Something Strange Happened On The Way To The Parthenon
Copyright © 2002, Ernest P. Moyer
I was strolling along, enjoying the day, minding my own business, when suddenly there dropped out of the sky a papyrus scroll, kerplop, right at my feet. More or less accustomed to these kinds of things I assumed it came from Spock who beamed it down from the Enterprise. I picked up this curious document, looked it over, and decided to unroll it to determine its contents.
It was a brief affair, written in English, with the title “Earth Dimensions From WGS84.” More or less informed about scientific matters I knew that WGS84 was a system of earth measurements established in 1984 and that WGS meant World Geodetic System. Agreement among official agencies of many countries had defined a standard set of geographical data for earth survey, air traffic control, earth positioning, and so on. (1) (2)
Below the title was a set of tabular data, showing equatorial and polar earth radii, circumferences, and radii of curvature of the earth surface.
The several rows of the Table gave values in metric dimensions, in English feet, and in inches. I felt that the persons responsible for the scroll probably wanted me to discern the values that appeared.
Refer to attached table below.
I sat down by the side of the road, whipped out my handy zip-bang calculator, and proceeded to investigate.
Results from the Parthenon
Since my immediate interest was the Greek Parthenon located on the hill just north of Athens, where I was headed until this unusual event took place, I considered the dimensions reported for that striking temple.
Different people have measured the Parthenon in recent centuries. See a reprint of: James Stuart, The Antiquities of Athens, Measured and Delineated by James Stuart and Nicholas Revett, (3).
The Parthenon was also measured by Michael Gabriel Paccard, an outstanding French university professor who, along with Jacques Balmat on August 8 in 1786, became famous as the first historic person to climb Mt. Blanc in the French Alps, while people in the valley below observed this feat with telescopes.
Paccard was architect of the Palace of Fontainebleau. His reputation was great among other architects, who eagerly sought him for his wise counsel and clear and methodical instruction. Unfortunately, I have been unable to locate his work on the Parthenon. The United States Library of Congress does not show him in their catalog. Hence I cannot know his precise measurements, except as gleaned from secondary sources. (4)
where you may find the following remarks by Kim Veltman:
In archaeology, ever more detailed measurement of ancient architecture led to a new awareness of optical adjustments made by the ancients. Hoffer (1838) made a preliminary report on new measurements of the Parthenon. Pennethorne (1844) offered a larger framework for understanding these adjustments. His ideas were taken up by Penrose (1846) and in an unpublished work of Paccard in the same period. Not everyone was convinced that these curvatures in the Parthenon had been intentional. The possibility that they might have come about through a sagging of the foundations was considered by Bötticher (1862) and Ziller (1865). Further studies of Durm (1871) and Burnouf (1875) gradually established that these curvatures had indeed been intentional. Hauck (1879), in an important study related these curvatures to Greek optical theories and in so doing prepared the way for Goodyear’s (1912) studies and Panofsky’s (1927) claims.
I found Paccard’s measurements (5).
See also Francis Penrose, An Investigation of the Principles of Athenian Architecture, W. Nicol, London, 1851, (6), with a subtitle: The results of a recent survey conducted chiefly with reference to the optical refinements exhibited in the construction of the ancient buildings at Athens, by Francis Cranmer Penrose. Pub. by the Soc. of Dilettanti.
As Penrose wrote:
. . . no two neighboring capitals correspond in size, diameters of columns are unequal, inter-columnar spaces are irregular, the metope spaces are of varying width, none of the apparently vertical lines are true perpendiculars, the columns all lean towards the center of the building, as do the side walls, and antae at the angles lean forward, the architrave and frieze lean backward, the main horizontal lines of construction are in curves which rise in vertical planes to the center of each side, and these curves do not form parallels . . .
A. W. Lawrence, in Greek Architecture, (7), reviewed the optical sophistication that had to be reduced to architectural design before implementing in actual structures. He emphasized the subtleties of Doric innovations found in the Acropolis. Many ancient writers, later speaking about the wonders of the Athenian designs, failed to grasp the truly unique nature of that construction era.
The ‘refinements’ in the Hephaisteion, Parthenon, and Propylaia extend far beyond the mere optical corrections which these late authors (ancient writers) recommend: very few portions of the buildings are actually straight, perpendicular or horizontal. Broadly speaking, the lines which should be horizontal are curved upwards convexly while those which should be perpendicular slope inwards. All the distortions are too small dimensionally to be easily noticed.
The remarkable nature of the Parthenon structure is seen from the fact that if extended 1.5 miles upward the columns would all meet at a common point directly over the head of the Athena Parthenos! The angle was a ratio of about 50 ft/7920ft, or about 22 minutes or arc. (Some disagreement exists among different measures. Others say the projection of the columns would meet at 11,500 feet.) Overall error in construction was as low as a fiftieth of an inch.
Clearly the Greek architects went to great pains to erect an object of beauty and precision.
Livio Stecchini wrote about the Greek designs. Educated at Harvard, and with a vast knowledge of the ancient literature, he presented his theories of the intent of the designers. Unfortunately, his repetitious writing and his theorizing about ancient geodetic knowledge adversely affected his scholarship and reduced his contribution to our understanding of the past. He never published his survey work except as an appendix to Secrets of the Great Pyramid, and through papers now maintained by others on a web site. (Stecchini has since died.) He mentioned the work of other Parthenon surveyors, including B. H. Hill, and Nicolas Balanos, Curator of the monuments of the Acropolis in the nineteenth century. Unfortunately, the work of the last two men did not add to the refinement of measurements necessary to verify the Parthenon dimensions. See Stecchini references below.
Stuart and Revett measured the width of the original Parthenon platform (stylobate) as 101.141 feet, with the length at 227.587 feet (1213.69 and 2731.04 inches).
Paccard measured the width at 101.233 feet (1214.80 inches).
Penrose reported the short base at 101.341 feet (1216.09 inches). He gave the long base dimension at 228.141 feet (2737.69 inches). His measures were slightly larger than those of Stuart and Revett.
The ratio of the length to the width is 2:25, not 2:1, because the Greek builders added one more column along the length for an 8 X 17 colonnade.
Relationship to Earth Geodetic Measure
I knew from other reading that the Nautical Mile had been defined to be equal to one minute of arc on the earth’s surface. This differed somewhat among countries until an international meeting in Monaco in 1929 set the standard value at 1852 meters (6076.103 feet). Because the earth is an oblate spheroid it varies somewhat with latitude, from 59.7 to 60.3 nautical miles per degree. See http://www.unc.edu/~rowlett/units/dictN.html (8)
When I divided the Monaco value by 1/60 to determine the length of an average second of arc I found that the result was 101.268 feet.
In summary, these sources give us:
|SOURCE||WIDTH (FT)||LENGTH (FT)||WIDTH (IN)||LENGTH (IN)|
|Stuart and Revett||101.141||227.587||1213.69||2731.04|
|Defined Nautical Mile/60||101.268||1215.22|
This surprised me. The length of one second of arc on the earth’s surface as defined by modern measure was the same as the short stylobate dimension of the Parthenon, within measurement error. In other words, the short stylobate dimension of the Parthenon was set by the builders to be equal to the length of one second of arc of the earth’s surface.
Of course, the builders may not have known that they achieved this unusual geodetic feat. See discussions below.
Penrose published this fact when he presented his measurements. However, it seems he obtained this idea from Paccard. Wilbur F. Creighton, Jr. and Leland R. Johnson also reported this fact in The Parthenon in Nashville, Athens of the South, J.M. Press, Brentwood, TN, in 1991 (9). (The Nashville Parthenon was intended to be an exact duplicate of the Athenian Parthenon.) But apparently not too many people paid attention. Refer to the Lyre Magazine Online (10).
The difference of the measures of the two extremes of Stuart and Revett, and Penrose, from the defined nautical distance was minus 0.127 feet and plus 0.073 feet, -1.5 and +0.88 inches, or -0.13% and +0.07%.
Some persons think the difference between these two measures is due to the curvature of the Parthenon stylobate that was designed to provide an illusion of a level surface. According to this view Stuart and Revett measured directly from corner to corner, while Penrose followed the curvature. However, the difference in height from the end to the center to produce this disagreement is nearly 40 inches, much more than the curvature created by the builders, which is about four inches. Hence I conclude that the differences are due to measurement methods, determining structure end-points, or simple errors. (Refer to extended discussion by Stecchini, who analyzes these structural and measurement nuances in considerable detail.)
Being somewhat of a believing nature I felt that the Parthenon stylobate base width was not a coincidence. If the builders truly intended to follow this design criteria it meant that they knew the earth was a sphere. They also knew the measure of the circumference with enough precision to produce this feat, at 21,600 Nautical Miles, (360 degrees X 60 minutes), or their equivalent dimension. In fact,
the Greeks held the stylobate dimension so close to the geodetic value that modern measures have difficulty distinguishing their construction resolution.
I now faced a major dilemma. According to our modern views of the ancients they mostly did not believe the earth was a sphere. Even more, they could not have known its dimension with such exactness.
Imagine what it must mean if they had actually measured the dimensions of the earth, with optical refinement necessary to such feat. In modern times, and with refined sighting equipment, it took laborious effort to determine those dimensions – after the French decided how it should be done.
Refer to terse mention at the web site of the National Institute of Standards and Technology (11).
More lively discussion may be found at
For the actual historical account see P. F. A. Méchain and J. B. J. Delambre (13).
Plutarch in Pericles wrote about the construction of the Parthenon and the Acropolis.
Phidias had the oversight of all the works, and was surveyor-general, though upon the various portions other great masters and workmen were employed. For Callicrates and Ictinus built the Parthenon; the chapel at Eleusis, where the mysteries were celebrated, was begun by CorÉbus, who erected the pillars that stand upon the floor or pavement, and joined them to the architraves; and after his death Metagenes of Xypete added the frieze and the upper line of columns; Xenocles of Cholargus roofed or arched the lantern on the top of the temple of Castor and Pollux; and the long wall, which Socrates says he himself heard Pericles propose to the people, was undertaken by Callicrates.
The project created great controversy among the Athenians because Pericles embezzled funds raised from allies to finance war against the Persians “to adorn Athens like a whore.” Work began in 447 BC, and the building itself was completed by 438 BC.
But I found no historical mention that the ancient Greeks had such advanced knowledge as a measure of the size of the earth. Until the discovery by Paccard and Penrose in modern times no one seemingly was aware of this design and construction feat.
Here was the hook: I was convinced that the ancient Greeks used an architectural measure that was based on the spherical dimensions of the earth, a geodetic measure. It certainly was not based on the length of the human forearm or any other mythical human body parts. Furthermore,
some unknown person or persons in antiquity had defined the earth circumference into the divisions of degrees, minutes and seconds we use yet today. The ancient Greeks used this definition.
But this was not the end of the difficulty. The division of the stylobate short length was in 100 parts of a Greek foot. The Greeks used a “foot” measure similar to ours. If the stylobate short length was not an accident, then the number 100 also could not be an accident. This meant that:
the Greek “foot” derived from geodetic measure and was equal to 1/100 part of a second of earth surface arc!
Perhaps the Parthenon stylobate design width of 100 Greek feet was a mere coincidence to one arc second of earth measure. Perhaps this coincidence was merely the result of the definition of the length of one Greek foot devoid of any geodetic intent. Perhaps such length may have been inherited from the more ancient past, and no Greek remembered.
Even more, their measure of 12.15 English inches was only slightly larger than our English foot, (101.27/100 X 12), or thereabouts. Since the modern English foot and the ancient Greek unit of measure are so close to one another it probably meant that the English foot was also based on geodetic measure, not upon some mythical body part, and that it had shifted slightly in value from much earlier times. This meant that views of our ancestors as primitive savages using a distance from the tip of their noses to the ends of their fingertips as their standard were based strictly on folk tales. Our forefathers were perceptive and rational people who were careful to maintain their measuring standards, whether in the dense forests of northern Europe, or the farming villages of the Anglo-Saxons.
For those who are concerned about drift from standards and different values for the “foot” from the ancient Mediterranean and Europe you may consult, for example, the report by Flinders Petrie in Ancient Weights and Measures, (14). He shows
- Punic (Phoenician) 10.9-11.1 inches
- Northern (Europe) 13.1-13.3
- Naples (Bronze standards) 11.5-11.6
- Statilus (Architect standard) 11.42
- Aebutius (standard) 11.63
- Lapis Capponianus (standard) 11.67
While such reports show variations of the “foot,” comparison with the Parthenon foot shows that the present English foot is much closer than any of these values to the geodetic nominal value — which makes me wonder how each generation maintained their standards. Why would the Anglo-Saxons fare better than their European or Mediterranean counterparts? As we can easily observe, it would appear that all the nations of the Mediterranean and Europe used a “foot” that must have come from some common, more ancient source, but that they forgot the geodetic origin. They did not maintain it correctly because they could no longer measure the geodetic value. We should not infer that Pericles could make such measure but rather that he also had inherited from a forgotten past.
Except for precision provided by modern technology, the ancient Greeks had a better handle on their units of measure than we modern people. They knew things we don’t commonly know. How many people today know that the present English foot is equal to one second of arc of the earth’s surface, divided by 100 (or thereabouts within 1.25%)?
Because this knowledge was unknown to modern man, and he had inherited a hodgepodge of confused measurements, he attempted to recapture a sensible system by resorting to a new standard. We call it the meter — which is also based on geodetic values. Modern man replaced the ancient measures because he forgot them. Knowledge of his measurement system had deteriorated with time. It had run down because it had lost the hallowed tradition of sacredness derived from the gods. Or, at least, that is what ancient people believed.
A very few modern people recognize that the current metric system is lacking in many substantial elements that were preserved in the old systems, such as linking the measurements of time, weight, volume, and so on. The old system was based on nature; the new system derived out of man’s intellect.
Some idea of how this loss came about may be obtained by examining measurement history.
I knew that the Greek Eratosthenes, (circa 276 to 196 BC), living in Alexandria, is commonly credited with the first scientific measure of the size of the earth. If so, he came along more than 150 years after the Parthenon was built but apparently was unaware of the geodetic measure used by his ancestors. He obtained a value based on Egyptian distances between Syene and Alexandria but, since his assumptions are not known with certainty, modern estimates of his accuracy vary between 1 and 15% too large.
Eratosthenes did not use refined optical equipment. He did not have such technical power. His methods were crude compared to modern abilities. Could we then claim that Pericles had such power? Or is it more sensible to assume that the Greeks were unaware of the unique nature of the measure they used? Were they as ignorant of the source of their units of measure as we are today?
I knew from my reading that many ancient persons believed Greek mathematical knowledge derived from the Egyptians. In his Metaphysics Aristotle wrote:
“Thus the mathematical sciences originated in the neighborhood of the Egypt, because there the priestly class was allowed leisure.”
If the ancient Greeks used this unique measure, and they had acquired it from the Egyptians, my curiosity would not be satisfied until I located the source of that ancient knowledge.
At that point I decided to continue my journey, gaze in awe at the local attractions, and then go on to further adventures.
I soon headed for Egypt, and the pyramid complex at Giza.
Results from the Great Pyramid
My interest was piqued by remarks made by Agatharchides of Cnidus, a Greek who lived in the time of Ptolemy Philometor (181-146 B.C.) He was born just after Eratosthenes died. Agatharchides was a philosopher, geographer, historian, traveler and naturalist. He lived in Alexandria and spent much of his life on expeditions of exploration. He is cited by Athenaeus, Strabo, Plutarch, Diodorus Siculus, Artemidorus, Lucian and Photius. (15).
As many modern persons know, he said that the length of one side of the base of the Great Pyramid was equal to 1/8 minute of degree of the earth’s surface.
Note what this means. He certainly believed the earth was a sphere. Further, he believed the Egyptians some 2500 years earlier also viewed the earth as a sphere. Even more, he gave them credit for knowing not only that the earth was a sphere, but also the size of it. He could not speak in such technical terms unless both conditions hold. But then where did we moderns get the idea the ancients did not know the earth was a sphere? And why have we pinned our evolutionary schemes on Eratothenes as the first discoverer of the size of the earth? Still further, the ancient Greeks believed the more ancient Egyptians divided the earth into the same units of measure of degrees, minutes, and seconds of arc. But this sexagesimal system is not known from the records of ancient Egypt.
We test the numbers given by Agatharchides by multiplying the base dimension by 4 (one perimeter) and then by 2 to obtain the value he stated.
The measurements of Flinders Petrie (16) and J. H. Cole (17) gave a perimeter value of 3022.93 feet and 921.405 meters, (3022.98 feet, only one-half inch different from Petrie). If one doubles this number to obtain 6045.86 feet we can then compare with the 6076.08 feet of a modern Nautical Mile. The values differ by 30.22 feet or 0.5%. This amount of error certainly is nothing to sneeze at, although it would suggest considerable inaccuracy in the assumed size of the earth or in the design. Since the stylobate of the Parthenon was held to an accuracy to provide 6076.2 feet (60 X 101.27), the Greeks of the fifth century BC did far better than the Egyptians of 2700 BC in representing that ancient source of geodetic measure.
(If we use the polar radius obtained from modern satellite measures, the value of 1213.32 inches, 101.11 feet X 60, gives us 6066.6 feet, to reduce the Egyptian error to less than 0.3%.)
Did the Egyptian measure improve over the following two millennia to give the Greeks a more precise value for the measure of the earth? Or was the report by Agatharchides inaccurate, thus showing the Old Kingdom Egyptians were not quite as good as the (much) later Greeks?
Of course, by the time of Agatharchides the Egyptians and the Greeks had the measurements of Eratosthenes; he could make such claim if someone in his day had measured the Khufu pyramid. However, this does not explain how the Egyptians two thousand years earlier had this knowledge.
When I discussed these problems with a friend of mine, he said that since we moderns do not believe either of those ancient people knew the size of the earth we should not hold them accountable. I felt he was cavalier about the whole matter.
At this point I was not going to lay down any wagers about who knew what.
I was convinced that something was going on in our ancient past that escapes virtually all modern scholars, academic experts, and the common man.
Results from the Royal Egyptian Cubit
Petrie’s exquisite work in Egypt, and that of many other surveyors, revealed that the Egyptian standard of measure was the Royal Cubit, with a value of 20.62 or 20.63 inches. Refer to Petrie’s determination from the King’s Chamber in the Great Pyramid, at 20.632, page 178 of Pyramids and Temples. Petrie also gave a mean measure for the Great Pyramid at 20.620 +/- 0.005. For all measurements from the Fourth to Sixth Dynasties he gave 20.63 +/- 0.02. Refer to cubit rods and discussion published by Marshall Clagett (18), who borrowed his illustrations from Lepsius (19), and others. While measuring rods of different working lengths, such as the “small” cubit, and with considerable variability, have been found, most Egyptologists accept 20.62 to 20.63 as an “official” value. See Petrie’s variations from 20.3 to 20.8 inches (14). Other studies show similar variation.
Calculating the numbers we find that each side of the Great Pyramid was nearly, but not quite, 440 cubits. Petrie measured 9069.4, 9067.7, 9069.5, and 9068.6 inches respectively for the north, east, south and west sides. These add to 36,275.2 inches, or 1759.2 (1758.4) cubits with an average of 439.8 (439.6) cubits per side. From this slight deficiency one might question the intent of the Egyptian builders of the Fourth Dynasty. Of course, some might say that being off by only 0.2 or 0.4 cubits out of 440, less than 0.1%, might be big fuss over nothing. However, the reader certainly understands that major insights might hang on such small differences.
(If we add the deficient amount to make each side equal to the nominal value of 440 cubits we obtain 9075 inches per side. Multiplying by 4 and 2 we obtain 72,600 inches. This compares with 1215.22 X 60 for 72, 913.2 inches from the defined Nautical Mile. Thus the pyramid would have been deficient by only 156 perimeter inches or 39 inches per side, 0.4 %, to make the perimeter equal to one-half a Nautical Mile — if the builders had held to this ideal. The satellite measures reduce this to less than 0.3%.)
In attempt to get a better handle on the intent of those ancient people I began to examine the roots of these phenomena. At this point a startling fact appeared.
If I take a circle and divide it into 360 degree, 60 minutes, and 60 seconds I obtain 1,296,000 seconds of arc. One can obtain radian values by dividing the circle into parts of 2 Pi. This is the famous circle relationship: C = 2 Pi R. I found that the value for the radius of the earth is 206,264.81 seconds of arc.
At first I did not recognize the result I had. Then it suddenly appeared to me.
This was most curious. This is the length of the Royal Cubit in English inches multiplied by 10,000. (10,000 X 20.625 = 206,250.)
I was perplexed. How could such a coincidence in numbers exist?
Arc seconds are a mathematical concept, they do not have dimension. They only express relationship. They take on dimension when we divide a real distance, such as the surface circumference of the earth, by 1,296,000 arc seconds. I did this above by showing that the stylobate of the Parthenon was built to one arc second of earth surface circumference expressed in (100) Greek feet.
Furthermore, the calculation of 206,264 is a radian value, not a linear dimension. It is a measure of the number of arc seconds in the earth radius.
The numbers must be a mere coincidence. But I did not like such strange coincidences that agreed within less than 0.1%.
Peter Thompkins described this coincidence in Secrets of the Great Pyramid, but did not have the insight or perception to penetrate the cause (20).
The Dimension of the Earth Radius in English Inches
How could I relate the earth radius in linear measure to the number of arc seconds in that radius?
Simple. Just as we determined the length of the Greek foot in arc seconds of surface measure, we can determine the number of English inches (or Greek feet) in arc seconds of radial measure. Then there would be 206,264+ units expressed in inches (or Greek feet) for the radius of the earth. What would be the result?
If 100 Greek feet were equal to one arc second of surface measure, there would be 206,264+ X 100 Greek feet in the radius of the earth. In English inches this would be equal to 206,264+ X 100 X 1215.22 = 250,656 million. (I use the value assumed from the Nautical Mile.)
I pulled out the WGS84 satellite data Spock had so kindly sent me, and which I carefully kept with me, and began to examine its numbers. I saw that the radial measure of the earth varied between 249,426,220 inches for the equatorial radius of curvature, and 251,951,975 inches for the polar radius of curvature. (Because the earth is an oblate spheroid the curvature at the poles is different than the curvature at the equator. The polar and equatorial radii of curvature express those values.) I saw from my handy-dandy Spock table that all of the different radii of the earth were about 250 million inches, more or less. This confirmed my calculation using Greek feet. If I divided those numbers by 250 million I obtained 1.0044289, 1.0010613, 1.0033064, 1.0078079, and 0.9977049 respectively. In other words the radius of the earth was very nearly 250 million English inches (within 4 parts per thousand). In fact, if I were a surface observer I might not be able to make up my mind whether the radius of the earth was more than 250 million inches or less.
Now that is what I would call a strange round number.
I could express it differently. 500 million inches were in one earth diameter. Or two earth diameters had one billion English inches.
How in heaven’s name did such a number come about? Would these strange coincidences never cease?
When calculated, an ideal 250 million inches divided by 206,264.81 gives 1212.03 inches per arc second. This compares to the 1215.22 inches from the defined Nautical Mile, and the 1214.86 inches from the average of the Parthenon stylobate measurements. The range of values from Spock’s Table for the different radii was from 1209 to 1221 inches.
Relationship Among the Systems of Measurement
My cavalier friend wanted to know where this was all headed. Pericles had built the Parthenon according to the dimensions of the earth, the Egyptians might have built the Great Pyramid of Khufu likewise except they did not quite make it, there was a strange coincidence of numbers in the English length of the Royal Egyptian cubit and the number of arc seconds in the radius of the earth, and the radius of the earth in English inches was a nice round number. He wanted to know how this all tied together.
Furthermore, he felt confused with all the possible variations of the numbers I was tossing around.
I replied that perhaps we should assume the ancients thought as we. Perhaps they used a measure of the earth along a polar axis, as Erastothenes determined, and as Méchain and Delambre determined. Then we could limit ourselves to the Polar Radius column in the Tabulation. Of course, if the ancient measurers used such value we would be assigning to them more knowledge than possessed by Erastothenes or the Frenchmen in that they were not limited to any one latitude. This limit is one of the major assumptions made by all modern analysts who try to understand the work of the ancients. Perhaps the ancients were a lot better than we give them credit, and had a measure of the earth polar radius similar to what we obtained from satellites, not limited by geographical latitude. I explained that this was a great leap in assumption but for the sake of presentation I would proceed along those lines. Later we could be more definitive about the source.
I invited him to return to further examination of those measures. If the polar radius of the earth is 250,265, 313+ inches from the WGS84 data, and this is divided by the radial seconds of arc of 206,264.81 we obtain 1213.3204+ inches. Now we had a fourth measure of the length of earth arc seconds in addition to the three listed above. I emphasized how these values all agreed within measurement error of one another. These were:
- 1212.03 inches from an ideal 250 million inches in the earth radius;
- 1213.32 inches from satellite data, (assuming the measured polar radius);
- 1214.86 inches from average Parthenon stylobate measure;
- 1215.22 inches from the defined Nautical Mile;
Now! I ran into a lethal problem, lethal to our modern philosophies of human evolution. In order to obtain a unit length of 100 Greek feet, (one arc second), expressed in English inches, with the radius of the earth divided by the number of radians, a defined mathematical entity, I had to have 250 million inches, more or less, total radius length in order to arrive at that result. I could not have 240 or 260 million, or any other number that was very far from 250. (The range from actual observation and Nautical definition is from 250.266 to 250.658 million inches.)
Such number could not have originated from pure chance. To propose pure chance violates rational thinking. If this value were not pure chance then,
the western unit of measure we use in modern times, the English inch, had to be defined in some ancient past that is now unknown to us.
Otherwise, we could not define 1215 inches in 1/60 Nautical Mile.
This unit of length was designed to fit an earth spherical model devised by some ancient mind.
Otherwise, we could not observe 1213 inches in one arc second from satellite data.
No other logical alternatives exists. A round number of 250 million did not develop out of pure happenstance.
The difference between the ideal and the satellite data might be due to a slight drift in the standard inch length over the centuries. This would produce a difference of about 1.5 inches in every 100 English feet. But this difference is not any greater than that between the satellite data and the defined Nautical Mile.
Furthermore, none of this makes logical sense unless
that ancient mind knew the exact C = 2 Pi R relationship. He had to know not only a practical value of Pi, he also had to know it theoretically.
This conclusion is true because all the elements of the relationship are demonstrated by earth measure in the number of surface arc seconds, the length of the surface units, the length of the radius units, and the relationship of 2 Pi R to obtain the surface length from the radius length.
But this was not the end of the lethal impact of my investigation.
If I take the number of English inches in one Greek foot, as determined from modern satellite data, and divide it into the length of the Royal Cubit of 20.62 or 20.63 English inches, as obtained from modern measurements of ancient Egyptian monuments and ancient cubit rods, I obtain a ratio of 1.69947 or 1.70029. If I take the median of 20.625 I obtain 1.6999. In other words,
within the possible error of modern measures the ratio of the length of the Egyptian Royal Cubit to the length of the Greek foot is a mathematically simple 1.7/1.0.
The fact that I performed this calculation in English inches does not alter the ratio; I could have performed the calculation in any other consistent system of measure.
The length of the Egyptian Royal Cubit is 1.7/100 arc seconds of earth measurement.
To propose that the ancient Egyptians came at this simple ratio and this length measure out of some random chance voids plain human sense. Thus,
if I use the Greek foot value of 12.1332046 calculated from the WGS84 satellite data and multiply it by 1.7 the value of the Royal Egyptian cubit would be 20.6264+ inches.
When I was done with all this my cavalier friend was no longer so cavalier. In fact, he became quite pale. If I had not walked him a few times around the Great Pyramid I fear he would have fainted.
a direct mathematical relationship exists between the length of the ancient Greek foot and the length of the Egyptian Royal Cubit.
The two measures of length are tied together mathematically in a manner never before recognized in western history.
Which means that ancient measures from different countries were connected to one another by some profound relationship, the knowledge of which has not come down to modern man. In other words, the evidence strongly suggests that the ancient Greeks did, indeed, get their length measures from a source that was Egyptian, or both may have obtained them from some more ancient common source.
After he had recovered from these disturbing revelations, my friend wanted to know if I could state the evidence in a more formal manner, and if we could more clearly follow the definitions on which the ancient measures were based.
I replied that we must first forget the words we modern people assign to measurement lengths. Forget English inches and Greek feet and concentrate on the facts.
Some ancient mind arbitrarily defined units that provide a means of manipulating calculations regarding the earth. For CIRCULAR measure he set this to 360 degrees, 60 minutes, and 60 seconds. There would be 360 degrees, 21,600 minutes, and 1,296,000 seconds in one earth circumference.
That definition is sexagesimal, not decimal, nor any other number system. But we do not know sexagesimal mathematics from the ancient Egyptians. This fact naturally leads one to believe the definition must predate the Egyptian culture, as far back as Badarian days, circa 6,000 BC. If it had been brought into the evolving culture of Egypt from another culture after that date, it surely would have left some affect, since the Egyptians then would have included such important information into their social customs of calendar keeping and land measure.
We do know such sexagesimal system from the Babylonians. The Greeks borrowed that system, and it came down to us in western culture. Does this mean our hero lived in a geographic center that later gave the Babylonians the sexagesimal system, but left only the cubit measure in Egypt?
Since the Babylonian system of time keeping was based on similar principles we could say that in like manner our hero set units of time for human convenience at 24 hours, 60 minutes, and 60 seconds. This makes 15 surface degrees in each hour. In this manner time and distance are directly connected to one another. However, these do not concern us in this discussion; I do not wish to digress into the history of world length measure or time keeping. Therefore, I shall ignore them.
I retain the sexagesimal global definitions with words familiar to us because they do not influence our discussion of length measure.
If that ancient mind had a measure of earth surface distance he might have given it a length dimension. However, this length definition was inherently a measure of circular distance because it was tied to the surface of our mother sphere.
In order to remove this dependence on circular distance he used another definition. From the evidence demonstrated here we know he was cozily familiar with the mathematical rules for circular and spherical geometry. He decided to set the LINEAR measure of the earth by the radius, not the circumference. He defined the length of the radius of the earth by a practical unit. He set this at 250,000,000.
Note: If he had set the earth radius at 100 million units, or 500 million, and so on, he would have altered the usefulness of the length of the unit. More than likely he set this to a distance that would be convenient to remember, such as a “thumb’s breadth” or “finger’s breadth.” Later, as men forgot the original definitions, they assumed this measure derived from a convenient human body part while they continued to cling to the original unit length.
This linear length could be defined at a convenient size only if that ancient mind had a good working knowledge of the actual dimensions of the earth, within millionths of resolution.
Earlier we saw that the definition of the earth units depended upon a close familiarity with earth surface dimensions. Similar familiarity with the radius would naturally follow from that other knowledge.
Our hero then had two definitions with which to work: units of circular measure from the circumference, and units of linear measure from the radius.
The Greek foot is based on units of circular measure; the English foot is based on units of linear measure. (Again, the designation of “foot” is a convenient reference back to human body parts, but has no relationship to the original definition, except for the help of human memory.)
In the course of time, and forgotten past, these two measures became confused. In an effort to untangle them modern investigators then devised systems of ratios, 24/25, and so on, to relate them. As an example, refer to the work of Livio Stecchini in an appendix to the book by Peter Thomkins (20) and the web page at http://www.metrum.org/ (21).
Then a decision had to be made of which measure should be used for practical work, circular or linear.
We know from historic use (the Greeks as example) that one could devise length measures based on the circular definition. While such use would make no practical difference for small-scale man-made structures, it could make a major difference for large land measures.
(Before we assume that our historical information shows that the linear definition found its way into Egyptian land measures we must examine the secret behind the Egyptian Royal Cubit.)
From these definitions we can see that if linear measure were to be used it had to be based on the measure of the radius of the earth, and not circumferential length measure. We should be able to reach the logical deduction that:
All linear measures were originally based on the definition of the number of units in the radius of the earth.
All circular measures were originally based on the definition of the number of units in the circumference of the earth.
The linear measure (radial length) could be translated into circular measure (surface length) through the circular relationship of C = 2 Pi R.
If the dimension of 250 million is multiplied by 2 and again by Pi we obtain 1,507,796+ million nominal units of linear measure in the circumference of the earth. Dividing this linear measure by 1,296,000 arc seconds we obtain the length of the now familiar 1212.034+ nominal units in every arc second of the surface. For convenience of construction on a practical scale this can be divided into 100 parts. This is the nominal length of the Greek foot, a circular measure, expressed in the linear definition of units, and now assigned the designation of English inches.
Then someone, either the person who devised the original definitions, or someone else, decided to incorporate both the circular and linear definitions into a new practical measure. In doing so he accomplished two objectives:
a. he developed a new standard measure, and
b. he forced attention to the fact that this new measure was consciously designed around the size of our mother earth.
It was not derived from human body parts.
My use of the singular third person is not accidental. I do not accept that these profound definitions were derived from a committee, nor do I believe that they gradually evolved over cultural time. They demonstrate a conscious decision to define the earth by a person who knew the dimensions, and then decided on how future men should communicate with one another about such important matters. A definite superior intelligence was the source. But it was a human intelligence, or an intelligence that could relate on human terms.
This conclusion can be illustrated by the length of the Royal Egyptian Cubit. A conscious choice made the numerical value of the Royal Cubit in defined earth linear units (20.62+) equal to the numerical value of arc seconds in the defined circular units radius of the earth (206,264+). This was not an unconscious, random, nor accidental happenstance. If he set out to make these two numerical values equal he could easily decide the amount to add to the 1/100 arc second unit to achieve that equality. It turned out to be 1.7, which, by itself, might be considered fortuitous.
But it was not fortuitous.
We can follow his logic.
We know that the value of the Royal Cubit of 20.625 inches is numerically equal to the number of arc seconds in the radius of the earth, 206,265.81+ divided by 10,000. This is the condition we are searching for.
We shall formalize the expressions.
We know the Royal Cubit is composed of
A: (one arc second/100) plus (Y arc seconds/100), or (1 + Y) X (arc seconds/100).
Using the linear definition of an arc second we know that this is equal to
B: (R X 2 Pi) divided by 1,296,000 where R is the linear radius of the earth.
We can substitute into expression A the linear definition of the arc sec we determined in expression B. We then reach a new expression
C: (1 + Y) X