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I just know that I'm going to look like a crackpot, but here goes.
The number $\pi$ is defined as the ratio of the circumference of a circle to its diameter. So there is an assumption here that all circles are similar. Using undergraduate calculus (Moise), I can analytically convince myself that this is true. But how would a 5th century geometer prove this synthetically?

Steven Alexis Gregory
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    Similarities preserve distances up to a constant factor. If this factor is $t$ then each point on a circle of radius $r$ is sent to a point of distance $rt$ from some fixed point... – Angina Seng Aug 16 '17 at 10:17
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    This is only true in the plane, of course. On a sphere or a saddle, $\pi$ varies with the radius. (On a sphere $\pi$ is decreasing and on a saddle it's increasing as a function of $r$. In fact, given a point on a surface, $\pi''(0)$ for circles centered at that point is (proportional to) the Gaussian curvature at that point.) – Arthur Aug 16 '17 at 10:26
  • @Arthur Actually very interesting. But I don't think that would have occurred to Pythagoras. – Steven Alexis Gregory Aug 16 '17 at 11:04
  • it is not constant https://math.stackexchange.com/questions/17366/why-is-pi-equal-to-3-14159/17384#17384 – jimjim Aug 16 '17 at 11:25
  • @Arjang: please? – Francesco Polizzi Aug 16 '17 at 11:28
  • @FrancescoPolizzi yes, what is the question? – jimjim Aug 16 '17 at 11:29
  • I do not see how the question you are linking says that $\pi$ is not constant – Francesco Polizzi Aug 16 '17 at 11:30
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    This does not look to me like a crackpot question. – MJD Aug 16 '17 at 12:15
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    It is upsetting when commentators jump on a question and mark it as a duplicate when there are obvious differences. The OP 's question was tagged with "Math History". They should take a deep breath and count to 10! – CopyPasteIt Aug 16 '17 at 14:35
  • @MikeMathMan Sorry but I am the one that deserves your wrath. I allowed this to happen. – Steven Alexis Gregory Aug 16 '17 at 15:15
  • OK, but my comment still stands as a generality. I liked your question - the path from straight lines to curves is fascinating - Descartes thought it was still spooky stuff in 1637, and he was not a crackpot! – CopyPasteIt Aug 16 '17 at 15:58
  • @MikeMathMan Thanks. I'm retired now but I still tutor. It always bothers me, when I define $\pi$, that students never wonder why $\pi$ is a constant. – Steven Alexis Gregory Aug 16 '17 at 16:05
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    @CopyPasteIt - I wish that you had written $10!$ as $10!.$ – Steven Alexis Gregory Apr 30 '20 at 19:24

4 Answers4

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Up to translations, you may assume that all the circles are centered at the same point $p$. Now the existence of a dilation sending each circle into another is obvious, because the property "being equidistant from $p$" is invariant under dilations with center $p$.

Francesco Polizzi
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Ancient geometers took the similarity of circles (in the plane) as self-evident. They knew that if you created shapes with ruler and compass following a formula, then by scaling up or down the new shape would be congruent to the first.

Euclid used this to show a relationship between the area of two circles without knowing about $\pi$. He might have taken his ruler and compass and thrown it against the wall when contemplating the 'perimeter' of a circle.

Years later Archimedes took on the challenge. For more, see

Who First Proved that C / D is Constant?

and for the in-depth historical details,

CIRCULAR REASONING: WHO FIRST PROVED THAT C/d IS A CONSTANT?

where, amazingly, you'll find,

Aristotle’s belief that curves and line segments can not be compared persisted. Most famously, in his 1637 La Géométrie, René Descartes (1596–1650) wrote:

Geometry should not include lines [curves] that are like strings, in that they are sometimes straight and sometimes curved, since the ratios between straight and curved lines are not known, and I believe cannot be discovered by human minds, and therefore no conclusion based upon such ratios can be accepted as rigorous and exact.

CopyPasteIt
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They're not!!

Circles have circumferences which depend on the radius, except if you happen to be in Euclidean space (a space with zero curvature). For instance, in hyperbolic space: $$\frac{C}{r}=2\pi \frac{\sinh r}{r}\neq 2\pi$$

So really, the question is how we know (or the ancients knew) that in Euclidean space the ratio is constant. This can be seen by observing triangles, and noting that as long as the angles are the same on a plane, the side lengths scale linearly. Subdividing a circle into many small triangles, shows that this logic then must apply to circles as well - if you increase the radius, the circumference must scale by a constant factor, which happens to be $2\pi$.

Nathaniel Bubis
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Take a unit cube. To make a cube doubling the length of each dimension requires 8 unit cubes.

Now take the inverted pyramid with a perfectly square base and unit height. Now compare volumes when the height of the inverted pyramid is doubled. The internal volume increases 8 times.

Now take an inverted cone, size it so the inverted pyramid just fits exactly within it (then approximately the radius of the cone at any height equals $\sqrt{2\left( \frac{x}{2}\right)^2}$ where x is the length of one of the sides of the inverted pyramid at that particular height.

Doubling the height of the inverted pyramid will increase the volume 8 times. If we double the height of the inverted cone (with the pyramid giving an inscribed square at all heights) the volume increases 8 times. Double the height again and the volume will increase by 8 times again. This is a clearly uniform and repeatable process analogous to doubling all the dimensions of a unit cube.

The simplest conclusion from these studies is that the ratio of the area of any given circle to the area of its inscribed square, is always a constant.

James Arathoon
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