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|
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.
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