Archimedes of Syracuse
The ratio of the circumference to the diameter of a circle was a subject of ancient interest. Around 2000 BC the Babylonians made the observation that the circumference was roughly 3 times as long as its diameter. It was Archimedes of Syracuse who made a real start on the mathematical theory of π in around 225 BC. Archimedes is right up there with the greats. Mathematicians love to rate their co- workers and they place him on a level with Carl Friedrich Gauss (The ‘Prince of Mathematicians’) and Sir Isaac Newton. Whatever the merits of this judgment it is clear that Archimedes would be in any mathematics Hall of Fame. He was hardly an ivory tower figure though – as well as his contributions to astronomy, mathematics and physics, he also designed weapons of war, such as catapults, levers and ‘burning mirrors’, all used to help keep the Romans at bay. But by all accounts he did have something of the absent-mindedness of the professor, for what else would induce him to leap from his bath and run naked down the street shouting ‘Eureka’ at discovering the law of buoyancy in hydrostatics ?
How he celebrated his work on π is not recorded.
Given that π is defined as the ratio of its circumference to its diameter, what does it have to do with the area of a circle? It is a deduction that the area of a circle of radius r is πr 2 , though this is probably better known than the circumference/diameter definition of π. The fact that π does double duty for area and circumference is remarkable.
How can this be shown ?
The circle can be split up into a number of narrow equal triangles with base length b whose height is approximately the radius r. These form a polygon inside the circle which approximates the area of the circle. Let’s take 1000 triangles for a start. The whole process is an exercise in approximations. We can join together each adjacent pair of these triangles to form a rectangle (approximately) with area b × r so that the total area of the polygon will be 500 × b × r. As 500 × b is about half the circumference it has length πr, the area of the polygon is πr × r = πr 2 . The more triangles we take the closer will be the approximation and in the limit we conclude the area of the circle is. πr 2 . Archimedes estimated the value of π as bounded between and . And so it is to Archimedes that we owe the familiar approximation 22/7 for the value o f π. The honour for designating the actual symbol π goes to the little known William Jones, a Welsh mathematician who became Vice President of the Royal Society of London in the 18th century. It was the mathematician and physicist Leonhard Euler who popularized π in the context of the circle ratio.
The ratio of the circumference to the diameter of a circle was a subject of ancient interest. Around 2000 BC the Babylonians made the observation that the circumference was roughly 3 times as long as its diameter. It was Archimedes of Syracuse who made a real start on the mathematical theory of π in around 225 BC. Archimedes is right up there with the greats. Mathematicians love to rate their co- workers and they place him on a level with Carl Friedrich Gauss (The ‘Prince of Mathematicians’) and Sir Isaac Newton. Whatever the merits of this judgment it is clear that Archimedes would be in any mathematics Hall of Fame. He was hardly an ivory tower figure though – as well as his contributions to astronomy, mathematics and physics, he also designed weapons of war, such as catapults, levers and ‘burning mirrors’, all used to help keep the Romans at bay. But by all accounts he did have something of the absent-mindedness of the professor, for what else would induce him to leap from his bath and run naked down the street shouting ‘Eureka’ at discovering the law of buoyancy in hydrostatics ?
How he celebrated his work on π is not recorded.
Given that π is defined as the ratio of its circumference to its diameter, what does it have to do with the area of a circle? It is a deduction that the area of a circle of radius r is πr 2 , though this is probably better known than the circumference/diameter definition of π. The fact that π does double duty for area and circumference is remarkable.
How can this be shown ?
The circle can be split up into a number of narrow equal triangles with base length b whose height is approximately the radius r. These form a polygon inside the circle which approximates the area of the circle. Let’s take 1000 triangles for a start. The whole process is an exercise in approximations. We can join together each adjacent pair of these triangles to form a rectangle (approximately) with area b × r so that the total area of the polygon will be 500 × b × r. As 500 × b is about half the circumference it has length πr, the area of the polygon is πr × r = πr 2 . The more triangles we take the closer will be the approximation and in the limit we conclude the area of the circle is. πr 2 . Archimedes estimated the value of π as bounded between and . And so it is to Archimedes that we owe the familiar approximation 22/7 for the value o f π. The honour for designating the actual symbol π goes to the little known William Jones, a Welsh mathematician who became Vice President of the Royal Society of London in the 18th century. It was the mathematician and physicist Leonhard Euler who popularized π in the context of the circle ratio.
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