Thanks for the mention of perimetron from which pi comes. I’ve always thought that pi is a fundamental aspect of mathematics. As a fun aside, though, it represents the ratio of circumference to diameter.
If we’re getting into mnemnonics, you can also use “May I have a large container of coffee”, and count the letters in each word, for 31415926, better than CurtC’s ratio.
You mean, define it by something like pi = 4/1 - 4/3 + 4/5 - 4/7 + … , and then have it as a theorem that pi times diameter equals circumference? I suppose you could do it that way, but it makes sense to use the most obvious application as the definition.
Read “Contact” for one reason to figure out pi to a zillion digits.
Bragging rights is probably the main reason.
Brian
Or my favorite,
How I wish I could recollect of circle round
The exact relation Archimede unwound
I know you have to cheat on Archmedes name, but hey: how else are you going to the number nine in there.
As to the application of pi, Chronos, I was thinking more along the lines of how it crops up in power series and Gaussian distributions and a number of other places.
It is a fundamental number in math, as I’m sure you know, and a feature of that essential nature is the ratio of circumference to diameter. But it is not the definition of the the number. It is rather one of the key bricks at the bottom of the structure.
By the way, love your posts, **Chronos[/]. Getting cold up there yet?
Here’s an impressive…well, I don’t think mnemonic is quite the right word, since I’m doubtful that it could serve much use as a memory aid…piece of constrained writing is a better description. It’s constrained by the first (nearly) 4000 digits of pi (there are other constraining devices in it as well, I believe):
Another mnemonic is the Pi Round. It’s sung to the tune of Frere Jacques:
Three point one four (three point one four)
One five nine (one five nine)
Two six five three five eight (two six five three five eight
Nine seven nine (nine seven nine)
Besides being defined as being equal to C/D, another way to define pi is to say that it is (ln -1)/sqrt(-1). In some ways, that is a more natural and useful definition.
amore ac studio:
Your quote does present an example of circular (heh) reasoning, but it can avoided by performing the integral in polar coordinates.
Is ln(-1)/sqrt(-1) really a more useful defintion as both numerator and denominator are imaginery?
I prefer the sequence as you can obtain a value with arbitary error from it.
Polar coordinates, eh? How would you know the value of the Jacobian without knowing the derivatives of the trig functions? Maybe with some handwaving you could convince yourself that the polar area element is r dr d(theta), but Goldberg explicitly wanted to avoid assuming formulae from geometry. His definition of \pi is in terms of a definite integral whose integrand has the antiderivative arcsine. He defines almost all the elementary functions in terms of integrals, but he is able to get the exponential function almost for free, as the sum of cosh and sinh. It’s not the most intuitive presentation, but it does avoid circular reasoning like the example quoted above.
Then at least do the complete joke! 
"Teacher: The area of a circle is equal to pr[sup]2[/sup] [pi r squared].
Doofus student: Pie are not squared! Cake are squared! Pie are round."
Best. GQ rant. Ever.
The value of the Jacobian indeed . . .
Jacobian? What do you need the Jacobian for? The Jacobian is used in double integrals. This is single integral. Sure, you could look at this as a special case of a double integral, in which the integrand is an indicator function, but why? Take any wedge wi. Let li be the largest triangle contained in wi, and ui be the smallest triangle that contains wi. Let lt=sum (li), wt=sum (wi), and ut= sum (ui). Since for each i, li<wi<ui, lt<wt<ut. lt and ut both converge to pi r squared, so by the Sandwich Theorem, so does wt. Yes, this is a geometric proof, but I don’t see anything inferior about geometry.
MC
Actually, it is possible, with a little work, to get a series expression for pi from the definition I gave. It’s not as nice as the normal one, though.
But of course, there are many other methods for finding the formula for area of a circle. Archimedes didn’t know trig, but he was still able to determine that a circle has the same area as a triangle with base equal to the circle’s circumference and height equal to the radius. He did so by showing that an inscribed polygon always has an area less than this amount, and a circumscribed polygon always has area greater than this amount (today we would do this using limits, but the Wizard of Syracuse didn’t (quite) have that tool at his disposal).
And I would hardly say that ln(-1) / sqrt (-1) is superior as a definition; it depends either on the plane interpretation of complex numbers, in which case you are just measuring a circle (the circle of roots of unity), or on the power-series definition of the exponential function (and hence of the log function), which ends up just being the 4 - 4/3 + 4/5 … series I mentioned earlier, in an uglier form.
If you wanted to do the integral in polar coordinates to avoid circular reasoning, it would have to be a double integral. I’ve never heard of a polar coordinate system for a one-dimensional space.
Seems to me that the polygon method is a good start, but one must still show that the inner and outer areas converge to the same amount. Did Archimedes do so? (Without limits, I don’t see how he could).
I don’t know what you mean when you say that it depends on the plane interpretation of complex numbers. pi can be defined as the number such that e^pi*sqrt(-1)=-1, regardless of how one interprets sqrt(-1).
I remember a historical note in one of the calculus textbooks of my father’s generation, which mentioned that if Archimedes had not been killed by the Romans, he probably would have gone on to invent the integral calculus. He had already established many area and volume formulas using techniques that we would classify today as calculus, but in order to convince his contemporaries who were trained in the Euclidean method of geometry, he had to derive his results again without the calculus. The same is true of Newton, who in his Principia did not employ any techniques of differential calculus, but instead proved his results with Euclidean geometry.
It sounds too good to be true, almost like the notion that Aristotle set science back 2000 years by marginalizing the beliefs of forward-thinking men like Democritus.
Perhaps you meant evaluating the one-dimensional integral \int_{0}^{2\pi} \frac{1}{2} r^{2} d heta = \pi r^{2}. That will avoid circular reasoning, as you intended.
The case with Newton is more that, having pretty much spectacularly self-educated himself in mathematics, he invents calculus, but that in then learning more about what other people had done, he’s impressed by the classical Greek style. So that when he comes to develop and write Principia he thinks that the latter is the best way of doing things. In particular, he doesn’t rewrite most of the results into the more traditional style. Those are probably mainly worked out in the Greek style in the first place, rather than using calculus.
There’s also the factor that he believed that everything in Principia was a rediscovery of things known to the ancients. He didn’t think he was doing anything the Greeks hadn’t.
Furthermore, there are parts of the book that do use techniques that can only be described as calculus. (And, goodness knows, Newton was prepared to fight down and dirty to defend these techniques as innovative in the 17th century.) It’s however striking that much of Principia is written in such a way as to segregate these sections. Later propositions often include qualifications like “granting the quadratures”. He’s shown how you can work out areas at the start, but he’s thus careful to formulate the later sections in such a way that they don’t depend on the acceptance of this particular method.
“If you wanted to do the integral in polar coordinates to avoid circular reasoning, it would have to be a double integral. I’ve never heard of a polar coordinate system for a one-dimensional space.”
No, you could integrate theta against r, or r against theta. Either way, it would be one integral. Yes, it’s two dimensional space, that’s why we only need one integral. Double integrals are for three dimensional spaces. In one integral, the integrand ranges over one dimensional, and the dummy variable varies over another. One integral, two dimensions.
Some more mnemonics:
“How I need a drink - alcoholic of course! - after the heavy chapters involving quantum mechanics”, and:
"Now I, even I, would celebrate
"In rhymes unapt the great
"Immortal Syracusan rivaled* nevermore
"Who in his wondrous lore passed on before
“Gave men his guidances how to circles mensurate”.
*Is this the US spelling? In English we would spell it “rivalled”, but then the mnemonic doesn’t work - as can be seen by reference to the previous mnemonic.
AAAAAAAAAGH!
hits head on desk
WACK-WACK-WACK-WACK