Chapter 14. Equations and Algorithms
In this chapter we take up the history of algebra from the point at which equa-
tions appear explicitly and carry it forward along two parallel lines. In one line of
development the emphasis is on finding numerical approximations to the roots of
an equation. In the second line the emphasis is on finding an algorithm involving
only the four operations of arithmetic and the extraction of roots that will yield the
solution. The second line of development reached its highest point of achievement
in sixteenth-century Italy, with the arithmetical solution of equations of degree 4.
That is the point at which the present chapter ends. Standing somewhat to one side
of both lines of evolution was the work of Diophantus, which contains a mixture of
topics that now form part of number theory and algebra.1. The Arithmetica of Diophantus
The work of Diophantus of Alexandria occupies a special place in the history of
algebra. To judge it, one should know something of its predecessors and its in-
fluence. Unfortunately, information about either of these is difficult to come by.
The Greek versions of the treatise, of which there are 28 manuscripts, according to
Sesiano (1982, p. 14), all date to the thirteenth century. Among the predecessors
of Diophantus, we can count Heron of Alexandria and one very obscure Thymari-
das, who showed how to solve a particular set of linear equations, the epanthema
(blossom) of Thymaridas. Because the work of Diophantus is so different from the
Pythagorean style found in Euclid and his immediate successors, the origins of his
work have been traced to other cultures, notably Egypt and Mesopotamia. The his-
torian of mathematics Paul Tannery (1843-1904) printed an edition of Diophantus'
work and included a fragment supposedly written by the eleventh-century writer
Michael Psellus (1018-ca. 1078), which stated that "As for this Egyptian method,
while Diophantus developed it in more detail,... ." It was on this basis, identify-
ing Anatolius with a third-century Bishop of Laodicea originally from Alexandria,
that Tannery assigned Diophantus to the third century. Neugebauer (1952, p. 80)
distinguishes two threads in Hellenistic mathematics, one in the logical tradition of
Euclid, the other having roots in the Babylonian and Egyptian procedures and says
that, "the writings of Heron and Diophantus... form part of this oriental tradition
which can be followed into the Middle Ages both in the Arabic and in the western
world." Neugebauer sees Diophantus as reflecting an earlier type of mathemat-
ics practiced in Greece alongside the Pythagorean mathematics and temporarily
eclipsed by the Euclidean school. As he says (1952, p. 142):
It seems to me characteristic, however, that Archytas of Tarentum
could make the statement that not geometry but arithmetic alone
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