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62. Canonical Numbers

Updated: Dec 15, 2023


“Thus was the principle of harmony revealed as an unseen and unheard principle of order and concord, identical with a system of numbers bound together by interlocking ratios.”1
F.M. Cornford

Bust of Pythagoraa. Roman copy of the original Greek. Musei Capitolini, Rome, Wikimedia Commons

The Greek word κόσμος (kósmos) refers to both the universe and to a concept of order and harmony. In the Greek philosophical tradition, Pythagoras is believed to have been the first to use it. Underlying this concept is a view according to which a structured principle is found on a large scale in the entire universe, but also on a small scale, in individuals and all living things, macrocosm and microcosm. Individuals are thus parts of the whole which includes them, but also analogous to the whole. This view presents a fundamental community of nature, which contrasts with the Olympian religion, and other more modern forms of religion, in which the divine is separate from the rest of the world. The macrocosm itself is seen as a living creature, with a soul, or some sort of life principle of its own, and this soul is a harmony or system of numbers. This world soul, or soul harmony, provides the framework against which the individual soul can be tuned. According to this philosophy, we might expect to find the principles, or numbers that govern nature in any part of the universe.

This philosophy may not have been specific to the Greeks, and may be very old. Long ago, many aspects of human activity were governed by principles obtained from an understanding of the world soul. We might begin to look for signs of such harmony in the sizes and proportions of the stone structures, and systems of measure that have survived since ancient times. Indeed, many researchers have found echoes of astronomical cycles, irrational numbers, and musical ratios expressed in stone circles, megaliths and pyramids all over the world, from Europe to Asia, and from Africa to the Americas. The structure and geometry can be found to match the local orientation of a site, or properties of latitude, the visual perspective of the celestial dome above, or astronomical cycles. This would imply that a system of number designed to express the laws of the universe was in use and applied to human constructions. The ground breaking work of researchers like Alexander Thom, Gerald Hawkins, John Michell and Hugh Franklin, and many others present engaging and intriguing understandings of these ancient sites, which offer a connection to a world in which everything could be an expression of the harmony of the universe.

The technology that we have at our disposal today is invaluable in piecing together the various elements of an ancient site, or accessing numbers related to astronomy. What they seem to be revealing is that, at some point in the distant past, knowledge of how the various heavenly bodies move was very sophisticated, and this is not true of only one particular region but across the world. A study of ancient measure can show that the systems in place rely on numbers that are consistent with sophisticated mathematics and astronomy, and can be backed up by references to literature from the oldest known traditions. In particular, some of these numbers can be found in Plato's work. A study of early weight and volume units, and very early coinage, reveals the existence of a number system, based on a certain key values. Many of these numbers were referred to by Plato in his Republic, Timaeus and Laws, such as 5040, a number connected to an ideal city, and 2160, a Platonic “year”. The highest branch of education for Plato was arithmetic, and knowing how the heavenly bodies moved was a foundation of religion, of the economy, and of the state.

Soul harmony

Cleinias: And what are the laws about music and dancing in Egypt? Athenian Stranger: You will wonder when I tell you: Long ago they appear to have recognized the very principle of which we are now speaking-that their young citizens must be habituated to forms and strains of virtue. These they fixed, and exhibited the patterns of them in their temples; and no painter or artist is allowed to innovate upon them, or to leave the traditional forms and invent new ones. To this day, no alteration is allowed either in these arts, or in music at all. And you will find that their works of art are painted or moulded in the same forms which they had ten thousand years ago;-this is literally true and no exaggeration-their ancient paintings and sculptures are not a whit better or worse than the work of to-day, but are made with just the same skill. 2
Plato, Laws

Plato tells us that ten thousand years before his time there were laws and principles that governed art, music and architecture, which informed a virtuous life, and were not allowed to change. Plato believed in a principle of harmony at work in the universe. It seems that in his day the ancient Egyptian system was only a memory. Yet, for Plato, and Pythagoras before him, various aspects of the universe and our experience of it could be explained by this harmony, expressed in numbers and ratios. The reasons we like or value certain things could be explained by simple numerical proportions. In essence, the foundation of knowledge was number. The whole universe was number. This view holds true for many today still, and perhaps an essential human characteristic is the love of number. In Plat o's Timaeus, Timaeus says:

And God lighted a fire in the second orbit from the earth which is called the sun, to give light over the whole heaven, and to teach intelligent beings that knowledge of number which is derived from the revolution of the same.
Plato, Timaeus, Section 1

The soul harmony was present in all things and could be tuned into consonance with the celestial harmony. The macrocosm itself was understood as a living creature with a soul, or principle of life, and a body. The whole world being harmony and number, various parts of it could be compared in ways that are surprising to us, for example a particular song might have notes that existed in the same ratio as the distances of the various heavenly bodies from the Earth. It was in relation to this harmony of the heavenly bodies that the individual soul could be attuned. Number was therefore central to all aspects of human existence.

While the focus of research has often been on the ritualistic aspect of the people who designed and used ancient sites from Stonehenge to the Great Pyramids of Giza, often with an emphasis on the quality of life of the living, and the nature of their after-life beliefs, there is a whole area of research, focusing on an examination of the orientations, locations, measurements and proportions found, that can add a great deal to our understanding. John Michell was key in bringing the ancient systems of what he termed "canonical" numbers, and gematria back to popular consciousness. Through a study of this system we can see that many of the key numbers and ratios that underpin it also hold up the workings of the cosmos. For Plato, especially in Timaeus, something integral to nature is mathematics, and it is through number that we can become aware of the properties of objects, and also define human laws and morality.

Plato put forward the importance of the concept of a great cycle within which the cycles of the motions of the planets could fit.


[39d] that the “wanderings“of these bodies, which are hard to calculate and of wondrous complexity, constitute Time. Nevertheless, it is still quite possible to perceive that the complete number of Time fulfils the Complete Year when all the eight circuits, with their relative speeds, finish together and come to a head, when measured by the revolution of the Same and Similarly-moving. In this wise and for these reasons were generated all those stars which turn themselves about as they travel through Heaven, to the end that this Universe might be as similar as possible to the perfect and intelligible Living Creature in respect of its imitation of the Eternal 23

Within this framework, the concept of time itself is linked to astronomy.

Stradivarius guitar, National Music Museum in Vermillion, SD, Photo Larry Jacobsen, Wikimedia Commons

Pythagoras was said to have discovered that:

“the concordant intervals of the musical scale or harmony could be expressed exactly in the terms of 'simple' ratios, : 2 (octave), 3:2 (fifth), and 4 : 3 (fourth), and that, if the smallest whole numbers having these ratios to one another (viz. 6 : 8 : 9 : 12) are taken, the internal terms are the means (arithmetic and harmonic) between the extremes.”3

These ratios are still important today in music theory, but were also once part of a system of interpreting the world around us, from macrocosm to microcosm. The Pythagorean world view was quite different in many aspects to ours, and it is difficult to re-assemble the various elements we have of this doctrine into a cohesive whole that is compatible with a contemporary a mind set, and with a different sense of evidence. Often, ancient world quests for understanding the universe as number are seen as as simply mystical, and primitive.

The influence of Pythagoras’s doctrine of ψυχη αρμονια [soul harmony] was far reaching, both geographically, and in time. It belongs to a tradition which most probably existed long before Pythagoras’s time. The mystical and cosmological symbolism of numbers for the Pythagoreans was developed further by philosophers such as Philo and Nicomachus, who relied to a large extent on Plato’s Timaeus. The belief that mathematics provided the key to understanding the world carried on through the Middle Ages, and is still central to our beliefs today. There have been changes in approach of course. One particular aspect that has largely disappeared from today’s understanding of the world as number is the secret and the symbolic.

The proportions in which the soul of the world and the human soul are divided up correspond to two number series associated with Pythagoras and starting with the number 1: 1, 2, 4, 8, and 1, 3, 9, 27. The first consists of multiples of 2 and the second of multiples of 3, 2 and 3 being for the Pythagoreans male and female numbers. Since there are only four numbers in each series, we can see that they represent the three dimensions of our world, starting with a point, then a line, a square (or area) and a cube (or volume). These numbers could also be taken to represent properties of the solar system, and properties of musical scales, in other words, they represent important ratios. There could perhaps also be a reference to the four elements of fire, water, earth and air, which, for Plato, are characterised by rectangular triangles, combined to form regular solid figures.

God placed water and air in the mean between fire and earth, and made them to have the same proportion so far as was possible (as fire is to air so is air to water, and as air is to water so is water to earth); and thus he bound and put together a visible and tangible heaven. And for these reasons, and out of such elements which are in number four, the body of the world was created, and it was harmonized by proportion, and therefore has the spirit of friendship; and having been reconciled to itself, it was indissoluble by the hand of any other than the framer.
Timaeus, Plato

At the very start of Plato's dialogue, Timaeus, the eponymous speaker says

"God took of the unchangeable and indivisible and also of the divisible and corporeal, and out of the two he made a third nature, essence, which was in a mean between them, and partook of the same and the other, the intractable nature of the other being compressed into the same. Having made a compound of all the three, he proceeded to divide the entire mass into portions related to one another in the ratios of 1, 2, 3, 4, 9, 8, 27, and proceeded to fill up the double and triple intervals thus—
- over 1, 4/3, 3/2, - over 2, 8/3, 3, - over 4, 16/3, 6, - over 8: - over 1, 3/2, 2, - over 3, 9/2, 6, - over 9, 27/2, 18, - over 27;
in which double series of numbers are two kinds of means; the one exceeds and is exceeded by equal parts of the extremes, e.g. 1, 4/3, 2; the other kind of mean is one which is equidistant from the extremes—2, 4, 6. In this manner there were formed intervals of thirds, 3:2, of fourths, 4:3, and of ninths, 9:8. And next he filled up the intervals of a fourth with ninths, leaving a remnant which is in the ratio of 256:243. The entire compound was divided by him lengthways into two parts, which he united at the centre like the letter X, and bent into an inner and outer circle or sphere, cutting one another again at a point over against the point at which they cross. The outer circle or sphere was named the sphere of the same—the inner, the sphere of the other or diverse; and the one revolved horizontally to the right, the other diagonally to the left. To the sphere of the same which was undivided he gave dominion, but the sphere of the other or diverse was distributed into seven unequal orbits, having intervals in ratios of twos and threes, three of either sort, and he bade the orbits move in opposite directions to one another—three of them, the Sun, Mercury, Venus, with equal swiftness, and the remaining four—the Moon, Saturn, Mars, Jupiter, with unequal swiftness to the three and to one another, but all in due proportion."
Plato, Timaeus, Section 1

Rather than just number, used on its own, as a value or even symbolically, it is ratio which is important, in that it describes not only the process of creation, but also relationships between the orbits of the planets in the solar system. The Pythagorean limma is formed by the fraction 256/243, and we can also see here the importance of dividing by 7, and ratios formed by 2, 3 and 4.

Researcher Jim Alison has written about the number 7 in relation to Plato and the motions of the planets:

By what proportion gravity decreases by receding from the planets the ancients have not sufficiently explained. Yet they appear to have adumbrated it by the harmony of the celestial spheres, designating the sun and the remaining six planets, Mercury, Venus, Earth, Mars, Jupiter and Saturn, by means of Apollo with the lyre of seven strings, and measuring the intervals of the spheres by the intervals of the tones. Thus they alleged that seven tones are brought into being, which they called the harmony diapason, and that Saturn moved by the Dorian phthong, that is, the heavy one, and the rest of the planets by the sharper ones (as Pliny, bk. I, ch. 22 relates, by the mind of Pythagoras) and that the sun strikes the strings. Hence Macrobius, bk. I, ch. 19, says: ‘Apollo’s lyre of seven strings provides understanding of the motions of all the celestial spheres over which nature has set the sun as moderator.’ And Proclus on Plato’s Timaeus, bk. 3, page 200, ‘The number seven they have dedicated to Apollo as to him who embraces all symphonies whatsoever, and therefore they used to call him the God of the Hebdomagetes’, that is the prince of the number seven. Likewise in Eusebius’ Preparation of the Gospel, bk. 5, ch. 14, the sun is called by the oracle of Apollo the king of the seven sounding harmony.31

While we might be tempted to interpret the ancient system of seven "planets" as a purely symbolic, even whimsical view, there could well be scientific research propping up this ancient system, which contains within it an understanding of some of the main workings of the solar system. Researcher Richard Heath has found that many musical ratios in fact do closely define astrnomical patterns, such as a 9/8 interval between the lunar year and the Jupiter synod, or the 15/16 interval between the lunar year and the Saturn synod. In his paper "The Harmonically Numerical Form of Creation (and why it used to matter)", Richard Heath writes:


Mythology is presumed to have come from an oral world, and temple designs are often known to have been images of the cosmos, but the modern reading of both of these is as works of imagination, art or belief. Persistent notions of a music of the spheres have, since Plato, regularly failed to provide a plausible organising principle for the planets using harmonic ratios. Yet Plato’s tuning theory, as recovered by Ernest G. McClain, has proved to be my key for revealing, in ancient monuments and textual allusions, a pattern of Just intonation in the heavens that corresponds to our knowledge of these same astronomical invariants today; an actual music of the spheres. 32

It may be that Plato was part of a long tradition of describing the workings of the universe in terms of musical ratios.


Platonic numbers


Many of the early Christian theologians such as Clement of Alexandria were neo-Pythagoreans, adopting their ascetic values, as well as their love of number. In a chapter called “Reasons for veiling the truth in symbols”, Clement of Alexandria discusses the historical facts around secrecy with the philosophers:

“They say, then, that Hipparchus the Pythagorean, being guilty of writing the tenets of Pythagoras in plain language, was expelled from the school, and a pillar raised for him as if he had been dead. Wherefore also in the Barbarian philosophy they call those dead who have fallen away from the dogmas, and have placed the mind in subjection to carnal passions.”4

Clement of Alexandria also writes:

" It was not only the Pythagoreans, and Plato, that concealed many things, but the Epicureans, too, say that they have things which may not be uttered, and do not allow all to peruse those writings. The Stoics also say, that by the first Zeno things were written which they do not readily allow disciples to read, without their first giving proof whether or not they are genuine philosophers. And the disciples of Aristotle say, that some of their teachings are esoteric and others common and exoteric. Further, those who instituted the mysteries, being philosophers, buried their doctrines in myths, so as not to be obvious to all."5

If Plato and the Pythagoreans concealed many things, as Clement of Alexandria suggests, we might attempt to interpret the use of a number such as 5040, or 216, in the light of such a concealement. Number was understood as a central part of life, or at least the ideal life. As Plato said in Epinomis:

“ In our discussion, the other courses of these bodies that we have previously described are not easily understood: we must rather prepare our faculties, such as they may possibly be, for these matters; and so one must teach the pupil many things beforehand, and continually strive hard to habituate him in childhood and youth. And therefore there will be need of studies: the most important and first is of numbers in themselves; not of those which are corporeal, but of the whole origin of the odd and the even, and the greatness of their influence on the nature of reality.7

The understanding of the world presented by Plato and Pythagoras has its roots in science and number. Within this framework, certain numbers were especially important. We can see many of these at work in systems of measure. Looking at Plato's texts confirms the importance of number in his work, but it can be hard to determine exactly in what way, or what these numbers actually are.

In the Laws, it is clear, at least, that the number 5040 is paramount for Plato. The ideal state would have precisely 5040 citizens, divided into four classes.

Let us now proceed to the distribution of our state, and determine the size of the territory and the number of the allotments. The territory should be sufficient to maintain the citizens in moderation, and the population should be numerous enough to defend themselves, and sometimes to aid their neighbours. We will fix the number of citizens at 5040, to which the number of houses and portions of land shall correspond. Let the number be divided into two parts and then into three; for it is very convenient for the purposes of distribution, and is capable of fifty-nine divisions, ten of which proceed without interval from one to ten. Here are numbers enough for war and peace, and for all contracts and dealings. These properties of numbers are true, and should be ascertained with a view to use.

Plato, The Laws, Book V, translated by Benjamin Jowett


The number of citizens in a state, the way the land in a state is divided up, and the way religion is organised are related to this special number 5040. Far from allowing a city to grow organically, Platos suggests various ways of keeping the number of citizens at 5040.

When parents have too many children they may give to those who have none, or couples may abstain from having children, or, if there is a want of offspring, special care may be taken to obtain them; or if the number of citizens becomes excessive, we may send away the surplus to found a colony. If, on the other hand, a war or plague diminishes the number of inhabitants, new citizens must be introduced; and these ought not, if possible, to be men of low birth or inferior training; but even God, it is said, cannot always fight against necessity.
Wherefore we will thus address our citizens:—Good friends, honour order and equality, and above all the number 5040.

Plato, The Laws, Book V, translated by Benjamin Jowett.


This is a decidedly odd way of imagining an ideal state, to us today. Platos clearly states, in Book V of the Laws, that "the number 5040 has many convenient divisions", and it is because of the importance of the number 12, corresponding to the number of tribes, that the number 5040 is important, being 12 x 12 x 35, or 12 x 12 x 70/2. It is also 7!, or 1 x 2 x 3 x 4 x 5 x 6 x 7. This must be a key part of the importance of this number, as Plato writes that "every divisor is a gift of God, and corresponds to the months of the year and to the revolution of the universe. "(Book V). Also:


And now, let us proceed to legislate with a view to perfecting the form and outline of our state. The number of our citizens shall be 5040—this will be a convenient number; and these shall be owners of the land and protectors of the allotment. The houses and the land will be divided in the same way, so that every man may correspond to a lot. Let the whole number be first divided into two parts, and then into three; and the number is further capable of being divided into four or five parts, or any number of parts up to ten. Every legislator ought to know so much arithmetic as to be able to tell what number is most likely to be useful to all cities; and we are going to take that number which contains the greatest and most regular and unbroken series of divisions. The whole of number has every possible division, and the number 5040 can be divided by exactly fifty-nine divisors, and ten of these proceed without interval from one to ten: this will furnish numbers for war and peace, and for all contracts and dealings, including taxes and divisions of the land. These properties of number should be ascertained at leisure by those who are bound by law to know them; for they are true, and should be proclaimed at the foundation of the city, with a view to use.
Plato, The Laws, Book V

At all times, the state should strive towards an ideal, and it is an ideal rooted in number. Another number that comes up in the Laws is 360, which is the number of members in council, and 12 x 30.


The council shall consist of 30 x 12 members—360 will be a convenient number for sub-division. If we divide the whole number into four parts of ninety each, we get ninety counsellors for each class.
Book VI

One number which is important to Plato he alludes to mysteriously but doesn't actually state, as if to underline the mystery of the universe itself. This number is alluded to in a section of the Republic (8.546b)and it has been suggested by commentators that it may be 216, or 1296, or perhaps 3600, or some other number altogether. As with 5040, this number is portrayed as crucial to human life, and neglecting to abide by its laws could incur terrible consequences, affecting births, marriages, and the state itself. Below are several versions of this passage.


Barton, George A. “On the Babylonian Origin of Plato’s Nuptial Number.” Journal of the American Oriental Society, vol. 29, 1908, pp. 210–19. JSTOR, https://doi.org/10.2307/592627.

Barton, George A. “On the Babylonian Origin of Plato’s Nuptial Number.” Journal of the American Oriental Society, vol. 29, 1908, pp. 210–19. JSTOR, https://doi.org/10.2307/592627.

This is the Benjamin Jowett translation as found in the Project Gutenberg text:

For whereas divine creations are in a perfect cycle or number, the human creation is in a number which declines from perfection, and has four terms and three intervals of numbers, increasing, waning, assimilating, dissimilating, and yet perfectly commensurate with each other. The base of the number with a fourth added (or which is 3:4), multiplied by five and cubed, gives two harmonies:—the first a square number, which is a hundred times the base (or a hundred times a hundred); the second, an oblong, being a hundred squares of the rational diameter of a figure the side of which is five, subtracting one from each square or two perfect squares from all, and adding a hundred cubes of three. This entire number is geometrical and contains the rule or law of generation. When this law is neglected marriages will be unpropitious; the inferior offspring who are then born will in time become the rulers; the State will decline, and education fall into decay; gymnastic will be preferred to music, and the gold and silver and brass and iron will form a chaotic mass—thus division will arise.

The text is opaque. The translations differ. The four terms, with three intervals, could refer to either of the number series 1, 2, 4, 8 and 1, 3, 9, 27, in which, starting from 1, each term is multiplied by 2 or 3. In this case the base of these series is 1, and adding a third to it gives 1¹/³, or 1.33333. "When combined with 5 and raised to the third power" could mean adding 5, or multiplying by 5, before cubing. In which case we obtain either (1¹/³ x 5)³ = 296.296296, or (1¹/³ + 5)³ = 254.037037037. However the second translation clearly suggests 100 for the base, not 1. The adding of a third or fourth (the two translations differ on this) is perplexing, as without it we could obtain a nice round 216, or 125. Adding a fourth instead of a third would produce, with a base of 1, 244.140625, and curiously, this works whether we add or multiply by 5. With a base of 100, and a fourth added, the results are (125 x 5)³ = 2 197 000, and (125 x 5)³ = 244 140 625, which is 5¹², and with a third added, (133¹/³ + 5)³ = 2 647 162.037 (133¹/³ x 5)³ = 296 296 296.296 296 = (2/3)³ = 8/27

There is an interesting geometric connection which lends itself to astronomy in the number 254.037037: A circle with a diameter of 1000 x 29.53059 (average lunation in days) has a circumference of very close to 254.037037037 x 365.242, the last number being the days in a year. So that this number, 254.037037, is very close to approximating the relationship between the solar and lunar cycles via the geometry of the circle (for an exact match, using 254.037037037, you would need a value of 3.1419977 for pi; or even better, by using simply 254, 365.242199 and 29.53059, the value of the diameter / circumference ratio becomes 3.1415396, which is very close to pi).

A hundred cubes of 3 must mean 100 x 3³ = 2 700. What does it mean for something to consist of a hundred numbers squared upon rational diameters of a square? The perfect square of an irrational diameter could refer to any square, as a diameter is always the square root of 2 multiplied by the side, and is therefore necessarily irrational. The length of a diagonal of a square with sides of 5, will be 5√2, though, in order to avoid irrationality, we can use the fraction 99/70 for √2. The "hundred squares of the rational diameter of a figure the side of which is five" must be 100 x 5 x 99/70 = 707.142857. If then this number is divided by 3 cubed, we obtain 5 x 99/70 x 1/(3 x 3 x 3) = 0.261904762. This is a familiar number to anyone interested in ancient metrology, as it is one of several aproximations of the Phi squared produced by combining Fibonacci numbers, Phi being the golden number. Indeed, the two Fibonacci numbers 21 and 55 combine to make 55/21 = 2.619047619. Whether this geometrical figure that Plato mentions is a special golden rectangle (with sides of 1 and Phi squared) is open to debate. The golden ratio is present in every living thing, in an approximate form, so can be associated with life, or regeneration itself. Perhaps when pertaining to the divine, the golden ratio must be expressed as (√5 + 1)/2, or for Phi squared, (√5 + 3)/2, with the irrationality of the square root of 5 intact, that is, no in an papproximate or applied form. Whereas when the golden ratio expresses something pertaining to the imperfect world of humans, then it is preferable to use an approximation, such as one with the Fibonacci numbers, like 34/21 or 55/34 for Phi, and 55/21 or 144/55 for Phi squared. As Plato says at the beginning of this passage: "For whereas divine creations are in a perfect cycle or number, the human creation is in a number which declines from perfection."And indeed, just as we can never actually measure out √2 or √5 in a geometric shape, but only ever a number which is close to it, we will not find the golden ratio as (√5 + 1)/2 or (√5 + 3)/2 in the natural world, but something close to it, which tends towards this ideal. In this world, there is no number which squared produces √2 or √5, and so these so called irrational numbers can be understood as belonging to an ideal, or divine world. However, maybe Plato is not refering to this golden ratio at all, but to the number 8.

The geometric shape which is the most associated with births (and deaths) is the octagon, as it is often found in baptistries and mausoleums, as well as in the centres of religious buildings generally, such as the floor pattern of many cathedrals. Athena's temple, the Parthenon, in Greece, has eight huge columns to the front, and sixteen to the side, echoing the octagon, as well as the number 12, as the total number of columns is 12 x 4. A reference is made in Plato's text to uniting bride and groom in the correct season, which would support the presence of the number 8, in that it could be taken to symbolise the year, as a circle divided into eight sections, these being the marked by the quarter days and cross quarter days. And if we divide 8000 by 27, the last number of one of the number series (1, 3, 9, 27), we obtain 296.296296296, which is what we saw above, (1¹/³ x 5)³. So possibly, if the octagon is the geometric figure being referred to, it is represented by (1¹/³ x 5)³ x 3³/1000, or, simplified, (1¹/³ x 15)³ /1000. And we know from Stonehenge that, to within a small margin of error, the side of an octagon can double up as the short side of a 5:12:13 Pythagorean triangle, the 12 side being close to the medium diagonal of the octagon, and the 13 side being close tot he longest diagonal. A 5:12:13 triangle has a perimeter of 30, and has a side of 5, which could match what Plato refers to. Also, 13 is half of 27, the last number in the number series 1, 3, 9, 27 put orward by Plato. Perhaps the oblong referred to is a 5 x 12 rectangle, though it is difficult to know how to relate it to Plato's text. Such a triangle, being Pythagorean, would indeed have a rational diameter and a side of 5. This particular Pythagorean triangle is of particular interest in an astronomical context, as Robin Heath has demonstrated. In his book The Lost Science of Measuring the Earth, co-written with John Michell, he shows how not only can such a triangle be understood to fit precisely within the Aubrey Circle at Stonehenge, but that 12 and 13 represent the number of possible lunations in a solar year. Moreover, the average number of lunations in a solar year (12.3683) is very closely approximated by the hypotenuse of a smaller triangle within it, with sides of 3, 12, and 12.36932. The numbers 5, 8 and 13 are Fibonacci numbers so, when combined, provide a connection to the golden ratio. Also the number13 can be combined with pi and √2, the ratios of a circle's diameter and circumference, and a square's side and diagonal respectively, and the ratio 2/3. (π x 2/3)³ x √2 = 12.99244, almost 13; and using 22/7 and 99/70 for pi and the square root of two, the result is 13.00880.



And, though this is not mentioned by Plato, it is fitting that (1¹/³ x 15) x 3³/10 = 54, one of the central numbers of the canonical system, and the value of an angle within an octagon.



The canonical system

The numbers which described the cosmos, and the ratios of geometry and music, were central to a society and kept within its temples. While the contents of these temples are often long gone, the temples themselves can be interpreted in terms of their dimensions and locations. The dimensions can be read in terms of ratios to the rest of the site, as well as in particular units of measure, some of which are still in use today.

Vestiges of the canonical system would continue to survive throughout the remaining centuries up until the present era, but with time the connecting linkages to it were hard to discern, as the system’s workings were forgotten. A study of weights and volumes in addition to linear units reveals a system that was used across three dimensions. Moreover, both time and space, and small and large scales could be expressed by these numbers, perhaps multiplied or divided by powers of ten. As John Michell wrote:

“The measurement of time, cosmic distances and the body of the canonical man were formerly united in one scientific system by the canon of numbers which was common to them all. The beauty of the ancient units, some of which still survive as the English inch, foot, furlong, mile, etc, is that they occur naturally in the measurement of all classes of phenomena from human to the astronomical scale.“8

William Stirling’s The Canon illustrates how numbers came to be assimilated to letters of the alphabet, and that for example, he interprets the story of Noah, and the seven members of his family, who took a year's voyage in the Ark, together with all the animals of the world, as a celestial metaphor. In the traditional cosmological view there were seven ‘planets’, which have survived in our week day names (the sun and the moon are of course not planets).

“If this explanation be correct, we must conceive, by the proportions of the Ark, the vast figure of a man, in the likeness and image of God, whose body contains the measure of the sun's path in the ecliptic, the circuit of the earth, and the orbits of the seven planets. We are, in fact, to imagine the whole material universe, accurately epitomized in a human body, symbolizing the Creator reflected in creation. That the human body was used as a means of illustrating the parts of the universe, is evident from the figures surrounded by the 12 signs, depicted on old almanacks, and by the custom of astrologers, who allocate the signs and planets to the various members of the body.”6

John Michell was key in reviving knowledge of this way of thinking.

Ancient science was based like that of today on number, but whereas number is now used in the quantitative sense for secular purposes, the ancients regarded numbers as symbols of the universe, finding parallels between the inherent structure of number and all types of form and motion. Theirs was a very different view of the world from what that which now obtains . They inhabited a living universe , a creature of divine fabrication, designed in accordance with reason and thus to some extent comprehensible by the human mind. The special regard paid to mathematical studies in the ancient world arose from the understanding that number is the mean term in the progression from divine reason to its imperfect reflection in humanity. At some very early period, by a process quite beyond explanation, certain groups of numbers were brought together and codified. Thus was created that numerical standard, or canon of proportion , which was at the root of all ancient cultures, and was everywhere attributed to some form of miraculous revelation . It was taken to be the nucleus and activating principle of number generally, a summary of all the types of progressions and relationships which occur within the field of number, and thus a faithful image of the numerically created universe .”
One day rotation with Thuban as north star. No stellar proper motion included. en:Axial_precession#Changing_pole_stars Date 26 October 2016 Source,Tomruen, Wikimedia Commons

A remarkable aspect of the system was the repeated presence of key numbers, such as 12, 54, 108, and 25920. In this system, a Great Year is 25930 (solar) years, and corresponds to a very long period of time named the precessional cycle. According to an 18th century French astronomer, named Le Gentil, who travelled to India to witness the transit of Venus, the Indian Brahmins had calculated that precession could be counted as 54 seconds a year.9 The number 54 is perhaps the most important number in this system, the foundation, and corresponded precisely to the observations taken by the Indian astronomers. At the time these observations were made,

the obliquity of the earth was 24 degrees, and 54 x 24 = 1296, or 64, so this would go back many centuries, probably to when Thuban was the pole star. When Le Gentil arrived in India, he found that the astronomical tables and methods the Brahmins were using to predict lunar eclipses and other celestial events were surprisingly accurate.10

For Ptolomy the precession of the equinoxes was a westward shift of approximately one degree every 72 years, which is 54 x 4/3. The time for an equinox to make a complete revolution through all the zodiac constellations and return to its original position would be approximately 25,920 years, or 480 x 54. Both 72 and 54 are numbers found in a pentagon.





A noteworthy feature of many of the most important numbers is the the sum of their digits is 9, eg 54, with 5+4 = 9, or with 72, 7 + 2 = 9, with 5040, 5 + 0 + 4 + 0 = 9, or with 25920, 2 + 5 + 9 + 2 + 0 = 18, and 1 + 8 = 9, similarly with 1296, 1 + 2 + 9 + 6 = 18, and even with 19 595 520, the sum of the digits is 36, and the sum of 3 and 6 is 9.

These numbers are found in many measures. For example the moon has a radius of 1080 miles. A Megalithic Yard can be regarded as 2.7216 feet, or 32.6592 inches. One value for the Egyptian Royal cubit is 1134 / 55 inches. 12.96 inches is a Persian or Assyrian foot. One side of the Great Pyramid of Giza measures close to 756 feet. A Roman foot of 11.664 inches is 54 x 6 x 6 x 6 /1000. 54 Roman digits (1/16th of a Roman foot) are a metre of 39.375 inches. The Ninevah constant is said to contain many astronomical cycles. The Ninevah constant taken as a measure in inches gives: 195 955 200/ 6 000 000 = 32.6592, Megalithic Yard 195 955 200/ 21 600 = 9072, Great Pyramid side 195 955 200/ 168 000 00 = 11.664, Roman foot 195 955 200/ 9 504 000 = 20.6181818, Egyptian royal cubit 195 955 200/ 7560 = 25 920, which divided by 20 gives Persian / Assyrian foot, 700 of which in GP base side.

195 955 200 / 6 480 000 = 30.24, Aragon vara

195 955 200 / (20 x 12⁵) = 39.375, ancient metre, or 54 Roman digits These numbers: 80 x 54 = 4 320 and 6! x 54 x 8 x 109 = 311 040 000 000 000 are significant in the ancient Indian system of calculating ages. A Kalpa is 4.32 billion years, as defined in the Puranas.11 A Kalpa is one day of Brahmā, which is 1000 cycles of four yugas. Brahmā lives one hundred "years" and then dies, and this period is equivalent to 311,040,000,000,000 earth years, or 54 x 3 x 4 x 6 x 8 x 1010 years.

Using the number 54, a lunation of 29.53059 days can be counted as 29.56125 days, as this is 540 x 7 /128.

An example of the use of the number 25 920 in measuring time can be found in the traditional Hebrew calendar. Each day is divided into 25 920 parts or “chalakim”, of 3 1/3 seconds, each hour is divided into 1080 parts. In our time systems today a day is divided into 24 hours, or 86 400 seconds.

An example of the use of the number 25 920 in weights is seen with the metric gram value of 7000 grams, which, divided by 453.6 is 15.43209877 grams, divided again by 4 is 3.85024691 grams. 25 920 x 7 / 400 = 453.6.

In linear measure, 25.92 inches are a a royal Assyrian cubit. A royal Egyptian cubit of 20 34/55 inches is 25920 x 175 / 220 000 inches, a royal Egyptian cubit of 20.625 inches is 25920 / 480 x 55/144. A shusi of 0.66 inches is 25920 / 55 x 2 160 00, a Roman foot is 25920 x 9 / 20 000 inches, and an imperial foot is 25920 / 2 160 inches. A megalithic yard as 2.7216 feet is made up of 40 megalithic inches of 0.81648, or 18 units of 1.8144 inches, equivalent to 25920 x 7/100 000. A unit of 1.8144 inches would go 5000 times into a side of the Great Pyramid, if taken as 9072 inches. A 2.7216 MY is also 2.8 Roman feet of 11.664 inches.

181440 is a key number, being 25920 x 7, or 36 x 7!.

The number 12 is also central if only because of the 12 or so lunar months in a year. 12 is 54 x 2 / 9. The system relies up to a point on this number 12, in the way that the imperial system of measure does, but also on multiplication and division by 10.



Other numbers such as 3168 and 792, multiplied or divided by powers of ten, are also found as multiples of units in ancient sites. John Michell found that:

“Earth diameter = 7920 miles
Moon diameter = 792 megalithic miles
Diameter of square containing the circle of the earth = 31680 miles
Diameter of square containing the circle of the moon = 3168 megalithic miles
Sun diameter = 316800 megalithic miles
The significance of these facts is unambiguous in proving the cosmological origins of the English and megalithic miles.” 12

There is a certain fractal quality to this way of defining the universe in terms of number, which must reflect the need for order in the minds of these designers of this system of measurements, an attempt to do justice to a sense of order within the universe. This could rely on the view that throughout the universe the same laws of physics apply. The diameter of the sun is generally thought of as 864 000 miles. 864 is already a number which fits well within the canonical system, being 36 x 4!, or 80 x 10.8. The moon's mean radius is 1079.57 miles, but the equatorial radius is exactly 1080 miles, and so the corresponding diameter is 2160 miles. The mean radius of the moon is 0.2727 times that of the earth, implying a 9/33 ratio. The megalithic mile John Michell refers to is 90/33 = 2.727272 miles long, or 14 400 feet, or 172 800 inches. This last number is 12³ x 100, and multiplied by 33/300 produces another key number, 19 008 = 3168 x 6.

Comparing the diameters of the sun and moon, 864 000 / 2 160 = 400. There is a clear 1:400 ratio between the actual sizes of the moon and sun. Because their distances from earth are also within a 1:400 ratio, they look more or less identical in size to us, in the sky. From these coincidences alone, we can infer that the circle and the square might be of some importance in symbolising the harmony of the universe.

In his ideal city, Plato explains that the city and the whole country must be divided into twelve parts, each of which is divided into 5040 allotments.

“The only way in which this division can be represented is by a circle of radius 5040, a hundred times larger than that of Stonehenge measured in feet; the perimeter of this circle is 31,680.”13

The number 1575 is also a key number. As researcher Stephen Dail has observed, the Earth's circumference can be interpreted as 1 575 000 000 inches. 157.5 is also the grain weight of a cubic inch of barleycorns, these grains being a staple of metrology, and three lined up in a row making an inch.

An ancient system based on science

The metre was created in the hope that it would represent exactly a 10,000,000ᵗʰ part of a quarter of the meridian circumference of the earth. The desire to base a unit of measure on a particular aspect of the natural world makes sense, be it something large, like a planet, or something small, like a barely grain. The desire to base a unit of measure on extrapolations from linear, volumetric, mass, and even time measurements and principles of geometry, and then apply the numbers back to express as many aspects of the universe as possible, even to language, is something else all together. What if we had a main standard of length that related not just to the earth’s circumference, meridian or equatorial, but to the dimensions of the sun and the moon, to their distances from earth, to their individual cycles, and many other measurable parts of reality, in all its dimensions?

It is well known, from the work of Hero of Alexandria for example, that ancient units of measurement were derived from the meridian circumference of the earth. Where do the units of measure we use today originally come from? How old are they? The foot and the mile are widely acknowledged to be very old units of measure, derived from a reasonably accurate estimate of the circumference of the earth. This is because the polar circumference is very close to being 12 x 12 x 12 x 12 x 12/10 = 24,883.2 miles. Robin Heath has suggested the equatorial circumference was divided by the number of days in a solar year, and that figure was then divided by 360,000, to produce one small unit of measurement, the foot. Hence, the equatorial circumference of the earth is 365.242199 x 360,000 =131,487,191.64 feet, or 24,902.877 miles. The exact distance around the equator is estimated today to be exactly 24,901.461 miles, and around the poles is 24,859.734 miles (producing an average of 24,880.5975 miles, which is very close to 24,883.2). It seems that units of time (days) have been combined with the girth of the earth and used to produce units of space (feet). If the foot itself is a product of temporal and spatial measurements - the girth of the Earth and it's path round the sun - perhaps something could be said of an understanding of the interdependence of space and time on behalf of its creators. Robin Heath describes the foot as ‘an astonishing enduring artefact bequeathed to us from the prehistoric world.’27 At some stage, this circumference was measured and reconfigured with the units we use today. In the case of the mile we can see that this may have been part of a recurring division into twelve parts. Though equally, you could take a theoretical value of not 24,883.2 miles for the circumference, but 24,857.954545 miles, = 25,000 x 175/176 miles. This corresponds to 1,575,000,000 inches, or 40,000,000 ‘ancient’ metres of 39.375”. Such an ancient metre is close to the modern metre, multiplied by 8,001/8,000. 28 This same value for the circumference also gives 6,000,000 Roman or Egyptian digits of 0.729166667” per degree, so 1,575,000 divided by 360, and then by six million. Jim Alison explains:

Given .5238 meters, or 20.62 inches, for the length of the royal cubit, the length of the remen is .3704 meters or 14.58 inches. The meridian circumference of the earth is 40,008,000 meters or 111,133 meters per average degree of latitude. 300,000 remen times .3704 equals 111,120 meters. 5000 remen equals one minute of latitude and 500 remen equals one tenth of one minute of latitude. Given 20.62 inches for the length of the royal cubit, the remen expresses the length of an average degree of latitude with greater accuracy than the modern meter, that was fixed before the exact length of the meridian circumference was known to its creators, and unlike the meter, the remen is in unity with minutes and degrees of latitude. Archaeological and textual evidence from throughout ancient Egyptian history, as well as textual evidence from ancient Greek and Roman sources, support a conclusion that the correspondence between the length of the remen and the royal cubit, and the meridian length of Egypt and the earth, was known to their creators.29

This suggests that accurate dimensions of the earth were known in the past. Can we infer from this that time cycles were also well known, and measured as part of this system?

The Pythagorean understanding of the motions of the planets was not heliocentric, unlike Aristotle’s, who described it as so:

It remains to speak of the earth, of its position, of the question whether it is at rest or in motion, and of its shape. As to its position, there is some difference of opinion. Most people–all, in fact, who regard the whole heaven as finite–say it lies at the center. But the Italian philosophers known as Pythagoreans take the contrary view. At the centre, they say, is fire, and the earth is one of the stars, creating night and day by its circular motion about the center. They further construct another earth in opposition to ours to which they give the name counterearth.18

This suggests a theory of astronomy which was scientific and relied on observation and reason. The Copernican revolution was a reaction to Christian Aristotelian views, and does not necessarily imply that no-one before Aristotle’s time understood that the earth and other planets revolved around the sun. The anti-scientific spirit of large sections of Christian history should not be confused with the sum total of human history and prehistory.

Stephen Dail has observed that the Chinese I Ching mentions a 13 Month calendrical system composed of 384 days, which seems fairly inaccurate, but when rectified by the number of changes in the I Ching structure of x 4095/4096, it reveals the same Lunar Synodic Month of 29.53125 days used in Egypt and elsewhere in the ancient world system.

In 1775, Jean-Sylvain Bailly, wrote a series of letters to Voltaire, in which he described an ancient, long forgotten, highly competent civilisation. He pointed out that the Chinese believed that their first three emperors had a perfect knowledge of astronomy. Bailly wrote two large volumes on astronomy, one dealing with ancient astronomy and the other the astronomy of his day. He noted that in his own time the Chinese were convinced that to regain some of this knowledge you had to look for it in monuments and the I Ching.

Bailly wrote that when you look at the state of astronomy in China, in India, in Chaldea, "we find the debris rather than the elements of science", and he adds that if you see the cottage of a labourer made of stones, but mixed in with fragments of beautiful columns, would you not conclude that this was the debris of a palace, built by an able architect, from a time before the builders of the cottage? He goes on to say that what accurate knowledge there is has been inherited from a long time before. Voltaire wrote to Bailly that his book was not only a master-piece of science and genius but also one of the most probable systems, and would bring him infinite honour. Since then, the clues left to us by Bailly about an ancient advanced civilisation have not been very much examined or discussed.

“Man, temple and cosmos were therefore seen to be identical, and on this understanding the entire philosophy and science of the ancient world was founded. Unless the justification for this point of view is appreciated, it is impossible to gain any deep insight into the history of the past or to sympathise with the re-establishment of mystical perception at the present. Scholars who like to trace the progress of science from the first manufactured tool to the most recent agent of world destruction have cruelly misunderstood the geocentric cosmologies of antiquity, which were adopted for philosophic reasons, not as models of physical reality. The medieval Church declared the earth to be at the centre of the universe as a matter of actual fact, to the disadvantage of those who knew otherwise; but it is obvious from out knowledge of their achievements that the early Chinese and Chaldean astronomers could never have thought such a thing and the subtleties recently discovered in the megalithic yard...”19

Alexander Thom, who discovered the megalithic yard and studied hundreds of megalithic structures, observed:

“As the investigation advanced it became evident that I was dealing not with monuments orientated for some ritualistic purpose but rather with the remnants of a scientific study of the Moon’s motion. When this is recognised, it will be found that a great mass of material falls into place.”20


One ring to rule them all


John Michell wrote:

The measurement of time, cosmic distances and the body of canonical man were formerly united in one scientific system by the canon of numbers which was common to them all. The beauty of the ancient units, some of which still survive as the English inch, foot, furlong, mile, etc, is that they occur naturally in the measurement of all classes of phenomena from the human to the astronomical scale. 30

Whoever designed this system of numbers must have this over-arching goal in mind, a system to measure everything, based partly on the number 54, partly on the factorials of the numbers from 1-10, and partly on other significant numbers such as 3168, and 12. Whether they actually occur naturally as Michell claims, or whether this system involves a little tweaking here and there to make the various natural phenomana and objects fit, is open to question. Ratios such as 4375/4374, 3025/3024, 176/175 are very helpful in refining these values, and are signifcant in themselves as they are connected to music theory, or to useful approximations of pi, or Phi, such as 22/7 and 25/8, or 144/55 and 55/21.

Natural cycles such as a lunation, which is estimated as 29.53059 days, can be adjusted slightly to fit into this system of numbers, to 29.53125 in the moon's case, or 181440 / (8³ x 12). Multiplied by 25/27 we get 27.34375, which approximates the sidereal lunar month. Likewise, a solar year can become 25920 x 70 x 7875 / (7776 x 5040) = 364.583333 days. Stephen Dail has put forward some very interesting combinations of these numbers to express the dimensions of the planet, and even the speed of light, taking John Michell's work to a new level:

Earth’s polar circumference = 29.53125 x 400,000,000 x 1224/1225 x (77/1800) x (560/561) x (22/7) x (175/176) inches.

Speed of light = 29.53125 x 400,000,000 x 1224/1225 inches per second

The beauty of this system is that cycles of time and dimensions in space can be related by simple fractions, even including the speed of light.


We might even express pi and Phi in terms of these numbers.

Phi as 55/34 = (22/7) x 29.53125 / 57.37

Pi as 22/7 = 55/34 x 1224/1225 x 210 x 39.375 / (144 x 29.53125)


The canon as a code of number to live by


John Michell writes about "a cosmology or model of the universe", adding:


Plato refers to it in the Laws (656) as a canon of sacred music. By control of music, he says, and by licensing only such compositions as create harmony in the soul and in society, the ancient Egyptians preserved their civilization from corruption for ten thousand years. In the first instance, the canon was a code of number, from which were derived the psychologically beneficial types of music and the geometric and architectural proportions that are most appealing to the aesthetic eye. The ancient philosophers understood the structure of number to be analogous with the structure of creation, and they realized also that number is basically duodecimal, being naturally governed by the number twelve. By reference to their sacred canon of number, they ordered their societies and all their institutions within a duodecimal framework. Through their science, and by constant vigilance against the corruption of its standards, they cast a spell over whole countries and spread a golden-age air of enchantment across many generations.

From "Twelve-Tribe Nations: Sacred Number and the Golden Age" by John Michell.


It could be that because these numbers were considered so important, they became part of the fabric of daily life for a long time, and that is why we find so many of these numbers in ancient structures as units of length as well as in myths and religions, and some ancient texts, including Plato's work. These numbers would have been considered essntial to create harmony, be it in the soul or in society. We can see that the canon formed a code, by which all things important to a good life must conform.


Empirical and thoretical metrology


Perhaps we can make a distinction between empirical and theoretical metrology. We could define theoretical metrology as the process of building a model, and empirical metrology as the process of testing that model, to see if it actually explains the phenomena we are trying to understand. If the canonical numbers constitute the backbone of a theoretical system of measure, it is interesting to see how closely this fits what is found in historical measures and ancient sites. This would reflect the difference between the ideal world, in which it might be possible to know the value of an irrational number, and the world that we actually live in.


Conclusion


The canonical system of numbers is mysterious in terms of its origins, but allows us to question received notions of the beginnings of science




Many thanks to Stephen Dail for his help with researching canonical numbers



Notes



1 Cornford, F. M. “Mysticism and Science in the Pythagorean Tradition.” The Classical Quarterly, vol. 16, no. 3/4, 1922, pp. 137–50. JSTOR, http://www.jstor.org/stable/636499. Accessed 23 June 2023. Mysticism and Science in the Pythagorean Tradition on JSTOR

2 Plato, Laws, 360 BC, Book II, translated Benjamin Jowett, The Internet Classics Archive | Laws by Plato (mit.edu)

3 F.M. Cornford, 1922, “Mysticism and Science in the Pythagorean Tradition”, The Classical Quarterly, vol 16, no 3/4 *Jul/Oct) pp 139-140

7 Plato. Plato in Twelve Volumes, Vol. 9 translated by W.R.M. Lamb. Cambridge, MA, Harvard University Press; London, William Heinemann Ltd. 1925.

8 City of Revelation p 129

9 Le Gentil, Voyage dans les Mers de l’Inde, Tome 1

10 See Appendix for an account of his education by the Brahmins.

11 Vishnu Purana and Bhagavata Purana

12 Michell, John, City of Revelation p 129

13 Ibid p 94

14 William Stirling, The Canon,. p96

15 Ibid. p 104

16 William Stirling, The Canon, p 137

17 Ibid. p 161

18 Aristotle. "13". On the Heavens. Vol. II.

19 Michell, City of Revelation, p 23

20 Alexander Thom, quoted in Kenworthy, David. Stonehenge: Cracking the Megalithic Code.: Book Three: The Eclipse (Kindle Locations 102-106). Kindle Edition.

23 Plato. Plato in Twelve Volumes, Vol. 9 translated by W.R.M. Lamb. Cambridge, MA, Harvard University Press; London, William Heinemann Ltd. 1925.Plato, Timaeus, section 39d (tufts.edu)

27 Heath and Michell, 2006: 43

28 There is of course a difference between the polar and equatorial circumferences of the earth. It is open to debate whether people in megalithic times knew the difference, or thought of the earth as a perfect sphere. At any rate, both a 24,883.2 mile and a 25,857.954545 mile estimate are close to the current value of 40,007.863 km or 24,859.734 miles.

29 ” The measure of the remen and the royal cubit and the meridian of Egypt and the Earth”, by Jim Alison, 2020, http://home.hiwaay.net/~jalison/blu5.PDF

How is it that the moon’s radius is 1080 miles? According to William Stirling, the “word uttered” is (nPO- ^OPIKON, 1,080)14, adding that “TO HNETMA 'AFION (the Holy Ghost) has the value of 1,080, which is the number of miles in the moon's radius. She is thus also a personification of the moon, whom the ancients regarded as the wife or sister of the sun.”15

30. John Michell, City of Revelation, p 129

31.Alison, Jim, Earth Measures, C:\art\R28.wpd (hiwaay.net)

32. Heath, Richard, 2014, The Harmonically Numerical Form of Creation (and why it used to matter), (8) HEATH_Harmonically Numerical Form of Creation_ICONEA_2014_v1-4.pdf | Richard Heath - Academia.edu


Appendix 1


For a brilliant in depth explanation of music theory and Pythagoras, watch Norman Wildberger's videos, especially this one:


Appendix 2


Article from The Guardian


Plato's stave: academic cracks philosopher's musical code


Historian claims Plato's manuscripts are mathematically ordered according to 12-note scale

Jay Kennedy, a historian and philosopher of science, described his findings as "like opening a tomb and discovering new works by Plato."

Plato is revealed to be a Pythagorean who understood the basic structure of the universe to be mathematical, anticipating the scientific revolution of Galileo and Newton by 2,000 years.

Kennedy's breakthrough, published in the journal Apeiron this week, is based on stichometry: the measure of ancient texts by standard line lengths. Kennedy used a computer to restore the most accurate contemporary versions of Plato's manuscripts to their original form, which would consist of lines of 35 characters, with no spaces or punctuation. What he found was that within a margin of error of just one or two percent, many of Plato's dialogues had line lengths based on round multiples of twelve hundred.

The Apology has 1,200 lines; the Protagoras, Cratylus, Philebus and Symposium each have 2,400 lines; the Gorgias 3,600; the Republic 12,200; and the Laws 14,400.

Kennedy argues that this is no accident. "We know that scribes were paid by the number of lines, library catalogues had the total number of lines, so everyone was counting lines," he said. He believes that Plato was organising his texts according to a 12-note musical scale, attributed to Pythagoras, which he certainly knew about.

"My claim," says Kennedy, "is that Plato used that technology of line counting to keep track of where he was in his text and to embed symbolic passages at regular intervals." Knowing how he did so "unlocks the gate to the labyrinth of symbolic messages in Plato".

Believing that this pattern corresponds to the 12-note musical scale widely used by Pythagoreans, Kennedy divided the texts into equal 12ths and found that "significant concepts and narrative turns" within the dialogues are generally located at their junctures. Positive concepts are lodged at the harmonious third, fourth, sixth, eight and ninth "notes", which were considered to be most harmonious with the 12th; while negative concepts are found at the more dissonant fifth, seventh, 10th and 11th.


Appendix 3


Voyage dans les Mers de l'Inde

by Le Gentil

Extract from tome 1, translated from the French

I amused myself also during my days in Pondicherry, by gaining some knowledge of the Astronomy, Religion, Manners & Customs of Tamil Hidians, whom we very improperly call Malabars. What I had heard of their Astronomy had piqued my curiosity; but what spurred me on on was the ease with which I saw calculating in front of me, at one of these Indians, an eclipse of the Moon which I proposed to him, the first which came to my mind. This eclipse, with all the preliminary elements, did not cost him three quarters of an hour of work. I offered to put me in a position to do as much, & to give me an hour of his time every day. He conferred there; & having asked him in how how long I could hope to be able to calculate an eclipse of the Moon, according to his method, he answered me, with an air that would reflect my self-love a little, that with training, I could do as much as he did after a few weeks. This response did not put me off, it only made me even more curious. I decided to take every day, for about an hour, my Indian Astronomy lesson. Either it was my Master's fault, or it was mine; or it was that of the Interpreters, (I changed it up to three times) I needed more than a month of work, an hour a day, to be able to calculate an eclipse of the Moon, although the method has since seemed to me very simple & very easy. The solar eclipse gave me so much more trouble, because the calculation is much more complicated. As for the accuracy of this method, the agreement with the observation seemed to me very regular in lunar eclipses: the error, in several that I have calculated, does not go up to more twenty-five minutes an hour. For solar eclipses, the calculation deviates more, which does not come so much from the time of the true conjunction, that of the method of calculating the apparent conjunction & the other phases of the eclipse. They have agronomic Tables of the true & daily movement of the Moon, which seemed to me to be made with great art: it is a result, or a combination of the true motion of the apogee of the Moon & of its proper motion.

This combination produces a true period of two hundred and forty-eight days. It is easy to learn about it by consulting our Astronomical Tables of the mean motion of the Moon at its apogee. It is generally believed that the Brames, Brahmins or Bramines, descend from the Bracmanes, ancient Philosophers of Asia. If this fact were well confirmed, and if the knowledge that the Brahmims posed today were reflections of those of the Bracmanes, these refects would be very precious for the history of Astronomy; there would be no reason to doubt that

"The Bracmans are not very versed in this science & they had not made some very interesting discoveries. We see nothing in antiquity to prove to us that the Egyptians ever knew the precession of the Equinoxes; but we find it known among the Bramins. They suppose that the Stars advance annually by 54 seconds from West to East; that's it, not only the base or the foundation of their astronomical calculations, but also of their belief during the time of creation. By means of this movement of 54 seconds, they formed periods of several million years; they introduced them into their religion, as indicating the age of the world, what it must still last;- The Brahmins take great pains to teach these daydreams to children in schools. It does not seem easy to me to know from where the Brahmins have drawn this precession from the Equinoxes of 54 seconds per year, all the more so because they do not know practical Astronomy. If they observe the eclipses of the Sun and the Moon, it is solely for a reason of religion; but if we suppose that this precession of the Equinoxes of 54 seconds their comes from the Brahmins, and that these recognized this movement by a long flight of observations, the annual movement of the Stars would be slower today than it would not have been then, since it is only found 50 seconds; but we can't guess anything on a subject as obscure as that one seems to me to be. Here however some thoughts that have come to me since I wrote this, and which I submit to the judgement of my readers. The principal periods which the Brahmins use, & from which their other periods seemed to me to derive, make ten years Tome L F 42 Travel & of three thousand fixed hundred years; but I find in Berose, a Chaldean author, two similar periods; the neros of fifty years, & the saros of three thousand six hundred. But both periods of sixty & of three thousand six hundred years, are exactly contained in that of twenty-four thousand years, coming from the annual movement of the Stars of 54 seconds. I conjecture that the neros & the saros of Bérose have the same movement in principle, and that the ancient Chaldeans knew the precession of the Equinoxes! I will detail this idea in my Astronomy of the Brahmins. With these acquaintances worthy of our attention, remarkable for their antiquity, the Brahmins do nothing. touching the Comets: the Indians believe that these make kinds of vines of the wrath of Heaven. They were amazed at see me spend part of the nights to observe the comet which appeared in 1769; they asked me many questions about the cause of this phenomenon. What finished astonishing them was to see this Comet again at the end of the month of October in the first days of November, in accordance with what I had predicted to them, as well as to all of Pondicherry. Although the Slabs are not fervent, they have to trace there Meridian line by means of the gnomon; they are used every time they build a pagoda, because their religion teaches that the temples are oriented according to the four cardinal points; strong as the four faces of the pyramids that fervent entrance & portal to their pagodas, are exactly North & South, East & West. So the Egyptians don't make the fires we should admire for having oriented their pyramids according to the four cardinal points; maybe they won't even be the first who have practiced this method. The Brahmins calculate in a very ingenious way (in supposing the length of the gnomon's shadow on the day of Equinox), sunrise & sunset times for in a given day. This calculation, which is indispensible to them for that of the eclipses of the Sun &: of the Moon, supposes the obliquity of the ecliptic to be more than 24 degrees. The use of the gnomon among them goes back to great antiquity, if they are always fervent in it, as there is reason to presume, to orient their pagodas.

The little knowledge that I was able to draw from religion of the Talmouts Indians, made me read in Pondicherry the sixth volume of Religious Ceremonies of the Different Peoples of the Earth, Paris edition, by M-'"' l'Abbé Banier

& the Maserier; I perused this volume pen in hand, & I made some remarks which will be seen widespread in the first part of this volume. On my return to Paris, Mr. Pingré lent me from the Sainte-Geneviève Library, a Dutch edition made in 1723, of this same book, which I still have in my hands. This edition, besides its beauty has the advantage of being more in conformity with the truth, at least in many things that I have been able to check. The Bramins make the repositories of the Astronomy of Indians & their religion: it is a type of reference referred to only by this Cast. My Interpreter brought me one day a Brahmin who lived at Karîcal, a town thirty leagues in the south of Pondicherry. He told me that this Brahmin had come on purpose to see me: perhaps he imagined that I was some French Brahmin. He asked me very few questions, and seemed to me not very curious; I felt the thing for which he showed admiration, was to see through the telescope of my quarter-circle, the roadstead in an inverted position.

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