How Pi Was Computed

Multiply the diameter by 3.14. That gives 376.8, so 380 centimetres will do with a little to spare.

The 3.14 is pi. And what it is is simpler than it sounds.

Try it. Wind a string around the base of a cup, unwind and measure it, then measure the cup's width and divide. You will get something near 3.1. Shaky hands make it hard to get past two decimal places.

Why does it never end?

School says 3.14; a calculator shows 3.141592… How much of it is real?

All of it, and it never stops. 3.14159265358979… runs on forever with no repeating block — not even the way a third repeats as 0.333… It cannot be written as a fraction at all.

Which leaves the question: how did anyone pin down a never-ending number to three or four places without a ruler that bends? People managed it two thousand years ago.

A little further in

How do you measure a circle's rim? A ruler is straight and a circle is not. Wrap a string and unwind it, and the string stretches and the hand shakes. You will not get past two decimal places.

Ancient civilisations mostly used something near three. Babylon used three and an eighth; an Egyptian papyrus preserves a calculation amounting to 3.16. Good enough in practice, but nobody knew how far it could be trusted.

Trapping the curved between the straight

Archimedes' idea was not to measure the circle at all. Draw a polygon inside it and another outside. The inner perimeter is shorter than the circle, the outer is longer.

A polygon's perimeter is a sum of straight lines, so it can be computed. That fixes pi between two numbers you can name. Add sides and the two values close in, with pi trapped between them.

The digits are not the point. He produced not an estimate but a range that cannot be wrong — and, with it, a method for narrowing that range as far as anyone likes.

The same method, in several places

1. 250 BCArchimedes · 96-gon · two decimal places
2. 263Liu Hui · 3,072-gon · 3.1416
3. c. 480Zu Chongzhi · 12,288-gon · six decimal places, a record that stood for 900 years
4. 1424al-Kashi · Samarkand · sixteen decimal places
5. 1630the last record set by polygons · thirty-eight places

Note that these calculations were carried out independently, far apart. No text survives describing Zu Chongzhi's method, but the precision of his result suggests polygons pushed to an extreme.

All of this required place-value notation. To handle six decimal places, zero has to hold the column. The precision of a civilisation's pi was the precision of its number system.

Abandoning polygons

In the seventeenth century the approach changed entirely. Pi could be written as an endless sum, each further term bringing the total closer.

Euler was especially good at such series. Add the reciprocals of all the squares — one, a quarter, a ninth, and so on forever — and you get pi squared over six. Pi had emerged from a sum with no circle anywhere in it.

A proof that it cannot be written

In 1761 it was proved that pi cannot be written as a fraction. In 1882 something stronger followed: pi cannot be the solution of any polynomial equation with whole-number coefficients.

That closed a problem two thousand years old. Constructing, with straightedge and compass, a square equal in area to a given circle is not difficult. It is impossible.

The question that remainsArchimedes did not know the answer, but he knew exactly which two numbers it lay between. Where else could we use a method for not-knowing precisely?