Sorry for the clunky thread title. I’m assuming that several cultures with no exposure to Greek mathematics derived formulas for calculating the area and circumference of a circle. Were their formulas significantly different? Are they still in use today (i.e., not useable, but actually used in practice)?
Well, if we’re talking about accurate formulas, they can’t have been significantly different… Perhaps some cultures thought the most salient (one-dimensional) measure to start from was the radius (circ. = 2πr, area = πr^2), while others used the diameter (circ. = πd, area = πd^2/4), or such things, but what other kinds of variations could one expect to see? Or are we talking about mere approximations?
(What would be interesting to me would be if any culture singled out 2π as the fundamental constant rather than π.)
This is an article from a German newsmagazine (in German) about formulas used by the Aztecs for land tax purposes in maps from the 16th century. The article mentions (about two-thirds down) that a popular method for computing the area of a non-rectangular quadrangle was to multiply the average of two opposing sides by a third side. A similar formula for the same problem (where a, b, c and d are the sided of the quadrangle) was (a + c)/2 * (b + d)/2.
The article doesn’t mention problems involving circles, though.
Oh, and then of course there’s the famous biblical formula in 2 Chronicles 4:2:
(King James Version)
This obviously gives you a value of 3 for pi.
And, as always, Wiki knows it all.
You could have some silly formula, like calculating the area of a circle given the area of the circumscribed/inscribed square, or any other value that lets you calculate the radius. It seems strange to us, but that might only be because we’re so used to our formulae
As an added bit of background, I’m editing a lesson plan (6[sup]th[/sup] grade) and came across a sentence I objected to. Here’s the sentence I was handed:
“Ask students to research ways that students in other countries find measures of circles in order to determine if the origins of those methods are rooted in other parts of the world.”
My first objection is to the overall logic of the sentence. (If a student in Canada learns the same formula, Greek origins don’t necessarily follow, nor does a different method necessarily imply a Canadian origin—Canadians could have been influenced by clever Laplanders, who could have gotten them from the Greek, could have developed them independently, or could have gotten them from Luxemburg.)
But I get the spirit of the author’s intent, and while wondering about whether kids in China (or elsewhere) used different methods, I thought of the Dope. Hence the GQ, and hence my love of the Dope in the replies so far.
This may not be what you’re after, but I discussed some ancient Mesopotamian and Egyptian formulas of area and volume in the first half of this staff report: Who invented pi?
The ancient Egyptians and some early modern Western mathematicians singled out the number we would call pi/4.
I thought we called it maize?
Thanks for the link—I remember when it came out, and enjoyed rereading it. I loves me some number theory, which is a bit odd given that I’m somewhat mathematically inept.