Āryabhaṭa’s Saṅkhyāvinyāsa

Āryabhaṭa devised a unique saṅkhyāvinyāsa[1] (numeration system) that uses the saṃskṛtavarṇamālā, i.e., the svaras (vowels) and vyañjanas (consonants). He presents this in his work Āryabhaṭīya (6th century CE). The saṅkhyāvinyāsa employs vyañjanas as numeric digits and vowels as positional markers, allowing very large numbers to be expressed within poetic verse. Āryabhaṭa devised this saṅkhyāvinyāsa as large numbers were part of regular calculations in Siddhāntajyotiṣa, the branch of traditional vaidika astronomy focused on astronomical calculations, planetary positions, and mathematical modeling of the cosmos. Siddhāntajyotiṣa also covers the mechanics of time, eclipses, and orbital positions and is often inseparable from Gaṇitaśāstra (mathematics).

The second verse in the Gītikāpāda of Āryabhaṭa’s Āryabhaṭīya is as follows -

वर्गाक्षराणि वर्गेऽवर्गेऽवर्गाक्षराणि कात् ङ्मौ यः। खद्विनवके स्वरा नव वर्गेऽवर्गे नवान्त्यवर्गे वा॥१-२॥

vargākṣarāṇi varge'varge'vargākṣarāṇi kāt ṅmau yaḥ| khadvinavake svarā nava varge'varge navāntyavarge vā||1-2||

“The varga letters in the varga (places) and the avarga letters in the avarga (places). From k onwards, y is equal to ṅ plus m. In the places of the two nines of zeros, the nine vowels should be written. In the varga places beyond the nine vowels too.”

It is normal that this verse may seem terse to anyone who reads it for the first time. Every śāstra has its own paribhāṣā (terminology, technical phraseology, or technical terms). Any beginner of a śāstra must first learn the paribhāṣā to get acquainted with the contents of the śāstra. Paribhāṣā may be defined in the following way -

अव्यक्तानुक्तलेशोक्तसन्दिग्धार्थप्रकाशिकाः। परिभाषाः प्रवक्ष्यन्ते दीपीभूताः सुनिश्चिताः ॥[1]

avyaktānuktaleśoktasandigdhārthaprakāśikāḥ| paribhāṣāḥ pravakṣyante dīpībhūtāḥ suniścitāḥ ||

“Paribhāṣās which are full of light (and) are definite (i.e., without doubt) will be said to be those which throw light upon avyakta (unmanifest/unrevealed), anukta (unsaid), leśokta (less said), and sandigdhārtha (doubtful meaning).”

Bhāskara I, the earliest commentator to the Āryabhaṭīya, calls this specific verse a paribhāṣāsūtra by the ācārya (Āryabhaṭa) and goes on to explain this saṅkhyāvinyāsa. In vyākaraṇaśāstra, there are six types of sūtras, which are applicable to other śāstras too, usually remembered in the form of the following verse -

संज्ञा च परिभाषा च विधिर्नियम एव च । अतिदेशोऽधिकारश्च षड्विधं सूत्रलक्षणम् ॥[2]

saṃjñā ca paribhāṣā ca vidhirniyama eva ca | atideśo'dhikāraśca ṣaḍvidhaṃ sūtralakṣaṇam ||

These six types of sūtras are -

Saṃjñāsūtra - gives a name to some entity,
Paribhāṣāsūtra - clarifies the meaning of the formula,
Vidhisūtra - performs a new task or order,
Niyamasūtra - limits the received rule,
Atideśasūtra - attributes one thing to another, that is, describes one as similar to the other, and
Adhikārasūtra - indicates where subsequent clauses will apply

Bhāskara I, in his Āryabhaṭīyabhāṣya, explains Āryabhaṭīya 1.2 systematically. Here’s a simplified version.

The vyañjanas in Saṃskṛta are grouped into vargīya/varga (grouped) and avargīya/avarga (ungrouped). The vargavyañjanas contain the five vargas (groups) with five vyañjanas each. The avargavyañjanas contain the remaining eight vyañjanas. Each of these vyañjanas is assigned a specific numerical value. All vargavyañjanas are assigned numerical values from 1 to 25 in increments of 1. All avargavyañjanas are assigned numerical values from 30 to 100 in increments of 10. It is to be noted here that even though the text says increments of ten, while multiplying with the exponents of ten from the svara positions explained below, the values of the avargavyañjanas must be divided by 10 for correct calculation purposes. (See Fig. 1)

Fig. 1: Vyañjanas in Saṃskṛta with their assigned values according to Āryabhaṭa’s Saṅkhyāvinyāsa.

There are 13 svaras in Saṃskṛta that are used for regular usage outside of the Veda. (See Fig. 2)

Fig. 2: Commonly used Svaras in Saṃskṛta outside of the Veda.

Due to the well-known rule in Gaṇitaśāstra - “अङ्कानां वामतो गतिः” “aṅkānāṃ vāmato gatiḥ”, i.e., numbers must follow a right-to-left rule, these svaras are written from right to left in pairs to give 9 pairs. The svaras being hrasva or dīrgha do not matter for this arrangement. The svaras at the positions of even exponents of ten are called varga places and those of odd exponents of ten are called avarga places. This gives a total of 18 places starting from the zeroth exponent of ten to the seventeenth exponent of ten in the resultant number-chronogram. (See Fig. 3)

Fig. 3: The number-chronogram indicating the different positions by arranging pairs of svaras in Saṃskṛta.

An encoded word indicating large numbers may be decoded using the following method. Only guṇitākṣaras (vyañjana and svara combinations) in the encoded word can be decrypted in this way. The vyañjana and svara in the guṇitākṣaras must be understood by the varga and avarga notations assigned to both vyañjanas and svaras as given above. The vyañjanas are assigned the position of the svara that follows it. A vargavyañjana must be assigned to the corresponding vargasvara and an avargavyañjana to the corresponding avargasvara.

For example, in Āryabhaṭīya 1.3, the encoded word ”ख्युघृ” “khyughṛ” is given as the measure of the eastward revolutions of the Sun. This may be understood by first performing ānupūrvī, i.e., breaking down the word into the components in the exact order - ख् य् उ घ् ऋ (kh y u gh ṛ). Thus, kh (varga, of value 2) and y (avarga, of value 3) are at the (varga and avarga) positions of u and gh (varga, of value 4) is at the (varga) position of ṛ. So the final decrypted value can be obtained from the sum - (2 x 104) + (3 x 105) + (4 x 106) adding up to 43, 20, 000. (See Figs. 4 and 5)

Fig. 4: The ānupūrvī of ”ख्युघृ” “khyughṛ”.

Fig. 5: Decoding the measure of the eastward revolutions of the Sun from Āryabhaṭīya 1.3.

Another example can be the word कृष्णमूर्ति kṛṣṇamūrti whose ānupūrvī would be - क् ऋ ष् ण् अ म् ऊ र् ति (k ṛ ṣ ṇ a m ū r ti). Thus, k (varga, of value 1) is at the (varga) position of ṛ, ṣ (avarga, of value 40, i.e., 4 for calculation) and ṇ (varga, of value 15) is at the (avarga and varga) positions of a, m (varga, of value 25) is at the (varga) position of ū, and t (varga, of value 16) is at the (varga) position of i. So the final decrypted value can be obtained from the sum - (1 x 106) + (25 x 104) + (4 x 103) + (16 x 102) + (8 x 10) + (15), adding up to 1255695. (See Figs. 6 and 7)

Fig. 6: The ānupūrvī of ”कृष्णमूर्ति” “kṛṣṇamūrti”.

Fig. 7: Decoding the value of कृष्णमूर्ति kṛṣṇamūrti using Āryabhaṭa’s Saṅkhyāvinyāsa

Āryabhaṭa’s Saṅkhyāvinyāsa is used for encoding all large numbers within the Āryabhaṭīya. This method was taken as inspiration to develop other numeration methods, such as the Kaṭapayādi (also known as Paralpperu in Malayalam.