Spiral Mechanics. The Mesopotamian Arithmetic and Geometry of antiquity. Discovered in 3200 BC. Demonstrably true for all values of A, B, x and y. First applications were the invention of writing and base 60. Introduced to ancient Greece by Pythagoras, used by Plato and Archimedes, and erased from history by Aristotle and Euclid.
4. - 3 -
Table of Contents
Spiral Mechanics 1
Introduction 4
The Discovery of Heaven 11
The Trojan from Babylon
Egalitarian versus Elitist
Aristotle’s Stratagem: the Final Solution
The Elements of Euclid: the Implementation
The Mesopotamian Identity and the Sexagesimal System
Conclusion
Bibliography
Endnotes
16
30
39
49
54
66
69
70
5. - 4 -
Spiral Mechanics
There are only two mistakes one can make along the road to truth; not going all the way, and not
starting.
– Shakyamuni Buddha
The right to search for the truth implies also a duty; one must not conceal any part of what one has
recognized to be the truth.
– Albert Einstein
Introduction
The most widespread principle among scholars of classical antiquity - the dogma of the time of
Pythagoras and Plato – was the "The Truth" had been fully discovered in earlier times and since
then had only been reformulated by more recent writers. The most widely held belief because it
was not based on faith but on evidence. A dogma because the evidence was mathematical from
start to finish and therefore irrefutable. Everything that is comes from one ultimate truth and all
that is can be traced back to this truth.
The distinction between the writers who reformulated the truth and the writers who distorted
the truth was also easily made. The reformers revealed their source and honored the memory of
the Ancients by generously sharing and embracing the wisdom. The perverts obscured their
existence and selfishly used all the benefits of their own conversion to the exclusion of others. Of
course "everything for me and nothing for the other" is only the despicable maxim of the false
prophets and the corrupt rulers of humanity.
Three thousand years before the birth of Archimedes, the Sumerians discovered the
Archimedean point ("Punctum Archimedis") of arithmetic and geometry. The ideal of "moving
away" from the study object so that it can be viewed in relation to all others. The hypothetical
vantage point from which an observer can observe the subject of research with a view of the
totality. At the beginning of the second half of the fourth millennium BC, this truth found its first
applications in southern Mesopotamia.
6. - 5 -
The Sumerians invented writing and arithmetic. Both applications are more practical and
timeless monuments of this Truth than the enormous monolithic works of stone from before and
after that were erected for her honor. From the very beginning the Sumerians used a
sexagesimal number system for computation. In the following millennium this sexagesimal
number system was expanded with a positional-place value number system. A sexagesimal
positional numerical system four thousand years before Simon Stevin in 1585 published "De
Thiende" and introduced Europe to the Hindu-Arabic numerals of our modern decimal place
value number system. The classical sciences of arithmetic, geometry, astronomy and music and
the classical arts such as literature and speech are all directly derived from this truth. The
Sumerians believed with appropriate justification they had discovered the Gate of Heaven,
because the truth is precisely expressed in a demonstrable number theory of everything. A
perfect theory which exactly expresses (all) the facts. At the end of the third millennium BC the
Sumerian civilization in the south came to an end. The knowledge that had given rise to the
power of the ancient cities of Ur and Larsa was transferred north where a new city had been
founded Bābilim. Babylon meaning ‘The Gate of God.’
Written in cuneiform 𒆍𒀭𒊏𒆠 (KA2.DINGIR.RAki/bāb ili/, “gate of god”).
Pythagoras studied with the priests in Egypt, when in 525 BC Egypt was conquered by the
Persian king Cambyses II (son of Cyrus II) and was taken to Babylon. In Babylon, Pythagoras
learned the truth of the ancients and was taught all its components: arithmetic, geometry,
astronomy, and music. Upon his return from Babylon, Pythagoras founded a commune called
the "mathematikoi" in the Greek colonies of southern Italy. In this commune women were
admitted on equal terms and slavery and private property were prohibited. The essence of
Pythagoras' philosophy is equality thinking or egalitarianism.
Bertrand Russell at the beginning of The History of Western Philosophy says about Pythagoras:
“It is only in quite recent times that it has been possible to say clearly where Pythagoras was
wrong. I do not know of any other man who has been as influential as he was in the sphere of
thought. I say this because what appears as Platonism is, when analysed, found to be in essence
Pythagoreanism. The whole conception of an eternal world, revealed to the intellect but not to the
senses, is derived from him. But for him, Christians would not have thought of Christ as the Word;
but for him, theologians would not have sought logical proofs of God and immortality. But in him
all this is still implicit. How it became explicit will appear.”1
The Pythagorean philosophy is entirely based on the mathematical Identity of the ancient
Mesopotamians. The mathematical truth which Pythagoras introduced in the Hellenic world
was not his famous theorem. No, A2 + B2 = C2 is only the equation that remains when the
Mesopotamian Identity is limited to right angled triangles.
7. - 6 -
The Discovery of Heaven
The Mesopotamian Identity is in modern notation:
𝐴 𝑥
· 𝐵 𝑦
= (
𝐴 𝑥+𝐵 𝑦
2
)
2
− (
𝐵 𝑦 −𝐴 𝑥
2
)
2
the product van unequal factors (Ax · By) = a difference of
squares (Ax < By).
----------------------------------------------------------------------------------------------------------------------
𝐴1
· 𝐴1
= 𝐶2
− 𝐵2
the product of equal factors (A · A) = a difference of squares.
𝐴2
+ 𝐵2
= 𝐶2
The product of Ax and By = is equal to = the square of the half-sum of Ax and By
Minus
the square of the half-difference of Ax and By.
Discrete multitude = difference of two continuous magnitudes.
A rectangle = a gnomon, the difference of two squares.
Above the line is a mathematical Identity. It is always demonstrably true for every value of A, B, x
and y.
---------------------------------------------------------------------------------------------------------------------
Below the line is a mathematical equation. It is always demonstrably true for specific values of
A, B and C.
The Identity explains the dogma about the truth in antiquity as well as the emphasis of the
Pythagoreans on the monad (singularity), dyad (duality), and triad (trinity). It is completely
pointless to philosophically challenge the truthfulness of the Identity. It is demonstrably true for
all variables. This mathematical identity cannot be improved upon. Nobody in antiquity
therefore took a different starting point. Pythagoras, Plato, Archimedes all used the science of
the Identity of arithmetic and geometry. With the notable exception of one person, no one
thought it worthwhile to argue about the Identity or mathematical proof.
8. - 7 -
The difference between the Identity and the Pythagorean Theorem can be illustrated with a
simple example. Fill the Identity with values A = 1, B = 9, x = 1 and y = 1.
𝐴 𝑥
· 𝐵 𝑦
= (
𝐴 𝑥
+ 𝐵 𝑦
2
)
2
− (
𝐵 𝑦
− 𝐴 𝑥
2
)
2
11
· 91
= (
11
+ 91
2
)
2
− (
91
− 11
2
)
2
11
· 91
= (
10
2
)
2
− (
8
2
)
2
11
· 91
= (5)2
− (4)2
-------------------------------------------------------------------------------------------------------------------
32
= 52
− 42
𝐴2
= 𝐶2
− 𝐵2
The values 1.9.1.1 for the Identity look overly simple and insignificant. For the comparison and
the history of mathematics, however, these specific values are fundamental. With these values,
the Identity provides the smallest possible integers that satisfy the Pythagorean theorem. After
all, the 1 by 9 rectangle can also be represented as a 3 by 3 square.
At the same time, filling the Identity with A and B as equal values - that is, making the unequal
factors equal - naturally does not make it counter-factual. The Identity is of course always
correct.
31
· 31
= (
31
+ 31
2
)
2
− (
31
− 31
2
)
2
31
· 31
= (
6
2
)
2
− (
0
2
)
2
31
· 31
= (3)2
− (0)2
An example without a value of 1. Fill the Identity with values A = 7, B = 9, x = 2 and y = 3.
9. - 8 -
𝐴 𝑥
· 𝐵 𝑦
= (
𝐴 𝑥
+ 𝐵 𝑦
2
)
2
− (
𝐵 𝑦
− 𝐴 𝑥
2
)
2
72
· 93
= (
72
+ 93
2
)
2
− (
93
− 72
2
)
2
49 · 729 = (
49 + 729
2
)
2
− (
729 − 49
2
)
2
49 · 729 = (
778
2
)
2
− (
680
2
)
2
49 · 729 = (389)2
− (340)2
49 · 729 = 151.321 − 115.600
49 · 729 = 35.721
A simple example with A = 1 and B = 7 to further illustrate the limitation of the Identity to the
Pythagorean theorem and give a two-dimensional representation of a prime number:
𝐴 · 𝐵 = (
𝐴+𝐵
2
)
2
− (
𝐵−𝐴
2
)
2
1 · 7 = (
1+7
2
)
2
− (
7−1
2
)
2
1 · 7 = (
8
2
)
2
− (
6
2
)
2
1 · 7 = (4)2
− (3)2
Below the geometric figure of a rectangle with a width of 1 and a length of 7, the geometric
representation of the prime number 7 based on its two divisors.
↑
Y
5
4
3
2
1 ⑦
1 2 3 4 5 6 7 X →
10. - 9 -
The difference of the two squares (of 16 and 9) is another two-dimensional representation of the
prime number 7.
The figure that is the difference of two squares is called a gnomon.
MINUS
=
↑
Y
5 42
4
3
2
1
1 2 3 4 5 6 7 X →
1 + 7
2
2
↑
Y
5
4 32
3
2
1
1 2 3 4 5 6 7 X →
7 − 1
2
2
↑
Y 42
-32
=
5
4 ⑦
3
2
1
1 2 3 4 5 6 7 X →
11. - 10 -
𝐴 𝑥
· 𝐵 𝑦
= (
𝐴 𝑥
+ 𝐵 𝑦
2
)
2
− (
𝐵 𝑦
− 𝐴 𝑥
2
)
2
1 · 7 = (
1+7
2
)
2
− (
7−1
2
)
2
Of course, the two squares can - trivially - be laid down as two (of the three) squares on the side
of one rectangular triangle. The larger square on the diagonal and the smaller square as one of
the legs of the triangle. The length of the remaining leg of the triangle will then exactly
correspond to the square root of the surface of the gnomon.
A 1 by 7 rectangle with area ⑦ can be represented by a square with sides of √7.
1 · 7 = ( 4 )2
− ( 3)2
-------------------------------------------------------
7
2
= (4)2
− (3)2
(𝐵)2
= (𝐶)2
− (𝐴)2
↑
Y
5
4
3
2
1 ⑦
1 2 3 4 5 6 7 X →
↑
Y 42
-32
=
5
4 ⑦
3
2
1
1 2 3 4 5 6 7 X →
4
3 ⑯
3 ⑨ 4
√7 ⑦
√7
↑
Y 42
-32
=
5
4 ⑦
3
2
1
1 2 3 4 5 6 7 X →
12. - 11 -
Everything from One
Mathematics concerns provable truth and accordingly the Mesopotamian Identity leaves nothing
to the imagination. The Identity gives a complete and consistent axiomatization of mathematics
and two-dimensional space. Whomever can grasp the arithmetic of the Table of one, can grasp
the mathematical configuration of the Universe.
Beginning with the arithmetic of “The table of 1,” the Mesopotamian Identity delivers:
1. The tree of all Pythagorean triples (ABC), with 1 · n2 = Primitive Pythagorean Triple,
see Table I.A .
2. The tree of all quadratic equations, with roots 1 and n, see Table I.B .
Even though the Identity is neutral as regards the number system and is, of course, as true in
decimal as it is in binary, the Mesopotamian Identity grants additional advantages to the
sexagesimal number system (base 60). When A and B are a pair of reciprocal numbers, that is
A=
1
𝐵
and B=
1
𝐴
, the left side of the Identity -A times B- gives the multiplicative identities:
3. A ·
1
𝐴
= 1 and/ or B ·
1
𝐵
= 1, thus one has an identity within the Identity, see Table I.C.
We usually start learning multiplication when we are in 2nd or 3rd grade. Table I.A and I.B cover
the large majority of high school mathematics. Table I.C covers the arithmetic of time, how two
times half an hour makes one hour, four times fifteen minutes makes one hour, eight times 7
minutes and 30 seconds makes one hour, and the square of one minute makes one hour.
Table I.A. The Mesopotamian Identity and the Theorem of Pythagoras.
The Mesopotamian Identity is unparalleled as regards a recipe for Pythagorean triples. Line 7 of
Table I.A features the geometry of ‘1 · 7’ from the previous pages. From A= 1, B= {1, 2, 3, ..},
x= 1and y= 1, i.e. the Multiplication Table of One, the Mesopotamian Identity generates the tree
of (all primitive) Pythagorean triples.
1 · (number)2 = a primitive Pythagorean triple.
1 · (odd number)2 = 1 · 32 = 1 · 9 = ((1+9)/2)2 – ((9-1)/2)2 = 52-42
1 · (even number)2 = 1 · 42 = 1 · 16 = ((1+16)/2)2 – ((16-1)/2)2 = 8.52 – 7.52
14. - 13 -
The even is that which can be divided into two equal parts without a unit intervening in the middle;
and the odd is that which cannot be divided into two equal parts because of the aforesaid
intervention of a unit.
Now this is the definition after the ordinary conception; by the Pythagorean doctrine, however,
the even number is that which admits of division into the greatest and the smallest parts at the
same operation, greatest in size and smallest in quantity, in accordance with the natural
contrariety2 of these two genera; and the odd is that which does not allow this to be done to it, but
is divided into two unequal parts. [Please note definition ‘B’ of Aristotle’s Stratagem!]
In still another way, by the ancient definition, the even is that which can be divided alike into
two equal and two unequal parts, except that the dyad, which is its elementary form, admits but
one division, that into equal parts; and in any division whatsoever it brings to light only one species
of number, however it may be divided, independent of the other. The odd is a number which in any
division whatsoever, which necessarily is a division into unequal parts, shows both the two
species of number together, never without intermixture one with another, but always in one
another’s company.
By definition in terms of each other, the odd is that which differs by a unit from the even in either
direction, that is, toward the greater or the less, and the even is that which differs by a unit in either
direction from the odd, that is, is greater by a unit or less by a unit. [..]”3
The Pythagorean definition of an odd number is the exact opposite of the definition of an odd
number given by Euclid in the Elements:
VII.7 An odd number is that which is not divisible into two equal parts, or that which differs by a
unit from an even number.
Suffices to say here the Pythagorean definition is logically coherent and the Euclidean is not.
Table I.B. The Mesopotamian Identity and Quadratic equations versus the ABC-formula.
From the Identity quadratic equations are as easily constructed as they are solved. The degree
of an equation is given by the sum of x and y, thus x= 1 and y= 1 make a quadratic equation.
17. - 16 -
1
Bertrand Russell, The History of Western Philosophy , p. 37.
2
The natural contrariety of magnitude and quantity:
1, 2, 3, 4, 5, .., ∞
1, ½, ⅓, ¼, ⅕, .., ∞
3
Nicomachus of Gerasa, Introduction to Arithmetic, vertaling van Martin Luther D’Ooge, London:
MacMillan and Company, 1926. pp. 182-191.
