Varga

Varga refers to squaring. It is one of the parikarmas or basic operations as reckoned in Indian treatises on mathematics. Squaring is a simple operation - multiply a number by itself. It is also well known. However, it is also one of the most heavily used operations. It is required at a fundamental level in geometry to find the area of various figures. Practically speaking, it occurs in numerous problems in Indian texts related to mathematics and astronomy, such as the computation of planetary locations and that of latitude, to name a few. These problems have geometric solutions, hence this is not surprising.

The geometric connection of squaring was fundamental to the understanding of this operation by Indian authors. The Āryabhaṭīya (499 CE), for example, states:

वर्गः समचतुरश्रः फलं च सदृशद्वयस्य संवर्गः ।

vargaḥ samacaturaśraḥ phalaṃ ca sadṛśadvayasya saṃvargaḥ |  (Gaṇitapāda 3a, Shukla, 1976)

“Varga means the area of a square. It is the product of two like numbers.”

In the verse, “two like numbers” refers to the number with itself, so, , or  and so on. We see in the verse that Āryabhaṭa presents a geometric picture of what squaring means. The area of a square is equal to the square of the side.

For small numbers, multiplying the two numbers directly is the easiest way of finding the number’s square. However, with large numbers, which are common in astronomy, this can be cumbersome. As this was a very common operation, there was a need for simplifying operations when dealing with very large numbers.

Yuktibhāṣā (c. 1530 CE) enumerates a few different methods of simplifying the squaring operation. As an example, one of these has been given below:

അനന്തരം ഖണ്ഡഘാതത്തേ നാലിൽ ഗുണിച്ചിട്ട് അതിൽ ഖണ്ഡാന്തരവർഗവും കൂട്ടൂ. ഏന്നാലും ഈ വർഗമുണ്ഡാകും.

anantaram khaṇḍaghātatte nālil guṇicciṭṭ atil khaṇḍāntaravargavuṃ kūṭṭū. ennālum ī vargamuṇḍākum. (Sarma et al., p. 332)

“Multiply the portions with each other and then by 4. To that, add the square of the difference of the portions. This [procedure] also gives the square.”

Let’s say we want to compute the square of . The steps are as follows:

• Split this into 50 and 52

• Multiply these numbers together. This gives 2600

• Multiply this by 4. We get 10400.

• Add to this the square of the difference of the original numbers. The difference is 2. Its square is 4. Add it to the above number.

The final answer is 10404. These methods enable quick calculation of the varga.

References

• Sarma, K. V., et al. (2008). Gaṇita-Yukti-Bhāṣā of Jyeṣṭhadeva (Vol. 1). Hindustan Book Agency.

• Shukla, K. (1976). Āryabhaṭīya of Āryabhaṭa, with the commentary of Bhāskara I and Someśvara. Indian National Science Academy.