422 14. EQUATIONS AND ALGORITHMSor quartic equations, it must be found by the solver's own ingenuity" (Colebrooke,
1817, pp. 207-208). That ingenuity includes some work that would nowadays be
regarded as highly inventive, not to say suspect; for example (Colebrooke, 1817, p.
214), how to solve the equation
(0(÷+|÷))^2 + 2(è(÷+^)) _ ic
0 ~^15 '
Bhaskara warns that multiplying by zero does not make the product zero, since
further operations are to be performed. Then he simply cancels the zeros, saying
that, since the multiplier and divisor are both zero, the expression is unaltered.
The result is the equation we would write as |x^2 + 3x = 15. Bhaskara clears the
denominator and writes the equivalent of 9x^2 + l2x = 60. Even if the multiplication
by zero is interpreted as multiplication by an expression that is tending to zero, as
a modern mathematician would like to do, this cancellation is not allowed, since the
first term in the numerator is a higher-order infinitesimal than the second. Bhaskara
is handling 0 here as if it were 1. Granting that operation, he does correctly deduce,
by completing the square (adding 4 to each side), that ÷ = 2.
5. The Muslims
It has always been recognized that Europe received algebra from the Muslims; the
very word algebra (al-jabr) is an Arabic word meaning transposition or restora-
tion. Its origins in the Muslim world date from the ninth century, in the work
of Muhammed ibn Musa al-Khwarizmi (780-850), as is well established.^12 What
is less certain is how much of al-Khwarizmi's algebra was original with him and
how much he learned from Hindu sources. According to Colebrooke {1817, pp.
lxiv-lxxx), he was well versed in Sanskrit and translated a treatise on Hindu com-
putation^13 into Arabic at the request of Caliph al-Mamun, who ruled from 813 to
- Colebrooke cites the Italian writer Pietro Cossali^14 who presented the alter-
natives that al-Khwarizmi learned algebra either from the Greeks or the Hindus
and opted for the Hindus. These alternatives are a false dichotomy. We need not
conclude that al-Khwarizmi took everything from the Hindus or that he invented
everything himself. It is very likely that he expounded some material that he read
in Sanskrit and added his own ideas to it. Rosen (1831, p. x) explains the difference
in the preface to his edition of al-Khwarizmi's algebra text, saying that "at least
the method which he follows in expounding his rules, as well as in showing their
application, differs considerably from that of the Hindu mathematical writers."
(^12) Colebrooke (1817, p. lxxiii) noted that a manuscript of this work dated 1342 was in the Bodleian
Library at Oxford. Obviously, this manuscript could not be checked out, and Colebrooke com-
plained that the library's restrictions "preclude the study of any book which it contains, by a
person not enured to the temperature of apartments unvisited by artificial warmth." If he worked
in the library in 1816, his complaint would be understandable: Due to volcanic ash in the at-
mosphere, there was no summer that year. This manuscript is the source that Rosen (1831)
translated and reproduced.
(^13) It is apparently this work that brought al-Khwarizmi's name into European languages in the
form algorism, now algorithm. A Latin manuscript of this work in the Cambridge University
Library, dating to the thirteenth century, has recently been translated into English (Crossley and
Henry, 1990).
(^14) His dates are 1748-1813. He was Bishop of Parma and author of Origine, trasporto in Italia,
primi progressi in essa dell' algebra (The Origins of Algebra, and Its Transmission to Italy and
Early Progress There), published in Parma in 1797.