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This question already has an answer here:

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?

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marked as duplicate by Hans Lundmark, Dando18, user99914, Community Aug 16 '17 at 14:16

This question has been asked before and already has an answer. If those answers do not fully address your question, please ask a new question.

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    $\begingroup$ 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... $\endgroup$ – Lord Shark the Unknown Aug 16 '17 at 10:17
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    $\begingroup$ 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.) $\endgroup$ – Arthur Aug 16 '17 at 10:26
  • $\begingroup$ @Arthur Actually very interesting. But I don't think that would have occurred to Pythagoras. $\endgroup$ – steven gregory Aug 16 '17 at 11:04
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    $\begingroup$ This does not look to me like a crackpot question. $\endgroup$ – MJD Aug 16 '17 at 12:15
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    $\begingroup$ 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! $\endgroup$ – CopyPasteIt Aug 16 '17 at 14:35
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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 $p$.

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  • $\begingroup$ Does that also imply that the ratio of circumference to diameter is a constant? $\endgroup$ – steven gregory Aug 16 '17 at 11:03
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    $\begingroup$ Yes, because you can approximate the circumference by inscribed and circumscribed polygons (for which the ratio perimeter/radius is clearly constant) and then apply Archimedes' exhaustion argument. $\endgroup$ – Francesco Polizzi Aug 16 '17 at 11:10
  • $\begingroup$ Yes. That will do it just right. $\endgroup$ – steven gregory Aug 16 '17 at 11:13
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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 $2\pi$.

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    $\begingroup$ +1 This answer is worth elaborating. The existence of similar triangles (hence of similarity that isn't congruence) is equivalent to Euclid's parallel postulate, hence to Euclidean geometry. $\endgroup$ – Ethan Bolker Aug 16 '17 at 13:34
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    $\begingroup$ I did mention that I wanted a 5th century point of view. $\endgroup$ – steven gregory Aug 16 '17 at 14:10
  • $\begingroup$ @stevengregory - Triangles are pretty 5th century $\endgroup$ – nbubis Aug 16 '17 at 15:33
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    $\begingroup$ But hyperbolic geometry is not... $\endgroup$ – Francesco Polizzi Aug 16 '17 at 15:41
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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.

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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.

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