Category Archives: Ancient geometry

Magnificence 3, 6, 9,

The 3-6-9 and 6-3-9 Cycle in the Vedic Square can be thought of as Clockwise and Counter-Clockwise, or as Electricity and Magnetism. They are transport-channels, Nikola Tesla used ancient Sanskrit terminology in his descriptions of natural phenomena. As early as 1891 Tesla described the universe as a kinetic system filled with energy which could be harnessed at any location. His concepts during the following years were greatly influenced by the teachings of Swami Vivekananda. Swami Vivekananda was the first of a succession of eastern yogi’s who brought Vedic philosophy and religion to the west. After meeting the Swami and after continued study of the Eastern view of the mechanisms driving the material world, Tesla began using the Sanskrit words Akasha, Prana, and the concept of a luminiferous ether to describe the source, existence and construction of matter.

If you only knew the magnificence of the 3, 6 and 9, then you would have a key to the universe.

(3,6,9) is a Triangular Cycle that repeats itself until Infinity.

One of the most important Digital Root Patterns is the Vedic Square. It is the Digital Root of the Multiplication Table of the numbers 1 to 9.

The Second Part is about the Vedic Square. It is called the Vedic Square because this Square is one of the most important tools in Ancient Vedic Mathematics.


Vedic Mathematics was used in many Ancient Cultures (China, Egypt, Greece) with different names. The Chinese art of Feng Shui was called Vaastu Shastra in India.

Pythagoras, trained in Egypt (Heliopolis), used the same principles and used the same Patterns the Ancient Vedic Scientists were using.

The patterns of the Vedic Square Rotate. The End of a Horizontal and a Vertical Pattern connects with the Beginning of the Pattern. This means that the Vedic Square is a Torus.

This Torus is called the Rodin Torus. The Rodin Torus is a Coil that produces a Uniform Electro-Magnetic Field.

The last part is about the Game of Chess. This game is just like many other Ancient Games a Simulator of the Game of the Universe

This Table contains the Harmonics of the Numbers 1 to 9. These Harmonics are highly related to the Harmonic Pattern behind the Cycles in our Universe The Vedic Square was used to build the Pyramids,

Further Reading –

General –

india In

Consideration; –   Anyway, to me, it does not appear surprising that when one tries to put mathematical ideas of modern physics in human languages, they look similar to the philosophical ideas of ancient Rishis obtained after deep meditations. In fact it would be shocking if they did not agree. This would be true if e.g. the world was described by classical physics. I think the nature of reality is such that both parties were forced to adopt these ideas. As for Rishis, it is not clear when that classical to quantum transition took place or indeed if there was a sharp transition like modern physics.

In some parts of Vedas and many Upanishads, we see clearly concept of abstract, omnipresent, invisible, eternal, transcendent and immanent Brahman who has qualities unfamiliar in our everyday life. Just as many things in everyday world are described by classical physics, [ ] There is nothing wrong with that. Concept of Brahman would correspond to quantum concept.

”There is a story about two yogis. One meditated all day. The other one read scriptures all day. At the end of every day the second one always said “I do not understand. I do not understand”. Then one day, to the great surprise of the first one, the second one said loudly, “I understand. I understand”. The first yogi asked him with astonishment as to what suddenly happened. “Now you understand everything suddenly!” The second yogi replied “I now understand that this cannot be understood!”

Similarly, in physics, one sees only the material objects around and does not see strange quantum fuzzy world underlying all the matter. Moreover, the particles of modern physics are believed to be in some kind of suspended state devoid of any specific properties until they are measured.

Ishopanishad: “It moves and it moves not; it is far and it is near; it is within all this and it is also outside all this.”

Quote;  ”There are amazing similarities between implications of theories of modern physics and ancient Hindu philosophy as expressed in Vedas and Upanishads. The basis of Hindu philosophy is the mystical idea of Brahman. The Brahman is usually described by the words “Neti, Neti “meaning, not this, not this! When I went into study of physics, I realized that, as far as knowledge of ultimate reality is concerned, physicists are in exactly same situation as the ancient Rishis. Both cannot describe it in everyday language. In Mundak Upanishad, knowledge is divided into two parts: Para Vidya which deals with the eternal truth that can lead to self realization and Apara Vidya which deals with knowledge about material world. Under this classification, Physics might come under Apara Vidya. But, I would like to convince you that Modern Physics is also Para Vidya!”  – 

 – KashyapVasavada

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Interconnecting Geometry of ALL…

Were these such sites around the world ”interconnected” through much precise geometry to leave a testimony or statement of an understanding of our place and true nature,? one that is only now being uncovered with the growing awareness of quantum science and the interconnecting geometry of ALL things, the quantum world is the invisible yet indivisible tapestry that provides all information for life including all seemingly ”solid” matter.

You, and everything else, is, 99.999% empty space,

These sites are in astounding accurate alignment with each other, a worldwide grid that aligns with star constellations the cosmos, a seeming puzzle, suggestive of the holographic nature of all things we perceive, us included, ”as above so below” the larger, layered universe above mimicked also in the atomic and sub atomic realm, is ”consciousness the master key” ?

Was ancient knowledge-history subverted or ‘undermined’ from consciousness as a means of control over the human belief system?, were the immense liberties of a rich and powerful worldwide knowledge catastrophically reduced to the chains of a global political-economic-religious consciousness? human concepts – that solicited control through ”dependency” of mind, to be strictly enforced, the very essence of  ”slavery”

For example, the systematic annihilation  of cultures, records, documents, books, practiced during most wars and coups, presently and historically, whereas the acts of violent slaughter are rarely erased from living memory,  rather, these psychological;y traumatic events were/are meticulously ”inculcated lest we forget”, methodically reenacted in classrooms and cultures of both ancient and modern world history, is it to inform, warn, remind, imprint, or to beguile impressionable young minds?, make no mistake here, societal indoctrination of the young is an age old practice, for every young mind and peer represents a cultural blank canvas – of  political opportunity.

It is incredulous, a matter of great shame in purpose, that those of a position to effect great moral change, historically ”do not”  ”innocents”  minds are indoctrinated and rationalized so to justify sacrificing of life, inclusive of present day subterfuge, as well as that sustained in historical violence on and off the battlefields, and yet may again, by force of law be required to ‘once more’ engage, as and when called upon to do so, to entrust to the ”dogs of war” and to hell with the carnage  – for to serve the contrivances of nationalistic propaganda’s with it’s hollow ”cry’s of freedom”  – peace is conveniently despised…

”Methods of mass mind control have fundamentally never changed”.

 Truth in peace with the once universal knowledge of  perpetual life, was replaced with denial, pain and death and the sinner complex, this was the conspired lie imposed on mankind, this insatiable desire to chain and ‘dominate’ the human mental domain and forego the spirit or self, is a selfish and murderous assail on our true ”being” 

Evidence is still gathered in our present paradigm as to what is life – us.  One thing is becoming clear, monoliths mathematically-geometrically placed in a worldwide formation make a ”huge” statement in itself, the employed accuracies of such ”markers” are awe inspiring. Disconnection of  present understanding or ignorance certainly does not excuse their existence, though quantum science is now pointing toward ancient insights, such measures of technological skills exhibit a knowledge base not so easily dismissed, ”rediscovery” is actually more likely the outcome – it is emergent that the ancient world knew far more about the origin and connectivity of consciousness or life, and who we really are, than we do at this juncture in time…


In these holograms of reality, and the myths within, Tiwanaku is another ancient civilization whose timeline came and went, leaving behind megalithic monuments that go without explanation as to their mathematical design and construction. Considered by some the oldest city in the world, much of Tiwanaku’s creation defies the laws of physics and mathematics even by today’s standards. Many monuments bear close resemblance to those created by other ancient cultures all over the planet, truly an overlap if one were to place hologram over hologram, to define the journey of humanity in time.

As with many other sacred sites throughout the planet Tiwanaku remains an enigma allowing researchers to speculate on its origins and purpose then parallel that with other ancient civilizations left behind by unknown beings surviving in time with great stone markers which bear clues to humanity’s creation story. Gods, temples, idols, metaphors, all clues in a puzzle humanity is unraveling at this time of conscious awakening




The ancient monument at Nabta Playa, in the desolate Sahara desert of southwestern Egypt, has been called one of the earliest aligned structures of mankind’s ancient past.  Consensus opinion currently indicates that usage of the site may have begun around 4500 BC (based on carbon dating analysis described in this 2007 article), although some analysts have provided arguments which suggest that the site may have been designed even earlier than that.

napta pl



July 10, 2017 – Easter Island not victim of ‘ecocide,’ analysis of remains shows –

Analysis of remains found on Rapa Nui, Chile (Easter Island) provides evidence contrary to the widely-held belief that the ancient civilization recklessly destroyed its environment –

“The traditional story is that over time the people of Rapa Nui used up their resources and started to run out of food,” said Binghamton University Professor of Anthropology Carl Lipo. “One of the resources that they supposedly used up was trees that were growing on the island. Those trees provided canoes and, as a result of the lack of canoes, they could no longer fish. So they started to rely more and more on land food. As they relied on land food, productivity went down because of soil erosion, which led to crop failures…Painting the picture of this sort of catastrophe. That’s the traditional narrative.”

Lipo and a team of researchers analyzed human, faunal and botanical remains from the archaeological sites Anakena and Ahu Tepeu on Rapa Nui, dating from c. 1400 AD to the historic period, and modern reference material. The team used bulk carbon and nitrogen isotope analyses and amino acid compound specific isotope analyses of collagen isolated from prehistoric human and faunal bone, to assess the use of marine versus terrestrial resources and to investigate the underlying baseline values. Similar isotope analyses of archaeological and modern botanical and marine samples were used to characterize the local environment. Results of carbon and nitrogen analyses independently show that around half the protein in diets from the humans measured came from marine sources; markedly higher than previous estimates. These findings point to concerted efforts to manipulate agricultural soils, and suggest the prehistoric Rapa Nui population had extensive knowledge of how to overcome poor soil fertility, improve environmental conditions, and create a sustainable food supply. These activities demonstrate considerable adaptation and resilience to environmental challenges — a finding that is inconsistent with an ‘ecocide’ narrative.

“We found that there’s a fairly significant marine diet, over time, throughout history and that people were eating marine resources, and it wasn’t as though they only had food from terrestrial resources,” said Lipo. “We also learned that what they did get from terrestrial resources came from very modified soils, that they were enriching the soils in order to grow the crops. That supports the argument we’ve made in our previous work, that these people came up with am ingenious strategy in enriching the soils by adding bedrock to the surface and inside the soil to crate, essentially, fertilizer to support their populations, and that forest loss really isn’t a catastrophe as previously described.”

Lipo said that these new findings continue to support the idea that the story of Easter Island is more interesting and complex than assumed.  – “The Rapa Nui people were, not surprisingly, smart about how they used their resources,” he said. “And all the misunderstanding comes from our preconceptions about what subsistence should look like, basically European farmers thinking, ‘Well, what should a farm look like?’ And it didn’t look like what they thought, so they assumed something bad had happened, when in fact it was a perfectly smart thing to do. It continues to support the new narrative that we’ve been finding for the past ten years.” –

Story Source: Materials provided by Binghamton University, State University of New York


 Ahu Akivi is a sanctuary and celestial observatory built about 1500 AD which was the subject of the first serious restoration accomplished on Easter Island by archaeologists William Mulloy and Gonzalo Figueroa, with excellent results. As in the case of many religious structures on Easter Island, it has been situated with astronomical precision – it’s seven statues look towards the point where the sun sets during the equinox.

Easter Island Eclipse New Moon Total Solar Eclipse July 11, 2010

Aligned with the moon

Rongorongo is the hieroglyphic script of Easter Island. It has remained a mystery since its discovery. For over a hundred years, controversy has raged over the meaning and source of these enigmatic characters.



Chapter4  Hidden Stonehenge – Gordon R Freeman – An enormous amount of energy and ingenuity was spent on the construction of Stonehenge and the associated Barrows.


Excellent Source:

Spring Equinox 2017:

Quote: ”In reality, the realm of the invisible and infinite is not “somewhere else,” in some far-removed “heaven,” but present and available at every point in this seemingly-material life: the material realm in fact proceeds from and is sustained by it everywhere and at every moment”.

And, we are working with our spiritual self at every moment even as we go through this incarnate life. As Alvin Boyd Kuhn declares in a passage from the same chapter which was quoted at greater length in the preceding post:

The ancient Messiah was a representative figure coming from age to age, cycle to cycle. He came “each day” in the Ritual; he came periodically; he came “regularly and continuously.” He came once through the cycle; but his solar and lunar natural types came cyclically and in eternal renewal. The Egyptian Messiah was one whose historical coming was not expected at any date, at any epoch. The type of his coming was manifest in some phenomenon repeated as often as the day, the year, or the lunation came around. The constant repetition of type was the assurance of its unfailing fulfillment. [ . . . ] The coming was taking place in the life of every man at all times. 546 (Italics as in the original).

David Mathisen –



Russian Megalithic


megalithic ruins at Baalbek, Lebanon…



The longitudinal circumference passing through Giza crosses the most landmass above sea level of any longitude, as does the site’s latitudinal circumference from 30 ° N to 30° S. These unique site specifications for the Orion pyramids would last for millennia before the inevitable changes due to catastrophic tectonic subduction events that occur repeatedly, perhaps in concert with the cycles of glaciation and magnetic polar reversal. Giza is 16.7% or about 1/6 of the Earth’s circumference from the N pole along this ancient prime meridian.

The work of Carl Munck provides astonishingly detailed numerical analyses of the geometry expressed in the forms of the largest ancient limestone and granite architecture, revealing a geometric design code for specifying the exact global position in relation to Giza. Once Munck made the 31°prime meridian adjustment his efforts in decoding the numbers of presented by pyramid faces and other features were successful when using the same set of simple rules.The great complexity of the numerical codes contained in the specified geometries of ancient sites necessitates a thorough review of Munck’s convincing demonstration that all ancient sites are oriented in relation to the Orion pyramids of Giza

The golden ratio of phi is precisely expressed in the global circumference distances between Giza, Angkor and Nazca. Because of this perfect ratio any three consecutive numbers of the Fibonacci series can be used to precisely describe their proportionate distances, the unit of measure itself defined by the specific Fibonacci triplet chosen. The Fibonacci triplet beginning with prime Fibonacci #137 accurately expresses the Sine Wave circumference distances as a percent of the whole circumference.

The Fibonacci triplet beginning with the next prime Fibonacci number, #359, accurately expresses the Sine Wave circumference distances in miles. This profound underlying relationship informs the application of the Mandelbrot set, which has been suspected by many as being a fundamental construct of the universe. The high degree of accuracy of these alignments forces the conclusion that ancient Sanskrit society’s capacity for aerial surveillance far exceeds that achieved today.

The Sine Wave alignment is a cross- section through the sphere of the Earth along which sacred ancient sites conform to the Fibonacci – ordered complex plane of the Mandelbrot set [ zn+1 = zn2+c ] (4.4). This circular network of phi – proportioned sites can also be visualized as a cross- section through the Magnetic Resonance fractal ( 4.5). Global positioning data for each site is provided in the final section of this chapter. The abbreviated name of the site is given with the percent of the circumference distance from Giza (An = Angkor, at 19.1% from Giza). Vibratory energy from Giza is simultaneously transduced at each resonant site after the 1.45 Hz oscillations of the Great Pyramid encircle the globe, taking 4 hours to return to their source. Piezoelectric stone structures worldwide transduce a broad – spectrum of vibratory energy into a magnetic field by virtue of the angular geometries specifically designed to focus infrasound.


The Sine Wave of ancient sites is presented as a circle in red, overlaid with a cross- section of the Magnetic Resonance structure in white. The sacred sites conforming to this planar alignment display Fibonacci- ordered distances from Giza that are provided in percent of the circle’s circumference of 24,892 miles

Yonaguni #48 is a gigantic geometrically carved platform over 450’ in length with megalithic arches and a cave opening that are hewn from a single massive outcropping of the Yaeyama mudstone formation off Iseki Pt. These marine sites are all less than 100’ of depth and must have been constructed over 9,000 years ago, when more of the Earth’s water was trapped in the polar glacial icecaps lowering sea levels. The design layout of the Orion’s belt constellation is observed at both the Xi’an pyramids #47 and the Borobudur limestone Buddhist temples #51 – 57. Several of Java’s Buddhist sites have clearly been designed using the same underlying geometry evident in the stepped pyramids of Mesoamerica, with a steep central ascending stairway.

Temple sites mapped in the eastern hemisphere represent the foundation of many contemporary religions including Islam, Hinduism and both Theravada and Mahayana traditions of Buddhism. Hindu sacred temples are spread throughout India and described here only by a few prime sites including Mahabalipuram #34 , where temples on the present shoreline sit adjacent to those now submerged off the coast and recently discovered by G. Hancock.

A spherical rendering presents the infrasound standing wave resonance pattern in the western hemisphere, where the resulting tectonic shifting generates earthquakes along ‘the Ring of Fire” ( 4.17 ). The relative geopositions of ancient sites reveal that the sacred mandala symbol encodes the unified structure of the cosmos by which the pyramids focus the ambient energy of planetary resonance. While the infrasound oscillations are most distinctly seen by infrared observations, the resonance also becomes visible to Doppler radar due to the suspension of wat er vapor in the multi – layered standing waves. A composite image of stationary vapor cover over the United States from January 23, 2003 ( 4.18 ) provides a clear view of the fixed network of infrasound focal points. The correlation between the IR measurements and vapor reflectivity demands a thorough comparison with ancient Native American sacred sites beyond the few earthworks and mounds mentioned here. A detail image of the Magnetic Resonance mapping of Mesoamerica (4.19 , left ) shows synchronicity with the sacred maps of the Mayan Elders presently involved in the activation of their pyramid sites beginning at Calakmul Pyramid

Sacred Sites and Magnetic Portals

Ancient Mysteries traditions around the world share one peculiar aspect: they maintain how certain places on the face of the Earth possess a higher concentration of power than others. These sites, named “spots of the fawn” by the Hopi, eventually became the foundation for many sacred sites and temple structures we see today. What is interesting is that each culture maintains that these special places are connected with the heavens by a hollow tube or reed, and by this umbilical connection the soul is capable of engaging with the Otherworld during ritual. However, it also allows a conduit for the spirit world to enter this physical domain

Mereux’s study of Carnac shows how its dolmens amplify and release telluric energy throughout the day, with the strongest readings occurring at dawn. The voltage and magnetic variations are related, and follow a phenomenon known as electric induction . According to Mereux, “The dolmen behaves as a coil or solenoid, in which currents are induced, provoked by the variations, weaker or stronger, of the surrounding magnetic field. But these phenomena are not produced with any intensity unless the dolmen is constructed with crystalline rocks rich in quartz, such as granite.”

His readings of menhirs reveal an energy that pulsates at regular intervals at the base, positively-and negatively-charged, up to thirty-six feet from these upright monoliths, some of which still show carvings of serpents. Extreme pulsations recycle approximately every 70 minutes, showing that the menhirs charge and discharge regularly.

Mereux also noticed how the voltage of standing stones in the Grand Ménec alignment diminished the farther away they lay from the stone circle, which itself behaved as a kind of condenser or concentrator of energy. – Freddy Silva


The rainforests of the Amazon are slowly revealing their ancient civilizations; M. Heckenbecker has documented the solar and lunar alignments of complex ancient set tlements of indigenous peoples of the Xingu River #61 and its tributaries including the Cuelene River #64 . This region is also well known for its advanced indigenous traditions of trance entrainment using psychoacoustic whistles in all forms and materials, including stone, bone, wood, shell and ceramic. The most complex psychoacoustic instruments developed were the tri – harmonic stone flutes of the Bolivian Andes discussed earlier, and the most widespread cultural practices of psychoacoustic entrainment observed of any region in the world.

One submerged limestone architectural complex is also plotted in resonant Fibonacci alignment: the sand- covered pyramids of the Guanahacabibes Gulf #71 at a depth of over 2,000’ off the west coast of Cuba.The dark oxidize d limestone blocks confirm that the stepped pyramid structures were built when the area was above sea level, having been catastrophically lowered by massive subduction events likely linked with the global catastrophe that abruptly ended the Pleistocene gla cial period and began the global warming trend defining the Holocene. Remote sensing and mapping of the submerged Cuban pyramid complex will provide details of the cosmic alignments of its buildings in the years to come, yet its placement within the structure of Magnetic Resonance can be determined by its geoposition

These megalithic structures stand among only a few others like Japan’s Yonaguni marine site that have been so completely preserved by their inundation as to have captured the very moment of devastation, to be archived along the continental shelf, silted upon for millennia. Sites of subterranean discoveries are also included in this analysis, namely at Sutatausa, Colombia and at La Maná and Cuenca in Ecuador, though of vast number of temple sites include uncharted subterranean chambers.

Another category of sacred sites built within the Magnetic Resonance alignment are the earthen mound formations of North America that also display resonant properties as clay is pi ezoelectric, like limestone and quartz. The locations of circular kivas and the adobe cliff dwellings of the American southwest are also presented. The largest rock formation symbolically designed as a sacred site by ancient people is the limestone Nazca Desert floor #98 . Lines were dug into the dark bedrock to expose white underlying limestone, forming paths likely used in processional group meditation. The various geoglyphs mark the prime locality where infrasound heart beat entrainment frequencies are f ocused and become amplified by the resonant calcite formations. Indeed, the very rock substrate of the area was apparently used as a psychoacoustic transducer of ULF signals from Giza, as symbolized by the sacred animals depicted including the spider, monk ey, killer whale and various birds, even one with a zigzag neck

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stone gaps

Babylonians Tracked Jupiter With Advanced Tools

Swirling gas clouds on Jupiter


Ancient tablets describe math that was thought to have been invented over 1,000 years later, rewriting the history books.

The Babylonians’ techniques outshine those used by contemporary Greek and Egyptian astronomers—and shockingly mirror the mean speed theorem, a mathematical description of motion developed by a 14th-century English group known as the Oxford Calculators.

And the findings probably represent the tip of a mathematical iceberg. “There are thousands of tablets in various museums that were never translated,” says Ossendrijver, “and often we can translate a tablet but don’t understand what is going on until much later.”Researchers have long known that the Babylonians, who lived in what is now Iraq, had considerable mathematical skill: They successfully approximated the square root of 2 and understood the Pythagorean theorem nearly 4,000 years ago—more than a millennium before Pythagoras was born.

They were also talented astronomers, maintaining nightly catalogs detailed enough to record the passage of Halley’s comet. Babylonians regularly used arithmetic to boost their astronomical predictions.

jupiter res

But no one had ever found a Babylonian astronomical calculation that leveraged their impressive knowledge of pure geometry, until Mathieu Ossendrijver of Germany’s Humboldt University of Berlin spent 13 years deciphering what he described as a “small bunch of four weird trapezoid computations” between 2,000 and 2,400 years old.

Ossendrijver was the first to notice that the tablets—stored in the British Museum since the 1880s—had something to do with the planet Jupiter. However, they didn’t make much sense without knowing how the Babylonians encoded aspects of Jupiter’s motion, such as its appearances on the horizon.

Such a find also speaks to the human spirit of discovery—both of the ancient astronomers who gazed at the heavens, and the modern researchers who seek to reconstruct their understanding of the cosmos.


Babylonian Science

Like in Egypt, priests encouraged much of the development of Babylonian science. Babylonians used a numeral system with 60 as its base, which allowed them to divide circles into 360 degrees. The use of 60 as a base of a mathematical system is not a minor issue: 60 is a number that has many divisors (1, 2, 3, 4, 5, 6, 10, 12, 15, 20, 30, 60), which simplifies the representation of fractions: 1/2 (30/60), 1/3 (20/60), 1/4 (15/60), 1/5 (12/60), 1/6 (10/60), and so forth. As early as 1800 BCE, Babylonian mathematicians understood the properties of elementary sequences, such as arithmetic and geometrical progressions, and a number of geometrical relationships. They estimated the value of pi as 3 1/8, which is about a 0.6 percent error. It is highly likely that they also were familiar with what we today call the Pythagorean Theorem which states that the square of the longest side of a right triangle equals the sum of the squares of the other two sides. However, we have no evidence that the Babylonians proved it formally, since their mathematics rested on empirical knowledge rather than formal proof.

It was in astronomy where Babylonians showed a remarkable talent, and magic, mysticism, astrology, and divination were its main drivers. They believed that the movement of the heavenly bodies forecasted some terrestrial event. Since the reign of Nabonassar (747 BCE), the Babylonians kept complete lists of eclipses and by 700 BCE, it was already known that solar eclipses could only be possible during new moons and lunar eclipses only during full moons. It is possible that by this time Babylonians also knew the rule that lunar eclipses take place every six months, or occasionally every five months. By the time Nebuchadnezzar ruled Babylon, the priests had also calculated the courses of the planets and plotted the orbits of the sun and the moon. –  Cristian Violatti


Examining all of the tablets at the British Museum, Ossendrijver figured out that the trapezoid calculations were a tool for calculating Jupiter’s displacement each day along the ecliptic, the path that the sun appears to trace through the stars. The computations recorded on the tablets covered a period of 60 days, beginning on a day when the giant planet first appeared in the night sky just before dawn.


During that interval, Jupiter’s motion across the sky appears to slow. (Such erratic apparent motion stems from the complex combination of Earth’s own orbit around the sun with that of Jupiter.) A graph of Jupiter’s apparent velocity against time slopes downward, so that the area under the curve forms a trapezoid. The area of the trapezoid in turn gives the distance that Jupiter has moved along the ecliptic during the 
60 days. Calculating the area under a curve to determine a numerical value is a basic operation, known as the integral between two points, in calculus. Discovering that the Babylonians understood this “was the real ‘aha!’ moment,” Ossendrijver says.

Although elated, Ossendrijver wasn’t ready to publish, because a second part of the trapezoid prescription remained unclear. By delving into older, purely mathematical Babylonian texts written between 1800 B.C.E. and 1600 B.C.E., which also described computations with a trapezoid, he realized that the astronomers who made the tablets had gone a step further. To compute the time at which Jupiter would have moved halfway along its ecliptic path, the astronomers divided the 60-day trapezoid into two smaller ones of equal area. The vertical line dividing the two trapezoids marked the halfway time; because of the different shapes of the trapezoids, it indicated not 30 days but slightly fewer.

The Babylonians had developed “abstract mathematical, geometrical ideas about the connection between motion, position and time that are so common to any modern physicist or mathematician,” Ossendrijver says.

Indeed, compared with the complex geometry embraced by the ancient Greeks a few centuries later, with its cycles and epicycles, the inscriptions reflect “a more abstract and profound conception of a geometrical object in which one dimension represents time,” says historian Alexander Jones of New York University in New York City. “Such concepts have not been found earlier than in 14th century European texts on moving bodies,” he adds. “Their presence … testifies to the revolutionary brilliance of the unknown Mesopotamian scholars who constructed Babylonian mathematical astronomy.”