Lilavati (Bhaskaracharya)

Formula for Determining the Proportion of Gold in a Mixture

Verse 17:
Subtract the lower fineness from the target fineness, and subtract the target fineness from the higher fineness. These differences, when swapped and multiplied by the desired total weight This refers to the rule of alligation, used to find how much of two different purities of gold must be mixed to reach a specific average purity, will provide the respective weights of the lower and higher purity gold pieces.

Example 18:
O friend, there are gold pellets of 10 varna: a unit of gold purity, similar to karats and 16 varna. When combined, they result in a mixture of 12 varna. Tell me the weights of these two types of gold used.

Statement of the Problem:
Target fineness: 12
Lower fineness: 10
Higher fineness: 16

Calculation:
Target fineness (12) minus the lower fineness (10) = 2.
Higher fineness (16) minus the target fineness (12) = 4.
The weights of the gold pieces are in the ratio of these differences. In this example, the resulting measures for the lower and higher purities are 6 and 5 The text suggests a specific total weight was intended in the missing context to reach these specific integers.

General Rule for Combinations

Verse 19:
Write down the numbers starting from one and increasing by one, up to the desired limit, in reverse order. Beneath them, write the same numbers in direct order. Divide the top numbers by the bottom numbers and multiply the results successively—the first by the second, that result by the third, and so on.

Verse 20:
This provides the variations for combinations of one, two, or more items. This is recognized as a general rule and is used by experts in the study of poetic meters original: "chanda-citi", the architecture of prosody to determine the number of possible variations in a verse.

Verse 21:
Variations of "faces" likely referring to the sides of geometric shapes or dice, the "Broken Meru" Khaṇḍameru: the Indian mathematical name for what is known in the West as Pascal's Triangle, and variations found in medical preparations calculating how many ways different tastes or herbs can be combined are not described here for fear of being too lengthy.

Now, let us look at some examples regarding the variations of poetic meters.