The Keys of Atlantis
A Study of Ancient Unified Numerical and Metrological Systems
by Peter Wakefield Sault
Copyright © Peter Wakefield Sault 1973-2008
All rights reserved worldwide
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The Keys of Atlantis
Copyright © Peter Wakefield Sault 1973-2008
All rights reserved worldwide
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|
|
The Parthenon was constructed between 447 and 438 BC, the embellishments being completed in 432 BC, to replace the Old Temple of Athena known as the Hekatompedon, destroyed by the Persians during the sack of Athens in 480 BC. Initiated by Perikles, the overall controller of the work was Pheidias, the most celebrated artist and sculptor of his time, the architect was Iktinos and the mason was Kallikrates. Unlike the usual Greek temple, roofed with terra cotta tiles supported by timbers, the Parthenon was built almost exclusively of Pentelic marble - twenty-two thousand tons of it - the greatest single expense of the construction being its transportation from Mount Pentelicos, some ten miles from Athens.
The temple was built in Doric order according to a peripteral scheme, the building being surrounded by forty-six columns in an eight by seventeen arrangement (counting the corner columns twice - see Figure 2-2). The central enclosure, or kella was divided into two non-communicating rooms. The eastern, or naos (‘temple’), was dedicated to Athena Polias and sheltered Pheidias' statue of Athena, and the western, or opisthodomos (‘backroom’), to Athena Parthenos - the Maiden - whence the building gets its name. The backroom had reinforced doors and was used as a strongroom, housing a treasury. Its construction was one of the few exceptions to the Doric order, the ceiling being supported by four Ionic columns.
The Parthenon became a Christian church around 600 AD and a mosque was built in the kella in 1458 after Athens was occupied by the Turks. It was in good condition until September 26th 1687 when it was bombarded by a Venetian fleet under Admiral Morosini during a siege of the Acropolis, Athens being at that time still under Turkish occupation. Impacts from Venetian cannonballs can be seen in the surviving masonry. The Turks had been using the temple as a gunpowder magazine and the whole lot went up, destroying the roof and interior walls and bringing down the friezes.
During the 18th century, the broken masonry became the victim of stone-robbers, lime-burners and the weather. In 1801 Thomas Bruce, the seventh Earl of Elgin and British ambassador to the Ottoman Empire, purchased a collection of sculptures from the Turks and shipped them to England. Parliament bought the collection from Bruce in 1816 and the ‘Elgin Marbles’ are nowadays stored in the British Museum at London. Today there is a growing international movement for the complete restoration of the Parthenon. The people of Greece and many sympathizers worldwide have been campaigning since the 1970s for the return of all the removed Parthenon Marbles*.
* Needless to say these are
sentiments with which this author agrees wholeheartedly.
It is the very least that modern Britain could do to acknowledge its immeasurable cultural and
historical debt to the City of Athens, without which there would be no British Museum.
Unless otherwise qualified the terms foot, cubit and stadion will refer here specifically to the units exhibited in the physical dimensions of the Parthenon. These units, although internally consistent, correspond in length neither to English units of the same name nor to the Doric units in day-to-day use by the general population of Periklean Athens1. However, like the Doric, the overall system is Solonian as laid out below in Table 2-1.
| Name of Linear Unit | Equivalent Length | ||
|---|---|---|---|
| English | Greek | ||
| Digit | DAKTULOS (pl. DAKTULOI) | daktulos, daktuloi | – |
| Foot | POUS (pl. PODES) | pous, podes | 16 digits |
| Cubit | PHCUS | pekhus | 24 digits |
| Plethron | PLEQRON (pl. PLEQRA) | plethron, plethra | 100 feet |
| Stadion | STADION | stadion | 600 feet |
Consideration will here be restricted to the top step, or stylobate, of the platform of the Parthenon. It will be assumed that the intended length and width were 100 and 225 feet respectively, in keeping with existing widely held opinion that the intended ratio of the short to the long sides was 4:9. This is the signature ratio of the building and is expressed in various ways throughout the structure. For example, the ratio of the height of the façade, without pediment, to the width of the stylobate2. The Parthenon was called the Hekatompedon Neos, or ‘new hundred-footer’, although a true Hekatompedon would have been 100 feet square. The number 100, the square of ten which is the tetraktys of the Pythagoreans, seems to have held some special, possibly sacred, significance to the Greeks.
Livio Stecchini tells us3 that the Smith Tablet employs a great cubit equal to 1½ common cubits, where the latter is equal to 1½ feet. Although this is a Babylonian document it reveals nonetheless the practice of deriving an additional cubit that bears the same relation to the common cubit as the latter does to the foot; a practice of which the builders of the Parthenon were undoubtedly aware. Hence the length of the stylobate can be expressed as 100 great cubits.
In 1846 the stylobate was measured by Francis Cranmer Penrose (see Table 2-2 below), whose meticulous precision and rigorous adherence to scientific principles place him in general high regard. Penrose thought the measurement of the better preserved east side to be the more reliable indication of the width of the stylobate, and after compensation for separation of the blocks he gave an adjusted figure of 101.335 English feet for that side5. Livio Stecchini,6 however, observes that Penrose did not take into account the weathering of the blocks over two and a half millenia and that more recent measurements give 30.889 metres (101.342 English feet) for the length of the east side. The fact that the measurements of both long sides give longer derived stadia than those of the short sides cannot be ignored. The addition of about a half-inch all round the stylobate (one inch, or 25mm, to each horizontal dimension) is needed to bring the ratio of the sides to exactly 4:9. Certain allowances for error, on the parts of both the constructors and the surveyor, must be made.
| Side | Length | Derived Stadion (metres) | |
|---|---|---|---|
| English feet |
metres | ||
| East | 101.341 | 30.8887 | 185.3322 |
| West | 101.361 | 30.8948 | 185.3688 |
| North | 228.141 | 69.5373 | 185.4328 |
| South | 228.154 | 69.5413 | 185.4435 |
| Means* | – | – | 185.3943 |
| Median | – | – | 185.4008 |
| *Arithmetic, geometric and harmonic. | |||
The range of the derived stadia is 0.1113m (4.3819 English inches), or 0.06% of the mean stadion. The best that can be done is to take the arithmetic mean, which equals both the geometric and harmonic means to eight significant digits, hence the derived stadion will be made equal to 185.3943 metres and some very small allowances will be made for the inevitable inexactitudes that will be encountered. As every engineer knows, there is no such thing as a perfect fit.
Viewed from above the stylobate is a simple geometric figure, a rectangle of theoretical proportions 4:9, or 2²:3² (see Figure 2-3 below). This proportion represents the minimum tiling, or regular tessellated division, of the rectangle of the stylobate. Each ‘tile’ is 25 feet along each side and has, therefore, an area of 625 square feet and a perimeter 100 feet - a plethron exactly - in length.
Twenty-five, the square of 5, has the peculiar property of being the sum of the preceding two squares, those of 3 and 4, and is the square on the hypotenuse of the {3-4-5} Pythagorean right-angled triangle, six of which can be constructed on the horizontal surface of the stylobate. The whole area of the stylobate comprises 36 such tiles, amounting to 22,500 square feet, equal to a myriad square cubits. This set of tiles can be rearranged to make a 6 × 6 square, as shown in Figure 2-4 below, 150 feet, or 100 cubits, along each side (the ratio of the foot to the cubit being 2:3). Thus do we obtain an invisible 100 × 100 square base - the ideal base, the foursquare counterpart of the real stylobate, the two having a 1:1 relationship through their equal areas. The ratio of the short axis of the stylobate to a side of the square is equal to the ratio of a side of the square to the long axis of the stylobate, each ratio being 2:3. This forms the perfect proportion 4:6::6:9 - a side of the square being the geometric mean of the dimensions of the stylobate and comprising the ‘doubly-perfect’ number six, so called because it is both the sum and the product of its aliquot parts. The perimeter of the square is 600 feet - a stadion exactly - in length.
The ratio of the respective perimeters of the equivalent square and the stylobate is 12:13, these numbers forming two sides of the {5-12-13} Pythagorean right-angled triangle and they sum to 25, the square of 5. This particular triangle has the unique property that its perimeter is numerically equal to its area. The ratio of the area of a true Hekatompedon, being a myriad (10,000) square feet, to that of either the stylobate or its equivalent square is 4:9.
Regarding the significance of equal areas in sacred monuments, F. Bligh Bond says7
“There is evidence of the highest antiquity for the practice of obtaining equal areas with diverse proportionals. It is found in the ancient Indian ‘Shilpashastras’, or rules of religious art; and Professor Petrie notes such a custom as controlling methods of the Egyptian builders.”
The Shilpashastras to which Bligh Bond refers are also known as the Sulbasutras. In the Baudhayana Sulbasutra (circa 800BC, named for its author) we find a geometrical method for constructing a square of area equal to that of a given rectangle. Baudhayana's main concern was the construction of sacrificial altars. In Figure 2-5 below the construction is shown for a rectangle, ABCD, of similar proportions to the stylobate of the Parthenon. The steps for constructing the square, DNQR, of equal area are shown in the key to this figure.
| Key to Figure 2-5 | ||
|---|---|---|
| 1. | Draw line EF perpendicular to AD such that DE = CD, making CDEF a square. | |
| 2. | Bisect AE at H and draw line HJ perpendicular to AD, produced to K such that JK = BJ. | |
| 3. | Draw line KL parallel to BC, intersecting line CD produced to L. | |
| 4. | Describe arc KM, centre L, intersecting line BC at M. | |
| 5. | Draw the perpendicular to BC at M, intersecting AD at N and KL at P. | |
| 6. | Draw line QR parallel to BC such that NQ = DN, making DNQR a square. | |
Baudhayana gives no proof of the veracity of the above construction but it is, however, easily demonstrated by the application of Pythagoras' theorem:-
| I. | LP2 | = | LM2 – MP2 | (Pythagoras) | |||
| but | LM | = | KL | ||||
| and | MP | = | JK | ||||
| II. | \ | LP2 | = | KL2 – JK2 | |||
| = | |||||||
| but | LP2 | = | DNQR | ||||
| and | CFGL | = | ABJH | ||||
| III. | \ | ABCD | = | DNQR | Q.E.D. |
Interestingly, in the case of the proportions of the stylobate of the Parthenon, LMP (congruent with CLM) is a Pythagorean {5-12-13} triangle.
Dinsmoor is rightly dismissive of certain types of geometrical speculation, saying1.
“It seems necessary here to insert a word of warning against the validity of the numerous modern attempts to derive the plans of Greek temples, and of the Parthenon in particular, from more or less intricate geometrical diagrams such as interrelated concentric circles and squares, pentagons or pentagrams, hexagons or hexagrams, octagons, decagons, ‘whirling squares’, or the ‘golden section’.”
However, this does not mean that all geometrical analysis is invalid. Stecchini has the following to say on the matter8.
“...since Ictinos wrote a book on the proportions of the Parthenon, it follows that these proportions constituted a system that was per se of intellectual significance and appeal.”
In other words, while Dinsmoor is largely correct there is still plenty of scope for exploration and discovery and it is highly likely that at least a part of that will be geometric.
Insofar as both the actual stylobate and its square of equal area can be geometrically constructed within the same circle, it is possible that the horizontal area of the stylobate was ceremonially laid out in a sacred geometric construction. There is neither record nor any substantial evidence to indicate that this was the case, let alone that such a construction, if used, was the one shown in Figure 2-7. However, the present absence of original architect's plans does not mean that they were never drawn up. It would be virtually impossible to raise such a complex structure without working from detailed drawings. Moreover, in view of the nature of Hellenic thought, it would seem likely that such a construction was at the very least employed in the production of drawings of the groundplan. What is important here is that the layouts can be constructed geometrically and not whether that of the Parthenon actually was, although it will be shown that such a construction could indeed have been used in practice upon the Acropolis. None of the dimensions shown in Table 2-2 would need to have been measured out by the builders and, moreover, this may account for the discrepancy between the length and the width. A measured half-stadion line starts off the geometric figure. The construction then proceeds with arcs of one stadion diameter circles, struck from each end of the line. The word stadion can be taken to mean ‘standard’ and this is exactly what it will be seen to have been - the standard, in fact. The square of equal area will be dealt with first as it is the simpler of the two (see Figure 2-6 for the square and Figure 2-7 for the stylobate).
| Key to Figure 2-6 | ||
|---|---|---|
| 1. | Draw line AB, ½ stadion in length. | |
| 2. | Describe arc CBD, centre A. | |
| 3. | Describe arc CAD, centre B. | |
| 4. | Draw line CD, the perpendicular bisector of AB at O. | |
| 5. | Describe circle AEBF from centre O. | |
| 6. | Describe arc HOJ, centre A. | |
| 7. | Describe arc KOL, centre B. | |
| 8. | Describe arc MON, centre E. | |
| 9. | Describe arc POR, centre F. | |
| 10. | Draw line HJ, the perpendicular bisector of OA. | |
| 11. | Draw line KL, the perpendicular bisector of OB. | |
| 12. | Draw line MN, the perpendicular bisector of OE. | |
| 13. | Draw line PR, the perpendicular bisector of OF. | |
| Key to Figure 2-7 | ||
|---|---|---|
| 1. | Draw line AB, ½ stadion in length. | |
| 2. | Describe arc CBD from centre A. It must be noted that point D could not have been realized in practice as it is located some 50yds beyond the Acropolis wall (see Figure 2-8). Other arcs with shorter radii would have had to have been struck in order to complete the bisection of line AB. Alternatively, Step 2 could be displaced by moving the description of circle ATBV from Step 5. | |
| 3. | Describe arc CAD, centre B. | |
| 4. | Draw line CD, the perpendicular bisector of AB at O. | |
| 5. | Describe circle ATBV, centre O. | |
| 6. | Describe arc EOF, centre A. | |
| 7. | Describe arc HOJ, centre B. | |
| 8. | Draw line EF, the perpendicular bisector of OA at K. | |
| 9. | Draw line HJ, the perpendicular bisector of OB at L. | |
| 10. | Describe arc MKN, centre A. | |
| 11. | Describe arc MAN, centre K. | |
| 12. | Describe arc PLR, centre B. | |
| 13. | Describe arc PBR, centre L. | |
| 14. | Draw line MN, the perpendicular bisector of AK at S. | |
| 15. | Draw line PR, the perpendicular bisector of BL at U. | |
| 16. | Draw line ST, intersecting line EF at W. Triangle OST is a {3-4-5} Pythagorean. | |
| 17. | Draw line SV, intersecting line EF at Z. | |
| 18. | Draw line TU, intersecting line HJ at X. | |
| 19. | Draw line UV, intersecting line HJ at Y. | |
| 20. | Draw line WX, produced to meet lines MN and PR at a and b respectively. | |
| 21. | Draw line YZ, produced to meet lines MN and PR at d and c respectively. | |
In Figure 2-8, the largest arcs belong to circles whose diameters are exactly a stadion in length. The shape enclosed by these arcs forms a sacred figure known as a Vesica Piscis (‘fish bladder’), or Mandorla (‘almond’), an ancient yonic symbol of the fertility of the Goddess and the figure of Euclid's first proposition. The overlapping of the two circles represents the union of spirit and matter, male and female, Sun and Moon.
In Figure 2-9, a 4 × 9 rectangle, EFGH, is raised above a 6 × 6 square, JKLM, such that their edges and planes are parallel and they share a common normal, PQ, through their centres. This forms an irregular hexahedron, or cuboid, with four trapezoidal faces, e.g. EHMJ, in a ring, each congruent with its opposite, connecting the rectangle to the square and making it a trapeza (‘table’), standing foursquare to the ground. When the planes of these faces are produced to their intersections, a demisemi-regular tetrahedron, ABCD, results. This tetrahedron is not quite semi-regular because not all four faces are congruent. Rather, there are two interlocking pairs of congruent isosceles triangles, hence the resulting tetrahedron is described as demisemi-regular. This process is reversible and the trapeza can be created from the tetrahedron by truncating those edges, AB and CD, which connect each pair of congruent faces.
The configuration described above has some interesting properties. Within the trapeza, if the elevation of the rectangle above the square is a whole number of units, where those units divide both the dimensions of the rectangle and the side of the square a whole number of times, then the volume of the trapeza is also a whole number. This property extends to the axeheads comprising the truncated edges of the tetrahedron and thereby to the whole tetrahedron. Whatever divisions of the sides of the rectangle and square are employed, the volume of the trapeza is always a multiple of the whole number 38. So, with a unit derived from the minimum number of divisions, i.e. 1/6th of the side of the square, a unit elevation gives the trapeza a volume of 38. Each unit increment in elevation increases the volume by 38.
The elevation of each axehead is always twice that of the trapeza, hence the elevation of the tetrahedron is always five times that of the trapeza. The length of the extreme edge of each axehead is always the sum of the lengths of the two parallel sides of those opposing trapezoid faces of the trapeza whose non-parallel sides meet at that extreme edge. Thus the length of the upper edge of the tetrahedron is 15 (= 6 + 9) and that of the lower 10 (= 6 + 4) and, since they are in constant proportion, the ratio of their lengths is always 2:3. None of the remaining edges of the tetrahedron has a whole number length when the elevation of the trapeza is a whole number.
For a unit elevation of the trapeza, the volume of each axehead is 44. Altogether, therefore, for (and only for) the same elevation, the tetrahedron has a volume of 126. All these volumes rise in direct proportion to the elevation of the trapeza.
Our imaginary trapeza is at once a pedestal for the Parthenon and the altar upon which the temple is consecrated, therefore symbolizing, or perhaps idealizing, the Athenian Acropolis. Since the elevation of the Acropolis above the Plain of Attica is a half stadion, or 300 feet, the corresponding elevation of the trapeza would be 12 (units of 25 feet), making the elevation of the tetrahedron 60 (= 1500 feet). Hence the volume of the trapeza is 456 (= 7,125,000 cubic feet). Each of the corresponding axeheads has a volume of 528, making the volume of the tetrahedron 1,512 or, as it can be written in the Greek alphabetic numeral system, APOKALUYIS – a revelation. This is an example of gematria, or isopsephia as it is also known. Although the word ‘apokalypse’ belongs to a different era and a different mystery cult it reveals a common use of the same numeric value. The number 1,512 belongs to the series of multiples of the number 216, the cube of six (6 × 6 × 6), which is the value by Hebrew gematria of the Dbir, or ‘Holy of Holies’4. Subtracting 216 from 1,512 gives us the number 1,296, the square of 36 and, for example, the number of square English inches in a square English yard. Adding 216 to 1,512 gives us 1,728 which is the cube of 12 and the value by gematria of TO QUSIASTHRION – the altar4. The mystical symbolism of the cube and the literary use of alphabetic numeral systems are explored in detail in Chapter 4.
Now, 1,512 of our cubic cells, each of whose edges is 25 feet long, are equal to 23,625,000 cubic feet, or 7,000,000 cubic cubits exactly. Dividing through by 7, we find that 216 cubic cells have a combined volume of 3,375,000 cubic feet or 1,000,000 cubic cubits. Hence we can equate the volume of the tetrahedron to the sum of the volumes of seven identical cubes, each of whose edges is 100 cubits (150 feet) long, making the area of each and every face equal to that of the stylobate; a square of equal area.
Each of the 7 cubes described above has 6 faces of area 36 (that is to say, 36 square tiles of 25 feet along each side) giving it a surface area of 6 × 36 = 216. It is of necessity a unique property of the cube of edge 6 that its surface area is numerically equal to its volume, 6 being the number of faces on a cube. Since there are 7 cubes the total surface area is 7 × 216 = 1,512.
A method for calculating the volumes of the trapeza and axeheads is given in Appendix F.
As already mentioned in Section 2-3, the ratio of the height of the façade, without pediment, to the width of the stylobate is 4:9, matching the proportions of the stylobate. This implies a rectangular block whose proportions are 16:36:81, representing the minimum three-dimensional tessellation into cubic cells. Each edge of the cells is therefore 2 7/9 feet long. This takes into three dimensions the process applied previously in two dimensions to the stylobate. The volume of the block in terms of such cells is 16 × 36 × 81, giving 46,656 or the sixth power of six (i.e. six sixes multiplied together). Hence the cells can be rearranged as either the square of 216 or the cube of 36. With respect to the former each side is 600 feet long, making the area a square stadion exactly. With respect to the latter, the cube of equal volume, since each edge is equal in length to the width of the stylobate – 100 feet, or 1 plethron, giving it a volume of 1,000,000 cubic feet, or 1 cubic plethron, exactly – each of its six faces is a true Hekatompedon. Hence we have the image of the sacred space, a plethron cube, standing at the centre of the sacred area, or precinct, a stadion square.
In terms of the square tiles with 25 foot sides (whose perimeters are, of course, exactly one plethron in length) and cubic cells with 25 foot edges, the square stadion is equal to 576 tiles and the cube of equal volume is equal to 64 cells.
* First observed by Mr. Chris Graves during the course of our
correspondence on the subject of cubes and mentioned in his wide-ranging study
of the role of the cube in ancient architecture,
Citadel of the Gods.
F.C.Penrose, in his survey of the Parthenon in 1846-1847, found the magnetic compass bearing of the long axis of the Parthenon to be 2½° N. of E., and gives the magnetic declination at that time as W.11½°, saying5
“...in determining their [the principal buildings'] actual bearings, the variation of the compass was assumed to be 11 degrees, 30 minutes.”
Although it was possible in those days to measure magnetic declination to within a few arcminutes, Penrose did not actually measure it for himself, estimating the value from a chart, thus reducing the accuracy of the figures he gives. He seems to be operating to a precision no better than the nearest ½° in respect of both the bearing and the magnetic declination. Hence his value for the azimuth must be stated as 14° N. of E., ±1°.
Now, Penrose himself came under a lot of stick from the archeological community for his level of precision. Not, as you might expect, that he was not precise enough in his measurements. Far from it, and hardly credible though it is, he was actually criticized for being too precise. Stecchini sums up this rank attitude very nicely9
“There are many problems in the architecture of the Parthenon that cannot be solved, because archeologists prefer to go on building fanciful theories rather than establish the facts by an accurate survey. It is a basic principle of epistemology that our ability to reject erroneous theories increases in proportion with the precision and accuracy of the measurements; the converse is true, and this is what archeologists like, because, as they put it, it permits the spirit to soar.”
According to Penrose's friend J.N.Lockyer, the Parthenon is oriented towards the rising of the Pleiades (M45 in Taurus). Lockyer says10
“I was fortunate enough to find that he [Penrose] had already determined the orientation of the Parthenon with sufficient accuracy to enable him to agree in my conclusion that that temple had been directed to the rising of the Pleiades. He has subsequently taken up the whole subject with regard to Greece in a most admirable and complete way and has communicated papers to the Society of Antiquaries (February 18, 1892), and more recently to the Royal Society (April 27, 1893) on his results.”
This claim was based on tables, created by the German Astronomical Society, of the amplitudes of various stars going back to 2000 BC. Lockyer does not mention a figure but, however, in 447 BC at Athens the Pleiades rose at an azimuth of 18° N. of E. (altitude 0°). This has been separately indicated by two different astronomical computer programs11,12. Lockyer, however, is saying that the Pleiades rose at the azimuth measured by Penrose, 14° N. of E. Perhaps he is making some unspoken allowance for the mountains to the east of Athens, as a result of which the Pleiades would not be visible at altitude 0° and might not become so until they reached an altitude of 5°. As the Pleiades rise, they also move southwards, their path across the sky making an apparent angle of very roughly 45° to the horizon. Hence the azimuth of first sighting could indeed have been close to 14° N. of E. Until a reputable survey establishes the exact azimuth of the Parthenon and the elevation of the mountaintops in that direction from the Acropolis there is little anyone can do except speculate about precisely which, if any, particular star or star cluster it might have been. If an alignment to the rising of the Pleiades can be reasonably demonstrated then it could equally have been to that of the red giant star of the archangel Michael, ‘The Watcher in The East’, Aldebaran, The Follower (so called because it appears to follow the Pleiades across the sky), called Torkh, The Guiding Light, by the Greeks. Aldebaran is the eye, or head, of Taurus, The Bull, according to Greek mythology the heavenly manifestation of Zeus, from whose brow Athena sprang fully clad in armour when, at dawn, He was struck on the head with an axe by Hephaistos.
Stecchini describes Penrose's efforts to discover the ‘Parthenon Star’ thus13
“Penrose assumed that the orientation of the Parthenon was established by the point of the heliacal rising of some heavenly body on a day particularly sacred to Athena. In his numerous publications on the subject, Penrose could not arrive at any definite conclusion, because he considered too wide a range of possibilities – the heliacal rising of a star or planet, and even the point of the rising of the sun on given dates.”
The reader may, however, set all such fanciful notions aside, for there is a simpler, better and more forceful explanation. Unlike the Egyptian temples described by Lockyer, which had to be rebuilt periodically due to the continual shifting of a star's ascendence resulting from the Earth's various wobbles, the Parthenon will remain almost perfectly aligned to its correct and intended orientation until the day the Earth is swallowed by the Sun.
In order to understand how this can be, the reader must first become familiar with great circles of the Earth. These are the largest circles that can be described around the Earth, hence their planes are always coincident with its centre. However, such circles need not coincide with the Equator nor with any meridian but can be at a tilt to both, their respective planes making dihedral angles. In Figure 2-10 below, just such a circle is shown coloured blue and passing through points V and W. Its plane makes the dihedral angle q (i.e. ÐEOV) with that of the Equator, shown coloured red and passing through points E and W. This angle is equal to the vertex latitude.
The vertices of any great circle are its northernmost and southernmost points. In great circle navigation of the globe, the vertex is the point at which the north/south compass bearing reverses. For example, in sailing eastward along great circle VW (see Figure 2-10), from point W the bearing is decreasingly northward until the vertex at point V is reached, after which the bearing becomes increasingly southward until the Equator is crossed again diametrically opposite point W. The bearing is at its most northward or southward while crossing the Equator.
In Figure 2-11 below, the Parthenon is treated as though it were rotatable around its central vertical axis, from a position where the naos would face due east, through 90° so as to face due north. As it is rotated from the the former to the latter position, the vertex of the resulting great circle aligned with the building's principal axis (i.e. the ridge of the roof) travels northward from the latitude of the Parthenon to the North Pole (Axis of Rotation) respectively.
As can be seen, the vertex of the great circle resulting from the actual orientation, the Parthenon Great Circle, azimuth of 14° N of E at 37.97°N, is to all intents and purposes on the 40th parallel. The 40th parallel is 4/9ths of the way along the meridian quadrant. Four and nine comprise the signature ratio of the building.
So there we have it; the vertex latitude, itself determined by the signature ratio 4:9, of the Parthenon Great Circle determines the orientation of the building. We can now address the matter of the circumference of the Parthenon Great Circle which, as will be seen, determines the size of the building.
It must be understood that the stadion being discussed here
is specifically that exhibited in the dimensions of the Parthenon.
There appear to have been other stadia of different lengths in use
elsewhere in Greece and even within Athens itself. About the Attic stadion
A.E.Berriman says14
“It is only in the light of the metric system that the
first and second of these coincidences are visible; they could not
have been discovered in antiquity and there is no record of any
traditional knowledge of such an origin. Certainly the Greeks were
unaware that the circumference of the Earth measures 216000 Greek stades.
“The Greek stade measures 600 Greek ft., and the best evidence for the
length of the Greek foot is the platform of the Parthenon, which is
100 Greek ft. in width by 225 Greek ft. in length. It was measured in
1750 by Stuart and a century later by Penrose: the mean of their
measurements gives a Greek foot that corresponds in length to
one-hundredth of the sexagesimal arcsecond on any great circle
of the Earth regarded as a sphere.*
The first thing to note is that the Earth is not a true sphere
(see Appendix D), its extreme
circumferences varying by some 67km, or roughly 362 stadia.
The closest regular approximation to its true form is the oblate
spheroid (or ‘flying saucer’). With the exception of the
equatorial great circle, all other great circles are in fact ellipses
which more correctly have perimeters and not circumferences.
However, for historical reasons (i.e. they didn't know any better in the 16th Century AD)
navigators conventionally refer to these ellipses as great circles and
to their perimeters as circumferences and that great tradition will be recycled here.
Nonetheless, it should be noted the reference ellipsoid, upon which all our
calculations are based, is still not an exact model of the geoid, which has
a few odd dips and bulges here and there, as revealed by the deviations of artificial
satellites from their previously calculated orbits. Once again, we do the best we can,
this time accepting the best mathematical approximation that we have to the actual reality.
In the early stages of our present culture Sir Isaac Newton was the first
to question the general assumption that preceded him, that the Earth is a sphere.
Stecchini says15 that the ancient Egyptians,
however, were aware that degrees of latitude get longer towards the Poles and
“The Egyptians calculated that the polar flattening is 1:298.6.”
Stecchini further tells us that “a degree by definition is 600 stadia” although,
strictly speaking, this applies only around a true circle.
A stadion is a unit of length, which must remain constant within a given context,
here an elliptically eccentric meridian, whereas a degree of angle is a dimensionless unit of rotation.
The two coincide all the way around, and only around, the Equator because of its essential circularity.
Elsewhere, and most markedly around the meridians, the two must necessarily diverge.
Furthermore, the only way that anyone can compare the curvilinear distance
between degrees of latitude on the surface of the Earth is with units of length which
remain constant along the length of the curve, that being the context of the comparison.
Since the geoid is not spherical, a practical stadion must be a 1/216,000th
part of a particular great circle. In figure 2-12 below, the full range of possible terrestrial stadia is laid out.
Identified on this graph are the stadia derived from the lengths of the sides of the stylobate.
“* The geodetic appearance of the Greek linear scale was
noticed by Jomard in 1812 and by Watson in 1915; see XV(1). Hitherto
it has seemed an isolated coincidence.”
Secondly, Berriman's assumption that the Greeks were unaware of the lengths of the Earth's circumferences cannot be allowed. The only valid assumption that can be made is that they apparently did not generally possess the means to accurately measure the circumferences, as revealed by Eratosthenes' figure of 250,000 stadia, using the method of parallax at Alexandria, for the polar circumference. Some authors believe that the stadion used by Eratosthenes differed from the Parthenon stadion, being that of the length of the stadium at Athens which measures 184.98 metres from start line to finish line. Since the circumference of the Earth stated in such stadia ranges from 216,493 (polar) to 216,856 (equatorial), this stadion cannot be geodetic in the manner described by Berriman. It makes little difference here which of the two Eratosthenes used as his error remains at roughly 15%. It was perhaps the case that the precise circumferences, or means of accurately measuring them, were known only to a closed circle of Greeks but, however, it will be shown that they must have been available to the builders of the Parthenon, along with some knowledge of the true shape of the Earth.
Now, let us examine the number 216,000. It is the cube of 60, which immediately identifies it as belonging to the Babylonian dual-radix sexagesimal counting system. Using Arabic numerals it is written 1|00|00|00 in that system – it is a sexagesimal ‘thousand’ hence is as rational a division of anything as is our contemporary metre in being a thousandth part of a kilometre using the denary counting system. It is also equal to 360 × 600, which is to say that each degree of arc on the circumference of a great circle is 600 stadia in length. It has already been demonstrated that the degree of arc is no arbitrary division (see Section 6 of Chapter 1). The stadion is itself further subdivided into 600 feet. It is surely no accident that a stadion should be divided into as many parts as it itself divides a degree of arc on a great circle. In the light of this it must be questioned whether it is mere coincidence that the difference between the polar and equatorial circumferences of the Earth is very nearly one six-hundredth of the latter, at 1/597. Each arcminute on a (theoretically circular) great circle is ten stadia, or 6,000 feet, in length and, therefore, each arcsecond is 100 feet long or exactly the intended width of the stylobate of the Parthenon. This distance is also known as a plethron.
Great circle circumferences of the Earth range from 40,007,857 metres around the poles to 40,075,010 metres around the equator. Table 2-3 below shows circumferences of the Earth in both metres and Parthenon stadia, using the conversion factor of 185.3943 metres per stadion obtained earlier (see Table 2-2), from the equatorial (0°) to the polar (90°) vertices in latitudinal steps of 5°. The method used to approximate these circumferences is given in Appendix D.
| Vertex Latitude (q) |
Circumference | Vertex Latitude (q) |
Circumference | |||
|---|---|---|---|---|---|---|
| Metres | P. Stadia | Metres | P. Stadia | |||
| 0° | 40,075,010 | 216,160.96 | 45° | 40,041,342 | 215,979.36 | |
| 5° | 40,074,498 | 216,158.20 | 50° | 40,035,514 | 215,947.92 | |
| 10° | 40,072,975 | 216,149.98 | 55° | 40,029,869 | 215,917.47 | |
| 15° | 40,070,489 | 216,136.58 | 60° | 40,024,576 | 215,888.93 | |
| 20° | 40,067,117 | 216,118.39 | 65° | 40,019,797 | 215,863.15 | |
| 25° | 40,062,962 | 216,095.98 | 70° | 40,015,674 | 215,840.91 | |
| 30° | 40,058,153 | 216,070.04 | 75° | 40,012,332 | 215,822.88 | |
| 35° | 40,052,837 | 216,041.36 | 80° | 40,009,871 | 215,809.61 | |
| 40° | 40,047,175 | 216,010.82 | 85° | 40,008,364 | 215,801.48 | |
| 41.73° | 40,045,167 | 215,999.99 | 90° | 40,007,857 | 215,798.74 | |
As shown in Table 2-3 above, the circumference of the Earth is precisely 216,000 Parthenon stadia at 41.73°. This is too close by far to the vertex latitude of the Parthenon Great Circle to be considered a product of chance.
Let us look again at the orientation of the Parthenon. In Figure 2-13 below, the Parthenon Great Circle PV (shown in blue) is aligned with the long axis, or ridge of the roof.
Key to Figure 2-13
| N | North Pole (Axis of Rotation) | 90° N. |
| O | Centre of the Earth | – |
| P | Parthenon | 37.97° N. × 23.725° E. |
| V | Northern Vertex of the Parthenon Great Circle | To be found |
Let:–
| A | Azimuth (NPV) of long axis of Parthenon (P) from True North (N) | 76° E. of N. |
| B | Angle (NVP) between the Parthenon Great Circle (PV) and the meridian (NV) through its vertices | 90° |
| a | Complementary angle (NOV) of latitude of northern Parthenon Great Circle vertex (V) | To be found |
| b | Complementary angle (NOP) of latitude of Parthenon (P) | 52.03° |
| According to the Law of Sines (Spherical Analogue)16, |
sin A / sin a | = | sin B / sin b |
| Therefore . . . . . . . . . . . . . . . . . | sin a | = | sin A × sin b / sin B |
| For an azimuth of 14° N. of E., | A | = | 90° - 14° |
| = | 76° E. of N. | ||
| Therefore . . . . . . . . . . . . . . . . . | sin a | = | sin 76° × sin 52.03° / sin 90° |
| = | 0.9703 × 0.7883 / 1.0000 | ||
| = | 0.7649 | ||
| Therefore . . . . . . . . . . . . . . . . . | a | = | 49.8995° |
| Therefore . . . . . . . . . . . . . . . . . | V | = | 90° - 49.8995° |
| = | 40.1005° N. |
Circumferences in Parthenon stadia are shown below in Table 2-4 for azimuths between 12° N. of E. and 16° N. of E. in increments of ½°.
| Parthenon Azimuth (° N. of E.) |
Great Circle | |
|---|---|---|
| Vertex Latitude (° N.) |
Circumference (P. stadia) | |
| 12.0 | 39.55 | 216,013.62 |
| 12.5 | 39.68 | 216,012.81 |
| 13.0 | 39.81 | 216,012.01 |
| 13.5 | 39.95 | 216,011.13 |
| 13.66 | 40.00 | 216,010.82 |
| 14.0 | 40.10 | 216,010.20 |
| 14.5 | 40.25 | 216,009.26 |
| 15.0 | 40.41 | 216,008.26 |
| 15.5 | 40.57 | 216,007.26 |
| 16.0 | 40.73 | 216,006.26 |
From Table 2-4 above it can be seen that the great circle PV in Figure 2-13 has a circumference between 216,014 and 216,006 stadia for azimuths of the Parthenon between 12° and 16° N. of E. respectively. It has already been shown that a stadion of 185.3943 metres is exactly 1/216,000th part of the circumference of a great circle with vertices at latitudes 41.73° N. and S., corresponding to a Parthenon azimuth of 18.8° N. of E.. The 40th parallel great circle circumference (i.e. the Parthenon Great Circle) shown in Table 2-4 above (216,011 stadia) varies from this by 0.005%, or a quarter of the variation between the intended and actual Napoleonic metres, that being 0.02% (the intended metre being a ten-millionth part of the meridian quadrant through Paris).
If it is allowed, for the sake of argument, that the long axis of the Parthenon was intended to be aligned with a great circle of the Earth, the Parthenon Great Circle, whose circumference is a multiple of both the length and the width of the building then it follows that the overlapping stadion-diameter circles which form the Vesica Piscis containing the Parthenon are but two of a virtual chain of 432,000 such circles which girdles the globe. A segment, with the Parthenon at centre, is shown below in Figure 2-14.
The length and width of the Parthenon's stylobate divide into the circumference of the Parthenon Great Circle exactly 576,000 and 1,296,000 times respectively. Each side of the Square of Equal Area divides exactly 864,000 times into the same circumference. In Table 2-5 below the various angles of the Circle of Natural Intervals (See Figure 1-2) are divided by the sides of the stylobate, equivalent square and other related units.
| Interval Class | Angle | Stylobate | Equivalent Square | Geodetic | ||||
|---|---|---|---|---|---|---|---|---|
| Pitch | Tonal | Lengths | Widths (Plethra) |
Perimeters (Stadia) |
Sides | Cubits | Feet | |
| 1 | Semitone | 24° | 38,400 | 86,400 | 14,400 | 57,600 | 5,760,000 | 8,640,000 |
| 2 | Tone | 45° | 72,000 | 162,000 | 27,000 | 108,000 | 10,800,000 | 16,200,000 |
| 3 | Minor 3rd | 72° | 115,200 | 259,200 | 43,200 | 172,800 | 17,280,000 | 25,920,000 |
| 4 | Major 3rd | 90° | 144,000 | 324,000 | 54,000 | 216,000 | 21,600,000 | 32,400,000 |
| 5 | Perfect 4th | 120° | 192,000 | 432,000 | 72,000 | 288,000 | 28,800,000 | 43,200,000 |
| 6 | Diminished 5th | 152° | 243,200 | 547,200 | 91,200 | 364,800 | 36,480,000 | 54,720,000 |
| 7 | Perfect 5th | 180° | 288,000 | 648,000 | 108,000 | 432,000 | 43,200,000 | 64,800,000 |
| 8 | Minor 6th | 216° | 345,600 | 777,600 | 129,600 | 518,400 | 51,840,000 | 77,760,000 |
| 9 | Major 6th | 240° | 384,000 | 864,000 | 144,000 | 576,000 | 57,600,000 | 86,400,000 |
| 10 | Minor 7th | 280° | 448,000 | 1,008,000 | 168,000 | 672,000 | 67,200,000 | 100,800,000 |
| 11 | Major 7th | 315° | 504,000 | 1,134,000 | 189,000 | 756,000 | 75,600,000 | 113,400,000 |
| 12 | 8ve | 360° | 576,000 | 1,296,000 | 216,000 | 864,000 | 86,400,000 | 129,600,000 |
| Degree | 1° | 1,600 | 3,600 | 600 | 2,400 | 240,000 | 360,000 | |
| Minute (Time) | ¼° | 400 | 900 | 150 | 600 | 60,000 | 90,000 | |
| Hour | 15° | 24,000 | 54,000 | 9,000 | 36,000 | 3,600,000 | 5,400,000 | |
| Hour (Vedic) | 6° | 9,600 | 21,600 | 3,600 | 14,400 | 1,440,000 | 2,160,000 | |
| Day (= 8ve) | 360° | 576,000 | 1,296,000 | 216,000 | 864,000 | 86,400,000 | 129,600,000 | |
Some of these numbers crop up in the most unexpected places, indicating that they were part of a widespread tradition. In Table 2-6 below we see the durations of the Vedic Yugas.
| Yuga | Duration (years) |
Correspondence (per revolution) |
|---|---|---|
| Krita | 1,728,000 | (The cube of 120) |
| Treta | 1,296,000 | Geodetic 100-foot rope = Arcseconds |
| Dvapara | 864,000 | Geodetic 100-cubit rope |
| Kali | 432,000 |
What has been demonstrated here firstly is that the underlying design matrix of the Parthenon is neither arbitrary nor accidental but follows and expresses an even more ancient tradition of sacred geometry predicated upon the mystical significance and symbolism of the cube in both form and number.
Secondly, its greater significance notwithstanding, comes the discovery that the principal axis of the Parthenon is oriented along the path of a great circle through the Acropolis, the Parthenon Great Circle, whose vertex is clearly intended to touch the 40th parallel, thereby cutting off 4/9ths of the meridian quadrant, 4 and 9 constituting the signature ratio of the building, and that the dimensions of the stylobate are equally intended to be precise subdivisions of the circumference of the selfsame circle, with the Parthenon stadion, a 1/216,000th part of it, being the fundamental unit of linear measure.
Had it been that only the size of the Parthenon's platform, or only its orientation, was dependent upon the corresponding characteristic of this circle then it might have been possible meaningfully to attribute it to a chance coincidence. However, the occurrence in the construction of the Parthenon of both dependencies upon the same Great Circle leads inescapably to the conclusion that this correlation was intended by the builders. An intention they could not possibly have possessed without first having had a clear knowledge of the precise shape and size of the Earth.
This thesis is reinforced by the fact that the dimensions map directly, as shown in Table 2-5, onto the system of natural proportions that gives us both the Babylonian unit of rotation and western harmony, as described in Chapter 1.
Such a purpose indicates a knowledge of both geodesy and mathematics that the Greeks of the classical period are generally supposed not to have possessed but which, however, at least one of them certainly did, and which, furthermore, has not been matched until very recent times.
That any coincidental orientation to the heliacal rising of a star was also intended by the builders cannot be confirmed at this time. However, one must consider the possibility, however remote, that the story of Athena's birth was invented to explain a merely fortuitous alignment to the rising (i.e. first sighting) of Aldebaran, the Eye of Zeus.
The Parthenon Great Circle can be seen to represent the Ouroboros itself, that great symbol of virginal self-creation and perpetual self-fertilization, of life itself, which connects Heaven and Earth, its head and tail meeting at the Athenian Acropolis, directly below the Parthenon. Hence the emblem of the khthonic serpent under the shield of Athena in Pheidias' statue.
Figure 2-15. Okeanos
| EN TO PAN |
References
| 1. | W.B.Dinsmoor, The Architecture of Ancient Greece, p.161 (footnote)
“The [Parthenon] stylobate dimensions, while generally recognized as forming the ratio 4:9, are often interpreted as 100 by 225 ‘Greek’ feet - but of a foot unit (12 1/8 [English] inches) which no Greek ever employed.”* | |
| 2. | ibid., p.161
“The total height of the order, that is, of the column and entablature together, was made 3 1/5 times the axial spacing or 7 1/5 lower diameters, that is, exactly 42 Doric feet, and again forms the proportion of 4:9 with the width of the stylobate. This consistency in proportions is most unusual and suggests the care with which the entire design [of the Parthenon] was studied.” | |
| 3. | Livio Stecchini†, Units of Length from A History of Measures | |
| 4. | F. Bligh Bond and T.S.Lea, Gematria, 1917, p.9 | |
| 5. | F.C.Penrose, An Introduction to the Principles of Athenian Architecture‡, 1888. | |
| 6. | Livio Stecchini, Francis Penrose from The Athenian Acropolis | |
| 7. | F. Bligh Bond, The Geometric Cubit, included as a supplement to Gematria by F. Bligh Bond and T.S.Lea, 1917, p.106 | |
| 8. | Livio Stecchini, The Dimensions of the Parthenon from The Athenian Acropolis | |
| 9. | Livio Stecchini, Notes on the Relation of Ancient Measures to the Great Pyramid, included as an appendix to Secrets of the Great Pyramid by Peter Tompkins, 1971 | |
| 10. | J.N.Lockyer, Dawn of Astronomy, 1894 | |
| 11. | Coeli, Stella 2000 | |
| 12. | Patrick Chevalley, Cartes du Ciel | |
| 13. | Livio Stecchini, Orientation of the Parthenon from The Athenian Acropolis | |
| 14. | A.E.Berriman, Historical Metrology, 1953, p.1 | |
| 15. | Livio Stecchini, Egyptian Estimates of the Size and Shape of the Earth | |
| 16. | Eric W. Weisstein, Spherical Trigonometry, from MathWorld – A Wolfram Web Resource |
* In reality much closer to 12 1/6 English inches,
although this in no way invalidates the point Dinsmoor is making.
† The sole printed work of Livio Stecchini is the appendix to Secrets of the Great Pyramid (ref. 9).
Stecchini's remaining work is presently being transcribed by anonymous volunteers
and published on the World Wide Web.
‡ I am most grateful to Mr. Joel Dias-Porter of Washington D.C.
for visiting the Library of Congress on my behalf in order to obtain Penrose's figures
for the azimuth of the Parthenon from its copy of this very rare book, the British Library's
copy being described in the catalog as “Missing”. – PWS
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Copyright © Peter Wakefield Sault 1973-2008
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