The Kālacakra world model is not mechanical
The following is a paper prepared for presentation at the Central Institute of Higher Tibetan Studies in Sarnath, India.
The short version of this paper would read: the Kālacakra system possesses a model of the world system, but it is qualitative, and not a mechanical model that could be used to derive calculations of solar, lunar and planetary positions. The Kālacakra system in fact lacks such a model, and the calculations used stem from older models that are not now preserved in the Kālacakra literature.
I should describe here first just what I mean by a mechanical model in this context. In modern physics, the model could be called dynamic. The Sun, Moon, Earth and other planets are described in terms of their sizes, masses, positions and velocities at any moment in time. The gravitational forces between them are described in terms of mathematical formula, and from all these, their accelerations (changes in velocity) can be calculated and therefore their later positions.
This has now been achieved to a very high degree of accuracy, and systems are in place for calculating planetary positions several thousand years, both in the distant past and in the future. These modern systems are very complex, and as one who has computerized the most up-to-date formula for deriving planetary positions, I can testify to their complexity – the longitude of the Sun alone contains over a thousand trigonometric expressions. It is these calculations that I have used in my interpretation of the Kālacakra calendar on this website.
Basically, given any one planet, from the dynamic information of its position, mass, velocity and the gravitational forces acting on it, we can calculate its future position.
It has been recognized by many writers that many of the early astronomical systems in India are derived from Greece, specifically the pre-Ptolemaic system of simple epicycles. These early Greek systems conceived of physical invisible spheres or circles that determined the positions of the planets. In these systems, the planets were considered to move in circles around points that themselves moved in circles around the Earth. The relative radii of these circle made a reasonably good model for the motion of the planets, and based on these, mathematics could be constructed that enabled the calculation of the real, as opposed to mean, motions of the planets. To a first degree of approximation, this can work pretty well.
The accuracy that the Greeks achieved was due to a combination of two things. They combined their geometric models of the solar system with the Babylonian sexagesimal (base 60) number system sometime in the 2nd century B.C.E. On this, Glen Van Brummelen writes: "The capacity afforded by efficient numeration transformed their astronomy, from the essentially qualitative model-building science of Eudoxus and Autolycus to the more familiar quantitative one of Hipparchus and Ptolemy. The explanatory models now became capable of predicting positions, not merely mimicking behaviour."
Many consider that this was the birth of true science. A model representing an aspect of the universe was constructed from which predictions could be made of planetary positions. The model was then adjusted in the light of observation, and quite reasonable accuracy achieved.
These pre-Ptolemaic calculations are the eccentric and epicycle-on-deferent techniques of Hipparchus and Apollonios. For what we would nowadays call the heliocentric component of a planet's motion, the planet is conceived as rotating in a circle, the centre of which is offset from the Sun. This is also exactly equivalent to having a point move around a circle centred on the Sun, and the planet move around this point in a smaller circle. If the centre of this smaller circle rotates once around the Sun every time the planet itself rotates around that point, then the planet describes a circle that is offset from the Sun. (See the diagram and explanation below.)
The deferent refers to the simple circle or sphere that is centred on the Sun or Earth and the epicycle is the smaller circle around which the planet moves. With the period of revolution of the deferent being equal to that of the period of the planet, and the epicycle having a period equal to that of the Earth, this model creates the important variation in the apparent speed of the planet, including the retrograde motion when the planet appears to move backwards against the stars.
The combination of these two – eccentric and epicycle-on-deferent – gives a good first approximation to the motions of the Sun, Moon and planets. Ptolemy later enhanced these concepts and created a much more elaborate system with many further cycles within cycles, but it is clear that the Kālacakra system uses mathematics that matches closely these pre-Ptolemaic systems – this shall be demonstrated later in this paper – but the Kālacakra contains no reference whatsoever to any such underlying model.
There is a model of the world system given in the Kālacakra, but it bears no relationship to the calculations; it cannot be used as the basis for any calculation. Interestingly, this model combines elements that are to be found in both Hindu and Jain systems as well as Buddhist. At the bottom are the disks of the four elements, with earth sitting at the top, and water, fire and wind beneath, each one having a larger diameter than the one above it. In the middle of the 100,000 yojana-diameter disk of earth sits a colossal five-peaked Mt. Meru, circular in cross-section, and 100,000 yojanas in height; it is wider at the top (50,000) than at the bottom (16,000).
Meru is surrounded by six alternating sets of continents, oceans, and mountain ranges. First is the circular continent of Candra, then the sweet ocean surrounded by the circular Nīlābha mountains. These are surrounded then by the continent of Sitabha, and so on. The diameter of the outermost Sīta mountains is 50,000 yojanas. This leaves a ring of ground, 25,000 yojanas wide, on which sit the 12 main continents, including our Jambudvīpa, sitting in oceans covering the rest of the upper surface of the disk of the earth. These comprise the seventh major continent. The rivers on these 12 continents flow into these oceans, which in turn flow into the great salty ocean that rises up from the protruding lower disk of the water element; this is the seventh great ocean. Finally, fire rising up from the disk of fire to a level somewhat above the level of the upper surface of the disk of earth forms the seventh great mountain range.
I have described this at some length in order to put the following in context: at a level approximately two thirds of the way up Mt. Meru circle the Sun and Moon, in circular orbits that are offset from the centre, that themselves also slowly rotate around Meru. This motion of the Sun around Meru in one day models two natural phenomena; the first being the daily change between day and night. However, in order to make this even close to reasonable, the idea has been introduced that the light from the Sun can only travel a certain distance – otherwise darkness would only exist in the shadow of Meru.
I hope the computer generated image of this makes the point clear. In this first image the Sun is over the ocean beyond the semi-circular eastern continent of Videha. From the point of view of each continent, Meru is to the north, and so the Sun is well to the south of Videha – it is winter in Videha, and approximately midday. (Notice the Moon behind the Sun in this image.) In the western continent of Godānīya – barely visible on the left – it is midnight during the summer. (Click on this link for an animation illlustrating this.)
The second image below shows the orbit of the Sun in two different positions as seen from above. Towards the bottom of the image is our triangular continent of Jambudvīpa. The blue orbit gives winter in Jambudvīpa – when the Sun is at the lowest point in that orbit from the point of view of this image, it is over the ocean, far to the south of Jambudvīpa. When it is in an equivalent position in the red orbit, six months later, the Sun passes overhead of Jambudvīpa, and it is summer.
In this way, the orbit is considered to rotate around Mt. Meru, producing the changes of the seasons, and the Sun (together with the Moon) travels around the orbit bringing day and night to the continents. The planets, stars and lunar mansions (rgyu skar, nakṣatra) are not described clearly, but are considered to revolve above and behind the orbit of the Sun.
In addition to the problem I described regarding the need to have the Sun's rays only reach a certain distance, there are three other immediate problems with this model. For a start, such a Mt. Meru would be clearly visible in the sky. The Sun and Moon simply go out of sight because of their movement beyond the distance their light can reach, and they do not actually rise and set. Finally, at the moment of full Moon, the Sun and Moon are on opposite sides of Mt. Meru, and so the Moon would be in the shadow of Meru – and as a consequence, invisible. The Moon is hardly invisible at the time of full Moon!
It is easier to see these problems nowadays with modern computer simulations, but this is not a difficult scene to visualise, and there is no reason to assume that the Indian Kalacakra experts took it literally. It seems more likely that embodying the symbolism and the concepts associated with Meru and the continents and fitting in with these long existing traditions was of the greatest importance. This is clearly not a mechanical model. So, the only cosmological model that we have in the Kālacakra system is qualitative, and not suitable as a basis for calculations.
There have been many interactions between different societies, but one relevant thread is clear. The Babylonians with their advanced calculation system (derived from the Assyrians) became quite adept at predicting the motion of the planets, based simply on repeated observations over long periods of time. The Greeks took this further by combining these methods of calculation with their concept of models describing the motion of the planets, and basing their mathematics on the geometry of those models.
These calculation methods found their way to India, but the notion of the underlying models was not accepted or not considered necessary and did not take hold in Indian systems. Indian astronomers had to fit their inherited calculation systems with the observed motion of the Sun, Moon and planets, and the Hindu astronomers ended up by creating increasingly complex calculation systems as a result. The Buddhist Kālacakra system rejected this approach, accepts the small error in the motion of the Sun, and insists on adjusting the solar longitude simply in the light of observation. It was not considered necessary to construct complex calculations; just make regular adjustments. The relative motion of the Moon was very accurate in the system, and no doubt the planetary calculations seemed similarly satisfactory. It was the Sun that really mattered.
The calculations for the five planets as given in the Kālacakra literature can be shown to be derived from the Saura Siddhānta. If one takes the data given for this siddhānta by Varāhamihira (in his Pañcasiddhāntikā) for his epoch of 21st March 505 C.E., and calculates forward to the base Kālacakra epoch of 23rd March 806 C.E., you get the same data as given in the Tantra, including the replication of an error in Varāhamihira's data for Venus.
It was mentioned earlier that the Kālacakra system uses mathematics that matches closely pre-Ptolemaic systems. In the illustration above, the eccentric motion of a planet or the Sun is depicted. In the diagram below, on the left, the Earth (E) is offset from the centre (C) of the path of the planet (P). This is the eccentric model. The diagram on the right has the planet revolving in an epicycle around a point (C) that travels around the circular deferent. In this epicycle-on-deferent model, the period of rotation of the point C around the deferent is the same as the period of rotation of the planet about C. This means that the line CP is always pointing in the same direction – upwards in this diagram. This is exactly equivalent to the eccentric model, and the planet will travel in a circle that has its centre offset from the Earth by the same amount.
To take the Sun as an example, in the diagram on the left, the point at the top of the orbit is equivalent to the birth-sign (skyes khyim, janmarāśi) of the Sun, and is the same as the Sun's apogee when it is at its furthest from the Earth. In order to calculate the true solar longitude from the mean, the difference between the longitude of the mean Sun and the birth-sign is first derived. Longitude is measured in a clockwise direction, and so this is equivalent here to the obtuse angle between CP and the vertical line. In the diagram on the right, we can take the angle PCE (the geometry is symmetrical about the vertical line.)
In the calculations, the first step is to convert from the mean position of the planet or Sun to the eccentric position – for the Sun this is the "true longitude" (nyi ma dag pa) and for a planet its "slow motion" (dal ba dag pa). We can see that the figures used in the Kālacakra literature are taken from an epicycle model by re-creating the basic calculations. All that is needed is to choose suitable radii for the deferent (EC) and epicycle (CP). We then use the cosine and sine rules to determine the angle PEC – this is the angle that has to be added to or subtracted from the mean motion to find the true slow motion. Using the cosine rule to find PE:
- PE2 = EC2 + CP2 − 2.EC.CP.cos PCE
Having determined PE, we can then use the sine rule to find the angle PEC:
The results of these calculations are shown in the following tables for the five planets and then for the Sun. The first two figures are the values chosen for the radii of the deferent and the epicycle. The first column gives the value in degrees of the angle between the mean planet and the birth-sign. The difference between each is the size of one zodiac sign, 30°. The next column gives the calculated value of PE, and the next is the calculated value, in degrees, of PEC. In the next column, this is converted into the units used in the Kālacakra literature, nāḍī (chu tshod), correct to two decimal places. In the final column are the equivalent figures given in the Kālacakra literature.
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In every single case, the calculated values agree closely with those in the Kālacakra system. We can do something similar for the Moon, this time taking steps of 13°.33, the length of a lunar mansion, given that there are 27 lunar mansion in a full circle:
The fit is not quite as good, but I do not know for certain that the originators of these numbers would have used exactly the methods that I am using. The Greeks, for example, used the chord function rather than sine and cosine. Also, the symmetry in the Kālacakra figures cannot be right – the same value 5 for the last row – half a mansion from a half-circle. This has to be wrong. If instead we take 28 mansions in a full circle, we get:
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