jAtavedase… and shredhI vyavahAra
The hymn RV 1.99 is a peculiar hymn in many ways:
jātavedase sunavāma somam-arātīyato ni dahāti vedaḥ |
sa naḥ parṣadati durgāṇi viśvā nāveva sindhuṃ duritātyaghniḥ ||
It is addressed to agni jAtavedas: We extract soma for jAtavedas Jatavedas; may he consume the wealth of the evil enemies. May agni tide us over our troubles, through difficulties as in a boat across the river.
The hymn is the one sUktaM in the R^ig that comprises of a single R^ik. It also re-appears in the tantras as the celebrated invocation of durgA. It may be combined with other tantric incantations in the prayoga of shUlinI. Likewise, in the triple prayoga combining sharabha, shUlinI and pratya~NgirA, it is in fact the joint attacking and protecting module which forms the left wing, with the ugra-kR^ityA R^ik forming the right-wing of the same prayoga with a principal role in back-hurling. The stuhi shrutaM R^ik form the central destroying module.
However, both shaunaka’s bR^ihad-devata (3.130) and kAtyAyana’s sarvAnukramaNI have a peculiar tradition regarding this hymn jAtavedase…: It is supposed to be the only listed suktaM of a special khila known as the kashyapa khila, which comprised of 1000 suktaM-s composed to agni-jAtavedas and also known as the grand glorification of agni. shaunaka then quotes shlokas of the ancient authority on vedic particulars, shAkapUNi that the 1000 sUktaM-s followed an arithmetic progression including the jAtavedase hymn of having 1 to 1000 R^iks in them. Then shAkpUNi calculates the value of the series 1+2+3+…+1000=(1000*1001)/2=500,500. Thus, the shadowy vedic sage shAkpUNi was one of the earliest to provide the correct demonstration of the formula for sum of an arithmetic series. This is clearly older than Pythagoras by any stretch of imagination and at the lower bound the same age as the Rhind Papyrus of Egypt.
It should be Hindus had a long-standing fascination with series (known in sanskR^it as shredhI) of various types from the earliest vedic period. In the taittirIya saMhita (7.2.11-20) there are numerous simple arithemetic series associated with the “multiplicator”, ascending and descending offerings. One of these is a power series of 10: 10^2, 10^3, 10^4, 10^5, 10^6, 10^7, 10^8, 10^9, 10^10,1 0^11, 10^12. This series also occurs in the taittirIya brAhmaNa 3.8.16 and is repeated verbatim in several hindu works like the rAmAyaNa to exaggerated size of the rAkShasa or vAnara army (e.g. yuddha kANDa 3.24-28). The pa~nchavimsha brAhmaNa 18.3 gives the series: 12, 24, 48, 96, …196608, 393216 showing that the early Hindus also worked on geometric series. pi~Ngala in the Chandas shAstra provides a formula for the sum of a geometric series. pi~Ngala also provides early accounts of the binomial expansion series. This early fascination with the series for its own sake is unique to the Hindus, beginning well before any Egyptian mention of it and developing in an algebraic sense well beyond that of their peers. This suggests that it was an original development of the Hindus. In the classical period of Hindu mathematics, beginning with halAyuddha, passing through the pinnacle of the illustrious AryabhaTa, this fascination for the series was formalized, with the term shredhI meaning progression being adopted. shredhi-phala or saMkalita is used for sum of series the formulae for it and nth terms of the series were all obtained. The formulae are also given form composite series of the form: tn=r*tn-1+b or r*tn-1-b
The binomial expansions were provided as meru-prasthas or what were known in the West as Pascal’s triangles. Finally, the great Nambuthiri mathematicians developed the trignometric series expansions and the concept of series with limit n->infinity (kha-hara) to obtain definite integrals.