Archimedes' Limits to Value of Pi/Trigonometric Proof
Theorem
The value of $\pi$ lies between $3 \frac {10} {71}$ and $3 \frac 1 7$:
- $3 \dfrac {10} {71} < \pi < 3 \dfrac 1 7$
Proof
Let $O$ be a circle with diameter $AB = 1$
Let $Q$ be the circumference of $O$.
By the definition of $\pi$:
- $\dfrac Q {AB} = \pi$
Since $AB = 1$ then $Q$ is exactly $\pi$.
Using regular polygons inscribed and circumscribed about the circle $O$, we aim to demonstrate that the bounds on $\pi$ are:
- $\dfrac {223} {71} < \pi < \dfrac {22} {7}$
Lemma 1
- $\ds \cot \dfrac \phi 2 = \cot \phi + \csc \phi$
Lemma 2
Let:
- $\cot \phi = \dfrac p q$
Then:
- $\csc \phi = \dfrac 1 q \cdot \sqrt {p^2 + q^2}$
Lemma 3
- $\pi < \dfrac {22} 7$
Lemma 4
- $\pi > \dfrac {223} {71}$
Combining the two results from the fifth iteration of Lemma $3$ and Lemma $4$, we have:
| \(\ds 3 \tfrac {10} {71}\) | \(<\) | \(\, \ds \pi \, \) | \(\, \ds < \, \) | \(\ds 3 \tfrac 1 7\) | ||||||||||
| \(\ds \leadsto \ \ \) | \(\ds \dfrac {223} {71}\) | \(<\) | \(\, \ds \pi \, \) | \(\, \ds < \, \) | \(\ds \dfrac {22} 7\) |
Expressing this in decimal:
- $3.1408 < \pi < 3.1429$
$\blacksquare$
Historical Note
Archimedes demonstrated the limits to the value of $\pi$ in his Measurement of a Circle.
He does not say where the two estimates $\dfrac {265} {153}$ and $\dfrac {1351} {780}$ for $\sqrt 3$ come from.
They can be easily derived from the continued fraction representation, namely, $\sqbrk {1, \sequence {1, 2} }$, but whether he knew this, or had another method, is unclear.