The sexagesimal system of measurement is based on the number sixty. There are sixty seconds in a minute, sixty minutes in a hour. When we measure angles, we use the sexagesimal to express units in degrees, minutes and seconds. This method of measurement familiar to both the Indian and Mesopotamian cultures. It may be that one culture borrowed from the other or that both developed the system independently. Or it could be of such antiquity that both cultures shared a common origin. Whatever the case, it seems quite evident that the sexagesimal system may be based in large part upon the observation of the planets, specifically Jupiter and Saturn.
After every sixty years, Jupiter and Saturn will return to the same relative place in the zodiac. Even though they conjoin every twenty years, it is every third conjunction that they will be in the same zodiacal position as they were sixty years before. Jupiter takes twelve years to complete one circuit of the zodiac. It takes thirty years for Saturn to complete a similar circuit.
Consider the following:
- Jupiter takes twelve years years to transit the zodiac. The zodiac has twelve signs. Jupiter travels an average thirty degrees or one complete sign in one solar year.
- Saturn takes an average thirty years to transit the zodiac. Each zodiacal sign has thirty degrees and Saturn travels one degree per month. It takes thirty months for Saturn to travel one sign. It takes three hundred and sixty months for Saturn to transit the entire zodiac. Jupiter divides the zodiac into twelve parts or signs. The number derived from Saturn’s motion suggests the division of each sign into thirty parts or degrees.
- Jupiter and Saturn take sixty years between conjunctions to reach the same place in the zodiac. This joint motion suggests the third and fourth division of the degree into sixty minutes and each minute into sixty seconds.
The sixty years cycle of Jupiter and Saturn gives rise to another interesting number. In a sixty year period, Jupiter will complete five circuits of the zodiac and Saturn will complete two circuits. The combined individual cycles equal seven which is also the total number of visible planets plus the two luminaries.
The Seven Day Week
The primacy given to Saturn and Jupiter becomes apparent by the study of the origination of the seven day week which, contrary to common opinion, was not followed by everyone in the ancient world. Vedic Indians had a six-day week.
The most ancient usage of day names used in India was that of the Nakshatra. There are twenty seven lunar asterisms or constellations in the old lunar zodiac. This number was derived from the average number of days it took the Moon to complete one circuit of the heavens in relation to any particular star (one sidereal revolution). Since the Hindus didn’t use hours to divide their day, the natural consequence of using a seven day week would not follow. Instead they divided a day into 60 equal parts called ghatikas. Each ghatika is equal to 24 minutes. The word “ghatika” means little jar and thus the use of water clocks suggest itself. A ghatika is further divided into 60 vinadikas. So between the two cultures, it was the Hindus who made direct use of the sexagesimal system; whereas the Chaldeans used an indirect method of 24 hours.
It was not until much later in the third century AD where we find the first usage of the seven day week in India. Indeed much of the rest of the world had not adopted it until after the first century AD. The 7 day week was was introduced into the Christian world by an edict of the Roman emperor Constantine, about 323 AD, who changed the Sabbath to the Lord’s Day (Sunday), the week day after the Jewish Sabbath. Its introduction into India is about the same time and from the same sources. The week days are not found in earlier scriptures like the Vedas of the classics like the great epic Mahabharata. They occur only from 484 AD, but not in inscriptions of 300 AD or earlier. Even now, they form but an unimportant part in the religious observances of the Hindus which are determined by the Moon’s phases and lunar asterisms.
In the schema of the Moon’s phases we see a repeated pattern to that of Jupiter and Saturn. A lunar month is made up of 30 tithis. Each tithi is determined when the moon moves in advance of twelve degrees ahead of the Sun. Here we see the numbers 30 and 12 that are common with Saturn and Jupiter. A complete synodic period (a complete revolution around the zodiac in relation to the Sun) of the Moon, however, takes only 29 civil days. (A civil day for the Indians is reckoned from sunrise to sunrise). It is quite a regular occurrence for a tithi to be expunged from the consecutive civil day count. This characteristic of the Hindu calendar is not found in the Greek, Chinese or Mesopotamian calendars. Other cultures, without exception, use solely a civil day count of 28, 29 and 30 days for their lunation cycles and had not even considered a pure lunar day count independent from the civil reckoning. The consistency of the Hindu astronomical methods make it unlikely that they borrowed their knowledge from other sources. And the repeated usage of the sexagesimal measurement makes it more like that they were the inventors of the system.
The lunar asterisms (nakshatras) are derived from the average daily motion of the Moon’s mean sidereal cycle, which is 13° 20′ of arc. In a circle of 360 degrees this would make twenty seven nakshatras. Each nakshatra has a planetary ruler and they are shown as follows:
Nakshatra | Ruler | |
---|---|---|
1 | Ashvini | Ketu |
2 | Bharani | Venus |
3 | Krittika | Sun |
4 | Rohini | Moon |
5 | Mrigashia | Mars |
6 | Aridra | Rahu |
7 | Punarvasu | Jupiter |
8 | Pushya | Saturn |
9 | Aslesha | Mercury |
10 | Magha | Ketu |
11 | Purva Phalguni | Venus |
12 | Uttara Phalguni | Sun |
13 | Hasta | Moon |
14 | Chitra | Mars |
15 | Svati | Rahu |
16 | Vishakha | Jupiter |
17 | Anuradha | Saturn |
18 | Jyeshtha | Mercury |
19 | Mula | Ketu |
20 | Purva Ashadha | Venus |
21 | Uttara Ashadha | Sun |
22 | Shravana | Moon |
23 | Dhanishtha | Mars |
24 | Shatabhisha | Rahu |
25 | Purva Bhadra | Jupiter |
26 | Uttaea Bhadra | Saturn |
27 | Revati | Mercury |
In the above table you may have noticed the two strange words “Rahu” and “Ketu”. These are the nodes of the moon and their usage in astronomy is important for predicting the eclipses of the Sun and the Moon. Although they possess no mass or density, they are treated as planets in the sense that they have an effect on human affairs.
Nakshatras
Whatever constellation (nakshatra) the Moon was in at sunrise, the entire day was named after it. The nine rulers of the nakshatras are repeated three times in sequence. So in this sense you can say that the Hindus followed a nine-day week. The same effect as a weekday is thereby achieved in terms of socio-religious significance. Whereas the Chaldeans used an unbroken consecutive day count of hour and day rulers, the Hindus used a more concrete system of the observable Moon in a group of stars. But there is a common thread that is stitched between both the seven-day week and the twenty seven nakshatras. Putting the stars (Gods) of the 7 day week on a circle with Sunday at the top, we get:
Now in Figure 4, we may make room for two more planets:
This figure appears in a book by Chiero the famous palmist in which he refers to this as the “Seal of Solomon.” Other writers assert that ther are many different Seals of Solomon and that this is only one. Whatever the case may be, if we were to add two more planets to this seal the obvious place would be in the areas vacated by the points of Jupiter and Mercury. Their placement in the seal is shown in Figure 5.
Hindu Astrology treats the nodes as planets even though they possess no mass and density. They are sensitive points where the path of the Moon crosses the path of the Sun. Being invisible, it is quite fitting that they should not have a point of the star aiming at them in the diagram shown above.
But note the general position of the planets and see how they are unchanged from the Chaldean star order. The next diagram, Figure 6, will show the connection with the Hindu nakshatra order.
Compare this diagram with Table 2, Nakshatras and Rulers. Here the order is clockwise starting with Ketu, going to Venus, the Sun, down to the Moon, back up to Mars, over to Rahu, Jupiter, Saturn and finally bypassing over to Mercury. The detour down to the Moon doesn’t seem so strange when you consider that the nakshatra reckoning is entirely dependent upon the Moon’s position in the constellations. In fact, this diagram may taken as a single pointed star with the significant planet being at the single point.
It seems that the similarities between Hindu astronomy and Chaldean magical seals are too great to be a mere coincidence. It is quite likely that the two share a similar origin or that one was derived from the other. But since ancient Hindu tradition and current usage make more consistent usage of a direct base sixty system of counting, and the other cultures use a derivative of that, it seems more likely that the sexagesimal is of Hindu origin.
It may be argued that the sexagesimal was not founded upon the joint cycles of Jupiter and Saturn but upon some other measurement such as the average number of three hundred and sixty days in a year. This, of course, is assuming that ancient man was incapable of counting the correct number of days in a year and that he was infinitely unclever. For the sake of argument, let us accept that position. There still remains the problem of dividing the year into parts that would yield a base sixty system of counting. The Babylonians divided their year into three seasons. The Hindus, however, divided their year into six seasons. Of the two cultures which would arrive at a base sixty system? (60 is convenient too for division by 2,3,4,5 and 6.)
But judging from the rest of Hindu astronomical techniques it is clear that they knew precisely how long was the solar year. All the other planetary cycles were also studied with great scrutiny. From this it may be tempting to think that the sexagesimal was arrived at in order to provide the great average mean of measurement for celestial phenomena just as today the binary mode of counting is most convenient for computer science. Three hundred and sixty, which is a multiple of the sexagesimal, is the midpoint number between the 365 day solar year and the 354 day lunar year. One synodic period of Mars is 780 days which is equally divided by 60 thirteen times. Between two consecutive conjunctions of Saturn and Jupiter in a twenty year period, Mercury will go retrograde a little over sixty times. The Hindus also based their knowledge of breath control, Pranayama, upon the sexagesimal system. In one twenty four hour period, or between two consecutive sunrises, a person takes an average of 21600 breaths, each breath being four seconds long. This number, 21600 divided by 60 equals 360.
Cosmological Cycles
The structure of Hindu astronomy is built upon the foundation of an unique concept of cosmological time cycles. No other culture on Earth has or is known to have such a unique system of Cosmology. The only other culture to come close to the vast scale of time conceived by the Hindus are the Mayan.
The astronomical quantities derived from these cosmological time cycles are vastly more accurate than anything achieved by the Greeks. And they were in use at a time when the Britons were still living a neolithic lifestyle.
Outline of Time Scales
Prior to the creation of the universe, Lord Vishnu lies asleep on the ocean of all causes. He rests upon a serpent bed with thousands of cobra-like hoods. While asleep, a lotus sprouts from His navel. Upon this lotus is born Brahma the creator of the universe. Lord Brahma lives for a hundred years and then dies, while Lord Vishnu remains. One year of Brahma consists of three hundred and sixty days. At the beginning of each day Brahma creates the living beings that reside in the universe and at the end of each day the living beings are absorbed into Brahma while he sleeps on the lotus. One day of Brahma is known as a Kalpa. Within each kalpa there are fourteen manus and within each manu are seventy one chatur-yugas. Each chatur-Yuga is divided into four parts called yugapadas.
From the first chapter of Surya-Siddhanta, the most revered authoritative source of Hindu astronomy, we have the following passage:
- That which begins with respirations (prana) is called real. Six respirations make a vinadi, sixty of these a nadi:
- And sixty nadis make a sidereal day and night. Of thirty of these sidereal days is composed a month; a civil (savana) month consists of as many sunrises;
- A lunar month, of as many lunar days (tithi); a solar (Saura) month is determined by the entrance of the Sun into a sign of the zodiac; twelve months make a year. This is called a day of the gods.
- The day and night of the gods and of the demons are mutually opposed to one another. Six times sixty of them are a year of the gods, and likewise to the demons.
- Twelve thousand of these divine years are denominated a chatur-Yuga; of tena thousand times four hundred and thirty two solar years is composed that chatur yuga, with its dawn and twilight. 16 The difference of the krita yuga and the other yugas, as measured by the difference in the number of the feet of virtue in each is as follows:
- The tenth part of a chatur-yuga, multiplied successively by four, three, two, and one, gives the length of the krita and the other yugas: the sixth part of each belongs to its dawn and twilight.
- One and seventy chatur-yugas make a manu; at its end is a twilight which has the number of years of a krita-yuga, and which is a deluge.
- In a kalpa are reckoned fourteen manus with their respective twilights; at the commencement of the kalpa is a fifteenth dawn, having the length of a krita yuga.
- The kalpa, thus composed of a thousand chatur-yugas, and which brings about the destruction of all that exists, is a day of Brahma; his night is of the same length.
- His extreme age is a hundred, according to this valuation of a day and a night. The half of his life is past; of the remainder, this is the firsts kalpa.
- And of this kalpa, six manus are past, with their respective twilights; and of the Manu son of Vivasvat, twenty seven chatur yugas are past;
- Of the present, the twenty eighth chatur yuga, this krita yuga is past.
Now to make plain what is stated above. Commentaries are very clear on the fact that in verse 12 the “sidereal day” refers to a revolution of the Earth relative to any fixed star and is the true revolution reference point of the Earth. Verse 13 refers to “a day of the gods” means one sidereal year. A night of the gods is half a sidereal year. Verse 21 mentions “his extreme age is a hundred refers to the lifespan of Brahma and consists of one hundred years of 360 days. Each of these days being two kalpas long. Verse 23 shows that the Surya-Siddhanta was composed right after krita-yuga and during the treta-yuga. The present yuga we are in right now is the kali-yuga which is said to have begun on Friday February 18th 3102 BC of the Julian calendar. This becomes clearer when represented in a tabular form.
Divine Years | Solar Years | |
---|---|---|
Krita-Yuga Period | ||
Dawn | 400 | 144,000 |
Krita-Yuga | 4,000 | 1,440,000 |
Twilight | 400 | 144,000 |
Subtotal | 4,800 | 1,728,000 |
Treta-Yuga Period | ||
Dawn | 300 | 108,000 |
Treta-Yuga | 3,000 | 1,080,000 |
Twilight | 300 | 108,000 |
Subtotal | 3,600 | 1,296,000 |
Dvapara-Yuga Period | ||
Dawn | 200 | 72,000 |
Dvapara-Yuga | 2,000 | 720,000 |
Twilight | 200 | 72,000 |
Subtotal | 2,400 | 864,000 |
Kali-Yuga Period | ||
Dawn | 100 | 36,000 |
Kali-Yuga | 1,000 | 360,000 |
Twilight | 100 | 36,000 |
Subtotal | 1,200 | 432,000 |
Total | 12,000 | 4,320,000 |
Derivation of Astronomical Values
The first is the measurement of the day which begins with the breath. One respiration is a prana. Six prana equal one vinadi. Sixty vinadis equal one nadi, (also known as a ghatika). Sixty nadis equal one sidereal day. A sidereal day equal the time it takes for the earth to make one complete rotation on its axis in relation to a fixed star. A sidereal day is slightly shorter than a civil day of 24 hours. A sidereal day is equal to 23 hours, 56 minutes and 3.4446 seconds.
one sidereal day | 23h 56m 03.4446s |
---|---|
sixty nadis | 23h 56m 03.4446s |
one nadi (ghatika) | 23m 56.06s |
vinadi | 23.93s |
one prana | 3.99s |
It is clear in the text of Surya-Siddhanta and the current practice of Indian astrology that sidereal measurements are of primary importance. Tropical measurements are also used but in a secondary way.
Three Mean Motions of the Sun
The three mean motions of the Sun used to construct the Cosmological Time Cycles shown above are as follows:
one sidereal year | = | 360 sidereal days + 6 sidereal days + 0.2563795 sidereal days |
= | 366.2563795 sidereal days |
Remember we are not talking about civil solar days here, we are talking about the total number of times the Earth rotates on its axis in relation to a single star during the course of one year. This happens to be one greater than the mean solar days in a year which is.
One sidereal year | = 365.2563795 mean solar days |
These three mean motions of the Sun may be compared to the hour, minute and second hands of a clock. Each cycle is counted and completed separately. Using this system of the three mean motions, the ancients reckoned time that put the day, year and longer periods of time into exact correspondence with each other.
Proof of the Sexagesimal Number System
The first two mean solar motions, that of 360 + 6 Earth revolutions, generate the sexagesimal number system completely. A count of six for every 360 is the same as one for every 60. This is the basis of the six seasons of the year observed by the Hindus. Counting six days per year, the second mean motion of the Sun completes a cycle of 360, the number of degrees in a circle, after 60 years which correlates with the Babylonian sossos period and the cycles of Jupiter and Saturn.
Year | First Mean Motion |
Second Mean Motion |
Total |
---|---|---|---|
1 | 360 | 6 | 366 |
2 | 720 | 12 | 732 |
3 | 1080 | 18 | 1098 |
4 | 1440 | 24 | 1464 |
5 | 1800 | 30 | 1830 |
10 | 3600 | 60 | 3660 |
(sossos) 60 | 21,600 | 360 | 21,960 |
(neros) 600 | 216,000 | 3600 | 219,600 |
(saros) 3600 | 1,296,000 | 21,600 | 1,317,600 |
In the same interval that the first mean motion completes a count of 21600, it has done so at a rate 60 times greater than the second mean motion 360 × 60 and represents the number of arc minutes in a circle. The number 21600 is also the same average number of breaths (prana) a person will make in a 24 hour period.
Vedic Evidence of the Sidereal Year
The Rig Veda, the earliest of the Hindu scriptures says the following:
Twelve spokes, one wheel, navels three. Who can comprehend this? On it are placed together three hundred and sixty like pegs. They shake not in the least^{}
[1].
A seven-named horse does draw this three-naved wheel. Seven steeds draw the seven-wheeled chariot. Wise poets have spun a seven-strand tale around this heavenly calf, the Sun^{[2]}.
The number seven related to the Sun has much significance when understanding the third mean solar motion (0.2563795). The Kali-yuga of 432,000 years is the unit of reference for determining the length of the sidereal year in Hindu cosmological time cycles. During the course of 10,000 years there are seven rotations of the third mean solar motion. For a single year the count is 0.2563795 diurnal revolutions of the earth. For two years it is .512759 and so on. One complete rotation (to equal 366.2564.) of the third motion takes 1428.571429 sidereal years. Or you can reduce it to a fraction of 14284/7 sidereal years.
366.2563795.
0.2563795 |
= | 10000
7 |
= | 1428 and 4/7 sidereal years |
The integer of this sidereal interval, 1428 years, multiplied by the number of years in a Kali-yuga and then further multiplied by seven equals the number of years of fourteen Manus. (see table 4).
1428 × 432,000 × 7 = 4,318,272,000 = 14 manus
The fractional part of this sidereal interval, 4/7 years, multiplied by seven and further multiplied by the number of years in a Kali-yuga equals the time of an introductory dawn (see table 4).
4/7 × 7 × 432,000 = 1,728,000 years = introductory dawn.
Relating the Vedic verses above to what we have just demonstrated it is clear that the “navels three” refer to the three mean motions of the Sun and “seven-wheeled chariot” to the rate of precession of the equinoxes. Thus, there can be no doubt that the cosmological time cycles were already an established conclusion at the time of the Vedic era and not in the formative stages.
Precessional Constant
The precession is clearly derived from the cosmological time cycles as shown below. The chatur-yuga of 4,320,000 years is the unit of reference for determining the rate of precession used in the construction of the Hindu cosmological time cycles.
The constant rate of precession is 50″.4 = 0°.014 = 7/500 degrees of precession per sidereal year.
This is the same as one degree of precession in 71³/7 = 71.42857 sidereal years.
This correlates to the cosmological time cycles as follows:
One manu | = | 71.4 chatur-yugas |
1/14^{th} of an introductory dawn | = | 0.02857. chatur-yugas |
1/14^{th} Kalpa | = | 71.42857 chatur-yugas |
In the interval of 1/14th kalpa there are:
(71^{³}/_{7}) × 4,320,000 × 0°.014 = 4,320,000 degrees of precession = 12,000 precessional years
From table one we see that a period of one chatur yuga is 4,320,000 years and is equivalent to 12,000 divine years.
Is it just a happy coincidence that the Cosmological Time Cycles agree with the precession? Burgess and Whitney would probably think so.
Other related values of interest are:
1 precessional year | = | 25,714 ^{²}/_{7} sidereal years |
7 precessional years | = | 180,000 sidereal years |
= | 7 × 18 (126) cycles of the 3^{rd} mean mothion of the Sun | |
7 × 24 (168) precessional years | = | 1 chatur-yuga |
168,000 precessional years | = | 1 kalpa |
(4,320,000 ÷ 168) × 0°.014 = 360°
Derivation of the Tropical Year
In a chatur-yuga there are: 4,320,000 sidereal years = 4,320,000 + 168 precessional years. Therefore:
1 tropical year | = | 4,320,168 |
= | 365.2421756… mean solar days |
It has been shown conclusively that the Hindu Cosmological time cycles are based upon the diurnal motion of the Earth in reference to any particular fixed star, hence it is purely of sidereal origin. The later practice of adopting the ahargana or “heap of days” is based upon solar and civil day reckoning which is of obvious practical value for calendrics. The sidereal basis of the cosmological time cycles is without question the oldest known positive proof of the origin for the sexagesimal number system.
Comparison With Modern Science
The standard values for the tropical year and annual precession in longitude determined by Simon Newcomb for the epoch 1900.0, mean noon at Greenwich December 31st 1899 are:
One tropical year | = | 365.2421988 |
Precession in one year | = | 50″.2564 |
The sidereal year and its precessional constant may be derived from these values.
1 sidereal year (1900.0) | = | 360° – 50″.2564 |
× 365.2421988 + 1 |
= | 366.2563627 | diurnal revolutions of the Earth |
Precession in longitude in one year | = | 50″.2564 × | 365.2563627
365.2421988 |
= | 50″.2583 |
The following shows the astronomical quantities used in the construction of Hindu cosmological time cycles with those of Simon Newcomb for the epoch 1900.0
Hindu | Newcomb | Difference | |
---|---|---|---|
Constant of Precession | 50″.4 / year | 50″.2583 / year | 0″.1417 / year |
Sidereal Year (Solar) | 365.2563795 | 365.2563627 | 1.4 seconds / year |
Tropical Year | 365.2421756 | 365.2421988 | -2.0 seconds / year |
The sidereal year in the above table refers to the number of solar civil days it takes for the earth to orbit the sun in relation to any particular star. The former is a sidereal-diurnal relation and the later is a sidereal-solar relation. The very close agreement between the length of the year as measured by Hindu cosmological time cycles and that determined by modern science, together with the demonstrated great antiquity of the cycles, shows that the rotation of the Earth is not being sensibly retarded by “tidal friction” or any other cause.
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References
- This article is taken in large part from “The Six Thousand Year Barrier-An Essay on Hindu Astrology“, by by Glen R. Smith