Chapter on Mean Motion — 1

Page 6-7

As the Ahargana total number of civil days elapsed since the start of the era is 159,316, this is placed in a second position. One instance of 159,316 is multiplied by "the Universe" original: "vishvai"; a code word for the number 13, representing the 13 classes of gods which equals 2,071,108. This is divided by "Three, Void, and Numerals" 903; where Tri=3, Kha=0, and Anka=9, read from right to left in the Indian positional system. The resulting quotient is 2,293 degrees. The remainder is 529. Multiplied by sixty, it becomes 31,740; divided by the divisor 903, the quotient is 35 minutes of arc. The remainder 135, multiplied by sixty, is 8,100; divided by the divisor, the quotient is 8 seconds of arc. This procedure for calculating minutes and seconds should be understood as the standard method everywhere.

From the Ahargana 159,316, the degrees and minutes 2293;35;8 are subtracted, leaving 157,022. By taking one degree and converting it to sixty minutes, then subtracting 35, the remainder is 25. By taking one of these and converting it to sixty seconds, and subtracting 8, the remainder is 52. Thus, the position in degrees and further units is 157,022° 24' 52".

Now, the annual correction for the year:
The elapsed years since the book's epoch are 436. Divided by "the Vedas and Numerals" 64, the resulting quotient in minutes and seconds is 6' 48". This is subtracted from the previously obtained position 157,022;18;4 Note: There is a slight variation in the manuscript's intermediate numbers here compared to the previous line. Dividing the degrees by thirty gives 5,234 with a remainder of 2 degrees. Dividing the 5,234 signs by twelve gives 436 revolutions bhagana; one full circle of the zodiac with a remainder of 2 signs. Thus, the count of revolutions is 436, and the position is 2 signs, 2 degrees, 18 minutes, and 4 seconds.

Adding the Kshepaka the constant position of a planet at the start of the epoch of 10 signs, 26 degrees, 12 minutes, and 0 seconds, the result is 12 signs, 29 degrees, 13 minutes, and 4 seconds. Since 30 degrees make a sign, we add 1 to the signs, making 13. Since 12 signs make a full revolution, we add 1 to the revolutions, making 437. The final position is 1 sign, 29 degrees, 13 minutes, and 4 seconds. This is the Mean Sun at sunrise in Lanka the theoretical city on the equator used as the prime meridian. The same method should be known for calculating the positions of Mercury and Venus.

Now, the method for verifying the arithmetic, as stated by the masters of the Bija corrections:

original: "gunye gune navahrte parisheshaghate..."
"When the multiplicand and the multiplier are each divided by nine, and their remainders are multiplied together and then divided by nine, the final remainder must equal the remainder of the product when it is divided by nine. If they are equal, the calculation is correct."

For example: the multiplicand (Ahargana) is 159,316; divided by nine, the remainder is 7. The multiplier is 13; divided by nine, the remainder is 4. The product of the two remainders (7 × 4) is 28; divided by nine, the remainder is 1. Now, the actual product 2,071,108, when divided by nine, also leaves a remainder of 1. Since both are equal, the product 2,071,108 is correct. This applies everywhere.

Now, the verification for division as taught in Patiganita traditional arithmetic:
The remainder of the dividend divided by nine must equal the remainder of the product of the divisor's remainder and the quotient's remainder (added to the final remainder) divided by nine.

Example: The dividend 2,071,108 divided by nine leaves 1. The divisor 903 divided by nine leaves 3. The quotient 2,293 divided by nine leaves 7. The product of the divisor and quotient remainders is 21. Adding the remainder of the original division (529), which leaves 7 when divided by nine, we get 21 + 7 = 28. Divided by nine, this leaves 1. Since this equals the dividend's remainder of 1, the quotient 2,293 and the remainder 529 are correct. Usually, the Sun's revolutions and the elapsed years will match, though occasionally there is a difference. ॥ 7 ॥

Commentary of Sudhakara:
This is for the sake of clarity. Here is the mathematical derivation: The daily motion multiplied by the Ahargana results in the planet's position in degrees. The author has simplified the daily motion into fractions to make the calculation easier. For example, the Sun's daily motion is calculated using the ratio of total solar revolutions in a cycle to total civil days.
Sudhakara provides the ratio: 432,000,000 / 1,577,917,828 approx.
This simplifies to 384,000 / 389,609. The expanded form of this fraction using continued fractions is:
1 / (1 + 1 / (68 + 1 / (9 + 1 / (2 + 1 / (5 + 1 / (1 + 1 / (42 + 1 / (3 + 1 / 3))))))))

From this, the closest approximations are:
1/1, 68/69, 937/939, 753/764, 890/903, and so on. Here, the author has chosen the approximate value 890/903. This is transformed into the form (903/903) - (13/903), which equals 1 - (13/903). This is the daily motion in degrees...