0

In the title, $A$ refers to the area and $r$ to the radius of the circle, respectively. Assume that there are two constants $\alpha$ and $\beta$, so that the area of the circle is given by $$ A = \alpha r^2 $$ And the circumference of the circle is $$ C = \beta d=2\beta r $$ Where $d$ is the diameter. Of course, from elementary geometry we know that $\alpha= \beta = \pi$. A very elegant way of showing this is found in this answer to a highly related question, but of course, there is a limiting process implicitly involved. But can you show that you always have to use limits in some way to show this rigorously? Is it related to the transcendentality of $\pi$? Note that I am not asking about showing that $\frac{A}{r^2} = \pi$ or $\frac{C}{2r} = \pi$ on their own, but only about the equality $$ \frac{A}{r^2} = \frac{C}{2r} $$

paulina
  • 731
  • 2
    How are you defining $A,C$ without limits? – lulu Feb 12 '25 at 21:00
  • To stress: the "usual" way to proceed would be to consider the regular $k-$gon inscribed in the circle of radius $r$ and establish a parallel claim. Then, of course, we allow $k$ to go to $\infty$. That gives the various definitions and the claim you want drops out. But, if you want to avoid taking any limits, you'll need to find a non-limit definition of both $A$ and $C$. – lulu Feb 12 '25 at 21:11
  • @lulu yes, I am aware of the canonical way to derive the equality. Is there no purely geometric/axiomatic way to define $A$ and $C$ without appealing to integration or measure theory? Not that I could come up with one off the top of my head, anyway... – paulina Feb 12 '25 at 21:18
  • 1
    Not aware of any. And, of course, even informal limit definitions can lead to apparent paradox. – lulu Feb 12 '25 at 21:21
  • 3
    Even defining the real numbers themselves without limits (or something “limit-like”, like cuts) is challenging, if not impossible. – Malady Feb 12 '25 at 22:50
  • I'm not sure how to define the kind of method you are seeking to prove doesn't exist, but one way to prove that it doesn't exist would be to show that it leads to a problem that is unsolvable. For instance, squaring the circle. – Matthew Leingang Feb 12 '25 at 23:40
  • I might be missing something, but what is wrong with using the fact that all circles are similar to note that the ratio of a circle's area to its squared radius must be constant? (Or the ratio of a circle's circumference to its diameter). The similarity of circles follows easily from the definition of one as being the locus of points which are equidistant from a given point and the application of scaling to a given circle's radius – H. sapiens rex Feb 12 '25 at 23:45
  • @H.sapiensrex how do you know that those constants are the same? All that tells us is that the area must be proportional to $r^2$ and the circumference is proportional to $2r$. – Malady Feb 12 '25 at 23:50
  • @MatthewLeingang That is also what I believe is the most promising way to go about this. A "purely geometric" proof would probably have to involve a compass-and-straightedge construction, which is impossible since $\pi$ is transcendental. But I have no idea how to actually do the proof. – paulina Feb 12 '25 at 23:56
  • 1
    The idea of length as well as area requires the notion of real numbers. Eudoxus gave a definition based on ratio of length of line segments and that's good enough and matches Dedekind's definition of reals. But this also involves dealing with infinite sets and essentially a kind of limit based definition. There is just no way to deal with with length of segments incommensurable with segment of unit length except using some form of limits. Dealing with curve length and areas goes much deeper but involves limits. – Paramanand Singh Feb 13 '25 at 13:17
  • @ParamanandSingh if you wrote this as an answer, I'd accept it. – paulina Feb 13 '25 at 21:25

1 Answers1

2

Eudoxus gave a theory of magnitudes which is essentially same as Dedekind's theory of real numbers but uses geometric notions. You should have a look at the "Mathematics" section of the linked Wiki article for more details.

The length and areas of plane figures were then essentially handled in terms of magnitudes. Eudoxus also developed the method of exhaustion in a rigorous fashion and proved that ratio of areas of two circles is same as ratio of squares of their radii. All of these developments essentially involved use of infinite sets and a kind of limiting procedure. In fact the very notion of comparing ratios of magnitudes requires the quantifier "for every ($\forall$)" which is a key aspect of the definition of limit.

All of these ideas were further subsumed into the theory of real numbers, analytic geometry and integral calculus. Before concluding let me just add that your key concern about the link between area of a circle and its circumference is already handled (using integral calculus) in this answer of mine.

Paramanand Singh
  • 92,128
  • I don't know if this is an answer to the question OP was really asking. – Cameron L. Williams Feb 14 '25 at 01:24
  • @CameronWilliams: well, I am trying to show that these ideas need limits in an essential way. I had added a comment to the question and OP requested to put that as answer. I then posted this answer. – Paramanand Singh Feb 14 '25 at 01:26
  • Ope I see it. Fair enough! My apologies. – Cameron L. Williams Feb 14 '25 at 01:27
  • @CameronWilliams it answers my question in the sense that the very quantities involved in the equality I seeked to prove need limits to even properly define, so that the concept of a limit was baked into the question at a much deeper level than I realized. To try to prove it without limits is therefore impossible. – paulina Feb 14 '25 at 02:08