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134
CHAPTER 2. CURVES
(d)
x
2
+ 4
x
−
16
y
2
+ 32
y
+ 4 = 0
Theory problems:
3.
Show
that Equation (
2.2
) (the statement of Prop. 13, Book VI of
the
Elements
) is equivalent to the standard equation for a circle.
History notes:
Spiral of Archimedes:
Archimedes in his work
On Spirals
[
3
], studied
the curve with polar equation
r
=
aθ
(
a
a positive constant) (see p.
150
).
4.
Quadrature of the Circle:
According to Heath [
24
, vol. 1, p.
230] and Eves [
13
, p. 84], Archimedes is said to have used the spiral
to construct a square whose area equals that of a given circle. This
was one of the three classical problems (along with trisecting the
angle and duplicating the cube) which the Greeks realized could not
be solved by rulerandcompass constructions [
24
, vol 1, pp. 218F],
although a proof of this impossibility was not given until the
nineteenth century. However, a number of constructions using other
curves (
not
constructible by compass and straightedge) were given.
Our exposition of Archimedes’ approach follows [
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This note was uploaded on 10/20/2011 for the course MAC 2311 taught by Professor All during the Fall '08 term at University of Florida.
 Fall '08
 ALL
 Calculus

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