Calculations for the Positions of the Five Planets, Sun, Moon and Nodes
The course of Tibetan astro studies at the Astro Division of the Tibetan Medical and Astro Institute in Dharamsala, India takes five years. Everything is calculated by hand in the traditional manner, on a wooden board covered with soot upon which one writes with a stylus. There is no actual ephemeris compiled in which to look up figures. One of the main aspects of the training is the mathematics involved in all these calculations.
The Kalachakra system, like those of the Hindu traditions, gives formulas for determining what are known as the five planets and five inclusive calendar features (gza’-lnga lnga-bsdus). The latter set are:
- The lunar weekday (gza’)
- The date of the lunar month (tshe)
- The moon’s constellation (skar)
- The combination period (sbyor-ba)
- The action period (byed-pa).
For the sake of the ephemeris, almanac and horoscopes, the five inclusive features, as well as the positions of the sun, moon, Rahu and Kalagni are calculated in terms of lunar date days, whereas the five planets, namely Mercury, Venus, Mars, Jupiter and Saturn, in terms of solar days.
The position of the sun, as that of the other planets, is calculated according to a mathematical model, as was also the case in the ancient Greek system. Thus, it is unlike the Chinese side, which derived the position and motion of the heavenly bodies based mostly on observation. Chinese mathematics, when sometimes applied, is primarily algebraic, as is Arabian. Observation was also a prominent feature of Arabian astronomy, and it was through the Arab influence that observatories were built in India in the 18th century. In earlier Hindu astronomy, as with the Tibetan, except for the examination of the shadow of a gnomen-stick (thur-shing) for calculating the ascendant (tat-kal) for use in certain prognostications, observation did not play a part. A gnomen-stick is a stick placed vertically in the ground. Based on the length of shadow it casts and how far north one's position is on the earth, one can derive figures that can substitute for those from a clock in order to determine the ascendent.
Comparison with the Hindu Calculations — The Four World Ages
The Kalachakra and Hindu approaches to the calculations for the positions of the planets are somewhat different. In the Hindu system, the universe begins with a universal conjunction, in other words, with all the planets at the same point. Eons are divided into four world ages, called “yugas” in Sanskrit, the names for each of which derive from a game of dice. These are the full, third, second and bankrupt or “kaliyuga,” according to whether the die cast in the game comes up with four, three, two or only one pip, in which latter case one lost. These four ages repeat, each lasting, according to the most popular version as found in the Sun System of Tenets (Suryasiddhanta), 1,080,000 years. The universal conjunctions recur at the beginning of each of these four eons, with the most recent kaliyuga having begun, according to this system, in 3102 BCE, the year of the war that is the subject of the Indian Mahabharata epic.
Furthermore, the number of revolutions each planet makes around the zodiac during an eon is given. Each Hindu system of astronomy differs slightly as to the length of an age, when the last one began and how many revolutions each planet makes during one. But in general, by working out the proportion of how many days have passed in the eon compared to the number of days in the eon, one can calculate the number of revolutions each planet has made, its position and the mean value of how far each travels in a day. The numbers involved in such calculations are thus quite large and unwieldy.
Also, since discrepancies in the positions of the planets were later observed, occasionally correction factors, called “seminal corrections,” were made in the form of adding or subtracting a few revolutions per eon for each of the planets. The position of the sun was always taken to be correct. One such correction was undoubtedly added to this Sun System of Tenets in around the 10th or 11th century.
In the Buddhist Kalachakra system, the calculations are also made in terms of daily motion constants (rtag-longs) for the planets, in other words, how far they move each day. No mention is made, however, of the derivation of these constants, since there is no assertion of recurring universal conjunctions. As was mentioned at the beginning of this lecture, there are two Kalachakra systems of mathematical calculations, the full tenet system (grub-rtsis) deriving from the reconstructed Root Kalachakra Tantra, based on fragments quoted in other texts, and the precis system (byed-rtsis) from the Abridged Kalachakra Tantra and its commentary, Stainless Light. Both Stainless Light and the Abridged Kalachakra Tantra appeared in India in Sanskrit by the beginning of the 11th century. In the Stainless Light, it says that a “seminal correction” must be made to the old system and, in fact, the sun’s daily motion constant in the Abridged Kalachakra Tantra is different from that reconstructed from the fragments of the Root Kalachakra Tantra. Thus, it would seem that corrections appeared in both the Buddhist and main Hindu systems at approximately the same time, although in the Hindu one, it was not the constant for the sun that was adjusted, but that of the other planets. Therefore, at least historically, it is not strange that there are two versions of the sun’s daily motion constant (nyi-rtag) in the Kalachakra system.
Since the Buddhists do not assert universal conjunctions at the start of world ages, the calculations for the positions of the planets are made in a different manner from that of the Hindus. 60-year cycles are employed, in accordance with the Jupiter “prominent” cycles mentioned before, and one must take into account the origin-sign positions of where each planet first arose, since this was the starting point for their revolutions, and also the position where each is at the end of the previous 60-year cycle. These calculations will be discussed in more detail in a moment.
The Hindu scheme of four world ages, ending in a kaliyuga, although usually not found in other Buddhist texts, is employed in the Kalachakra system. The duration of each, however, is much shorter than in any of the Hindu systems. According to Kalachakra, the teachings of the Buddha will last 5104 years, starting from 880 BCE, which is when it dates the Buddha’s parinirvana or passing away. In all the non-Kalachakra Buddhist systems, this figure is 5000 years. This period of 5104 years is divided into two cycles of the four world ages, with each of them ending in a kaliyuga. At present, we are in the kaliyuga of the first cycle, and this began in 1598 CE. In Tibetan history, that was at the start of a period of great civil war.
At the end of the current kaliyuga, in 2424 CE, the predicted war will occur with the forces of the non-Indic invaders (kla-klo, Skt. mleccha), during which help will come from Shambhala to defeat them and then a new full or golden age will begin. What is quite curious is that the year of this war and subsequent start of a new golden age coincides very closely with the beginning of the so-called Age of Aquarius as calculated from the precession of the equinox. Although the two systems derive a date for the so-called “new age” quite independently, they work out to be almost the same.
Greek, Hindu and Kalachakra Models for Planetary Motion
Let us now explore briefly the mathematical models used for the calculation of the position of the planets. Even from the point of view simply of the history of science, this is very interesting. The ancient Greeks used primarily geometry, namely different geometric proportions, to determine and describe the motion of the planets. Epicycles, a series of smaller circles along the circumference of a larger circle, were used to correct for discrepancies in the planet’s speed. The epicycles revolve in the opposite direction from that of the circle so that on one side of the circle the motion of the revolving epicycle is in the same direction and so adds to the motion along the circle, while on the opposite side it is in the opposite direction and must be subtracted.
In the Hindu systems, the sine function was developed so that the same type of correction with revolving epicycles could be made using trigonometric rather than solely geometric methods. But whereas in the ancient Greek system of Ptolemy the planets are conceived as traveling in an orbit of an eccentric circle, which is the circle described by motion along an epicycle as it revolves around the circumference of a larger circle and which has a center shifted from that of the larger circle, in the classical Hindu systems this shift from simple circular motion is not the case. Corrections for planetary speed using revolving epicycles are made, but without planetary motion being in eccentric circles described by these epicycles. Furthermore, these revolving epicycles vary in size as they are located around the circumference of the larger circle they circumscribe, so that they describe approximately elliptical rather than simple circular orbits for the planets.
The calculations in the Tibetan system, on the other hand, involve neither geometric proportions nor trigonometric functions, but are purely arithmetic, as was the case with ancient Babylonian methods. They accomplish the same corrections as in the Greek and Hindu systems, but without the explicit use of either revolving epicycles or even the conceptual framework of changing planetary speed. Rather, corrections are made in terms of variable position. Planets are sometimes ahead of or behind the mean positions predicted from their daily motion constants, and this discrepancy is what is calculated.
Planetary motion in general is circular. The orbit of the sun is again an eccentric circle described by epicycles (go-la), but here the epicycles do not revolve, affecting the speed, nor do they vary in size. Rather, the epicycles are arranged around Mount Meru, the mean center of the orbits of all the heavenly bodies around a flat earth, in an oblique or slanting fashion. More precisely, each epicycle slopes downward with its highest point being the closest to Mount Meru and its lowest the furthest away, as if arranged around the surface of a point-up cone projected up in the air and surrounding Mount Meru. The orbit of the sun, then, is not only an eccentric circle described by these epicycles but is itself oblique or slanting. This is to explain the varying height and arc of the sun across the sky at different locations and times of the year. The exact motion of the sun during the different months of the year will be described a little later.
In the Hindu systems of astronomy, as found in the 6th century CE, the earth is conceived as an immovable sphere with the sun, moon, planets and constellations revolving around it. This concept of a spherical earth might well be an indication of ancient Greek influence, since in Plato’s Timaeus, the earth is already portrayed as an unmoving sphere with the heavenly bodies revolving around it. Based on the logical argument for a spherical earth by Aristotle in the mid-4th century BCE in his On the Heavens, this theory was accepted throughout the ancient Greek and later medieval European world. On a popular level, however, the earth was conceived as flat before Columbus discovered America at the end of the15th century. This conception of a spherical earth was never found in the Buddhist systems. Nor did they conceive of the planets and stars being located in eight concentric spheres of increasing diameter around the earth, as did Ptolemy.
In European thought, it was only in the early 16th century that the theory of a stationary sun with the earth and planets revolving around it was proposed by Copernicus. In the early 17th century, it was proven by telescopic observation by Galileo, and then Kepler derived the mathematical formulas to describe planetary motion not in eccentric circles, but in ellipses.
In the Kalachakra system as preserved in Tibet, the planets and constellations are each described as having two gaits (‘gros) or types of motion. They each move by means of an orbit of wind clockwise or to the right, that is from east to west, in their daily course across the sky. This is called their wind-gait (rlung-‘gros) and, in European terms, is their apparent motion due to the perspective of the earth’s counterclockwise rotation on its axis. This motion is at a constant speed, never faster or slower.
Each planet and constellation also has its own-gait (rang-‘gros) which, except for the motion of the nodes and retrograde phases, is counterclockwise or to the left, that is from west to east. This refers to how each night at the same hour Mars, for instance, appears progressively further to the east, and how the constellations at the ascendant and descendent at sunrise and sunset also appear to have moved further to the east each night. In European terms, this is their apparent motion due to the perspective of the earth’s clockwise revolution around the sun. As my teacher, the late master astrologer of the Tibetan Medical and Astro Institute in Dharamsala, India, Gen Lodro Gyatso, used to describe it, the motion of the planets is like that of a fly walking counterclockwise on a record rapidly spinning clockwise.
Calculations for the Positions of the Planets
When daily motion constants of the sun and the other planets are discussed, these refer to their own-gait motion. According to it, each planet has a mean position in relation to the constellations each day. To determine the mean position of the sun (nyi-bar), for instance, on a particular date, one calculates how many years, months and days have passed since the beginning of the current “prominent” 60-year cycle, taking into account extra doubled months (zla-zhol), which will be explained below. Then based on the monthly and daily motion constants of the sun and taking into consideration the sun’s birth zodiac-sign position (skyes-khyim) and the leftover position (rtsis-lhag) of where the sun was at the end of the last cycle, one calculates the mean position.
But the speed of each planet’s own gait is affected by other variables, and thus most of the time the planets travel either faster or slower, and consequently are mostly ahead of or behind their mean position. For the most part, the rate of change for their going more ahead of their mean position is the same as that for their going less ahead. The same rates of change also hold true for when they are going increasingly or decreasingly behind their mean position. Thus, calculations are made by quadrants (rkang) of the planets’ orbit by using tables (re’u-mig) of quadrant-motion constants (rkang-longs), such as that for the sun (nyi-rkang). The appropriate constant is then either added or subtracted in accordance with whether the planets are increasing or decreasing their discrepancy from their mean position, and whether this is ahead of or behind it, to get, for instance, the sun’s corrected position (nyi-dag).
Thus, although neither revolving epicycles nor trigonometric functions are explicitly employed, the results derived purely by arithmetic describe the same type of accelerating and decelerating motion of the ancient Greeks and Hindus which, after all, is descriptive of elliptical motion from the European point of view. In the derivation of the tables of quadrant-motion constants, however, several steps of trigonometric calculations may have been combined.
The sun’s position is calculated as described above. The position of the moon is derived simply and directly from that of the sun. It takes into account its own daily motion constant during the lunation cycle from new moon to new moon in each successive sign as it travels further away from the sun each day after the new moon conjunct with the sun. Furthermore, additional correction must be made for how the moon itself accelerates and decelerates ahead of and behind its mean position as it goes through its lunation cycle. The lunation cycle is not the approximately 30-day one from new moon to new moon, such as from new moon in Aries to new moon in Taurus, but rather is the 28 and one-fourteenth day cycle required for the moon to return to the same point in the zodiac. The moon’s position at the dawn of a lunar date day is known as the corrected moon’s conjunct constellation (tshe-‘khyud zla-skar dag-pa) for that date.
The positions of the five planets are determined in a manner somewhat similar to that of the sun, except that they entail two position corrections rather than one. The five planets are the two peaceful ones (zhi-ba) of Mercury and Venus, and the three forceful ones (drag-pa) of Mars, Jupiter and Saturn. Once one calculates their mean position from the count of the general day (spyi-zhag) of the 60-year cycle and their individual daily motion constants within their own cycle, one does both a slow-quarter motion correction and a fast-quarter motion correction. The slow-quarter one employs a brake-point quadrant-motion constant (dal-rkang) and corrects for how the planet sometimes travels ahead of or behind its mean position, accelerating and decelerating alternately in each quarter as it orbits around the zodiac. A planet is only located at its mean position and travelling at its own uncorrected speed when it is conjunct with its own birth-sign position. Since Mercury and Venus can never be more than a relatively small number of degrees away from the sun, they can only travel around the zodiac linked along with the sun. Thus, the sun's mean position is used when making the slow quarter correction for the two peaceful planets. This is not necessary for the three forceful planets as they can be travelling anywhere around the zodiac regardless of where the sun is located.
The fast-quarter correction employs a speed-point quadrant-motion constant (myur-rkang) and compensates for how a planet travels sometimes ahead and sometimes behind the sun – in other words, faster or slower than the sun, accelerating as it rises or goes further in front of or behind the sun, and decelerating as it sets or goes less in front or behind. The only time there is no need for this fast-quarter correction, then, is when the planet is conjunct the sun. Since Mars, Jupiter and Saturn reach the extreme limit of this discrepancy with the sun's motion when they are in opposition to the sun, for them this calculation is linked with the sun's mean position. The two peaceful planets, on the other hand, can never be in opposition to the sun, and so for them this linking with the sun's mean position for the fast quarter correction is not necessary.
When a planet, the sun and that planet's birth-sign position are all three conjunct, there is no need for either a slow or a fast quarter correction. This occurs only when the planet has finished the cycle of days of its table of daily constant motion. For the forceful planets this is the time it takes for the planet to make one orbit around the zodiac, for Venus ten orbits around the sun and Mercury one hundred around the sun.
Together with the position of each of the five planets, one of the four cardinal directions is specified, referring to the planet’s directional orientation with respect to the sun. The direction also indicates which of the four gaits (‘gros-bzhi) or types of motion the planet has – brake-point or slow gait (dal-‘gros), speed-point or fast gait (mgyogs-‘gros), origin or steady gait (‘byung-‘gros), and turn-bending, lateral or zigzag (‘khyog-‘gros). This is rather complicated, but it should be noted that retrograde motion is not one of them. Although retrogradation, when a planet appears to be moving backwards from its normal motion, is known, it is neither noted nor considered important astrologically. In modern European astrology, however, it is quite significant and affects the interpretation. The positions of the north and south node planets, Rahu and Kalagni, are calculated from their daily motion constant without recourse to a quadrant-correction.
The trans-Saturnian planets Uranus, Neptune and Pluto are not discussed in Tibetan astrology, nor does there seem to be any reason for introducing them. As was explained, the Tibetan system does not conceptualize that influences actually come from the planets. Furthermore, the planetary positions calculated from the traditional mathematical formulas do not exactly correspond to the planets’ observed positions. And it still must be researched whether the original system gives the most accurate astrological information, and if not, whether the modern Hindu solution of subtracting a standard precession factor from their positions derived by European methods gives a better result. If the traditionally calculated positions are the astrologically most useful, then how could one fit in the trans-Saturnian planets? It would be very difficult to decide where to locate their positions.
It seems illogical to correct everything to the European system. No one is claiming to be able to send a rocket to Mars by relying on the Tibetan astronomical information. One must always look at the purpose of any system of information. The position of the planets is symbolic, like a map for reading the situation of the karmic potentials with which one is born. The addition of further planets into the system would not really fit.
Astrology of the Planets
Let us examine some of the features of the astrology of the sun, moon, nodes and five planets. When interpreting the periods within a lifespan ruled by the various planets, Rahu, Kalagni, Mars and Saturn are generally considered harsh, while the sun, moon, Mercury, Venus and Jupiter gentle. In a natal chart, it is slightly different. The planets are not interpreted so much in and of themselves, although there are some general characteristics for each. It is not exactly like in a European horoscope in which Mars in Aries in the first house has a certain interpretation, and this must be modified by considering the other planets it has aspects with, where aspects are defined in terms of specific numbers of degrees separating two planets.
In Tibetan astrology, for the interpretation of natal charts, each planet has, first of all, a field (zhing) of signs that it rules:
- Mars – Aries and Scorpio
- Mercury – Gemini and Virgo
- Jupiter – Sagittarius and Pisces
- Venus – Taurus and Libra
- Saturn – Capricorn and Aquarius
- The moon – Cancer
- The sun – Leo
- Rahu – co-rules Virgo
- Kalagni – co-rules Cancer.
These correspond to the rulerships in European astrology, except that the north and south nodes are not assigned rulerships in the European system, and the Trans-Saturnian planets are given co-rulerships: Uranus Aquarius, Neptune Pisces and Pluto Scorpio.
The planets are classified as peaceful or forceful:
- Mercury, Venus and Kalagni – peaceful
- Mars, Jupiter and Saturn – forceful.
This is different, then, from their classification as peaceful or forceful when ruling periods in a progressed chart, in which case Kalagni is forceful and Jupiter peaceful.
Pairs of peaceful and forceful planets are linked with four of the five Indian elements: earth, water, fire and wind. These are, in fact, the same as the four elements Aristotle postulated as composing the universe and which are then found in ancient Greek and modern European astrology. They are found in the Hebrew mystical tradition as well:
- Mercury and Mars – fire
- Venus and Jupiter – water
- Kalagni and Saturn – earth.
This is different from the element assignments of the days of the week, which are also equivalent to the names of the planets:
- Sunday and the sun – fire
- Monday and the moon – water
- Tuesday and Mars – fire
- Wednesday and Mercury – water
- Thursday and Jupiter – wind
- Friday and Venus – earth
- Saturday and Saturn – earth.
And all this is yet different again from the five Chinese elements associated with the five planets in the classical Chinese system, in which:
- Mars – fire-planet
- Mercury – water
- Jupiter – wood
- Venus – metal
- Saturn – earth.
In an astrological chart, when one of these planets is in its own field and the other planet of its same element enters that field, they become enemies and clash regardless of how many degrees separate the planets. All that matters is that they are in the same sign. The same is true if one of the pair is in its own birth-sign and the other enters that sign. The birth-sign of the planets are:
- Mars – Leo
- Mercury – Scorpio
- Venus – Gemini
- Jupiter – Virgo
- Saturn – Sagittarius
- The sun – Cancer
- The moon – Aries
- Rahu – Virgo
- Kalagni – Pisces.
If these pairs of planets are together in any other sign, however, they do not clash. It is not so much that each planet and element, then, has a certain significance. This is a different system of interpretation from the European one.
Furthermore, the planets are interpreted not only in personal horoscopes in this manner but are also involved in the predictions given in the Tibetan almanac for the general conditions of a country in terms of weather, floods, droughts, famines, epidemics, wars and so on.
- For the situation in Tibet, the transiting positions of the nine heavenly bodies in the 12 signs and 27 constellations for the 8th, 15th, 22nd and 30th of each month are regularly examined and interpreted based on the 18th-century Tibetan text White Aquamarine by Desi Sanggye Gyatso.
- For that of India, the transiting position of all the planets at the exact moment the sun enters Aries is analyzed in terms of four charts deriving from the “arising from the vowels” system to predict for an entire year.
- The situation for China, on the other hand, is predicted according to the yellow Chinese-style system from calculations involving the spring nadir-point – namely, the nadir-point of the first Chinese-style month, in a scheme called the “earth-bull” and does not involve the five planets. Nadir-points will be explained shortly.
There is an oral tradition among the Tibetan nomads, and especially those of the northern high plateau, of predicting weather, famines, epidemics, wars and so on, by direct observation of the stars during the autumn and winter nights. They also predict weather from the wind, the clouds, the behavior of the animals and birds, and so on.