Chapter 12. Modern Geometries
In geometry, as in number theory, the seventeenth century represents a break with
the past. The two main reasons for the sudden surge of mathematical activity are
the same in both cases: first, the availability of translations from the Arabic, which
stimulated European mathematicians to try to recover and extend the fascinating
results achieved by the ancient Greeks and medieval Muslims; second, the devel-
opment of algebra and its evolution into a symbolic form in the Italian city-states
during the sixteenth century. This development suggested new ways of thinking
about old problems. The result was a variety of new forms of geometry that came
about as a result of the calculus: analytic geometry, algebraic geometry, projective
geometry, descriptive geometry, differential geometry, and topology.
1. Analytic and algebraic geometry
The creation of what we now know as analytic geometry had to wait for algebraic
thinking about geometry (the type of thinking Pappus called analytic) to become a
standard mode of thinking. No small contribution to this process was the creation
of the modern notational conventions, many of which were due to Frangois Viete
(1540-1603) and Descartes. It was Descartes who started the very useful convention
of using letters near the beginning of the alphabet for constants and data and those
near the end of the alphabet for variables and unknowns. Viete's convention, which
was followed by Fermat, had been to use consonants and vowels respectively for
these purposes.
1.1. Fermat. Besides working in number theory, Fermat studied the works of
Apollonius, including references by Pappus to lost works. This study inspired him
to write a work on plane and solid loci, first published with his collected works
in 1679. He used these terms in the sense of Pappus: A plane locus is one that
can be constructed using straight lines and circles, and a solid locus is one that
requires conic sections for its construction. He says in the introduction that he
hopes to systematize what the ancients, known to him from Book 7 of Pappus'
Synagdge, had left haphazard. Pappus had written that the locus to more than six
lines had hardly been touched. Thus, locus problems were the context in which
Fermat invented analytic geometry.
Apart from his adherence to a dimensional uniformity that Descartes (finally!)
eliminated, Fermat's analytic geometry looks much like what we are now familiar
with. He stated its basic principle very clearly, asserting that the lines representing
two unknown magnitudes should form an angle that would usually be assumed a
right angle. He began with the equation of a straight line:^1 Z^2 - DA = BE. This
equation looks strange to us because we automatically (following Descartes) tend
(^1) Fermat actually wrote "Z pi — D in A aequetur  in E."
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