CLS 15, MATH 15 — Spring 2020Indian Mathematics: Overview

The most important idea in ancient Indian mathematics is the simplest of all: zero. Our familiarnumerals are directly descended from the 10 digits in use in South Asia from as early as the 6thcentury CE, adopted by Arabic mathematicians and then, between 1000 and 1200 CE, by Europeans.

Harappan periodThe Harappan civilization, or Indus Valley culture, existed in the north-western part of the SouthAsian subcontinent (more or less modern Pakistan and adjacent parts of India and Afghanistan)roughly between 3300 and 1500 BCE. It is therefore contemporary with the Egyptian and Babylo-nian cultures we looked at earlier. Harappa and Mohenjo Daro were two of the largest cities in thisculture.

We do not know what language the Harappans spoke;there are many extant carved seal-stones with what appearsto be writing on them, but the language has not yet beendeciphered. It’s possible that the language is related to themodern Dravidian family (which includes Tamil, Kannada,and Telugu).

The Indus Valley civilization built large cities, with watersystems, warehouses, and fortifications. They used bronze,but not iron. Naturally, there was agriculture; they grewwheat and barley and had domesticated animals includingcattle.

Each city had a standard system of weights and mea-sures; scale weights have been found at several sites, andthere is a good set in the MFA collection. A set of weights

contains carved blocks of 1 unit, 5, 10, 20, 50, and so on, along with 1/2, 1/5, 1/10, and 1/20 of a unit.There was trade between the various cities.

Eventually the Harappan culture disappeared and its cities were abandoned; we don’t know why,though it’s possible that speakers of proto-Sanskrit, coming in from the north, displaced them.Vedic periodThe Vedic period is roughly 1500 to 600 BCE. “Veda” is a Sanskrit word meaning “knowledge”; it iscognate with English “wit” and “wisdom.” The Rig Veda is an ancient sacred text in Sanskrit, firstcomposed orally, in the early part of the Vedic period, and then fixed and memorized. It is a collectionof over 1,000 hymns to the various gods, in a variety of poetic forms. Along with the Rig Veda are theYajur Veda, Sama Veda, and Atharva Veda, all roughly contemporary; the Yajur Veda and Sama Vedaprovide background for the rituals in which the Vedic hymns are used, and the Atharva Veda givesmagic spells and other types of ritual.

Other Vedic texts come along a bit later: the Brahman. asand the Upanis.ads. These are more philosophical, and theyare in prose.

The basic academic disciplines of the Vedic period, besidesstudy of the Vedas themselves, are language (specificallyphonetics, grammar, etymology, poetic meter), ritual prac-tice, and star-science (which means calendar-making and as-tronomy). Astronomy is mathematical, as we have seen al-ready. Ritual practice brings in geometry, as altars were con-structed in complex shapes and arrangments.

Some hymns in the Rig Veda and associated texts showoff an early system of names for powers of ten. For exam-ple, Yajur Veda 7.2 gives the mythological background to aparticular type of sacrifice. Section 11 of this hymn says:

Hail to one! Hail to two! Hail to three! Hail to four! Hail to five! Hail to six! Hailto seven! Hail to eight! Hail to nine! Hail to ten! Hail to eleven! Hail to twelve! Hailto thirteen! Hail to fourteen! Hail to fifteen! Hail to sixteen! Hail to seventeen! Hail toeighteen! Hail to nineteen! Hail to twenty-nine! Hail to thirty-nine! Hail to forty-nine!Hail to fifty-nine! Hail to sixty-nine! Hail to seventy-nine! Hail to eighty-nine! Hail toninety-nine! Hail to a hundred! Hail to two hundred! Hail to all!

The next section hails odd numbers, the next even numbers; others hail multiples of four, five, ten,and twenty. Finally, in section 20:

Hail to a hundred, hail to a thousand, hail to ayuta (10,000), hail to niyuta (100,000), hailto prayuta (1,000,000, million), hail to arbuda (10,000,000), hail to nyarbuda (100,000,000),hail to samudra (1,000,000,000, billion), hail to madhya (10,000,000,000), hail to anta(100,000,000,000), hail to parardha (1,000,000,000,000, trillion), hail to the dawn, hail tothe daybreak, hail to him who will rise, hail to the rising, hail to the risen, hail to heaven,hail to the world, hail to all.

Classical and medieval periodsIn the second half of the first millennium bce, the Sanskrit language evolves to its classical form.Linguistics develops at this time so that priests and worshippers can understand and preserve thelanguage of the Rig Veda; the differences between Vedic Sanskrit and classical Sanskrit are similarto the differences between Middle English and Modern English. The modern vernacular languagesalso start to develop during the classical period, but scholarly texts (including mathematics) and highliterature continue to be written in Sanskrit for many hundreds of years.

The well-known mathematicians of Indian history are allin this period:

• Aryabhat.a 476–550, astronomer

• Brahmagupta 598–668, mathematician doing algebra,geometry, and astronomy

• Bhaskara II 1114–1185, author of the Lılavatı

The famous inscription in the temple at Gwalior that in-cludes a zero was carved in 876 CE.

Bhaskara’s Lılavatı is a relatively accessible mathemati-cal treatise addressed to a woman named Lılavatı (“Playful”),perhaps Bhaskara’s daughter.