12 1. THE ORIGIN AND PREHISTORY OF MATHEMATICStype, "If 3 bananas cost 75 cents, how much do 7 bananas cost?" occur in the work
of Brahmagupta from 1300 years ago. Brahmagupta named the three data numbers
argument (3), fruit (75), and requisition (7). As another example, cuneiform tablets
from Mesopotamia that are several thousand years old contain general problems
that we would now solve using quadratic equations. These problems are stated
as variants of the problem of finding the length and width of a rectangle whose
area and perimeter are known. The mathematician and historian of mathematics
B. L. van der Waerden (1903-1996) claimed that the words for length and width
were being used in a completely abstract sense in these problems.
In algebra symbolism seems to have occurred for the first time in the work of
the Greek mathematician Diophantus of Alexandria, who introduced the symbol ς
for an unknown number. The Bakshali Manuscript, a document from India that
may have been written within a century of the work of Diophantus, also introduces
an abstract symbol for an unknown number. In modern algebra, beginning with the
Muslim mathematicians more than a millennium ago, symbolism evolved gradually.
Originally, the Arabic word for thing was used to represent the unknown in a
problem. This word, and its Italian translation cosa, was eventually replaced by
the familiar χ most often used today. In this way an entire word was gradually
pared down to a single letter that could be manipulated graphically.
4. Mathematical inference
Logic occurs throughout modern mathematics as one of its key elements. In the
teaching of mathematics, however, the student generally learns all of arithmetic
and the rules for manipulating algebraic expressions by rote. Any justification of
these rules is purely experimental. Logic enters the curriculum, along with proof,
in the study of Euclidean geometry. This sequence is not historical and may leave
the impression that mathematics was an empirical science until the time of Euclid
(ca. 300 BCE). Although one can imagine certain facts having been discovered by
observation, such as the rule for comparing the area of a rectangle with the area
of a square unit, there is good reason to believe that some facts were deduced from
simpler considerations at a very early stage. The main reason for thinking so is
that the conclusions reached by some ancient authors are not visually obvious.
4.1. Visual reasoning. As an example, it is immediately obvious that a diagonal
divides a rectangle into two congruent triangles. If through any point on the diago-
nal we draw two lines parallel to the sides, these two lines will divide the rectangle
into four rectangles. The diagonal divides two of these smaller rectangles into pairs
of congruent triangles, just as it does the whole rectangle, thus yielding three pairs
of congruent triangles, one large pair and two smaller pairs. It then follows (see
Fig. 2) that the two remaining rectangles must have equal area, even though their
shapes are different and to the eye they do not appear to be equal. For each of
these rectangles is obtained by subtracting the two smaller triangles from the large
triangle in which they are contained. When we find an ancient author mentioning
that these two rectangles of different shape are equal, as if it were a well-known
fact, we can be confident that this knowledge does not rest on an experimental or
inductive foundation. Rather, it is the result of a combination of numerical and
spatial reasoning.
Ancient authors often state what they know without saying how they know
it. As the example just cited shows, we can be confident that the basis was not