Thursday, December 26, 2013

Why 108 ?




            In the Hindu-Buddhist civilizational sphere, the number 108 is among the most sacred and appears as the true or fictitious cardinal number of all manner of philosophical sets and religious series.  What makes this number so special?  We will try to position ourselves as best as possible into the minds of profundity-oriented symbolists in order to extract their kind of meaning from this unusual number.



Insufficient reasons 


            Since matters of religious symbolism typically attract self-styled esotericists who have a dislike for serious logical ratiocination, the usual explanations of the sacred status of the number 108 don’t amount to much.  Just as those people “explain” the status of the number 12 by merely enumerating “the 12 months, the 12 apostles” etc., they will merely enumerate a number of instances where the number 108 is in evidence.

Thus,  I have seen it claimed by Western esotericists that 108 = 6² + 6² + 6², the sum of the squares of the three equal numerals making up the Biblical-Apocalyptic “number of the Beast”, 666.  The calculation isn’t incorrect, but it remains unclear why squares should be counted, or why 666 should be of any importance.  This number has some strange numerical properties in its own right, but they are not the reason for its popularity among the mystery-minded.  The real reason is that through a numerical-alphabetical manipulation best known by its kabbalistic name gematria, the number refers to a  Roman emperor disliked by the first Christians who called him “the Beast”.  Emperor or “Caesar” seems to be the meaning of the component 60, this being the numerical value of the Greek letter Ksi, shorthand for its Greek rendering Kaisar.  This probably doesn’t refer to Nero, as has been assumed for long, but to “Divus Claudius”, the 6th emperor, who came after Julius Caesar, Octavian August, Tiberius, Germanicus (who received the imperium maius but was murdered; his caesarian status is shown by the fact that the succession devolved to his son:), Caligula.  Even Claudius’ mother considered Claudius ugly like a beast, and his initials DC were read in Latin as the number 500 + 100 = 600.  So, probably 666 means “the 6th Caesar/60, named D.C./600”. 

Admittedly, there are other explanations, but whatever the details, the reference to “the number of the Beast” is merely an expression of the abysmal hatred of Rome by some early Christians, a mere footnote in history.  At any rate, the abusive religio-political slogan “666” is totally irrelevant to any serious philosophy.  It is equally irrelevant to Hindu-Buddhist culture, and relating it to the status of 108 is anachronistic since it appeared on the scene centuries after the number 108 gained its aura of sacredness.

            Even where the explainers try to prove their point with proper Hindu-Buddhist examples, they fail to get to the bottom of the matter.  Thus, the Upanishads (philosophical texts completing the Vedic corpus) are classically counted as 108, eventhough their actual number, depending on which ones you include, can range from 13 to more than 200.  Rosaries east of the Indus have 108 beads, the Nepalese parliament has 108 seats.  The poses of Bhârata Natyam dancing, the number of gopî-s (cowgirls) enamoured of Krishna, the number of sacred sites in the tradition of Vishnu worship, the number of Buddhist arhat-s (realized saints), and many other sacred sets are all conventionally counted as 108.  Likewise with derived numbers, e.g. the Rg-Vedic verses are conventionally counted as very approximately 10800, the Purâna-s as 18, the Bhagavad-Gîtâ has 18 chapters, and many Hindu monks carry titles like “Swâmi 1008 Padmânanda”. 

All very fine, but that list doesn’t explain anything.  The number 108 has been chosen in these instances, and often forced upon rather unwilling sets as their cardinal number, because it was already a sacred number to begin with.  We need something more objective as a basis for the special status of this number.

Among attempts to find a more solid basis, we still have to be wary of cheap and easy proposals.  Thus, I have seen it claimed that “108 x 20 = 2160, the number of years spent by the equinox in each Zodiac sector”, on the assumption that the total precessional cycle takes 2160 x 12 years, i.e. 25,920 years or neatly 1° per 72 years (and the extra assumption that the Vedic seers knew and cared about the precessional cycle).  But in reality, the cycle takes ca. 25,791 years, which doesn’t yield any round number when divided by 108.

It is already better to note that 108 lurks in a corner of the Hindu (or actually Indo-European) number 432 with any number of zeroes added.  Thus, 432,000, the number of years sometimes attributed to a Yuga, a world age, which happens to be equal to the number of guardians of the Germanic Walhalla or heaven (viz. 800 for every one of its 540 gates), can be analysed as 108 x 4000.  Or as 18 x 24000, for that matter.  This is true, but is it important?  Many numbers are related to other numbers.  Is it relevant to anything?

So, we will try to do better than that and give correct data which underlie the special status of our sacred number in a more compelling manner.  We will distinguish between a pair of contingent astronomical facts singling out the number 108 and four mathematical properties of the number 108, two of these conditional and two unconditional.



Solar and lunar distances


It could have been otherwise, but it so happens that the distance between the earth and the sun equals about 108 (actually 107-odd) times the sun’s diameter.  Likewise, it so happens that the distance between the earth and the moon equals about 108 (actually 109-odd) times the moon’s diameter.  That sun and moon look equally big in the earthly sky is the immediate result of their having the same ratio between distance and diameter.  Moreover, it so happens that the sun’s diameter approximately equals 108 times the earth’s diameter.

These are contingent data, which means that they could have been different.  And they are subject to change, meaning that if you look deep enough into the past or the future, you find values considerably different from the present ones of ca. 108.  While the distance between the sun and its planets is fairly stable, the distance between the earth and the moon is subject to steady and ultimately very sizable changes.  In the times of the dinosaurs, the moon was so close to the earth that a lunar revolution (i.e. a month) took only a few earthly days, with the days themselves also being shorter than today.  In the future, the lunar revolution will take thirty days, forty days, etc.  Its distance from the earth will then equal 110 lunar diameters, 120 etc. 

It is a cosmic stroke of luck that the solar and lunar distances happen to match the number 108, a remarkable number for non-contingent reasons we will discuss below, right at the time when life on earth was reaching a level of intelligence sufficient to start astronomical observations and wonder at this coincidence.  Just as it is a cosmic stroke of luck that in this same age, the moon is at such a distance from the earth that its annual number of revolutions is approximately 12, another number with unique non-contingent properties.

Can we be sure that this remarkable astronomical state of affairs has played a role in the selection of 108 as a sacred number?  Did the ancient Indians know about the moon’s diameter or its distance from the earth?  According to Richard L. Thompson (Mysteries of the Sacred Universe, Govardhan Hill Publ. 2000, p.16, p.76), the medieval Sûrya-Siddhânta gives an unrealistically small estimate for the distance earth-sun, but the estimate for the distance earth-moon and the lunar diameter differs less than 10% from the modern value.  The ratio between distance and diameter of the moon is implicitly given there as 107.5, admittedly a very good approximation. 

However, I have never heard of any text, whether from the Vedic or the medieval period, that explicitly derives the importance of the number 108 from these or any other astronomical data.  But this is merely an argument from silence, with limited proof value.  For on the other hand, the estimation of the relative distance of sun or moon isn’t that difficult to calculate even without any instruments: “Take a pole, mark its height, and then remove it to a place 108 times its height.  The pole will look exactly of the same angular size as the moon or the sun.” (Subhash Kak: “Shri 108 and Other Mysteries”,, 27 Nov. 2001)  Also, in some respects the Vedic-age astronomers were more advanced than their medieval successors, who had jettisoned part of their own tradition in favour of Hellenistic import. 

So, it remains speculative but quite possible that the solar and lunar data were estimated with a good degree of accuracy at the time when 108 was selected as a sacred number.  But it is also possible that the selection was made purely on the basis of the non-astronomical considerations discussed in the following sections.   



Big 1, little 8


            One of the arithmetical properties of 108 is dependent on the choice of counting system.  In the near-universally used decimal counting system, the quantity 108 is expressed as “108”, meaning “1 hundred, 0 tens, 8 units”.  In other counting systems, it would look different, e.g. in a duodecimal (12-based) system, it would be written as “90”, and in the binary system, it is written as “1101100”.  Assuming the conventional decimal system, what is remarkable about “1-0-8”?

            Like 18, it brings together the numerals 1 and 8, with the former in the leading and the latter in the lowly position.  The main difference (valid even more in subsequent numbers like 1008) is merely that an abyss of worshipful distance is created between the regal 1 and the servile 8.  So, let us briefly focus on this symbolism of 1 and 8.  It is chiefly remarkable as a reference to yet other important symbols.


                        8  1  6

                        3  5  7

                        4  9  2


            In the magic square of 9, there is 1 little square in the middle and 8 on the periphery.  Also, the 1 central number is 5, the sum of the 8 peripheral numbers is 40, yielding a ratio of 1:8.  The magic square itself, with equal sums of the three numerals on every line, is an important symbol of cosmic order, balance and integration.  Painted on walls or wrought into little metal plates it is used as a luck-charm. 

Moreover, consider the sums in the magic square, adding the central number and a number in the middle of the sides, yielding the number in an adjoining corner (counting only the units):  5 + 1 = 6; 5 + 7 = 2 (12 modulo 10, as it were); 5 + 9 = 4 (14 modulo 10); 5 + 3 = 8.  If you draw lines following the numerals in these sums, you get the Swastika, yet another lucky symbol in Hindu-Buddhist culture.

            The “nine planets” of Hindu astronomy are also often depicted in a square arrangement for ritual purposes such as the Navagraha Agnihotra (nine-planet fire ceremony), with the sun in the middle and the 8 others around it: moon, Mercury, Venus, Mars, Jupiter, Saturn, Râhu (northward intersection of lunar orbit and ecliptic, “Dragon’s head”) and Ketu (southward intersection, “Dragon’s tail”).  So, 18 or 108 may add some detail to the symbolism of 9 as representing the planets.  This is, however, of lesser importance than the magic square because the number of planets is contingent and changeable whereas mathematical properties are intrinsic and forever.



The Golden Section and 108°


            A conditional geometrical property of 108 is dependent on the conventional division of the circle into 360°.  This division is arithmetically very practical, it also alludes to the division of the year in ca. 365 days, it is now universally accepted, yet it is contingent and essentially only the result of a human convention.  At least one alternative division is known, viz. the division into 400° introduced during the French Revolution on the assumption that the division into religion-tainted numbers like 7 and 360 was less “rational” than the division into 10 or 100 or their multiples.  Hence also the Revolutionary replacement of the 7-day week by a 10-day week and the definitional choice of the meter as one hundred-thousandth of one “decimal degree” measured on the earth’s equator of 40,000 km.

            But for now, we may settle for the division in 360°.  In that case, the angle of 108° has a unique property: the ratio between the straight line uniting two points at 108° from each other on a circle’s circumference (in effect one of the sides of a 10-pointed star) and the radius of that circle equals the Golden Section.  Likewise, the inside of every angle of a pentagon measures 108°, and the pentagon is a veritable embodiment of the Golden Section, e.g. the ratio between a side of the 5-pointed star and a side of the pentagon is the Golden Section.  So, there is an intimate link between the number 108 and the Golden Section.  But why should this be important?

            The Golden Section means a proportion between two magnitudes, the major and the minor, such that the minor is to the major as the major is to the whole, i.e. to the sum of minor and major.  The general equation yielding the Golden Section is A/B = (A + B)/A, or alternatively but equivalently, X = 1 + 1/X.  In numbers, X = (1 + square root 5)/2; or decimally, X = 1, 618…  This infinite series of decimals can be replaced with a more predictable infinite series of numbers, viz. X equals the limit of the series G/F in which F is any member and G is the very next member of the Fibonacci series, i.e. the series in which every member equals the sum of the two preceding members: 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144,…  This means that every next fraction G/F, i.e. 1/1, 2/1, 3/2, 5/3, 8/5 etc. forms a better approximation of the Golden Section, whose value can be approximated to any desired degree of precision if fractions of sufficiently highly-placed members of the Fibonacci series are considered.

            In art and architecture, it is found that the Golden Proportion is naturally pleasing to our inborn tastes.  In living nature, there are plenty of sequences where every member stands to the preceding member in a Golden Proportion or its derivatives (square root etc.), e.g. the distances between or the sizes of the successive twigs growing on a branch, the layers of petals on a flower, the rings of a conch, the generations of a multiplying rabbit population, etc.  What this symbolizes is the law of invariance: in every stage of a development, the same pattern repeats itself.  The son is to the father as the father was to the grandfather.  Wheels within wheels: every whole consisting of parts is itself likewise part of a larger whole.  And the principle of order: the underling obeys the orders of his master to the same extent that the master obeys the requirements of the whole.  Or with a pre-feminist maxim: “he for God alone, she for God in him”, i.e. the wife serves the husband because (and to the extent that) the husband serves the cosmic order defining his duties.  As Confucius said, the authority of the ruler, his capability of making the people willingly obey him, is that he himself obeys the Laws of Heaven.

            So, the Golden Section is a meaningful symbol in the cosmological, aesthetical and ethical realms.  And somewhere in a corner of Golden Section lore, in the pentagon and decagon, we see the number 108 participating.  This is meritorious though perhaps a bit too indirect to count as sensational.



Sacred 9 times sacred 12


An intrinsic and ever-unchangeable property of 108 is that it equals 9 times 12, the product of two smaller sacred numbers.  It is the number of divisions in the Zodiac in the so-called Navamsha horoscope, a horoscope which Hindu astrologers always calculate along with the basic horoscope, and in which all original positions expressed in angular distance from the beginning point of the Zodiac are multiplied by 9.  This implies, for example, that a planet at 8° Aries is projected to 72°, meaning 12° Gemini.  In effect, the whole sector between 6°40’ and 10° Aries is projected onto Gemini (i.e. between 60° and 90°) and given a Gemini colouring, just as the sector between 10° and 13°20 Aries is navamsha-projected onto Cancer, etc.  This way, every one of the 12 signs is subdivided into 9 sectors, or 108 in total.  But of course, this doesn’t explain the status of 108, as the idea of subdividing the Zodiac this way apparently results from the awe in which 108 or 9 x 12 was already held.

As we have seen, 9 is the Hindu number of planets, and 12 is the Zodiac, so 108 is the total number of planet-in-Zodiacal-sign combinations.  This makes it into the total set of all possible planetary influences taken separately, or in a more generalized symbolism, the matrix containing all possibilities.  However, to purists, 9 as the number of planets isn’t good enough.  For one thing, the Hindu definition of a planet is pre-heliocentric, counting sun and moon and their two eclipse points as planets all while failing to count the earth as a planet (though it so happens that planets by the modern astronomical definition are again counted as 9, from Mercury to Pluto including the earth).  Also, planets may be added through empirical discovery, as some have indeed been in astronomy and in Western astrology; and in some rare but not-impossible catastrophe, a planet may disappear. They are only creatures, born from dust and returning to dust.  If we’re looking for intrinsic properties of numbers, we should not settle for contingent data such as the present number of Hindu “planets”.

To the unique properties of 12, revealing it to be a logical symbol of cosmic order, we have devoted a separate paper, Why Twelve?, where we have focused on its unconditional properties, both arithmetical and geometrical.  We may add that, conditional upon the choice of the decimal system, 12 is structured as 1 tenfold plus 2 units: 1 big, 2 small.  Which to a freely-associating symbolist mind can mean: 1 precedes 2 (as indeed it does, in fact it is the most elementary observation to be made in the number series, e.g. long preceding the realization that there must be a zero), unity is superior to division, oneness precedes and underlies polarity, the odd/yang dominates the even/yin.  Note however that for all their inequality, the numbers 1 and 2 and all that they symbolize are at any rate united and synthesized in the number 12 and in whatever the latter symbolizes.  Meanwhile, this conditional arithmetical property of 12 must remain inferior to the unconditional properties of 12, especially those of 12 conceived geometrically as the regular dodecagon, e.g. the fact that its construction uniquely flows automatically from the construction of the circle, keeping the same compass width; and the fact that it bridges the gap between straight/radius and round/circumference by dividing both rationally in a single move (the radius into 2, the quarter-circumference into 3).

Like 12, the number 9 has its unusual properties.  Once more, we cannot be satisfied with simply enumerating instances where 9 has been used in a sacred context: the 9 worlds of Germanic cosmology, the 9 Muses, etc.  We want objective properties, and when looking for these, we must again distinguish between conditional and unconditional properties.  Thus, it is often remarked that 9 is the highest among the decimal numerals, and hence symbolizes anything that is highest, including God, who, in comparison with anything you may propose, is always Greater.  However, this property is conditional upon our choice of numeral system, i.c. decimal rather than binary or any other: in the binary system, the number 1 would have this property, and in the duodecimal one, the number eleven would.  Likewise, 9’s property of equalling the sum of the numerals in its own multiples (e.g. in 9 x 8 = 72, we find that 7 + 2 = 9) is again dependent on our choice of the decimal system. 

For unconditional properties, we might look at some characteristics of the enneagon (9 x 40°).  This is the first polygon with a non-prime base (as distinct from the heptagon, 7 being a prime number) that eludes construction with ruler and compass.  This contrasts sharply with the division of the circle into 4 or 6 or 12, which is so simple and natural, or with its division into 5 or 10, which is more complicated but very rewarding (yielding the Golden Section) and at any rate possible.  So, 9, though analysable as 3 x 3, is elusive.  Does anyone care to read some symbolism into this property?  If 7 represents the mystical eluding the rational, what should be represented by 9, which is more structured yet equally eludes rational construction?  Let’s see: how about God, who always eludes our concepts?  Allahu Akbar, God is greater!

But we need not look that far.  Whatever else 9 may be, its most immediate arithmetical property is certainly that it equals 3², or 3 x 3.  Unlike the neat balance of even numbers like 2 or 4, suggesting stability and a waiting matrix of potentialities, the number 3 expresses motion, as even the most vulgar book on number symbolism will tell you.  The number 9, therefore, is a movement affecting the movement, i.e. acceleration.  It is dynamic par excellence, Shakti as the dynamic expression of static Shiva.  The primal form of acceleration is the change from rest to motion, i.e. setting things in motion, starting the whole process from zero.  This, of course, is the doing of the Creator who is Greater.  Or as the Scholastics used to say: God is not Potency, God is pure Act.

So, whereas 12 represents synthesis of opposites within an ordered cosmos (3 x 4, time-space, motion-structure) and harmonization of self and non-self, 9 represents unfettered dynamism, pure self-expression riding roughshod over the non-self, the joy of being entirely oneself.  It transcends and leaves behind all compromise in favour of purity and absorption.  Its structure as 3 x 3 actually explains its elusiveness: whereas divisions of any angle into 2 and its multiples are always feasible and simple, the division into 3 is impossible, though very good approximative techniques have been developed.   Even folding a letter into three requires a manual jump, an approximation rather than a slow but sure technique guaranteeing an exact division into equal parts.  The enneagon is the first regular polygon which requires for its construction the trisection of a known angle (120°, itself easily constructed though not by trisecting the “angle” of 360°), an impossible operation with ruler and compass.  Likewise, according to poets, the Absolute cannot be caught in a conceptual net but can only be approximated, hinted at, spoken of in parables and metaphors.

Let’s put ourselves into the mood of god-seekers in order to understand this.  As 9 x 12, the number 108 infuses the cosmic order represented by 12 with the god-drunkenness, the enthusiasm free of all doubts, the pure dedication represented by 9.  That makes it an excellent number for the prayer-wheel or rosary, which is used in a disciplined and systematic manner in order to lift up the spirit towards god-absorption.



Square times cube


Among other intrinsic and ever-unchangeable properties, it may be hard to choose which one is sufficiently relevant.  Thus, 108 equals the sum of the first 9 multiples of 3, viz. 0, 3, 6, 9, 12, 15, 18, 21 and 24.  This reconfirms its intimate relation with the richly symbolic number 9, but then, so what?

Slightly more remarkable is that 108 equals the product of the second power of 2 and the third power of 3, i.e. the first non-trivial even and odd numbers multiplied by themselves as many times as themselves.  In figures: 108 = 2² x 3³, or 108 = 2 x 2 x 3 x 3 x 3.  This way, it unites on their own terms the polar opposites of even and odd, the numerical counterparts of female and male, yin and yang, etc.  If nothing else, at least it’s cute, is it not?  That may well be the most we can expect of number symbolism.



(copyright: author, October 2003)


(in case we have missed some important symbolically-charged properties of the number 108, we welcome feedback)

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Why twelve ?

(Upon request, I republish an old article.)


     The present paper deals with a question of symbolism: what is so special about the number 12?  Historically, the preference for the number 12 goes back to the Zodiac.  Thus, the twelve-star flag of the European Union was designed, in a public contest, by a devotee of the Virgin Mary who thought of the Apocalypse passage where a celestial virgin appears in a circle of twelve stars.  And in their turn, these "twelve stars" (in Hebrew mazzalot, whence mazzel!, "good luck", originally "lucky star", "beneficial stellar con­figuration"), were a standard expression referring to the Zodiac, the division of the Ecliptic in twelve equal parts, each one of them represented by a symbol: Aries, Taurus, Gemini, Cancer, Leo, Virgo, Libra, Scorpio, Sagittarius, Capricorn, Aquarius, Pisces.

     Though we acknowledge the intimate connection between astrology and the symbolic structure of the Zodiac, it is outside the scope of this paper to comment on the merits claimed for astrology.  Indeed, we assume that stellar lore including the Zodiacal constellations’ names precedes its use as a tool for divination, and that it is worth analyzing purely as a symbolic construct, regardless of its use by diviners.  Contrary to what some astrologers claim, astronomy is very much older than astrology.  But unlike astrology, the natural tendency to read "faces in the clouds", or in this case images in the stellar groupings, is probably as old as star-gazing itself. 



Not-so-special properties of twelve


     The relationship of the number 12 with other numbers is interesting, but not really unique.  Thus, it is said that 12 = 3 x 4, with the added explanation that "3 represents time" while "4 represents space".  All very good, but then 10 = 2 x 5, which is not bad either and just as pregnant with number symbolism.  And note that in both cases equally, the factors when added (rather than multiplied) yield 7, that mystical number.  So, for a unique property, we must look elsewhere.

     In number theory, we do meet 12 in intriguing places.  It is the sum of the first three natural numbers satisfying Pythagoras's (actually Baud­hayana's) theorem, 3² + 4² = 5², and also figures in the next Pythagorean threesome: 5² + 12² = 13².  In Fibonacci's series, the 12th number happens to be 12², or 144; it is the only number to have this property except for 1 (for the first power, the property is shared by the numbers 1 and 5, which stands at the 5th place; for the third power, there is none).  There are twelve multiplications of natural numbers equaling 360 (1x360, 2x180, 3x120, 4x90, 5x72, 6x60, 8x45, 9x40, 10x36, 12x30, 15x24, 18x20).  All very interesting, but less telling and unique than the properties of 12 conceived as a geometrical entity, viz. as the division of the circle into 12 equal parts. 



How to divide the Ecliptic?


     The Ecliptic can be divided into any number of zones.  Well-known is the division of 27 or 28 moon-stations of about 13° each, marking the angular distance covered daily by the moon.  The division in lunar mansions links an astronomi­cal phenomenon, the moon's movement, with a division of space.  The same principle probably underlies the division in twelve: it seems to be based on the approximately twelve lunation cycles in the solar year (whose quarter-periods of roughly seven days may also be related to the division in weeks). 

     But why should immutable space be subjected to divisions suggested by the coincidental and highly impermanent data of the moon's motion?  There could well have been no moon at all (as is the case for Venusians), or mankind could have come into existence and designed a Zodiac millions of years ago, when the moon was closer to the earth and its cycle as expressed in earthly days or fractions of earthly years much shorter.

     Freeing ourselves from the suggestions emanating from accidental circumstances, we want to construct a division of the circle based on nothing but the abstract circle itself, con­sidered as a geometrical figure, hence part of a continuum of geometrical constructions.  Which division of the circle is intrinsically most meaningful to the whole project of symbolical­ly representing the diverse aspects of the universe with the sections of the ecliptical circle?



World models


     The Zodiac is devised and understood as a world model, a simplification of the infinite complexity of the phenomenal world to a scheme with a finite number of elements, which nonetheless approximates the structure of the real world in that it embodies fundamental worldly relations, starting with oppositions.  In a rational world model (e.g. the four/five elements), if we have an element meaning "big" or "cold", then we must have one which means "small" c.q. "hot", just as in real-life natural cycles, a sunrise is counterbalanced with a sunset.  The symmetry of a circle and of its rational divisions is already a good metaphor for this general symmetry requirement of credible world models.

     A world model replaces the practically infinite multiplicity of phenomena with a finite set of classes, just as a regular polygon inscribed in a circle replaces the infinite division of the circle into infinitely small sections with a finite division into discreet and finite sections.  If we study the surface of these polygons, we find that practically all of them, just like the circle itself (surface = pi, assuming radius = 1), have a surface numerically represented by a number reaching decimally into infinity (in practice represented by a finite sum involving at least one irrational root), although the surface values of the polygons, unlike those of the circle, are not transcen­dent numbers (meaning numbers which can only be analyzed into infinite sums, e.g. pi = 4/1 - 4/3 + 4/5 - 4/7... ad infinitum).

     For our project of replacing unmanageable infinity with more manageable finiteness, we find polygons with surface values consisting of a finite combination of roots and rational numbers a great improvement vis-à-vis transcendent numbers, but we would prefer polygons with even more finite and manageable measuremen­ts, viz. those which have a rational surface value.  Best of all are those with a natural number as magnitude of its surface. 

     There are three of these: the bronze medal is for the inscribed square with surface = 2, silver for the cir­cumscribed square with surface = 4, and gold for the inscribed dodecagon with surface = 3, the natural surface value most closely approaching pi.  This way, the division into twelve is not just one in a series, it is quite special and corresponds in a neat metaphori­cal way with the whole project of devising a world model.



Squaring the circle


     The second unique property of the division into twelve is that it somehow "squares the circle".  At least, it bridges the gap between straight and circular, radius and circumference.  As anyone who studied trigonome­try knows, the sinus of 30° is 1/2.  This means that, alone among the angles into which a circle can be divided, the angle of 30° combines a rational division of the circle (into 12, or of the quarter-circle into 3, the dynamic number befitting the orbit of motion) with a rational division of the radius (into 2, the static number befitting the radius as the “skeleton” of the circle).  This is a truly unique intrinsic property of the division into 12. 



Effortlessly dividing the circle


     A third special property of the division in 12 is that it is the most natural division of the circle, i.e. the one which does not require any other data (c.q. geometrical instruments and magnitudes) to get constructed except those already used in the construction of the circle itself, viz. compas width equal to the radius.  If one constructs a same-size circle with any point of the first circle's perimeter as the centre, one obtains a perimeter passing through the first circle's centre and intersec­ting its perimeter twice at 60° of the new circle's centre.  Next, these two intersection points become the centres of new same-size circles, and so on.  The result is a set of six same-size circles symmetrically distributed around the original circle, with 13 intersection points: one in the original centre, six on the original perimeter at 60° intervals, and six outside the circle.  The straight lines connecting the latter six with the original centre intersect the original perimeter exactly halfway in the said 60° intervals.  This way, the circle is neatly divided in 12 x 30°.

     Moreover, this entirely natural construction reveals a specific structure: the division in alternating "positive" and "negative" signs, being the intersec­tion points on c.q. outside the original perimeter.  The twelve intersec­tion points can also be connected to form the pattern known in India since millennia as the Sri Cakra (circle of the Goddess of Plenty) or in Jewish circles since a few centuries as the Magen David (David’s shield), i.e. a straight-standing triangle intertwined with an inverted triangle. 

     One of the side-effects, as it were, of this natural construction of the division into 12 is that it somehow achieves three trisections.  Though some hare-brained “trisectors” refuse to accept it, the ancient Greek dream of dividing a given angle into three equal angles with ruler and compass has been proven to be an impossibility.  In the present case, we have therefore not tried to divide the square angle (which wasn’t given in the first place) into three equal angles of 30°, nor that of 180° into 3 x 60°, nor even that of 360° into 3 x 120°.  Yet, it so happens that we have ended up with the angles of 120°, 60° and 30° explicitated in our construction.  What was otherwise impossible, here it’s been done.



An open-ended appeal


     We welcome feedback, as we may have overlooked something.  Possibly more special geometrical facts can be mustered to show that the division into 12 (which a temporary coincidence may be credited with suggesting, viz. through the moon's cycle as a fraction of the solar year) has a more profound, more stable and more universal mathematical basis.  But the three properties outlined here are, in our opinion, decisive.



Copyright: author, 1989. 


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Yoga in Transformation


            On 19 to 21 September 2013, a conference took place in Vienna on the subject of "Yoga in Transformation: Historical and Contemporary Perspectives on a Global Phenomenon." The organizers announced the conference thus: 

            Yoga type practices and associated teachings emerged for the first time in South Asia around the middle of the first millennium BCE. Ever since the phenomenon yoga has shown a protean flexibility and creativity. It constantly produced new forms of practice and theories depending on the changing social, religious and philosophical contexts with different uregional variations. Thus, the history of yoga is a complex and multifaceted one, and still remains far from having been exhaustively investigated.

„Furthermore, yoga today has become a phenomenon of global mass culture, which influences the everyday lives of millions of people. Against this background the investigation of yoga in both the past and the present is a task of high topical value that affects several academic disciplines.

„The past two decades in particular have brought new insights, methodological approaches and questions concerning the history of premodern yoga traditions, the interpretation of yoga-related literature, and the  impact on other Asian cultures. Furthermore, the investigation of modern transnational yoga has established itself as a multidisciplinary field of study in its own right. The motives and experiences of contemporary practitioners and their global networks are being investigated with methods of the social sciences and cultural studies. The startling results of studies on the history of modern yoga have not only caused scholarly discussions but also public debates on the relation of traditional and modern yoga, which sometimes have been politically charged, especially in India.

„The conference will explore yoga from a broad perspective: it will examine different strands of South-Asian yoga in the premodern period and forms of modern yoga, the changes that occurred within the premodern yoga practices and theories, as well as more recent developments and the current transformation of transnational modern yoga.

„For this purpose, outstanding specialists in South Asian studies, the study of religions, sociology, cultural studies, theology and history of religions have been invited to contribute their research papers. The conference will give them an opportunity to intensify their mutual communication. For those who are interested in yoga in general, it will provide convenient access to information on recent high-level research.

„Vienna has been a fertile ground for yoga studies, as is evidenced by the seminal work of the indologists Erich Frauwallner (1898-1974) and Gerhard Oberhammer at the University of Vienna, and by more recent initiatives such as the international conference on ‚Yogic Perception, Meditation and Altered States of Consciousness‘ which was held at the Austrian Academy of Sciences in 2006. The organizers of the present conference are happy to continue this tradition.“


Ideological angle

                Let us decode this introduction. „Around the middle of the first millennium BCE“ means that yoga does not predate the age of the Buddha. Anything of value should be denied to Hinduism, and if it exists, it has to be borrowed from „another religion“, viz. Buddhism. In reality, the Buddha himself  already learned at the feet of several yoga teachers, who in turn did not claim to be innovative. Asceticism was an established institution. Centuries earlier, the Rg-Veda already describes sky-clad wandering muni-s with muddy long hair, or what we now know as Naga Sadhu-s. The Upanishads describe the fundamental ideas of yoga, starting with the attention for the breathing process. The fundamental definition of yoga and the analysis of the human person (compared with a chariot) is given in the Katha Upanishad. In order to belittle Hinduism, all this history is either denied or chronologically put as late as possible, and the role of the Buddha, deemed a non-Hindu, is magnified.

                „Complex“ has acquired a specific connotation. It is often used when established scholars are confronted with hard but inconvenient evidence for a position not their own. Thus, when Romila Thapar is presented with hundreds of Muslim testimonies for wilful temple destruction, she says that communal relations in India were „highly complex“. When Michael Witzel is faced with the fact that the early part of the Rg-Veda (roughly books 6, 3,7, 4, 2) describes inner India and its animals and geography, while the later part (roughly books 5, 1, 8, 9, 10) describes western India and Afghanistan and their animals and geography, he understands the implication that the Vedic Aryans moved from East to West, and consequently declares that Rg-Vedic tradition is a „complex issue“.

“Complex“ is a tactic against something straightforward and simple. In this case, it is directed against the „essentialist“ notion that yoga is an authoritative tradition practised by specialists and respected by the common Hindu. It is meant to free Hindus from the illusion that -- perish the thought! -- yoga is a Hindu contribution to human civilization. indu scholars



Welcome Address

            Nonetheless, most participants were not aware of any political agenda, which turned out not to be so intrusive after all.
            The welcome address
 was given by Dominik Wujastyk, currently staff member of the Asian Studies department at Vienna University. After the usual polite platitudes, he said one memorable thing. As an academic who worked for long and till recently in London, he said he was very happy now to work in a free university. This means that in Austria, like in some other countries of the European continent, university education costs no more than a nominal fee. He knew grimly well that in the UK, as also in the US, people have to borrow astronomical sums for a university education.
My comment on this point: it is the Continental view that society as a whole, including those who don’t have children, has a stake in educating the next generation. Therefore, it is deemed reasonable to use taxpayer’s money for financing the universities. I, as a product of this system of low-cost schools and socialized education, support that view.
In subsequent instalments, we will discuss the important papers read at this conference.

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Wednesday, December 25, 2013

Ayodhya interview 2013

(This interview was conducted by e-mail, rendered here exactly as in my correspondence with the editor of India Facts. It was published there on 8 January 2014.)



1. It has been 21 years since the Babri Masjid has been demolished, and the issue has all but been forgotten. As one of the experts on the issue, what do you think has happened? Have the Hindus lost that zest that characterized the Ram Janmabhoomi movement?


Firstly, the Hindu masses have seen that even their leaders who led this movement have practically abandoned the issue. And those who are committed to the temple, have channeled it through official procedures that don’t requie participation of the masses.


Secondly, it is a fact that the rise in consumerism and rampant westernization have made the Hindus less Hindu, less passionate about Rama. Christian schoolmasters have this as their explicit goal: Hindu pupils are not forcibly converted to Christianity, but are taught to get estranged from and indifferent to Hinduism, to look as outsiders upon it. The same strategy is, consciously or de facto, followed by the present media, the educational and the cultural sector: to estrange the Hindus from Hinduism by feeding them Sufi music, Christian concepts of religion and general westernization. In Mumbai films, Hindu priests are subtly but systematically slandered or ridiculed to familiarize the Hindu audience with the idea that there is nothing venerable about Hinduism.



2. If a new Government comes to power in 2014--headed by whichever party--do you think it would have the political will to rebuild the temple? If yes, and if the temple indeed begins to get built, do you foresee riots and/or violence akin to the ones witnessed in the 1990s? 


That Ayodhya is a far less important issue than in 1992, makes the atmosphere more conducive to a peaceful settlement. It is less prominent in the media thanks to the fact that the dominant intelligentsia have suffered a judicial defeat, so they are more muted. So, normally the days of the Ayodhya riots should be over. The Court verdict of 2010, though disappointing for the Muslim side, already caused no riots.



3. From a historical perspective, it was relatively easy for Sardar Patel to rebuild the Somanath temple. Why did Ayodhya become such a huge problem? 


Because the secular intelligentsia felt so self-confident that they could blow this issue out of all proportion. They could reasonably have taken the position that a temple was indeed demolished to make way for a mosque but that we should let bygones be bygones. Instead, they went out of their way to deny facts of history. Rajiv Gandhi thought he could settle this dispute with some Congressite horse-trading: give the Hindus their toy in Ayodhya and the Muslims some other goodies, that will keep everyone happy. But this solution became unfeasible when many academics construed this contention as a holy war for a frontline symbol of secularism. Now that the evidence and the judicial decision have put them in the wrong, they are not so loud anymore.


4. Hypothetically, had the Ayodhya movement occurred in today's milieu, would it garner a similar kind of fervour? 


Two things have changed: the Hindu masses don’t care as much about Ram, and the artificially created doubt about the history of the site has been cleared by the excavations and the verdict of the Allahabad High Court. It should never have caused such fervor or even a controversy in the first place. After all, it is a Hindu sacred site, Hindus go on pilgrimage there but not Muslims, and functionally the mosque already was a temple. Any sensible person would have awarded the site to the Hindus without further ado. Instead, the secular intellectuals raised the issue of history, falsely alleging that it was not what people had always thought. It is not just that they were wrong on the history, it is also that this wasn’t an issue of history in the first place. The site is venerated as Ram’s birthplace, and therefore deserves protection by a state that calls itself secular, not because something happened there hundreds of years ago, but because this belief is alive right now.



5. Has the BJP and/or Sangh Parivar all but abandoned the Ram temple issue? This question is not from an election issue perspective, but from the perspective of a party which claims to speak for, and is seen as perhaps the only hope for Hindu-related causes and issues. 


The BJP had already abandoned the issue after reaping the electoral harvest in the 1991 elections. From then on they treated it as a hot potato and preferred the Courts to handle it. That is why LK Advani was in such distress when he witnessed the demolition of the temple/mosque: he was there to show that the BJP could master the Hindu emotions about Ayodhya and make the masses toe the line scripted by the elite. He didn’t expect this much of Hindu activism and certainly didn’t side with it. Today, the broader Hindu movement, not just the Sangh, feels confident that it will henceforth have its way on Ayodhya through official channels.



6. Political parties apart, has there been a gradual build up of a sort of apathy even among large sections of the Hindu society towards the Ram temple rebuilding? 


Yes. But that apathy has also developed inside the “militant” Hindu movement. The logical consequence of the Ayodhya agitation would have been a systematic look at the history of Muslim-Hindu hostility, but in reality nothing of the sort has happened. In the 1990s a few retired historians were working on Muslim history, chiefly Harsh Narain, KS Lal and of course SR Goel, but that school is long dead, and nothing has replaced it, except maybe the Shivaji museum in Pune. The Sangh has simply given this issue away to the secularists, who have filled the textbooks with their version of history, downplaying Islamic destruction and generalizing Islam’s intolerance to all religions including Hinduism.


In the very long run, of course, truth will be restored. If you can learn anything at all from history, it is that everything changes. So, the present power equation that has made these distortions possible, won’t endure forever. It is a foregone conclusion that one day, the negative role of the secularist historians will be seen for what it was. Western Indologists who chose to toe the secularist line, even against their own research findings, will not look good either. But that will only happen after they are safely dead, after enjoying a life of prestige and positions. For there is no one in sight who could threaten them, certainly not the Hindu movement.  



7. You had in an earlier interview mentioned about the moderate Muslims (for e.g. Ali Asghar Engineer) who were willing to come to a reasonable settlement. Given how aggressive Islamism has crept into India over these two decades--for e.g. the Owaisis--would you reconsider this position? 


Frankly, I know little about the internal trends in the Muslim community. I have the impression that they are investing their energy in more important concerns than this purely symbolic issue. They may have understood that the Muslim stand was unreasonable: it is a Hindu site, of great significance to Hinduism and not to Islam, so insisting on re-Islamizing a Hindu sacred site wouldn’t win them friendship or goodwill. But if they care less about Ayodhya, it means they care more about issues involving tangible power and privileges, such as reservations for Muslims.  



8. You were someone who mounted a scholarly & bold opposition to secularist historians during the Ayodhya evidence phase. However, as we notice, the same breed of historians have returned to the academia and we observe the same distortions in school and college textbooks. And this despite Arun Shourie's expose. What happened? Is it merely that the political equations returned to status quo after 2004? 


Nothing happened, that is precisely the problem. Against the great offensive by secularist historians to whitewash Islamic rule and to deny that the mosque was built on a Hindu site, the Hindu movement did nothing at all. There were some private Hindu historians, all long dead now, but they were given no organized support. That the BJP seems set to win the general elections gives little hope: they already were in power in 1998-2004 and did nothing to implement any part of their Hindu agenda, though they did provide good governance on the economic front. The one thing they did try, viz. to change the history textbooks, was a horror show of incompetence.


But maybe Narendra Modi will prove different. He has been denounced systematically for over ten years by the secularists and slandered no end in the media, to the extent that the US has denied him a visa. Usually this sort of hounding by the secularists leads a Hindu to take extra secularist positions, but in this case Modi might really remember just how vicious the secularists can be. Perhaps he will do something really Hindu for once. Only time will tell.




9. In the end, do Hindus really have any hope at all to see the Ram temple getting built? 


 The more marginal the temple becomes, the better its chances of being built. This shrill controversy wasn’t very Indian anyway.  Better to work discreetly and achieve your goal, than this banging your head against the wall.

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