A History of Pi

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A little known verse of the Bible reads

And he made a molten sea, ten cubits from the one brim to the other: it was round all about, and his height was five cubits: and a line of thirty cubits did compass it about. (I Kings 7, 23)

The same verse can be found in II Chronicles 4, 2. It occurs in a list of specifications for the great temple of Solomon, built around 950 BC and its interest here is that it gives π = 3. Not a very accurate value of course and not even very accurate in its day, for the Egyptian and Mesopotamian values of 25/8 = 3.125 and √10 = 3.162 have been traced to much earlier dates: though in defence of Solomon’s craftsmen it should be noted that the item being described seems to have been a very large brass casting, where a high degree of geometrical precision is neither possible nor necessary. There are some interpretations of this which lead to a much better value.

The fact that the ratio of the circumference to the diameter of a circle is constant has been known for so long that it is quite untraceable. The earliest values of π including the ‘Biblical’ value of 3, were almost certainly found by measurement. In the Egyptian Rhind Papyrus, which is dated about 1650 BC, there is good evidence for 4 × (8/9)2 = 3.16 as a value for π.

The first theoretical calculation seems to have been carried out by Archimedes of Syracuse (287-212 BC). He obtained the approximation

223/71 < π < 22/7.

Before giving an indication of his proof, notice that very considerable sophistication involved in the use of inequalities here. Archimedes knew, what so many people to this day do not, that π does not equal 22/7, and made no claim to have discovered the exact value. If we take his best estimate as the average of his two bounds we obtain 3.1418, an error of about 0.0002.

Here is Archimedes‘ argument.

Consider a circle of radius 1, in which we inscribe a regular polygon of 3 × 2n-1 sides, with semiperimeter bn, and superscribe a regular polygon of 3 × 2n-1 sides, with semiperimeter an.

The diagram for the case n = 2 is on the right.

The effect of this procedure is to define an increasing sequence

b1 , b2 , b3 , …

and a decreasing sequence

a1 , a2 , a3 , …

such that both sequences have limit π.

Using trigonometrical notation, we see that the two semiperimeters are given by

an = K tan(π/K), bn = K sin(π/K),

where K = 3 × 2n-1. Equally, we have

an+1 = 2K tan(π/2K), bn+1 = 2K sin(π/2K),

and it is not a difficult exercise in trigonometry to show that

(1/an + 1/bn) = 2/an+1   . . . (1)

an+1bn = (bn+1)2       . . . (2)

Archimedes, starting from a1 = 3 tan(π/3) = 3√3 and b1 = 3 sin(π/3) = 3√3/2, calculated a2 using (1), then b2 using (2), then a3 using (1), then b3 using (2), and so on until he had calculated a6 and b6. His conclusion was that

b6 < π < a6 .

It is important to realise that the use of trigonometry here is unhistorical: Archimedes did not have the advantage of an algebraic and trigonometrical notation and had to derive (1) and (2) by purely geometrical means. Moreover he did not even have the advantage of our decimal notation for numbers, so that the calculation of a6 and b6 from (1) and (2) was by no means a trivial task. So it was a pretty stupendous feat both of imagination and of calculation and the wonder is not that he stopped with polygons of 96 sides, but that he went so far.

For of course there is no reason in principle why one should not go on. Various people did, including:

Ptolemy (c. 150 AD) 3.1416 Zu Chongzhi (430-501 AD) 355/113 al-Khwarizmi (c. 800 ) 3.1416 al-Kashi (c. 1430) 14 places Viète (1540-1603) 9 places Roomen (1561-1615) 17 places Van Ceulen (c. 1600) 35 places

Except for Zu Chongzhi, about whom next to nothing is known and who is very unlikely to have known about Archimedes‘ work, there was no theoretical progress involved in these improvements, only greater stamina in calculation. Notice how the lead, in this as in all scientific matters, passed from Europe to the East for the millennium 400 to 1400 AD.

Al-Khwarizmi lived in Baghdad, and incidentally gave his name to ‘algorithm’, while the words al jabr in the title of one of his books gave us the word ‘algebra’. Al-Kashi lived still further east, in Samarkand, while Zu Chongzhi, one need hardly add, lived in China.

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