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Sometime around third grade, I was first introduced to pi and the basic circle equations for circumference, area and volume. I could see how the circumference could be measured, to check the equation, but area left me stumped. I wondered about filling a cylinder to a specific depth with water or clay, and using the volume divided by the known height to calculate the area. This seemed like a terribly sloppy method to me even then, but it was the best I could think of. I asked a teacher how we knew the area of a circle, and I was more or less brushed off with a non-answer. The teacher didn't know! So that question joined the throng at the back of my mind, waiting for more information.
(I'm not sure why I didn't ask my dad, who certainly would have known. Kids are weird.)
The answer finally came almost a decade later when I got to calculus in senior year. Integration over an area! It was a beautiful revelation. So clean and elegant, and always a bit of a wonder to see the depth of the abstraction being used, from limits to derivatives to definite integrals, yet in the end my old friend π * r2 pops out. Glorious... yet I was disappointed that my teacher hadn't known, back in third grade. Not that I expected them to be able or willing to walk a 9 year old through calculus, but they should have at least been able to say, "It can be shown using this thing called calculus, which you'll learn about in high school." That's not too much to ask.
Now, two decades after that, I'm reading a lot of pre-modern mathematics for the book I'm working on. As I noted in a post here awhile back, I found it very interesting that Euclid didn't include any equations for the quantitative area or volume of circles, cylinders and spheres, just ratios between them. A cone is 1/3 the volume of a cylinder that just contains it, etc. Which made perfect sense, as calculus was still a long way in the future. Except when I was reading Leonardo of Pisa (better known as the son of Bonacci, AKA filius Bonacci, AKA Fibonacci), something caught my eye. In a problem he plainly states that the area of a circle is 1/2 * radius * circumference. Which, if you convert c to being 2 * pi * r, comes out as π * r2! How could he have known that, so long before Leibniz and Newton?
It turns out, for some problems anyway, the ancient Greeks beat calculus by 2,000 years. And somehow I had gone 40 years without knowing this! Archimedes had solved the area of a circle, using the method of exhaustion. This looks an awful lot like an epsilon-delta limit proof, but done using geometry instead of algebra. He postulates that the area of the circle is equal to that of a triangle with a base the length of the circle's circumference, and a height equal to the circle's radius. He is them able to show, through a series of contradictions, that the circle's area cannot be either greater or less than the area of this triangle. The error of a polygon fit inside/outside the circle can get arbitrarily small, and thus can always get smaller than the error of the triangle if it isn't actually the correct size. Yet those polygons can't actually be bigger/smaller than the triangle, if you do the math. The only option left if for them to be equal to each other, and thus the triangle is the correct area! It's a lovely proof, particularly for how clever it is at avoiding the explicit use of any mention of infinity.
So it turns out my high school annoyance at my grade school teacher was still incomplete. Calculus can give the answer, and it's definitely a superior tool, but it turns out the area of a circle is knowable using nothing more than geometry. I wonder how my uneasy-at-times relationship to mathematics would have changed, had I been shown Archimedes' proof back in 1986.
(I'm not sure why I didn't ask my dad, who certainly would have known. Kids are weird.)
The answer finally came almost a decade later when I got to calculus in senior year. Integration over an area! It was a beautiful revelation. So clean and elegant, and always a bit of a wonder to see the depth of the abstraction being used, from limits to derivatives to definite integrals, yet in the end my old friend π * r2 pops out. Glorious... yet I was disappointed that my teacher hadn't known, back in third grade. Not that I expected them to be able or willing to walk a 9 year old through calculus, but they should have at least been able to say, "It can be shown using this thing called calculus, which you'll learn about in high school." That's not too much to ask.
Now, two decades after that, I'm reading a lot of pre-modern mathematics for the book I'm working on. As I noted in a post here awhile back, I found it very interesting that Euclid didn't include any equations for the quantitative area or volume of circles, cylinders and spheres, just ratios between them. A cone is 1/3 the volume of a cylinder that just contains it, etc. Which made perfect sense, as calculus was still a long way in the future. Except when I was reading Leonardo of Pisa (better known as the son of Bonacci, AKA filius Bonacci, AKA Fibonacci), something caught my eye. In a problem he plainly states that the area of a circle is 1/2 * radius * circumference. Which, if you convert c to being 2 * pi * r, comes out as π * r2! How could he have known that, so long before Leibniz and Newton?
It turns out, for some problems anyway, the ancient Greeks beat calculus by 2,000 years. And somehow I had gone 40 years without knowing this! Archimedes had solved the area of a circle, using the method of exhaustion. This looks an awful lot like an epsilon-delta limit proof, but done using geometry instead of algebra. He postulates that the area of the circle is equal to that of a triangle with a base the length of the circle's circumference, and a height equal to the circle's radius. He is them able to show, through a series of contradictions, that the circle's area cannot be either greater or less than the area of this triangle. The error of a polygon fit inside/outside the circle can get arbitrarily small, and thus can always get smaller than the error of the triangle if it isn't actually the correct size. Yet those polygons can't actually be bigger/smaller than the triangle, if you do the math. The only option left if for them to be equal to each other, and thus the triangle is the correct area! It's a lovely proof, particularly for how clever it is at avoiding the explicit use of any mention of infinity.
So it turns out my high school annoyance at my grade school teacher was still incomplete. Calculus can give the answer, and it's definitely a superior tool, but it turns out the area of a circle is knowable using nothing more than geometry. I wonder how my uneasy-at-times relationship to mathematics would have changed, had I been shown Archimedes' proof back in 1986.