history of mathematics - National STEM Centre

M
Figure 4.10
F L
Figure 4.11
This result, in the words of Apollonius is:
I 'if
4 More Greek mathematics
we apply the square LPto a straight line length k, we obtain the
abscissa FL', or'the exact parabola of the square of the ordinate on the
line segment k gives the abscissa'.
This is based on a much earlier use of the word 'parabola', possibly introduced by
the Pythagoreans. Hence Apollonius named this curve the parabola.
2 In the second case, FL cuts AB and AC at the points F and J (Figure 4.10).
a Use an argument like the one in question 1 to show that LP 2 = LM x LN is
proportional to LF x LJ.
b Suppose that the constant of proportionality is the ratio of some length FK to the
length FJ (Figure 4.11).
Show that there is a point Q on KJ such that LP 2 is equal to the rectangle FLQR.
In Apollonius's words, this is to say that 'the square of LP is applied to the line FJ
in ellipsis by a rectangle LJSQ, similar to the rectangle FJTK'. Here, the term
'ellipsis' means 'deficiency'. Hence this conic section was named the ellipse.
3 In the third case, FL cuts AB at F, and cuts CA produced at J.
The Greek word 'hyperbola' means 'in excess' or 'a throwing beyond'. Use an
argument similar to that in question 2 to explain why Apollonius named the curve in
this third case the hyperbola.
The way in which Apollonius worked with the conies, using abscissa and ordinates,
suggests that he was very close to developing a system of coordinates. Many of his
results can be re-written almost immediately into the language of coordinates, even
though no numbers were attached to them. As a result, it has sometimes been
claimed that Apollonius had discovered the analytic geometry which was developed
some 1800 years later. However, the fact that the Greeks had no algebraic notation
for negative numbers was a great obstacle to developing a coordinate system.
Greek mathematicians classified construction problems into three categories. The
first class was known as 'plane loci', and consisted of all constructions with straight
lines and circles. The second class was called 'solid loci' and contained all conic
sections. The name appears to be suggested by the fact that the conic sections were
defined initially as loci in a plane satisfying a particular constraint, as they are
today, but were described as sections of a three-dimensional solid figure. Even
Apollonius derived his sections initially from a cone in three-dimensional space, but
he went on to derive from this a fundamental property of the section in a plane,
which enabled him to dispense with the cone. The third category of problems was
'linear loci', which included all curves not contained in the first two categories.
Hypatia
Hypatia of Alexandria was born in about AD 370 towards the end of the Greek era.
She was the daughter of Theon of Alexandria, known for his commentary on
Ptolemy's Almagest and his edition of Euclid's Elements.
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