Langi Platforms of Tonga

THE LANGI PLATFORMS OF TONGA

AN OPEN-AIR UNIVERSITY FOR TEACHING NAVIGATION

In September 1920, archaeologist William Carleton McKern, a member of the Bayard Expedition, commenced a survey of the langi platforms and mounds of Tonga. His primary focus was in measuring structures on Tongatapu Island, especially around the township of Mua, along with other structures on outlying islands of the Kingdom of Tonga.

These many stone-walled or earthmound structures had intrigued early explorers of the Pacific Ocean region, as the laboriously built, multi-tiered truncated pyramids or symmetrical mounds were found from Hawaii to French Polynesia to the Marquesas Islands, to Samoa and generally across the entire expanse of the Pacific ocean to Melanesia and Micronesia.

The question remains these 100-years after McKern: Who built these huge structures and for what purpose?

More recent archaeological dabbling has failed to provide answers of any real consequence to these questions and, despite the increasing availability of advanced aerial photography, Lidar scan technology, precise gps and laser surveying tools, we're still stuck in the same, none-the-wiser, time-warp of a century ago.

One recent-era publication sums up the 100-year old archaeological stalemate:

'Liller (2000) makes a brief commentary about the orientations of the langi, indicating the impossibility of finding a unique orientation because the right-angled corners are rare in these monuments. This was also noticed by McKern (1929). I visited Tongatapu in October 1999 and made direct measurements of 12 langi at Mu'a with a magnetic compass'.

It comes as no surprise that the analysis fell-flat and provided no answers, as magnetic-compass measurements are highly inaccurate. All of McKern's orientation measurements were also done with a magnetic-compass and his true-north arrows are well off the mark on the survey plans he provided. McKern also limited his length measurements to only 2-sides of a langi, but that was undoubtedly due to having to work through a tangle of bush and overgrowth on long-abandoned structures.

Note: This researcher, over several years, has written to the government of the Kingdom of Tonga, as well as the so-called leading-lights, supposedly conversant with the langi structure archaeology, asking for survey plans showing precise lengths and angles, but has never received an answer. The only measurements available, limited to two sides of the langi's, and no reliable degree angles, are found in WC McKern's book, The Archaeology of Tonga ... partially surveyed over 100-years ago.

The Mua, Tongatapu, landscape has been subjected to advanced LIDAR photography, and the langi platforms or mound structures are clearly detectable using the latest, advanced technologies. Many of the structures exposed by lidar mapping cannot be seen in Google Earth, or are only partially visible.

The stadium looking structure to the right of the photo, with gateways on each side, has the appearance of a henge, where, in the Northern Hemisphere, students of celestial navigation could stand centrally through the night watching star rise and set angles on the surrounding earthen bank. The embankment itself acted like an artificial sea horizon and, by long vigils of stellar observation, students could memorise the time-sequence for stars rising from or alighting onto the embankment, as well as the angle.

To get some appreciation of the importance of memorising the positions of Oceania's regional stars from a henge structure, then applying that sky-map knowledge to inter-island, stellar navigation, see:

https://openresearch-repository.anu.edu.au/bitstream/1885/114975/2/b13540300.pdf

The henge-like structure seen in the lidar scan is, these days, categorised as a fort and is fully obscured by heavy forestation in topographical imagery.

Precise data-sets related to the dimensions and angles of the langi platforms are seemingly unavailable to the worldwide research community and not so much as one modern, reliable or useful measurement of a langi terrace side-length or its angle of orientation is available to the public. Also, from past experience, if any measurements did become available from that quarter, they would undoubtedly be expressed in "metres" and "rounded" to a tolerance of maybe half a metre, rather than being expressed in "old-measure" (inches & feet), which ancient civilisations are known to have used.

Our archaeologists tend to not-believe that there could be anything significant to find amidst the lengths and angles of the multi-tiered langi's and were built solely as regal platforms, where royalty could stand above their minions and survey all under their dominion. The present archaeological conclusion is that they were built only about 900-years ago to serve largely as a necropolis for the royal dead.

That's an incredible amount of human labour to create a "city of the dead"!

Were it not for a sparse few measurements done by WC McKern 100-years ago in his comprehensive identification of "pigeon-mounds", "langi-platforms" and "fort structures", etc., we would have little or nothing of any real use to consult amidst the available archaeological literature. Again, inasmuch as McKern's angle measurements are limited to hand-held compass-bearings, all are widely off the mark and mere approximations of true north.

In order to explore the theory that the langi platforms may have been carefully-built, code-bearing structures for teaching navigation, the only useful resource available for length and angle measurements is Google Earth.

Whereas our archaeologists, to this day, can't find even one compelling clue that would demonstrate accurate-surveying in the orientations of all the hundreds of terrace-alignments, let's see if we can get-the-ball-rolling and find a few, before launching into a comprehensive study of "what's left" of the langi complex at Mua.

We'll start with a simple, 1-terrace high langi called Tauhala:

McKern commenced his survey of these langi in September 1920 and published his results in 1929 in his book, The Archaeology of Tonga.

For this langi he measured the northern wall to be 177-feet and the eastern wall to be 170-feet. These are, in fact, potentially, code-bearing lengths at code-bearing angles, which will be explained in full as we proceed.

Based upon McKern's measurements and verified by the accurate length & angle tool of Google Earth, an image of Tauhala was taken into the exacting-environment of AutoCAD and scaled over the remnant walls of the structure.

It seems very obvious that there is reference to the lunar-year in the orientation of Tauhala langi, as well as the intention to encode 240 ancient Greek feet of 12.6-inches each into the north-to-south length ... (more analysis available below).

From Katoa langi's high epicentre of the top tier, it achieves a TRUE NORTH line-of sight onto the northeast corner of Namoala Tapa'itea langi, 1200-feet distant. The Mua landscape is very flat and, without trees, Namoala Tapa'itea would be perfectly visible.

At the same time, the line runs over the epicentre position of Tuofefafa Fo'ou Fofa 'Ua and marker poles on all 3 langi of this north-south line would provide the ability to astronomer-surveyors to determine all angles within 360-degrees, using this dissecting alignment.

Further to the above, the true north alignment, extending from the epicentre of Katoa langi, should finally have resolved upon a marker near the water's edge at this peninsula jutting into Fanga 'uta Lagoon. This researcher would be looking for evidence of an obelisk boulder or former Nui-pole at or near 21-deg 10' 21.02" S 175-deg 06' 59.4"W

The true north line falls upon the northeastern corner of Namoala Tapa'itea langi's top tier, after passing over Paepae'otelaea langi. However, near the southwestern corner of Namoala Tapa'itea there is a huge marker-stone (see arrow) lying hard up against the bottom terrace. It was obviously placed there to mark something and McKern measured the boulder and wrote about it in his survey.

WC McKern, a century ago, noted the existence of this large block on the SW side of Namoala Tapa'itea langi.

A surveying alignment from the epicentre of Katoa langi resolves upon the large marker-boulder at 354.375-degrees ... (354 & 3/8ths), again highlighting the lunar year @ 354.375-days.

Here's another surveying alignment running from the epicentre of Tuofefafa Fo'ou Fofa 'Ua langi to Leka Puipui Mau at 52.5-degrees. It then brushes the length of the bottom terrace edge, which marks that alignment. The ancient Greek mile was 5250-feet (5000 Greek feet of 12.6-iches each) and their short Stadium measurement 525-feet? Equally represented along the Leka Puipui Mau terraces could well be code-bearing numbers 52 and 51.84-degrees, respectively.

The alignment extending from the epicentre of Tuofefafa Fo'ou Fofa 'Ua langi brushes the northwest corner of Lekamakatuituioha langi's top tier, then hits the epicentre of Tauhala langi, then brushes the top tier terrace-edge of Fa 'apite Malu o 'otonga langi through its full-running length, before brushing the bottom tier edge, full-length, of Leka Puipui Mau langi.

This is a purpose-built surveying alignment encoding very specific numbers.

THE PACIFIC REGION'S, ANCIENT STRING-KNOT LANGUAGE.

Usage of the ancient string-knot language (a mathematical language that preserved very special numbers) ranged from Peru (quipu) to Hawaii to New Zealand (kupu) to the Western Pacific, Austronesia and beyond to India and China, etc.

The surveyors of ancient Egypt were called "harpedonaptae" or rope-stretchers, as they used knotted ropes, some of very considerable length, to do very accurate measurements. The measuring cords for Hawaii were called 'ahaku or 'aha hele honua, connected with measuring height or length ('ahaku), squaring corners to sacred houses, or finding the middle of a circumscribed area.'

Quipu or Kupu, on the other hand, were generally bunches of short-length cords or strings with complex, special knots that represented a number or numbers per string. In essence they were "strings of numbers".

'Only in one instance is there comparability between the quipu and hipu'u, and that is in the existence of a decimal system in the calculations of Quechuan and Hawaiian numbers in the use of a base ten: '... The single most important breakthrough in understanding quipus remains that of Leland Locke, who, in 1912, demonstrated that the quipus he studied were evidently not language, but rather purely numerical in nature. He also discovered that they used a decimal counting system of a positional nature, and that the concept of zero was present" [Conklin, 1982: 267: *Note: The old Hawaiian counting system was decimal using place value & assumption of the zero position].' (See: Essays in Hawaiian Literature, The Cord in the World, pp. 257 - 265, by Rubellite Kawena Johnson, University of Hawaii-Mama, 2000 AD).

Ancient civilisations, ranging across the Pacific Ocean and settling permanently, brought their advanced mathematical concepts with them, encoded into number-strings. These whole number and fraction expressions related directly to accurate methods of navigation on the featureless oceans, cyclic astronomy, the lunisolar calendar and counts within the very important 18.61-year (6804-day) Lunar Nutation Cycle.

These are the code-bearing lengths and angles that they built into the langi platforms of Tonga, Hawaii, French Polynesia and Easter Island, ranging to the standing stone structures of New Zealand.

NOW LET'S ATTEMPT TO IDENTIFY THE IN-BUILT, CODED LENGTHS AND ANGLES OF EACH TERRACE ... OF 8 LARGELY VISIBLE AND INTACT LANGI PLATFORMS AT MUA.

Namoala Tapa 'itea langi platform, composed of 3-tiers or levels. The third or top level is especially impressive, as it is walled on all 4-sides by very large, cut-blocks of heavy coral stone. Whereas one would reasonably expect the builders to want the structure to be a perfectly symmetrical rectangle, this was clearly not the intention.

In consideration of all the langi platforms throughout Tonga, why are all so different in length, size, angle-per-side and general orientation?

This in-built departure from symmetry was clearly deliberate and affects all the langi platforms of the Mua-Lahana district, which lie on differing angles of orientation in relation to each other, as well as within the dimensions and angles, tier-by-tier, of each individual langi itself.

Were the builders installing coded lengths and angles within every retaining wall or was it all just a haphazard exercise that was based on no specific plan or outcome, as our archaeologists suppose?

Let's try to find out!

NAMOALA TAPA 'ITEA LANGI

In the Google Earth imagery the latest photo of Namoala Tapa ítea shows the northeastern side to be largely obscured by overhanging trees. For this reason a more detailed image had to be sourced and the Google photo of October 2022 was chosen as the best one available for this analysis.

With very careful work and reference to some ground-level photographs taken at points all around the structure, the above results were achieved in AutoCAD. Lengths and angles measurements were done using the Google Earth tool and, from past experience, as well as reference to the experience and commentary of other researchers, it can be concluded that the Google Earth length & angle tool is very good. So here (below) are the "best-guess"* code-bearing results for Namoala Tapa'itea Langi:

* Note: WC McKern measured the SW base of Namoala Tapa'itea as 86-feet, but that value, in-and-of-itself has no real significance in the ancient parcel of highly-specialised factorable numbers, whereas the number 86.4-feet would have huge significance.

It would be a futile exercise to proceed with this analysis by including arbitrary numbers that have no significance, as they can't really shed-light on what the true intention of the builders was. For this exercise, therefore, if McKern stated that such-and-such a terrace was 170-feet long, then that value will be adjusted to 170.1-feet, as the number would then have an ancient pedigree and be very historically significant. The sum of 170.1-feet of 12-inches is 162 ancient Greek feet or 81 ancient Assyrian cubits.

Also, McKern's length measurement of 86-feet for Namoala's SW base terrace is not one of the special parcel of numbers that got included in ALL the ancient Weights, Measures & Volumes standards of the great-civilisations of antiquity, whereas 86.4 features very prominently. For example, the ancient Hebrew Cab volume under the Desert Liquid standard, was 86.4 cubic inches. Amongst the Greek liquid capacities, their Metretes volume was 864 Cyathus.

The objective of this exercise will be to see if the langi platform length and angle numbers are taken from that special parcel of numbers. ALL number-strings seen herein come from the special parcel of navigational, cyclic-astronomical, lunisolar Sabbatical Calendar, Precession of the Equinoxes or Lunar Nutation Cycle values used by the ancient civilisations, continent-to-continent. The numbers are derived from Egypt's Great Pyramid or Khafre Pyramid and their code-bearing dimensions & angles.

To the uninitiated, unfamiliar with the "ancient codes of civilisation", the numbers shown above, expressed in decimals, will look quite complex and cumbersome. However, they translate to reasonably simple fractions. For example, here are the more complex looking lengths and degree angles for Namoala Tapa'itea langi, as shown above, expressed as whole numbers and fractions:

Top: 114.583333' @ 51.84-degrees (114 & 7/12ths @ 51 & 21/25ths degrees); 515.625-degrees is simply 515 & 5/8ths; 51.42857143 degrees is simply 51 & 3/7ths.

Right: The degree angle 138.24 is simply 138 & 6/25ths; 53.76-degrees is simply 53 & 19/25ths.

Encoding all of these whole number & fractions expressions on a quipu-kupu was well within the capacity of ancient mathematicians.

WAS THERE ONCE A STANDARD METROLOGY & ANGLE SYSTEM SHARED BY THE GREAT CIVILISATIONS, FROM CONTINENT-TO-CONTINENT, ACROSS THE EARTH?

As it turns out, deciphering code-bearing lengths and angles on the langi's poses less of a mystery than most people realise, as highly-developed metrology systems of the great civilisations, living, from remote-antiquity around the Mediterranean Sea Basin, across the vastness of Continental Europe, the British Isles and elsewhere, all the way to the Americas and beyond, are reasonably well-known and well-documented. However, through neglect and declining knowledge over the past thousand years or so, many standards “drifted” slightly off the mark resulting in the very significant, original numbers being only half-remembered.

This ancient, universal parcel of special numbers certainly found its way into the Pacific Ocean region, carried there and encoded into the dimensions and angles of "mnemonic-recall" structures by early explorers and settlers of remote-antiquity.

Sadly, the ages-old measures, weights & volume standards, of Europe were assailed by the predations of early Roman-Christianity, followed-on centuries later by the introduction of metrification, resulting in the ages-old sciences, encoded by numbers, being lost from memory or obscured.

IT'S ALL ABOUT NUMBERS.

The dimensions and angles encoded into Great Pyramid of Egypt and other pyramids of the Gizeh Plateau acted as a Bureau of Standards from which the civilisations of antiquity, extending to modern eras, based their Weights, Measures and Volume Standards.

It was obviously realised in remote antiquity that if one is to enjoy the abundance and benefits of "CIVILISATION", then there is a body of essential, prerequisite, scientific knowledge that must first be gained. Once a civilisation has that knowledge, then its “wise-ones” (the few set aside to bless the many) must religiously preserve it and hand it on intact to each ensuing generation. Failure to do so means that one has just cursed one's children and, to whatever degree the knowledge is lost, so too is lost one's grasp on civilisation.

One truth that will come as a surprise to some is that the "so-called" British Standard "inch" and "foot" are not British at all, but are far older. All of the cubits or feet of the great civilisations of recorded history in the Mediterranean-Continental European-British Isles theatres were based upon the self-same inch. Moreover, all of the measurement standards of one civilisation were either the same, or precise ratio expressions to the standards of the surrounding confederation of cousin civilisations.

Everyone used the same integrated parcel of factorable numbers to describe their cyclic-astronomical or navigational sciences and, the way they eternally recorded those sciences was by incorporating those special numbers, by code-bearing lengths and angles, into public buildings/ edifices or portable memory devices.

Even much earlier in history, extending from the Neolithic Age, the sprawling marker-mound complexes and multitude of standing-stone circles were eternal repositories of the special numbers, (extractible by length and angle calculations from a hubstone to an outlier stone). Added to that, specialist teachers carried mnemonic devices into which the numbers were encoded and this, as it turns out, included the carefully fabricated “Volva staffs” of the traveling, wise women of Europe.

In the 19th century J Ralston Skinner, American archaeologist, did extensive investigations and measurements of Ohio's expansive, ancient, geometric earth-embankments. Skinner's conclusion was that the mound builders of the Ohio Valley had positively used the increment that today is known as the British standard inch in the design and construction of the geometric complexes. See: Skinner, J. Ralston. "Identification of the British Inch as the Unit of Measure of the Mounds of the Ohio Valley Builders." Journal of the Cincinnati Society of Natural History Vol. 9 (July 1886), pp. 51-63.
This journal could formerly be accessed online by going to: Cincinnati Historical Society Digital Journals.

Likewise, Sir William Flinders Petrie deduced that the "inch" used on the Great Pyramid was within 1/100th of an inch (visually undetectable) to the British Imperial Inch. Other researchers had come to similar conclusions, one of which (Taylor) put the Pyramid Inch to within 1/1000th of the Imperial Inch.

The finalised, big numbers that had to be remembered by ancient civilised society might look reasonably unfriendly. This is because nature has a bad habit of running cycles to its own schedule, without consulting accountants. A solar year is 365.2422-days and a lunar year is 354.3643519-days. The Lunar Nutation Cycle, where the Moon goes to a position called major standstill, returns to minor standstill, then returns to major standstill again, takes 6798.36-days or 18.61329277 solar years. The Earth is 24902.44523-miles in equatorial circumference. The polar circumference is 24816.55084-miles. The cycle of the Precession of the Equinoxes takes 25776-years to complete.

So, with these cumbersome, non-factorable numbers to contend with, how did ancient scientists turn this chaotic set of difficult values into an ordered, integrated and functional mathematical system?

The solar year can be called 365 & 1/4-days, which is a nice rounded number equating to 8766 hours.
The lunar year can be called 354 & 3/8th-days, which is 8505 hours.
If a calendar is produced that runs for 7-solar years, then that equates to 2556 & 3/4th-days.
If the lunar period within that calendar runs for 7 & 1/5th lunar years (one tenth of 72-years), then that equates to 2551 & 1/2-days.
If a measurement rule is produced, based upon 5 & 1/4 feet, then 487 X 5 & 1/4 = 2556 & 3/4 feet/ days.
The same measurement rule can be used for the Moon and 486 X 5 & 1/4 = 2551 & 1/2 feet/ days.
If an auspicious mistletoe culling ceremony is performed on the 6th day after the calendar commences at the Summer Solstice, then the solar and lunar cycles can end on the same day 7-solar years later.
If the world is described as 12 X 12 X 12 X 12 X 1 & 1/5th-miles (note: 1 & 1/5th is a tenth of 12), then that's 24883 & 1/5th-miles...only 18.8 miles short of the true equatorial circumference.
If one wishes to navigate using a "6&7" series of numbers, those miles can be 5250' each.
If one wishes to navigate using an "11" series of numbers, those miles can be 5280' each (for a world with a circumference of 24750-miles).

A wonderful little window of opportunity arises when a straight-line diameter, using an "11" number is converted to a "6" based circle using PI @ 1728/550ths. The resultant circumference is fluidly divisible by 360°.

A wonderful little window of opportunity arises when a straight-line diameter, using a "6&7" number is converted to an "11" based circle, using PI @ 22/7. Therefore 525 X 22/7ths = 1650, which is divisible by 330, 660 or 360°.

A wonderful little window of opportunity arises when 1/7th of a 360° circle (51 & 3/7ths°) is multiplied by 1.75 = 90. A 630° compass will work very well using "7" series values.

If the cycle of the Precession of the Equinoxes is described as 25920-years, then that's 72 X 360.

If the Lunar Nutation Cycle is described as 6804-days, then that's 19 & 1/5th lunar years, which is nicely divisible by "6&7".

On the basis of these "big" numbers, which had been "factored" to be in very close tolerance to the true size of the Earth or duration of cycles, the ancient civilisations then produced their "Weights, Measures & Volume" standards in direct compliance to the highly factorable numbers generated.

Therefore:

What (much later) became known as a "Greek" foot has an overall length of 12 & 3/5ths British standard inches (one tenth of 126"). A Greek foot is simply half of an Assyrian cubit or a tenth of a Hebrew Reed (10 & 1/2 British standard feet). Babylonian-Sumerian or Egyptian standards, as well as ancient Swedish, Germanic or many others preceding the fledgling Greek & Roman civilisations, comply to this same integrated system. All are in direct ratio to the standards of their cousin nations or more distant forebears of remote antiquity.

SO, WERE THE LANGI PLATFORMS LAID OUT IN ACCORDANCE WITH ANCIENT, INTERNATIONALLY-DISTRIBUTED, METROLOGY STANDARDS?

In consideration of the great open-air universities of ancient megalithic Europe, the Mediterranean as elsewhere, where principles of navigation, cyclic-astronomy, the lunisolar calendar, weights-measures-volumes, pi & phi ratios and the 360-degree angle system were taught, there would be a need for similar schools in Oceania.

The Pacific Ocean is the most expansive of all the oceans of the world and is dotted with hard-to-find tiny islands or bordered by continents and large land masses, like the Americas to the east, Australia or New Zealand to the west & southwest and Japan to the northwest, etc.

The evidence shows that groups like the very ancient Lapita people could successfully navigate back & forth across this incredibly wide expanse, as their pottery and other evidence of their presence, ranges through all the island groups of Tonga, then all the way to New Guinea (3000-miles), then SE from there to New Caledonia (2000- miles).

It's very evident that these people had advanced navigational skills, from as late as 3600-years ago. Did they establish an advanced school of navigation at Mua, Tongatapu? Are the stone langi platforms at Taputapuatea marae, Raiatea, French Polynesia (1700-miles ENE of Tongatapu) indicative of similar ancient schools of navigation? What about the Ahu platforms that circuit Easter Island, 2700-miles East of Raiatea?

The distribution of Ahu platforms, extending around the coastline of Easter Island. Earlier analysis by this researcher shows that, from one Ahu on the coastline, the next adjacent one sits at a code-bearing distance and angle from its neighbour. This code-of-position relationship occurs in a continuing line-of-sight, Ahu-after-Ahu, around the coast. The evidence shows that Easter Island was once a very developed school for teaching navigation and cyclic astronomy.

AN ANALYSIS OF THE APPARENT LENGTHS & ANGLES FOUND UPON NAMOALA TAPA 'ITEA LANGI, TONGATAPU, AND THE SIGNIFICANCE OF THE NUMBERS.

The true orientation of the langi.

TERRACE 1 SW.

The length is 86.4-feet and the angle is 145.8-degrees.

A number string based upon 86.4 goes:

86.4, 172.8, 259.2, 345.6, 432, 518.4, 604.8, 691.2, 777.6, 864, 9504, 1036.8 ... 1209.6, 1296, 1382.4 ... 1555.2 ... 1728 ...1900.8 ... 2073.6, 2160 ... 2332.8 ... 2592 ...2851.2 ...3024, 3110.4 ...3628.8 ... 3888 ... 4147.2 ...4665.6 ... 5433.2 ... 6048 ... 6220.8 ... 6480 ... 8294.4 ... 9072 ... 9331.2 ... 10800 ... 11664 ... 12441.6 ...24883.2

This number-string was used anciently to identify the equatorial circumference of the Earth, or as Julius Caesar stated concerning what the Druids taught in their open-air universities:

"The magnitude of the Earth and its divisions"... or ... "astronomy, mathematics, geometry, medicine, jurisprudence, poetry and oratory were all proposed and taught-natural philosophy and astronomy with severe exactitude". (See: De Bello Gallico, VII, 15, 16.. See also: Strabo I IV, page 197. Caesars Comm. Lib V. Sueotonius, V Calegula. E. Campion, Accounts of Ireland, pg. 18.).

In the above line-up of generated numbers, based upon 86.4', are the two final values of 12441.6 & 24883.2 (miles). The ancient method for arriving at the Earth's equatorial circumference was to multiply 12 X 12 X 12 X 12 X 1.2 miles of either 5250' (Greek miles) or 5280' (British miles) to arrive at the 24883.2-mile figure ... the half value of which is 12441.6-miles. This number-string above refers directly to the navigational and Earth division method which is encoded into the literal base dimension circuit of the Great Pyramid of Egypt, based upon a 5250-feet (of 12-inches each) Greek mile.

If one circumnavigates the Great Pyramid twice (@ 756' per side length) then one has traveled 6048-feet or 1-minute of equatorial arc under the 5250-feet mile method of navigation. The slope angle of the Great Pyramid is 51.84-degrees and the base area that the Great Pyramid covers is 5184 square "Reeds" of 10.5-feet each (10 Greek-feet of 12.6-inches each).

In the above line-up the value 259.2 is generated and 25920-years is the duration of the Precession of the Equinoxes for one complete cycle of Precession (360-degrees X 72-years). Under this slow Earth-wobble or shift, affecting the true day of the equinox, the calendar date of the equinox needs to be changed by 1-day every 72-years to remain accurate. Also, in the 25920-year cycle of precession, 1/5th part is 5184-years.

The 756-feet length of the Great Pyramid is also 9072-inches ... or 7.5 seconds of equatorial arc (100.8-feet X 7.5).

The sum of 11.664-inches was 1 Roman foot.

The distance of 3110.4-miles was 1/8th of the 24883.2-mile equatorial circumference.

The sum of 9331.2-feet would equate to 1/1400th of the 130636800-feet (24883.2-miles of 5250-feet) equatorial circumference. Also the sum of 933120-grains was the ancient Babylonian Talent weight.

An interesting value generated is 1036.8 and this tells us how fast the Earth spins in a 24-hour day. Therefore 24883.2-miles ÷ 24-hours = 1036.8 mph.

A number-string based upon 145.8 goes:

145.8, 291.6, 437.4, 583.2, 729 ... 1166.4, 1312.2, 1458 ...2332.8 ... 4665.6 ...5103... 9331.2 ... 26244, etc.

This degree angle introduces yet another important method of navigation, which was adopted by the Romans and based upon the Roman foot of 11.664-inches ... which for practical computation usage in overland marching was 11.66666-inches (11 & 2/3rds).

In the above line-up 58.32-inches was the Roman Pace measurement or 5-Roman feet of 11.664-inches. Similarly, the value 7.29-inches was 1/8th of a Roman Pace.

The Roman foot was found to be 11.664 inches by John Greaves, professor of geometry, who in 1639 went to Rome specifically to ascertain the length of an ancient Roman foot. Greaves located a monument of Roman architect, Stalius Asper and measured bas-relief instruments used by him in the first century A. D. Greaves concluded, after careful investigation, that the Roman foot, 'contained 1944 such parts as the English foot contains 2000'. This means a Roman foot of 11.664 inches. See: http's://en.wikipedia.org/Vikki/John_ Greaves

Well yes, it was certainly 11.664” in number-strings tracking lunar cycles or standard volume capacities, etc., but the simpler value of 11.66666” (11 & 2/3rds) was used for overland distances, with a Pace of 58.333333” (5 Roman-feet). The same foot, in doubled value, was used in Sweden for the Rydaholmsalm length (1.94444-feet of 12-inches).

One of the more ancient concepts that survived into Mediaeval times was that 75-Roman miles was 1-degree of arc in the equatorial circumference of the Earth. Given that the Roman foot was represented as 11.664-inches, that would equate to a Roman-mile (5000 Roman-feet) being 4860 feet of 12-inches. Therefore, 75 such miles, if representing 1-degree of arc in the equatorial circumference, would equate to a world of 27000 Roman-miles

The degree angle 145.8 is also represented as 145.833333 (148 & 5/6ths).

The Roman overland mile was 5000 Roman-feet or 4861.11111 feet of 12-inches (4861 & 1/9th) X 75 X 360 = 131250000 feet or 125000000 Greek-feet or 135000000 Roman-feet or 67500000 Swedish-Rydaholmsalm or 27000 Roman-miles or 3750 Swedish-Mil or 25000 Greek-miles.

Note: The ancient Druidic bronze calendar plaque unearthed at Coligny, Ain, France in 1897 is 4.86-feet long (58.32-inches or 1 Roman Pace or 4 X 14.58-inches).

WC McKern, a century ago, noted the existence of this large block on the SW side of Namoala Tapa'itea langi.

Although the block, undoubtedly, served as a step up onto the first terrace, compelling evidence suggests it was a part of the original langi design. From the epicentre of Katoa langi, it represents a "milestone-type" marker placed 1/5th of a Scottish mile (1188-feet) @ a lunar-coded angle of 354.375-degrees (the lunar year is 354.375-days).

THIS IS THE SAME LUNAR CODING THAT ONE FINDS AT AVEBURY HENGE IN SOUTHERN ENGLAND.

This same coding of the 354.375-day lunar year occurred at Avebury Henge, Southern England, in a distance and angle relationship between the former giant obelisk (17-feet tall) of the southern circle and the doorway entry into West Kennet Longbarrow (1.35 Greek-miles to the south).

A yellow code-bearing survey line extends from the position of the giant obelisk marking the centre of Avebury Henge's southern circle and resolves perfectly on the standing stone alignment marking the entrance into West Kennet Long Barrow. The angle northwards is 354.375-degrees and the distance is 7087.5-feet (Note: 7087.5 is 3543.75 X 2).

The survey line runs from the precise position of the former obelisk marking the centre of Avebury Henge's southern circle.

The survey line extending from Avebury's giant obelisk position to the obelisk alignment at the doorway of West Kennet Long Barrow.

The former giant obelisk sat at the position of this modern-day plinth marker. Avebury Henge remained largely intact until about the 14th century, when zealous Christians began systematically destroying the stones (because they considered the "Pagan" site to have been a place of former devil worship). The general method used by the fanatics to destroy marker stones was to pull them down over a pit, light fires under them, then dowse the stones with water to crack and break them up. Thankfully, these eradicated positions have been restored by concrete plinths, so the codes-of-position are still intact and decipherable.

BACK TO MUA & Namoala Tapa'itea langi.

The position of the block would suggest coded lengths along the terrace to each side of the block. WC McKern measured the block to be about 7-feet of length and from the southeastern corner to the somewhat misshapen block edge, the distance appears to be about 66-feet.

If this was the case, then the 66-feet length would be the "chain" measurement in the ancient link, fathom, perch, chain, furlong, mile and league system of navigation (based upon the "11" number family).

The opposite, short side, corner to the block edge, might therefore achieve a coded length of 13.44-feet. These are the kinds of "up-close & personal" measurements we need to know, but aren't being supplied by our archaeologists.

Also, the northeastern, top tier corner area of Namoala Tapa'itea langi marks "True North" from the epicentre of Katoa langi.

TERRACE ONE, SE SIDE.

WC McKern recorded, in about 1920-21 that this terrace was 122.5-feet long for the eastern wall, which would mean the SE side. The Google Earth measuring tool indicates that it is about 3.5-feet longer and seemingly achieves 126-feet.

The angle of orientation seems to positively achieve 51.84-degrees for the first 2 terraces, corner-to-corner, although the second terrace has bulged out slightly through its centre section after many centuries or millennia of settling and slumping ground above it.

The weight of the retaining wall blocks of this second terrace was seemingly insufficient to counter this slight outwards distortion ... due, undoubtedly to wet tropical conditions and root pressure from encroaching jungle growth through long periods of abandonment.

So, for this analysis we'll consider it on the basis of 126-feet @ 51.84-degrees.

A number-string based upon 126 goes:

126, 252, 378, 504, 630, 756, 882, 1008, 1134, 1260, 1386, 1512, 1638, 1764, 1890, 2016 ... 2268 ... 3024, 3150...3402 ... 3654 ... 4032 ... 4536 ... 5670 ... 6048 ... 6300 ... 6804 ... 9072 ... 12 096 ... 13608 ... 130636800

In the above lineup 12.6-inches is a Greek foot and 25.2-inches is an Assyrian cubit; 630-feet is a Greek Stadium (Stadia ... pl.) and the length of the sprint race in the Greek Olympics; 756-feet is the length of the Great Pyramid; 100.8-feet would equate to 1-second of equatorial arc for a world that was 24883.2 Greek miles in circumference; 1512-feet would be the length of two sides of the Great Pyramid, with 2268-feet and 3024-feet being the 3rd & 4th sides; 6048-feet would be two circumnavigations of the Great Pyramid & 1-minute of equatorial arc; 6804-days would be the number of days for the duration of the Lunar Nutation Cycle; 9072-inches would be the length of the Great Pyramid and 9072-feet would be 1.5 minutes of equatorial arc.

In addition to the above, 1360.8-days would be 1/5th of the Lunar Nutation Cycle and 1360800-feet would be 1/96th of the 130636800-feet equatorial circumference of the Earth (24883.2 Greek miles of 5250-feet each).

The 51.84-degree angle number-string goes:

51.84, 103.68, 155.52, 207.36, 259.2, 311.04, 362.88, 414.72, 466.56, 518.4 ... 622.08 ... 777.6 ...933.12 ... 1244.16, 1296 ... 24883.2

In the above lineup 51.84-degrees is the slope angle of the Great Pyramid; 1036.8 is the speed the Earth spins in miles per hour and the "Stirling Jug" of Scotland, dating to 1457 AD and possibly much earlier, had a capacity of 103.68 cubic inches, which was the same as the ancient Hebrew liquid volume called the Jerusalem Cab (predating the Babylonian conquest); In the 24883.2-mile equatorial circumference, the sum of 1555.2-miles would be 1/16th, 3110.4 would be 1/8th & 12441.6 would be 1/2; The length of 20.736-inches was the longest Egyptian Royal cubit and 20.736-miles would be 1/1200th of the Earth's equatorial circumference according to the literal navigational method encoded into the base dimensions of the Great Pyramid; 12960-years is half the duration of the Precession of the Equinoxes; 933120-grains was the weight of the ancient Babylonian Talent.

TERRACE ONE, NW SIDE.

The northwestern side of Namoala Tapa'itea is quite overgrown and, in many of the Google Earth images, it's hard to detect beginning and end points of these terraces, mostly enshrouded by overhanging trees. McKern provided no measurement for this northwestern end, but it appears to comply to 86.4-feet at a code-bearing angle of 138.24-degrees.

A number-string based upon 138.24-degrees goes:

138.24, 276.48, 414.72, 552.96, 691.2, 829.44 ... 1244.16 ... 2073.6 ... 24882.2.

In the 24883.2-mile equatorial circumference, 138.24-miles would be 1/180th; 141.72-miles would be 1/60th; 829.44-miles would be 1/30th and 12441.6, half.

Amongst the values generated in this number-string is 2073.6. The largest Egyptian Royal Cubit found by William Flinders Petrie was 1.728-feet long or 20.736-inches. This was obviously used as a mnemonic device for remembering the equatorial circumference of the Earth, as 20.736 X 1200 = 24883.2.

TERRACE ONE, NW SIDE.

A view down the dilapidated, leaning stone, retaining wall blocks on the NW side of Namoala Tapa'ítea langi.

This 4th side, at its base, complies to a code-bearing length of 114.583333-feet (114 & 7/12ths) at an angle of 51.84-degrees (already explained).

A number-string based upon 114.5833333 goes:

114.5833333, 229.166666, 343.75, 458.33333, 572.9166666, 687.5 ... 1375 ... 1718.75 ... 2062.5 ... 2750 ... 4125 ... 5156.25 ... 5500 ... 6187.5 ... 6531.25 ... 7562.5 ... 8250 ... 16500 ... 24750.

This is a navigational-string for positional-plotting and dead-reckoning when voyaging across the vast featureless oceans, with no landmarks to refer to and only the celestial-bodies by night for reference.

When using the mile of 5250-feet for sea legs of travel, the PI value used for converting a leg into a circumference on the plotting chart was 22/7ths or 3.142857143. The circle generated was 5250 X 3.142857143 = 16500-feet (1-English league or 3.125-miles of 5280-feet).

The circle produced was divisable by 360-degrees and one degree of arc on 16500-feet was 45.8.333333-feet (45 & 5/6ths). Using calipers on the plotting-chart circle, one could calculate the angle back to the port of departure or onwards to the destination, as well as one's exact position in the ocean, after zigzagging across a leyline during many legs of travel on the wind.

The Greeks, like the English, undoubtedly preferred a larger sea-leg or league of travel when crisscrossing a leyline to a destination at sea and 3 Greek miles (15750-feet) would be very mathematically useful. The sum of 15750-feet X PI @ 22/7 = 49500-feet. A plotting-chart circle of this circumference would be 137.5-feet per degree of arc.

The two main systems of navigation used were (1) a mile of 5250-feet (which was adopted from older civilisations by the Greeks) and (2) a mile of 5280-feet (which was adopted by the English).

The increments in the above number-string are divisions of the English circumference of the Earth under the 5280-mile system.

In a literal reading of the Great Pyramid's side length the distance is 756-feet of intended design length and this value X 8 (2 circumnavigations of the pyramid) = 6048-feet for 1-minute of equatorial arc X 60 = 362880-feet for 1-degree of arc X 360 for the full equatorial circumference (130636800-feet or 12 X 12 X 12 X 12 X 1.2 Greek miles of 5250-feet = 24883.2-miles).

Under this second system, the Great Pyramid was considered to be 3-inches longer (756.25-feet). Therefore, 8 X 756.25-feet = 6050-feet for 1-minute of equatorial arc X 60 = 363000-feet for 1-degree of arc X 360 = 130680000-feet for the equatorial circumference of the Earth.

The sum of 130680000-feet equates to 24750-miles of 5280-feet.

In the above number-string, the sum of 114.5833333-miles would be 1/216th of the 24750-mile equatorial circumference. It would therefore represent 1 & 2/3rds degrees of arc.

Remember: the 360-degree compass system is divided into degrees X 60 = minutes or 360 X 60 = 21600 ... and if one wishes to reduce this to seconds of arc X 60 = 1296000.

In the above progression the value 687.5 occurs. In the 24750-mile reading of the Earth's circumference the sum of 68.75-miles would be 1-degree of arc.

Also the value 2062.5 occurs. The second longest Egyptian Royal Cubit located by William Flinders Petrie was 20.625-inches, and that value, read as miles, X 1200 = 24750 ... in other words, it was a mnemonic device for remembering the size of the Earth under this navigational system.

The value 5156.25 occurs and "Noah's ark" was described as 300-cubits long. If the 20.625-inch cubit was used, then 300 such cubits would equate to 515.625-feet.

The value 6187.5, read as miles, would be 1/4th of the 24750-mile equatorial circumference.

TERRACE 2, SW SIDE.

The length is 75.6-feet and the angle is 148.5-degrees.

The sum of 75.6-feet is 1/10th of the intended design length of the Great Pyramid and 3/4ths of 1-second of equatorial arc (100.8-feet).

A number-string based upon 75.6 goes:

75.6, 151.2, 226.8, 302.4, 378, 453.6 ... 604.8, 680.4, 756 ... 907.2 ... 1134, 1209.6 ... 1360.8 ... 1512 ... 1890 ... 2268 ... 2721.6 ... 3024 ... 3402 ... 3628.8 ... 3780 ... 5443.2 ... 5670 ... 6048 ... 6804 ... 7182 ... 7560 ... 9072 ... 9450 ... 11340 ... 17010 ... 34020 ... 51300 ... 68040 ... 362880 ... 130636800.

This number-string provides the Great Pyramid number for the perimeter of the Great Pyramid in feet as 756, 1512, 2268, & 3024; it also provides 6048-feet for 1-minute of equatorial arc or 362880-feet for 1-degree or arc and 130636800-feet for a full equatorial circumnavigation, The length of the Great Pyramid is 9072-inches and its height to the top surface of its altar floor is 453.6-feet

The number-string also provides the values for the Lunar Nutation Cycle in days, 1701 (1/4th), 3402 (1/2), 5130 (3/4ths), 6804 (full-term).

A number-string based upon 148.5-degrees goes:

148.5, 297, 445.5, 594, 742.5, 891, 1039.5, 1188 ... 1485 ... 1782 ... 2376 ... 3564 ... 5940 ... 6534 ... 7128 ... 11880 ... 130680000.

This is a progression based upon another "mile" that became the ancient Scottish mile of 5940-feet. This mile was 1-furlong longer than the English mile (5280 + 660 = 5940-feet). Because it was based within the same "11" family of numbers as the English mile it generated the same number (130680000-feet) for the finalised, equatorial circumference of the Earth, Therefore, the Earth could be described as 24750 English miles (5280-feet) in circumference or 22000 Scottish miles of 5940-feet.

The ancient Irish-mile was 6720-feet.

TERRACE 2, SE SIDE.

The length is 118.125-feet and the angle is 51.84-degrees.

The 51.84-degree angle had already been explained. for this langi.

The length of 118.125 is lunar coding found upon the Khafre Pyramid of the Giza Plateau, Egypt. This pyramid is 15/16ths the side length of the Great Pyramid. Therefore, 756 ÷ 16 = 47.25 X 15 = 708.75-feet.

The Khafre Pyramid of Egypt was built as two 3, 4, 5 triangles placed back-to-back. The half-base horizontal was 354.375-feet or 3 X 118.125-feet; The centre vertical was 472.5-feet or 4 X 118.125 and the diagonal hypotenuse side was 590.625-feet or 118.125 X 5.

Note: The lunar year is 354.375-days (354 & 3/8ths). Also, the lunar month is 29.53125-days (29 & 17/32nds). Half the 590.625-feet face diagonal is 295.3125-feet. The sum of 118.125-days would equate to 4-lunar months of 29.53125-days (29 & 17/32nds).

The Khafre Pyramid is a pyramid of the moon and gives us all the essential information for tracking the lunar cycle. Four side lengths @ 708.75-feet per side = 2835-feet or 34020-inches. The Lunar Nutation Cycle endures for 6804-days and the half period is 3402-days.

TERRACE 2 NE SIDE.

The length is 75.6-feet and the angle is 138.24-degrees, both of which have been explained earlier.

TERRACE 2 NW SIDE.

The length is 105-feet and the angle is 51.5625-degrees.

A number-string based upon 105-feet goes:

105, 210, 315, 420, 525, 630, 735, 840, 945, 1050 ... 1260 ... 1470, 1575, 1680... 1890 ... 2520, 2625 ... 2835 ... 3360 ... 3780 ... 5250 ... 5670... 6720 ... 7560 ... 10080 ... 13125 ... 130636800.

The sum of 105-feet is 10 Hebrew Reeds or 100 Greek-feet of 12.6-inches each. In the equatorial size of the Earth, under the Great Pyramid's literal navigational method, there would be 12441600 Reeds of 10.5-feet each (24883.2 Greek miles).

In the above lineup the value 3.15 was often used as a close approximation of PI. One cross-measure on the inside of the slightly elliptical Sarsen Circle at Stonehenge is 100-feet, inferring a circle of 315-feet or 1/2 a Greek Stadium of 630-feet. At 90-degrees opposed to this inner Sarsen Circle measurement, the diameter is 100.8-feet, or 1-second of equatorial arc.

The sum of 189-feet is 1/4th the length of the Great Pyramid and 2835-feet is the perimeter length around the Khafre Pyramid of Egypt; 5250-feet became a Greek-mile & 6720-feet became an ancient Irish-mile.

The sum of 567-days would be 1/12th of the 6804-day Lunar Nutation Cycle.

The angle is 51.5625-degrees and a number-string based upon this goes:

51.5625, 103.125 ... 206.25 ... 412.5 ... 618.75 ... 825 ... 1650 ... 2475 ... 3300 ... 24750 ... 130680000.

This number-string is based upon the mid-range Egyptian Royal cubit of 20.625-inches and there would be 30 such cubits in 51.5625-feet. There would be 9600 such cubits in the English league of 16500-feet (3.125-miles of 5280-feet). There would be 1200 X 20.625-miles in the 24750-mile equatorial circumference, under that navigational method.

TERRACE 3 SW SIDE.

The length is 64.8-feet @ 148.5-degrees.

A number-string based upon 64.8-feet goes:

64.8, 129.6, 194.4, 259.2, 324, 388.8, 453.6, 518.4, 583.2, 648, 712.8, 777.6 ... 907.2, 972, 1036.8 ... 1166.4 ... 1296 ... 1555.2, 1620 ... 2073.6 ... 3110.4 ... 36288 ... 12441.6 ... 24883.2

This number-string carries Great Pyramid navigational values, as well as others that relate to the duration of the Precession of the Equinoxes.

In the 24883.2-miles equatorial circumference 64.8-miles would be 1/384th part; 194.4-miles 1/128th, 777.6-miles 1/32nd, 1555.2-miles 1/16th, etc.

The height of the Great Pyramid (a truncated pyramid, never known in all recorded history to have a pointed top) is 453.6-feet. This is 264 of the shortest Egyptian Royal cubits @ 20.61818182-inches 20 & 34/55ths. An example of this one is found in the Turin Museum. The ancient formula for calculating the length of this cubit was 9072-inches (the length of the Great Pyramid @ 756-feet per side) ÷ 440 = 20.61818182.

Again, in this progression is found the speed at which the Earth spins (1036.8 mph); the slope angle of the Great Pyramid (51.84-degrees) and the duration of the Precession of the Equinoxes (25920-years, 1/5th part of which is 5184-years.

TERRACE 3 SW SIDE.

This is 110-feet @ 53.76-degrees.

A number-string based upon 110-feet goes:

110, 220, 330, 440, 550, 660, 770, 880, 990, 1100, 1210, 1320 ... 1650, 1760 ... 1980 ...2640, 2750 ... 2970 ... 3960...5280 ...5940 ...6050 ... 7920 ... 16500 ... 130680000.

In this lineup, 5.5-feet was the original "fathom" measurement; 66-feet was the chain and 660-feet was the furlong; 16.5-feet was the "rod" or "perch" (3 fathoms), 1760-"yards" (3-feet) was a mile of 5280-feet and 5940-feet was a Scottish-mile.

Also, the value 6050 occurs and under the navigational method that used the mile of 5280-feet the sum of 6050-feet was 1-minute of equatorial arc.

The radius of the Earth is 3960-miles of 5280-feet each, producing an overall diameter of 7920-miles.

The degree angle is 53.76 and a number-string based on this goes:

53.76, 107.52, 161.28 ... 2688 ... 537.6 ... 806.4 ... 1344 ... 6720 ... 12096 ... 130636800.

This is yet another navigational method, based upon the Irish mile of 6720-feet. It works within the Great Pyramid's circumference of the Earth reading of 130636800-feet. The sum of 53.76-feet would be 1/125th of an Irish mile and an Irish mile is 1/54th of 1-degree of equatorial arc (362880-feet).

TERRACE 3 NE SIDE.

The distance is 69.12 @ 137.5-degrees.

A number-string based upon 69.12 goes:

69.12, 138.24, 207.36, 276.48, 345.6, 414.72 ... 622.08, 691.2 ... 829.44 ... 1036.8 ... 1244.16 ... 24883.2.

With the world 's equator configured to be 24883.2-miles in circumference, 1-degree of arc would equate to 69.12-miles. Therefore, this line-up of values simply enumerate 1-degree, 2-degrees, 3 degrees, etc.

The sum of 207.36-miles would be 3-degrees, 622.08-miles would be 9-degrees, 1036.8-miles would be 15-degrees, etc. In a more usual sense, this progression would refer to miles of 5250-feet.

With the angle set at 137.5-degrees, the number-string goes:

137.5, 275, 412.5, 550, 687.5, 825, 962.5, 1100, 1237.5, 1375, 1512.5, 1650, 1787.5 ... 2062.5, 2200 ... 2475 ... 2750 ... 3025 ... 3300, 3437.5 ... 9075... 24750

The angle code mirrors the distance code in that 1-degree of arc, in a world configured to be 24750-miles (of 5280-feet) in circumference, would be 68.75-miles, the double value of which would be 137.5-miles.

The progression refers directly to the second navigational system encoded into the Great Pyramid, elongated slightly to be 756.25-feet. Under this 3-inch increase, two side values are 1512.5-feet, and a full circumnavigation is 3025-feet, making 1-minute of arc jump from 6048-feet to 6050-feet. A degree of arc jumps from 362880-feet to 363000-feet.

TERRACE 3 NW SIDE.

The distance is 97.2-feet and the angle is 51.42857143-degrees.

A number-string based upon 97.2-feet goes:

97.2, 194.4, 291.6, 388.8, 486, 583.2, 680.4, 777.6 ... 972 ... 1166.4...1360.8, 1458, 1555.2 ... 1944 ...2332.8, 2430 ... 3110.4 ... 3402 ...4665.6 ...54432 ... 6804 ...7290 ... 9331.2 ...12150 ...12441.6 ... 24883.2.

Traditionally through the ages, the 360-degree compass was halved, quartered and read in divisions of 8ths, 16ths, 32nds, 64ths, 128ths and 256th divisions. The sum of 97.2-miles is 1/256th of 24883.2-miles, or 1.40625-degrees.

This is equivalent to 360-degrees reducing to 180, 90, 45, 22.5, 11.25, 5.625 (5 & 5/8ths), 2.8125 (2 & 13/16ths), 1.40625 (1 & 13/32nds)-degrees respectively.

The progression is also useful in simultaneously tracking the divisions of the 6804-day Lunar Nutation Cycle, within a 360-degree circuit being halved, quartered, etc.

The degree angle is simply 51 & 3/7ths. Under the ancient lunisolar Sabbatical Calendar system, 7-solar years were tracked daily alongside counts within 7.2-lunar years. This was due to the solar-year being 365.25-days and the lunar-year being 354.375-days. However, 7 solar years is 2556.75-days and 7.2 lunar years is 2551.5-days, for a difference of only 5.25-days. For this reason the Druids and others would commence their solar count, then six days later commence their lunar count. By this means the 2 periods would end on the same day 7-solar years or 7.2-lunar years later.

PAEPAE 'O TELEA LANGI

Sitting to the south of Namoala Tapa'itea langi is very impressive Paepae'otelea langi, which has a second terrace featuring some of the largest and most impressive cut coral-stone blocks to be found in Tongatapu.

This langi is composed of 3 tiers or terraces.

Here are the results of identifying the lengths and angles of each terrace edge, using an older Google Earth image that provided a better topographical view..

The correct orientation of Paepae 'otelea langi.

TERRACE 1 SW.

WC McKern, a century ago, recorded this side length to be 92.5-feet.

The code-bearing length is 92.16-feet and the angle is 138.24-degrees.

A number-string based upon 92.16 goes:

92.16, 184.32, 276.48 ... 460.8 ... 829.44 ... 1013.76 ... 1382.4 ... 2304 ... 2488.32 ... 4147.2 ... 4608 ... 11520 ... 12441.6 ... 24883.2.

The above values are divisions of the 24883.2-mile (of 5280-feet) equatorial circumference, with samples showing 1/270th, 1/54th, 1/18th, 1/6th, etc.

The true equatorial circumference of the world is 24902 English miles of 5280-feet but, unfortunately, it is very hard to work with that number when configuring a fluid, easy-to-compute, navigational system. For this reason ancient savant-astronomer-navigators chose close-proximity values that were highly divisible ... that could be halved, quartered, reduced to eighths, sixteenths etc., right down to a tiny expression of the original. Thus, one could reduce a value like 24883.2-miles, very easily, describing the entire equatorial circumference as 12 X 12 X 12 X 12 X 1.2-miles, reducible down by degrees, minutes and seconds to one inch, then fractions thereof.*

*Note: This adjustment of a number into a factorable or easily manipulated value was also used to describe the duration of the Lunar Nutation Cycle, which is, in actuality, 6798.36-days. In the ancient edifices, like Khafre Pyramid, it is encoded as 6804-days and thus easily described as 19.2 lunar years or 230.4 lunar months. It then becomes easy to daily track on a lunisolar calendar, parapegma plaque, like the bronze Calendar of Coligny artefact, throughout the entire lunar cycle, with less than one minute (time) of error each lunar-month.

In a general sense, the mile chosen for easy navigation was 5250-feet in length, as it could convert to a circle that was easily divisible by 360-degrees under one of several adequately precise versions of the PI ratio.

For the 5250-feet mile, PI @ 22/7ths was used ... or 3.142852143 X 5250 = 16500-feet (1-English league ... 3.125 English miles). On the plotting chart, calipers scaled to 45 & 5/6ths, identified each degree of arc.

Alternatively, one could use a longer sea-leg (league) like 3-miles of 5250-feet = 15750-feet ... converting, using PI @ 22/7ths, to 49500 ÷ 360 = 137.5-feet per degree of arc on the plotting chart.

Equally easy to use for navigation was the 5280-feet mile, adopted by the English and Germanic-Teutonic peoples.

To use a memory-key for remembering the equatorial size of the Earth under the 5280-mile navigational system, one could use the "link" value of 7.92 X 5 X 5 X 5 X 5 X 5 = 24750.

Because of convenience in the way that these "miles" (either 5250' or 5280') worked with PI (either 22/7ths or 1728/550ths), they appear to have been the most popular of several methods used. However, they fell short of the "true"equatorial size of the Earth by a relatively small margin ... The 24883.2 Greek-mile reading is 159 English miles short and the 24750 English mile reading is 152 miles short.

If navigation was limited to the Mediterranean, Atlantic or even the Pacific oceans, the two systems were sufficient for very good relative-accuracy in positional-plotting, but the savant-astronomers-navigators needed to have a more accurate figure for the true equatorial circumference and chose 12 X 12 X 12 X 12 X 1.2-miles of 5280-feet or 24883.2 English miles ... only 18.8-miles short of the 24902-mile circumference we use today.

In the above lineup, the number 101.376 is represented at an enlarged value.

In a world configured to be 24883.2-miles of 5280-feet each, the segment for 1-second of equatorial arc would be 101.376-feet, a minute of arc 6082.56, a degree of arc 364953.6, and the equatorial circumference 131383296-feet.

In a world configured to be 24883.2-miles of 5250-feet each, the segment for 1-second of equatorial arc would be 100.8-feet, a minute of arc 6048, a degree of arc 362880, and the equatorial circumference 130636800-feet.

In a world configured to be 24750-miles of 5280-feet each, the segment for 1-second of equatorial arc would be 100.833333-feet, a minute of arc 6050, a degree of arc 363000, and the equatorial circumference 130680000.

THE DEGREE ANGLE.

The angle of orientation for this terrace on Paepae 'o Telea langi is 138.24-degrees ( Note: 138.24 = 92.16 X 1.5).

A number-string based upon 138.24-degrees goes:

138.24, 276.48, 414.72, 552.96, 691.2, 829.44 ... 1244.16 ... 2073.6 ... 24882.2.

In the 24883.2-mile equatorial circumference, 138.24-miles would be 1/180th; 141.72-miles would be 1/60th; 829.44-miles would be 1/30th and 12441.6, half.

TERRACE 1 SE.

WC McKern, a century ago, recorded this side length as 135.5-feet.

The code-bearing length is 134.4-feet and the angle is 44-degrees. No further terrace side lengths were recorded by McKern for this langi.

A number-string based upon 134.4-feet goes:

134.4, 268.8, 403.2, 537.6, 672, 806.4 ... 1209.6, 1344 ... 2016 ... 2688 ... 3360 ... 3628.8 ... 6048 ... 10080 ... 130636800.

In the above line-up the sum of 1344-feet would be 1/97200th of the Earth's circumference; 2688-feet would be 3 side lengths of the Great Pyramid; 362880-feet would be 1-degree of arc; 6048-feet would be 1-minute of arc and 100.8-feet would be 1-second of arc. A value of 1344-feet would be 11.25-seconds of equatorial arc under the navigational method encoded into the Great Pyramid's literal dimensions.

WC McKern's measurement of 135.5-feet would also be in a close-proximity value to a significant number. It would work within a number-string progression of 135.

A number-string based upon 135 goes:

135, 270, 405, 540, 675, 810, 945, 1080, 1215, 1350, 1485, 1620 ... 1890 ... 2160 ... 2430 ... 2700 ... 2835 ... 2970, 3105 ... 3240 ... 3375 ... 3780 ... 4185 ... 4320 ...4725 ... 4860 ... 5130 ... 5670 ... 5940 ... 7290 ... 7560 ... 8505 ... 8910 ... 9720 ... 10125 ... 10800 ... 11340 ... 11880 ... 12960 ... 13500 ... 14175 ... 14580 ... 14850 ... 15120 ... 17010 ... 17280 ... 19440 ... 25515 ... 25920.

It can be readily seen that the number 135 is very versatile as an increment for use in Great Pyramid navigational calculations; divisions of the Khafre Pyramid lunar-related dimensions; divisions within the equatorial circumference of the Earth; intervals in years within the Lunar Nutation Cycle; intervals within the lunar year or counts within the lunisolar Sabbatical Calendar; intervals within the duration if the Preccession of the Equinoxes and one of the secondary divisions of the 360-degree compass (i.e.) 11.25, 22.5, 33.75, 45, 56.25, 67.5, 78.75, 90, 101.25, 112.5, 123.75, 135 ... 270.

TERRACE 1 NE.

The length is 89.1-feet and the angle is 137.5-degrees.

The "891" mathematical coding recurs on ancient open-air-university sites from Avebury and Durrington Walls henges in England to Ring o' Brodgar in Scotland to Nazca in Peru to Easter Island and, thus, had very obvious importance to ancient navigators. It is part of an "11" progression (11 X 9 X 9 = 891) and 24750 (the value used to describe the equatorial circumference under the "11" system) X 36 = 891000. Also, the Scottish mile @ 5940' X 15 = 89100.

The ancient Scottish mile of 5940-feet was 1-furlong longer than the English mile (5280 + 660 = 5940-feet). Because it was based within the same "11" family of numbers as the English mile it generated the same number (130680000-feet) for the finalised, equatorial circumference of the Earth, Therefore, the Earth could be described as 24750 English miles (5280-feet) in circumference or 22000 Scottish miles of 5940-feet.

Let's look at some ancient stone discs used for tracking astronomical cycles or making computations, excavated from the North American mounds.

The ancient Mound-builder discs, with calibration notches or pattern segments around the perimeter. These discs could then act as mnemonic devices or aids in remembering the "times-table" progression of numbers associated with particular principles of cyclic-astronomy or navigation, etc., counts. Let's review how the above stone discs would have been used:

a. This disc has 15 segments and 15 notches. A dot on one of the segments probably relates to alternative counts using progressions of, say, 1512 (Earth's equatorial circumference), etc., The 15 segments work fluidly in a 360-degree environment, where each segment represents 24-degrees of arc. If this disc was considered to represent a circumference of 89100, then 1/15th of that value would be 5940 or the value of 1-Scottish mile. Likewise, If the 4-sides of the Khafre Pyramid @ 708.75-feet per side (2835-feet) was used as a diameter and turned into a circle using PI @ 22/7ths, then the circle produced would be 8910-feet.

b. This calibration (15) duplicates disc a.

c. This disc has 13 segments and notches and would have been used for calendar computations. Under a Sabbatical Calendar count, used by many ancient civilisations, there were 13 months of 28-days each in a year (364-days). This count ran for 7 solar years, wherein the missing 9-days (8.75 actual) were added back in during a festival and the count (now fully corrected) recommenced.

d. This disc has 22 segments and notches and would have been used for counts within the "11" family of navigational values, like the 5280' mile (22 X 240). The disc would have been useful for remembering the 24750-mile circumference of the Earth as 22 X 1125-miles.

e. This disc has 17 segments and notches and would have been used for lunar counts within the 6804-day Lunar Nutation Cycle (1701-days X 4). The use of intertwined dual snakes in the design or the "all-seeing eye" is very reminiscent of ancient Egyptian cultural symbolism or that of Scandinavia.

f. This disc has 23 segments and pointers and would have been used to remember the 24883.2 Greek mile equatorial circumference of the Earth (130636800'). One of the very important number-strings of antiquity is the 1152 one, which goes 1152, 2304, 3456, 4608, 5760, 6912, 8064, 9216, 10368, etc. At Stonehenge the outer rim of the Sarsen Circle measures 345.6' or 11.52 feet (average length) per lintel stone (30 overhead lintel stones) or 23.04' per two lintel stones. The disc would have been used on the basis of 230.4 X 108 = 24883.2.

g. This disc also has 23 segments and notches and works to 230.4.

h. This disc has 20 segments and notches, each representing 18-degrees of arc. It could also have served as an aid in calculating minutes of arc if configured to work to the value 20.16. Note: 2016' is 1/3rd of 6048' (1-minute of equatorial arc).

i. This disc is septimal, with seven segments and notches. It would have been used for calendar and navigational computations and especially for computing the lunar cycles and solar-year counts under the Sabbatical Calendar system.

j. This disc has 13 segments and notches and would have been used in calendar computations. Ancient agrarian calendars had 13-months of 28-days in a year (364-days) ... allowed to run for 7-years until corrected by adding 8.75 days

k. This disc is another "open-eye" design, like the "all seeing eye of Horus", as found on other of the North American mound discs.

l. This disc is yet another that works to progressions of 23.04.

See: Vol. XII, American Antiquity, July 1946, No 1, pg. 9, Article title: Stone Disks As Treaty Suns.

With the angle set at 137.5-degrees, the number-string goes:

137.5, 275, 412.5, 550, 687.5, 825, 962.5, 1100, 1237.5, 1375, 1512.5, 1650, 1787.5 ... 2062.5, 2200 ... 2475 ... 2750 ... 3025 ... 3300, 3437.5 ... 9075... 24750

TERRACE 1 NW.

The distance is 133.33333-feet and the angle is 45.36-degrees.

A number-string based upon 133.33333 goes:

133.33333, 266.66666, 400, 533.33333, 666.66666, 800 ... 933.33333, 1066.66666, 1200 ... 1466.66666, 1600 ... 2800 ... 2933.33333 ... 3600 ... 5200 ...5600 ... 21600 ... 1296000 ... 2592000.

As can be seen, 133.33333 is simply 1/3rd of 400. In the above line-up the value 1066.66666 occurs and the old French Pied du Roi measurements was 1.0666666-feet or 12.8-inches. The progression can also be used for mnemonic recall of the 25920-year duration of the Precession of the Equinoxes. Also, the 360-degree compass is 21600-minutes and 1296000 seconds of arc.

A number-string based upon 45.36 goes:

45.36, 90.72, 136.08, 181.44, 226.8, 272.16 ... 362.88 ... 453.6 ... 544.32 ... 680.4 ... 1134 ... 3402 ... 5670 ... 6531.84 ... 130636800.

The terrace angle is 45.36-degrees and the height of the Great Pyramid to the surface of its altar floor is 453.6-feet. The length of the Great Pyramid is 9072-inches (453.6" X 2). In a world configured to be 130636800-feet, the sum of 4536-feet would be 1/28800th part.

Note: based upon the information Greek Historian Herodotus received from Egyptian priests in circa 454 BC we can calculate the Egyptian Pyramid acre to be 28000-square feet. The priests also told him that the Great Pyramid was 800-feet long, which refers to a lunar foot of 11.34-inches. Three side lengths of the Great Pyramid are 2268-feet (1134-feet X 2).

In the above line-up, 1360.8-days would be 1/5th of the 6804-day Lunar Nutation Cycle; 36288-inches would be 1/2 a minute of equatorial arc and 362880-feet would be 1-degree of equatorial arc; 653184-feet would be 1/200th of the 130636800-feet equatorial circumference.

TERRACE 2 SW.

The length is 82.5-feet and the angle is 137.5-degrees.

A number-string based upon 82.5-feet goes:

82.5, 165, 247.5, 330, 412.5, 495 ... 660 ... 825, 907.5, 990 ... 1320 ...1485 ... 1650 ... 1980 ... 2062.5 ... 2640 ... 2970 ... 3630 ... 3960 ... 4537.5 ... 5280 ... 5940 .. 7920 ... 16500 ... 12375 ... 24750.

In the above line-up, 82.5-feet would be 1/64th of a 5280-feet English mile and 1/72nd of a 5940-feet Scottish mile; 16.5-feet would be 1-rod or perch, as well as 3 Merchant Navy fathoms of 5.5-feet each; 660-feet would be 1-furlong or furrowlong; 9075-inches would be considered to be the length of the Great Pyramid and 453.75-feet (half of 9075) the height to the top surface of the altar floor (truncated pyramid); The sum of 453.75-feet would be 264 Egyptian Royal cubits of 20.625-inches each (20 & 5/8ths); 363000-feet would be 1-degree of arc; 3960-miles of 5280-feet would be the radius of the Earth at the equator and 7920-miles the diameter; 16500-feet (3.125-English miles) would be a league; half the equatorial circumference of the Earth would be 12375-miles of 5280-feet for a full circumference of 24750-miles.

Note: for tacking on the wind in a sailing ship, doing sea-legs of 16500-feet back and forth across the leyline to the destination, the circle generated on the plotting chart at the termination of a leg (league), using PI @ 1728/550ths = 51840-feet or 144-feet per degree of arc.

TERRACE 2 SE.

The length is 126-feet and the 44-degrees.

A number-string based upon 126-feet goes:

In the above lineup 12.6-inches is a Greek foot and 25.2-inches is an Assyrian cubit; 630-feet is a Greek Stadium (Stadia ... pl.) and the length of the sprint race in the Greek Olympics; 756-feet is the length of the Great Pyramid; 100.8-feet would equate to 1-second of equatorial arc for a world that was 24883.2 Greek miles in circumference; 1512-feet would be the length of two sides of the Great Pyramid, with 2268-feet and 3024-feet being the 3rd & 4th sides; 6048-feet would be two circumnavigations of the Great Pyramid & 1-minute of equatorial arc; 6804-days would be the number of days for the duration of the Lunar Nutation Cycle; 9072-inches would be the length of the Great Pyramid and 9072-feet would be 1.5 minutes of equatorial arc.

The angle of this terrace is 44-degrees and a number-string based upon 44 goes:

44, 88, 132, 176, 220, 264 ... 352, 396 ... 440 ... 528 ... 660, 704 ... 792... 1056, 1100 ... 1584 ... 1760 ... 1980 ... 2112 ... 2376 ... 2640 ... 3168 ... 3300 ... 16500 ... 29700 ... 36300 ... 52800 ... 59400 ... 89100 ... 22000 ... 247500.

All of these increments are very recognisable as belonging to the 5280-feet English-mile or 5940-feet Scottish mile, 16500-feet English league, 7920-mile diameter of the Earth or 24750 English mile equatorial circumference and 22000 Scottish mile equatorial circumference.

TERRACE 2 NE.

The length is 79.2-feet and the angle is 137.5-degrees.

A number-string based upon 79.2 goes:

79.2, 158.4, 237.6, 316.8, 396, 475.2 ... 633.6, 712.8, 792 ...950.4 ... 1188 ...1900.8, 1980 ... 2851.2 ... 3564 ...3960 ... 5940 ... 7920 ... 9900... 10137.6 ... 13068 ... 130680000.

This line-up provides the numbers for calculating the radius (3960-miles), diameter (7920-miles) and equatorial circumference of the Earth under the English mile (5280-feet) navigational system reading of 24750-miles (130680000-feet).

The rendition of PI used for converting the 7920-mile (of 5280-feet) diameter of the Earth into the 24750-mile circumference was 3.125.

The rendition of PI for converting the 7920-mile diameter of the Earth into the 24883.2-mile (of 5250-feet) circumference was 1728/550ths or 3.141818182.

The true equatorial diameter of the Earth is 7926-miles and for the "True" reading that ancient savant-astronomer-navigators used 24883.2-miles (of 5280-feet) or 131383296-feet was the 22/7 rendition of PI (3.142857143). This would have achieved 7917.3818-miles ... only 8.6-miles short.

In this line-up the Scottish mile (5940-feet) is generated. Also, the value 10137.6 occurs and under the ancient "True" reading of 24883.2-miles of 5280-feet, one second of equatorial arc was 101.376-feet.

THE DEGREE ANGLE.

With the angle set at 137.5-degrees, the number-string goes:

137.5, 275, 412.5, 550, 687.5, 825, 962.5, 1100, 1237.5, 1375, 1512.5, 1650, 1787.5 ... 2062.5, 2200 ... 2475 ... 2750 ... 3025 ... 3300, 3437.5 ... 9075... 24750

TERRACE 2 NW.

The length is 126-feet and the angle is 46.08-degrees.

A number-string based upon 126 goes:

In the above lineup 12.6-inches is a Greek foot and 25.2-inches is an Assyrian cubit; 630-feet is a Greek Stadium and the length of the sprint race in the Greek Olympics; 756-feet is the length of the Great Pyramid; 100.8-feet would equate to 1-second of equatorial arc for a world that was 24883.2 Greek miles in circumference; 1512-feet would be the length of two sides of the Great Pyramid, with 2268-feet and 3024-feet being the 3rd & 4th sides; 6048-feet would be two circumnavigations of the Great Pyramid & 1-minute of equatorial arc; 6804-days would be the number of days for the duration of the Lunar Nutation Cycle; 9072-inches would be the length of the Great Pyramid and 9072-feet would be 1.5 minutes of equatorial arc.

THE DEGREE ANGLE.

A number-string based upon 46.08-degrees goes:

46.08, 92.16, 138.24, 184.32, 230.4, 276.48 ... 414.72, 460.8 ... 691.2 ... 829.44 ... 1013.76 ... 1152 ... 1244.16 ... 2073.6... 12441.6 ... 24883.2.

This is an enlargement of the 11.52 progression and 46.08 is 11.52 X 4.

At Stonehenge, the outer faces of the 30 lintels, that lie horizontally atop the huge Sarsen Circle stone-uprights, average 11.52-feet each ... X 30 = 345.6-feet of circumference. In a world having an equatorial circumference of 24883.2-miles, the sum of 345.6-miles would be a 1/72nd division.

Again the 691.2 occurs and in a world of 24883.2-miles, 1-degree of arc would be 69.12-miles and 101.376-feet would be 1-second of arc.

TERRACE 3 SW.

The length is 68.04-feet and the angle is 137.5-degrees.

A number-string based upon 68.04 goes:

68.04, 136.08, 204.12, 272.16, 340.2 ... 544.32 ... 604.8 ... 1701 ... 3402 ... 5103 ... 6531.84 ... 6804 ... 130636800.

The foremost determining factor in weather conditions is the gravitational attraction shared between the Earth and Moon. Added to that, the Earth & Moon seasonal position around the Sun, in either cold, wintry or hot summer conditions significantly affects localised weather.

Highly-trained individuals, conversant with cyclic-astronomy, knew how to predict, with reasonable accuracy, such things as famines or other potential set-backs to agricultural abundance They understood the predictable effects of the conjoined cycles of the Sun, Earth and Moon over the 18.613-year lunar nutation period, where these celestial bodies spin out of synchronised coordination with each other, then drift back into unison to repeat the whole, tri-body, cycle-dance anew.

Longstanding experience, handed down for millennia from forebears, could give a reasonably fair estimation as to when the droughts or heavy snowstorms, etc., would occur during the 6804-day cycle’s duration, as well as how long adverse conditions may endure; what seas were too dangerous to venture onto at certain times or whether the Moon was going to be a good (benevolent) one or a bad (dangerous) one in a particular season, place or time.

An adept, highly trained savant-astronomer-navigators could predict the expected weather for a region, based upon oral-traditions handed down through countless generations, concerning repetitive weather conditions encountered through each phase of the Lunar Nutation Cycle. An individual allotted three-score-&-ten years of life would experience the cycle 3 & 3/4ths times during their sojourn on Earth, so there'd be generations of experience to draw from.

From this accrued experience, such techniques as "Monsoon-sailing" were devised, where, from the Indian Ocean, the seasonal winds or currents would carry one either towards the Americas to the west or to Australia and the Pacific Ocean to the southeast.

The actual Lunar Nutation Cycle period is 6798.36-days, but that number is not easily halved, quartered or further subdivided down to a single day or hour. Ancient astronomer-mathematicians therefore settled upon 6804-days to describe the cycle ... an error of only 5.64-days in 18.6133-years

The values 1701, 3402, 5103 & 6804, seen above, are 1/4th, 1/2, 3/4ths and the whole cycle in days.

TERRACE 3 SE.

The length is 113.4-feet and the angle is 43.2-degrees.

A number string based upon 113.4 goes:

113.4, 226.8, 340,2, 453.6, 567, 680.4 ... 970.2 ... 1134... 1360.8 ... 1701 ... 2268 ... 2721.6, 2835... 3402 ... 3628.8 ... 4536 ... 5103 ... 5443.2 ... 5670 ... 6804 ... 8505... 14175 ... 130636800.

Again, we have strong lunar coding, especially in relation to the Lunar Nutation Cycle wherein 850.5-days would be 1/8th of the period, 1701-days 1/4th, 3042-days 1/2 and 6804-days the full cycle being monitored and measured.

The sum of 2268 (1134 X 2) is 3-sides-lengths of the Great Pyramid @ 756-feet per side; 36288-feet is half of 1-minute of equatorial arc on the Great Pyramid and 362880-feet is 1-degree of arc.

Also in the line-up is 2835 ... the perimeter value of the Khafre Pyramid (Egypt's pyramid of the moon). 2835-feet is 34020-inches.

THE DEGREE ANGLE.

A number-string based upon 43.2 goes:

43.2, 86.4, 129.6, 172.8, 216, 259.2, 302.4, 345.6, 388.8, 432 ... 518.4, 604.8, 648, 691.2 ... 777.6 ... 864, 907.2, 950.4 ... 1036.8, 1080... 11664, 1209.6 ... 1382.4 ... 1512, 1555.2 ... 1900.8, 1944... 2073.6 ... 2376 ... 259.2 ... 2851.2 ... 3024 ... 3110.4 ... 3240 ... 3628.8 ... 5400... 5443.2 ... 5832 ...6048 ...7560 ... 8294.4 ... 9072 ... 9331.2 ... 9720 ... 11880 ... 12441.6 ... 24883.2.

As can be seen, there is an absolute banquet of Great Pyramid numbers in this progression, all of which have been discussed so far.

TERRACE 3 NE.

The length is 66-feet and the angle is 137.5-degrees.

A number-string based upon 66 goes:

66, 132, 198, 264, 330, 396 ... 528, 594, 660 ... 792 ... 990, 1056 ... 1188 ... 1452 ... 1584 ... 1650 ... 1782 ... 1980 ... 2112 ... 2376 ... 2970 ... 3168 ... 3564 ... 3630 ... 5280 ... 5940 ... 6534 ... 6600 ... 7128... 7920 ... 9504 ... 130680000.

TERRACE 3 NW.

The length is 113.4-feet and the angle is 44 & 19/64ths.

The number string for 113.4 has been explained.

THE DEGREE ANGLE.

A code-bearing degree angle assigned to a terrace wall could be very close in numerical value to another close-proximity angle, and no opportunity would be overlooked by a tutor to mention the others. For example, an angle of 52.8 is very close to 52.5, 51.84 and 51.5625, stretching to 50.4. In close-proximity, higher values there is 53.76 or 54.

For terrace 3 NW the more obvious choice is to call the angle 44-degrees, but a close proximity angle with code-bearing significance is 44.296875-degrees or 44 & 19/64ths.

A number-string based on this goes:

44.296875, 88.59375, 132.890625, 177.1875, 221.484375, 265.78125, 310.078125, 354.375.

In the above progression the lunar year of 354.375-days is reduced in 1/8th divisions. The half value is 177 & 3/16ths-days.

LEKA PUIPUI MAU

Leka Puipui Mau is a large langi platform of 4-tiers retained by impressive stone blocks.

Leka Puipui Mau langi platform sits northeast of Fa'apite Malua 'otonga.

WC McKern, a century ago, measured the bottom, southwest terrace to be 134-feet in length and the southeast terrace to be 159 feet long. In-and-of-themselves, both 134 & 159 fall marginally off the significant numbers 134.4 or 158.4 used in ancient navigation systems.

TERRACE 1 SW.

134.4-feet @ 144-deg.

A number-string based upon 134.4-feet goes:

134.4, 268.8, 403.2, 537.6, 672, 806.4 ... 1209.6, 1344 ... 2016 ... 2688 ... 3360 ... 3628.8 ... 6048 ... 10080 ... 130636800.

A number-string based upon 144 goes:

144, 288, 432, 576, 720, 864, 1008, 1152, 1296, 1440, 1584, 1728 ... 2016, 2160, 2304 ... 2592, 2736, 2880, 3024, 3168 ... 3456, 3600... 3888, 4032 ... 4320 ... 4608 ... 5040, 5184 ... 5760 ... 6220.8 ... 6048 ... 6480 ... 6912 ... 7776, 7920 ... 8064 ... 9072, 9216 ... 9504 ... 10368 ... 10800 ... 11664 ... 12096 ... 13824 ... 14400 ... 15552 ... 18000 ... 19008 ... 19440 ... 20736 ... 25200 ... 25920 ... 27216 ... 28512 ... 28800 ... 31104 ... 32400 ... 36288 ... 39600 ... 41472 ... 45360 ... 124416 ... 248832.

It would be difficult to find another number-string that was more power-packed with navigational numbers, or others related to the Precession of the Equinoxes, than this one.

Let's discuss the merits of some of these numbers:

The geomancer's mile, (an archaic distance increment used repeatedly around Taunton, England and equating to 14400-feet) was rediscovered by English antiquarian, John Michell. Based upon information gleaned from Egyptian priests by Greek historian Herodotus (circa 454 BC), we can calculate the Egyptian pyramid acre to be 28800-square feet.

In the line-up are the values related to 1-second of equatorial arc (100.8-feet), as well as a minute (6048-feet), a degree (362880-feet) and the full circumference (24883.2-miles),

There are also the divisions of the Earth, like 388.8-miles (1/64th), 777.6-miles (1/32nd), 1555.2-miles (1/16th), 3110.4-miles (1/8th), 6220.8-miles (1/4th), 12441.6-miles (1/2) and the whole circuit @ 24883.2-miles.

The value 1728 cubic inches is a cubic foot and 1.728-feet was the length of the largest Egyptian Royal cubit (20.376-inches). For sea-leg navigation by the English league the PI version used was 1728/550ths or 3.14 1818182 = 51840 feet of circumference.

The essential values are present to calculate divisions in the 25920-year duration of the Precession of the Equinoxes as, 2160-years (1/12th of 25920-years ...the sun spends 2160-years in each house of the zodiac during the cycle), 4320-years (1/6th), 5184-years (1/5th), 12960-years 1/2). Also, the Old English bushel measure under the Winchester standard was 2160-cubic inches and the diameter of the Moon is 2160-miles.

Generating this string of numbers and committing them to memory, then learning how to manipulate them would be essential knowledge to navigation and positional-plotting at sea.

TERRACE 1 SE.

158.4-feet @ 52.5-degrees

A number-string based upon 158.4 goes:

158.4, 316.8, 475.2, 633.6, 792, 950.4 ... 1900.8 ... 2376, 2851.2 ... 3960 ... 7128 ... 7920 ... 10137.6 ... 11880 ... 19800 ... 28512 ... 43560 ...79200.

The sum of 158.4-miles would be 1/50th of the 7920-mile diameter of the Earth; 1980-miles would be 1/4th

The value 10137.6 occurs and 101.376-feet would be 1-second of arc in an equatorial circumference of 24883.2-miles of 5280-feet each. The number 158.4-feet would be 1/64th of 10137.6.

A number-string based upon 52.5 goes:

52.5, 105, 157.5, 210, 262.5, 315, 367.5, 420, 472.5, 525 ... 630 ...787.5 ... 840 ... 945 ... 1050 ... 1260, 1312.5 ... 1417.5, 1470 ... 1680 ... 1890 ... 2520 ... 2625 ... 2835 ... 3360 ... 3780 ... 4147.5 ... 5040 ...5250 ... 5670 ... 6300 ... 6720 ... 7087.5 ... 7560 ... 7875... 8400 ... 8505 ... 10080 ... 10500 ... 11340 ... 11812.5 ... 12600 ... 13440 ... 15120 ... 17010 ... 19687.5 ... 20160 ... 22680 ... 25515 ... 2488320 ... 130636800.

In the above string, 2625-feet would be 1/2 a Greek mile of 5250-feet; 10.5-feet would be a Hebrew Reed or 1/3500th of a Greek mile; 525-feet would be a Greek short-Stadium and 630 feet the long-Stadium or the length of the sprint race at Olympia.; 189-feet would be 1/4th of the side length of the Great Pyramid (756-feet or 7.5-seconds of equatorial arc circumference of the Earth.); 1701-days would be 1/4th of the 6804-day Lunar Nutation Cycle; 2551.5-days would be 7.2-years years (354.375-days per lunar year and the lunar period monitored (alongside 7-solar years ... 2556.75-days) in the lunisolar Sabbatical Calendar counting system; 6720-feet would be the ancient Irish mile; 118.125-days would be 4-lunar months of 29.53125-days; 708.75-feet would be 15/16ths the length of the Great Pyramid and the length of the Khafre Pyramid (354.375-feet X 2); 2016-feet would be 1/3rd of 1-minute of equatorial arc (6048-feet) according to the navigational method encoded into the Great Pyramid's equatorial circumference of the Earth navigational system. Under that system the Earth would be 24883.2 Greek miles round at the equator (130636800-feet or 12 X 12 X 12 X 12 X 1.2 Greek miles).

TERRACE 1 NE.

128-feet @ 141.75-degrees.

A number-string based upon 128 goes:

128, 256, 384, 512, 648, 768, 896, 1024, 1152, 1280 ... 1536 ... 1920 ... 2304 ... 2688 ... 3200 ... 3456, 3584 ... 4608 ... 5376 ... 5760, 5888 ... 6400 ... 6912, 7040 ... 9600 ... 10368 ... 11520 ... 12160 ... 13440 ... 13824 ... 17280 ... 20736 ... 1306800000.

This string of numbers represent divisions of the 24883.2 equatorial circumference, where 1/128th would be 194.4-miles, 10125 seconds of arc, 168.75 minutes of arc or 2.8125 (2 & 13/16 ths) degrees of arc.

In the above string , 115.2-miles would be 1/216th, 230.4-miles would be 1/108th part, 460.8-miles would be 1/54th, etc. Again, the number 10368 is generated and 1036.8 mph is the speed that the 24883.2-mile circumference world spins. The 138.24 number would, in miles, be 1/180th of the circumference; A cubic foot is 12 X 12 X 12-inches or 1728 cubic inches; 69.12-miles would equate to 1-degree of equatorial arc.

TERRACE 1 NW.

153.6-feet @ 55-degrees.

A number-string based upon 153.6 goes:

153.6, 307.2, 460.8, 614.4, 768, 921.6... 1382.4, 1536 ... 2304 ... 3840 ... 4147.2 ... 5376 ... 6912 ... 7680 ... 8294.4 ... 10137.6 ... 11520 ... 12441.6 ... 19200 ... 20736 ... 23040 ... 24883.2.

Again, a slight variation on the foregoing string, generating many of the division numbers of the 24883.2-mile equatorial circumference. An expansion of the number 101.376 occurs and in a world of 24883.2-mines of 5280-feet the value for 1-second of arc would be 101.376-feet.

A number-string based upon 55 goes:

55, 110, 165, 220, 275, 330, 385, 440, 495, 550, 605, 660 ... 770, 825, 880 ... 990 ... 1100 ... 1320 ... 1485 ... 1650 ... 1760 ... 1980 ... 2475 ... 2640 ... 2750 ... 2970 ... 4125 ... 5280 ... 5500 ... 5940 ... 6050 ... 6875 7040 ... 7920 ... 8910 ... 9075 ... 9900 ... 10560 ... 16500 ... 24750 ... 130680000.

These are, of course, divisions of the Earth using the link (7.92-inches), Merchant Navy fathom (5.5-feet), Rod or Perch (16.5-feet), Chain (66-feet), Furlong or Furrowlong (660-feet), Mile (5280-feet) and League (16500-feet or 3.125-miles).

In the above line-up, 1760-feet would be 1/3rd of a mile or 1760-yards (3-feet) would be a mile; 6050-feet would be 1-minute of the Earth's equatorial arc; 68.75-miles would be 1-degree of arc for a world that was 24750-miles (of 5280-feet) in equatorial circumference ... 130680000-feet.

TERRACE 2 SW.

123.75-feet @ 144-degrees.

A number-string based upon 123.75 goes:

123.75, 247.5, 495, 618.75, 742.5 ... 990 ...1485 ... 1980 ... 2475 ... 2970 ... 3960 ... 5940 ... 7920 ... 8910 ... 9900 ... 12375 ... 15840 ... 17820 ... 24750.

Again, a very simple and useful progression related to the 24750-mile equatorial circumference. The values generated include the 1/4th (1980-miles), 1/2, (3960-miles) 3/4ths (5940-miles) and full diameter of the Earth @ 7920-miles, as well as the half and full equatorial circumference.

A number-string based upon 123.75 goes:

123.75, 247.5, 495, 618.75, 742.5 ... 990 ...1485 ... 1980 ... 2475 ... 2970 ... 3960 ... 5940 ... 7920 ... 8910 ... 9900 ... 12375 ... 15840 ... 17820 ... 24750.

Simultaneously to the above, the number-string for 124.416 would also have been taught:

24.416, 248. 832 ... 622.08 ... 1244.16 ... 1866.24 ... 2488.32 ... 3110.4 ... 6220.8 ... 9331.2 ... 12441.6 ... 24883.2.

This of course refers to the 12 X 12 X 12 X 12 X 1.2-mile equatorial circumference of the Earth, with 3110.4-miles @ 1/8th; 6220.8 @ 1/4th and 12441.6 @ 1/2.

The sum of 933120-grains became the ancient Babylonian Talent and the majority of numbers used in this study of the langi platforms graduated into the weights, measures and volumes standards of the great civilisations, from remote antiquity.

The number-string for 144 has already been explained for this langi.

TERRACE 2 SE.

151.2-feet @ 52.8-degrees.

A number-string based upon 151.2 goes:

151.2, 302.4, 453.6, 604.8, 756, 907.2 ... 1209.6, 1360.8, 1512, 2268, 2721.6 ... 3024 ... 3628.8 ... 3780 ... 5443.2 ... 6048 ... 6804 ... 9072 ... 11340 ... 362880 ... 130636800.

This string of numbers refers directly to the equatorial circumference of the Earth coding that is built into the literal dimensions of the Great Pyramid, where 1-side length (756-feet) is 1/8th of 1-minute of arc; 1-perimeter circumnavigation (3024-feet) is 1/2 a minute of equatorial arc and 2 (6048-feet) are full minute of arc; 362880-feet is 1-degree of arc and 130636800-feet (24883.2 Greek miles of 5250-feet) is the full circumference.

A number-string based upon 52.8 goes:

.52.8, 105.6, 158.4, 211.2, 264, 316.8 ... 475.2, 528 ... 633.6 ... 792 ... 9504 ... 1056 ... 1320 ... 1900.8 ... 2376 ... 2851.2 ... 3960 ... 5280 ... 6600 ... 7128 ... 7920 ... 9504 ... 10137.6 ... 11880 ... 2475000.

This refers to the mile of 5280-feet and divisions thereof. The sum of 3960-miles is the radius of the Earth, for a full diameter of 7920-miles.

Under the "true"equatorial circumference of the Earth reading of 24883.2-miles of the Earth, the diameter (7920-miles of 5280-feet) is multiplied by the rendition of PI @ 1728/550ths = 3.141818182. This meant an Earth equatorial measurement of 131383296-feet.

Under the second navigational method encoded into the Great Pyramid's dimensions that read the circumference of the Earth as 24750-miles of 5280-feet the 7920-mile diameter of the Earth was multiplied by PI @ 3.125 = 24750. This meant an equatorial circumference of 130680000-feet

TERRACE 2 NE.

120.96-feet @ 141.75-degrees.

A number-string based upon 120.96 goes:

120.96, 241.92, 362.88, 483.84, 604.8 ... 1209.6 ... 3024 .... 3628.8 ... 5443.2 ... 6048 ... 6531.84 ... 9072 ... 12096 ... 13063.68 ... 15120 ... 18144 ... 27216 ... 36288 ...75600... 362880 ... 65318400 ... 130636800.

This progression relates to the Great Pyramid's perimeter value and the 130636800-feet equatorial circumference that is generated therefrom.

Also, the distance for one circumnavigation around the pyramid is 3024-feet or 36288-inches (half of 1-minute of equatorial arc); 1-degree of arc is 362880-feet; Half the equatorial circumference is 65318400-feet.

This number-string and side length of the langi would have been simultaneously read as 121.5-feet and, thereby, relate to 1-perimeter circumnavigation around the pyramid being 3025-feet (36300-inches); a degree of arc 363000-feet and half the equatorial circumference as 6534000-feet.

The number string for 141.75 has already been explained for this langi.

TERRACE 2 NW.

146.66666-feet @ 54-degrees:

A number-string for 146.66666 goes:

146.66666, 293.33333, 440, 733.33333, 880 ... 1320 ... 1760 ... 2200 ... 2640 ... 3080 ... 3520 ... 3960 ... 4400 ... 5280 ... 6600 ... 7040 ... 7920 ... 8800 ... 10560 ... 11000 ... 11880 ... 15840 ... 17600 ... 19800 ... 22000 ... 33000 ... 130680000.

This is another variation on divisions of the 5280-feet mile or the 7920-mile diameter of the Earth and its divisions.

A number-string based upon 54 goes:

54, 108, 162, 216, 270, 324, 378, 432, 486, 540, 594, 648 ... 756 ... 864 ... 972 ... 1080 ... 1134, 1188 ... 1296, 1350 ... 1458, 1512 ... 1728 ... 1782 ... 1890, 1944 ... 2160 ... 2268 ... 2376, 2430 ... 2592 ... 2700 ... 2916 ... 2970 ... 3024 ... 3402, 3456 ... 3780 ... 3888 ... 4050 ... 4536 ... 5130, 5184 ... 5832 ... 5940 ... 6048 ... 6534 ... 6750 ... 6804 ... 25920 ... 362880 ... 130680000.

This provides numbers for two navigational methods, the 130680000 circumference (based upon a mile of 5250-feet) and the 130636800 circumference (based upon a mile of 5280-feet), both divisible by 54.

It primarily works with the navigational method using the Greek-mile of 5250-feet.

The Great Pyramid side and perimeter numbers in feet are represented as 756, 1512, 2268 & 3024 and the value for 1-minute of equatorial arc (6048-feet), as well as 1-degree of arc (362880-feet) are also generated.

TERRACE 3 SW.

115.2-feet @ 144-degrees.

A number-string based upon 115.2 goes:

115.2, 230.4, 345.6, 460.8, 576, 691.2, 860.4, 921.6, 1036.8, 1152... 1382.4... 1728 ... 2073.6 ... 2304 ... 2880 ... 3110.4 ... 3456 ... 4032, 4147.2 .... 4608 ... 5184 ... 5760 ... 8294.4 ... 8640 ... 9331.2 ... 10137.6 ... 12096 ... 12441.6 ... 24883.2

Again, navigational coding for a world configured to represent 24883.2-miles of equatorial circumference. This progression-string is essentially the 11.52-code where 11.52-miles is 1/216th of the equatorial circumference, or 1 & 2/3rds degrees of arc. the sum of 345.6-miles would be a 1/72nd division or 5-degrees of arc.

It's of interest that Stonehenge's horizontal lintel stones, lying atop the Sarsen circle of giant stone pillars, average or were configured to be, 11.52-feet each in length, or 138.24-inches. Because they originally were 30 in number, the circumference is 345.6-feet, coding 1/72nd of the Earth's circumference. The outer face of the Sarsen Circle would be 1/378000th of the equatorial circumference under this navigational method.

Under this system, 1-degree of arc would be 69.12-miles; 138.24-miles would be 1/180th; 3110.4-miles would be 1/8th, etc.

Again, we have the value 1036.8 represented and 1036.8 is the speed the Earth spins in miles per hour. The "Stirling Jug" of Scotland, dating to 1457 AD and possibly much earlier, had a capacity of 103.68 cubic inches, which was the same as the ancient Hebrew liquid volume called the Jerusalem Cab (predating the Babylonian conquest).

The number-string for 144 has already been explained for this langi.

TERRACE 3 SE.

144-feet @ 52.8-degrees.

The number-string for 52.8 goes:

52.8, 105.6, 158.4, 211.2, 264, 316.8 ... 475.2, 528 ... 633.6 ... 792 ... 9504 ... 1056 ... 1320 ... 1900.8 ... 2376 ... 2851.2 ... 3960 ... 5280 ... 6600 ... 7128 ... 7920 ... 9504 ... 10137.6 ... 11880 ... 2475000.

This refers to the mile of 5280-feet and divisions thereof. The sum of 3960-miles is the radius of the Earth, for a full diameter of 7920-miles.

Under the literal navigational method encoded into the Great Pyramid's dimensions that read the circumference of the Earth as 24883.2-miles of 5250-feet (adopted by the late era Greeks from the Egyptians) the 7920-mile diameter of the Earth was again multiplied by PI @ 1728/550ths, but in this case the lesser mile length produced an equatorial circumference of 130636800-feet.

Whenever the angle of 52.8-degrees was encoded, the close-proximity angle of 52.5-degrees would have been automatically inferred and taught to initiates learning the navigational arts across all the langi platforms.

The number-string for 144 has been explained for this langi.

TERRACE 3 NE.

112.5-feet @ 141.75-degrees.

A number-string for 112.5 goes:

112.5, 225, 337.5, 450, 562.5, 675, 787.5, 900, 1012.5, 1125, 1237.5, 1350 ... 1575, 1687.5 ... 1800 ... 2025 ... 2250 ... 2475 ... 2700 ... 2812.5 ... 3037.5, 3150 ... 3600 ... 4050 ... 4500 ... 4725 ... 4950, 5062.5 ... 5400 ... 6075 ... 6300 ... 7087.5, 7200 ... 8100 ... 9000 ... 9900 ... 10800 ... 11812.5 ... 12150 ... 12375 ... 12600 ... 14062.5, 14175 ...14400 ... 14850 ... 16200 ... 18000 ... 18900 ... 19687.5 ... 19800 ... 21600 ... 23400 ... 130636800 ... 130680000.

Again, this number-string is compatible in supplying divisions within both the 130636800-feet equatorial circumference navigational system, as well as the 130680000 reading.

For the most part it represents divisions within the 360-degree compass system of 5.625, 11.25, 22.5, 45, 90, 180, 270, 360-degrees respectively.

It also introduces divisions of the Earth under the 24750-mile reading and 112.5-miles would be 1/220th.

Important lunar numbers like 708.75 or 118.125 or 141.75 in expanded values also occur, along with increments associated with the Greek system of metrology.

The value 196.875, which became the Beqa, gold standard grain-weight for the Egyptians, Greeks & Romans is also represented as an expanded value.

A number-string based upon 141.75-goes:

141.75, 283.5, 425.25, 567, 708.75, 850.5 ... 1134 ... 1417.5 ... 1701... 2268 ... 2551.5 ... 2835 ... 3402 ... 3543.75 ... 4536 ... 5103 ... 6804 ... 9072 ... 13608 ... 17010 ... 130636800.

This string is excellent for lunar counts within the lunisolar Sabbatical Calendar system, where 7.2 lunar years @ 354.375-days (2551.5-days) are counted alongside 7-solar years (2556.75-days).

Also, in the 6804-day Lunar Nutation Cycle there would be 48 periods of 141.75-days.

TERRACE 3 NW.

140-feet @ 54-degrees.

A number-string based upon 140 goes:

140, 280, 420, 560, 700, 840, 980, 1120, 1260, 1400, 1540, 1680, 1820, 1960, 2100, 2240 ... 2520 ... 3360, 3500 ... 3780 ...4200 ... 5040 ... 6300 ... 6720 ... 7000 ... 7560 ... 10080 ... 10500... 11340 ... 13440 ... 15120 ... 20160 ... 22680 ... 31500 ... 60480 ... 68040 ... 362880 ... 130636800.

This progression was especially good for following the solar and lunar cycles under the lunisolar Sabbatical Calendar system, as well as throughout the 6804-day lunar nutaion cycle, The sum of 6804-days would be 972-weeks or 486 bi-weeks.

The progression also features Great Pyramid navigational numbers, divisible by the number 7, all the way through to the equatorial circumference of the earth @ 130636800-feet or 12 X 12 X 12 X 12 X 1.2 Greek -miles of 5250-feet each.

The number-string for 54 has already been explained for this langi.

TERRACE 4 SW.

105-feet @ 144-degrees.

A number-string based upon 105-feet goes:

The sum of 567-days would be 1/12th of the 6804-day Lunar Nutation Cycle.

The number-string based upon 51.5625 goes:

51.5625, 103.125 ... 206.25 ... 412.5 ... 618.75 ... 825 ... 1650 ... 2475 ... 3300 ... 24750 ... 130680000.

This number-string is based upon the mid-range Egyptian Royal cubit of 20.625-inches and there would be 30 such cubits in 51.5625-feet. There would be 9600 such cubits in the English league of 16500-feet (3.125-miles of 5280-feet). There would be 1200 X 20.625-miles in the 24750-mile equatorial circumference, under that navigational method.

The 144 code-bearing merits have been covered for this langi.

TERRACE 4 SE.

134.4-feet @ 52.8-degrees.

Both of these code-bearing numbers have been explained for this langi.

TERRACE 4 NE.

103.125 @ 141.75-degrees.

A number-string based upon 103.125 goes:

103.125, 206.25, 309.375, 412.5, 515.625, 618.75 ... 825 ... 1237.5 ... 1650 ... 2062.5 ... 2475 ... 3300 ... 4537.5 ... 4950 ... 6600 ... 9075 ... 9900 ... 12735 ... 13200 ... 24750 ... 130680000.

This is akin to using the 20.625-inch Egyptian Royal Cubit mnemonic-device ruler for remembering the Equatorial size of the Earth. The 20.625-inch rule X 1200 (read as miles of 5280-feet) = 24750.

TERRACE 4 NW.

The length is 130.68-feet and the angle is 53.76-degrees.

A number-string based upon 130.68 goes:

130.68, 261.36, 392.04, 522.72, 653.4 ... 1306.8 ... 65340 ... 130680000.

As a fraction, .68 would equate to 17/25ths

The tutor, teaching initiates the principles of navigation, would also represent this same terrace length as 130.6368-feet or 130 & 16/25ths.

A number-string based upon 53.76 goes:

53.76, 107.52, 161.28 ... 2688 ... 537.6 ... 806.4 ... 1344 ... 6720 ... 12096 ... 130636800.

FA 'APITE MALU O 'OTONGA LILI

TERRACE 1 SW.

126-feet @ 138.24-degrees.

A number-string based upon 126 goes:

A number-string based upon 138.24-degrees goes:

138.24, 276.48, 414.72, 552.96, 691.2, 829.44 ... 1244.16 ... 2073.6 ... 24882.2.

In the 24883.2-mile equatorial circumference, 138.24-miles would be 1/180th; 141.72-miles would be 1/60th; 829.44-miles would be 1/30th and 12441.6, half.

Amongst the values generated in this progression is 2073.6. The largest Egyptian Royal Cubit found by William Flinders Petrie was 1.728-feet long or 20.736-inches. This was obviously used as a mnemonic device for remembering the equatorial circumference of the Earth, as 20.736 X 1200 = 24883.2.

TERRACE 1 SE.

140.625-feet @ 52.8-degrees.

A number-string based upon 140.625 goes:

140.625, 281.25, 421.875, 562.5 ... 1125 ... 1687.5 ... 1968.75 ... 2250 ... 2953.125 ... 3937.5 ... 4500 ... 5062.5 ... 6750 ... 7875 ... 9000 ... 10125 ... 11250 ... 11812.5 ... 13500 ... 18000 ... 22500 ... 24750.

A distance of 140.625-miles would be 1/176th of the 24750-miles (of 5280-feet) equatorial circumference; 562.5-miles would be 1/44th; 1125-miles would be 1/22nd; 4500-mile 2/11ths, 9000-miles would be 4/11ths; 18000-miles would be 4/55ths; 3600-miles 8/55ths.

The line-up infers divisions of the 360-degrre compass, with 11.25, 22.5, 45, 90, 180, 360. The value 135 is half of 270.

It also contains values related to the lunar cycle of 29.53125 (the lunar month) and four lunar months (118.125). Also 101.25-days would be 1/7th of 708.75-days. The Khafre Pyramid is 708.75 feet long and (in days) that's 2 lunar years.

Of interest is 196.875, which became the ancient grain-weight for 1-Beqa of gold dust, used by the Egyptians, then adopted by the Greeks and later, Romans. Two Beqa of gold weighed 393.75-grains and this became the gold standard uncia of the Romans. Also, in the Greek Attic weight standard 393750-grains was 1 Talent.

A number-string based upon 52.8 goes:

This refers to the mile of 5280-feet and divisions thereof. The sum of 3960-miles is the radius of the Earth, for a full diameter of 7920-miles.

TERRACE 1 NE.

133,33333-feet @ 145.8-degrees.

A number-string based upon 133.33333 goes:

A number-string based upon the angle of 148.5-degrees goes:

145.8, 291.6, 437.4, 583.2, 729 ... 1166.4, 1312.2, 1458 ...2332.8 ... 4665.6 ...5103... 9331.2 ... 26244, etc.

In the above line-up 58.32-inches was the Roman Pace measurement or 5-Roman feet of 11.664-inches. Similarly, the value 7.29-inches was 1/8th of a Roman Pace.

Well yes, it was certainly 11.664” in number-strings tracking lunar cycles or standard volume capacities, etc., but the simpler value of 11.66666” (11 & 2/3rds) was used for overland distances, with a Pace of 58.333333” (5 Roman feet). The same foot, in doubled value, was used in Sweden for the Rydaholmsalm length (1.94444-feet of 12-inches).

One of the more ancient concepts that survived into Mediaeval times was that 75-Roman miles was 1-degree of arc in the equatorial circumference of the Earth. Given that the Roman foot was represented as 11.664-inches, that would equate to a Roman mile (5000 Roman feet) being 4860 feet of 12-inches. Therefore, 75 such miles, if representing 1-degree of arc in the equatorial circumference, would equate to a world of 27000 Roman miles

The degree angle 145.8 is also represented as 145.833333 (148 & 5/6ths).

The Roman overland mile was 5000 Roman feet or 4861.11111 feet of 12-inches (4861 & 1/9th) X 75 X 360 = 131250000 feet or 125000000 Greek feet or 135000000 Roman feet or 67500000 Swedish Rydaholmsalm or 27000 Roman miles or 3750 Swedish Mil or 25000 Greek miles.

Note: The ancient Druidic bronze calendar plaque unearthed at Coligny, Ain, France in 1897 is 4.86-feet long (58.32-inches or 1 Roman Pace or 4 X 14.58-inches).

TERRACE 1 NW.

157.5-feet @ 50-degrees.

A number-string based upon 157.5 goes:

157.5, 315, 472.5, 630, 78.75, 945 ... 1260, 1417.5, 1575... 1890 ... 2520 ... 2835 ... 3150 ... 3780 ... 5040 ... 5670 ... 6300 ... 7087.5 ... 7560 ... 8505 ... 10080 ... 11340 ... 11812.5 ... 15120 ... 17010 ... 18900 ... 19687.5 ... 20160 ... 25515 ... 28350 ... 30240 ... 35437.5 ... 51030 ... 68040 ... 130636800.

This line-up is composed of values that were incorporated into the so-called Greek metrology system, which had existed for thousands of years before there was a Creek civilisation There would be 3&1/3rd increments of 157.5-feet in a Greek mile of 5250-feet. The sum of 630-feet would be a Greek Stadium and was the distance of the Olympic sprint race.

The length of 12.6-inches was the Greek foot and 126-inches was a Hebrew Reed, used by the Greeks (10.5 English feet).The Stadium length is 60 Reeds or 1.2 Greek miles.

The line-up is also very representative of Great Pyramid and Khafre Pyramid numbers. There would be 72 Reeds of 10.5-feet in the 756-feet length of the Great Pyramid or 67.5 Reeds in the 708.75-feet length of the Khafre Pyramid. The lengths of each pyramids are, therefore, 720 Greek feet and 675 respectively. On the Khafre Pyramid,inch value per side is 8505-inches, with two sides amounting to 17010-inches and the full circuit equaling 34020 -inches. This is mnemonic coding to remember the 1701-day (1/4th), 3402-day (1/2), 5103-day (3/4ths) duration of the 6804-day Lunar Nutation Cycle. There would be 432 intervals of 15.75-days (378-hours) in the 6804-day cycle.

The Great Pyramid numbers, in inflated values are represented as, 189, 378, 756, (1/4th, 1/2 and one full side value). Similarly, the Khafre Pyramid numbers are represented in inflated value for 354.375, 708.75 & 2835 (1/2, full side length and full perimeter).

The value 2016-feet would be 1/3rd of 1-minute of equatorial arc; The lunar month @ 29.53125-days is 1.5 X 19.6875-days and the number of 196.875-grains was the "Beqa" grain-weight standard of the Egyptians, Greeks and Romans.

The progression leads to the number 130636800-feet, which was the equatorial circumference of the Earth under the Great Pyramid's base dimensions of 100.8-feet for a second of arc, 6048-feet for a minute of arc and 362880-feet for a degree of arc X 360.

TERRACE 2 SW.

115.2-feet @ 138.24-degrees.

A number-string based upon 115.2 goes:

Under this system, 1-degree of arc would be 69.12-miles; 138.24-miles would be 1/180th; 3110.4-miles would be 1/8th, etc.

A number-string based upon 138.24-degrees goes:

138.24, 276.48, 414.72, 552.96, 691.2, 829.44 ... 1244.16 ... 2073.6 ... 24882.2.

In the 24883.2-mile equatorial circumference, 138.24-miles would be 1/180th; 141.72-miles would be 1/60th; 829.44-miles would be 1/30th and 12441.6, half.

TERRACE 2 SE.

130.68-feet @ 52.8-degrees.

A number-string based upon 130.68-feet goes:

130.68, 261.36, 392.04, 522.72, 653.4 ... 1306.8 ... 65340 ... 130680000.

As a fraction, .68 would equate to 17/25ths

The tutor, teaching initiates the principles of navigation, would also represent this same terrace length as 130.6368-feet or 130 & 16/25ths .

TERRACE 2 NE.

121.5-feet @ 145.8-degrees.

A number-string based upon 121.5 goes:

121.5, 243, 364.5, 486, 607.5, 729, 850.5, 972 ... 1215 ... 1458 ... 1701 ... 1944 ... 2551.5 ... 2916 ... 3402 ... 3888 ... 5103 ... 5832 ... 6804.

In the 6804-day Lunar Nutation Cycle the sum of 121.5-days would be 1/56th, 486-days would be 1/14th, 850.5-days would be 1/8th, 1701-days would be 1/4th, 3402-days would be 1/2 and 5103-days would be 3/4th.

Also in the line-up, 7.29-inches would be 1/8th of a Roman Pace of 58.32-inches (4.86-feet).

Under the lunisolar Sabbatical Calendar system, the lunar period being counted daily was 2551.5-days (7.2 lunar years of 354.375-days). Consecutive to that count being done within peg holes of a parapegma plaque was the solar count of 7-solar years @ 365.25-days ( 2556.75-days) With the solar period being 5.25-days longer, the Druids and their forebears commenced the solar count 6-days before the lunar count commenced, so that the two periods would end on the same day 7-solar years later (487 X 5.25-days = 2556.75) or 7.2-lunar years (486 X 5.25-days =2551.5).

The number-string based upon 145.8 has been explained for Terrace 1 NE

TERRACE 2 NW.

145.8-feet @ 50.4-degrees.

A number-string based upon 50.4 goes:

50.4, 100.8, 151.2, 201.6, 252, 302.4 ... 403.2, 453.6, 504 .. 604.8 ... 756, 806.4 ... 907.2 ... 1008 ... 1209.6, 1260 ... 1512 ... 2016 ... 2268 ... 2721.6 ... 3024 ... 3628.8 ... 3780 ... 4032 ... 6048 ... 6300 ... 6804 ... 7560 ... 11340 ... 130636800.

This progression begins with 1/2 of 1-second of arc (50.4-feet) and goes on to identify the values of 1-minute of equatorial arc (6048-feet), a degree of arc (362880), all the way to the full equatorial circumference (130636800-feet). It also identifies the circuit measurement for sides of the Great Pyramid (756, 1512, 2268, & 3024-feet respectively).

The duration value for the Lunar Nutation Cycle is also shown and there would be 135 X 50.4-days in the 6804-day cycle. There would also be 6 X 1134-days in 6804-days.

TERRACE 3 SW.

105.6-feet @ 138.24-degrees.

A number-string based upon 105.6 goes:

105.6, 2112, 316.8, 422.4, 528, 633.6 ... 950.4, 1056 ... 1584 ... 1900.8 ... 2640 ... 2851.2 ... 5280 ... 7920 ... 10560 ... 13200 ... 130680000.

This string identifes divisions of the mile of 5280-feet, as well as the value for the diameter of the Earth (7920-miles) and the equatorial circumference at 130680000-feet or 24750-miles.

TERRACE 3 SE.

124.416-feet @ 52.8-degrees.

A number-string based upon 124.416 goes:

124.416, 248.832 ... 622.08 ... 1244.16 ... 1866.24 ... 2488.32 ... 3110.4 ... 6220.8 ... 9331.2 ... 12441.6 ... 24883.2.

This of course refers to the 12 X 12 X 12 X 12 X 1.2-mile equatorial circumference of the Earth, with 3110.4-miles @ 1/8th; 6220.8 @ 1/4th and 12441.6 @ 1/2.

TERRACE 3 NE.

112-feet @ 145.8-degrees.

A number-string based upon 112 goes:

112, 224, 336, 448, 560, 672 ... 896 ... 1008, 1120 ... 1344 ... 1680 ... 2016 ... 2240 ... 2688 ... 2800 ... 3024 ... 3584 ... 5040 ... 5376 ... 5600 ... 6048 ... 8064 ... 9072 ... 12096 ... 13440 ... 14000 ... 15120 ... 130636800.

Through the ages the 112 value has been very important in, especially, weights standards of the British Isles, wherein 112-pounds was 1-CWT (hundredweight), 20 CWT was the Ton @ 2240 pounds. One pound was 16-ounces and there were 14-pounds in a stone. A stone, therefore, was 224-ounces (112 X 2). The English ounce is 437.5-grains, the same weight as a Roman silver ounce, attesting to its great age. There are 7000-grains in an pound

Essentially, the English weights system followed a progression of numbers applicable to the dimensions of the Great Pyramid or, divisions of the Earth.

Or as Julius Caesar stated concerning what the Druids taught in their open-air universities:

TERRACE 3 NW.

138.24-feet @ 50-degrees.

A number-string based upon 50-degrees goes:

50, 100, 150, 200, etc., and simply taught students base 10 counting. Ancient civilisations used all the number families in their calculations and most certainly had knowledge of a decimal system.

KATOA LANGI.

WC McKern (circa 1920-21) measured the SW side of Terrace 1 as 120-feet and the SE side as 140.4-feet.

TERRACE 1 SW.

This is 120-feet long @ 120-degrees.

A number-string based upon 120 goes:

120, 240, 360, 480, 600, 720, 840, 960, 1080, 1200, 1320, 1440, 1560, 1680, 1800, 1920, 2040, 2160, 2280, 2400, 2520, 2640 ... 2880, 3000 ... 3120 ... 3240, 3360, 3480, 3600 ... 3960 ... 4200, 4320 ... 4800 ... 5040, 5160 ... 5280 ... 5400 ... 5760 ... 6000 ... 6480, 6600, 6720, 6840... 7560,7680, 7920 ... 8640 ... 10080 ... 10560 ... 11520 ... 13440 ... 15120 ... 16200 ... 19440 ... 19680, 19800 ... 20160 ... 30240 ... 60480 ... 130636800 ... 130680000.

Both the "Greek-mile" (5250-feet) generated equatorial circumference of the Earth (130636800-feet ... 24883.2-miles) and the "English-mile" (5280-feet) equatorial circumference (130680000-feet ... 24750-miles), navigational methods are divisible by 120. This means that both can be calculated within a 360, 3600, 36000 circuit, being read as seconds, minutes, degrees or the full circumference.

The fact that this terrace is 120-feet long and its orientation is 120-degrees is very clear evidence of carefully in-built coding and design.

The progression carries essential Great Pyramid navigational numbers: 756, 1512, 3024, 6048; It carries the numbers for the diameter of the Earth as 1980 (1/4th), 3960 (1/2) and 7920-miles, respectively.

In an age lacking electronic calculators, apprentices of the navigational arts had to develop photographic memories, learning by rote and grueling repetition. Years of training allowed them to become thoroughly conversant with the essential numbers, before being entrusted with a boat, the lives of its crew and the precious cargo, made it possible to traverse the featureless, vast oceans and find the destination. The sea is no respecter of persons and will kill the novice or unskilled intruder.

TERRACE 1 SE.

The length is 140.625-feet and the angle is 31.5-degrees.

A number-string based upon 140.625-feet goes:

140.625, 281.25, 421.875, 562.5 ... 1125 ... 1687.5 ... 1968.75 ... 2250 ... 2953.125 ... 3937.5 ... 4500 ... 5062.5 ... 6750 ... 7875 ... 9000 ... 10125 ... 11250 ... 11812.5 ... 13500 ... 18000 ... 22500 ... 24750.

The line-up infers divisions of the 360-degrree compass, with 11.25, 22.5, 45, 90, 180, 360. The value 135 is half of 270.

Of interest is 196.875, which became the ancient grain-weight for 1-Beqa of gold dust, used by the Egyptians, then adopted by the Greeks and later, Romans. Two Beqa of gold weighed 393.75-grains and this became the gold standard uncia of the Romans. Also, in the Greek Attic weight standard 393750-grains was 1 Talent.

A number-string based upon 31.5 goes:

31.5, 63, 94.5, 126, 157.5, 189 ... 252, 283.5, 315 ... 378 ... 472.5, 504... 567... 630 ... 756, 787.5 ... 850.5 ... 945... 1008 ... 1134 ... 1260 ... 1417.5 ... 1512 ... 1701 ... 1890 ... 2016 ... 2268 ... 2551.5 ... 2835 ... 3024 ... 3150 ... 3402 ... 3937.5 ... 4032 ... 4536 ... 5103 ... 6048 ... 6804 ... 7087.5 ... 8064 ... 9072 ... 11812.5 ... 12096 ... 130636800.

This progression contains the essential Great Pyramid numbers for navigation by the Greek-mile. It also contains the essential, lunar-cycle numbers found encoded into the dimensions and slope angle of the Khafre Pyramid of Egypt.

The numbers needed for tracking the 6804-day Lunar Nutation Cycle, within the lunisolar Sabbatical Calendar are also present in the line-up, including the value 2551.5, which in days = 7.2-lunar years (Anciently tracked daily alongside 7-solar years on parapegma counting plaques.

Added to the above, the angle of 31.5 relates to one of the versions of the PI ratio as 3.15 and tutorials for 3.125 (3&1/8th), 3.15 (3 & 3/20ths), 22/7, as well as 1728/550ths would have been taught. A similar modus operandi was being followed at Stonehenge, where one inner cross-measure of the slightly elliptical Sarsen circles (lintel to lintel face) was 100-feet . This could therefore be read as generating circumferences of 3.125, 3.15 or the more accurate renditions of 22/7 or 1728/550, which, depending on the correct choice, generated significant, whole number circumference codes for the Greek-mile or English league, etc.

Therefore 5250-feet X 22/7ths = 16500-feet (1-English league of circumference); Also, 16500-feet X PI @ 1728/550ths = 51840-feet. Alternatively, 7920-miles (the diameter of the Earth) X 3.125 = 24750-miles (the circumference of the Earth under the 5280-mile circumference reading) and 7920-miles X 1728/550ths = 24883.2-miles.

The 100-feet cross-measure of the slightly elliptical Sarsen Circle at Stonehenge has another cross-measure of 100.8-feet running 90-degrees opposed and this length is 1-second of arc for the Earth.

The sum of 31.5-feet would equate to 1/20th of the Greek Stadium measurement of 630-feet of 12-inches or 600 Greek feet of 12.6-inches.

TERRACE 1 NE.

The distance is 115.2-feet and the angle is 114.5833333-degrees.

A number-string based upon 115.2 goes:

Under this system, 1-degree of arc would be 69.12-miles; 138.24-miles would be 1/180th; 3110.4-miles would be 1/8th, etc.

A number-string based upon 114.5833333 goes:

This is a navigational progression for positional-plotting and dead-reckoning when voyaging across the vast featureless oceans, with no landmarks to refer to and only the celestial bodies by night for reference.

The circle produced was divisable by 360-degrees and one degree of arc on 16500-feet was 45.8.333333-feet (45 & 5/6ths). Using calipers on the circle, one could calculate the angle back to the port of departure or onwards to the destination, as well as one's exact position in the ocean, after zigzagging across a leyline during many legs of travel on the wind.

The two main systems of navigation used (1) a mile of 5250-feet (which was adopted by the Greeks) and (2) a mile of 5280-feet (which was adopted by the English).

The increments in the above progression are divisions of the English circumference of the Earth under the 5280-mile system.

The sum of 130680000-feet equates to 24750-miles of 5280-feet.

In the above progression, the sum of 114.5833333-miles would be 1/216th of the 24750-mile equatorial circumference. It would therefore represent 1 & 2/3rds degrees of arc.

Remember: the 360-degree compass system is divided into degrees X 60 = minutes or 360 X 60 = 21600 ... and if one wishes to reduce this to seconds of arc X 60 = 1296000.

In the above progression the value 687.5 occurs. In the 24750-mile reading of the Earth's circumference the sum of 68.75-miles would be 1-degree of arc.

Also the value 2062.5 occurs. The second longest Egyptian Royal Cubit located by William Flinders Petrie was 20.625-inches, and that value, read as miles, X 1200 = 24750 ... in other words, it was a mnemonic device for remembering the size of the Earth under this navigational system.

The value 5156.25 occurs and "Noah's ark"was described as 300-cubits long. If the 20.625-inch cubit was used, then 300 such cubits would equate to 515.625-feet.

The value 6187.5, read as miles, would be 1/4th of the 24750-mile equatorial circumference.

TERRACE 1 NW.

The length is 129.6-feet and the angle is 34.02-degrees.

A number-string based upon 129.6 goes:

129.6, 259.2, 388.8, 518.4, 648, 777.6, 907.2, 1036.8, 1166.4, 1296 ... 1555.2 ... 1944, 2073.6 ... 2332.8 ... 2592, 2721.6, 2851.2 ... 3110.4, 3240... 3628.8 ...4536 ... 5443.2 ... 5832 ... 7128 ... 8294.4 ... 9072 ... 9331.2 ... 9720 ... 12441.6 .... 24883.2.

The sum of 12960-years would equate to half the duration of the Precession of the Equinoxes, with the full duration being 25920-years. The sum of 5184-years would be 1/5th of the precessional cycle.

This progression homes in on the equatorial circumference of the Earth, of which 129.6-miles = 1/192nd; 518.4-miles = 1/48th; 777.6-miles = 1/32nd; 3110.4-miles = 1/8th, etc.

The sum of 362880-feet would be 1-degree of arc; 20.736-inches would be the length of the largest Egyptian Royal cubit; 58.32-inches would be a Roman Pace; 933120-grains would be the weight of an ancient Babylonian Talent, etc.

A number-string based upon 34.02 goes:

34.02, 68.04, 102.06, 136.08, 170.1 ... 272.16 ... 340.2 ... 510.3 ... 544.32 ... 680.4 ... 850.5 ... 1701 ... 2551.5 ... 3402 ... 5103 ... 6804.

This progression is, of course, devoted to the lunar nutaion cycle duration of 6804-days, with 850-days being 1/8th; 1701-days being 1/4th; 3402-days being 1/2; 5103-days being 3/4ths and finally 6804-days being the full cycle.

Honorable mention is given to 2551.5-days or 7.2 lunar years ... the count used for the moon, alongside 7-solar years (2556.75-days) in the lunisolat Sabbatical Calendar.

TERRACE 2 SW.

The distance is 110-feet and the angle is 120-degrees.

A number-string based upon 110-feet goes:

In this lineup, 5.5-feet was the original "fathom" measurement; 66-feet was the chain and 660 feet was the furlong; 16.5-feet was the "rod" or "perch" (3 fathoms), 1760-"yards" (3-feet) was a mile of 5280-feet and 5940-feet was a Scottish mile.

Also, the value 6050 occurs and under the navigational method that used the mile of 5280-feet the sum of 6050-feet was 1-minute of equatorial arc.

The radius of the Earth is 3960-miles of 5280-feet each, producing an overall diameter of 7920-miles.

TERRACE 2 SE.

The length is 130.68-feet and the angle is 31.5-degrees.

This length was for mnemonic reference to 130680000-feet or 24750-miles of 5280-feet for the equatorial size of the Earth. The same length would have simultaneously meant 130636800-feet in tutorials to apprentice navigators.

TERRACE 2 NE.

The length is 105.6-feet and the angle is 114.58333-degrees (114 & 7/12ths).

A number-string based upon 105.6 goes:

105.6, 2112, 316.8, 422.4, 528, 633.6 ... 950.4, 1056 ... 1584 ... 1900.8 ... 2640 ... 2851.2 ... 5280 ... 7920 ... 10560 ... 13200 ... 130680000.

This string identifes divisions of the mile of 5280-feet, as well as the value for the diameter of the Earth (7920-miles) and the equatorial circumference at 130680000-feet or 24750-miles.

TERRACE 2 NW.

The length is 120.96-feet and the angle is 34.02-degrees.

A number-string based upon 120.96 goes:

This progression relates to the Great Pyramid's perimeter value and the 130636800-feet equatorial circumference that is generated therefrom.

This side-length would have been simultaneously read a 121-feet and thereby relate to 1-perimeter circumnavigation being 3025-feet (36300-inches); a degree of arc 363000-feet and half the equatorial circumference as 6534000-feet.

TERRACE 3 SW.

The length is 101.25-feet and the angle is 120-degrees.

A number-string based upon 101.25 goes:

101.25, 202.5, 303.75, 405, 506.25, 607.5, 708.75, 810, 911.25, 1012.5 ... 1215 ... 1417.5 ... 1620 ... 2126.25 ... 2430, 2531.25 ... 2835 ... 3240 ... 3543.75 ... 4860 ... 5670 ... 7290 ... 8100 ... 8505 ... 8910 ... 9720 ... 11340 ... 12960 ... 1701 ... 34020 ... 51030 ... 68040.

In the lunar year of 354.375-days there would be 35 intervals of 10.125-days (243-hours) and 672 such intervals in the 6804-day Lunar Nutation Cycle.

In the 25920-year cycle of the Precession of the Equinoxes, there would be 256 intervals of 101.25-years.

TERRACE 3 SE

The distance is 120-feet and the angle is 31.5-degrees.

Both of these codes have been explained for this langi.

TERRACE 3 NE.

A number-string based upon 97.2-feet goes:

This is equivalent to 360-degrees reducing to 180, 90, 45, 22.5, 11.25, 5.625 (5 & 5/8ths), 2.8125 (2 & 13/16ths), 1.40625 (1 & 13/32nds)-degrees respectively.

The progression is also useful in simultaneously tracking the divisions of the 6804-day Lunar Nutation Cycle, within a 360-degree circuit being halved, quartered, etc.

A number-string based upon 113.4 goes:

Also in the line-up is 2835 ... the perimeter value of the Khafre Pyramid (Egypt's pyramid of the moon). 2835-feet is 34020-inches.

TERRACE 3 NW.

The length is 100.8-feet and the angle is 33-degrees.

A number-string for 100.8 goes:

100.8, 201.6, 302.4, 403.2, 504, 604.8 ... 806.4, 907.2, 1008 ... 1209.6 ... 1512 ... 2016 ... 2520 ... 3024 ... 3628.8 ... 4536 ... 5443.2 ... 6048 ... 7560 ... 9072 ... 10080 ... 130636800.

The 100.8-feet length is 1-second of equatorial arc in the Earth's circumference and this terrace would, therefore, be 1296000th of the Earth's equatorial circumference under the navigational method encoded into the base dimensions of the Great Pyramid.

A number-string based upon 33 goes:

33, 66, 99, 132, 165, 198 ... 264 ... 297, 330, 363, 396 ... 495, 528 ... 594 ... 660 ... 792, 825 ...891... 990 ... 1056 ... 1188 ... 1320 ... 1485 ... 1584 ... 1650 ... 1980 ... 2475 ... 3960 ... 5280 ... 5940 ... 7920 ... 24750,

Under the 5280-feet mile, all the divisions are based upon the number 11. These are the link (7.92-inches), Merchant Navy fathom (5.5-feet), Rod or Perch (16.5-feet), Chain (66-feet), Furlong or Furrowlong (660-feet), Mile (5280-feet) and League (16500-feet or 3.125-miles).

Under this navigational system, the diameter of the Earth is 7920-miles and the circumference 24750-miles. Also the Scottish mile of 5940-feet (1-furlong or 660-feet longer than the English mile) is included as an increment.

TERRACE 4 SW.

The length is 92.16-feet and the angle is 118.8-degrees.

A number-string based upon 92.16 goes:

92.16, . 184.32, 276.48 ... 460.8 ... 829.44 ... 1013.76 ... 1382.4 ... 2304 ... 2488.32 ... 4147.2 ... 4608 ... 11520 ... 12441.6 ... 24883.2.

The above values are divisions of the 24883.2-mile (of 5280-feet) equatorial circumference, with samples showing 1/270th, 1/54th, 1/18th, 1/6th, etc.

The true equatorial circumference of the world is 24902 English miles of 5280-feet but, unfortunately, it is very hard to work with that number when configuring a fluid, easy to compute, navigational system. For this reason ancient savant-astronomer-navigators chose close proximity values that were highly divisible ... that could be halved, quartered, reduced to eighths, sixteenths etc., right down to a tiny expression of the original. Thus, one could reduce a value like 24883.2-miles, very easily, describing the entire equatorial circumference as 12 X 12 X 12 X 12 X 1.2-miles, reducible down by degrees, minutes and seconds to one inch, then fractions thereof. *

*Note: This adjustment of a number into a factorable or easily manipulated value was also used to describe the duration of the Lunar Nutation Cycle, which is, in actuality, 6798.36-days. In the ancient edifices, like Khafre Pyramid, it is encoded as 6804-days and thus easily described as 19.2 lunar years or 230.4 lunar months. It then becomes easy to daily track on a lunisolar calendar, parapegma plaque, like the bronze Calendar of Coligny artefact, throughout the entire lunar cycle, with less than one minute (time) of error each lunar month.

Alternatively, one could use a longer sea-leg (league) like 3-miles of 5250-feet = 15750-feet ... converting, using PI @ 22/7ths, to 49500 ÷ 360 = 137.5-feet per degree of arc on the plotting chart.

Equally easy to use for navigation was the 5280-feet mile, adopted by the English and Germanic-Teutonic peoples.

To use a memory key for remembering the equatorial size of the Earth under the 5280-mile navigational system, one could use the "link" value of 7.92 X 5 X 5 X 5 X 5 X 5 = 24750.

In the above lineup, the number 101.376 is represented at an enlarged value.

A number-string based upon 118.8 goes:

118.8, 237.6, 356.4, 475.2, 594, 712.8 ... 950.4 ... 1188, 1306.8 ... 1900.8 ... 2851.2, 2970 ... 5940 ... 6534 ... 8910 ... 13068 ... 17820 ... 130680000.

The sum of 118.8-feet is 1/50th of a Scottish mile of 5940-feet. In the 130680000-feet equatorial circumference, 11880000-feet would be 1/11th part.

TERRACE 4 SE.

The length is 105.6-feet and the angle is 31.5-degrees.

Both of these codes have been explained for this langi.

TERRACE 4 NE.

The length is 90-feet and the degree angle is 295.3125.

The length is accentuating the quarter value of the 360-degree compass.

A number-string based upon 90 goes:

90, 180, 270, 360, 450, 540, 630, 720, 810, 900, 990, 1080 ... 1260, 1350, 1440 ...1620 ... 1800, 1890, 1980 ... 2160, 2250 ... 2430, 2520 ... 2700 ... 2880 ... 2970 ... 3150, 3240 ... 3600 ... 5940 ... 130680000.

The sexagesimal, 360-degree compass system is said to have been invented by the ancient Sumerians, but is obviously far older. It has been the mainstay, angle system for navigation through all recorded history and students of navigation would most certainly have needed to become very conversant and expert in its usage. In navigation, while sailing on the wind, and the necessity of calculating new angles of attack, back-&- forth across a leyline when zigzagging a course to a destination, there would be constant mental-arithmetic-awareness of distance covered and angle for each leg (league).

The progression provides increments for voyaging using the Scottish mile of 5940-feet, of which 90-feet would be 1/66th part. Note: 66-feet was the ancient "Chain" measurement and 660-feet was the ancient "Furlong" or "Furrowlong". As a sideline point of interest 66-feet X 660-feet = 43560 square feet, which is the Acre.

The angle this terrace edge sits at is 295.3125-degrees, which, 180-degrees opposed, is 115.3125 (a value that has no code-bearing relevance).

A number-string based upon 295.3125 goes:

295,3125, 590.625 ... 1181.25 ... 1771.875 ... 3543.75 ... 4725 ... 5906.25 ... 7087.5 ... 9450 ... 14175 ... 18900 ... 28350 ... 34020 ... 68040 ... 130636800.

The sum of 29.53125-days (29 & 17/32nds) is a lunar month to a tolerance of under 1-minute in time. Under the 3-4-5 triangle-coding for half the base length of the Khafre Pyramid of Egypt, the increment used was 118.125-feet X 3 (base), 118.125-feet X 4 (vertical height to the apex) and 118.125-feet X 5 (diagonal face hypotenuse).

The sum of 118.125-days is 4 lunar months of 29.53125-days each.

In the above progression we have expanded values for 29.53125, 354.375, 472.5 and 590.625. Under the design architecture of the Khafre Pyramid, half the base length is 354.375-feet, the vertical height is 472.5-feet and the hypotenuse face is 590.625-feet. Note: 590.625-feet is 295.3125-feet X 2.

The design-length of the Khafre Pyramid is 708.75-feet (15/16th of the 756-feet length of the Great Pyramid). The full perimeter-circuit length for Khafre is 2835-feet @ 708.75-feet per side.

In the above progression we have 3402 & 6804 represented in enlarged values. This of course refers to the half-period and full period of the Lunar Nutation Cycle of 6804-days.

Also occurring is the number 18900. Under the lunisolar Sabbatical Calendar system, when doing calculations for the duration of a solar month compared to a lunar month the formula was: 1890 ÷ 63 = 30 (the solar month); 1890 ÷ 64 = 29.53125 (the lunar month). Note: the ancient Calendar of Coligny bronze plaque of France was a parapegma device, with peg holes to track the daily calendar positions of both the sun & moon, daily.

Under this lunisolar Sabbatical Calendar system, the lunar period was tracked throughout 7.2 lunar-years alongside 7 solar years. The larger 6804-day count for the Lunar Nutation Cycle was simultaneously tracked.

All of the named segments seen were divided into two parts, and all had small peg holed on the vertical dividing lines. The peg for the solar count and the peg for the lunar count were moved to a new slot daily.

TERRACE 4 NW

The distance is 100.8-feet and the angle is 33-degrees.

A number-string for 100.8 goes:

100.8, 201.6, 302.4, 403.2, 504, 604.8 ... 806.4, 907.2, 1008 ... 1209.6 ... 1512 ... 2016 ... 2520 ... 3024 ... 3628.8 ... 4536 ... 5443.2 ... 6048 ... 7560 ... 9072 ... 10080 ... 130636800.

The angle progression of 33-degrees has been explained for this langi.

TERRACE 5 SW.

The distance is 79.2-feet and the angle is 118.8-degrees.

A number-string based upon 79.2 goes:

The rendition of PI used for converting the 7920-mile (of 5280-feet) diameter of the Earth into the 24750-mile circumference was 3.125.

The rendition of PI for converting the 7920-mile diameter of the Earth into the 24883.2-mile (of 5250-feet) circumference was 1728/550ths or 3.141818182.

The 118.8 angle code has been explained for this langi.

TERRACE 5 SE.

The length is 94.5-feet and the angle is 31.5-degrees.

A number-string based upon 94.5 goes:

94.5, 189, 283.5, 378, 472.5, 567 ... 756, 850.5, 945 ... 1134 ... 1417.5, 1512 ... 1701 ... 1890 ... 2268 ... 2551.5 ... 2835 ... 3024 ... 3402 ... 3780 ... 4536 ... 5103 ... 6048 ... 6804 ... 7087.5 ... 9072 ... 11812.5 ... 12096 ... 13608 ... 130636800.

This progression produces essential Great Pyramid navigational numbers simultaneously to Khafre Pyramid lunar-cycle numbers.

For the Great Pyramid we have: 189 (1/4th of a side length), 378 (1/2), 756 (one side length), 1512 (two side lengths), 2268 (three side lengths), 3024 (four side lengths), 6048 (two full perimeters and 1-minute of equatorial arc), etc.

For the Khafre Pyramid we have : 472.5 (its height in feet), 708.75 (its side length), 8505 (its side length in inches and, in days, 1/8th of the Lunar Nutation Cycle), 1701 (the number of days in 1/4th of the Lunar Nutation Cycle), 3402 (1/2 the cycle), 5103 (3/4th of the cycle) and 6804 (the full cycle in days).

In the lineup is 453.6 (the height of the Great Pyramid to its flat altar floor) ... also we have 9072 ( the side length of the Great Pyramid in inches).

The expanded value of 11812.5 occurs and 118.125-days would equate to 4-lunar months of 29.53125-days.

Ultimately, the progression grows to 130636800-feet ... the equatorial circumference under the Greek-mile and lunar count numbers.

The angle of 33-degrees has been explained for this langi.

TERRACE 5 NE.

The length is 79.2-feet and the angle is 295.3125-degrees.

The codes related to both of these values have been explained for this langi.

TERRACE 5 NW.

The length is 89.6-feet and the angle is 31.68-degrees.

A number-string based upon 89.6-goes:

89.6, 179.2, 268.8, 358.4 ... 537.6 ... 716.8 ... 806.4 ... 896 ... 1344 ... 2240 ... 4032 ... 6720 ... 12096 ... 130636800.

It seems that the ancient architects wished to have a tutorial that made reference to the Irish mile of 6720-feet and, 89.6-feet would be 1/75th of an Irish mile. Similarly, there would be 2.5 X 2268-feet (3-side lengths of the Great Pyramid) in an Irish mile.

In the 24883.2 Greek mile equatorial circumference there would be 19440 Irish miles, which is 672-feet larger than the value for 1-minute of equatorial arc (6048-feet).

If the Irish were using their mile for sea-leg navigation, then 6720-feet would convert to a circumference of 21120-feet. using PI @ 22/7ths.

The English preferred a larger league @ 16500-feet (3.125 English-miles).

In a purely mathematical sense it would be convenient to use legs of 3 Irish miles for their league of travel at sea, as that works out at 20160-feet. Remember ... 1/3rd of a minute of equatorial arc is 2016-feet, so an Irish mile is 3&1/3rd minutes of arc, resulting in a circumference of that sea-leg length (20160 X 22/7ths) = 63360-feet or (when divided by 360) = 1056-feet per scaled degree on the plotting chart. Remember, 1056-feet is 1/5th of an English mile.

From this exercise, we can see that the Irish method of navigation is simply a ratio expression of the Greek method. Various nations may have preferred slightly different systems, but all were drawing from the self-same parcel of special numbers and using the same "inch" or grain-weight for all of their weights, measures and volumes standards ... the numbers of civilisation..

A number-string based upon 31.68 goes:

31.68, 63.36, 95.04 ... 158.4, 190.08 ... 285.12, 316.8 ... 475.2 ... 792 ... 950.4 ... 1013.76... 1584 ... 2376 ... 3168 ... 3960 ... 4752 ... 6336 ... 7128 ... 7920 ... 130680000.

In the 7920-mile diameter of the Earth 31.68-miles would be 1/250th part and there would be 7920 X 16500-feet (the English league) in the 130680000-feet equatorial circumference.

LEKAMAKATUITUIOHA LANGI.

The huge blocks forming the four-tiered langi's southwestern side.

This is the true orientation of Lekamakatuituioha langi. One hundred years ago, archaeologist, WC McKern measured the southwestern base as 140-feet and the northwestern base as 166-feet.

TERRACE 1 SW.

The length is 140.625-feet and the angle is 312.5-degrees.

Careful analysis within Google Earth seems to indicate that the langi's southwestern base slightly exceeds 140-feet. Although that length would be an excellent code-bearing value, for the purposes of this analysis we'll consider it to be 140.625-feet. At this position, the merits of both 140 & 140.625-feet would have been taught to student astronomer-navigators.

A number-string based upon 140.625 (140 & 5/8ths) goes:

The line-up infers divisions of the 360-degrre compass, with 11.25, 22.5, 45, 90, 180, 360. The value 135 is half of 270.

Alternatively, a number-string based upon 140-feet would go:

A number-string based upon 312.5 goes:

312.5, 625, 937.5, 1250, 1562.5, 1875, 2187.5, 2500, 2812.5, 3125, 3437.5, 3750 ... 4375 ... 5000 ... 5625 ...6875 ... 7000 ... 9375 ... 10000 ... 11250 ... 12500 ... 13125 ... 15000 ... 171875 ... 17500 ... 178125 ...19687.5 ... 20625 ... 22500 ... 45000 ... 51562.5 ... 24750 ... 295312.5 ... 130680000.

The number 3.125 (3 & 1/8th) was used as a ratio or form of PI. Thus, the 7920-mile diameter of the Earth X 3.125 = 24750-miles. Similarly, the English mile @ 5280-feet X 3.125 =16500-feet ... the English league.

In the above line-up we have increments, in expanded values, that relate to the 360-degree calendar (11.25, 22.5, 45).

Also, the sum 437.5 grains is the English ounce and 7000 grains is the English pound; 196.875-grains (a lunar number) was the Beqa weight for gold dust under Egyptian, Greek and Roman standards; 68.75-miles was 1-degree of equatorial arc under the 24750-mile circumference of the Earth; 515.625-feet is 300 Egyptian Royal cubits @ 20.625-inches each (cubit found via Sir William Flinders Petrie's a measurement of the King's Chamber of the Great Pyramid).

TERRACE 1 SE.

The length is 170.1 @ 40.5-degrees.

A number-string based upon 170.1 goes:

170.1, 340.2, 510.3, 680.4, 850.5 ...1360.8 ... 1701 ... 2551.5, 2721.6 ... 3402 ... 5103 ... 6804 ... 130636800.

This, of course provides us with the values related to the 1/8th, 1/4th, 1/2 & full duration of the Lunar Nutation Cycle (6804-days). The sum of 2551.5-days is 7.2 lunar years @ 354.375-days per lunar year. Under the lunisolar Sabbatical Calendar 7.2-lunar years was daily monitored alongside 7-solar years (2556.75-days @ 365.25-days per annum).

A number-string based upon 40.5 goes:

40.5, 81, 121.5, 162 ... 243, 283.5, 324, 364.5, 405 ... 486 ... 567, 607.5, 648 ... 729 ... 810 ... 850.5, 891 ... 972, 1012.5 ... 1134 ... 1215 ... 1296 ... 1417.5, 1458 ...1701... 1944 ... 2268 ... 2551.5, 2592 ... 2835 ... 2916 ... 3037.5 ... 3402 ... 3888 ... 4050 ... 4536 ... 5062.5, 5130 ... 5184 ... 5832 ... 6075 ... 6804 ... 7087.5, 7128 ... 7776 ... 9072 ...11664 ... 13122 ...130636800.

This progression is a veritable banquet of Khafre Pyramid lunar, Great Pyramid navigational, Precession of the Equinoxes, Lunar Nutation Cycle and equatorial circumference of the Earth values.

In the 2551.5-day (7.2 lunar years) count being done under the lunisolar Sabbatical Calendar method, there would be 63 intervals of 40.5-days (972-hours). In the 6804-day Lunar Nutation Cycle count there would be 168-intervals of 40.5-days (Note: there are 168-hours in a 7-day week).

One of the more ancient concepts that survived into Mediaeval times was that 75-Roman miles was 1-degree of arc in the equatorial circumference of the Earth. Their mile was 5000 Roman feet of 11.664-inches for the foot they used in lunar calculations and in their volume measures. Under that foot, their mile was 4860-feet of 12-inches, which means an equatorial circumference of 4860 X 75 X 360 = 131220000-feet. However, the overland or marching Roman foot was 11.66666-inches (simply 11 & 2/3rds feet), meaning that their overland mile was actually 4861.111111 (4861 & 1/9th-feet of 12-inches) X 75 X 360 = 131250000 feet or 125000000 Greek feet or 135000000 Roman feet or 67500000 Swedish Rydaholmsalm or 27000 Roman miles or 3750 Swedish Mil or 25000 Greek miles.

TERRACE 1 NE.

The length is 134.4-feet and the angle is 131.25-degrees.

A number-string based upon 134.4 goes:

134.4, 268.8, 403.2, 537.6, 672, 806.4 ... 1209.6, 1344 ... 2016 ... 2688 ... 3360 ... 3628.8 ... 6048 ... 10080 ... 130636800.

A number-string based upon 131.25 goes:

131.25, 262.5, 393.75, 525, 656.25, 787.5 ... 1050, 1181.25, 1312.5 ... 1575 ... 1968.75, 2100 ... 2625 ... 3150, 3281.25 ... 3543.75 ... 4200 ... 4725 ... 5250 ... 5906.25 ... 6300 ... 7087.5 ... 8400 ... 9450 ... 12600 ... 14175 ... 16800 ... 17718.75 ... 18900 ... 19687.5 ... 25200 ... 28350 ... 29531.25 ... 130636800.

This progression refers to navigation by the Greek-mile of 5250-feet. Their short Stadium was 525-feet and their long Stadium, 630-feet. Their reed was 10.5-feet or 126-inches (10 Greek feet of 12.6-inches). The equatorial circumference of the Earth was 130636800-feet or 12 X 12 X 12 X 12 X 1.2 Greek miles or 24883.2-miles.

Simultaneously in this string are (in inflated values) the numbers related to the Khafre Pyramid of Egypt (15/16ths the size of the the Great Pyramid or 708.75-feet) with half its base length being 354.375-feet (the number of days in a lunar year), along with its vertical height 472.5-feet and its sloping face 590.625-feet. Note: half of 590.625 is 295.3125 and there are 29.53125-days in a lunar month.

Yet again, the Egyptian-Greek-Roman value for the Beqa of gold dust is present ... 196.875-grains.

TERRACE 1 NW.

The length is 166.66666-feet and the angle is 42.525-degrees.

A number-string based upon 166.66666 goes:

166.66666, 333.33333, 500, 666.66666, 833.33333, 1000, 1166.66666, 1333.33333, 1500. 1666.66666, 1833.33333, 2000, 2166.66666, 2333.33333. 2500 ... 3000 ... 3500 ... 4000 .... 4500 ... etc.

In a 3-4-5 triangle, one side will be 1.25 less in length than the hypotenuse and the other side will be 1.66666 less.

In this progression we can see what the Romans based their 11.66666-inch overland foot on, culminating in a Pace of 58.33333-inches (5-Roman feet). 5000 such feet became their overland mile. Also, the ancient Swedish people had a Rydaholsalm measurement of 1.944444-feet or 23.33333-inches (2-Roman overland feet),

An ancient Swedish Mil equated to 35000 English feet of 12-inches or 18000 Swedish Rydaholsalm of 1.9444444-feet or 36000 Roman feet.

TERRACE 2 SW.

The length is 126.5625-feet and the angle is 312.5-degrees.

A number-string based upon 126.5625 ( 126 & 9/16ths) goes:

126.5625, 253.125 ... 506.25 ... 1012.5 ... 1771.875 ... 3037.5 ... 3543.75 ... 4050 ... 6075 ... 130636800.

This value @ 12 & 21/32nds would have been used as a means of dividing the lunar year (354.375-days) into 1/28th divisions or 303.75-hours. Likewise, there would be 201.6 intervals of 12 & 21/32nds-days in the 2551.5-day period (7.2 lunar years) being tracked within the lunisolar Sabbatical Calendar.

Alongside this number-string being taught in relation to the terrace length was the more simple and obvious one of 126-feet.

A number-string based upon 126 goes:

The merits of 312.5 have been explained for this langi.

TERRACE 2 SE.

The distance is 158.4-feet and the angle is 41.25-degrees.

A number-string based upon 158.4 goes:

158.4, 316.8, 475.2, 633.6, 792, 950.4 ... 1900.8 ... 2376, 2851.2 ... 3960 ... 7128 ... 7920 ... 10137.6 ... 11880 ... 19800 ... 28512 ... 43560 ...79200.

The sum of 158.4-miles would be 1/50th of the 7920-mile diameter of the Earth; 1980-miles would be 1/4th.

The value 10137.6 occurs and 101.376-feet would be 1-second of arc in an equatorial circumference of 24883.2-miles of 5280-feet each. The number 158.4-feet would be 1/64th of 10137.6.

A number-string based upon 41.25 goes:

41.25, 82.5, 123.75, 165, 206.25, 247.5 ... 330 ... 412.5, 453.75, 495 ... 618,75, 660 ... 9075 ... 990 ... 1237.5 ... 1320 ... 1485 ... 1650 ... 1980 ... 2475 ... 2640 ... 2970 ... 3630 ... 3960 ... 4125 ... 5156.25 ... 5280 ... 7920 ... 8910 ... 16500 ... 24750.

This progression relates directly to navigation by the English mile of 5280-feet and Scottish mile of 5940-feet.

The sum of 41.25-feet would be 1/128th of an English mile (5280-feet) and 144th of a Scottish mile (5940-feet).

The sum of 41.25-miles would be 1/192nd of the 7920-mile diameter of the Earth and 1/600th of the 24750 English mile equatorial circumference.

Also, 41.25-inches is 2 X 20.625-inches (the second Egyptian Royal cubit length found by Sir William Flinders Petrie). There would be 76032000 cubits of 20.625-inches in the 24750-English mile equatorial circumference.

Remember, 1-degree of equatorial arc under this navigational system was 363000-feet.

The increments of the "11" family of navigational numbers are mostly represented in this lineup: 7.92-inches (link), 5.5-feet (Merchant Navy fathom, 16.5-feet (rod or perch) 66-feet (chain), 660-feet (furlong or furrowlong), 5280-feet (mile), 16500-feet (league).

Also included: The sum of 453.75-feet would be the height of the Great Pyramid to the top surface of it's altar floor @ 264 cubits of 20.625-inches. The true design-height is 453.6-feet based upon the cubit of 20.61818182-inches.

The value 515.625-feet would be 300 X 20.625-inches ... note: Noah's arc was said to be 300 cubits long, and would attain this length if the 20.625-inch cubit was used.

TERRACE 2 NE.

The length is 123.75-feet and the angle is 130.68-degrees.

A number-string based upon 123.75 goes:

123.75, 247.5, 495, 618.75, 742.5 ... 990 ...1485 ... 1980 ... 2475 ... 2970 ... 3960 ... 5940 ... 7920 ... 8910 ... 9900 ... 12375 ... 15840 ... 17820 ... 24750.

A number-string based upon 130.68 goes:

130.68, 261.36, 392.04, 522.72, 653.4 ... 1306.8 ... 6534 ... 13068 ... 65340000 ... 130680000.

This string was for teaching the equatorial circumference of the Earth under the 5280-mile navigational system encoded into the dimensions of the Great Pyramid, with a side-length increased by 3-inches to 756.25-feet.

The literal navigational method encoded into the literal side-length of the Great Pyramid @ 756-feet would also have been taught at Terrace 2 NE ... (Equatorial circumference of 130636800-feet or 12 X 12 X 12 X 12 X 1.2 Greek miles).

TERRACE 2 NW.

The distance is 154.2857143-feet @ 42.1875-degrees.

These 7-based values can look horrible, until it is realised that they are whole numbers and simple fractions.

The ancient mathematicians used all of the number families in their calculations and the number 7 featured prominently.

The sum of 154.2857143 is simply 154 & 2/7ths.

A number-string based upon 154.2857143 goes:

154.2857143, (7-increases) ... 1080 ... 2160 ... 3240 ... 4320 ... 5400 ... 6480 ... 7560 ... 8640 ... 9720 ... 10800 ... 11880 ... 12960 ... 15120 ... 16200 ... 17280 ... 19440 ... 22680 ... 25920 ... 27000 ... 30240 ... 60480 ... 130636800.

.The sum of 15.42857143-feet would be 1/196th of the 3024-feet perimeter of the Great Pyramid @ 756-feet per side.

Alongside this coded value for Terrace 2 NW, there would also have been a tutorial based upon the number 155.52 . In the 24883.2-mile equatorial circumference the sum of 1555.2-miles would equate to 1/16th part or 22.5-degrees.

A number-string based upon 42.1875 goes:

42.1875, 84.375, 126.5625, 168.75 ... 253.125, 295.3125, 337.5 ... 506.25 ... 590.625 ... 675 ... 1012.5 ... 1181.25 ... 1350 ... 1771.875 ... 2700 ... 2953.125 ... 3543.75 ... 130636800.

The sum of 4.21875-days (4 & 7/32nds) or 101.25-hours is 1/7th of a lunar month or 1/84th of a lunar year.

TERRACE 3 SW.

The length is 115.2-feet and the angle is 131.25-degrees.

A number-string based upon 115.2 goes:

Under this system, 1-degree of arc would be 69.12-miles; 138.24-miles would be 1/180th; 3110.4-miles would be 1/8th, etc.

The 131.25 number has already been explained for this langi.

TERRACE 3 SE.

the length is 148.5-feet and the angle is 42-degrees.

A number-string base upon 148.5-degrees goes:

145.8, 291.6, 437.4, 583.2, 729 ... 1166.4, 1312.2, 1458 ...2332.8 ... 4665.6 ...5103... 9331.2 ... 26244, etc.

In the above line-up 58.32-inches was the Roman Pace measurement or 5-Roman feet of 11.664-inches. Similarly, the value 7.29-inches was 1/8th of a Roman Pace.

Well yes, it was certainly 11.664” in number-strings tracking lunar cycles or standard volume capacities, etc., but the simpler value of 11.66666” (11 & 2/3rds) was used for overland distances, with a Pace of 58.333333” (5 Roman feet). The same foot, in doubled value, was used in Sweden for the Rydaholmsalm length (1.94444-feet of 12-inches).

One of the more ancient concepts that survived into Mediaeval times was that 75-Roman miles was 1-degree of arc in the equatorial circumference of the Earth. Given that the Roman foot was represented as 11.664-inches, that would equate to a Roman mile (5000 Roman feet) being 4860 feet of 12-inches. Therefore, 75 such miles, if representing 1-degree of arc in the equatorial circumference, would equate to a world of 27000 Roman miles

The degree angle 145.8 is also represented as 145.833333 (148 & 5/6ths).

Note: The ancient Druidic bronze calendar plaque unearthed at Coligny, Ain, France in 1897 is 4.86-feet long (58.32-inches or 1 Roman Pace or 4 X 14.58-inches).

A progression based upon 42 goes:

42, 84, 126, 168, 210, 252, 294, 336, 378, 420 ... 504... 630, 672... 756... 840...1008, 1050 ... 1134 ... 1260 ... 1344 ... 1512 ... 1890 ... 2016 ... 2268 ... 3024 ... 3150 ... 3402 ... 3780 ... 5250 ... 6048 ... 6300 .. 6720 ... 6804 ... 7560... 9072 ... 362880 ... 130636800.

Again, we have the essential navigational numbers, encoded into the Great Pyramid's dimensions, for circumnavigating the world using the 5250-feet mile.

Also, values related to the 6804-day Lunar Nutation Cycle duration are represented. There would be 378 periods of 420-days in the 6804-day cycle.

TERRACE 3 NE.

The length is 115.2-feet and the angle is 130.68-degrees.

Note: Both of these values have been explained for this langi.

TERRACE 3 NW.

The length is 147.65625-feet (147 & 21/32nds) and the angle is 42.1875 (42 & 3/16ths).

A number-string based upon 147.65625 goes:

147.65625, 295.3125 ... 590.625, 1181.25 ... 1771.875 ... 2953.125 ... 3543.75, etc.

The progression is in homage to the lunar month and lunar year values of 29.53125-days (29 & 17/32nds) and 354.375-days (354 & 3/8ths).

The value 42.1875 (also a strong lunar code) has already been explained for this langi.

TERRACE 4 SW.

The distance is 103.68-feet and the angle is 131.25-degrees.

A number-string based upon 103.68 goes:

103.68, 207.36, 311.04, 414.72, 518.4, 622.08 ... 829.44, 933.12, 1036.8 ... 1244.16 ... 1555.2 ... 2488.32, 2592 ... 3628.8 ... 4665.6 ... 5184 ... 6531.84 ... 7776 ... 12441.6 ... 12960 ... 130636800 ... 24883.2 ... 25920.

These are the divisions of the Earth's equatorial circumference based upon the literal encoding of the Great Pyramid of Egypt, culminating in 24883.2-Greek miles or 130636800-feet.

The value 1036.8 tells us the speed at which the Earth spins in a 24-hour period, thus 24883.2-miles ÷ 24-hours = 1036.8 mph.

The "Stirling Jug" of Scotland, dating to 1457 AD and possibly much earlier, had a capacity of 103.68 cubic inches, which was the same as the ancient Hebrew liquid volume called the Jerusalem Cab (predating the Babylonian conquest).

In the 24883.2-mile equatorial circumference, the sum of 1555.2-miles would be 1/16th, 3110.4 would be 1/8th & 12441.6 would be 1/2; The length of 20.736-inches was the longest Egyptian Royal cubit identified. It was found to be represented in the lengths of the paving slabs at the base of the Great Pyramid as 1.728-feet or 20.736-inches.

The sum of 20.736-miles would be 1/1200th of the Earth's equatorial circumference according to the literal navigational method encoded into the base dimensions of the Great Pyramid.

The ancient Babylonian Talent weight was 933120-grains.

Also represented are divisions of the 25920-year Precession of the Equinoxes cycle, with 5184-years being 1/5th and 12960-years being 1/2.

The 131.25 coded value has already been explained for this langi.

TERRACE 4 SE.

The length is 135-feet and the angle is 42-degrees.

A number-string based upon 135 goes:

TERRACE 4 NE.

The length is 100.8-feet and the angle is 130.68-degrees.

A number-string for 100.8 goes:

100.8, 201.6, 302.4, 403.2, 504, 604.8 ... 806.4, 907.2, 1008 ... 1209.6 ... 1512 ... 2016 ... 2520 ... 3024 ... 3628.8 ... 4536 ... 5443.2 ... 6048 ... 7560 ... 9072 ... 10080 ... 130636800.

The 130.68 number has already been explained for this langi.

TERRACE 4 NW.

The length is 134.4-feet and the angle is 43.2-degrees.

A number-string based upon 134.4-feet goes:

134.4, 268.8, 403.2, 537.6, 672, 806.4 ... 1209.6, 1344 ... 2016 ... 2688 ... 3360 ... 3628.8 ... 6048 ... 10080 ... 130636800.

A number-string based upon 43.2 goes:

These are, of course, divisions of the 24883.2-mile equatorial size of the Earth.

TAUHALA LANGI.

TERRACE 1 SW

The length is 170.1-feet and the angle is 314.1818182-degrees.

A number-string based on 170.1 goes:

170.1, 340.2, 510.3, 680.4, 850.5 ...1360.8 ... 1701 ... 2551.5, 2721.6 ... 3402 ... 5103 ... 6804 ... 130636800.

The angle of 314.1818182 (314 & 2/11ths) relates to tutorials on the various forms of the PI ratio that had to be used under different navigational systems, when turning a leg (league) of straight-line travel into a circumference. The English league @ 3.125-miles of 5280-feet = 16500-feet ...X PI @ 1728/550ths (3.141818182) = 51840 ... or 144-feet per degree of arc on the 360-degree circuit.

Alternatively, if using a leg of travel @ 3-Greek miles as a league for a straight-line distance of travel at sea (15750-feet), the 22/7ths version of PI was used. Therefore, 15750-feet X 3.142857143 = 49500-feet for the circumference generated or 137.5-feet per degree of arc.

Both of these versions of PI would have been taught at Tauhala langi, as well as less accurate versions such as 3.15 to 1 or 3.125 to 1.

TERRACE 1 SE.

The length is 172.8-feet and the angle is 42-degrees.

A number-string based upon 172.8 goes:

172.8, 345.6, 518.4, 691.2, 864, 1036.8, 1209.6, 1382.4, 1555.2, 1728, 1900.8, 2073.6 ... 2592 ... 3110.4 ... 3628.8 ... 4147.2, 4320 ... 4665.6 ... 6048 ... 7776 ... 8294.4 ... 9331.2, 9504 ... 12441.6 ... 24883.2.

This is a very power-packed string that identifies divisions of the Earth, the speed at which the Earth spins in a 24-hour period (1036.8 mph), values for a minute of Earth equatorial arc (6048-feet), as well as a degree of arc (362880-feet) and the distance around the Earth's equator (24883.2-miles).

It was obviously included in Tauhala langi's tutorials for reference to the PI generating method of 1728/550ths. Also a cubic foot is 12 X 12 X 12-inches (1728 cubic inches) and the longest Egyptian Royal cubit was 1.728-feet or 20.735-inches.

A number-string based upon 42 goes:

Again, we have the essential navigational numbers, encoded into the Great Pyramid's dimensions, for circumnavigating the world using the 5250-feet mile.

Also, values related to the 6804-day Lunar Nutation Cycle duration are represented. There would be 378 periods of 420-days in the 6804-day cycle.

TERRACE 1 NE.

The length is 177.1875-feet and the angle is 314.1818182-degrees.

A number-string based upon 177.1875 goes:

177.1875, 354.375 ... 708.75 ... 1417.5 ... 1771.875 ... 2835 ... 3543.75 ... 4252.5 ... 5670 ... 8505 ... 11340 ... 17010 ... 22680 ... 25515 ... 34020 ... 51030 ... 68040 ... 130636800.

This string provides numbers related to the lunar year of 354.375, as well as the side length and perimeter length around the Khafre Pyramid of Egypt. It also provides the essential numbers (mostly in expanded values) for tracking the 6804-day Lunar Nutation Cycle or the 2551.5-day (7.2-lunar years) tracked alongside 7-solar years (2556.75-days) in the lunisolar Sabbatical Calendar system.

The merits of 314.1818182 have been explained for this langi.

TERRACE 1 NW.

The length is 172.8-feet (already explained) and the angle is 39.6-degrees.

A number-string based upon 39.6 goes:

39.6, 79.2, 118.8, 158.4, 198, 237.6 ... 316.8, 356.4, 396 ... 475.2 ... 594 ...633.6 ... 712.8 ... 792 ... 950.4 ... 990 ... 1306.8 ... 1782 ... 1900.8 ... 1980 ... 3960 ... 5940 ... 7920 ... 130680000.

This number-string relates to divisions of the Earth's diameter:in miles: 1980 (1/4th), 3960 (1/2), 5940 (3/4ths) & 7920 (full). The string ultimately arrives at 130680000-feet of equatorial circumference or 24750 English miles of 5280-feet or 22000 Scottish-miles of 5940 feet.

Added to the above, the dissecting distance from the SE corner of Tauhala langi to the NW corner is 252-feet (3024-inches or 240 Greek-feet of 12.6-inches each). The angle southwards is 177.1875-degrees. This is the half-value of 354.375-degrees, coding the lunar year of 354.375-days.

.TUOFEFAFA FO'OU FOFA'UA LANGI.

TERRACE 1 SW.

The length is 120-feet and the angle is 113.4-degrees.

A number-string based upon 120 goes:

The fact that this terrace is 120-feet long and its orientation is 120-degrees is very clear evidence of carefully in-built coding and design.

A number-string based upon 113.4 goes:

Also in the line-up is 2835 ... the perimeter value of the Khafre Pyramid (Egypt's pyramid of the moon). 2835-feet is 34020-inches.

TERRACE 1 SE.

The length is 121.5-feet and the angle is 22.5-degrees.

A number-string based upon 121.5 goes:

121.5, 243, 364.5, 486, 607.5, 729, 850.5, 972 ... 1215 ... 1458 ... 1701 ... 1944 ... 2551.5 ... 2916 ... 3402 ... 3888 ... 5103 ... 5832 ... 6804.

Also in the line-up, 7.29-inches would be 1/8th of a Roman Pace of 58.32-inches (4.86-feet).

A number-string based upon 22.5 goes:

22.5, 45, 67.5, 90, 112.5, 135, 157.5, 180, 202.5, 225, 247.5, 270, 292.5, 315, 337.5, 360.

The line-up is of course related to traditional readings, in 1/16th divisions, of the 360-degree compass. Mariners, through the ages, have shown a preference of dividing the 360-degree circuit into, at least, 32 or 64 divisions.

TERRACE 1 NE.

The length is 114.5833333-feet and the angle is 116.64-degrees.

A number-string based upon 114.5833333 (114 & 7/12ths) goes:

The Greeks, like the English, undoubtedly preferred a larger sea-leg or league of travel when crisscrossing a leyline to a destination at sea and 3 Greek miles (15750-feet) would be very mathematically useful. The sum of 15750-feet X PI @ 22/7 = 49500-feet. A circle of this circumference would be 137.5-feet per degree of arc.