Ancient Timekeepers

It seems that ancient people had knowledge about the astronomical cycles of the Earth equal to our modern knowledge.
Explore with us how the ancient astronomers could have discovered and precisely measured these cycles without any prior knowledge about them and without advanced technology we have today.
This article is presented in 3 separate parts (part 4 and 5 is coming soon)

1. Part 1: Movements of the Earth (this page)
   What modern astronomy tells us about movements of our planet (and the moon).
2. Part 2: Observing the Sky
   Learn how the discovery and precise description of basic astronomical cycles of our planet can be relatively easy (even in the ancient times).
   Although ancient astronomers could observe the sky only from a moving “point of view”, they managed to create calendars, predict seasons and eclipses, and precisely orient their monuments.
3. Part 3: Archaeoastronomy
   Ancient monuments and writings from around the world are undeniable proof that thousands of years ago people had advanced astronomical knowledge. Explore few examples providing the evidence for it.
4. Part 4: Ancient Calendars
5. Part 5: Ancient Metrology (units of measure)

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Part 1: Movements of the Earth

Modern astronomy tells us there are 3 basic astronomical cycles of the Earth’s movement:

• spinning on its axis – cycle period: 1 solar day
• orbital movement around the Sun – cycle period: 1 solar year
• precession (gyroscopic wobble of the rotation axis) – cycle period: 25,920 years

 
Looking at the solar system from above the Northern Pole we would see that the Earth is spinning on its axis in a counter-clockwise direction (like all other planets except Venus and Uranus), and rotating around the sun also in a counter-clockwise direction, while the earth’s spinning axis (very slowly) 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  1 degree/per 72 years,  or one full “wobble”  of the earth’s axis in 25,920 years
Did you know?
From a point high above the north pole of the solar system the planets are revolving about the sun and rotating about their axes in a counterclockwise direction. This holds true also for the asteroids.

There are 2 exceptions: Venus, which spins clockwise (retrograde motion) and Uranus, which rotates 90 degrees away from its orbital motion).  Venus rotates backwards compared to the other planets, likely due to an early asteroid hit which disturbed its original rotation. Venus travels around the sun once every 225 Earth days but it rotates clockwise once every 243 days. This peculiar combination gives it a day with respect to the sun of 117 Earth days. Uranus was also likely hit by a very large planetoid early in its history, causing it to rotate “on its side”.
The satellites of the planets also generally revolve and rotate in a counterclockwise direction. Of the thirty something satellites only six do not do so; they are said to have retrograde motion. Of the six exceptions five are outer satellites likely to be captured asteroids.
The Sun itself also rotates in a counterclockwise direction.  The entire Sun doesn’t rotate at the same rate. Because the Sun is not solid, but is instead a giant ball of gas and plasma, different parts of the Sun spin at different rates. We can tell how quickly the surface of the Sun is rotating by observing the motion of structures, such as sunspots, on the Sun’s visible surface. The regions of the Sun near its equator rotate once every 25 days. The Sun’s rotation rate decreases with increasing latitude, so that its rotation rate is slowest near its poles. At its poles the Sun rotates once every 36 days!
The Moon spins once around its axis in counterclockwise direction at the rate 27 .3 days which is also the time for it to complete full orbit around the Earth. As the result the Moon always shows us the same side.Although the Moon makes a complete orbit around the Earth with respect to the fixed stars about once every 27.3 days (its sidereal period), since the Earth is moving along its orbit about the Sun at the same time, it takes 29.5 days for the Moon to show the same phase to Earth.

1.1 Earth’s Axial Cycle: 1 Day  – (24 hours)

The axial period is the time taken for an object to make one complete rotation on its axis. The axial period of a planet is its “day”.  Earth spins around its axis in 1 day (24 hours = 1440 min = 86400 seconds). This axial rotation cycle is perceived as the cycle of day and night and it is responsible for the apparent (and constant) movement  of the sun, the moon and stars. This axial cycle is useful for defining units of time and calendar.
Note: The Earth makes 360 degrees rotation around its axis in relation to the distant stars in one “sidereal day”. However, in relation to the sun, due to the earth’s orbital movement, it rotates on its axis a bit more than 360° per (solar) day.  This way in 1 year the Earth makes 1 extra rotation in relation to stars. However, we define our 24 hour “day” relative to the Sun, not relative to the Earth’s actual rotation. If we didn’t do this, then “Noon” would occur a little later each day until “Noon” was happening at night! At present, Earth orbits the Sun once every 366.25 times it rotates about its own axis, which is equal to 365.25 solar days, or one sidereal year.
Earth spin axis is tilted at 23.44 deg in relation to the plane of its orbital movement around the sun (called ecliptic) Note: Simple geometry shows that the angle between the zenith and the celestial equator (i.e. the zenith’s declination) must also be the angle between the north celestial pole and the north horizon.  Since the zenith’s declination is equal to one’s latitude, one can determine his/her latitude by measuring the altitude of the celestial North Pole (currently well approximated by position of the star called Polaris).
The tilt of the Earth changes slightly, with a dominant cycle every 41,000 years. The change in angle of inclination is only about 1° from the present tilt, from 23.5 to 24.5°. However, Earth’s tilt is a critical factor in climate resulting in very large differences in solar radiation. Changes in Earth’s angle with respect to the Sun often go by the name “obliquity”.

1.2 Earth’s Orbital Cycle – 1 year (365.24 days)

This orbital cycle combined with the tilt of the earth’s spin axis ( 23.44 degrees) is perceived as the cycle of seasons and it is also responsible for the variable length of the daytime (throughout the year); on the northern hemisphere summer daytime is much longer than in winter.
The earth makes one complete rotation around its axis (360 deg) in relation to distant stars (not the Sun) and the time to complete this full rotation is called sidereal day.  In our daily lives it is practical to use “solar day” as a unit of time. Solar time is measured by the apparent diurnal motion of the sun, and local noon in solar time is the moment when the sun is at its highest point in the sky (exactly due south or north depending on the observer’s latitude and the season). The average time for the sun to return to its highest point is 24 hours.
In one sidereal day the earth moves a short distance (about 1°) along its orbit around the sun. So after a sidereal day has passed the Earth still needs to rotate a bit more before the sun reaches its highest point. A solar day is, therefore, nearly 4 minutes longer than a sidereal day.  4 minutes = 240 seconds
The stars are so far away that the Earth’s movement along its orbit makes nearly no difference to their apparent direction and so they return to their highest point in a sidereal day. Another way to see this difference is to notice that, relative to the stars, the Sun appears to move around the Earth once per year. Therefore, there is one less solar day per year than there are sidereal days.
This makes one sidereal day approximately 365.24 / 366.24 times the length of the 24-hour solar day, giving approximately 23 hours, 56 minutes, 4.1 seconds (86,164.1 seconds).
A mean sidereal day is about 23 h 56 m 4.1 s in length.
A tropical year has 365.242190402 days.

1.3 Precession of the Equinoxes (25,920 years)

In astronomy, the ecliptic is apparent great-circle annual path of the sun in the celestial sphere (the projection of objects in space into their apparent positions in the sky as viewed from the Earth) , as seem from the earth.
The plane of the path, called the plane of the ecliptic, intersects the celestial equator (the projection of the earth’s equator on the celestial sphere) at an angle of about 23.5º. This angle is known as the obliquity of the ecliptic and is approximately constant over a period of millions of years.             

The two points at which the ecliptic intersects the celestial equator are called nodes, or equinoxes. The sun is at the vernal equinox about March 21st and at the autumnal equinox about September 23rd.
 The equinoxes do not occur at the same points of the ecliptic every year, for the plane of the ecliptic and the plane of the equator revolve in opposite directions, respectively. The two planes make a complete revolution with respect to each other once every 25,920 years. This movement of the equinoxes along the ecliptic is called the precession of the equinoxes (also known as Platonic Year).

Summary

The earth wobbles like a gyroscope in space spinning once per day while it rotates around the sun during its annual course.


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 of the earth’s axis.
Although it is the Earth that moves on the orbit around the Sun, to the observer on the ground it appears that the Sun, the Moon, planets and stars are moving instead.

1.4 Other cycles

Modern science tells us that Earth’s orbit has few more subtle cycles with very long periods.

Eccentricity Cycle

The shape of Earth’s orbit becomes more elliptical on time scales of about 100,000 years. At present the maximum difference between Sun-Earth distances during the years is only about 3% but over the past several hundred thousand years this number has been as small as 1%( a nearly circular orbit) and as large as 11%.
Eccentricity is the change in the shape of the earth’s orbit around the sun. Currently, our planet’s orbit is almost a perfect circle. There is only about a 3% difference in distance between the time when we’re closest to the sun (perihelion) and the time when we’re farthest from the sun (aphelion). Perihelion occurs on January 3 and at that point, the earth is 91.4 million miles away from the sun. At aphelion, July 4, the earth is 94.5 million miles from the sun.
Over a 95,000 year cycle, the earth’s orbit around the sun changes from a thin ellipse (oval) to a circle and back again. When the orbit around the sun is most elliptical, there is larger difference in the distance between the earth and sun at perihelion and aphelion. Though the current three million mile difference in distance doesn’t change the amount of solar energy we receive much, a larger difference would modify the amount of solar energy received and would make perihelion a much warmer time of the year than aphelion.

Perihelion Precession (Apsidial Precession)

Planets orbiting the Sun follow elliptical (oval) orbits (most orbits in the Solar System have very small eccentricity, making them nearly circular). These orbits rotate gradually over time (apsidal precession).
The orbital ellipse of the Earth precesses in space (primarily as a result of interactions with Jupiter and Saturn). This orbital precession is in the same direction as the gyroscopic motion of the axis of rotation, shortening the period of the precession of the equinoxes with respect to the perihelion from 25,771.5 to ~21,636 years. Apsidal precession occurs in the plane of the Ecliptic and alters the orientation of the Earth’s orbit relative to the Ecliptic. In combination with changes to the eccentricity it alters the length of the seasons.

Orbital Inclination (precession of the ecliptic)

The inclination of Earth’s orbit drifts up and down relative to its present orbit. This movement is known as “precession of the ecliptic” or “planetary precession”.
In the solar system, the inclination of the orbit of a planet is defined as the angle between the plane of the orbit of the planet and the ecliptic — which is the plane containing Earth’s orbital path.
More recent researchers noted this drift and that the orbit also moves relative to the orbits of the other planets. The invariable plane, the plane that represents the angular momentum of the solar system, is approximately the orbital plane of Jupiter. The inclination of Earth’s orbit drifts up and down relative to its present orbit with a cycle having a period of about 70,000 years. The inclination of the Earth’s orbit has a 100,000 year cycle relative to the invariable plane. This is very similar to the 100,000 year eccentricity period. This 100,000-year cycle closely matches the 100,000-year pattern of ice ages.
It has been proposed that a disk of dust and other debris exists in the invariable plane, and this affects the Earth’s climate through several possible means. The Earth presently moves through this plane around January 9 and July 9, when there is an increase in radar-detected meteors and meteor-related noctilucent clouds

Milankovitch cycles

Milankovitch hypothesized that when some parts of the cyclic variations are combined and occur at the same time, they are responsible for major changes to the earth’s climate (even ice ages). He compared data about climatic fluctuations over the last 450,000 years (cold and warm periods) with mathematical theory of variations in eccentricity, axial tilt, and precession of the Earth’s orbit and found they were connected.
The Earth’s axis completes one full cycle of precession approximately every 26,000 years. At the same time the elliptical orbit rotates more slowly. The combined effect of the two precessions leads to a 21,000-year period between the seasons and the orbit. In addition, the angle between Earth’s rotational axis and the normal to the plane of its orbit (obliquity) oscillates between 22.1 and 24.5 degrees on a 41,000-year cycle. It is currently 23.44 degrees and decreasing.  These three orbital variations take place simultaneously. Like overlapping musical tones, these cycles create resonances that are not quite the same as the original cycles. The result is that the Earth’s climate is affected by these Milankovitch cycles on four different periods: 19,000, 23,000, 41,000 and 100,000 years.

 

Movements of the Moon

Eclipse Cycles

The term eclipse is most often used to describe either a solar eclipse, when the Moon’s shadow crosses the Earth’s surface, or a lunar eclipse, when the Moon moves into the Earth’s shadow.
Eclipses may occur when the Earth and the Moon are aligned with the Sun, and the shadow of one body cast by the Sun falls on the other.

An eclipse does not happen at every new or full moon, because the plane of the orbit of the Moon around the Earth is tilted (5°09′ ) with respect to the plane of the orbit of the Earth around the Sun (the ecliptic): so as seen from the Earth, when the Moon is nearest to the Sun (new moon) or at largest distance (full moon), the three bodies usually are not exactly on the same line.

When the Moon is in conjunction with the Sun (on the same plane) passing in front of the Sun, it casts its shadow on a narrow region on the surface of the Earth and cause a solar eclipse.
At full moon, when the Moon is in opposition to the Sun (and on the same plane as Earth’s orbit around the sun) the Moon has to pass through the shadow of the Earth, and a lunar eclipse is visible from the night half of the Earth.
Eclipses are separated by a certain interval of time. The orbital motions of Earth and Moon in relation to each other and the Sun generate repeating harmonic patterns. A particular instance is the saros, which results in a repetition of a solar or lunar eclipse every 6,585.3 days, or a little over 18 years.
An eclipse involving the Sun, Earth and Moon can occur only when they are nearly in a straight line, allowing one to be hidden behind another, viewed from the third. An important aspect of eclipses viewed from the Earth is that apparent mean diameter of the Moon and the Sun is almost the same:    32′ 2″ for the Sun and   31’37″  for the Moon (as seen from the surface of the Earth right beneath the Moon)  and: 1°23′ for the mean diameter of the shadow of the Earth at the mean lunar distance.
Eclipses can occur only when the Moon is close to the intersection of these two planes (the nodes). The Sun, Earth and nodes are aligned twice a year (during an eclipse season), and eclipses can occur during a period of about two months around these times. There can be from four to seven eclipses in a calendar year, which repeat according to various eclipse cycles, such as a saros.
Therefore, at most new moons the Earth passes too far north or south of the lunar shadow, and at most full moons the Moon misses the shadow of the Earth. Also, at most solar eclipses the apparent angular diameter of the Moon is insufficient to fully obscure the solar disc, unless the Moon is close to perigee. In any case, the alignment must be close to perfect to cause an eclipse.

Solar Eclipse
A solar eclipse occurs when the Moon passes in front of the Sun as seen from the Earth. The type of solar eclipse event depends on the distance of the Moon from the Earth during the event. A total solar eclipse occurs when the Earth intersects the umbra portion of the Moon’s shadow. When the umbra does not reach the surface of the Earth, the Sun is only partially occulted, resulting in an annular eclipse. Partial solar eclipses occur when the viewer is inside the penumbra.
Solar eclipses are relatively brief events that can only be viewed in totality along a relatively narrow track. Under the most favorable circumstances, a total solar eclipse can last for 7 minutes, 31 seconds, and can be viewed along a track that is up to 250 km wide. However, the region where a partial eclipse can be observed is much larger. The Moon’s umbra will advance eastward at a rate of 1,700 km/h, until it no longer intersects the Earth’s surface. During a solar eclipse, the Moon can sometimes perfectly cover the Sun because its size is nearly the same as the Sun’s when viewed from the Earth. A total solar eclipse is in fact an occultation while an annular solar eclipse is a transit.

Lunar eclipse

Note:
Our diagram is reflecting the fact that the angular diameter of the Sun viewed from the Moon is almost 4 times smaller than the angular diameter of the earth.
Most of the Internet resources about lunar eclipse (including NASA and Wikipedia) use wrong “eclipse diagram” to explain its geometry:

  Lunar Eclipses of Historical interest: http://eclipse.gsfc.nasa.gov/LEhistory/LEhistory.html

PS Interesting Facts about the Moon

The Moon has mass 81 times small than the Earth. It does not have a significant magnetic field, and it has no atmosphere.
The new moon rises and sets at approximately the same time as the sun.
The full moon rises at sunset and sets at sunrise.


The Moon angular size varies within 12%

Moon’s Path around the Sun
Orbital and Rotational Period  27.322 days
Synodic Period 29.5 days (between the same moon phase)
The moon moves toward the east in our sky by about 12 degrees each day.
In representations of the Solar System, it is common to draw the trajectory of the Earth from the point of view of the Sun, and the trajectory of the Moon from the point of view of the Earth. This could give the impression that the Moon circles around the Earth in such a way that sometimes it goes backwards when viewed from the Sun’s perspective. Since the orbital velocity of the Moon about the Earth (1 km/s) is small compared to the orbital velocity of the Earth about the Sun (30 km/s), this never occurs.
Considering the Earth-Moon system as a binary planet, their mutual centre of gravity is within the Earth, about 4624 km from its centre or 72.6% of its radius. This centre of gravity remains in line towards the Moon as the Earth completes its diurnal rotation. It is this mutual centre of gravity which defines the path of the Earth-Moon system in solar orbit. Consequently the Earth’s centre veers inside and outside the orbital path during each synodic month as the Moon moves in the opposite direction. [ Source: wikipedia]
 Phases of the Moon
Lunar libration
The Moon generally has one hemisphere facing the Earth, due to tidal locking. Therefore, humans’ first view of the far side of the Moon resulted from lunar exploration in the 1960s. However, this simple picture is only approximately true: over time, slightly more than half (about 59%) of the Moon’s surface is seen from Earth due to libration.
Libration is manifested as a slow rocking back and forth of the Moon as viewed from Earth, permitting an observer to see slightly different halves of the surface at different times.
Lunar libration results from the eccentricity of the Moon’s orbit around Earth, a slight inclination between the Moon’s axis of rotation and the normal to the plane of its orbit around Earth and finally,  from a small daily oscillation due to the Earth’s rotation, which carries an observer first to one side and then to the other side of the straight line joining Earth’s and the Moon’s centers, allowing the observer to look first around one side of the Moon and then around the other—because the observer is on the surface of the Earth, not at its center.

Other Important Moon Facts

Earth diameter = 12,742 km. Earth-Moon distance: 378,028 km (average, moon surface to Earth center) so Earth angle from the moon = arctan ( 12,742 / 378,028) = 1.93 degrees.
[Note: a more precise calculation, using the Apollo 11 site, gives an average value of 1.909 degrees, plus or minus 0.150 degrees as the moon orbits.]
Earth-Sun distance = 149,598,261 km so Earth angle from the Sun is arctan (12,742 / 149,598,261) = 0.00488 degrees = 17.57 arc seconds.
Formation
The prevailing hypothesis today is that the Earth–Moon system formed as a result of a giant impact: a Mars-sized body hit the nearly formed proto-Earth, blasting material into orbit around the proto-Earth, which accreted to form the Moon.

Part 2: Archaeoastronomy

Archaeoastronomy is the study of how people in the ancient past have understood and used the phenomena in the sky, and what role the sky played in their cultures.
Ancient monuments and writings from around the world are undeniable proof that thousands of years ago people had advanced astronomical knowledge. We propose that in fact all ancient sites of any major significance were constructed in order to allow astronomical observations and time tracking – and often, as the record/expression of their knowledge.
  In this article we explore connections of famous ancient sites to astronomy (Note:  In Part 4 we explain ancient calendars and units of measure.)

The heliocentric universe was known in antiquity

Thousands of years before Copernicus published De Revolutionibus Orbium Coelestium people considered the Sun as the center of our planetary system.
Aristarchus of Samos (ca. 310-230 BC) and others proposed that the sun was motionless in the center of an infinitely large sphere of fixed stars, and that the earth revolved about it, as well as rotating about its axis. The story is told in Sir Thomas Heath’s Greek Astronomy (1932, New York: Dover, republished 1991) and Aristarchus of Samos, the Greek Copernicus (1913). The lack of an adequate theory of motion and inertia as well as the position of the Church) caused the usual view of a fixed earth to prevail.
Aristarchus also estimated the distances of the sun and moon from the earth. The measurements were not accurate enough to give a correct result, but did show that the sun was much more distant than the moon. Eratosthenes, somewhat later, estimated the radius of the earth, obtaining a value that happened to be accurate, although his estimate was thought too large (which encouraged Columbus). These accomplishments are of the greatest significance, as we can see from our perspective. No other science or religion in history came within miles of a similar understanding.
Archimedes discusses Aristarchus in his Sand-Reckoner. A translation is: Aristarchus of Samos proposed certain hypotheses in writing, from which it followed that the universe was much larger than is now commonly believed. He proposed that the firmament and the sun were unmoving, and the earth described a circle about the sun, which was located at the centre of the orbit, and that the sphere of the fixed stars was as great relative to the orbit of the earth as a sphere is to its center.
Antikythera Mechanism is undeniable example of advanced astronomical knowledge possessed by astronomers thousands of years ago.

3.1 Astronomical Alignments of Ancient Monuments (Pyramids and Temples)

The evidence of  very good understanding by the ancient people of astronomical cycles of our planet can be found in their monuments (pyramids and temples) and  surviving written records (found in Mesoamerica, Egypt and India). Places like Stonehenge and Pyramids of Egypt and Mesoamerica reveal in their location, orientation, shape and dimensions incredible astronomical knowledge.
The easiest way to record spatial observations — other than by framing them against a prominent topographic features — was by erecting durable markers of their own in the landscape (including pyramids and temples)  to calibrate the rising or setting azimuths of the sun, moon, planets and stars.
Most of the ancient structures were used as astronomical markers/observatories and functioned as calendars. Below we are presenting just a few examples.

Nabta, Egypt

STANDING STONES at Nabta in the Nubian desert  predate Stonehenge and other astronomically aligned sites in Europe (site’s age estimate: between 6400 and 4900 BC)
They are the oldest dated astronomical alignment discovered so far and bear a striking resemblance to Stonehenge and other megalithic sites constructed a millennium later in England, Brittany, and Europe.
In Nabta, there are six megalithic alignments extending across the sediments of the playa, containing a total of 24 megaliths or megalithic scatters. Like the spokes on a wheel, each alignment radiates outward from the complex structure.
 These standing megaliths and ring of stones were erected from 6.700 to 7,000 years ago in the southern Sahara desert. Although more research needs to be done, many scientists, believe that the alignments had an astronomical significance.
Three hundred meters north of these alignments is the stone calendar circle. Compared to Stonehenge, this circle is very small, measuring roughly 4 m in diameter. The calendar consists of a number of stones, the main ones being four pairs of larger ones.


Each of these four pairs were set close together to form “gates.” Two of these pairs align to form a line very close to a true north-south line, and the other two pairs or gates align to form an east-west line. The east-west alignment is calculated to be where the sun would have risen and set from the summer solstice 6,500 years ago (4,500BC).  

 

Stonehenge, England

Archaeologists suggest that the iconic stone monument was erected around 3000BC – 2500 BC. Many Stonehenge alignments exist which undoubtedly indicate the astronomical significance of Stonehenge’s construction. For  example, the axis of Stonehenge aligns approximately to the midsummer rising sun azimuth.
Stonehenge is a megalithic monument on the Salisbury Plain in Southern England, composed mainly of thirty upright stones (sarsens, each over ten feet tall and weighing 26 tons), aligned in a circle, with thirty lintels (6 tons each) perched horizontally atop the sarsens in a continuous circle. There is also an inner circle composed of similar stones, also constructed in post-and-lintel fashion.
In the foregoing picture a line (blue) originating at Stonehenge’s centre, dissects the centre of the Heel stone. The azimuth angle of this line, off North, is 51.18333 degrees, which equates to 51 degrees, 11 minutes. This is the official latitude designation for Stonehenge (51 deg. 11 minutes).
Note how the line relates to the Avenue set of circles (magenta) and brushes the northern side of the large post marker adjacent to the Avenue circles. A nearby line of posts extends toward the Heel Stone, as if to indicate this “latitude” line.
Another circle of immense importance, which links Stonehenge to the Lunar codes of the Khafre Pyramid of Egypt. It will be noted that this circle (2nd inward red) brushes two component positions on the Avenue, one of which has the official designation “B”. The diameter of this circle is 472.5 feet, which is exactly the intended vertical height of the Khafre Pyramid of Egypt. The base measurement of Khafre was 15/16ths that of the Great Pyramid or 708.75 feet. It was also built to a 3,4,5 triangulation code, with 1/2 the base length acting as the adjacent (354.375 feet), the vertical height acting as the opposite (472.5 feet) and the diagonal face acting as the hypotenuse (590.625 feet).
Each of these values was in deference to the lunar month and lunar year (based upon 29.53125 days per lunar month or 345.375 days per lunar year).
The diameter of this Stonehenge circle is, therefore, coding the height of the Khafre Pyramid in increments of 16 X 29.53125-days/ feet. This value of 472.5 days was also integral to the ancient method of measuring the 18.613-year lunar nutation cycle, which was calibrated to endure for 6804-days (230.4 lunar months of 29.53125-days or 14.4 time periods of 472.5-days duration). Note also that 230.4 is an expression of the very important ancient number 11.52.

Chankillo, Peru

Possibly the oldest solar observatory in the Americas has been found, suggesting the existence of early, sophisticated Sun cults, scientists report.
It comprises a group of 2,300-year-old structures, known as the Thirteen Towers, which are found in the Chankillo archaeological site, Peru. The towers span the annual rising and setting arcs of the Sun, providing a solar calendar to mark special dates. The study was published in the journal Science.
Clive Ruggles, professor of archaeoastronomy at Leicester University, UK, said: “These towers have been known to exist for a century or so. It seems extraordinary that nobody really recognised them for what they were for so long.
Story from BBC NEWS:
http://news.bbc.co.uk/go/pr/fr/-/2/hi/science/nature/6408231.stm  © BBC 2011

Tiwanacu (Tiahuanaco), Bolivia

1,500 BC is the “official age of ruins near Tiwanacu, however there is strong evidence that this site could be as old as 15,000BC …The main archaeological site at Tiwanaku has five primary structures, including the Akapana pyramid and the Kalasasaya temple. Less than a kilometre to the south lies Puma Punku, with its man-made platform and megalithic ruins. For centuries the sites have been looted, vandalized and used as a quarry for stone, and all the walls now standing have been reconstructed. Many of the statues were smashed during a campaign by the Catholic church to wipe out idolatry.  Excavations of the Akapana have revealed a sophisticated, monumental system of interlinked surface and subterranean water channels. Alan Kolata comments that the system, ‘although superbly functional, is over-engineered, a piece of technical stone-cutting and joinery that is pure virtuosity’.

But it is an established fact that whatever calculation might be used to determine the age of the Temple of the Sun of Tihuanacu, on the basis of the variation of the obliquity of the ecliptic from those times until today, would demonstrate that [this] American solar observatory is more ancient than any monument of man in the world of which we know up to this time.“ 

Another stonework at Tiwanaku showing of almost identical image of the Sun-God Viracocha	This wonderful graphic by Ken Bakeman shows colored version of the Sun-god from the Gate of the Sun relief. Image source: http://www.aztectees.com/andes.html

Many people have stood in front of the massive stone monument known as the “Gate of the Sun” in the ancient city of Tiwanaku in the Bolivian Andes. They admire the craftsmanship of the small carved figures known as “Chasquis” or “Messengers of the Gods” and ponder over the use of the gate as an ancient calendar. But in fact, the gateway itself is not the calendar, the calendar is a little known row of 11 giant upright stones, now built into a wall which exists just behind the gateway. Today, there are only 10 stones in the wall and the missing 11th stone lies face down some distance out in the field behind the wall. When the 11 stones were in their original positions in a row, the sun would set each evening over the row of stones so that priests standing in the centre of the adjacent courtyard could easily calculate the time of the year in a remarkable calendar which divided the year into 20 periods of 18 days and also meshed with a lunar calendar so that three solar years equalled 40 sidereal lunar months which was 2 “zocam” years and even more remarkably it also meshed with a Muisca lunar period of 37 months. This meant that every 30 solar years was at the same time 20 Muisca Zocam years of 20 sidereal months of 27.32 days and 10 Muisca Acrotom years of 37 synodic months of 19.53 days. Additionally every thirty years an extra month had to be added to both lunar calendars to keep them synchronised with the solar calendar and this is commemorated in the Gateway of the Sun with thirty Chasquis marking the solar years and forty condor’s heads marking the lunar months. So the Gate of the Sun is not the actual calendar but the key to operating the calendar…

LOST CALENDAR OF THE ANDES, the calendar of Tiwanaku and of the Muisca

Background

In the year 2,000, I found myself in Tiwanaku with the Discovery Channel filming for “Atlantis in the Andes” and in the company of Oscar Corvison, a Bolivian Archeoastronomer who was keen to explain his interpretation of the vigesimal (base 20) system of the Tiwanaku calendar.  Oscar explained that it was not the Sun Gate which was the Tiwanaku calendar, but a wall which today is to one side of the Sun Gate, inside a courtyard called the “Kalasasaya”. He was particularly upset because he said, in the reconstruction of the western wall of the Kalasaya, one of the large stones which had originally been part of the calendar had not been restored, but left laying in a field a couple of hundred metres to the west of the wall.
Since my own fascination is for ancient measurements rather than calendars, I began to study Posnansky’s measurement of the wall, and discovered the wall was not simply a solar calendar as had been previously thought, but incorporated a sophisticated calendar based on sidereal lunar months.
The rest of this essay and the discovery of the sidereal lunar calendar follows on from Oscar Corvison’s pioneering discovery of the base 20 system in the Tiwanaku calendar.

Tiwanaku Calendar

 This is how it works. In the centre of the Kalasasaya courtyard there is a large block of stone which is said to represent the original observation point. From here the sun could be watched setting on the horizon over the pillars each night. When the sun set over the central pillar, the day would be the 22nd September (equinox) and Spring would begin (seasons reversed in southern hemisphere). . When the sun set over the next pillar to the left, one twentieth of a year would have passed and so on until reaching the pillar at the far left a quarter of a year later on the 21st December marking the Summer Solstice.
Posnansky seems to have considered the row of pillars as representing a calendar based upon a month of 30 days – probably because 30 small figures called “Chasquis” appear on the Gate of the Sun – and states that the solar year of twelve months was used with the sun showing through the gap between the pillars each month. But there’s a flaw with that. With eleven pillars, there are only 10 gaps or spaces, not 12 …
The Inca were sometimes said to be people of the sun, whereas the Aymara were sometimes said to be people of the moon, so I wondered whether in fact the pillars may also have been a soli-lunar calendar since what is called the ‘Saros’ cycle of lunar eclipses repeats itself every 20 ‘Inca’ years and 20 ‘Inca years’ of 12 months of 27.32 days is very close to 18 solar years of 365.24 days (Allen 1998 and Aveni 1990) and the people who built Tiwanaku were a race long before the Inca and possibly even before the Aymara. On the other hand, if used for agricultural purposes, it may simply have marked the winter and summer solstices with the appropriate pillar or space between the pillars marking the return of the sun to a suitable time for plantings crops, which is what Posnansky thought the purpose of the calendar was in the first place.
Although Corvison was correct in identifying the use of a solar calendar based on divisions of 20, (and this should not be a surprise since both the Aztec and Maya civilisations used a base 20 calendar) he does not seem to have considered the possibility it could also have been a lunar calendar.
On the above basis, when the sun reached the first pillar it would have travelled a 1/20th of a solar year which is 18.26 days. By the time it reached midway to the next pillar, it would have travelled half as much again, which when added to the first figure means 27.39 days would haved passed — virtually a sidereal lunar month —   every one and a half pillars would add another sidereal month and continuing the process would take us back to the central pillar after 13 and a third such sidereal lunar months (or divisions) had passed, completing a solar year and making it a dual purpose, soli-lunar calendar.
Now I wondered if this in some way tied in with the Saros cycle and since it takes thirteen and a third sidereal lunar months to circle round the calendar stones in order to complete one ‘lap’ and come back to a full year, how many ‘laps’ would it take to fulfil the Saros cycle?
Well, three ‘laps’ round the pillars would make the sun once more over the central pillar and represent 40 sidereal lunar months and since each lap around the pillars is a solar year, a total of 18 ‘laps’ round the pillars would complete the Saros cycle, the sun would be back again over the central pillar and the cycle would all begin all over again!
Maybe that’s why the Amautas (mathematicians) of the Aymara thought they had discovered the most perfect calendar in the world. Could this be the calendar of Atlantis? Some people thought so (Corvison 1996), but they failed to realise the Altiplano was Atlantis.
In addition to counting the Inca lunar year of 12 sidereal lunar months (328 days) the calendar also represents a year of 360 days as well as a year of 365.24 days. How it could do that may be something like this. From the centre to the centre of the end pillars is taken as 360 days (counting from one end to the other end then back again) then the distance from the outside to the outside of the opposite pillar (and back again) would represent 365.24 days. In this way, the calendar could mesh the Solar calendar with the Lunar calendar, the extra five and a quarter days being ‘lost’ (to view) when the sun reaches the end pillars and appears to stand still before returning in the opposite direction. Each division from pillar to pillar would be 18 days, which could be arranged in groups of 2 x 9 days.
It seems that in the Andes, a work period of six weeks of nine days was used, which would therefore correspond to three divisions of the pillar calendar and be two sidereal lunar months.
Beneath these chasquis there is a continuous row of smaller icons arranged so that eleven of them stand apart from the rest. We can assume that these eleven represent the pillars of the calendar. Now it has usually been wrongly assumed that because the upper chasquis in horizontal rows total fifteen on each side (not counting the outer ones) that the total of thirty chasquis represent a month of 30 days since a solar year of 360 days divided by 12 months would give a 30 day month. But as explained above, the actual calendar is divided by 20, which would make solar divisions of 18 days. And work periods of 18 days were used in the Andes.
The reason why people can’t see the correct number of chasquis on the lower freize of the Sun Gate is because the eleven chasquis in a row represent a circular or elliptical orbit, so the two end chasquis represent the solstices when the sun reaches the ends of the orbit, but the remaining nine chasquis conceal another chasqui behind them so to speak (if viewing the orbit in plan view) so the total is two end chasquis plus eighteen ‘double’ chasquis making 20 all told.
Many people have mistakenly thought that the Gate of the Sun was the calendar, but it isn’t. The pillar stones built into the west wall are the calendar and it could be instead, that the chasquis are telling you how to operate the calendar.
Instead of reading horizontally, if we read vertically, they seem to be saying, ‘count in blocks of three.’ But blocks of three what? When we studied the operation of the stones on the wall, we found that every one and a half pillars represented one sidereal lunar month. Therefore every half division between the pillars represented one fortieth of the year or a third of a sidereal lunar month, the month itself being the prime unit. Now on the Gate of the Sun there are a total of 48 Chasqui icons which could therefore represent 48 sidereal lunar months. Tahuantinsuyo, the empire of the Incas was ‘the land of the four quarters, or four divisions’ so dividing the 48 Chasquis by 4 results in 12 Chasquis — meaning 12 sidereal lunar months — which was the Inca lunar year of 328 days. In turn 328 days divided by 4 gave the 82 day (three month) period at the end of which the moon would be visible against the same group of stars etc and that I believe, is the message of the Chasquis — how to operate the calendar.
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.

Mayan Astronomy

The Maya were expert sky-watchers, careful observers of the motions of the celestial bodies. Proof of the Mayan fascination with astronomy is literally carved in stone in the grand architecture at sites such as Teotihuacan, Chichén Itzá, Uxmal, Uaxactun, Edzna, and dozens more. At many of these sites, hieroglyphic carvings refer to celestial bodies and cycles. Often, the buildings they adorn have been built to align with significant cyclical astronomical events — solstices, equinoxes, the shifting moon, or the rise of planets and stars.
Some ancient observatories were located in flat, desert-like areas (e.g. Teotihuacan), other (e.g. Chichen Itza,  Tikal) were built in a jungle and required very tall foundations for observation platforms and pyramids in order to observe  the sky from above the jungle canopy.

Teotihuacan, Mexico

The city of Teotihuacán is meticulously laid out on a grid which is offset 15º.5 from the cardinal points. Its main avenue, the “Street of the Dead,” runs from 15º.5 east of north to 15º.5 west of south, while its most impressive structure, the Pyramid of the Sun, is directly oriented to a point 15º.5 north of west — the position at which the sun sets on August 13.

Here, on the Plateau of Mexico, cityplanners had built a ceremonial center with a configuration commemorating a date whose calendrical importance had first been recognized 1000 km (600 miles) away to
the south more than a millenium earlier. And, in addition, they had located the city with such precision that it was also aligned to the highest mountain in all of Mexico at sunrise on the winter solstice –Citlaltépetl, or Orizaba (5700 m, or 18,700 ft. in elevation). To be sure, a low ridge obscures the mountain from the direct view of anyone standing atop the Pyramid of the Sun, but the alignment is so exact that in a paper published in 1978 I hypothesized that a “relay station” of sorts must have been constructed on the intervening ridge to alert the priests of the solsticial sunrise. In January, 1993, with the help of a GPS (i.e., global position-ing system) receiver I managed to confirm that such a “station” had in fact existed.  — Professor Vincent Malstrom.
There is no doubt that the Palace of Quetzalcoatl and the surrounding platforms functioned as an astronomical observatory (and calendar). The strong clues come from the direction of the sunrise and sun set on key days over the year. We suggest that the platforms were used for observing not only the sun but also the moon and the stars. Below are few undeniable examples.

Edzna, Mexico

This is a Maya archaeological site in the north of the Mexican state of Campeche. The most remarkable building at the plaza is the main temple. Built on a platform 40 meters high, it provides a wide overview of the surroundings. It was one of the Mayan astronomical observatories.

 Astronomical alignment on days of Solstice. Edzna was a luni-solar observatory. Click to enlarge.

 The astronomical importance of Edzná may be gauged from these facts:

1. only at its specific latitude could the beginning of the Maya new year be calibrated, here with the assistance of a remarkable gnomon;
2. the “day the world began” was commemorated in the “gun-sight” orientation between the doorway of Cinco Pisos and the small pyramid across the plaza; and
3. lunar cycles were measured by using the line of sight between Cinco Pisos and “La Vieja” on the northwestern horizon.

Although we cannot be certain when the Maya finally succeeded in working out the lunar eclipse cycle, it would seem that most of the basic “research” on the problem was carried out at Edzná. Located some 300 m (1000 ft) to the northwest of Cinco Pisos is the ruin of a lofty pyramid which Matheny has termed “La Vieja,” or the “Old One.”[...] Even in its dilapidated condition it is still high enough to intersect the horizon as seen from the top of Cinco Pisos; indeed, it is the only manmade construction which does so. This fact immediately prompted me [Dr. Vincent H. Malmström ] to measure its azimuth as seen from Edzná’s commanding edifice, and the value I obtained was 300º. This means that the summit of the pyramid lies exactly 5º beyond the sun’s northernmost setting position at the summer solstice. Because the moon’s orbit is just a hair over 5º off that of the sun, it seems very likely that the Northwest Pyramid, or “La Vieja,” had been erected as a horizon marker to commemorate the moon’s northernmost stillstand. Not only is “La Vieja” an eloquent testimonial to the patience and accuracy of Maya “science,” but because of its specialized function, it is also probably worthy of being designated as the oldest lunar observatory in the New World. (Indeed, if Matheny’s dating of “La Vieja” is accurate, then it is apparent that the Maya had succeeded in measuring the interval between lunar stillstand maxima at least by A.D. 300.)

Chichen Itza, Mexico

As practitioners of naked-eye, horizon-based astronomy, it is obvious that the Maya never recognized the existence of “nodes”, or points where the moon’s orbit intersected that of the sun; the simple reason was that they had no means of defining any celestial position other than the zenith. At least by marking the northernmost setting point of the moon against the horizon they obtained a fixed point from which they could count the number of days that elapsed between two lunar events.  For the Maya the easiest way to record spatial observations — other than by framing them against a prominent topographic features — was by erecting durable markers of their own in the landscape to calibrate the rising or setting azimuths of the sun, moon, planets and stars.

The Maya were known to be great mathematicians and are credited with the invention of the “zero” in their counting system. They were also great astronomers, and EL Castillo is a perfect marriage of their sciences with their religion. By far the most amazing aspect of the pyramid is the accuracy, significance, and relevance it has within the Mayan calendar and social system. There are many numerical details regarding the location of this structure that could not have all occurred by accident. Each side of the pyramid is made up of nine larger tiers or layers with a staircase in the center of each side leading to the temple at the top. Each stairway consists of ninety one steps, with one step at the top common to all four sides, for a total of three hundred and sixty five steps, the exact number of days in a solar year. Each side of the pyramid has fifty two rectangular panels, equal to the number of years in the Mayan cycle (at the conclusion of which they typically constructed a newer structure over an older one). The stairways divide the tiers on any given side into two sets of nine for a total of 18 tiers which corresponds to the 18 months in the Mayan calendar. The “square” that makes up the overall base of the structure is exactly 18 degrees from the vertical. Every aspect of the structure relates in some way to the Maya and their culture. The very physical presence of this structure and the shadows it casts, are also significant within the Mayan culture and are more fully explained in here the section detailing the Shadow Of The Equinox.
The Maya universe was comprised of 13 “compartments” in 7 levels with each compartment being ruled over by a different god. El Castillo reflects these beliefs as seen in the shadows it casts. 7 levels are shown in the 7 light triangles. 7 Triangles of light and 6 darker triangles give 13 triangles in all corresponding to the 13 overall levels of the underworld.
Suffice it to sum up here and say, the pyramid casts unique and identifiable shadows on the exact days of the year that represent the solstice and equinox that occur twice a year. This shows the Maya were aware of the rotation of the sun and the exact length of a year. Indeed, we know that the Mayan Calendar was more accurate than the one we use today.
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The phenomenon that El Castillo is famous for occurs twice each year, at the spring and fall equinoxes. (In fact, the effect is viewable for a week before and after each equinox.) As the equinox sun sets, a play of light and shadow creates the appearance of a snake that gradually undulates down the stairway of the pyramid. This diamond-backed snake is composed of seven or so triangular shadows, cast by the stepped terraces of the pyramid. The sinking sun seems to give life to the sinuous shadows, which make a decidedly snaky pattern on their way down the stairs.


Other features of El Castillo suggest astronomical understanding and intent on the part of the Mayan builders. The structure as a whole seems to be aligned with an important astronomical axis: The west plane of the pyramid faces the zenith passage sunset. Meanwhile, each of the four (exceedingly steep) stairways that climb the pyramid has 91 steps, with a final step at the top making a total of 365, the number of days in a solar year. Ninety-one is also the number of days that separate each of the four phases of the annual solar cycle: winter solstice, spring equinox, summer solstice, and fall equinox.
Using the patterns of light and shadow appearing on El Castillo throughout the year, the Maya could easily have tracked the seasons and marked these four annual solar events—the two solstices and two equinoxes. And so it seems the ancient Maya may have used this structure as, among other things, a calendar to signal appropriate times to plant, harvest, and perform ceremonies.



El Caracol seems to be carefully aligned with the motions of Venus. Venus had tremendous significance for the Maya; this bright planet was considered the sun’s twin and a war god.
The grand staircase that marks the front of El Caracol faces 27.5 degrees north of west—out of line with the other buildings at the site, but an almost perfect match for the northern extreme of Venus, Venus’s most northerly position in the sky. Also, a diagonal formed by the northeast and southwest corners of the building aligns with both the summer solstice sunrise and the winter solstice sunset.

In the half-ruined higher tower of El Caracol, three openings survive. These three openings are small, narrow, and irregularly placed, suggesting that they are actually viewing shafts. It turns out that these windows do in fact align with important astronomical sightlines. Looking through these windows a thousand years ago, observers could have watched for Venus rising at its northern and southern extremes, as well as the equinox sunset. The three window shafts that remain in the upper tower of El Caracol seem to align with various celestial events on the horizon.
At Chichén Itzá, the zenith passage is experienced on May 23 and July 20, give or take a day.
Parts of the above segment belong to http://www.exploratorium.edu/ancientobs/chichen/HTML/castillo.html

Tikal, Guatemala

While it may be of interest to know that some of the pyramids of Tikal also served as the final resting place of members of the Mayan elite, their primary function was to serve as observation platforms for priests working with the calendar.
Dr. Malmström discovered that pyramids had been constructed as an astronomical matrix whose purpose it was to calibrate the most important dates in the Maya year.
The sight-line between Temple I and Temple IV (the highest of the pyramids) marks the sunset position on August 13, whereas the sunrise position at the winter solstice is perpetuated in the sight-tine between Temple IV and Temple III. Because Temple I and Temple III are sited due east-west of each other, they mark sunrise and sunset alignments at the equinoxes. Although there was no star located directly above the earth’s pole of rotation in Maya times, a sight-line from Temple V to Temple II appears to have marked the most westerly position of the Maya’s equivalent to a polestar, Kochab.
The western horizon at Tikal as seen from Temple I. The low, squat structure in the middle foreground is Temple II, which serves not only as an architectural counterweight to Temple I as seen across the plaza of Tikal but also as a horizon marker for the enigmatic “8º west of north” orientation when viewed from Temple V. The latter orientation was present at La Venta about 1000 B.C., but also shows up at the Maya capital about A.D. 800. Farther to the left, Temple III defines the equinoctial sunset position as seen from Temple I, while the highest of the skyscraper pyramids — Temple IV, on the right — fixes the sunset position on August 13 as seen from Temple I.

 

 

The Giza Pyramids

“…it appears that there was drawn a plan of the Great Pyramid which included the calculation of the stars to be observed in order to obtain the direction of the north. After this plan was drawn, the ground of the Pyramid had to be cleared in order to proceed to the ceremony called “stretching the cord,” which for the Egyptians was the equivalent of our laying of the first stone. This ceremony had the purpose of establishing the direction of true north and, as the Egyptians saw it, suspending the building from the sky by tying the building with an imaginary string to the axis of rotation of the vault of heaven.” (Tompkins, Secrets of the Great Pyramid, pp. 380 and 381).
In ancient Egypt rope stretchers were surveyors who measured property demarcations and foundations using knotted cords which they stretched in order to take the sag out of the rope. When performed by kings during the initial stage of temple building the Stretching of the Rope was probably a religious ceremony rather than a surveying job.
 Egyptian architects, surveyors and builders are known to have used two specialised surveying tools, the merkhet (the ‘instrument of knowing’, similar to an astrolabe) and the bay (a sighting tool probably made from the central rib of a palm leaf). These allowed construction workers to lay out straight lines and right-angles, and also to orient the sides and corners of structures, in accordance with astronomical alignments. ( Read more about orienting pyramids here: http://www.world-mysteries.com/alignments/mpl_al4.htm )
It is clear that the Egyptians were using their knowledge of the stars to assist them in their architectural projects from the beginning of the pharaonic period (c.3100-332 BC), since the ceremony of pedj shes (‘stretching the cord’), reliant on astronomical knowledge, is first attested on a granite block of the reign of the Second-Dynasty king Khasekhemwy (c.2650 BC). Some of the Ancient Egyptian architecture was used for calendric (timekeeping ) purposes. 
It seems that Egyptian pyramids layout was designed to mark key dates in Egyptian calendar by position of the rising and setting sun ( just like in the case of the pyramid of Kukulcan in Chichen Itza).
It also appears that shadows of the Giza pyramids were designed to mark the key calendar dates.
From a very early time, the ancient Egyptians had a form of calendar based upon the phases of the moon followed by a solar calendar system of 360 days, with three seasons, each made up of 4 months, with thirty days in each month. The seasons of the Egyptians corresponded with the cycles of the Nile, and were known as Inundation (Akhet which lasted from mid June  to October ), Emergence (Perety which lasted from mid October to February ), and Summer (pronounced Shemu which lasted from mid February to June ).
 
To be continued — we are preparing much more material on this subject…

Ancient Timekeepers, Part 3: 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.
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).

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…

The Solar and Lunar Cycles

Although the “ideal” periods of the solar and lunar cycle are described by numbers very close to the current values, calendars must reflect the correct numbers in order to properly keep track of time and have seasons in sync.
At present the time for Earth to complete full orbit around the Sun in “Solar Days” is 365.242199 days* long (365 days 5 hours 48 minutes 46 seconds). Earth orbits the Sun once every 366.242 times it rotates about its own axis in relation to stars but in relation to the sun it turns only 365.242 times to complete its orbit.
The time between full moons is 29.530589 days.
So a month measured by the moon doesn’t equal an even number of days, and a solar year is not equal to a certain number of moon cycles (months or “moon”ths).

The Lunar Month

The moon repeats its cycle of phases every 29.53 days, and since it is a mysterious heavenly object, it is ideal for religious observances. Religious observances can be coordinated with the phases of the moon, and time divided into months (“moonths”), each month beginning when the thin crescent of the new moon is seen in the twilight.
The month can then be divided into days, which now are more memorable by being part of the longer cycle, and certain things are done on certain days.
The problem is that the month is not an integral number of days. This only worries the orderly mind, which wants some fixed number of days so the future can be precisely planned. One solution, which is actually used, is to alternate months of 29 and 30 days, which only slowly gets out of register with the moon.
What happens when we are in the evening of the 30th day of some month, and the new crescent is not seen as it should be? Well, we could start the month anyway, or we could add a day, called an intercalary day, hoping to see the crescent the next evening. Both methods have been used, but the latter is the most popular with the precise minds. This day we can make a holiday, or the occasion for an extra gift to the priests, or a day for athletic contests or drinking.

 The Solar Calendar

There is another great and very obvious cycle, the year. The observant notice that the sun moves northward and southward, through the equinoxes and solstices, as the cycle of nature in the temperate latitudes takes place. For an agricultural society, this cycle is of practical importance. Besides, it is also a good impetus for religious observances.
The year is about 365.24 days long, or 12.37 lunations, unfortunately. The cycles of day, month and year do not mesh evenly.
One alternative is simply to neglect the year, and base the calendar so that it stays in step with the moon, using an intercalary day now and again. 12 months are arbitrarily defined to be a year. This gives about a 354-day year, which is 11 days short of a real year. In about 33 years, the months would go completely through the seasons. People who don’t do much agriculture are happy with such a calendar, which is called lunar.
If you can’t do this, several shifts are possible. First, you can insert 11 extra days in every year, and devote them to a general festival or something. Or, you can insert a whole month every two or three years, as necessary, to keep the months in register with the seasons. King Numa Pompilius’ calendar of 713 BC was of the latter type, with 12 lunar months every year, and extra months thrown in now and then. It replaced Romulus’s calendar of 10 months and only 303 days which began on 1 March, and gave the ordinal names to the months (September, October, etc.) from the order they had in that calendar. These compromise calendars are termed luni-solar.
With good astronomy, it is perceived that the sun makes an annual journey through the stars on the path called the ecliptic. Of course, the average person can’t see the stars and the sun at the same time, so this takes a degree of scientific sophistication. One can divide the ecliptic into 12 segments of 30° each, the signs of the Zodiac. The passage of the sun through each segment is a zodiacal month. These months are not equal in length. Those of the winter are shorter than average, and those of the summer longer than average. It is possible to design a calendar with, say, 12 months of arbitrary but unvarying lengths that correspond roughly with the sun’s residence in each sign. The Egyptians logically took 30 days in each month, making 360 days in the year. Such a calendar is solar. The Egyptian year, being a little short, moved through the seasons slowly in a 72-year cycle. Since the actual year was of importance to them–it timed the rise of the Nile–5 intercalary days were added to get an alternative 365-day year. This ran through the seasons only once in 1,520 years or so. Since events were dated not only by these calendars, but also by the heliacal rising of Sirius, the differences allow precise calibration of Egyptian dates, something not possible anywhere else. Egyptians had little interest in the moon’s phases, since their religion was tied to the solar year and the Nile.

The Metonic Cycle

Meton of Athens was a Greek mathematician, astronomer, geometer, and engineer who lived in Athens in the 5th century BC. He is best known for calculations involving the eponymous 19-year (6,939.602 days) Metonic cycle which he introduced in 432 BC into the lunisolar Attic calendar. Meton approximated the cycle to a whole number (6940) of days, obtained by 125 long months of 30 days and 110 short months of 29 days. In the following century Greek astronomer Callippus developed the Callippic cycle of four 19-year periods for a 76-year cycle with a mean year of exactly 365.25 days.
Ironically, whereas the Metonic cycle overestimates the length of a solar year by 5 minutes, the Callippic cycle underestimates the length of a solar year by 11 minutes, and therefore produces results that are less accurate than those produced using the Metonic cycle.
The world’s oldest known astronomical calculator, the Antikythera Mechanism (2nd century BC), performs calculations based on both the Metonic, and Callipic calendar cycles, with separate dials for each.
It seems that Meton was not aware of  precession and did not make a distinction between sidereal years (currently: 365.256363 days) and tropical years (currently: 365.242190 days).

• 19 tropical years = 6939.602 days
• 235 synodic months (lunar phases) = 6939.688 days (Metonic period by definition)
• 254 sidereal months (lunar orbits) = 6939.702 days (19+235=254)
• 255 draconic months (lunar nodes) = 6939.1161 days

A lunar year of 12 synodic months is about 354 days on average, 11 days short of the 365-day solar year. Therefore, in a lunisolar calendar, every 3 years or so there is a difference of more than a full lunar month between the lunar and solar years, and an extra (embolismic) month should be inserted (intercalation). The Athenians appear initially not to have had a regular means of intercalating a 13th month; instead, the question of when to add a month was decided by an official. Meton’s discovery made it possible to propose a regular intercalation scheme. The Babylonians appear to have introduced this scheme well before Meton, about 500 BC.
The Metonic cycle is related to two less accurate sub-cycles:

• 8 years = 99 lunations (an Octaeteris) to within 1.5 days, i.e. an error of one day in 5 years; and
• 11 years ( i.e. 19 less 8 ) = 136 lunations within 1.5 days, i.e. an error of one day in 7.3 years.

By combining appropriate numbers of 11-year and 19-year periods, it is possible to generate ever more accurate cycles. For example simple arithmetic shows that:

• 687 tropical years = 250921.39 days
• 8497 lunations = 250921.41 days

giving an error of only about half an hour in 687 years (2.5 seconds a year), although this is subject to secular variation in the length of the tropical year and the lunation.

Calendars

A calendar is a system of organizing time. It is used for social, religious, commercial, or administrative purposes. Timekeeping is done by giving names to periods of time, typically days, weeks, months, and years. Periods in a calendar (such as years and months) are synchronized with the cycle of the sun or the moon (ancient astronomers also used planet Venus and/or star Sirius) .
The astronomical day had begun at noon ever since Ptolemy chose to begin the days in his astronomical periods at noon. He chose noon because the transit of the Sun across the observer’s meridian occurs at the same apparent time every day of the year, unlike sunrise or sunset, which vary by several hours. Midnight was not even considered because it could not be accurately determined using water clocks. Nevertheless, he double-dated most nighttime observations with both Egyptian days beginning at sunrise and Babylonian days beginning at sunset.

Julian Day

Probably most fundamental calendar is a count of days. A calendar that is a pure count of days is the Julian Day.
The Julian day number is based on the Julian Period proposed by Joseph Scaliger in 1583, at the time of the Gregorian calendar reform, but it is the multiple of three calendar cycles used with the Julian calendar:  15 (indiction cycle) × 19 (Metonic cycle) × 28 (Solar cycle) = 7980 years
Day 1 was January 1, 4713 BC Greenwich noon, and we are now up in the millions, but this count is reliable, and the best way to find the time between two events. The Julian Day is divided decimally, and begins at noon at Greenwich. It is the least confusing way to identify a particular day. Julian day is used in the Julian date (JD) system of time measurement for scientific use by the astronomy community, presenting the interval of time in days and fractions of a day.  For example, the decimal parts of a Julian date: 0.1 = 2.4 hours or 144 minutes or 8640 seconds, 0.01 = 0.24 hours or 14.4 minutes or 864 seconds, and so on…

The Julian Calendar

The Julian calendar began in 45 BC (709 AUC) as a reform of the Roman calendar by Julius Caesar. It was chosen after consultation with the astronomer Sosigenes of Alexandria and was probably designed to approximate the tropical year (known at least since Hipparchus).
The Julian calendar has a regular year of 365 days divided into 12 months with a leap day added to February every four years. The Julian year is, therefore, on average 365.25 days long. The motivation for most calendars is to fix the number of days between return of the cycle of seasons (from Spring equinox to the next Spring equinox, for example), so that the calendar could be used as an aid to planting and other season-related activities. The cycle of seasons (tropical year) had been known since ancient times to be about 365 and 1/4 days long.

The Gregorian Calendar

The more modern Gregorian calendar eventually superseded the Julian calendar: the reason is that a tropical year (or solar year) is actually about 11 minutes shorter than 365.25 days. These extra 11 minutes per year in the Julian calendar caused it to gain about three days every four centuries, when compared to the observed equinox times and the seasons. In the Gregorian calendar system, first proposed in the 16th century, this problem was dealt with by dropping some calendar days, in order to re-align the calendar and the equinox times. Subsequently, the Gregorian calendar drops three leap year days across every four centuries.

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.
This is not the first time people have used a 13 Moon calendar. The Druids kept a “tree” calendar, a count of 13 moons of 28 days each, plus one day. The Incas, ancient Egyptians, Mayans and the Polynesians all kept a 13 moon/28 day count. The Lakota Indians kept a 13 moon/28 day count based on the “keya”, or turtle, since the turtle has 13 scales on its back.
One of the great advantages of the 13 Moon Calendar is that day/date calculations are amazingly simple. The first day of every Moon is always a 13 Moon Dali. The last day of every Moon is always a 13 Moon Silio. (Note: the old paradigm names of the days in a week are replaced by galactic names which describe seven primary plasmas – electronically charged particles which activate our magnetic field. Thus, each week has the following 7 days: Dali / Friday, Seli / Saturday, Gamma / Sunday, Kali / Monday, Alpha / Tuesday, Limi / Wednesday, Silio / Thursday.)
The Gregorian calendar makes day/date calculations very difficult because the months are of unequal measure so the days and dates of the week vary from month to month and year to year. On the new 13 Moon Calendars, the Gregorian correlate dates are found at the bottom of each 13 Moon date. Find your birthday, every year it will always be on the same day of the same 13 Moon week.
Time is a frequency – the frequency of synchronicity.
The 13 Moon calendar is truly unique because it is synchronized with the Harmonic Module, the universal 13:20 timing frequency. Originally used by the Maya, the most sophisticated timekeepers ever known, the Harmonic Module consists of 20 icons or solar seals and thirteen galactic tones, 1-13. The resulting 260 permutations combined with the perfect harmony of the 13 Moon calendar give each day a unique quality. The two cycles – 13 Moons/28 days and the 260-day Harmonic Module perfectly mesh every 52 years! Each year your birthday moves up one tone and ahead five icons. In the center of the Harmonic Module is a black pattern of 52 “galactic activation portals.” See if you can find the sequence of 13 sets of four, counting from the four corners inward. Notice that the numbers of each set of four equals 28. 13 sets x 28 = 364, the number of days in the 13 Moon calendar!
In the 13 Moon calendar the obscurely named Gregorian months are replaced by names which correspond to a fourth-dimensional cosmology of time.
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.

The Intercalary days

The Julian Calendar lengthened some of the months to give a 365-day year without intercalary days, and in addition added an intercalary day every fourth year, now 29 February, to give an average 365.25 day year. The longer months were placed in the summer because the sun’s movement through the stars is slower in these months. A further correction of omitting the extra day on even century years, except every 400 years, keeps the calendar in close synchronization with the seasons. This is the Gregorian Calendar, but the difference from the Julian is piddling. No great scientific insight was necessary to devise this correction. This calendar, which is now all but universally used for civil purposes, is a purely solar calendar.
The week is a purely arbitrary grouping of days, mainly to give names to the days for convenience. Weeks go on oblivious of any astronomical happenings. The day of the week can be found from the Julian Day by simply dividing by 7. The Chinese week was five days, named for the five Chinese elements. The revolting French had a 10-day week. The revolting Russians had a 5-day week with a month of 6 weeks at first, then a 6-day week with 5 weeks in a month later. The Mayas and Aztecs used 13-day and 20-day divisions simultaneously.
It requires some intelligence to work out a system of arranging and naming days to stay in step with the moon and sun, but no more science than simply counting days. Errors will become evident after the passage of a sufficient interval of time. The lazy will merely intercalate days as necessary, the clever will think up formulas for adding the extra days in advance. No extra precision of measurement is necessary.
Because a calendar system counts years or days from some early date does not mean that the calendar existed at that date. Any calendar can be extrapolated backwards to create a proleptic calendar for expressing dates before the calendar actually existed. The Julian Day starts in 4,713 BC, but no one could have known it at the time, since the Julian Day was not devised until the 17th century.
References: Wikipedia,  M. Westrheim, Calendars of the World

Early Calendars

Why do we divide the circle into 360 degrees? Why the year is divided into 12 months? Why a week has 7 days?
Considering we use decimal system wouldn’t be easier to divide circle into 10, 100 and 1000 equal parts?
The answer could be related to the fact that our year has 365 days…Earth  rotates 365 times (366 in relation to stars) around its axis while completing 1 full orbit around the Sun.
The Babylonians were the first to recognize that astronomical phenomena are periodic and apply mathematics to their predictions.
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).

Calendars in Ancient Egypt

The Ancient Egyptians are credited with the first calendar of 12 months, each consisting of 30 days, comprising a year. They added 5 days at the end of the year to synchronize somewhat with the solar year. By making all their months an even 30 days, they abandoned trying to sync with lunar cycles and concentrated instead on aligning with the solar year. The Egyptians recognized that this calendar didn’t quite align with a actual year. Since the traditional Egyptian calendar of 365 days fell about one-fourth of a day short of the natural year, the ancients assumed that the helical rising of Sirius would move through the Egyptian calendar in 365 x 4 = 1,460 Julian years (that is, one Sothic peniod).
The earliest Egyptian calendar was based on the moon’s cycles, but the lunar calendar failed to predict a critical event in their lives: the annual flooding of the Nile river. The Egyptians soon noticed that the first day the “Dog Star,” which we call Sirius, was visible right before sunrise was special. The Egyptians were probably the first to adopt a mainly solar calendar. This so-called ‘heliacal rising’ always preceded the flood by a few days.
They eventually had a system of 36 stars to mark out the year and in the end had three different calendars working concurrently for over 2000 years: a stellar calendar for agriculture, a solar year of 365 days (12 months x 30 + 5 extra) and a quasi-lunar calendar for festivals. The later Egyptian calendars developed sophisticated Zodiac systems. According to the famed Egyptologist J. H. Breasted, the earliest date known in the Egyptian calendar corresponds to 4,236 B.C.E. in terms of the Gregorian calendar.
Source: wikipedia and  http://www.webexhibits.org/calendars/calendar-ancient.html

Calendars in Ancient Mesoamerica

Among their other accomplishments, the ancient Mayas invented a calendar of remarkable accuracy and complexity.
The Maya calendar was adopted by the other Mesoamerican nations, such as the Aztecs and the Toltec, which adopted the mechanics of the calendar unaltered but changed the names of the days of the week and the months.  The Maya calendar uses three different dating systems in parallel, the Long Count, the Tzolkin (divine calendar), and the Haab (civil calendar). Of these, only the Haab has a direct relationship to the length of the year.
The length of the Tzolkin year was 260 days and the length of the Haab year was 365 days. The smallest number that can be divided evenly by 260 and 365 is 18,980, or 365×52; this was known as the Calendar Round. If a day is, for example, “4 Ahau 8 Cumku,” the next day falling on “4 Ahau 8 Cumku” would be 18,980 days or about 52 years later.
For 365 and 260 the “least common multiple ( the smallest positive integer that is a multiple of two integers) is 18980 (365×260/5).  [ 365 x 1 = 73 x 5   and  260 x 1 = 52 x 5 ].

The Long Count is really a mixed base-20/base-18 representation of a number (see Maya numerals below), representing the number of days since the start of the Mayan era. It is thus akin to the Julian Day Number.

• The basic unit is the kin (day), which is the last component of the Long Count. Going from right to left the remaining components are:

• uinal    (1 uinal = 20 kin = 20 days)
• tun    (1 tun = 18 uinal = 360 days = approx. 1 year)
• katun    (1 katun = 20 tun = 7,200 days = approx. 20 years)
• baktun    (1 baktun = 20 katun = 144,000 days = approx. 394 years)

The kin, tun, and katun are numbered from 0 to 19.
The uinal are numbered from 0 to 17.
The baktun are numbered from 1 to 13.
Logically, the first date in the Long Count should be 0.0.0.0.0, but as the baktun (the first component) are numbered from 1 to 13 rather than 0 to 12, this first date is actually written 13.0.0.0.0.
The authorities disagree on what 13.0.0.0.0 corresponds to in our calendar. I have come across three possible equivalences:

• 13.0.0.0.0 = 8 Sep 3114 BC (Julian) = 13 Aug 3114 BC (Gregorian)
• 13.0.0.0.0 = 6 Sep 3114 BC (Julian) = 11 Aug 3114 BC (Gregorian)
• 13.0.0.0.0 = 11 Nov 3374 BC (Julian) = 15 Oct 3374 BC (Gregorian)

Assuming one of the first two equivalences, the Long Count will again reach 13.0.0.0.0 on 21 or 23 December AD 2012 – a not too distant future. This is not “the end of time” but a beginning of the new Long Count cycle.
Tzolkin
The Tzolkin date is a combination of two “week” lengths. While our calendar uses a single week of seven days, the Mayan calendar used two different lengths of week:

• a numbered week of 13 days, in which the days were numbered from 1 to 13
• a named week of 20 days

Haab – The Civil Calendar
The Haab was the civil calendar of the Mayas. It consisted of 18 “months” of 20 days each, followed by 5 extra days, known as Uayeb. This gives a year length of 365 days.
In contrast to the Tzolkin dates, the Haab month names changed every 20 days instead of daily.
The days of the month were numbered from 0 to 19. This use of a 0th day of the month in a civil calendar is unique to the Maya system; it is believed that the Mayas discovered the number zero, and the uses to which it could be put, centuries before it was discovered in Europe or Asia.
Aztec Calendar
The Aztec calendar is the calendar system that was used by the Aztecs as well as other Pre-Columbian peoples of central Mexico. It is one of the Mesoamerican calendars, sharing the basic structure of calendars from throughout ancient Mesoamerica. The calendar consisted of a 365-day calendar cycle called xiuhpohualli (year count) and a 260-day ritual cycle called tonalpohualli (day count). These two cycles together formed a 52-year “century,” sometimes called the “calendar round”. The Aztec Calendar is no less accurate than the Mayan Calendar and conversely.

First drawing in the book “Descripción histórica y cronológica de las dos piedras que con ocasión del nuevo empedrado que se está formando en la plaza principal de México, se hallaron en ella el año de 1790″(or “Historical and chronological description of the two stones found during the new paving of the Mexico’s main plaza.” in English ) by Antonio de Leon y Gama (1792). The book describes Aztec calendars and specifically the discovery of the Aztec calendar stone in México. The image depicts an aztec sun calendar.

Aztec Calendar Links:

• http://www.azteccalendar.info/Wiki/
• Aztec Calendar Handbook 

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.

Mesoamerican Archaeoastronomy

Using the summer solstice to calibrate the secular calendar

In reality, the formula was as simple as it was ingenious. The problem at San Lorenzo had been that the priests had no way of knowing when it was August 13, because in their part of the world the zenithal passage of the sun did not occur on that date. Thus, they had settled on using one of the solstices instead, because the date of the sun’s turning point was the same everywhere, they had discovered. Whereas at San Lorenzo they were obliged to use the winter solstice sunset to calibrate their calendar, when La Venta was founded it appears that they could once more think in terms of the summer solstice, as had originally been done in Izapa. Indeed, the only difference was that instead of marking the sunrise as they did at Izapa, they were obliged to use the sunset at La Venta.
Once back in the mental groove of using the summer solstice to calibrate the secular calendar, it would not have been long before some priest realized that the beginning date of the sacred almanac can itself be calibrated by reference to the summer solstice. In effect, he was recognizing that, if the solstice occurred on June 22 and the “beginning of time” occurred on August 13, there was a fixed interval of time between these two dates. Using our modern calendar to demonstrate his thought process, we would count 8 days to complete the month of June, add 31 more for the month of July, and then count 13 until the sunset of August 13, yielding a total of 52 days. (For anyone used to thinking in “bundles” of 20′s and 13′s, what a neat package this was — 4 rounds of 13 days = 52 days.) Thus, no matter where one wanted to build a ceremonial center, one could always find out when it was August 13. All that was required was to count 52 days from the time that the sun turns around in the north and mark the horizon at sunset!

Pre-Columbian Calendar of South America

The Gate of the Sun and Cracking the Muisca Calendar
by Jim Allen
The Muisca were a pre-Columbian people who lived in the territory now known as Columbia in South America.
In 1795, Dr Jose Domingo Duquesne, a priest of the church of Gachancipa in Columbia published a paper detailing the Muisca calendar, which information he claimed to have received from the Indians themselves. His paper was later ridiculed as being nothing but an invention of his.
Yet the figures given by Duquesne do in fact relate to a lunar calendar although Duquesne himself may not have fully understood the workings of it since it seems possible that the calendar was more sophisticated than might appear at first glance, and two types of lunar month may have been used, the Sidereal Lunar Month when the moon returns to the same position relative to the stars (27.32 days) and the Synodic Month which is the period between full moon and full moon (29.53 days).
Background
At Tiwanaku we found how the solar year was divided into 20 months of 18 days and also interlocked with the Inca calendar of 12 sidereal lunar months of 27.32 days (making 328 days) so that 3 x solar years also equalled 40 sidereal lunar months and the two calendars came together every 18 solar years which equalled 20 Inca years when the cycle started all over again (also known as the Saros Cycle).
Duquesne
At first difficult to read and understand, Duquesne’s  (shortcut to Wikipedia info about Duquesne) paper begins with a background about the Americas and the Egyptians and how the Muiscas counted by their fingers with names for each number up to ten, and then on to twenty.
He then relates their calendar to harvesting and sowing and begins:
The first thought on reading this, was that as at Tiwanku, they might have divided the Solar Year into twenty for their months, but the text implies that 20 lunar months made the year and it also implies that Synodic or phase months were intended. This year of twenty months he tells us was called a “Zocam” year. Now a period of 20 x 20 months which Duquesne mentions might seem worthy of fitting into an Aztec or Mayan calendar since 20 x 20 gives 400, but further down the text, if we read closely, Duquesne says that
Now, here is a question, not of translation, but of meaning. Because a little further along, Duquesne explains how the year of 37 months was a period of 36 months plus a “deaf” month so that the year adjusts to the solar year. This year of 37 months is called an “Acrotom” year. He also tells us that 20 x 37 of these months corresponds to 60 of our years, divided into four parts so that each part was ten Muisca years which equalled fifteen of ours.
From this we can easily work out that 60 of our solar years divided by 20 x 37 gives a month of 29.61 days suggesting that here, the synodic or phase month from full moon to full moon was intended since the synodic month has an average of 29.53 days.
But returning to the earlier statement
What I think is meant here, is that they counted in twenty times twenty then added an extra month in the same manner as they added an extra month to 36 months to make 37, so the real figure here is not 20 x 20 = 400 but 20 x 20 + 1 = 401.
There is also another difference.
I think they were running two calendars in parallel with each other, so the 37 month calendar was in Synodic Months of 29.53 days while the 20 month and 401 month calendar was in Sideral Lunar Months of 27.32 days, although at the same time Duquesne counts the 37 month year as being 20 months + 17 months (because the counting system was based on twenties) making 37 months when the solar and lunar calender synchronised, in this instance these 20 would be synodic months the same as the 17 months and he also explains this another way, as the extra month being inserted at the end of every three lunar years so they counted two x lunar years of 12 months then one of 13 months, the thirteenth month being the “sordo” (deaf) or extra month. So after 1 x Muisca year of 37 synodic months (3 solar years), sowing would begin again on the same day in January, while the intervening two years had a system of counting the months on the fingers as Duquesne puts it…
But returning to the calendar of 20 months running continuously as 20 x 20 months with an extra month inserted to give 401 months, we can check the figure of 401 Sidereal Lunar Months to see if it relates to a solar year and 401 x 27.32 days comes to a great period of 30 Solar Years, which in turn equals 10 Muisca Acrotom years of 37 x synodic months of 29.53 days….
Every three solar years equals the Muisca “Acrotom” year of 37 Synodic Months of 29.53 days and at the same time corresponds to 40 Sidereal Lunar Months of 27.32 days, and every one and a half solar years corresponds to a “sidereal lunar year” of 20 Sidereal Lunar Months which is the true “Zocam” year of the Muiscas.
So to sum up so far,
1 x Tiwankau Solar year = 20 “months” of 18 days (using a rounded-off 360 day year divided by 20).
1 x Tiwanaku Lunar year = 12 sidereal lunar months of 27.32 days (328 days) – also used by Incas.
1 x Muisca Zocam year = 20 sidereal lunar months of 27.32 days = 1½ solar years
2 x Muisca Zocam years of 20 sidereal months of 27.32 days = 1 Muisca Acrotom year of 37 synodic months
1 x Muisca Acrotom year = 37 x synodic months of 29.53 days
1 x Muisca Acrotom year = 3 x solar years = 40 x sidereal lunar months of 27.32 days = 2 x Muisca Zocam years
½ Muisca Acrotom year = 1½ solar years = 20 x sidereal lunar months of 27.32 days = 1 x Muisca Zocam year
18 solar years = 20 Inca years = 6 x Muisca years of 37 x 29.61 days = the Saros Cycle
10 Muisca Acrotom years = 30 solar years = 401 sidereal lunar months of 27.32 days = 20 Zocam years.
20 Muisca Acrotom years = 60 solar years = 2 x 401 sidereal lunar months of 27.32 days = 40 Zocam years.
It might appear that Duquesne made an error when stating that “the ‘century’ of the Muiscas consisted of 20 intercalcated years of 37 months each, which corresponded to 60 of our years, which comprised four revolutions counted in fives, each one of which equalled ten Muisca years, and fifteen of ours until completing the twenty….”
Since 1 x Muisca year of 37 months equals 3 solar years, then 10 x Muisca years should be 30 solar years as per the table above, and since Duquesne was talking about how they counted up to twenty in periods of fives which corresponded to five fingers, what he should have said here was that each of the five was five Muisca years of 37 months equalling fifteeen of ours. But in fact he is correct except it is 10 x Sidereal lunar month years of 20 x 27.32 days which equal the fifteen solar years…..
 5 Muisca Acrotom years of 37 synodic months of 29.53 days would be 15 solar years
10 Muisca Zocam years of 20 sidereal months of 27.32 days would be 15 solar years
I suspect therefore, and it is fairly clear, that the 20 month year which Duquesne called the “Zocam” year was actually the sidereal year of 20 sidereal months but the name may have mis-understood by Duquesne as a period of 20 synodic months if Duquesne were unaware of a different type of lunar month in use, otherwise there would have been little point in having years of 20 synodic months running continuously when they were actually grouped in 37 month years and by contrast 2 x 20 sidereal months mesh both with the Acrotom year and solar year at 3 year intervals and over longer periods.
To see how they compare at three year intervals,
37 synodic months of 29.53 days would be 1092.61 days
40 sidereal months of 27.32 days would be 1092.8 days
3 solar years of 365.2524 days would be 1095.72 days
Because of the small discrepency, over long periods of time some adjustments would probably be necessary such as the extra month inserted after 400 sidereal months on the Zocam calendar making
401 sidereal months of 27.32 days = 10955.32 days
 10 Muisca Acrotom years = 370 synodic months of 29.53 days = 10926 days
but if they added another month that would bring them to 10955.5 days and back into line with the Zocam sidereal lunar calendar and closer to the 30 solar years of 365.24 days = 10957.26 days
The Muisca “Acrotom” 37 month synodic month calendar with the phases of the moon was probably a more “user friendly” calendar for the man in the field, whereas the “Zocam” 20 month sidereal lunar calendar was probably of more interest to the time keeping priesthood and for bringing the other calendar into alignment periodically.
Duquesne also tell us that the Muisca “week” was a period of three days, and at face value, this would appear to have no relationship to the Muisca calendar whether using sidereal or synodic months, but then the calendar itself, in spite of Duquesne’s explanation as a usage for agriculture does not seem really practical for agriculture or at least not as practical as the Tiwanaku one but perhaps having the advantage that no construction of pillars or standing stones was required.
The calendar which is practical for agriculture is the one found at Tiwanaku where the solar year is divided by twenty and determined by the setting of the sun over a pillar, so it would be fairly easy to note the same pillar where the sun would return to each year, and this is the calendar which is easily divided into periods of three days, and period of nine days were also known to have been worked in that region.
So perhaps the Muisca also ran a solar calendar, undiscovered but in the same style as Tiwanaku, or perhaps their customs were left over from some forgotten era, based on the same mathematicas as Tiwanaku with it’s interlocking sidereal lunar calendar and counting in twenties.

Return to Tiwanaku
The Muisca calendar then, 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 Sun which gives the clues to the workings of the Tiwanaku calendar.
Many people have studied the icons on the Gate of the Sun at Tiwanaku and tried to relate them to a calendar. The icons are called “chasquis” or Messengers of the Gods and because there are fifteen of them on each side, some people have thought that they represented a thirty day month in a solar year of twelve months. But as explained earlier, 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.
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?
Above, detail of the “Gate of the Sun” at Tiwanaku, Bolivia showing the principal grouping of thirty “chasqui” figures with beneath them the freize showing eleven icons and forty condors heads arranged in two rows of  twenty heads.
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.

Units of Measurement – Introduction

Definitions

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 similar in value and closely related to each other. Ancient Metrology All ancient cultures used units of measures. The earliest known uniform systems of weights and measures seem to have all been created sometime in the 4th and 3rd millennia BC among the ancient peoples of Mesopotamia, Egypt and the Indus Valley. Indeed, people who have not previously been regarded as civilized in the literal sense, manifestly utilised this sophisticated measurement system to extraordinary degrees of accuracy.

Similarity of certain units used by ancient architects around the world raises these questions:

• Was there a system as a whole, which all civilizations used as reference, that predated them all?
• Were ancient people able to arrive to the identical systems of measure because they used nature to define such units?
• Was there re any direct cultural contacts between the disparate peoples who used the identical system?

The modern practice of dividing a circle into 360 degrees, of 60 minutes each, began with the Sumerians.

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”).

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) = 86400 seconds (24x60x60).
The Earth turns 15 degrees per hour.
Notes:

• The average (typical) resting heart rate in a healthy adult is 60–80 bits per minute  1-1.333 bits per second
• In 1 second light travels 299,792.458 km, in 1.000692286 milliseconds – 300 km in a vacuum

Ancients established value for the earth axis spin cycle (called day) and used this as measuring unit for the longer cycle of earth orbiting the sun (or the sun returning to the same constellation on the sky.) Earth axial speed is 360 deg/axial cycle (day). Earth orbital speed is 360 deg/solar cycle (year).
If we choose time units based on a solar day (86,400 seconds), sidereal day will be 365/366 x 86,400 seconds = 86,163.929 seconds. It means it will be shorter than solar day by 236.1 seconds = 3.9345 minutes = 3 min 56.1 seconds (rounded to 4 min)

Ancient Units of length

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 common cubit was the length of the forearm from the elbow to the tip of the middle finger. It was divided into the span of the hand (one-half cubit), the palm or width of the hand (one sixth), and the digit or width of the middle finger (one twenty-fourth) and the span or the length between the tip of little finger to the tip of the thumb.

The Sacred Cubit (aka Royal Cubit), which was a standard cubit enhanced by an extra span—thus 7 spans or 28 digits long—was used in constructing buildings and monuments and in surveying in ancient Egypt. The inch, foot, and yard evolved from these units through a complicated transformation not yet fully understood. Some believe they evolved from cubic measures; others believe they were simple proportions or multiples of the cubit.
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 = 0.7071 Royal Cubit =  0.3715 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 1/3 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
For RC=1 we get and  1MY=1.570795   and 1 Remen = 1/sqrt(2)=0.7071 
In whichever case, the Greeks and Romans inherited the foot from the Egyptians.
The Roman foot (~296 mm) was divided into both 12 unciae (inches) (~24.7 mm) and 16 digits (~18.5 mm).
The Romans also introduced the mille passus (1000 paces) or double steps, the pace being equal to five Roman feet (~1480 mm).
The Roman mile of 5000 feet (1480 m) was introduced into England during the occupation. Queen Elizabeth I (reigned from 1558 to 1603) changed, by statute, the mile to 5280 feet (~1609 m) or 8 furlongs, a furlong being 40 rod (unit)s (~201 m) of 5.5 yards (~5.03 m)each.
The introduction of the yard (0.9144 m) as a unit of length came later, but its origin is not definitely known. Some believe the origin was the double cubit, others believe that it originated from cubic measure. Whatever its origin, the early yard was divided by the binary method into 2, 4, 8, and 16 parts called the half-yard, span, finger, and nail. The association of the yard with the “gird” or circumference of a person’s waist or with the distance from the tip of the nose to the end of the thumb of King Henry I (reigned 1100–1135) are probably standardizing actions, since several yards were in use in Britain. There were also Rods, Poles and Perches for measurements of length. The following table lists the equivalents:

Ancient Metrology

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 fuelled 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:
No explanation is offered as to why no Egyptian pendulums have been found, despite the extremely rich archaeological material from this culture, nor to the question as to why none of the rich historic material from Egypt mentions this, or indeed why a divisor of 105 would have been chosen or measured.

Royal Cubit (Sacred Cubit)

Uniformity of royal cubits.
It is difficult to imagine how a supposedly anatomical measure could turn up in different nations with distinct subdivisions yet have a suspiciously similar length. If they were exaggerating in order to make their own king look the larger than life, why would the lengths be similar? There is even mention of English, Chinese and Mexican Aztec cubits within the range 518–531 mm (20.4 to 20.9 in).
Uniformity of royal / architectural cubits
Civilization        Length (mm)
Mesopotamia       522–532
Persia                       520–543
Egypt                        524–525
Mysterious royal cubit origin. ‘The anatomical length … cannot possibly be as long as the royal cubit of 525 mm.’24 (Unless, of course, it came from a people taller than the Egyptians.) Egyptian royal cubits had seven palms and 28 fingers in a cubit. The Babylonian had 30 divisions. Both numbers indicate astronomical origin of the cubit (28 or 30 day month with 4 weeks of 7 days).
Respect for the royal cubit. This indicates an important legacy, like a standard handed down from the ‘Gods’. The ‘Gods’ of certain cultures could be early post-flood founders a few generations after Noah. In Egypt, building overseers required the Royal Egyptian Cubit to be calibrated against a precision standard at regular intervals. Failure to do so was punishable by death.
http://www.worldwideflood.com/ark/noahs_cubit/cubit_references.htm

Charles Piazzi Smyth

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. A paper presented to the Royal Academy on the topic was rejected.

Taylor’s theories were, however, the inspiration for the deeply religious archeologist Charles Piazzi Smyth to go to Egypt to study and measure the pyramid, subsequently publishing his book Our Inheritance in the Great Pyramid (1864), claiming that the measurements he obtained from the Great Pyramid of Giza indicated a unit of length, the pyramid inch, equivalent to 1.001 British inches, that could have been the standard of measurement by the pyramid’s architects. From this he extrapolated a number of other measurements, including the pyramid pint, the sacred cubit, and the pyramid scale of temperature.
Smyth claimed—and presumably believed—that the inch was a God-given measure handed down through the centuries from the time of Israel, and that the architects of the pyramid could only have been directed by the hand of God. To support this Smyth said that, in measuring the pyramid, he found the number of inches in the perimeter of the base equalled 1000 times the number of days in a year, and found a numeric relationship between the height of the pyramid in inches to the distance from Earth to the Sun, measured in statute miles.
Smyth used this as an argument against the introduction of the metre in Britain, which he considered a product of the minds of atheistic French radicals.

The Grand Scheme

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:
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 defence 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):
This seems to bring pseudoscientific metrology to new heights, especially in view of the conclusion:
Megalithic Yard
A measuring unit defined by astronomical and/or geodetic properties of the Earth would have to contain information about the size of the Earth. Such unit was used by ancient builders of megaliths – it is called the Megalithic Yard (MY).
The MY turns out to be much more than an abstract unit such as the modern metre, it is a highly scientific measure repeatedly constructed by empirical means. It is based upon observation of three fundamental factors:

1. The orbit of the Earth around the sun
2. The spin of the Earth on its axis
3. The mass of the Earth

The Megalithic Yard is a unit of measurement, about 2.72 feet (32.4 in or 0.829 m), that some researchers believe was used in the construction of megalithic structures. The proposal was made by Alexander Thom* as a result of his surveys of 600 megalithic sites in England, Scotland, Wales and Brittany.
Christopher Knight and Alan Butler further develop the work of Smyth’s and Stecchini’s “Grand Scheme” in their Civilization One hypothesis, which describes a megalithic system of units. This system is claimed to be the source of all standard units used by civilization, and is so named after the Neolithic builders of megaliths. Knight and Butler contend the reconstructed megalithic yard (0.82966m) is a fundamental part of a megalithic system. Although the megalithic yard is the work of Alexander Thom, Knight and Butler make a novel contribution by speculating on how the MY may have been created by using a pendulum calibrated by observing Venus. It also explains the uniformity of the MY across large geographical areas. The accuracy claimed for this procedure is disputed by astronomers.
Knight and Butler describe a procedure for Neolithic astronomers to make a “Venus Pendulum“, using the transit of Venus across the sky to give both time and distance units.
Measures of volume and massare derived from the megalithic yard. It is divided into 40 megalithic inches. Knight and Butler claim that a cube with a side of 4 megalithic inches has a volume equal to one imperial pint and weighs one imperial pound when filled with unpolished grain. They also posit ratio relationships with the imperial acre and square rod.
A Megalithic Yard is a unit of measurement, about 2.72 feet (0.83 m), that some researchers believe was used in the construction of megalithic structures. The proposal was made by Alexander Thom as a result of his surveys of 600 megalithic sites in England, Scotland, Wales and Britanny. Thom additionally proposed the Megalithic Rod of 2.5 MY and suggested the Megalithic Rod could be divided into one hundred and the Meglithic Yard divided into forty, which he called the Megalithic Inch of 2.073 centimetres (0.816 in). Thom applied the statistical lumped variance test of J.R. Broadbent on this quantum and found the results significant while others have challenged his statistical analysis and suggested that Thom’s evidence can be explained in other ways, for instance the average length of a pace.
Source: Wikipedia
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 = 7920 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.

The Imperial System

Britain introduced Imperial Units, based on the yard, pound, and second, in the 19th century to resist the metric system and to uphold an alternative comprehensive system.
In engineering, English units were divided decimally just like metric ones, especially in the United States.
Both the Imperial units and US customary units derive from earlier English units. Imperial units were mostly used in the British Commonwealth and the former British Empire. US customary units are still the main system of measurement used in the United States despite Congress having legally authorized metric measure on 28 July 1866. Some steps towards US metrication have been made, particularly the redefinition of basic US units to derive exactly from SI units, so that in the US the inch is now defined as 0.0254 m (exactly), and the avoirdupois pound is now defined as 453.59237 g (exactly).
The basic English unit of length was the yard of three feet, or the fathom of six. Each English foot was divided into 12 inches, and each inch into 3 barleycorns or 12 lines.
Eventually, one inch was defined as exactly 25.4 mm, which tied the English and metric units together. In the United States, a meter was sometimes defined as exactly 39.37 inches, which gave 1 inch = 25.40005 mm, just enough different to be annoying in geodesy. 12 such inches made a survey foot, used by the Coast and Geodetic Survey. 5 feet, 6 inches, and 7 lines was written 5′ 6″ 7″‘. The single and double apostrophes have survived into modern times, but not the triple.
The modern feet are descended from the Roman measurement of the same name and approximate value. The Roman foot, 11.65 modern inches (29.6 cm), was usually divided into 16 inches, not 12, however (as four palms of four Roman inches, about 3 modern inches, each). Divisions by powers of 2 are specially useful, since they are binary, and much more adapted to computers than powers of 10. The English inch was later divided into halves, quarters, eighths, and so on, because of the utility and extendibility of this system, which completely replaced the use of lines.
Roman standards were relatively uniform, an interlude between times of confusion. Most Roman units of length survived in name or spirit in the English and other systems, even if changing somewhat in absolute value. For example, the stadium, which was 1/8 of a Roman mile, or 202 yards, became the furlong, 1/8 of an English mile, or 220 yards. The cubit, a forearm’s length, was 1-1/2 Roman feet or 6 palms, and typically used in building. Some ancient cubits seem to have been longer than this, up to about 22 inches. Hands of 4 inches are still used to measure the height of a horse (at the shoulders).
Note how length units were conveniently based on parts of the body used to measure distances.

• The Roman mile consisted of 1,000 double paces, or 5,000 Roman feet, or 1480 metre, or 1619 yards.
• Distances on Roman roads were measured by odometers attached to carriage axles, as described in Vitruvius, and marked on mile stones. 
• The English mile of 5280 feet is 1609 metre (a “metric mile” is, apparently, 1,500 metre). As explained in the article “Chaining,” it was defined as 80 chains of 66 ft each, and this is the reason for the odd number 5280. Gunter’s chain of 100 links was a successful attempt to create a portable length standard that was not as stretchy as a cord. The English mile happened to come out a little larger than the Roman mile, to which it was intended to be an approximation.
• The nautical mile is 1852 metre, which corresponds to one minute of arc of latitude, approximately.
• The ‘geographical’ mile was 7420 metre, and the Prussian mile 7532 metre. These long miles were about five Roman miles.
• The league was another measure of journeys, usually 3 English miles. France had an assortment of leagues: 2,000 toise for the lieue de poste, 3 Roman miles for the lieue de terre, 4 kilometers for the lieue kilométrique, and 3 nautical miles for the lieue marine.
• The Greeks had the stadium of 580-622 feet, and the plethron of 97-100 feet.
• The ancient Persian parasang was 3.25 to 3.3 miles, 30 Greek stadia. 

Any great accuracy in the size of old units is illusory unless a critical study is made. The standards have, of course, disappeared, and their magnitude can be determined only by remeasuring a distance in modern terms.
Source: History of units of measure: http://mysite.du.edu/~jcalvert/tech/oldleng.htm

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 as 5 130 738.62 toise, which was set equal to 10 000 000 metre. This was a bad choice for defining a unit of length, worse than simply making a couple of arbitrary scratches on a bar, since it involved a difficult and tedious procedure, especially when the news arrived in France that the earth was not a sphere to a sufficient approximation. At any rate, a scratched bar was later adopted as standard, and the earth’s quadrant allowed to be whatever it turned out to be in terms of metres.
A similar mistake was made in defining the kilogram as the weight of a cubic decimetre (a litre) of water, since the density of water changes with temperature, and weight can be measured more accurately than a decilitre can be measured. Thus, both the litre and the kilogram were defined arbitrarily. The clumsiness was reflected in the slight difference between a cubic centimetre and a millilitre that thereby arose.
The second of time remained 1/86400 of a mean solar day, with common units of time not related decimally, but by factors of 24 and 60. The decimal calendar was a ludicrous failure. The French also defined a right angle as 100 grads, another superfluity. The units of time and angles were already uniform, and required no work. As has been mentioned above, the French population ignored the metric system until it was forced upon them.
The circumference of the Earth
From the 18th century, inspired by the statement of Aristotle that the circumference of the Earth was calculated as 400,000 stadia, it became a belief among members of the French Académie des Sciences that ancient linear measures were all derived directly from the circumference of the Earth. Archaeologist Jean Antoine Letronne, in 1822, tried to show the connection to a supposed pre-Greek measurement of the Earth.
Ronald Zupko claims that since Gunter suggested the concept of division of the earth’s arc into length in the seventeen century, Cassini in 1720 suggested dividing the earth’s circle to 360 degrees, of 60 miles of 1000 fathoms of 6 feet, which again was the inspiration of the metric system, it is not all that unreasonable to suggest that this could have not happened in an earlier time. Indeed, he claims that the error in the Greek foot lies wholy in the range of the geographic measures (since the earth is not spherical), and that the multiples of it follow the sexagesimal division of the earth. Zupko, however, provides no evidence how the Greek would have actually measured the Earth’s actual circumference, if it was the basis of their units of measurement.
Table: Some milestones in measuring the Earth meridian.  stands for the length of the meter calculated as the 40,000,000th part of the meridian (for some important cases. In the results expressed in the form `  ‘ xxxxx stands for the length of one degree meridian arc (  in the text). Ancient estimates have to be taken with large uncertainty:
The Seconds Pendulum
In August 1790 the French National Assembly entrusted the reform to the Academy of Sciences. The Academy nominated a preliminary commission,which adopted a decimal scale for all measures, weights and coins. The commission presented its report on 27 October 1790. A second commission was charged with choosing the unit of length. The commission was set up on 16 February 1791 and reported to the Academy of Sciences on 19 March 1791. On 26 March 1791 the National Assembly accepted the Academy’s proposals of the decimal system and of a quarter of the meridian as the basis for the new system and the adoption of the consequent immediate unit.
The guiding ideas of the French scientists are well expressed in the introduction to the document presented to the Academy:
Three were the candidates considered by the commission:

• the seconds pendulum;
• a quarter of the meridian;
• a quarter of the equator.

The latter two units are based on the dimension of Earth. Indeed, Earth related units had had also quite a long history, though they were not as popular as the seconds pendulum, probably because their intrinsic difficulty to be determined.Mouton had suggested in 1670 the unit that we still use in navigation and call now nautical mile: the length of one minute of the Earth’s arc along a meridian, equal to 1852m. In 1720 the astronomer Jean Cassini had proposed the radius of Earth, a “natural” unit for a spherical object (he had also indicated the one ten-millionth part of the radius as the best practical unit). However, neither of these old, French proposals are mentioned in the report of the commission.
The quarter of the equator was rejected, mainly because considered hard to measure and somehow “not democratic”.
So, we believe we are bound to decide to assume this kind of unit of measure and also to prefer the quarter of the meridian to the quarter of the equator. The operations that are necessary to establish the latter could be carried out only in countries that are too far from ours and, as a consequence, we should have to undertake expenditures as well as to overcome difficulties that would be superior to the advantages that seem to be promised. The inspections, in case somebody would like to carry them out, would be more difficult to be accomplished by any nation, at least until the progress of the civilization reaches the peoples living by the equator, a time that still seems to be unfortunately far away. The regularity of this circle is not more assured than the similarity or regularity of the meridians. The size of the celestial arc, that corresponds to the space that would be measured, is less susceptible to be determined with precision; finally it is possible to state that all peoples belong to one of Earth’s meridians, while only a group of peoples live along the equator.
As far as the length of the seconds pendulum is concerned, during the 18th century its value was known with sub-millimeter accuracy in several places in France and around the world, often related to work of rather famous people like Isaac Newton, Mersenne, Giovan Battista Riccioli, Picard, Jean Richer, Gabriel Mouton, Huygens, Jean Cassini, Nicolas Louis de Lacaille, Cassini de Thury and La Condamine. For example, in 1740 Lacaille and Cassini de Thury had measured the length of the seconds pendulum in Paris (48 latitude), obtaining a value of 440.5597 lignes corresponding to 0.993828m (the metric conversion was fixed by the French law of 10 December 1799, that established the meter to be equal to 3 pied and 11.296 lignes; Metric equivalent: ligne [line] =  2.25583mm).
Newton himself had estimated the length of the seconds pendulum at several latitudes between 30 and 45 degrees: his value at 45 degrees was 440.428 lignes, i.e. 99.353cm. A measurement at the equator, made by La Condamine during the Peru expedition gave 439.15 lignes (99.065cm).
Space-time Connection?
The well known small angle formula that gives the period T of the simple pendulum (i.e. the elementary text book pendulum) as a function of its length L  is:
T= 2*Pi*sqrt(L/g)
where g is the gravitational acceleration, approximately 9.80665 m/s² at sea level on Earth
For L=1 m T=2.007 s, in other words each swing takes 1.0035 seconds
( more info: http://www.roma1.infn.it/~dagos/history/sm/node18.html)
 Source: http://www.roma1.infn.it/~dagos/history/sm/index.html

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.

The International System of Units (SI)

All systems of weights and measures, metric and non-metric, are linked through a network of international agreements supporting the International System of Units. The International System is called the SI, using the first two initials of its French name Système International d’Unités. The key agreement is the Treaty of the Meter (Convention du Mètre), signed in Paris on May 20, 1875. 48 nations have now signed this treaty, including all the major industrialized countries. The United States is a charter member of this metric club, having signed the original document back in 1875.
The SI is maintained by a small agency in Paris, the International Bureau of Weights and Measures (BIPM, for Bureau International des Poids et Mesures), and it is updated every few years by an international conference, the General Conference on Weights and Measures (CGPM, for Conférence Générale des Poids et Mesures), attended by representatives of all the industrial countries and international scientific and engineering organizations. As BIPM states on its web site, “The SI is not static but evolves to match the world’s increasingly demanding requirements for measurement.”
At the heart of the SI is a short list of base units defined in an absolute way without referring to any other units. The base units are consistent with the part of the metric system called the MKS system. In all there are seven SI base units:

•     m  – the meter for distance,
•     kg – the kilogram for mass,
•     s   -  the second for time,
•     A  -  the ampere for electric current,
•     K   – the kelvin for temperature,
•     mol – the mole for amount of substance, and
•     cd  -  the candela for intensity of light.

Definitions of 3 fundamental base units:

• 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 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 travelled 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

• second (s or sec or “) -  a fundamental unit of time in all measuring systems and the SI base unit of time. The name simply means that this unit is the second division of the hour, the minute being the first. The day is divided in 24 hours, each hour divided in 60 minutes, each minute divided in 60 seconds.The second was defined as 1/86,400 mean solar day until astronomers discovered that the mean solar day is actually not constant.  A second is 1 / (24 × 60 × 60) of the solar day. The definition was then changed to 1/86,400 of the specific mean solar day 1900 January 1. Since we can’t go back and measure that day any more, this wasn’t a real solution to the problem.  In 1967, scientists agreed to define the second as that “period of time which makes the frequency of a certain radiation emitted by atoms of cesium-133 equal to 9, 192, 631, 770 hertz (cycles per second)”. In other words, if we really want to measure a second, we count the duration of 9, 192, 631, 770 periods of the radiation corresponding to the transition between the two hyperfine levels of the ground state of the caesium 133 atom. This definition refers to a caesium atom at rest at a temperature of 0 K.” This definition allows scientists to reconstruct the second anywhere in the world with equal precision.

• kilogram (kg) – the base unit of mass in the SI and MKS versions of the metric system.
  The kilogram is defined as the mass of the standard kilogram, a platinum-iridium bar in the custody of the International Bureau of Weights and Measures (BIPM) near Paris, France. Copies of this bar are kept by the standards agencies of all the major industrial nations, including the U.S. National Institute of Standards and Technology (NIST). One kilogram equals exactly 1000 grams, or about 2.204 622 6 pounds. Original definition: The mass of one litre of water. A litre is one thousandth of a cubic metre. Since the redefinition of the metre in 1960, the kilogram is the only unit which is directly defined in terms of a physical artifact rather than a property of nature.

Other SI units, called SI derived units, are defined algebraically in terms of these fundamental units.
Currently there are 22 SI derived units.They include:

•     the radian and steradian for plane and solid angles, respectively;
•     the newton for force and the pascal for pressure;
•     the joule for energy and the watt for power;
•     the degree Celsius for everyday measurement of temperature;
•     units for measurement of electricity: the coulomb (charge), volt (potential), farad (capacitance), ohm (resistance), and siemens (conductance);
•     units for measurement of magnetism: the weber (flux), tesla (flux density), and henry (inductance);
•     the lumen for flux of light and the lux for illuminance;
•     the hertz for frequency of regular events and the becquerel for rates of radioactivity and other random events;
•     the gray and sievert for radiation dose; and
•     the katal, a unit of catalytic activity used in biochemistry.

Future meetings of the CGPM may make additions to this list; the katal was added by the 21st CGPM in 1999.
In addition to the 29 base and derived units, the SI permits the use of certain additional units, including:

•     the traditional mathematical units for measuring angles (degree, arcminute, and arcsecond);
•     the traditional units of civil time (minute, hour, day, and year);
•     two metric units commonly used in ordinary life: the liter for volume and the tonne (metric ton) for large masses;
•     the logarithmic units bel and neper (and their multiples, such as the decibel); and
•     three non-metric scientific units whose values represent important physical constants: the astronomical unit, the atomic mass unit or dalton, and the electronvolt.

The SI currently accepts the use of certain other metric and non-metric units traditional in various fields. These units are supposed to be “defined in relation to the SI in every document in which they are used,” and “their use is not encouraged.” These barely-tolerated units might well be prohibited by future meetings of the CGPM. They include:

•     the nautical mile and knot, units traditionally used at sea and in meteorology;
•     the are and hectare, common metric units of area;
•     the bar, a pressure unit, and its commonly-used multiples such as the millibar in meteorology and the kilobar in engineering;
•     the angstrom and the barn, units used in physics and astronomy.

The SI does not allow use of any units other than those listed above and their multiples. In particular, it does not allow use of any of the English traditional units (the horsepower, for example), nor does it allow the use of any of the algebraically-derived units of the former CGS system, such as the erg, gauss, poise, stokes, or gal. In addition, the SI does not allow use of other traditional scientific and engineering units, such as the torr, curie, calorie, or rem.
Sources: Wikipedia and Dictionary of Units of Measurement – by Russ Rowlett

Global Coordinates

The prime meridian divides the Earth into the Eastern and Western hemispheres. The prime meridian is the line of longitude that runs through Greenwich, England. Points on the globe are measured from Greenwich in an eastward or westward direction in units called degrees, from 0§ longitude at Greenwich to 180° east or west. All the lines of longitude converge at the two poles and are equally spaced from each other only at the equator.
Each degree of latitude and longitude is divided into 60 minutes, indicated by an apostrophe, and each minute is divided into 60 seconds, indicated by a quotation mark.
Every location on the Earth’s surface can be described in terms of its latitude and longitude, counting degrees north or south of the equator and east or west of Greenwich. When describing a position, the latitude is listed first. Minutes and seconds can be used to give a more precise location description. For instance, a part of Detroit, Michigan, is located at 42° 30′ N, 83° W.
Time is also measured from Greenwich. The time at Greenwich is referred to as Greenwich Mean Time (GMT). Universal Coordinated Time (UTC), or ZULU. The globe is divided into 24 time zones—12 to the east of Greenwich and 12 to the west of Greenwich. Time retreats by one hour as pilots fly every 15° westward from Greenwich and advances by one hour as pilots fly every 15°eastward until reaching the International Dateline halfway around the globe (or at 180§ longitude). Pilots use GMT to refer to the time of day rather than using local time zones.