodies" which disagreed with the Aristotlean "Law of Falling Bodies", accepted by The Church.

But the basis for resolving the "Achillean Paradox" (and the others) existed in those ancient times. Only prejudiced thinking prevented further development of this "basis".

The terms, 1, 1/2, 1/4, 1/16, 1/32, 164, form a "geometric progression", which -- for example -- was used by the Pythagoreans (along with "arithmetic progressions" and "harmonic progressions") to build THE CHROMATIC SCALE OF OUR WESTERN MUSIC. And discussion these progressions also appears in Euclid's Elements of Geometry.

Please notice. Starting with 1, the next term derives from MULTIPLYING by 1/2: 1 x 1/2 = 1/2. And the next term, 1/4, is deriveed again from MULTIPLYING by 1/2: 1/2 x 1/2 = 1/4. Similarly, the next term: 1/4 x 1/2 = 1/8. And the next: 1/2 x 1/8 = 1/16. And the next: 1/2 x 1/16 = 1/32. And the next term: 1/2 x 1/32 = 1/64. The "Etc." means that this MULTIPLYING by 1/2 CONTINUES.

Algebraically, we can write: 1, r, r², r³, r⁴, r⁵, r⁶, etc.

And we can sum this: 1 + r + r² + r³ + r⁴ + r⁵ r⁶ + ...., where r = 1/2.

Here's how to find a FORMULA for such a sum. Write (for n terms):
S = 1 + r + r² + r³ + ... + r^n-1.

Then MULTIPLY this by r, to obtain:
rS = S = r + r² + r³ + ... + r^n-1 + rⁿ.

Please note that these two SUMS are alike -- except for the FIRST TERM in the FIRST SUM, and the LAST TERM in the SECOND SUM. So, if we SUBTRACT THE FIRST SUM FROM THE SECOND, ALL THE MIDDLE TERMS CANCEL OUT, LEAVING ONLY THE TWO DIFFERENT TERMS:
rS - S = 1 - rⁿ. Factoring out S on the left, we have: S(1 - r) = 1 - rⁿ.

We can solve this for S by dividing both sides by 1 - r. But we mmust NOT DIVIDED BY ZERO. So we declare: r ¹ 1. Then we have: S = 1/(1 - r) - rⁿ/(1 - r).

We can now GIVE MEANING TO AN INFINITE SUM OF THIS KIND. Suppose (as in the "Achilles" case) that 0 < r < 0. THEN EXPONENTIATING r MAKES IT SMALLER AND SMALLER!!!! Why? FOR ANY NONZERO FRACTION LESS THAN 1, THE DENOMINATOR IS GREATER THAN THE NUMERATOR -- SO THE DENOMINATOR WILL GROW FASTER THAN THE NUMERATOR. You can MAKE IT LESS THAN ANY NUMBER NAMED. The mathematician calls this "passing to the limit", and writes (for this case):
lim(n -> inf) S = 1/(1 - r) -- because the SUBTRACTIVE TERM drops off.

Let's see what that tells us for the r = 1/2 of the "Achilles Paradox": S = 1/(1 - 1/2) = 1/(1/2) = 2. But that progression didn't have an initial 1, so we subtract that from 2: 2 - 1 = 1. So, that SUM DOES (IN THE LIMIT) ADD UP TO THE UNIT DISTANCE OF HEAD START THE TORTOISE WAS GIVEN. ACHILLES CAN CATCH UP AND BEAT THE TORTOISE. ARITHMETIC (IN THE LIMIT) CAN DESCRIBE MOTION!!!!!

Just two more things to note and we have our modern understanding.

You know that 1/3 = .333333.... That is a GEOMETRIC PROGRESSION: 1/3 = 3/10 + (3/10)² + (3/10)³ + (3/10)⁴ + (3/10)⁵ + (3/10)⁶ + ....

ASSIGNMENT: Show by the above formula that, IN THE LIMIT, .333333.... becomes the 1/3 of the left.

The point of this is that ANY RATIONAL NUMBER CAN BE PUT IN THE FORM OF A GEOMETRIC PROGRESSION SUMMED, STARTING TO REPEAT ITS "PERIOD" (3 WAS THE "PERIOD" ABOVE) INFINFINITELY.

But IMAGINE AN INFINITE SUM THAT DOESN'T HAVE A PERIODIC REPETITION -- YET IS BOUNDED BY SOME FINITE NUMBER. Such a SUM is known as a "CAUCHY SUM" or "SUM OF A CAUCHY SEQUENCE", after the great French mathematician, Augustin Cauchy (x-y). Then, IN THE LIMIT, IT IS A REAL NUMBER!!!!!

When we see a number listed thus, √2 = 1.41421..., we can rewrite this as a Cauchy sequence of rational numbers: 1, 1.4, 1.41, 1.414, 1.4142, 1.41421, ....

We can show that √2 is "pinched" between LOWER and UPPER BOUNDS:

1 < √2 < 2 -- since (by SQUARING) 1 < 2 < 4
1.4 < √2 < 1.5 -- since (by SQUARING) 1.96 < 2 < 2.25
1.41 < √2 < 1.42 -- since (by SQUARING) 1.9881 < 2 < 2.0164
1.414 < √2 < 1.415 -- since (by SQUARING) 1.99941 < 2 < 2.0022
1.4142 < √2 < 1.4143 -- since (by SQUARING) 1.99996 < 2 < 2.00024
1.41421 < √2 < 1.41422 -- since (by SQUARING) 1.99999 < 2 < 2.00002
.....

Please note that the (left) SEQUENCE OF LOWER BOUNDS IS INCREASING CONTINUOUSLY IN VALUE, while CORRESPONDINGLY the (right) SEQUENCE OF UPPER BOUNDS IS DECREASING CONTINUOUSLY IN VALUE. The real number, √2, is being "pinched more and more tightly" by these BOUNDS. You can carry the DECIMAL NUMBER OUT TO ANY NUMBER OF POSITIONS, EQUIVALENTLY EXTEND THE CAUCHY SEQUENCE TO ANY NUMBER OF TERMS, EQUIVALENTLY EXTEND THE SEQUENCE OF "PINCHES" TO ANY NUMBER. You can imagine that, "eventually" or IN THE LIMIT, ONLY ONE NUMBER IS DESIGNATED OR "PINCHED". This defines the given REAL NUMBER -- here, √2.

This defeats ZENO, justifies PYTHAGORAS, provides THE MATHEMATICLA TOOLS FOR BUILDING OUR CIVILIZATION.