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Mathematics of Army Organization in Ancient India

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Fact 1: There are 10 kinds of people: those who know binary and those who don’t.

Fact 2: There are 10 kinds of people: those who know ternary, those who don’t, and those who mistake it (and this joke) to be binary.

We all know that 18 ‘akshauhini’ (अक्षौहिणी) armies fought in the Mahabharata war (11 on Kaurava side, 7 on Pandava side). An ‘akshauhini’ is a large army with 21,870 chariots, 21,870 elephants, 65,610 horses, and 109,350 foot soldiers. If one carefully notices, one will realize the relations:

21,870 = 3^7 × 10
65,610 = 3^7 × 10 × 3
109,350 = 3^7 × 10 × 5

Note the high powers of 3 (7 or 8 in the numbers. This is not a coincidence, the fact is that all lower level of organizations of an ‘akshauhini’ were based on powers of 3, as if following a ternary system of counting. Here are all the levels of the army organization, from the smallest to the largest, as listed in the ‘Amarakosha’:[1]

Mathematics of Army Organization in Ancient India

‘patti’ (पत्ति) = 1 chariot + 1 elephant + 3 horses + 5 soldiers

‘senamukha’ (सेनामुख) = 3 ‘patti’s = 3 chariots + 3 elephants + 9 horses + 15 soldiers

‘gulma’ (गुल्म) = 3 ‘senamukha’s = 9 chariots + 9 elephants + 27 horses + 45 soldiers

‘gana’ (गण) = 3 ‘gulma’s = 27 chariots + 27 elephants + 81 horses + 135 soldiers

‘vahini’ (वाहिनी) = 3 ‘gana’s = 81 chariots + 81 elephants + 243 horses + 405 soldiers

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‘pritana’ (पृतना) = 3 ‘vahini’s = 243 chariots + 243 elephants + 729 horses + 1,215 soldiers

‘chamu’ (चमू) = 3 ‘prtana’s = 729 chariots + 729 elephants + 2,187 horses + 3,645 soldiers

‘anikini’ (अनीकिनी) = 3 ‘chamu’s = 2,187 chariots + 2,187 elephants + 6,561 horses + 10,935 soldiers

‘akshauhini’ (अक्षौहिणी) = 10 ‘anikini’s = 21,870 chariots + 21,870 elephants + 65,610 horses + 109,350 soldiers

While I am not aware of the reasons for this ternary system (I can only guess), it is clear that it was a highly mathematical organization. Such was ancient India, with mathematics followed in almost every sphere of life, including organization of armies.

Note:
[1] एकेभैकरथा त्र्यश्वा पत्तिः पञ्चपदातिका,पत्त्यङ्गैस्त्रिगुणैः सर्वैः क्रमादाख्या यथोत्तरम्,सेनामुखं गुल्मगणौ वाहिनी पृतना चमूः, अनीकिनी दशानीकिन्यक्षौहिणी … (अमरकोषे २.८.८०–८१)

Original Article can be read HERE.

Nityānanda Miśra is a renowned author based in Mumbai. Born in Lucknow, Misra is an MBA from IIM Bangalore. He writes on Indian literature, arts, and music. His famous books are KumbhaThe Om Mala(English & Hindi), and Mahaviri. He has edited and authored eleven books in Sanskrit, Hindi, and English.

Also read,

Know the Mathematics of the Kumbha Mela.

What Signifies the Triveni in Prayagraj?

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