Ancient Metrology – Numbers Don’t Lie

A. Sokolowski

There is an underlying order in Cosmos.  Our ancestors discovered it in ancient times and expressed it in their writings and monuments.  This article is an invitation to all our visitor to explore and expand this subject. Please share facts that should be mentioned here (via comments and/or e-mail) –  if you are aware of them. For example, many people today do not understand that ancient units of measure reflect advanced knowledge of mathematics and astronomy…

1,000 x 360 x 365.24

The result of multiplying above numbers is: 131,486,400
Equatorial Circumference of the Earth: 40,077km= 131,486,400 feet
The official value for equatorial circumference of the Earth: 40,075 kilometers. It reveals connection of the year ( 365.24 days) and the foot (unit of measuring length) with circumference of the Earth.

That is truly a “cosmic” coincidence !!!

“Philosophy is written in this grand book – I mean the universe — which stands continually open to our gaze. But it cannot be understood unless one first learns to comprehend the language and interpret the characters in which it is written. It is written in the language of mathematics, and its characters are triangles, circles, and other geometric figures, without which it is humanly impossible to understand a single word of it.”     — Galileo Galilei, Il Saggiatore (1623)

In order to reveal the “language” of the universe, we will explore numbers known by the ancients and compare them with numbers established by modern astronomy, geodesy and metrology (the science of measurement – do not confuse it with meteorology dealing with weather). It is important to realize that units of measure we use today are connected with the size and movements of our planet.  The same applies to the ancient units of measure.


   Units of Measure and Earth |  Ancient Units of Length and Time
Astronomy and Calendars
Cosmic Harmony |  Laws Written in Stone

Units of Measure and Earth

Geometry (Ancient Greek:  geo- “earth“, -metron “measurement“) was originally dealing with measuring of the earth. In our article “How ancient astronomers could have establish dimensions of the Earth?” we presented a simple method of establishing circumference (and diameter) of the earth that (most likely) was used by the ancient astronomers.

Measurement means the act of measuring or the size of something.

To Measure means to ascertain the dimensions, capacity, or amount (quantity) of something.A unit of measurement is a definite magnitude of a physical quantity, defined and adopted by convention and/or by law, that is used as a standard for measurement of the same physical quantity. Any other value of the physical quantity can be expressed as a simple multiple of the unit of measurement.People have always found it necessary to measure time, distance, area, volume and weight, and have devised units that measure these quantities. For time, there is an absolute standard in the motions of the heavens, but for the other quantities the units have had to be chosen arbitrarily.Official view is that only recently have we succeeded in creating system of measurement accepted all over the world as the standard system for use in science and trade: The International System of Units (SI).  However some researchers suggest that in ancient times people were commonly using units of measure related to dimensions of our planet and similar in value ( closely related to each other.)

It is important to notice that units of measure we use today are connected with the size and movements of our planet.
The same applies to the ancient units of measure.

Units of Length

Originally, the meter was designed to be one ten-millionth (1/10,000,000) of a quadrant, the distance between the Equator and the North Pole. In other words, meter was defined as  1/10,000,000 of the distance from the Earth’s equator to the North Pole measured on the circumference through Paris. Using this unit, the circumference of perfectly round Earth should be exactly 40,000, 000 meters (or 40,000 km).
Today, official value of the Earth’s circumference along the line of longitude is 40,007.86 km.


Nautical Mile
A nautical mile ( 1.852 km ) is based on the circumference of Earth. If you divide circumference of the Earth into 360 degrees and then divide each degree into 60 minutes you will get 21,600 minutes of arc.
1 nautical mile is defined as 1 minute of arc (of the circumference of Earth) is  This unit of measurement is used by all nations for air and sea travel. Using 40,007.86 km as the official circumference of our planet we get value of the nautical mile in kilometers:  1.852 km   (40,007.86/21,600 )

The metric system

The metric system, originating in the French Revolution and propagated widely in the 19th century, has brought a dreary but convenient uniformity to units of measurement. A number of metric systems of units have evolved since the adoption of the original metric system in France in 1791. The current international standard metric system is the International System of Units (SI). An important feature of modern systems is standardization. Each unit has a universally recognized size. In the establishment of the metric system, the quadrant of the earth was measured and set equal to 10,000,000 meters.

The guiding ideas of the French scientists are well expressed in the introduction to the document presented to the Academy:

The idea to refer all measures to a unit of length taken from nature has appeared to the mathematicians since they learned the existence of such a unit as well as the possibility to establish it: they realized it was the only way to exclude any arbitrariness from the system of measures and to be sure to preserve it unchanged for ever, without any event, except a revolution in the world order, could cast some doubts in it; they felt that such a system did not belong to a single nation and no country could flatter itself by seeing it adopted by all the others.
Actually, if a unit of measure which has already been in use in a country were adopted, it would be difficult to explain to the others the reasons for this preference that were able to balance that spirit of repugnance, if not philosophical at least very natural, that peoples always feel towards an imitation looking like the admission of a sort of inferiority. As a consequence, there would be as many measures as nations.

The International System of Units (abbreviated SI from French), established in 1960, is the modern form of the metric system and is generally a system of units of measurement devised around seven base units and the convenience of the number ten.

The SI, is the world’s most widely used system of measurement, which is used both in everyday commerce and in science. The system has been nearly globally adopted with the United States being the only industrialized nation that does not mainly use the metric system in its commercial and standards activities. The United Kingdom has officially partially adopted metrication, with no intention of replacing customary measures entirely. Canada has adopted it for all legal purposes but imperial/US units are still in use, particularly in the buildings trade.

meter or metre (m) – the metric and SI base unit of distance.

Originally, the meter was designed to be one ten-millionth (1/10,000,000) of a quadrant, the distance between the Equator and the North Pole. In other words, meter was defined as  1/10,000,000 of the distance from the Earth’s equator to the North Pole measured on the circumference through Paris. (The Earth is difficult to measure, and a small error was made in correcting for the flattening caused by the Earth’s rotation. As a result, the meter is too short by a bit less than 0.02%. That’s not bad for a measurement made in the 1790’s.)

For practical reasons, for a long time, the meter was precisely defined as the length of an actual object, a bar kept at the International Bureau of Weights and Measures in Paris.
In recent years, however, the SI base units (with one exception) have been redefined in abstract terms so they can be reproduced to any desired level of accuracy in a well-equipped laboratory.

The 17th General Conference on Weights and Measures in 1983 defined the meter as that “distance that makes the speed of light in a vacuum equal to exactly 299, 792, 458 meters per second”. In other words,  ”The metre is the length of the path traveled by light in vacuum during a time interval of 1/299, 792, 458 of a second.” The speed of light in a vacuum, c, is one of the fundamental constants of nature. Since c defines the meter now, experiments made to measure the speed of light are now interpreted as measurements of the meter instead.

The meter is equal to approximately:
1.093 613 3 yards,
3.280 840 feet, or
39.370 079 inches.

Its name comes from the Latin metrum and the Greek metron, both meaning “measure.” The unit is spelled meter in the U.S. and metre in Britain; there are many other spellings in various languages

Ancient Units of Length and Time

Megalithic Yard

When the late Professor Alexander Thom surveyed over a thousand megalithic structures from Northern Scotland through England, Wales and Western France he was amazed to find that they had all been built using the same unit of measurement. Thom dubbed this unit a Megalithic Yard (MY = 2.72 feet = 0.829m) because it was very close in size to an imperial yard, being exactly 2 feet 8.64 inches (82.966 cm). As an engineer he could appreciate the fine accuracy inherent in the MY but he was mystified as to how such a primitive people could have consistently reproduced such a unit across a zone spanning several hundreds of miles. The answer that eluded the late Professor lay not in the rocks, but in the stars.The MY turns out to be much more than an abstract unit such as the modern meter, it is a highly scientific measure repeatedly constructed by empirical means. It is based upon observation of three fundamental factors:  the orbit of the Earth around the sun, the spin of the Earth on its axis,  and the (constant) mass of the Earth.
[ Source: The Mystery of the Megalithic Yard Revealed ]

Making the Megalithic Yard

In the discussions leading up to the French adoption of the metric system in 1791, the leading candidate for the definition of the new unit of length, the metre, was the seconds pendulum at 45° North latitude. It was advocated by a group led by French politician Talleyrand and mathematician Antoine Nicolas Caritat de Condorcet. This was one of the three final options considered by the French Academy of Sciences committee. However, on March 19, 1791 the committee instead chose to base the metre on the length of the meridian through Paris. A pendulum definition was rejected because of its variability at different locations, and because it defined length by a unit of time.



where T is period of  simple pendulum (time of one full swing), L is length of pendulum, and g is  local acceleration of gravity ( g = 9.81 m/s2 = 32.2 ft/s2).

From this equation, for
L= 1 meter, the pendulum period is T= 2.006 seconds.

For L= 1 MY = 0.83 m, the pendulum period T= 1.827 seconds
(interestingly 666/1.827 = 364.5 , and for L=1 Royal Egyptian Cubit, period T = 1.45 sec )

Full 200 swings of this pendulum would take 365.4 seconds ( 6.09 minutes or 6 min and 5.4 sec). This is exactly the time period during which the moon (or the sun – since both have the same apparent angular diameter = 0.5 deg) moves by 1.5 degree on the sky (it is equivalent to its 3 diameters).

Here is how to make unit of measure equal 1 megalithic yard ( 1 MY)

Note: this is modification (by A. Sokolowski) of a method originally proposed by Lomas, Knight & Butler.

The apparent movement of the sky (caused by rotation of our  planet on its axis) provides an excellent astronomical clock.
The earth is rotating 1 degree per 4 minutes of time
(360deg/(24×60)min = 0.25 deg/min).
Since apparent angular moon diameter (= angular sun diameter) = 0.5 degree.
On the day of  equinox (daytime = night time) the circle of the moon fits exactly 360 times on its path above the horizon (180 degrees).


If we observe the moon, or a bright star (perhaps  with help of simple astrolabe with plumb-line), we can easily measure exactly 6 minutes by waiting for the star (or the moon) to move 1.5 degree on its path (this is the time taken by the moon to move on the sky by its 3 diameters). Pendulum with length (L) equal 1 MY would make exactly 200 full swings in 6 minutes (just like a pendulum 1 m long would swing exactly 180 times in 6 minutes).  Alternatively we can observe shadow of an obelisk and measure 6 minutes (using template of a half circle divided into 180 degrees).

When you see the upper edge of the moon  (or a first magnitude star) approaching the first marker on the astrolabe, swing the weight of a pendulum and count the pulses from one extreme and back (full swings) for 6 minutesuntil the moon/ star moves 1.5 degree (for the moon it would be its 3 diameters). There are only two factors that effect the swing of a pendulum; the length of the string and gravity – which is determined by the mass of the earth ( if you swing a pendulum faster it will move outwards further but it will not change the number of pulses.)

You need to adjust the length until you get exactly 200 swings during this period of 6 minutes. It is likely to take you several attempts to get the length right so be prepared to do quite a bit of moon/star gazing. To improve accuracy, you can increase the time (and accordingly number of pendulum’s swings).

It is likely that in ancient times, astronomers would keep the rod with 1MY length (once it was established this way) and use the described method for calibration.

One you have the correct length of pendulum, mark the distance from the  exact point of suspension to the centre of the weight (it would be best to use a spherical weight for the pendulum). Congratulations, you now have a stick that is exactly one Megalithic Yard long.

1 Megalithic Yard = 2.72 feet = 1 foot + 1RC = 0.82966m

1 m = 3.281 feet  |  1 foot = 12 inch = 30.48 cm  |  1 inch = 2.254 cm
1 Royal Egyptian Cubit = 20.62 inch = 0.531 m

1 mile = 1,609.344 meters = 1,941 MY = sqrt(44) MY
1 nautical mile = 1.15078 mile = 6,076 feet = 1.852 km = 2233 MY

Modern and accurate numbers describing the size of our planet:

Earth’s Circumference Between the North and South Poles:
21,602.6 nautical miles = 24,859.82 miles (40,008 km)

Earth Equatorial Circumference:
24,901.55 miles (40,075.16 kilometers)
Earth’s Diameter at the Equator:
6,887.7 nautical miles = 7,926.28 miles (12,756.1 km)

1 year = 365.25 days
1 day = 24 hours = 1,440 minutes = 86,400 seconds
24 hours = 360 degrees,
1 hour = 15 degrees,
1 minute (time) = 0.25 degree = 15 min of arc
1 degree = 4 minutes (time)
360 degrees = 21,600 minutes = 1,296,000 seconds (of arc)
1 degree = 60 min = 360 seconds (of arc)

Note:1/1,000th of a degree of arc around the equatorial circumference of the Earth is just 365.22 ft in length!

1 degree of arc = (1/360) * 40,075.16 km = 111.32 km = 365,223 feet (divided by 1000 = 365.22 which is number of days in a year)

Egyptian Cubit

The Egyptian cubit, the Indus Valley units of length referred to above and the Mesopotamian cubit were used in the 3rd millennium BC and are the earliest known units used by ancient peoples to measure length. The measures of length used in ancient India included the dhanus (bow), the krosa (cry, or cow-call) and the yojana (stage).

The Egyptian Royal Cubit rod,  from the Turin collection, has an official length of 20.618 inches. Its refined value, under the sexagesimal geodetic system, was calculated mathematically to be 20.61818182 inches.

Note: Royal Cubit consists of 28 units, digits, which is the same as 7 palms of 4 digits. The names of divisions of royal cubit may suggest anatomical origin, however the divisions indicate astronomical origin of the cubit (7 days per week, 28 days lunar calendar, 4 weeks per lunar month)…

Interesting Relationship between ancient units of lengths

From measurements of the King’s chamber and other dimensions in the Great Pyramid by John Greaves, Sir Isaac Newton realized that the King’s Chamber was 10 x 20 Royal Cubits (or Thoth Cubits) so that the Royal Cubit is determined as equal to 1.719 (1.72) feet.

Therefore, we can take the following as “ideal values” in metric system of 3 fundamental ancient units of length:

1MY = 1 foot + 1RC = 2.72 feet = 0.829m
1 Royal Cubit = 1.72 feet = 0.632 MY = 0.524m
1 Remen = Royal Cubit/sqrt(2) = 14.58 inch =  0.37 m

Notice the relationship between all 3 units  can be well approximated as follows:

1 Megalithic Yard = sqrt(5) x 1 Remen = 1 Royal Cubit x sqrt(5/2) = 1.5811388 RC

Let’s notice the relationship between all 3 units  can be well approximated as follows:

1 Megalithic Yard = sqrt(5) x 1 Remen = 1 Royal Cubit x sqrt(5/2) = 1.5811388 RC
1 Royal Cubit = sqrt(2) x 1 Remen = 0.525 m
1 Megalithic Yard = sqrt(5) x 1 Remen = 0.83 m

It may be seen that, from the basic square side of the Remen, the length of the Royal Cubit can be derived by multiplying the Remen by the square root of 2; similarly, the Megalithic yard can be derived by multiplying the Remen by the square root of 5.
Another geometric illustration of the relationship between Remen, Royal Cubit and Megalithic Yard, where1M.Y. is circumference of the circle (0.823m) inscribed in (1/2) R.C. square (with 99.3 % accuracy):

If Megalithic Yard was defined as equal to the circumference of the circle inscribed in ½ of Royal Cubit square:

1 MY=1/2 x RC x “Pi” so 1 MY = 1.570795 RC

1 MY (in RC) = Pi/2


For RC=1 we get and  1MY=1.570795   and 1 Remen = 1/sqrt(2)=0.7071

For the Great Pyramid:
Height = 280 Royal Cubits
Base Side = 440 Royal Cubits = 280 MY

GP_Blueprint_MY_RCClick to Enlarge

In whichever case, the Greeks and Romans inherited the foot from the Egyptians.

12 inches = 1 foot
36 inches = 3 feet =  1 yard
1760 yards = 1 mile

440 yards = quarter mile

Interest in ancient metrology was triggered by research into the various Megalith building cultures and the Great Pyramid of Giza.

In 1637 John Greaves, professor of geometry at Gresham College, made his first of several studies in Egypt and Italy, making numerous measurements of buildings and monuments, including the Great Pyramid. These activities fueled many centuries of interest in metrology of the ancient cultures by the likes of Isaac Newton and the French Academy.

The first known description and practical use of a physical pendulum is by Galileo Galilei, however, Flinders Petrie, a disciple of Charles Piazzi Smyth, is of the opinion that it was used earlier by the ancient Egyptians. Writing in an article in Nature, 1933 Petrie says:

If we take the natural standard of one day divided by 105, the pendulum would be 29.157 inches (0.7405878 m) at lat 30 degrees. Now this is exactly the basis of Egyptian land measures, most precisely known through the diagonal of that squared, being the Egyptian double cubit. The value for this cubit is 20.617 inches, while the best examples in stone are 20.620±0.005inches.

By the time measurements of Mesopotamia were discovered, by doing various exercises of mathematics on the definitions of the major ancient measurement systems, various people (Jean-Adolphe Decourdemanche in 1909, August Oxé in 1942) came to the conclusion that the relationship between them was well planned.

Livio C. Stecchini claims in his  A History of Measures:

The relation among the units of length can be explained by the ratio 15:16:17:18 among the four fundamental feet and cubits. Before I arrived at this discovery, Decourdemanche and Oxé discovered that the cubes of those units are related according to the conventional specific gravities of oil, water, wheat and barley.

Stecchini makes claims that imply that the Egyptian measures of length, originating from at least the 3rd millennium BC, were directly derived from the circumference of the earth with an amazing accuracy. According to “Secrets of the Great Pyramid” (p. 346), his claim is that the Egyptian measurement was equal to 40,075,000 meters, which compared to the International Spheroid of 40,076,596 meters gives an error of 0.004%. No consideration seems to be made to the question of, on purely technical and procedural grounds, how the early Egyptians, in defining their cubit, could have achieved a degree of accuracy that to our current knowledge can only be achieved with very sophisticated equipment and techniques.

Alexander Thom

Oxford engineering professor Alexander Thom, doing statistical analysis of survey data taken from over 250 stone circles in England and Scotland, came to the conclusion that there must have been a common unit of measure which he called a megalithic yard. This research was published in the Journal of the Royal Statistical Society (Series A (General), 1955, Vol 118 Part III p275-295) as a paper entitled A Statistical Examination of the Megalithic Sites in Britain.

As Professor Thom observed in his book Megalithic Sites in Britain (1967):

It is remarkable that one thousand years before the earliest mathematicians of classical Greece, people in these islands not only had a practical knowledge of geometry and were capable of setting out elaborate geometrical designs but could also set out ellipses based on the Pythagorean triangles.”

Robin Heath

Later, these ideas were further developed as defense for the Imperial units against the emerging metric system, and adopted by parts of the anti-metric movement. Robin Heath, in his book Sun, Moon & Stonehenge, connects the megalithic yard (and thus Stonehenge) to the imperial foot, and manages to connect a few astronomical phenomena, and the Egyptian Royal Cubit (and thus the Great Pyramid) into one grand equation (MY is an abbreviation for megalithic yard):

If the lunar year is represented by 12 MY then 1 ft corresponds precisely to the extra 10.875 days to coincide with the end of the solar or seasonal year. Furthermore, the period between the end of the solar year and 13 lunations – 18.656 days – is represented by another unit of length from antiquity, the ‘Royal Cubit’ of 20.63 inches or 1.72 ft.

Note: 1 solar year = 365.24 days,
1 lunar year = 12 lunar months = 354.36 days.

(1 lunar month = the time between full moons = 29.530589 days, therefore 1 lunar year = 12 * 29.53 = 354.36 and 365.24 – 354.36 = 10.88.
1 solar year = 1 lunar year (12 lunar months) + 10.88 days. 

If 1 lunar month is represented by 1 MY we get 29.53 days represented by 2.72 feet (or 10.8566 days are represented by 1 foot).
One lunar year (12 lunar month) = 354.36 days represented by 12 MY = 12* 2.72 feet= 32.64 feet

Also 13 lunar months = 13*29.53 = 383.89 days.
13 lunar months – 1 solar year = 383.89 – 365.24 = 18.65 days (representing 1.718 feet = 1 Royal Cubit).
18.65 /10.8566 = 1,718 feet = 1 Royal Cuibt

This seems to bring pseudo-scientific metrology to new heights, especially in view of the conclusion:

Hence the equally astonishing revelation that 1 MY = 1 ft + 1 RC. Assuming that the MY was the primary unit, then the derivative foot and cubit appear to have formed a logical and essential part of the astronomical and calendrical researches of our Neolithic ancestors. If, however, the foot preceded the MY in time – and here we must remember that 1/1,000th of a degree of arc around the equatorial circumference of the Earth is just 365.244 ft in length! – then knowledge of the roundness of the Earth must have predated use of the MY…i.e. well before 3,000BC. There are no other choices readily apparent!

J. Michell claims that all over the world traditional units of measurements are related.
He goes on to point out the value of the pu that still survives in Indo-China is given in L.D’A. Jackson’s Modern Metrology (available on the net) as 2.7272 miles with the fraction repeating. Without knowledge of the pu’s existence its former use in Britain was deduced by J. F. Neal, who called it the Megalithic Mile because the ratio is similar to that between the foot and the Megalithic Yard.

Since the ratio between the dimensions of the Earth and Moon is 10:2.7272 the following relationships unambiguously exist.

Earth’s diameter = 7,920 miles
Moon’s diameter = 792 megalithic miles
Perimeter of the square containing the circle of the Earth = 31,680 miles
Perimeter of the square containing the circle of the Moon = 3,168 megalithic miles.
Sun’s diameter = 864,000 miles = 316,800 megalithic miles.

More information:

Units of Time

Observing movement of the Sun and the stars suggested that Earth is spinning around its own axis and that the Sun is moving against the background of constellations. This suggested there are 2 cycles: earth axis spin cycle and earth around the sun orbital cycle.

In ancient times it was easy to observe the Sun in order to establish units of time so it is good assumption that such units would be “solar units of time”. We use them today and call them “solar days” or simply days (as opposed to stellar background based “sidereal days”). It was also easy to observe the moon and the stars at night. 

Hipparchus (190-120 BC) and his predecessors used various instruments for astronomical calculations and observations, such as the gnomon, the astrolabe, and the armillary sphere. Hipparchus is credited with the invention or improvement of several astronomical instruments, which were used for a long time for naked-eye observations. According to Synesius of Ptolemais (4th century) he made the first astrolabion: this may have been an armillary sphere (which Ptolemy however says he constructed, in Almagest V.1); or the predecessor of the planar instrument called astrolabe (also mentioned by Theon of Alexandria). With an astrolabe Hipparchus was the first to be able to measure the geographical latitude and time by observing stars. Previously this was done at daytime by measuring the shadow cast by a gnomon, or with the portable instrument known as a scaphe. [ — Wikipedia ]

Illustration below shows ancient Egyptians’ interest in astronomy.

The Northern [Bottom] and Southern [Top] Panel ‘Decan Chart’ from the Tomb of Senmut [c 1500 BC]. [In reality this panel is about 4 m long.] The decan system was actually used as a clock for time keeping in the night hours and through the year – modern Egyptologists call them Egyptian sidereal clocks.

After these cycles were noticed (discovered) the next step would be to quantify them (describe them using specific units of measurement).

Numbers like 360, 72, 30, 12 and multiples thereof were intentionally plotted in ancient myths. It was as if the storyteller were trying to convey a secret code. Here’s what the figures signify in the precession cycle:

  •  360 degrees = 12 x 30 degrees, or one full circuit through the zodiac constellations
  • 72 years = the time it takes for the stars to shift 1 degree
  •  30 degrees = one astrological age (a different zodiac constellation rises with the Sun every 2,160 years)
  •  12 = the total number of zodiac signs or astrological ages. 12 times 2,160 = 25,920 years, or one full precession cycle
  •  In Babylonia, the ancient scribe Berossus wrote that mythical kings ruled before the Great Flood for a total 432,000 years.
  • In India, the Rigvida contains exactly 432,000 syllables. And although the calculation has created some confusion of late, the Vedic Kali Yuga (representing the current world age) is said to be comprised of 432,000 years.
  • On the other side of the globe, Mayan calendar units parrot the precessional figures.
    For example: 1 tun (an astronomical year) = 360 days;
    6 tuns = 2,160 days;
    1 katun = 7200 days,
    6 katuns = 43,200.
    The standard Mayan base of 20 (ours is 10) is arrived at by dividing 43,200 by 2,160.

Today we use decimal system (multiples of 10) however when it comes to measuring angles, we use ancient convention: a circle is not divided into 100 (or 1000 parts), instead it  divided in 360 units (called degrees) and each unit is further divided into 60 equal pats called minutes and each one of these is further divided into 60 equal units called seconds.
A full circle has 360 degrees, 21,600 (360×60) minutes and 1,296,000 (360x60x60) seconds.

Ancient divided the whole sky into 12 equal parts called constellations.

  • Zodiac (circle division): 12 equal parts (or 4×3): 4 quarters each into 3 and further each third into 10 (4, 3, 10)
  • Each Constellation had 30 degrees (360/12)
  • 30 degrees = 1,800 minutes = 108,000 seconds.

Dividing the daily cycle into equal parts established units of time.
Ancient divisions were not decimal but based on 24 and 60:

1/24 of the Earth spin cycle was unit we call now 1 hour, Each hour is divided into 60 equal units (called minutes) and each minute is divided into 60 equal units (called seconds).

The finest unit of time in ancient times was one second:

1/(24x60x60) = 1/86400 part of one spin cycle (1 day). In other words 1 rotation cycle of the Earth (one day) has 24 hours, 1,440 minutes and 86,400 seconds.

Each day was divided into 24 parts called hours, each hour into 60 minutes and each minute in to 60 seconds (today we divide seconds using decimal system; 1/10, 1/100, 1/1000 (millisecond).

Each day has 24 hours = 1440 minutes (24×60) = 86,400 seconds (24x60x60).

The Earth turns 15 degrees per hour (1 degree each 4 minutes).

Astronomy and Calendars

All calendars began with people recording time by using natural cycles: days, lunar cycles (months), and solar cycles (years). Ancient peoples have attempted to organize these cycles into calendars to keep track of time and to be able to predict future events of importance to them, such as seasons (e.g. the annual Nile flood in ancient Egypt), eclipses etc. The main problem was that these natural cycles did not divide evenly.

Early people could either try to stay in sync with the moon, perhaps making months alternating combinations of 29 and 30 days, with special rules to re-sync occasionally with a solar year by adding leap months (such as the Jewish or Chinese calendar) or abandon lunar cycles and concentrate on the solar year (such as the Ancient Egyptian calendar of 12 same-sized months).

Today, the solar year is 365.242199 days long (or 365 days 5 hours 48 minutes 46 seconds) and the time between full moons is 29.530589 days.
Therefore in 1 year there are 12.37 moon cycles (365.24 / 29.53 = 12.37).

The Moon makes a complete orbit around the Earth with respect to the fixed stars about once every 27.3 days (sidereal period). However, since the Earth is moving in its orbit about the Sun at the same time, it takes slightly longer for the Moon to show the same phase to Earth, which is about 29.5 days (its synodic period).

Nature’s Nearly Perfect Calendar

The Moon could be “Nature’s perfect clock” if the solar cycle period were exactly divisible by the period of the lunar cycle.
For example if the solar year were exactly 364 days (instead of 365.24 ) and lunar cycle exactly 28 days (instead of  29.53), we would have a “perfect calendar” based on 13 months of 28 days per month, with each month having 4 weeks of 7 days. Such calendar was proposed as “13 Moon Calendar”  (discussed later in this article) with the 365th day called the “Day Out of Time”.

Another “perfect calendar” would require solar year to have 360 days and lunar cycle 30 days:

  • The “perfect” Earth would take 360 days to complete 360 degree circular solar orbit (1 deg per day).
  • The “perfect” Moon would take 30 days to complete 360 degree circular orbit around the Earth (12 deg per day).
  • In such case,  we could have a year based on 12 months of 30 days. Each month would have 5 weeks of 6 days each.

We can only wonder if these numbers were true for the Earth  in the the early period of the solar system…

13 Moon Calendar

A Culture of Peace through a Calendar of Peace

The 13 Moon Natural Time Calendar is based upon Dreamspell – a universal application of the mathematics and cosmology of the Classic Mayan Calendar and Prophecy of 2012 as deciphered and presented by Jose and Lloydine Arguelles.

Around the world, people of diverse beliefs and cultures are unifying with the 13 moon calendar as a global harmonic standard – thirteen moons of 28 days, with one day to celebrate “Peace Through Culture” before each new year (July 26).

Introduction to the 13 Moon Calendar:

The Thirteen Moon/28 day calendar is a perpetual, harmonic calendar. It is called a Moon Calendar because it is based on the female 28-day (average) menstruation cycle, which is also the average lunar cycle. The measure of the moon from new moon to new moon is called the synodic cycle and is 29.5 days in length.

However, the sidereal lunar cycle which measures the moon from where it reappears in the same place in the sky is only 27.1 days in length. So 28 days is the average lunar cycle.

In actuality the moon goes around the Earth thirteen times a year. This means that the 13 Moon calendar is a genuine solar-lunar calendar which measures the Earth’s orbit around the sun by the lunar average of 28 days.

Thirteen perfect months of 28 days = 52 perfect weeks of 7 days = 364 days.

The 365th day is called the “Day Out of Time” because it is no day of the week or month at all. This day which falls on the Gregorian correlate date of July 25 is a day for forgiveness and for the artistic celebration of life and freedom.

The synchronization, or new year’s date of the 13 Moon calendar is July 26. This corresponds to the rising of the great star Sirius. This makes the 13 Moon Calendar a tool for harmonizing ourselves with the galaxy.

Perfect Periodicity
For every one time we go around the Sun,the Moon goes around Earth 13 times.
The year has already been divided by Nature-13 ‘moon’ths of perpetual harmony.

Related Links

Cosmic Harmony

Earth Facts

Revolution Period around Sun: 365.2425 days

Rotation on Axis: 23 hours and 56 minutes and 04.09053 seconds.
Note: it takes an additional four minutes for the earth to revolve to the same position as the day before relative to the sun (i.e. 24 hours).

Axial Tilt: 23.4 o (23°26’21.4119″)

Earth’s Circumference at the Equator:
21,638.855 nautical miles = 24,901.55 miles (40,075.16 km)
Earth’s Circumference Between
the North and South Poles:

21,602.6 nautical miles = 24,859.82 miles (40,008 km)

Earth’s Diameter at the Equator:
6,887.7 nautical miles = 7,926.28 miles (12,756.1 km)
Equatorial Radius (1/2 of the diameter):
3,443.9 nautical miles = 3,963.14 miles (6,378.05 km)

Earth’s Diameter at the Poles:
7,899.80 miles (12,713.5 km)
Polar Radius: 3,950 miles (6,356.75km)

Average Distance from the Earth to the Sun:
93,020,000 miles (149,669,180 km)

Average Distance from the Earth to the Moon:
238,857 miles (384,403.1 km)

The closest distance between Earth and Sun is 147 x 106 km, which converted to Royal Egyptian Cubits is 280 x 109  ( 280 cubits is the height of the Great Pyramid).

Observe the measurements of the Moon:
Equatorial radius 1,738.14 km (0.273 Earths) so
the Moon Diameter is: 3,476.28 km or 2,160 miles.
Let’s make this “the cosmic unit” (one moon unit)
the “ideal” diameter of The Moon.

Diameter of The Sun =865,294 miles*.
If we divide 865,294 by 2,160
we get 400.6 …

*The SOHO spacecraft was used to measure the diameter of the Sun by timing transits of Mercury across the surface during 2003 and 2006. The result was a measured radius of 696,342 ± 65 kilometres (432,687 ± 40 miles). Staying within the margin of error,  measured radius of the Sun is 432,647 miles (or diameter 865,294 miles).

7 x 8 x 9 x 11 x 12 x 13 = 864,864 which is only 0.05%  (430 miles)  smaller than the measured value.
Also, 930 x 930 = 864,900 (only 394 miles less than measured value).

Equatorial Circumference of the Earth: 1,000 x 360 x 365.24 feet
= 131,486,400 feet = 40,077km

Official value for equatorial circumference of the Earth: 40,075 kilometers
That is truly a “cosmic” coincidence !!!

There is more… and the Great Pyramid has these values encoded in its proportions!

The radius of the Moon compared to the Earth is three to eleven, ie. 3:11.

 Radius of Moon = 1,080 miles = 3 x 360 = 1 x 2 x 3 x 4 x 5 x 6 x (3/2 )
(  1080.030 miles = 1738.1 km )

 Radius of Earth = 3,960 miles = 11 x 360 =  1 x 2 x 3 x 4 x 5  x 6 x  (11/2)
(3963.167 mi 6378.1 km )

 Radius of Earth plus Radius of Moon = 5,040 miles14 x 360 =
=1 x 2 x 3 x 4 x 5 x 6 x (14/2) = 7 x 8 x 9 x 10
(Actual 5043.197 miles )

1 x 2 x 3 x 4 x 5 x 6=720
1+2+3+4+5+6 = 21
21 x 240 = 5040

(720 x 21) / 3 = 5040
(720 x 21) / 14 = 1080

The ratio 3:11 is 27.3 percent, and the orbit of the Moon takes 27.3 days.
(Sidereal rotation period 27.321582 d (synchronous).
27.3 days is also the average rotation period of a sunspot.


Here is another unbelievable “coincidence”:

“Pi” approximated to the 9th decimal place is exactly

Pi = 3.141 592 654

3 x 1 x 4 x 1 x 5 x 9 x 2 x 6 x 5 x 4 = 129,600 

129,600 x 2 = 25,920  x 10
which is 10x Earth’s precession cycles ( 10 x period of precession of the equinoxes)

The Earth is spinning on its axis in a counter-clockwise direction, and rotating around the sun also in a counter-clockwise direction, while the earth’s spinning axis wobbles like a gyroscope in a clockwise direction. As the result,  as the twelve constellations appear to move clockwise along/around the horizon during the course of its annual rotation around the sun during a year, the constellations, from spring or fall equinox to equinox, appear to move counterclockwise at the rate of 72 years/per degree, or one full wobble in 25,920 years (also known as Great Year).of the earth’s axis.
The galactic year is the duration of time required for the Solar System to orbit once around the center of our galaxy (the Milky Way). Estimated length of the galactic year is close to 1000 precession cycles of the Earth.

Fibonacci Sequence, Phi and Pi

The golden ratio is an irrational mathematical constant, approximately 1.6180339887.

phi = 1.61803 39887 49894 84820 45868 34365 63811 77203 09179 80576

phi, 1.618 …  is the Golden Ratio in life and the universe.

phi + 1 = phi 2

Phi = 1/phi = 0.618034…
Phi = phi – 1

( 1 + sqrt(5) )/2 = 1.61803 3988…
which is an excellent approximation of Phi

The following formula connects “Pi” (3.14159…)  and “phi ” (1.6180…):

  (6/5 ) x  phi 2 = Pi
(12/5) x phi 2 = 2 x Pi

sqrt(phi) nearly equals 4/pi
( 1.272 vs 1.273 )

This Formula is expressed by the numbers describing Pyramids of Giza !!!
Here is how:

Proportions (base to height ratio) of the 1st Pyramid: 11/7
Proportions (base to height ratio) of the 2nd Pyramid: 3/2
Proportions (base to height ratio) of the 3rd Pyramid: 8:/5

(24/10) x phi2 = 2 x Pi

the same formula written with ratios of 3 pyramids:
2 x [ 2 x (11/7)] = (3/2) x (8/5) x phi

Note: the Great Pyramid is a Golden Pyramid: length of the slope side (356) divided by half of the side (440/2 = 220) height is equal to 1.6181818… which is the Golden Ratio Phi

The dimensions of the Earth and Moon are in Phi relationship, forming a Golden Triangle:

The Summation Series (Fibonacci)

The natural progression follows a “summation series” that is known today as the Fibonacci Series [ of course this Series was in existence before Fibonacci (born in 1179 CE) – he simply “re-discovered” it.] The Summation Series is a progressive series, where you start with the first two numbers, then you add their total to generate the next number, and so on. By definition, the first two Fibonacci numbers are 0 and 1, and each subsequent number is the sum of the previous two. The first summation progression:

1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, …

It is worth to note that the first summation (Fibonacci) progression generates  approximation of the golden ratio (PHI):

3/2 =1.5000
5/3 =1.666…
8/5 =1.6000
13/8 =1.6250

The second of Fibonacci progression (provides approximate geometry of the pentagram and also leads to the golden ratio ‘phi’ )

1, 3, 4, 7, 11, 18, 29, 47, 76, 123, 199, 322, …
just like in the above example:
199/123 = 1.61788…

The Summation Series is reflected throughout nature. The number of seeds in a sunflower, the petals of any flower, the arrangement of pine cones, the growth of a nautilus shell, etc…all follow the same pattern of these series.

The overwhelming evidence indicates that the Summation Series was known to the Ancient Egyptians. Throughout the history of Ancient Egypt, temples and tombs (including pyramids) show in their design  expression of the Summation Series: 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610, . . .

Pi = 3.141 592 653 5

3 x 1 = 3
3 x 1 x 4 = 12 (12/3=4)
3 x 1 x 4 x 1 x 5 = 60 (60/12=5)
3 x 1 x 4 x 1 x 5 x 9 = 540 (540/60=9)  :::  540/360=1.5
3 x 1 x 4 x 1 x 5 x 9 x 2 = 1080 (1080/540=2)  :::  1080/360=3 (3/1.5 =2)
3 x 1 x 4 x 1 x 5 x 9 x 2 x 6 = 6480 (6480/1080=6)  :::  6480/360=18 (18/3=6)
3 x 1 x 4 x 1 x 5 x 9 x 2 x 6 x 5 = 32400 (32400/6480=5)  :::  32400/360=90 (90/18=5)
3 x 1 x 4 x 1 x 5 x 9 x 2 x 6 x 5 x 3 = 97200  :::  97200/360=270 (270/90=3)
3  x 1 x 4 x 1 x 5 x 9 x 2 x 6 x 5 x 3 x 5 = 486 000 ::: 486000/360=1350 (1350/270=5)

Pi can be approximated also as sqrt(2) + sqrt(3) = 3.14626437…


Another important observation:

No matter what units of measure we use, the sides of the Giza Pyramids Rectangle (layout for all 3 pyramids)
generate the base of the 1st pyramid:

If we use Royal Egyptian Cubits:

1732/1417  =  440/360

This rectangle shows also relationship of
the Royal Egyptian Cubit 20.62 inch
to the Sacred Cubit 25.2 inch

1732/1417 = 25.2/20.62

Another interesting coincidence encoded in the Royal Cubit:

  • Royal Egyptian Cubit = 0.524 m
  • pi = 3.1416
  • 3.1416 feet = 0.957 m = 1.8263 RC
  • 2x (0.957 / 0.524 ) = 2x  1.8263 = 3.6526 = 365.26 x 10-2

Number of days in a year: 365.25

Another interpretation of the above:
circumference of a a circle with diameter equal 100 feet is
2 x pi x 100 feet = 2 x pi x 1200? =7539.8? = 365.74 RC
(or 356.25 RC measured in King’s Chamber)

Proportions of pyramids at Giza (height/base):

7/11 = 0.6363636363….
2/3 = 0.666666666….
5/8 = 0.625

7  = 5 + 2
11= 8 + 3


7 + 2 + 5 = 14
11 + 3 + 8 = 22
14/22 = 0.6363636363….
22/14 = 1.571428 = 0.5 * 3.142857

2+3+5+7+8+11 = 36

Laws Written in Stone

The Pyramid of Kukulkan in Chichen Itza

Chichen Itza Pyramid Calendar

The Pyramid of Kukulkan at Chichén Itzá, constructed circa 1050 was built during the late Mayan period, when Toltecs from Tula became politically powerful. The pyramid was used as a calendar: four stairways, each with 91 steps and a platform at the top, making a total of 365, equivalent to the number of days in a calendar year.

The ancient Mayan Pyramid at Chichen Itza, Yucatan, Mexico.
Copyright A. Sokolowski

Latitude of the pyramid of Quetzalcoatl: 20° 40? 58.44? N

20 x 40 x 58.44 = 46752

This pyramid is a precise calendar (it has 91 steps on each of 4 sides plus platform on top: 4×91 +1 = 365).
The calendar connection is also confirmed by pyramid’s orientation marking equinoxes and solstices.

The pyramid has 4 sides with 4 staircases dividing each side into 2 sections (in total 8 sections).
Using these numbers and accurate value for 1 year equal 365.25, the pyramid “number” will perfectly relate its latitude:

365.25 x 4 x4 x 8 = 46752

Note:  46752 = 365.25 x 128  (perhaps there is a better fit for the 128?)

For Munck’s  number crunching related to the Pyramid of Quetzacoatl in Chichen Itza go to:

Gate of the Sun at Tiwanacu

The Gate of the Sun is a very sophisticated calendar called “The Muisca Calendar”.


The Gateway of the Sun from the Tiwanku civilization in Bolivia

The Muisca calendar is another important piece in the jigsaw of the lost knowledge of the Andes. If the origins of the Muisca calendar were to be found at Tiwanaku, then perhaps they were also built into the Gate of the this calendar at Tiwanku is not based upon a divison of the solar year into twelve, but into twenty, and this is represented by the eleven smaller icons forming the freize at the bottom which represents the eleven pillars on the west side of the Kalasasayo which is the actual calendar. So if you count from the central icon or pillar out to the right hand end, then back past the central icon to the left hand end, then back to the centre, you will have effectively counted in twenty divisons and followed the path of the sun over a year.

This wonderful graphic by Ken Bakeman
shows colored version of the Sun-god
from the Gate of the Sun relief.
Image source:

So if the chasquis do not relate to the days in whichever number of days we choose for the months of the year, could it be that the chasquis represent the years themselves?

If each chasqui were to represent a solar year, then each column of three chasquis would represent three revolutions of the sun around the eleven pillar calendar wall and three solar years are equivalent to 1 x Muisca Acrotom year of 37 synodic months of 29.53 days and also equivalent to 2 x Muisca Zocam years of 20 sidereal months of 27.32 days.

There are fifteen chasquis on each side of the central figure and each block of 15 chasquis would represent fifteen solar years which would be
5 Muisca Acrotom years of 37 synodic months of 29.53 days or
10 Muisca Zocam years of 20 sidereal months of 27.32 days

The total number of chasquis is thirty chasquis representing thirty solar years which would be
10 Muisca Acrotom years of 37 synodic months of 29.53 days or
20 Muisca Zocam years of 20 sidereal months of 27.32 days

The choice of thirty chasquis as thirty solar years is no random figure, because after thirty solar years have gone by, it becomes necessary to add one sidereal lunar month to the Muisca Zocam calendar making it 20 x 20 + 1 = 401 sidereal lunar months to bring it back into line with the solar year.

At the same time of adding one sidereal month to the Zocam sidereal calendar, it also becomes necessary to add one synodic lunar month to the Muisca Acrotom calendar making it 10 x 37 + 1 synodic lunar months to also bring it into line with both the sidereal lunar calendar and the actual solar year.

Each of the sections with fifteen chasquis corresponds to the period of fifteen solar years which Duquesne tells us was one quarter of the great “century” of the Muiscas so to sum up, each block of fifteen chasquis represents fifteen solar years which is 10 Muisca Zocam years or 5 Muisca Acrotom years, the two blocks together make 30 chasquis representing 30 solar years which is 20 Muisca Zocam years or 10 Muisca Acrotom years and 2 x the 30 chasquis gives 60 chasquis representing 60 solar years completing the great “century” of the Muiscas which was therefore, 40 Muisca Zocam years or 20 Muisca Acrotom years.

Above, detail of the “Gate of the Sun” at Tiwanaku, Bolivia, the 30 Chasquis represent 30 Solar years, equal to 20 Zocam years of 20 sidereal lunar months or 10 Acrotom years of 37 synodic lunar months. At the end of this period, 1 x lunar month had to be added to the lunar calendars to bring them back into phase with the solar year..

Above, the “Gate of the Sun” at Tiwanaku, Bolivia, the 30 Chasquis represent 30 Solar years, equal to 20 Zocam years of 20 sidereal lunar months or 10 Acrotom years of 37 synodic lunar months. At the end of this period, 1 x lunar month had to be added to the lunar calendars to bring them back into phase with the solar year. Beneath the chasquis can be seen the freize with 11 smaller chasqui heads representing the 11 pillars on the calendar wall which in turn divide the solar year into 20 months of 18 days, and the 40 condor heads represent the 40 sidereal months which mesh with the solar calendar every three years.

Above, when the sun reached the end of the pillars, it appeared to “stand still” before beginning its journey back in the opposite direction.

The Great Pyramid of Giza

John Taylor, in his 1859 book “The Great Pyramid: Why Was It Built? & Who Built It?”, claimed that the Great Pyramid was planned and the building supervised by the biblical Noah, and that it was built to make a record of the measure of the Earth.”

Now let’s look at the coordinates of the Great Pyramid:

29°58? 45.03? N

Let’s multiply latitude numbers 29x58x45.03 = 75740.46

Now, this pyramid has 4 sides, its perfect slope angle is 51.8428° and there are 365.2425 days in a year:
4 x 51.8428 x 365.2425 =  75740.775

Note: Another very strange  “coincidence” was discovered by John Charles Webb Jr. : Precise latitude of the centre of the Grand Gallery (inside GP) is 29°58? 45.28? N = 29.9792458° N
The speed of light in vacuum, usually denoted by c, is a universal physical constant important in many areas of physics. Its value is 299,792,458 metres /s

This single, fundamental design principle: 11 : 7 Base to Height Ratio  generates ALL amazing mathematical properties of the Great Pyramid:

  • the Golden Ratio Phi=1.618 (the Great Pyramid is a Golden Pyramid: length of the slope side (356) divided by half of the side (440/2 = 220) height is equal to 1.6181818… which is the Golden Ratio Phi
  • squaring the circle ratio 1.571 (base/height = 44/28 = 1.571)
  • pi=3.14159… (2 x base/height = 2 x 44/28 = 3.14286 which is very close approximation of “pi” = 3.14159…)
  • Perimeter of the square base, 4×440=1760, is the same as circumference of the circle with radius = height: 2x ”pi” x height (2x 22/7 x 280=1760)
  • The ratio of the perimeter to height of 1760/280 cubits equates to 2x pi
    to an accuracy of better than 0.05%
  • Side of the base (440) plus double height (2x 280=560) = 1,000
  • Perimeter of the square base is equal 4×440=1760 RC = 0.5 nautical mile = 1/7,200th of the radius length of the earth
  • the slop angle 51°.843
  • The Pyramid exhibits in the design both pi and by Phi,  given the similarity
    of 2/ sqrt(phi) (2 divided by the square root of Phi) with pi/2 :
  • 11/ 7 equal 1.5714
  • 2/ sqrt(89/55) equal 1.5722
  • 2/ sqrt(Phi) equal 1.5723
  • pi/ 2 equal 1.5708
  • Royal Cubit = 0.5236 m,  pi – Phi2 = 0.5231

11 + 21 + 31 + 41 = 10

  1!+2!+3!+4! =33

1 + 2 + 3 + 4 + … + 36 = 666

For penulum with length L= 1 MY = 0.83 m,
the pendulum period T= 1.827 seconds

666/1.827 = 364.5

Another Intriguing properties of numbers

1Royal Egyptian Cubit (REC)=20.62” = 0.52375m

Pyramid complex at Teotihuacan reveals measuring units of length suggested by Hugh Harleston: Hunab (STU) = 1.059463 m
Another possible unit was suggested by Sugiyama: 1 TMU=32.8”=0.833m=1.591REC
1REC/1TMU = 0.62875 which is approximately = 2pi/10
1STU (1 HUNAB) =3.4757485’=41.71”=1.0594m=2.0228 REC     1REC/1STU=0.259


  • circumference of circle with R=1TMU=10 REC
  • circumference of circle with R=1 Hunab (using Sumerian Pi):
    2*3.146263*1.059463 = 6.6667=6*1.1111

“Perfect” Earth Circumference in km: 40,000/360 = 111.111111111

Sqrt(1.23456789)= 1.1111111 = 1+0.01+0.01+0.001+0.0001+0.00001+0.000001+0.0000001

0.55555555/1.0594575=0.524377 (geometric cubit in meters) = 20.645”

sqrt2+sqrt3=3.14626437 (Sumerian pi)



1x2x3x4x5x6x7x8x9=362880= 360x 1.008

Link: The Antiquity of the Sumerian PI Value

PS 1 Titus-Bode Law and Pascal’s Triangle

In 1768, Bode published his popular book, “Anleitung zur Kenntnis des gestirnten Himmels” [Instruction for the Knowledge of the Starry Heavens]. In this book, he described an empirical law on planetary distances, originally found by J.D. Titius (1729-96), now called “Bode’s Law” or “Titius-Bode Law”.

The original formula was:

a = ( n + 4 ) / 10
where n=0,  3,  6,  12,  24,  48,  96,  192, 384, …

To find the mean distances of the planets, begin with the following simple sequence of numbers:
0     3     6     12     24     48     96     192     384   …
These numbers (after 0) derive from the Pascal’s triangle as the sum of a row added to the sum of the previous row:

Row 1:  2+1=3
Row 2:  4+2=6
Row 3:  8+4=12
Row 4:  16+8=24
Row 5:  32+16=48
Row 6:  64+32=96
Row 7:  128+64=192
Row 8:  256+128=384

Next, add 4 to each number and you will get:
4     7     10     16     28     52     100     196     388
Then divide it by 10:
0.4     0.7     1.0     1.6     2.8     5.2     10.0     19.6     38.8

The resulting sequence is very close to the distribution of mean distances of the planets from the Sun
expressed in Astronomical Units (AU):

Body Actual distance (A.U.) Bode’s Law
Mercury 0.39 0.4
Venus 0.72 0.7
Earth 1.00 1.0
Mars 1.52 1.6
asteroid belt 2.77 2.8
Jupiter 5.20 5.2
Saturn 9.54 10.0
Uranus 19.19 19.6
Neptune 30.06 n/a
Pluto 39.44 38.8

1 AU is approximately the mean Earth–Sun distance equal AU = 149.597 *106 km

All planets (and asteroid belt) fit Titus-Bode Law except for Neptune!

Here is another way to connect Bode’s Law with Pascal’s Triangle

Planet k Pascal Triangle bin(k)
Mercury 1 1 0
Venus 2 1 + 1 1
Earth 3 1 + 2 + 1 2
Mars 4 1 + 3 + 3 + 1 4
Planet V 5 1 + 4 + 6 + 4 + 1 8
Jupiter 6 1+ 5+10+ 10 + 5 + 1 16
Saturn 7 1+6 +15+20+ 15 + 6 + 1 32
Uranus 8 1+7+21+35+35+ 21 + 7 + 1 64
Neptune 9 bin(7) + bin(8) 96
Pluto 9 1+8+28+56+70+56+ 28 + 8 + 1 128

The modern formulation of the Titus-Bode law is that the mean distance a of the planet from the Sun in astronomical units ( AU = 149.597 *106 km ) is:

a = 0.4 + 0.3 x k

where ”k’=0,1,2,4,8,16,32,64,128 (sequence of powers of two – from Pascal’s Triangle)

PS2 Spiral Solaris

.. For these reasons and from such constituents, four in number, the body of the universe was brought into being, coming into concord by means of proportion, and from these it acquired Amity, so that coming into unity with itself it became indissoluble by any other save him who bound it together. (Timaeus, 31b-32c, Plato’s Cosmology: The Timaeus of Plato, Trans. Francis MacDonald Cornford, Bobbs-Merrill, Indianapolis, 1975:44, emphases supplied)


“The three-fold number is present in all things whatsoever; nor did we ourselves discover this number, but rather nature teaches it to us”  — quotation from Ovid provided by Nicole Oresme in his major work, Le livre du ciel et du monde.

Commentaries of Proclus on the Timeus of Plato:

    The figure of the spiral likewise, is no vain, fortuitous things, but gives completion to the media between bodies that move in right lines, and those that are moved in a circle. For the circle alone, as we have said, is in the inerratic sphere, but the right line in generation. And the spiral is in the planetary region, as having a co-mixture of the periphery and the right line. The motions also according to breadth and according to depth, viz. of the upward and downward, and the oblique motions. Perhaps likewise, the theurgist [Julian] in celebrating time as a spiral form, as both young and old, directed his attention to this, conceiving that the temporal periods, were especially to be known through the motions of the planets. [Trans. Thomas Taylor, Vol. II, Book IV, p.239]

    With these things however, not only Plato as we have observed, but theurgists likewise accord. For they celebrate time as a mundane God, eternal, boundless, young and old, and of a spiral form. And besides this also, as having its essence in eternity, as abiding always the same, and as possessing infinite power. For how could it otherwise comprehend the infinity of apparent time, and circularly lead all things to their former condition, and renovate them, and also recall things which become old through it, to their proper measure, as being at once comprehensive both of things that are moved in a circle, and according to a right line. For a spiral is a thing of this kind; and hence, as I have before observed, time is celebrated by theurgists, as having a spiral form. [trans. Thomas Taylor, Vol. II, Book IV, pp.207-8, emphasis supplied]

Without going into the finer details we may nevertheless take an initial cue from the information provided by Archimedes that :

the area bounded by the spiral in the first revolution is a sixth part of that added in the second revolution”.

Simply stated, if the first area is 1 (i.e.,unity) then the second area will be 6, the third 12, the fourth 18, and so on, for according to Archimedes:

“generally the areas added in the later revolutions will be multiples of that added in the second revolution according to the successive numbers”,

thus the areas expand by successive multiples of 6.

But what does this have to do with the equiangular period spiral as derived in the previous section which concerns time and successive heliocentric revolutions?

Since the revolutions in the latter also proceed according to a fixed increment, i.e., by the fundamental period constant associated with the equiangular square and the construction of the spiral itself – Phi 2– we can also obtain successive “areas” from the corresponding periods (as “radii”) from the Phi-series planetary framework. And as it so happens, the “areas” pertaining to the planetary positions also expand in a uniform manner – not by 6 as given by Archimedes – but by a constant factor of 6.854101966 (Phi 4) instead, and thus 6 may (or may not) be considered a simplified approximation for the latter.

But however one looks at it, the number “6” is undoubtedly of importance in Pythagorean contexts while a similar weight appears to have been placed – either directly or indirectly – on the numbers 6, 12 and 18 by Plato in the Epinomis, in the Timaeus, and elsewhere. And here we might also recall that with respect to the equiangular period spiral (with notable exceptions) that the heliocentric distances occur at the 60 and 300-degree points, thus a six-fold division is also a feature of the basic configuration.

So what indeed is Archimedes referring to here? One can hardly be certain at this stage, but from what we have seen so far, one might begin to suspect that these seemingly simple operations are neither numerology nor primitive mathematics, but something quite different. It may well be that it is not common practice to treat time in radial form per se, but then again, we are not used to working with such an all-encompassing, complex entity as this particular equiangular spiral either. But if we are not dealing with numerology and elementary mathematics in this context, then what are dealing with? It may be too early to be definitive, but it is beginning to look more and more like highly condensed, competent methodology clothed in disarmingly simple terms.



Returning to Oresme’s diagram, by coincidence or chance, perhaps – but assuredly not by manipulation – even the direction of the circumscribed spiral matches our own:


Moreover, if we recall that our final product – the equiangular period spiral, is

  • (a) just that, i.e., equiangular,
  • (b) that this spiral includes all three mean parameters – Periods, Distances and Velocities – and significantly
  • (c), that the latter trio are delimited by three basic equiangular figures – the triangle, the square (more correctly the rectangle) and the hexagon

– we are then able to examine Oresme’s closing religious summations in Du Ciel from a more tightly focussed viewpoint:

Notwithstanding that He is everywhere, still is He absolutely indivisible and the same time infinite with respect to the three qualities that are divisible in living creatures, which we call duration, position, and power or perfection; for temporal duration of creatures is divisible in succession; their position, especially of material bodies, is divisible in extension; and their power is divisible in any degree or intensity. [Du Ciel, Book IV, Chapter 10, fols. 200a-200c, p.721]

… Besides the varieties of trinity noted there, there is another which is pertinent to our present discussion, because, in accord with what we said in chapter Twelve of Book II [see fol. 176b], there are three regular plane figures – the triangle, the square, and the hexagon – each of which we can imagine to be capable of filling so completely a flat area or surface that it is absolutely impossible that there could be more space to be filled; likewise there are three divine persons, each of whom fills all space. Isaiah the Prophet spoke of them thus: Holy, Holy, Holy, Lord God, etc. all the earth is full of thy glory. And there is one God, who spoke through His Prophet Jeremiah: I will fill the heaven and earth; and of who Virgil said: All things are replete with Jove.[Du Ciel, Book IV, Chapter 10, fols. 200d-201b, p.723]


The Double-formed Spiral k = Phi 4 and the Whirlpool Galaxy M51 [Color (200kb)]
Edge-on Whirlpool Galaxy M51 image by the Hubble Heritage Team (NASA/STScI/AURA) using data collected by
Principal Astronomer N. Scoville (Caltech) and collaborators.



In view of what follows next it seems necessary to remind the reader that the “phi-based” exponential planetary frameworks […] IV stemmed entirely from the rejection of Bode’s “law” and the resulting need to develop a more workable approach to the structure of the Solar System.
Simply stated, a mathematical problem concerning mean planetary periods resulted in the final determination of a constant of linearity (k) from the quadratic equation k 2– k – 1 = 0.
Thus the required constant for the periods (planetary and synodic) turned out to be the Golden Ratio Phi = 1.6180339887949.
Moreover, as a consequence of the applied methodology the final exponential planetary framework was in turn necessarily predicated on the larger (though intimately related) constant Phi 2= 2.6180339887949. This determination was in fact the direct result of dealing with Time before Distance and latter in turn before Velocity, although as we have seen, all three parameters were later combined in the final planetary frameworks in any event.
On completion of the latter, however, it became apparent from various philosophical writings and cues that the initial emphasis on Time, i.e., periods of revolution, was–to a certain extent at least–already present in ancient works (see below). Consequently, and also as treated in the previous sections, what follows next is a natural pursuit of the various related threads that have been passed down to us; thus basically a search for similarity, and hopefully further enlightenment.
   This said, however, the degree of understanding attained in earlier times is difficult to assess for many reasons; but if Spira Solaris and its integral parameters are in any way underlying features in certain philosophical writings, then it should be possible to focus on the subject matter with specific numerical values and mathematical concepts already in place. Thus although Plato’s emphasis in both the Epinomis and the Timaeus suggests that there is little likelihood of detailed understanding without lengthy and specific instructions:

     So much, then, for our program as a whole. But to crown it all, we must go on to the generation of things divine, the fairest and most heavenly spectacle God has vouchsafed to the eye of man. And: believe me, no man will ever behold that spectacle without the studies we have described, and so be able to boast that he has won it by an easy route. Moreover, in all our sessions for study we are to relate the single fact to its species; there are questions to be asked and erroneous theses to be refuted. We may truly say that this is ever the prime test, and the best a man can have; as for tests that profess to be such but are not, there is no labor so fruitlessly thrown away as that spent on them. We must also grasp the accuracy of the periodic times and the precision with which they complete the various celestial motions, and this is where a believer in our doctrine that soul is both older and more divine than body will appreciate the beauty and justice of the saying that ‘ all things are full of gods ‘ and that we have never been left unheeded by the forgetfulness or carelessness of the higher powers. There is one observation to be made about all such matters. If a man grasps the several questions aright, the benefit accruing to him who thus learns his lesson in the proper way is great indeed; if he cannot, ’twill ever be the better course to call on God. Now the proper way is this–so much explanation is unavoidable. To the man who pursues his studies in the proper way, all geometric constructions, all systems of numbers, all duly constituted melodic progressions, the single ordered scheme of all celestial revolutions, should disclose themselves, and disclose themselves they will, if, as I say, a man pursues his studies aright with his mind’s eye fixed on their single end. As such a man reflects, he will receive the revelation of a single bond of natural interconnection between all these problems. If such matters are handled in any other spirit, a man, as I am saying, will need to invoke his luck. We may rest assured that without these qualifications the happy will not make their appearance in any society; this is the method, this the pabulum, these the studies demanded; hard or easy, this is the road we must tread. (Epinomis, 989d-992a, Trans. A.E. Taylor, The Collected Dialogues of Plato, Princeton University Press, Princeton, 1982:1530-31; emphases supplied)

For these reasons and from such constituents, four in number, the body of the universe was brought into being, coming into concord by means of proportion, and from these it acquired Amity, so that coming into unity with itself it became indissoluble by any other save him who bound it together. (Timaeus, 31b-32c, Plato’s Cosmology: The Timaeus of Plato, Trans. Francis MacDonald Cornford, Bobbs-Merrill, Indianapolis, 1975:44, emphases supplied)

it is conceivable that the parameters and structure of Spiral Solaris might provide not only the necessary element of “luck”, but also some degree of understanding concerning the “binding.” Certainly there are enough parameters and associated concepts available (both past and present) as already touched upon in the last two sections. However, it would still be optimistic to expect that either the application or the precise details would be immediately apparent. In fact–after the manner of Orpheus, Pythagoras and Plato — it might well be that certain related matters were indeed: “promulgated mystically and symbolically (by the first); by the second, enigmatically and through images; and scientifically by the third.” Or, as Thomas Taylor put it: “conformably to the custom of the most ancient philosophers, (information) was delivered synoptically, and in such a way as to be inaccessible to the vulgar.” 

For example, although not especially “mystical” or “symbolic,” consider the following “poetic” dissemination in medieval scholar Nicole Oresme’s reference to Aristotle and “the three-fold number” according to Ovid:3

 Said Aristotle, prince of philosophers and never-failing friend of truth:
All things are three; The three-fold number is present in all things whatsoever…
Nor did we ourselves discover this number, but rather natures teaches it to us.

Here, historical preconceptions notwithstanding, it can undoubtedly be suggested that of all numbers the Golden Ratio is uniquely qualified to receive such an appellation, though it is not the only related choice in this regard. Equally applicable might be the reciprocal of the underlying constant of Spira Solaris, i.e., Phi  -2 = 0.381966011, which as will be seen in later sections, may be understood to represent many things, including the “fifth element”(Aether); “Venus philosophical” to some alchemists; in the same alchemical understanding Sir Isaac Newton’s aptly named “Quintessence” and also a key parameter associated with phyllotaxis to return us to Ovid and the link with Nature.
Then again, there is the more precise wording in the Chaldean Oracles, where Ovid’s “All Things are Three” is expanded to include intellection ( “for the Mind of the Father said, that all things can be cut into three, Governing all things by mind” ). This occurs in a larger passage that is also readily understandable in the present context with its thinly disguised references to the Golden Ratio and not least of all, “Fountain of Fountains, and of all Fountains, The Matrix containing all things”:

The Monad is enlarged, which generates Two.
For the Dyad sits by him, and glitters with Intellectual Sections.
And to govern all things, and to order all things not ordered.
For in the whole World shineth the Triad, over which the Monad Rules.
This Order is the beginning of all Section.
for the Mind of the Father said, that all things can be cut into three,
Governing all things by mind.
The Center from which all (lines) which way soever are equal.
for the paternal Mind sowed Symbols through the World.
Fountain of Fountains, and of all Fountains.
The Matrix containing all things . . .

The content of the above passage from the Chaldean Oracles may surprise some readers, but nevertheless the historical side of the matter is not that difficult, although it is clearly out of kilter. The Fibonacci series (and thereafter the Golden Ratio) have long been associated with natural growth from Fibonacci onwards for the moderns, through Kepler, and later via the efforts of a veritable host of investigators, as R.C. Archibald’s lengthy Bibliography 5 in Jay Hambidge’s Dynamic Symmetry (1920:146-156) 6 clearly attests. Though “to err is human,” noticeably absent from the latter list are the contributions of Samuel Coleman (Nature’s Harmonic Unity,1911)7 and those of Louis Agassiz 8 Essay on Classification, 1857)–but more on these omissions later. In passing, though, it is relevant to point out that the claim that the Fibonacci series was only discovered in the early part of the second millennium is surely invalid–a doubly ignorant assertion (in Thomas Taylor’s understanding of the term) that in any case was largely demolished by D’arcy Wentworth Thompson years ago as follows:

The Greeks were familiar with the series 2, 3 : 5, 7 : 12, 17, etc.; which converges to 21/2as the other (i.e., the Fibonacci series) does to the Golden Mean; and so closely related are the two series, that it seems impossible that the Greeks could have known the one and remained ignorant of the other. (Sir D’arcy Wentworth Thompson, On Growth an Form, Dover, New York 1992:923; unabridged reprint of the 1942 edition)

The latter also pointed out, however, that:

We must not suppose the Fibonacci numbers to have any exclusive relation to the Golden Mean; for arithmetic teaches us that, beginning with any two numbers whatsoever, we are led by successive summations toward one out of innumerable series of numbers whose ratios one to another converge to the Golden Mean ( (Sir D’arcy Wentworth Thompson, On Growth an Form, Dover, New York 1992:933; unabridged reprint of the 1942 edition)

This is true enough, but it also adds weight to his previous observation.

For example, consider the following Pythagorean reference (or mnemonic device, if one prefers) concerning the number 36 as explained by W. Wyn Westcott:

Plutarch, “De Iside et Osiride,” calls the Tetractys the power of the number 36, and on this was the greatest oath of the Pythagoreans sworn: and it was denominated the World, in consequence of its being composed of the first four even and the first four odd numbers; for 1 and 3 and 5 and 7 are 16; add 2 and 4 and 6 and 8, and obtain 36. (W.Wyn Westcott, Numbers: their Occult Power and Mystic Virtues, Sun Publishing Santa Fe, 1983:114).

Mere numerology? Elementary mathematics? Perhaps both, but also perhaps neither and an expansion in dimensional thinking that may or may not have historical precedents.
Either way, by adding the first four even and first four odd numbers to obtain 36 the stated intermediate values are also readily aligned for further use. Here I will leave it to the reader to observe that so aligned the numbers added vertically result in the Fibonacci and the Lucas Series with Phi also the limiting ratio in the other three columns. All of which reinforces Sir D’Arcy Wentworth Thompson’s point about the many routes and detours that lead to the Golden Ratio while also rendering historical claims concerning Fibonacci’s pre-eminent “discovery” of the series even less tenable than they already are. And here, one might note, we have not even factored in the main issue under consideration, which is the numerous pointers, guides and indicators leading to the ever-present Golden Ratio in Nature itself. 
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  1. says

    Just a couple of notes. Bucky Fuller noted that when man first travelled across the horizon and navigated by the stars (I say well over a million years ago.) we learned geometry and basic trigonometry. When we went on the ocean in watercraft to get to Flores Island almost a million years ago we learned carpentry and engineering. Add that to our higher attunement capability and geomancy will take a scholar deep into gematria. Zeno’s paradox recently re-surfaced as a non tenured amateur was backed by the great physicist Wheeler. It addresses what is known as infinity and contained in the oroborous or serpent eating it’s tail. That serpent can be seen going up the steps of Chichen Itza’s pyramid which you show, it happens in tune with astronomical events witnessed by Mayan telescopes in the nearby Caracol..These are the mathematicians who discovered what we use in modern math as zero – over a century before it was known in India – as the Mayans taught their brothers.

    • says

      Alfred North Whitehead is a panentheist luminary and philosophical genius with the heart of a true mystic like the great sages and alchemists of the dark and hidden past. I especially like these brief words of his. “Mathematical physics presumes in the first place an electromagnetic field of activity pervading space and time. The laws which condition this field are nothing else than the conditions observed by the general activity of the flux of the world, as it individualises itself in the events.”

      When he spoke to the students of Harvard in 1940 he identified the issues facing the difficulty of getting a real education just as Joseph Campbell did around the same time in a speech to his Sarah Lawrence students. I often quote his address titled Permanent Human Values and I highly recommend or insist that everyone read it. It can be summed up in these words often quoted from Whitehead by Ralph Nader. “Duty arises from our potential control over the course of events. Where attainable knowledge could have changed the issue, ignorance has the guilt of vice.”


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