Proportionality Principle Between Circumferences
The lengths of circumferences are directly proportional to their radii, expressed as $$ c:c' = r:r' $$
In essence, when comparing any two circumferences, their length ratio equals the ratio of their radii.
$$ \frac{c}{c'} = \frac{r}{r'} $$
Here, c and c' denote the circumferences' lengths, while r and r' stand for the radii.
Note: This principle reveals a fundamental relationship between a circle's circumference and its radius, highlighting the significance of what we recognize today as "pi" (π=3.1415...), which is the ratio of the circumference to its diameter (twice the radius).
Demonstrating the Principle
Let's consider two circles with circumferences c and c', and radii r and r'.
I'll illustrate this by inscribing regular polygons, such as hexagons, inside each circle.
Given these are regular polygons, and therefore similar polygons, their perimeters and sides maintain a consistent ratio.
Furthermore, a theorem on regular polygons states that the radii of the circumferences they inscribe or circumscribe are proportional as well.
This allows us to state:
$$ 2p : 2p' = r: r' $$
Where 2p represents the perimeter (and p the semiperimeter) of the regular polygon, and r the radius of the circle.
The proportionality ratio, therefore, remains consistent.
$$ k = \frac{2p}{2p'} = \frac{r}{r'} $$
Extending this logic to regular polygons circumscribed around the circle affirms the principle.
For example, using hexagons again but circumscribed, we find:
As these polygons are regular and similar, their perimeters, sides, and the circles' radii are proportionally related.
$$ 2P : 2P' = r: r' $$
With 2P as the perimeter (and P the semiperimeter) of the circumscribed regular polygon, and r the circle's radius.
Knowing the perimeter of the circumscribing polygon always exceeds that of the inscribed one, we can deduce:
$$ 2p < c < 2P $$
Meaning, the circumference's length (c) is bounded by the perimeters of the inscribed and circumscribed polygons.
Applying the proportionality ratio k across these relations:
$$ 2p \cdot k < c \cdot k < 2P \cdot k $$
Given 2p'=2p·k and 2P'=2P·k, we conclude:
$$ 2p' < c' < 2P' $$
Thus, a proportional relationship governs the lengths of the two circumferences, mir roring the ratio found in the inscribed and circumscribed regular polygons.
$$ c' = c \cdot k $$
or, equivalently,
$$ k = \frac{c}{c'} $$
Given k also represents the ratio between the two circles' radii.
$$ k = \frac{c}{c'} = \frac{r}{r'} $$
This confirms the proportion:
$$ c : c' = r : r' $$
Proving the direct proportionality between the circumferences' lengths and their radii.
This logic extends to the relationship between circumferences and diameters, where the diameter is double the radius.
$$ c : c' = 2 \cdot r : 2 \cdot r' $$
$$ c : c' = d : d' $$
By rearranging the means:
$$ c : d = c' : d' $$
In conclusion, the ratio between a circumference (c) and its diameter (d) remains constant, regardless of the circle's size, embodying the constant known as "pi" (π), valued at 3.141592...
And that's all there is to it.