history of mathematics - National STEM Centre

-a M
Figure 4.8
Other contributions by
iJ/iete, van Schooten and
Fermat are discussed
ater in the Descartes
4 More Greek mathematics
1 a Prove that QM = - \ ph 2
b Why is there a negative sign in this expression?
c Without doing further detailed calculations, show how you can say immediately
that UV = HK = -\ P(\hf = -^ph 2
2 a Now go back to the symmetric parabola shown in Figure 4.8. Show that a
suitable equation for it is f(x) = px 2 + r.
b Show that the area of triangle PRQ is ar . Apply the result of question Ic to
show that the area of each of the triangles QHR and PUQ is \ ar, so that the area of
the shape PUQHR is ar + 2 x { ar = ar + \ ar .
c Show that, if you continue this process by creating more triangles in the same
way, the areas of successive approximations to the area of the parabola are
ar
d Find the sum of this series, and so find the area under the parabolic arc.
Curves
The period of Greek history between about 300 and 200 BC is sometimes described
as 'The Golden Age of Greece'. During this time there was a remarkable flowering
of the arts and literature, and of mathematics. Three names are principally
associated with the development of mathematics in Greece at this time. The work of
two of these, Euclid and Archimedes, has been discussed already. The third of these
great mathematicians was Apollonius of Perga. Little is known of his life, and many
of his writings have been lost. It is thought that he lived from around 262 to 190 BC,
and he appears to have studied at Alexandria with the successors of Euclid, before
settling at Pergamum where there was an important university. Many of his writings
were briefly described by Pappus about 500 years later in a work called the
Collection. The descriptions given by Pappus and others were extensively used in
modern times, by European mathematicians of the 17th century including Viete, van
Schooten and Fermat, who devoted considerable effort to reconstructing some of the
lost works of Greek mathematicians.
As well as being an outstanding mathematician of his day, Apollonius was also
known for his work on astronomy. He proposed important ideas about planetary
motion, which were later used by Ptolemy (about AD 140) in his book, Almagest,
summarising the astronomy known to the ancient Greeks.
Apollonius of Perga is best known today for a major treatise on conic sections, the
only one of his major works substantially to have survived. This is a collection of
eight books, begun while Apollonius was in Alexandria, and continued from
Pergamum. The first four books draw together the theoretical work of earlier
mathematicians, including Euclid and Archimedes. The later books developed the
ideas substantially. This work was highly influential on later mathematicians in
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