The History of Mathematics: A Brief Course

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Geometry is a way of organizing our perception of shape. As in arithmetic, we can
distinguish levels of sophistication in the development of geometry. The first level
is that of measurement: comparing the sizes of objects having different shapes.
Measurement is arithmetic applied to space, beginning with figures having flat sides
or faces. The second level is the study of the proportions among the parts of
geometric figures, such as triangles, squares, rectangles, and circles. A good marker
for the beginning of this stage of development is the Pythagorean relation for right
triangles.


Although regular polygons and polyhedra can be measured using simple dissec-
tion techniques, algebra is needed to measure more complicated figures, such as the
portion of a disk remaining after three other disks, all tangent to one another and
to the original disk and whose radii are in given ratios, are removed from it. One of
the uses of geometry in many ancient treatises is as a source of interesting equations
to be solved. Although the problems are posed as problems in measurement, the
shapes being measured are so unusual that it is hard to think of them as a motive.
The suspicion begins to arise that the author's real purpose was to exhibit some
algebra.
The Pythagoreans, Plato, and Aristotle gave geometry a unique philosophical
grounding that turned it out of the path it would probably have followed otherwise.
Their insistence on a logical development based on a system of axioms made ex-
plicit many assumptions—especially the parallel postulate—that otherwise might
not have been noticed. As a result, there is a marked difference between the math-
ematical practitioners who learned geometry from Euclid (the medieval Muslims
and renaissance Europeans) and those who learned it from a different tradition.


At its highest level, elementary geometry employs algebra and the infinite pro-
cesses of calculus in order to find the areas and volumes of ever more complicated
curvilinear figures. Like arithmetic, geometry has given rise to many specialties,
such as projective, analytic, and differential geometry. As more and more general
properties of space became mathematized, geometry generated the subject known
as topology, from Greek roots meaning theory of position.
In the four chapters that constitute the present part of our history, we shall
look at all these aspects of geometry. In Chapter 9 we study the way space was
measured in a number of civilizations. Chapters 10 and 11 form a unit devoted
to the most influential form of elementary geometry, the Euclidean geometry that
arose in the Hellenistic civilization. Chapter 12 contains a survey of the variety of
forms of geometry that have arisen over the past three centuries.

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