## 31.8.10

### History of the Quadratic Equation

By Urmil Patel and Jason Page

The first form of quadratic equations discovered were used by the Babylonians at around 4000 years ago (Boyer, 32). In 1930 Otto Neugebauer documented a clay tablet that used two unknowns that involved a square root (Cooke, 401; Boyer, 31). These problems were used to find widths and diagonals of rectangles (Melville, Old Babylonian 'Quadratic' Problems). The solution they used would be equivalent to x=sqr((p/2)^2+q)+p/2 for a root of the equation x^2-px=q. Their equation used two constant unknowns, referred to as p and q. Both constants were positive and there was no known use of zero during Babylonian time (Boyer, 31).

The use of zero has its origins as a place holder rather than a number. The Greeks debated the issue “how could nothing be something” and they used alphabet of numbers not consisting of zero (Zero, Wikipedia).

The first known use of zero as a place holder in solving quadratic equations was found in the Bakhasheli manuscript (Pearce, The Bakhshali manuscript). The manuscript was discovered by a peasant in 1881 on the leaves of birchbark (Samponius, 147). The script was in an early form of the Sarada script which suggest its time of origin to be before 300 A.D. (Samponius, 147; Pearce, The Bakhshali manuscript)

The actual age of the manuscript had been disputed in the beginning of its discovery. G Joseph critizes the original author to first document the manuscript:

...It is particularly unfortunate that [G.R.] Kaye is still quoted as an authority on Indian mathematics.

Kay, according to Joseph dated the manuscript to justify Greek and Arabic influences on the text. Scholars such as Joseph contend that Kay was biased and his authority on Indian math history has been debunked. The continuity in the progress of quadratic equations is unknown...and it is possible the composer(s) of the Bakhshali manuscript were not fully aware of earlier works and had to start from 'scratch.' (Pearce)

Pearce addressed the section on quadratic equations in the Bakhshali as “development of remarkable quality.” The equation follows the form:

If the equation given is dn^2 + (2a - d)n -2s = 0

The equation consists of the constants d, a and s with variable n. The zero is used as a place holder that also represented unknown and is indicated by a dot.

Brahmagupta’s Siddhanta establishes zero as a number and establishes negative numbers (Wikipedia, Zero). He defines zero as “result of subtracting a quantity from itself.(Tanton, Brahmagupta)” Brahmagupta’s Brahmasphutasiddhanta (Opening of the Universe) in 628 AD used quadratic equations for astronomical calculations (Tanton, Brahmagupta) to predict the position of planets on certain lunar days. (Smith, 158).

The first complete solution of the quadratic equation is attributed to Abraham bar Hiyya Ha-Nasi (also known as Savasorda) in his 1145 AD book Hibbur ha-Meshihah ve-ha-Tishboret (Treatise on Measurement and Calculation) (Connor, Abraham bar Hiyya Ha-Nasi) His solution follows the methods for solution used today (formulalist.)