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Archaeological
survey indicates the Dark Ages of Greece extended from
the eleventh to the eighth Century BCE. Bizarre reasons including
an increase in population were attributed to be the cause of most of the
changes during this archaic period. It seems most of the
palaces
of the Mycenaean civilization were destroyed around 1200
BCE.
The only palace left un-destroyed was in Athens. But even Athens seems
to have had a hard time for the next several hundred years. There were
no more
kings, but
from time to time tyrants (the present equivalent of dictators) ruled.
Taxes
were not collected. The roads were neglected in an un-repaired state.
Trade and commerce suffered badly. The population fell from disease,
death. Greeks migrated mainly to the shores of and further inland into
Asia Minor (Anatolia) and Egypt. Because Greece was in such bad shape
during the Dark Ages, and could not defend herself, it also seems that
some of their neighbors to the north invaded Greece and began living in
some of the Greek cities. The Greeks called these invaders the
Dorians, and called the old Mycenaean Bronze Age Greeks, the
Ionians. Since a large number had settled along the coast of
Turkey people began to call that place Ionia (the Yonas of the
Iranian Golden Age). Here the knowledge of how to make tools and
weapons out of iron had spread from the Hittites around the
Mediterranean Sea.
It was during such
depressing times, about 569 BCE that Pythagoras was born in Samos, an
island off the western coast in the
Anatolian plateau of Turkey of a Greek
Father and an Asiatic mother. An ordinary Ionian of mixed blood he had
no opportunity to read or write. There has been not
a single record of Pythagoras having written anything. It was
the later, so-called Pythagoreans (his immediate followers and later new
conscripts), who did all the documentation and writing and, with it, the
claiming and the exaggerating of whatever he is believed to have
transmitted in his ‘School’. This resulted in all that they claimed to
be documented in history as the alleged Pythagoras’ original ideas and
knowledge. Let us inquire into this rather extraordinary historical
anomaly.
Not all that
Pythagoras is believed to have transmitted was readily accepted by the
Pythagoreans. Indeed, when the Pythagorean Society at Croton was
attacked by Cylon, a nobleman from Croton itself, about 480 BCE
Pythagoras escaped to Metapontium. Almost all authors say he died there;
some claiming that he committed suicide because of the attack on his
Society. After his death in about 480 BCE the ‘Pythagorean
Society’ expanded rapidly about 500 BCE and later. Knowledge,
traditionally being accepted synonymous with power (particularly, at the
time) the Society members in their euphoria (rather than in their
self-styled wisdom) decided to enter politics. As it often happens the
political nature of the, once, amicable members split them into a number
of political affiliations and factions. Such were the problems created
by these alleged intellectuals that the Society had to be
forcefully suppressed. Its meeting houses and halls were everywhere
violently sacked and burned. In particular, in the “House of Milo” in
Croton 50 to 60 Pythagoreans were trapped and slain. Those who managed
to escape or survive the wounds took refuge at Thebes and other places
in Egypt. That certainly was the end of their ambitions as they soon
went into obscurity.
Pythagoras, while in
Samos had somehow managed to remain tolerant of (though disillusioned
by) the dictatorial block on the freedom of speech imposed by the, then,
tyrant Polycrates, who had seized control of the city of Samos. Around
535 BCE Pythagoras, who had a natural bent towards intellectual
inquiries and learning, which he found gravely lacking in Greece at the
time, escaped to Egypt. According to Porphyry he visited many
of the temples and witnessed discussions but, as an alien, he was
refused admission to priesthood to all the Egyptian temples. It
was because there were strong links between Samos and Egypt at this time
that Polycrates dispatched a letter of recommendation that he be
permitted to enter the ‘inner circle’ of Egyptian priesthood. This
helped him gain admission into the Temple of Diospolis.
After completing the rigid temple rites necessary for admission he was
ultimately accepted into the priesthood.
In 525 BCE
Cambyses, son of Cyrus the Great, invaded Egypt, won the
Battle of Pelusium and captured Heliopolis and Memphis. Pythagoras was
taken as a prisoner of war, and, therefore as a slave to Babylon. In
Babylonia in the Zarathushti Maghavan Brotherhood Guild, which
was originally founded by Dai-aauku (Gk: Diocese) was
Gaomata, a rather ambitious Maghavan as the Principal. It was under the
tutelage, authority and guidance of the, then, Zarathsuhti Pontiff
Zarades (also called Zaradus in history). He was the supreme
Mobedan Mobed, equivalent to the present Roman Pope, both under Cambyses
and later, Daraius I) that Pythagoras attained deep
knowledge and understanding of the Gathic and Avestan Philosophy. It is
worthy of note that it was the name of this Royal Papacy, which prompted
some historians to incorrectly place the birth of Zarathushtra in
Babylonia and the incorrect date to 550 BCE. Pythagoras was instructed
in the sacred rites and learnt about mathematics, geometry, philosophy
of ideas, intellectual ideals and spiritual knowledge. He also reached
the acme of perfection in Arithmetic, Geometry and Music
and the other mathematical sciences taught by the Babylonians. It was
here that Pythagoras’s young mind was firmly impressed. Here he first
learnt the importance of numbers and the dependence on the
interaction of contraries or pairs of opposites -
good and evil; positive/negative; light/darkness; right/wrong;
rationality/irrationality; outwardness/inwardness…etc.
Pythagoras had also
borrowed from the teachings of the Buddha and the Egyptian priests
The concepts of the
transmigration of the soul (repeated cycles of rebirth in another
earthly form- Reincarnation) and Vegetarianism (eating
of flesh being abominable) were clearly learnt by Pythagoras in Egypt
and certainly borrowed from news emerging from India. The concept of
eating vegetable food only was prevalent, at the time, both in the
teachings of the Buddha and the priests of Egyptian Temples. The concept
of Reincarnation was clearly ‘borrowed’ from the contemporary
teachings of Gautama Siddhartha, the
Buddha (the Wise One)
born around 565 BCE in the city of Kapilavastu, India. The
extraordinary Greek claim that because of the
vast distance there could not possibly have been any contact whatsoever
between India and Egypt, is incorrect. Egypt (the modern version for
the Roman Aegyptus) was called Mizr, at the time. That there were trade
contacts and regular exchange of information with Mizr is
indicated in Vedic Texts (also termed Missr, Missra and Mishra). The
Indo-Iranian Sumerian rulers, Sharrukin (Sargon I circa 2334-2279
BCE) and son, Manish-tishu are mentioned in
Clay Tablets found in the Mohenjodaro diggings (The Indus Valley
Civlisation; now called the Saraswati-Sindhu Civilisation).
The
reason why Pythagoras could not possibly have picked up the
concept of Reincarnation (repeated cycles of rebirths in another
earthly form) from the Zoroastrian Pontiff, Zarades in Babylonia is
because of our unique belief in the presence of our individual Fravashis.
One’s Fravashi is, indeed, one’s own individualised ‘Guardian
Spirit’, which (like Ahura Mazda) is perfectly good and totally
incorruptible. It was present before the individual’s birth and
will continue to exist forever – in a state of perfect timelessness. It
follows that when one’s Soul (Urvan) is being judged at Chinvato Peretu
(the Bridge of the Separator of the good from the evil during the ‘first
Judgment’) one’s Fravashi, obviously, cannot be judged. Since each
person’s Fravashi was already present from the beginning of the Creation
before the person was born. At the birth of each person the individual
Fravashi forms part of the physical body and it will continue to exist
even after the person’s death, when the Fravashi of the dead moves back
into the Spiritual state and continues to exist eternally in the
spiritual world in the eternal glow of Garo-dēmāna (Gathic word). The
equivalent Avestan word is Garo-nmāna, Pahlavi word, Garosmān and
Gujarati word Garothmān.
Indeed, our Fravardigan Days (Gujarati: Muktad) are actually reserved
for the very invocation of the Fravashis of our departed beloved for
obtaining their blessings of good health and beneficent yearnings.
Zarathushtra’s extraordinary vision of ‘the end of Time’ as Frasha
(kar) / Fərashəm [Avestan /
Frashokereti; Pahlavi: Frashekart] – ‘making afresh / anew’ (Yasna
30.9 - Gathic Fəra is Avestan Fra) fully explains the reunification of
each individual Fravashi with the resurrected body at ‘the end of
Earthly Time/the Solar System’. The Fravashi is then fully reunited
with the resurrected physical individual body. This extraordinarily
amazing vision, in fact, not only explains fully, in the Religion of
Zarathushtra the total absence of the belief in Reincarnation (repeated
cycles of material rebirth in different forms, even other than human in
real life on Earth) but also induces the full impact of the Fravashi
being reunited with the resurrected body. Zarathushtra’s visionary
concept of ‘the end of Time’ has been improperly expurgated in
Christianity as the Apocalypse or the Doom’s Day, without really
explaining why a second judgment is at all necessary. The Christian
Apocalypse fails to explain the Zoroastrian concept of the total
annihilation of evil at that time, since in Christianity ‘God’ made
everything - the good as well as the evil.
It is
also rather strange that after the death of Pythagoras his original
teaching about the concept of Vegetarianism was distorted
by many of his followers to suit their own life style. They claimed they
practiced Vegetarianism only because the souls of all living creatures
pass after death into other living creatures. Others occasionally ate
flesh regularly and some refrained from eating meat only because of
their bizarre claim of a ‘taboo’ rather than Pythagoras’ philosophical
teachings. They also altered the concept of reincarnation claiming ‘it
was not the same in all
species’ of the animal kingdom, claiming that in some it occurs until
its ‘eventual purification’ is established.
In those days many a
slave earned his freedom from some thoughtful and humanistic master,
often because of a long faithful and industrial service to the master
and sometimes because of appreciable intellect. After earning his
freedom offered by the Pontiff Zarades in 520 BCE Pythagoras left
Babylon and returned to Samos. Polycrates, the tyrant had been
assassinated in early 522 BCE and Cambyses had died in the summer of 522
BCE. Pythagoras had a short stay in Crete shortly after his return to
Samos to study the system of laws there. Back in Samos he founded a
‘school’ which was called the Semicircle. He tried
to use his symbolic method of teaching, which was similar in all
respects to the lessons he had learnt in Egypt and Babylonia. The
Samians however were not very keen on this method and treated him with
utter ridicule.
Visiting Croton
(present name Crotone), a sea port on the east coast of southern
Italy (called Magna Graecia, because it was part of Greater Greece) in
about 518 BCE, he managed to attract many followers to form a
‘Society’. As the head of the ‘Society’ he formed an inner circle
of followers known as Mathematikoi, who
lived permanently with the ‘Society’. They had no personal possessions
and were strict vegetarians (as in the Temples of Egypt and Babylonia -
specifically they ate no beans). They were taught by Pythagoras himself
and obeyed strict rules. The beliefs that Pythagoras held were (a)
that at its deepest level, reality is mathematical in nature,
(b) that philosophy can be used for spiritual purification,
(c) that one’s soul can rise to the ultimate union with the
divine, (d) that certain symbols have a mystical
significance and (e) that all brothers of the order should
observe strict loyalty and secrecy.
Pythagorean teachings
Rite and rules:
They performed
purification rites and followed moral, ascetic, and dietary rules to
enable their souls to achieve a higher level in their subsequent lives
and thus eventually be liberated from the reincarnation ‘cycle of
rebirth.’ This also led them to regard the sexes as equal, to treat
slaves humanely, and to respect animals. The highest purification was,
of course, ‘philosophy’
Numbers Theory and Mathematics:
They believed that umbers constitute the true nature of all things – the
very essence of things was number. Even abstract ethical concepts like
justice–could be expressed numerically. That relationship between
musical notes could be expressed in numerical ratio. The later
Pythagoreans, after his death even elaborated a bizarre theory of
numbers. Pythagoras made remarkable contributions to the mathematical
theory of music. He was a fine musician, playing the lyre. He even used
music as a means to help those who were ill. He found that vibrating
strings produce harmonious tones when the ratios of the lengths of the
strings are whole numbers; these ratios could be extended to other
aspects of life.
Geometry:
The basics were: -
-
The sum of
the angles of a triangle is equal to two right angles. Also the
Pythagoreans knew the generalization which states that a polygon
with n sides has the sum of interior angles 2n - 4
right angles and sum of exterior angles equal to four right angles.
-
The Babylonian
theorem - in a right angled triangle the square on the
hypotenuse
is equal to the sum of the squares on the other two sides.
Astronomy:
The Earth was a sphere at the centre of the Universe. He also
recognized that the orbit of the Moon was inclined to the equator of
the Earth and he was one of the first to realize that Venus as an
evening star was the same planet as Venus as a morning star
Philosophy:
Primarily,
however, Pythagoras was a philosopher: His views: -
-
The dynamics of
world structure has dependence on the interaction of
contraries or pairs of opposites
-
The viewing of
the soul as a self-moving number experiencing a form
of rebirth and understanding.
-
All
existing objects were fundamentally composed of form
and not of material substance.
-
Further,
Pythagorean doctrine (probably conjured up by the Pythagoreans after
his death) identified the brain as the primary locus of the
soul.
 Ethics:
the Pythagorean were famous for their mutual friendship, unselfishness,
and honesty
but when the urge to enter Politics occurred (as noted above) there was
utter ruin
Archaeological findings:
Clay Tablets reveal this knowledge had been recorded 2 millennia before
the birth of Pythagoras. To show that Pythagoras obtained the above
knowledge in Babylonia here are some archaeological findings – Courtesy
– article by J J O’Connor and E F Robertson: - Pythagoras’s so-called
‘original’ work found in ancient Clay Tablets of Susa and the origins of
‘the Theorem’ in Babylonian and also in Indian Mathematics.
Plimpton 322
is the tablet numbered 322 in the collection of G A Plimpton housed in
Columbia University.
Pythagorean Theorem in
Mathematics of India
A brief outline of a study of the Theorem.
In all
early civilizations, the first expression of mathematical understanding
appears in the form of counting systems. Numbers in very early societies
were typically represented by groups of lines, though later different
numbers came to be assigned specific numeral names and symbols (as in
India) or were designated by alphabetic letters (such as in Rome).
Although today, we take our decimal system for granted, not all ancient
civilizations based their numbers on a ten-base system. In ancient
Babylon a sexagesimal (base 60) system was in use.
The
decimal system in Harrappa
[Saraswati-Sindhu Civilization, previously known as the Indus Valley
Civilization circa 2,500 BCE.
It extended over a large
land mass from Saraswati (Avestan Haraqvaiti - present southern
Afghanistan) eastwards through to the delta of the Indus River. Recent
diggings have shown even the North of Gujarat was involved].
In India a
decimal system was already in place during the Harappan period, as
indicated by an analysis of Harappan weights and measures. Weights
corresponding to ratios of 0.05, 0.1, 0.2, 0.5, 1, 2, 5, 10, 20, 50,
100, 200, and 500 have been identified, as have scales with decimal
divisions. A particularly notable characteristic of Harappan weights and
measures is their remarkable accuracy. A bronze rod marked in units of
0.367 inches points to the degree of precision demanded in those times.
Such scales were particularly important in ensuring proper
implementation of town planning rules that required roads of fixed
widths to run at right angles to each other, for drains to be
constructed of precise measurements, and for homes to be constructed
according to specified guidelines. The existence of a gradated system of
accurately marked weights points to the development of trade and
commerce in Harrappan society.
Mathematical knowledge and its application during the Vēdic period
In the Vēdic period, records of mathematical activity are
mostly to be found in Vēdic texts associated with ritual activities.
However, as in many other early agricultural civilizations, the study of
arithmetic and geometry was also impelled by secular considerations.
Thus, to some extent early mathematical developments in India mirrored
the developments in Egypt, Babylon and China. The system of land grants
and agricultural tax assessments required accurate measurement of
cultivated areas. As land was redistributed or consolidated, problems of
measurement came up that required solutions. In order to ensure that all
cultivators had equivalent amounts of irrigated and non-irrigated lands
and tracts of equivalent fertility - individual farmers in a village
often had their holdings broken up in several parcels to ensure
fairness. Since plots could not all be of the same shape - local
administrators were required to convert rectangular plots or triangular
plots to squares of equivalent sizes and so on. Tax assessments were
based on fixed proportions of annual or seasonal crop incomes, but could
be adjusted upwards or downwards based on a variety of factors. This
meant that an understanding of geometry and arithmetic was virtually
essential for revenue administrators. Mathematics was thus brought into
the service of both the secular and the ritual domains.
Arithmetic operations
(Ganit) such as addition, subtraction, multiplication,
fractions, squares, cubes and roots are enumerated in the Nārad
Vishnu Purānā attributed to Vēda
Vyās (pre-1000 BCE). Examples of the knowledge of ‘calculation
in lines’ (Rēkhā-ganit)
are to be found in the Sulva-Sutras of Baudhayana (800 BCE)
and Apasthamba (600 BCE), which describe techniques
for the construction of ritual altars in use during the Vedic era.
. Mathematics was classified either as Garna (Simple
Mathematics) or Sankhyān
(Higher Mathematics). Numbers were deemed to be of three types:
Sankhēya (countable), A-sankhēya (uncountable) and
A-nant (infinite). The Sanskrit word for Geometry was
‘Gēōmit’.
Although the Chinese were also using a decimal based counting system,
the Chinese lacked a formal notational system that had the abstraction
and elegance of the Indian notational system, and it was the Indian
notational system that reached the Western world through the Arabs (who
emerged after 641 CE with the defeat of the Sassanians coinciding with
the ousting from most of Europe of her Roman masters). Digits 1 to
9 are still called ‘hindsa’ in Arabic. Their alleged
‘first’ introduction of ‘zero’ was also ‘borrowed’ from the
Hindu ‘decimal system’. The true significance is perhaps best
stated by the French mathematician, Laplace: “The ingenious method
of expressing every possible number using a set of ten symbols (each
symbol having a place value and an absolute value) emerged in India.
The idea seems so simple nowadays that its significance and profound
importance is no longer appreciated. Its simplicity lies in the way it
facilitated calculation and placed arithmetic foremost amongst useful
inventions.”
Mathematics played a vital role in Āryābhatta's
revolutionary understanding of the solar system. His calculations
on ‘pi’, the circumference of the earth (62,832 miles) and the
length of the solar year (within 13 minutes of modern calculation) were
remarkably close approximations. In making such calculations, Āryābhatta
had to solve several mathematical problems that had not been
addressed before, including those in algebra (beej-ganit)
and trigonometry (trigonmiti - used
in India for
deciding the position, motion et-cetera of the spatial planets). It was
the Arab mathematician, al-Khwārizmi (780-850 CE) who expurgated the
Hindu text of Beej Ganit in his book, which he titled ‘al-Gebr’,
which soon became Algebra in Europe. It was even attributed to him as
the inventor.
Pythagorean Theorem in
Mathematics of India
- A brief outline of a study of the Theorem.
It is
likely that these texts tapped geometric knowledge that may have been
acquired much earlier, possibly in the Harappan period. Baudhayana's
Sutra displays an understanding of basic geometric shapes and
techniques of converting one geometric shape (such as a rectangle) to
another of equivalent (or multiple, or fractional) area (such as a
square). While some of the formulations are approximations, others are
accurate and reveal a certain degree of practical ingenuity as well as
some theoretical understanding of basic geometric principles. Modern
methods of multiplication and addition have clearly emerged from the
techniques described in the Sulva-sutras.
The Sulva-sutra
of Baudhayana is considered to be the oldest as well as the
most systematic and detailed version of the text. Scholars are not
agreed on the precise date of the sutra, but the text clearly pre-dates
Panini (c. 520-460 BCE) and is generally thought to have been written in
the 5th - 6th century BCE. His statement “in a
Deergh-chatursh
(Rectangle) the Chētra
(Square) of Rajju
(hypotenuse) is equal to sum of squares of
Parshvāmani
(base) and Triyangamani
(perpendicular)’ indeed sums up the so-called Pythagorus’
theorem.
The Taittiriya
Samhita quotes, “He who desires heaven may construct the
falcon-shaped altar; for the falcon is the best flyer among the birds;
thus he [the sacrificer] having become a falcon himself flies up to the
heavenly world.” It is described in Chapter 11 of Baudhayana's text.
The construction of the Fire altar needs a total of 200 bricks of
five different shapes in the first layer. The second layer is similar in
shape and also needs 200 bricks, but five additional brick types are
required. In constructing the altar, the bricks were laid in such a way
that no brick rested on another of the same size and shape. Generally,
there were five layers, the odd ones being replicas of the first layer
and the even ones of the second layer. Using the dimensions of the
bricks given in angulas in the text, and taking 1 ft = 16
angulas the span of the altar-falcon comes to 40.5 ft or
12.3 metres. The altar about knee-high would have an area 7 1/2 square
purushas (one purusha being the height of a
man with uplifted arms (120 angulas - 71/2 feet or 2.3 metres), which
comes to 56.25 sq. ft or 5.29 sq. metres.
This knowledge would have helped the Vedic people in the planning of the
construction of their intricate temple complexes.
The
Pythagoreans (in the same manner as they followed Gautama Buddha’s
teachings) were obviously familiar too with the Upanishads and applied
their basic geometry from the Sulva Sutras. An early statement of what
is commonly known as the Pythagoras theorem is to be found in
Baudhayana's Sutra: The chord which is stretched across the diagonal of
a square produces an area of double the size. A similar observation
pertaining to oblongs is also noted. His Sutra also contains geometric
solutions of a linear equation in a single unknown. Examples of
quadratic equations also appear.
Apasthamba's sutra
(an
expansion of Baudhayana's with several original contributions)
provides a value for the square root of 2 that is accurate to the fifth
decimal place.
Apasthamba
also looked at the problems of squaring a circle, dividing a segment
into seven equal parts, and a solution to the general linear equation.
Jain texts from the 6th Century BCE such as the Surya Pragyapti
describe ellipses. Modern-day commentators are divided on how some
of the results were generated. Some believe that these results came
about through hit and trial - as rules of thumb, or as generalizations
of observed examples. Others believe that once the scientific method
came to be formalized in the Nyāya-Sutras
- proofs for such results must have been provided, but these have either
been lost or destroyed, or else were transmitted orally through the
Gurukul system, and only the final results were tabulated in the
texts. Such study of Ganit i.e. mathematics was given considerable
importance in the Vēdic period.
The dissection,
however, is a Hindu achievement; the original so-called Pythagoras
drawing (of the right-angle triangle with the three squares) bears the
inscription 'Look' (behold), which must convince the reader better than
any verbal argument.
The
Vēdang Jyotish (1000 BCE) includes an extra-ordinary
statement pointing to the Hindu origin of Mathematics: "Just as
the feathers of a peacock and the jewel-stone of a snake are placed at
the highest point of the body (at the forehead), similarly, the position
of Ganit is the highest amongst all branches of the Vēdas and the Shāstras."
The
(so-called) Theorem of Pythagoras.
Its actual origin in much earlier times and its latter day
proofs
The Theorem of Pythagoras which most of us remember in
the form in which it was taught to us at school, states that the sum of
the squares on the two shorter sides of a right-angled triangle is equal
to the square on the hypotenuse.
Although algebraically expressed in the familiar form a2
+ b2 = c2, this equation really applies in a
geometrical sense to areas of all other well-defined shapes besides
squares. We could have, for instance, replaced the words ‘squares’
by ‘semi-circles, or parabolas, or similar
triangles, and still have been correct.
Take a square shaped piece of paper, and after folding it
along the lines shown dotted, as in Diagram ‘A’, reopen and examine the
shapes and sizes of the subdivisions you have created. You will find
that there are eight equal right-angled triangles.
 |
|
Diagram A |
Now, if you name the
junctions where the lines intersect by calling them A, B, C, D, E, F, G,
H and I, you will readily see that HDF is a right angled triangle with
the right angle situated at F, and that the sum of the triangles HGF and
DEF mounted on the shorter sides HF and DF are equal in area to the
similar larger triangle HBD mounted on the hypotenuse HD.
You have just
discovered the theorem of Pythagoras in one of its simplest ‘avatars’.
You should therefore not be surprised if anyone were to tell you that
the ancient civilizations also knew about such relationships.
There is ample
evidence to show that the Babylonians, the Indians, and the Egyptians
all knew about such relationships long before the theorem relating to it
was included by Euclid in his book called ‘ELEMENTS’. This came to be
better known after some clay tablets from the time of Hammurabi were
unearthed in Mesopotamia during archaeological excavations in the last
century, and the writings on them were deciphered and interpreted.
There is one called
the Tel Dhibayi Tablet, inscribed on which in cuneiform writing there is
the algebraic solution given to the problem of finding out the lengths
of the sides of a rectangle when its area and the length of its diagonal
are known. This indicates that the Babylonians of the time were also
well versed in algebra, and of course that they knew about the
relationships between the lengths of the sides of right-angled
triangles.
One of the other
such tablets is the Plimpton Tablet (see clay tablet above), which
actually lists 15 right-angled triangles in which the slopes of the
diagonal to the base range from about 45° to about 60°. This tablet was
difficult to decipher, because it was damaged and had certain parts
missing. But most of its contents are now fairly well determined and
several triangles which are definitely right angled from the dimensions
mentioned in the tablet can be easily recognized.
For instance, Serial
No. 1 corresponds to the triangle (119,120, 169) in which the slope is
very close to 45°. Serial No. 5 corresponds to (65, 72, 97), whereas
Serial No.6 relates to (319, 360, 481). No. 11 is the triangle (3, 4, 5)
with which every one has a nodding acquaintance, and No. 15 corresponds
to (28, 45, 53).
The Egyptians were
building pyramids using triangles of similar shapes. The Khafre Pyramid
at Giza has an angle of elevation of 53° 7.8’ compared to triangle (3
,4, 5), which has an elevation of 53° 5’. That is not the only pyramid
in which such matching angles can be traced. The Draco (Red) pyramid at
Dahshur and the upper portion of the Bent Pyramid, also at the same
place, both have angles of elevation corresponding to (20, 21, 29) that
is close to 45°.
 |
|
Diagram B |
To
ancient India we owe the diagram ‘B’ shown above. It is one of the
dissection proofs of the theorem we have been talking about. The
original drawing bears the inscription: “Behold” which carries greater
conviction than any verbal argument.
There
are many other such dissection proofs not requiring verbal arguments to
support them. Making a note of this we may now consider the proof
of the theorem attributed by Euclid (c. 300BCE) to Pythagoras, which
appears in the book called Elements. In a sense it is also Euclid’s
proof, because it appears twice in his book in two different forms,
which have many common features.
This
proof is definitely not of the dissection
type, and involves the comparison of congruent (and in the second
version, of similar) triangles. The first version bears the label
Proposition 47, Book I. while its converse is labelled Proposition 48.
The entire Book I consists of propositions included to pave the way for
these last two theorems.
What is
also important to note is that the Pythagoreans had at one stage come
upon an impasse in their thinking when they discovered, as they did
themselves, that in an isosceles right-angled triangle, where both legs
were of the same length, no integer could be found for the length of the
side facing the right angle within the triangle. This went totally
against their philosophy that all such relations, and everything in
nature could be expressed in whole numbers.
This
hurdle was crossed using the theory of proportionality, which Eudoxes
(408-355 BCE) had developed, and which is described by Euclid in Book V.
This theory did not depend on measurements. It is only thereafter in
Book VI that we find the second proof of the same theorem. My purpose in
mentioning all these details is to remove the confusion, which often
arises when the theorem of Pythagoras is discussed.
It
could be said in conclusion that Euclid confirmed, by a different line
of thinking, what others had also discovered earlier.
The
Silver Croton Crown
These unique coins
were Pythagoras’s physical symbolisms of the ideas of the inter
dependence on the interaction of contraries or pairs of opposites –
good/evil, right/wrong, inward/outward…etc he had been taught in
Babylonia. Born in the island of Samos, Greece, during her Dark Ages he
had not learnt to write. There are no documents known directly
attributed to his writing. It was the Pythagoreans after his death who
wrote whatever he would have taught and even elaborated on it.
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The
Croton Crown from my collection |
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Obverse |
Reverse
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The symbolism of
opposites is in the incuse/excuse molding of the coin metal.
Note that the front (Obverse side) of
my coin shows the raised portions (‘excuse’- outward molding of
the coin metal, akin to a rock bas relief) of a tripod holding a Fire
container at its top. Note that the feet of the three legs of the
tripod bear the' lion’s paw’ character found in the bas relief of the
Achaemenian thrones at Persipolis. Similar ‘lion’s paw’ bases are seen
on the reverse side of some Sassanian Coins depicting the heavy
Achaemenian type Fire altars. The back (Reverse side) of the
coin shows the opposite ‘incuse’ or ‘cut in’ depressed portions
in the metal of the coin, exactly opposite to and copying the raised
portions on the Obverse side of the coin. The Obverse and Reverse of
the coins, thus, form a perfect match (as one would notice on an
embossing of paper or tin, in modern times). Modern numismatists call
the coin a ‘Croton Crown’, since it was minted in ancient Croton (now
the port town of Crotone in Southern Italy), where he had established a
school. In those days the territory of Croton extended across the narrow
peninsula from sea to sea, and we note that some of its early incuse
coins are struck in the joint names of Croton and some neighboring town,
e. g. (Sybaris), (Temesa) and Pandosia.
The
equivalent Roman replica coins (see below) called ‘Staters’ were struck
during the reign of four Roman Emperors,
who were vigorously
antagonistic to Sassanian power. This same theme appeared on coins
struck for Septimus Severus (193-211 CE), Julia Mamaea
(231-235 CE), Trajan Decius and his wife, Herennia Etruscilla
(249-251 CE). The Severans (nine Roman Emperors ruling from 193 to 235
CE - from Septimius Severus 193-211 CE to Severus Alexander 222-235 CE)
relied even more heavily on astrologers than the average person. On
Trajan’s Stater the Reverse side has Pythagoras touching a globe with a
wand. He had been unable to win any battle against the Persians and the
assumption is that this caused him to honor Pythagoras as no one had
done before. Since the globe appears without a Roman deity such as Sol
or Jupiter or the ruling emperor, one can assume that the globe
Pythagoras is shown as divining represents our planet.
A
brief historical overview of the time gives a fair idea
It is
important to note that the Romans won battles during their conquest of
most of Europe but failed to win any battle against the Persians.
They were ousted twice from Asia by the conquest of their Eastern
Capital of Antioch by Khusru I, the Great (Anoush e Ravan e Adil), who
then forced them into costly treaties with a heavy tribute payable
annually in gold. In shear frustration, they had to move their Capital
further north to Constantinople. It is no wonder they struck coins to
commemorate the Pythagorean claims of the assumed originality of the
ideas.
Severus Allexander
had occupied the Parthian territory of Mesopotamia from the warring
Parthian brothers, Vologases VI (Valkash), who ruled from the Capital,
Ekbatana and his younger brother, Artabanus IV (Ardavan), who made his
Capital in exile in Ctesiphon. Ardeshir I (Parthian Governor in Pars)
took advantage of this split between the brothers. In 224 CE he
advanced towards Ctesiphon, killed Artabanus IV in the Battle of
Ctesiphon and occupied Ctesiphon. During the same year 224 CE, he
defeated and captured Vologases VI in the Battle of Hormus, thus
becoming the Overlord, the King of Kings of all Parthia. Vologases died
in prison in 226 CE, a date incorrectly attributed by some historians to
the beginning of the Sassanian dynasty.
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Roman Silver Staters |
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Obverse |
Reverse
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Ardeshir I then
turned his attention to the Roman occupation. He sent envoys to the
Roman Emperor, Severus Alexander (222-235 CE) with a message to
tell Severus to “vacate the lands of my Hakhamani ancestors”, which he
regarded as rightly his own, “by right of inheritance.” Severus replied
by treating the envoys as captives. Ardeshir, at once crossed the
Euphrates River with heavily reinforced divisions. Severus totally
misjudged Ardeshir’s determination. He deployed his army into three
detached divisions. First, Ardeshir annihilated the south division
aimed at Susa, Capital of Elam. He, then, routed the middle division
commanded by Severus himself. Severus’s third division tried to enter
via Armenia but Ardeshir crushed their advances. Alarmed by these
defeats and by further advance of Ardeshir Severus he pleaded for truce
in CE 232 at a costly price – the annexation of entire Mesopotamia to
the Sassanian Empire and an additional heavy annual payment of a tribute
from the Roman Capital, Antioch. Julia Mamaea was Severus’s
mother and co-ruler (231-235 CE) in Rome. She was a very dominant
mother to the point of choosing a wife of her fancy for her son. She
interfered constantly with the administration and the Senate during her
4 years as co-ruler. Both mother and son were assassinated in 235 CE.
When Trajan Decius (249-251 CE) tried to instigate a revolt in
Armenia, a Sassanian Province to secede from Sassanian Airan the revolt
was put down with a heavy hand by Shahpur I (240-271 CE). He quelled the
revolt, defeated the Roman army, drove them out of Armenia and appointed
his Army Commander as Ruler with reinforcements to quell any further
unrest.
References:
1) Vedic Mathematics, (Reprint), Jagadguru Swami Sri Bharati Krsna
Tirthaji Maharaja. Delhi, Motilal Banarsidass, 2004
2) Article by J. J. O’Connor and E F Robertson - Pythagoras’s Theorem in
Babylonian Mathematics.
3) H. Steinhaus, Mathematical Snapshots, 3rd American Edition, Oxford
University Press, New York, 1983.
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