**PUZZLE
OF ANCIENT MATHEMATICS, INDIA AND THE WORLD I**

**There is now a revision of
history of mathematics, recognizing Indian contributions in both geometry and
algebra.**

**NavaratnaRajaram**

In our study of
Indian contributions to the mathematics of the ancient world, we are faced with
two hurdles. First, extravagant claims in the name of Vedic Mathematics that
have no support in the Vedic literature. The famous book *Vedic Mathematics *by Swamy Bharati Krishna Tirtha (compiled by his
disciples) is a modern text by an accomplished mathematician (at a basic level)
and a capable Vedic scholar. It may have some pedagogical value in teaching
basic to intermediate mathematics, but is of no historical significance. Its
contents cannot be traced to the Atharva Veda or any Vedic texts as claimed by its proponents. [His book
*Vedic Metaphysics *is more sober and
useful but not as well known as it should be.]

The second hurdle is the attempt by some Western scholars and their followers to deny all Indian contributions, attributing them to foreign sources like the Greeks and the Chinese. Its perverse limit was reached probably by David Pingree in his creation of a pseudo-Greek text based on Indian sources which he later claimed to be the source of Indian mathematics. A prime example is the book on Indian mathematics by Pingree student Kim Plofker. But ancient authors including Greeks and Arabs make no such claims and freely acknowledge their indebtedness to other sources.

But a careful examination of the Bakshali Manuscript should have put an end these absurdities, but for reasons known only to herself Kim Plofker seems to take no note of its conents. Pingree too seemed to have been unaware of it when he created his pseudo-Greek source.

It is generally recognized that the so-called Arabic numerals, now used worldwide is a misnomer and they are of Indian origin. Arabs themselves called them Hindi numerals. The Italian mathematician Leonardo of Pisa (better known as Fibanocci) was among the first to introduce them in Europe as well as the decimal place value system (also from India). He states that he got it from the Arabs, then ruling Spain but also notes that the Arabs got them from the Indians. The Fibonacci numbers were also known to the Indians, centuries before they were discovered by Europeans.

More than 500 years before Fibonacci, we have more substantial evidence from Syria suggesting that as early as the seventh century, knowledgeable scholars recognized Indian mathematics to be superior to the Greek.Writing in 662 CE, Servius Sebokht (575-667 CE)., the Bishop of Qinnesrin in North Syria observed:

I will omit all discussion of the science of the Hindus
[Indians], *discoveries more
ingenious than those of the Greeks and the Babylonians:*

He
notes in particular their
valuable method of calculation [the decimal system]; their computing that
surpasses description. I wish only to say that this computation is done by
means of nine *signs.
If those who believe because they know Greek, that they have*

*reached the limits of
science should know these things, they would be convinced that there are also
others **(than the Greeks) **who know something.*

Aryabhata appears to have been the earliest to use the decimal system with zero extensively.

The origin of the modern decimal-based place value notation
can be traced to the *Aryabhatiya* (c. 500), which states *sthānātsthānaṁdaśaguṇaṁsyāt* ”
from place to place each is ten times the preceding

Even
though we take the decimal system as the standard, it is only a special case of
the place value system, to base 10. Now binary systems (base 2) are widely used
in computations, especially in computer science. Wilhem Leibnitz in the 17^{th}
century is usually credited with the discovery of the binary system, but he was
preceded by others going back centuries.

The ** Chandaḥśāstra** (c.
400 BCE or earlier), presents the first known description of a binary numeral system in
connection with the systematic enumeration of meters with fixed patterns of
short and long syllables.The discussion of the combinatorics of meter
corresponds to the binomial theorem. Halāyudha’s commentary includes a presentation of Pascal’s triangle (called

*meruprastāra*). Pingala’s work also includes material related to the Fibonacci numbers, called

*mātrāmeru*.

^{[7]}

Use of zero is sometimes ascribed to Pingala due to his
discussion of binary numbers, usually represented using 0 and 1 in modern
discussion, but Pingala used light (*laghu*) and heavy (*guru*)
rather than 0 and 1 to describe syllables. As Pingala’s system ranks binary
patterns starting at one (four short syllables—binary “0000”—is the
first pattern), the nth pattern corresponds to the binary representation of n-1
(with increasing positional values).

Pingala is
credited with using binary numbers in
the form of short and long syllables (the latter equal in length to two short
syllables). Pingala used the Sanskrit word *śūnya* explicitly to refer to zero. This and its
derivatives are still used in Indian languages and its derivatives.

Also the modern numeral notation from 0 to 9 are found explicitly presented in the 2000 year old Bakshali Manuscript.

**Large
numbers in ancient Indian literature**

Greeks and the
Indians approached mathematics in different spirit. Greeks are better known for
geometry, while Indians, while not ignoring geometry were focused more on the
representation and manipulation of numbers, excellent at working with large
numbers that seemed to beyond the capability of the Greeks . Evidence for this
is available from the *Mahabharata, *in
describing the armies engaged in the war. We are told that 18 akshauhanis (or
armies) battled it out at Kurukshetra battlefield.

An **akshauhini** (Sanskrit: अक्षौहिणी *akṣauhiṇī*)
is described in the Mahabharata as a battle formation (or army) consisting of approximately 5 lac.
warriors (500 K or half a million) i.e. 21,870 chariots
(Sanskrit *ratha*); 21,870 elephants (Sanskrit *gaja*);
65,610 horses (Sanskrit *turaga*) and 109,350 infantry
(Sanskrit *padasainyam*) as per the Mahabharata (AdiParva 2.15-23).^{[1]}^{[2]} (thus the total number of humans, warriors in
akshauhini is equal to 218,700). The ratio is 1 chariot : 1
elephant : 3 cavalry : 5 infantry soldiers. In each of these large
number groups (65,610, etc.), the digits add up to 18.

It is not our point here that such large armies actually took part in the Mahabharata War. But only ancient Indians were comfortable with large numbers and knew how to handle them. In this they were helped by a vastly superior number system based on place-values that remains the world standard even today, nearly 2000 years later if not more.