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      <text>6.4.2 (History of circle geometry)</text>
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      <text><![CDATA[<h3>History of circle mathematics</h3>
<p>The search for the mathematical properties of the circle extends into the past beyond recorded history. Think of the invention of the wheel - this arguably is the most important invention in human history, but it relies on the fundamental mathematical properties of the circle. Clearly circles appear throughout the natural world, from the head of a sunflower to the cross section of an orange, so understanding the laws which govern circles has been a topic that has interested humanity for millennia. The study of circles lies under the topic of geometry, which began to develop as a science firstly under Egyptians and then the ancient Greeks in the west and under the ancient Chinese in the west. As we have seen previously, most most important (defining) property of a circle is the ratio of its circumference to its diameter, i.e. \(\pi\). Much of the early study of circles was involved with trying to find accurate estimates of the value of \(\pi\) (this would then allow the accurate calculation of circular areas.)</p>
<p>One of the earliest discoveries regarding circles can be found in the Rhind papyrus ascribed to Ahmes who gave a rule for determining the area of a circle which corresponds to the estimate that \(\pi =3.16\). Prior to this, the best estimate came from the ancient Babylonians, who used the approximate value of \(pi = 3\). This estimate would later be improved by the ancient Greeks, for whom all mathematics must have foundations in geometry. Many of the early theorems relating to circles are ascribed to Thales (circa 650BCE), and of course Euclid's elements deals with many properties of circles and problems regarding the construction of polygons.</p>
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<p style="text-align: center;"><img src="@@PLUGINFILE@@/History%20of%20circles.png" alt="image shoing a sunflower and the RHind papyrus" width="710" height="266" /></p>
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<p>An early calculation of \(\pi\) was done by Archimedes of Syracuse (287–212 BCE). Archimedes approximated the area of a circle by using the Pythagorean Theorem to find the areas of two regular polygons: the polygon inscribed within the circle and the polygon within which the circle was enclosed. Since the actual area of the circle lies between the areas of the inscribed and circumscribed polygons, the areas of the polygons gave bounds for the area of the circle. This allowed Archimedes to approximate the value of \(\pi\) to be between around \(3 \frac{1}{7}\) and \(3 \frac{10}{71}\). One of the major problems studied by the Greeks was the problem of 'squaring the circle'. This problem involves constructing a square with the same area of a given circle, using a finite number of steps with a compass and ruler. This problem was first studied by Anaxagoras (circa 450BC) however no solution could be found. In fact squaring the circle is impossible but this was only shown in the 19th century and it is a direct result of the fact \(\pi\) is a transcendental number (meaning it cannot be expressed as the solution to some non-zero polynomial equation with rational coefficients). During the 18th century was when mathematicians began to adopt the symbol \(\pi\) for the ratio of a circles circumference to its diameter. This notation was popularised by Leonhard Euler, but it was first suggested earlier by William Jones.</p>]]></text>
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