mathematics at the turn of the fourteen century

1. Common Ideas of
Mathematics
Noemi M. Rosario
BSE III-4
Mathematics at
the turn of the
Fourteenth
Century

2. We begin with geometry. Practical geometry, that is,
the measure of fields, the determination of unknown
distances and heights, the calculation of volumes, and so
on, was performed by much the same techniques in the
four societies studied. All of them knew how to calculate
areas and volumes, at least approximately, and all knew
and used the Pythagorean Theorem when dealing with
right triangles. Even the techniques of determining the
height of a distant tower were nearly the same.

4. As far as theoretical geometry was concerned, it was in the
world of Islam that the heritage of classical Greek geometry was
preserved and studied and in which further advances were made.
It was there that questions were raised and answered about the
exact volumes of certain solids and about the locations of centers of
gravity, using both heuristic methods for arriving at answers and the
technique of exhaustion for giving proofs. It was there that
questions were raised and answers attempted about Euclid’s
parallel postulate. It was there that questions were raised and new
ideas developed about the classical Greek separation of number
and magnitude. And it was there that the Greek idea of proof from
stated axioms was most fully understood and developed.

6. Although Europe had always had at least a version of Euclid’s
Elements available, the beginning of the fourteenth century saw
only the bare beginnings of a renewed interest in Euclid and other
Greek geometers, stimulated by the appearance of a mass of
translations of this material in the twelfth and thirteenth centuries.
But although the idea of proof survived, there was still no new
work in theoretical geometry. Neither Indian or China had been
exposed to classical Greek geometry, as far as is known, but that is
not to say that they had no notion of proof.

8. In the works of the Chinese mathematicians and their
numerous commentators, there are always derivations
of results. These derivations are not, however, based on
explicitly stated axioms. They are, on the other hand,
examples of logical arguments. In India, written
derivations from early times have not survived; but
beginning in the fourteenth century, we see many
attempts to write out explicit derivations of mathematical
results.

9. In certain aspects of algebra, on the other hand, the
Chinese were the first to develop techniques that were later used
else where. For example, they had from early times constructed
efficient methods of solving systems of linear equations. By the
fourteenth century, they had developed their early root finding
techniques, which involved the use of the Pascal triangle, into a
detailed procedure for solving polynomial equations of any degree.
They also worked out the basis of what is today called the Chinese
remainder theorem, a procedure for solving simultaneous linear
congruences.

11. Linear congruence were also solved in India, by a
method different from that of the Chinese but still using
the Euclidean algorithm. Indian scholars were even
prouder, however, of the techniques they developed for
solving the quadratic indeterminate equations known
today as the Pellequations..

12. The Indian mathematicians were also familiar with the
standard techniques of solving quadratic equations, but
since there is no documentation of how they thought
about the method, we do not know whether they
developed the technique independently or absorbed it
from the ancient Babylonians. A third possibility is that
they learned it somehow from the work of Diophantus,
who in turn was probably aware of the methods at least
indirectly from the Babylonians.

14. For Islam, of course, there is copious documentation of an
interest in algebra. Not only did Islamic mathematicians study the
quadratic equation in great detail, giving geometric justifications for
the various algebraic procedures involved in the solution, they
studied cubic equations as well. For these equations, Islamic
mathematicians developed a solution method involving conic
sections and gained some understanding of the relationship of the
roots to the coefficients of these equations. In addition, they knew
a method of solving polynomial equations numerically, similar to
the Chinese method, ultimately based on the Pascal triangle.

15. The Pascal triangle also appeared in Islamic
mathematics in connection both with the binomial
theorem and with the study of combinatorics. Islamic
mathematicians who dealt with these two aspects of the
Pascal triangle also developed proof techniques closely
resembling our modern proof by induction. Such
techniques were further worked out in Europe by Levi ben
Gerson.

16. Furthermore, Islamic algebraists developed in great detail the
methods for manipulating algebraic expressions, especially those
involving surds, and thereby began the process of negating the
classical Greek separation of number and magnitude.
By the turn of the fourteenth century, algebraic techniques
were only beginning their appearance in Europe. Those
techniques that were available were clearly based on the Islamic
work, although Jordanus de Nemore considered the material from
a somewhat different point of view. He also introduced a form of
symbolism in his algebraic work, something missing entirely in
Islamic algebra but also present, in different form, in India and
China

18. On the other hand, European algebra of this time
period, like its Islamic counter part, did not consider
negative numbers at all. India and China, however, were
very fluent in the use of negative quantities in
calculations, even if they were still hesitant about using
them as answers to mathematical problems.