# In Theory

A beautiful animation of the apparent phases of the Moon.

The theory presented by Vedāṁga is that a unit of time called a “yuga” begins every five years when the Sun and Moon conjoin at a specific point in space; a point equivalent to what we nowadays call 0° Tropical Capricorn – the “winter solstice.”

For this to be true, the Moon most progress a certain amount each year, so that in five years it returns to the same point. The amount of the progression must therefore be some multiple of 360° divided by five.

The Vedāṁga itself says that the moon will progress 12 “tithi” each year.  A “tithi” is a measurement of the lunar phase. There are 15 tithi in the waxing period of the Moon, and another fifteen in the waning. Adding 12 to the 1st tithi is easy: it’s the 13th tithi. But adding 12 to the 13th tithi gets confusing at first, because after 15 it becomes the 1st tithi of the waning cycle. So instead of 25, the tithi is 25-15: 10. The 10th tithi of the waning cycle. Similarly, adding 12 is not the 22nd tithi of the waning cycle; because the waning cycle finished on the 15th tithi. Instead it is 22-15: 7; the 7th tithi of the waxing cycle.

In total there are 30 phases. So if we count the phases sequentially from 0, anything less than 15 is a waxing phase, anything else is a waning phase. Anything higher than 29 is a repeat. So the 36th phase, is identical to the 7th phase, etc.

Dividing the lunar cycle into 30 units means that each unit contains 12° of arc (360 divided by 30 is 12). So, by saying that there is a progressive difference of 12 tithis each successive New Year, the Vedāṁga tells us that the Moon will progress 144° each year (12 multiplied by 12 is 144).

Here is a theoretical presentation of how the New Years (winter solstices) would work within a yuga:

 Year Tithi Name Phase Ordinal Degrees Progressed One 1st waxing 0 0° Two 13th waxing 12 144° Three 10th waning 24 288° Four 7th waxing 36 432° Five 4th waning 48 576°

Where would the Moon be on the sixth New Year? Adding 12 tithi to the Moons location the previous New Year we would once again be at the 1st waxing tithi. It would be the 60th phase ordinal, and the Moon would have progressed a total of 720°. Since 720 is an exact multiple of 360, this means the Moon would join the Sun at the same exact place it did five years earlier.

Thus, a “yuga” ends, and the next one starts.

## In Reality

The average daily motion of the Moon is about 13.2°. The average daily motion of the Sun is about 0.986°. Thus it takes the Sun about 365.24 days to progress 360°. The Moon moves about 4,821° in the same amount of time. Let’s cast out all the 360° units from that number. There are 13 of them, with an extra .392, roughly. 39.2% of 360° is a bit more than 141°.

That is indeed very close to the theoretical 144° the Moon needs to progress relative to the Sun each year, to make the Vedāṁga’s yuga work.

The Vedāṁga’s “tithi” is not an exact conjunction by is the 12° span of various phases. So we have leeway, theoretically as much as 12° leeway, before the synchronicity of this yuga measurement breaks and needs to be fixed. Since there is about 3° inaccuracy per year: we have about four years before we need to adjust the calendar with some variation of a “leap year.”

But the problem becomes that although the yuga will work on a paper calendar that follows certain rules, gradually it will lose connection to the natural phenomenon that original marked it. As such the five year yuga very gradually fell out of vogue as an important time-keeping device.

- Vic DiCara

www.vicdicara.com

# The Oldest Definition of a Yuga (“Age”). It’s only FIVE years long and based on the Winter Solstice!

What is the Vedāṁga Jyotiṣa’s concept of a “yuga”? It is a period of time that starts and ends whenever the beginning of the year coincides with the beginning of a month.

What does Vedāṁga Jyotiṣa consider the beginning of the year? When the Sun ceases moving southward and begins moving north;  the “Winter Solstice.”

What does Vedāṁga Jyotiṣa consider the beginning of a month? When the Moon ceases waning and begins to wax;  the “New Moon.”

When the New Moon occurs in close proximity to the Winter Solstice, a new Yuga begins. This Yuga will last 5 years, because after that much time there will be another New Moon near the Winter Solstice.

The Vedāṁga Jyotiṣa gives details about the exact duration of the Yuga and how to keep it synchronized from solar and lunar perspectives. It also gives details about the stars and phases the Moon will be in when the yuga goes through all 20 of its important milestones: 5 Winter Solstices, 5 Summer Solstices, 5 Vernal Equinoxes, and 5 Autumnal Equinoxes.

The specific details reveal that the content of Vedāṁga Jyotiṣa really is very old, because it says the winter solstice occurs when the stars of “Dhaniṣṭhā” rise with the Sun. The last time that was true was in roughly the third millennium BCE.

Except for one text very obviously added at a later time, Vedāṁga Jyotiṣa does not divide the sky into the 12 divisions we are familiar with today – which are anchored to the solstices and equinoxes. Rather it divides the sky into 27 divisions anchored to the stars. The stars that rise with the Sun on the winter solstice gradually change over the centuries. Thus the data given in Vedāṁga Jyotiṣa regarding the stars at which the New Moon occurs to begin a yuga are about 5,000 years out of date. We could update them, but it is simpler to use the classical 12-sign tropical system and measure as the start of a yuga the New Moon occurring near the first degrees of Capricorn (the tropical marker for the winter solstice).

It is also rather unusual for readers familiar with later Sanskrit literature to learn that the Vedāṁga’s yuga is only five years long! We are used to hearing from the Purana that yugas are thousands, if not hundreds of thousands of years long. The very different definition we encounter in Vedāṁga Jyotiṣa should help us understand that “yuga” is a general concept for measuring a span of time longer than a year. There are different types of yuga for different spans of time.

Since “yuga” literally means “coupling”, and since Vedāṁga Jyotiṣa provides the oldest definition of yuga, it is safe to say that the five year yuga it defines – based on the coupling of the Sun and Moon with the Winter Equinox – is the original idea, and that a different type and duration are altogether different, and probably later as an expansion on the original idea.

- Vic DiCara

www.vicdicara.com

# Vedic Yugas… Five Years Long

This continues my presentation of Vedanga Jyotish – the oldest complete treatise on Indian calendrical astronomy.

# Yugas

A yuga contains 5 years, 1830 days, and 62 synodic (lunar) months. Knowing this, we can calculate many other ways of measuring the yuga.

• The star named Dhaniṣṭhā will rise 1835 times. [i.e. A yuga has 1835 sidereal days. A sidereal day is the duration between the rising of a star, in this case Dhaniṣṭhā. Every year there is one extra sidereal day than a normal day (measured as the duration between the rising of the Sun), and there are five years in a yuga.]
• There will be 67 lunar cycles. [62 synodic months in 5 years, plus one extra cycle for each year]
• The moon will rise 1768 times. [sidereal days minus one day per each lunar cycle]
• The Moon will cross the equator 134 times [the number of constellations traversed by the Sun in 5 years, minus one]

The yuga has 60+1 months of sunrises, 60+2 synodic months, and 60+7 lunar cycles. There are 30 sunrises in such a month. The solar month however has 30.5 days.

In a yuga, the Moon enters each constellation 67 times. The Sun stays in each constellation for 13 5/9 days.

Since Dhaniṣṭhā will rise 1,835 times, and since there are 27 constellations like Dhaniṣṭhā, there are 49,545 lagna [ascendants] in a yuga (1,835 x 27 = 49,545).

Since there are six ṛtu in a year, if we apply this to the Moon’s “year” (a lunar cycle) we find that there are 402 lunar ṛtu (“seasons”) in a yuga (67 x 6 = 402).

# Beginning of the Yuga

The five-year yuga begins when the Sun and Moon come together in Dhaniṣṭhā. It will be the lunar month of Maghā, the solar month of Tapas. The Moon’s waxing will begin, and both the Sun and Moon will begin moving northward in reference to the equator. The Yuga ends with the waning moon of the month of Pauṣa.

When the Sun and Moon are at the beginning of Dhaniṣṭhā’s area, they begin moving northward. When they are in the middle of Āśleṣā they begin moving southward. In the Sun’s case this always happens in the months of Māgha and Śrāvaṇa, respectively.

## Solstices during the Yuga

Tithi [lunar-phase-day] and nakṣatra of the beginning [winter solstice] and midpoints [summer solstice] of each year of the yuga:

• First Year:           1st waxing / Dhaniṣṭhā [“Winter”]             7th waxing / Citrā [“Summer”]
• Second Year:     13th waxing / Ārdrā                                          4th waning / Pūrva Bhādrapadā
• Third Year:          10th waning / Anurādhā                                                 1st waxing / Āśleṣā
• Fourth Year:       7th waxing / Aśvinī                                            13th waxing / Pūrva Aṣāḍhā
• Fifth  Year:          4th waning / Uttara Phālgunī                        10th waning / Rohiṇī

Comment: this translation should be fact checked. The details may be in wrong arrangement.

## Equinoxes during the Yuga

The equinoxes of the Yuga occur:

• First Year:           3rd waxing (“Vernal”)      9th waxing (“Autumnal”)
• Second Year:     Full Moon                            6th waning
• Third Year:          12th waning                         3rd waxing
• Fourth Year:       9th waxing                            Full Moon
• Fifth Year:           6th waning                           12th waning

The formulae for obtaining this information:

Double the ordinal of the equinox; subtract one; then multiply by six. This number is the number of waxing and waning periods of the Moon which have passed between the beginning of the yuga and the equinox in question. [odd numbers are waxing periods, even numbers are waning periods.]

If you half this number you will know on which tithi [lunar-phase day] the equinox will occur.

## Maintaining Synchronicity Between the Calendar and Reality

Every two synodic months and two synodic days, a new season begins. However, the eighth season begins on the 15th synodic day [not the 16th].

Comment: The addition of two and sometimes one synodic day is a method for synchronizing approximated and rounded cylindrical measurements.

If the end of the waxing or waning period [the syzygy of Earth, Moon and Sun] occurs before mid-day, you can omit that day from the calendar. If not, subtract 30 kāṣṭhā  from the kāṣṭhā of the syzygy  for each waxing or waning period elapsed [since the Yuga began].

Comment: I am not clear on the exact meaning of the second half of this statement.

## An Interpolated Verse

Comment: I will mention a verse without a number and found in only one of the two versions of the book, clearly added to the text at a later date:

Note the zodiac sign that Jupiter is in when the Yuga begins. Count it from Pisces. Divide that by five. Take the remainder. That is the number of the year in the five-year cycle.

- Vic DiCara

www.vicdicara.com

# Vedāṁga Jyotiṣa – Part 1

I will organize this presentation in leiu of the work of Professor T.S. Kuppanna Sastry and Dr. K.V Sarma. It combines the two versions of Vedāṁga Jyotiṣa found in Ṛg and Yajur Veda.

# Introduction

I purify myself by bowing my head to the Creator, whom I envision as being the power within the five-year cycles [“yuga”], and the body on which days, months, seasons, and half-years are limbs.

I also bow my head to the Goddess of Intellect, Sarasvati – so that I may write on the science of time, as explained by Sage Lagadha.  This science is meritorious and dear to the learned, because by it one can understand the perfect times to make auspicious endeavors.

The Vedas exist so that we can successfully obtain the results of our efforts and sacrifices, but doing so is very dependent on timing. Therefore one who understands the science of time, Jyotiṣa, also understands the science of successful efforts.  So they say that this science of Jyotiṣa is the foremost appendix to Vedic knowledge – much like the feathers of a peacock, or the treasures of a dragon.

Anyone who understands the Vedas and also understands the movements of the Sun and Moon will become prosperous in this world and afterwards will go to where the Sun and Moon move about in the heavens.

# Measuring Time

The time it takes to pronounce a long vowel is an akṣara. Five of them is a kāṣṭhā. Four groups of thirty-one kāṣṭhā are a kalā. 10.05 kalā is a nāḍikā. Two nāḍikā are a muhūrta. Thirty muhūrta are a day, which is equivalent to 603 kalā.

A year is 366 days. It has two ayana, six ṛtu, and twelve months.

A yuga is five years.

Also: a nāḍikā is three sixteenths of an āḍhaka, during which time a clepsydra will drain 50 pala of water. Four āḍhaka are a droṇa. And a Ṛtu equals 4.5 constellations.

Comment:

Assuming for now that a day is “24 hours” the approximate modern values of these time units are as follows.

Droṇa:                 512 minutes (>8.5 hours)

Āḍhaka:             128 minutes

Muhurta:           48 minutes

Kalā:                     2.4(-) minutes

Kāṣṭhā group: 4.5(-) second

Kāṣṭhā:                1(+) second

Akṣara:                ¼ of a second

Now, for the longer periods of time, approximately:

Yuga:                    5 years

Year:                     366 days

Ayana:                 183 days

Ṛtu:                       61 days

Month:                30(+) days

The text gives three ways to check the measurements in the real world. We can do it starting from the Akṣara, assigning it the amount of time it takes to pronounce two short-vowel syllables or one long-vowel syllable in normal speech. Or we can start from the āḍhaka, assigning it the amount of time it takes to drain a clepsydra (basically a pot with a pinhole in it) holding a fixed amount of water. Or we start from the ṛtu, assigning it the amount of time it takes the Sun to move 60 degrees of arc in reference to a zodiac star.

Since we have mentioned the zodiac stars, lets now pull together the definitive verses concerning the zodiac constellations.

# Zodiac Constellations

The zodiac constellations with their deities are:

1. Kṛttikā                           Agni (God of fire)
2. Rohiṇī                            Prajāpati (the Creator, Brahmā)
3. Mṛgaśīrṣā                     Soma (God of the immortal elixir)
4. Ārdrā                             Rudra (God of destruction)
5. Punarvasu                   Aditi (Goddess of space)
6. Puṣya                             Bṛhaspati (God of prayer)
7. Āśleṣā                            Naga (Dragons)
8. Maghā                           Pitṛ (Ancestral spirits)
9. Pūrva Phālgunī           Bhaga (God of love)
10. Uttara Phālgunī         Aryamā (God of vows)
11. Hasta                             Savitā (God of awakening)
12. Citrā                               Tvaṣṭā (God of design)
13. Svāti                               Vāyu (God of breath/air)
14. Viśākhā                         Indrāgñi (God of sacrificial fire)
15. Anurādhā                     Mitra (God of devotion/ friendship)
16. Jyeṣṭhā                         Indra (Chief of the gods)
17. Mūla                              Nirṛti (Goddess of destruction)
18. Pūrva Aṣāḍhā             Apa (Goddess of water)
19. Uttara Aṣāḍhā            Viśvadeva (All divinities)
20. Śravaṇa                         Viṣṇu (God of existence)
21. Dhaniṣṭhā                    Vasu (Gods of elements)
22. Śatabhiṣaj                    Varuṇa (God of night/ the underworld)
23. Pūrva Bhādrapadā    Ajaikapāt (Fire dragon)
24. Uttara Bhādrapadā Ahirbudhnya (Water dragon)
25. Revatī                            Pūṣan (God of protection)
26. Aśvinī                             Aśvini (Twin children of the Sun)
27. Bharaṇī                         Yama (God of death)

Those who understand the science of sacrifice recall that the names of these gods should be used in place of our own name, according to the constellation under which we are born, whenever we make serious sacrifices and efforts.

Ārdrā, Citrā, Viśākhā, Śravaṇa and Aśvinī have “fierce” qualities. Maghā, Svāti, Jyeṣṭhā, Mūla and Bharaṇī  have “harsh” qualities.

# Importance of Cross Multiplication

The following very common elementary equation will often be used in our calculations: a/b = c/x. Which is solved as: x = bc/a.

To be continued… stay tuned…

- Vic DiCara

www.vicdicara.com