How Newton Beat 1500 Years of Pi Calculations

For over a thousand years, mathematicians struggled to calculate the digits of π (pi), the mysterious number that relates a circle's circumference to its diameter. The ancient Greek genius Archimedes had started it all around 250 BC by drawing polygons inside and outside a circle, calculating their perimeters to trap π between two values. This method was ingenious but painfully slow.

For centuries, this "polygon method" was the best humanity had — with later minds like Ludolph van Ceulen calculating π to 35 digits using thousands of polygon sides. He dedicated his entire life to this — so much that his digits were engraved on his tombstone.

Then along came a man named Isaac Newton.

The Calculus Revolution

Newton didn’t need thousands of polygon sides.
He had something far more powerful: calculus — and a brilliant idea.

He thought: Why not express the geometry of the circle in a completely different language?
Not with shapes, but with algebra and infinite series.

Here’s how he did it.

Step 1: Start With a Circle

The equation of a circle is:

x² + y² = 1

Newton rearranged this to express y as a function of x — specifically for the top half of the circle:

y = √(1 - x²)

Now, the area under this curve from x = 0 to x = 1 is exactly one-quarter of the area of a unit circle, which is π/4.

So if he could integrate √(1 - x²) from 0 to 1, he’d get π/4.

But this integral was tricky to calculate directly — until Newton had a brilliant realization.

Step 2: Use the Binomial Theorem — With a Twist

Newton knew how to expand powers like (1 + x)ⁿ using the binomial theorem. Most people only used it for positive whole numbers like n = 2 or 3.

But Newton? He pushed it further.

He asked: What if n was a fraction? Or even negative?
This was radical thinking in the 1600s.

So he tried to expand:

(1 - x²)^(1/2)
(This is √(1 - x²))

Using the binomial theorem with n = ½, Newton created an infinite series of terms involving powers of x² — something no one had thought to apply like this before.

Step 3: Integrate to Find π

Now that he had expanded √(1 - x²) into a series, he integrated each term from 0 to 1.

And voilà — he had an expression that added up to π/4.

This method was much faster than the ancient polygon approach — but Newton wasn’t satisfied.

Step 4: A Speed Hack — Shrinking the Circle

Newton realized the series he had created converged slowly when integrating from 0 to 1.

So he came up with a trick:
He changed the limits of integration from 0 to 1 to 0 to ½.
This meant he was only calculating half of a quarter-circle (or one-eighth of the circle), but it dramatically improved the speed of convergence.

Now, instead of needing thousands of terms, Newton could get many digits of π with just 50 terms.

That’s compared to the millions of calculations required by older polygon methods.

A Genius at Work

While others spent their entire lives computing digits of π using geometric brute force, Newton showed that understanding beats effort.

He combined:

Binomial expansions with fractional powers
Infinite series
Integration
Creative limits
And a deep understanding of geometry and algebra

He did in hours or days what had taken others centuries.

Legacy

Newton never set out to find π specifically — he was more interested in how far calculus could go. But his work opened the door for a new kind of mathematics — one that could tame infinity and bring unreachable numbers like π within grasp.

Today, π is calculated using powerful algorithms and computers. But the first person to truly break the cycle of centuries-old methods… was a man sitting by candlelight with only a quill, paper, and the power of his mind.

-- Pavitra Kanetkar

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