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 pi = 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 sqrt10 = 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 pi 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 pi.
The first theoretical calculation seems to have been carried out by Archimedes of Syracuse (287-212 BC). He obtained the approximation
223/71
< pi < 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 pi 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 3.2 n-1 sides, with semiperimeter bn, and ascribe a regular polygon of 3.2 n-1 sides, with semiperimeter an.You can see the diagram for the case n = 2.
The effect of this procedure is to define an increasing sequence
Using trigonometrical notation, we see that the two semiperimeters are given by
b1, b2, b3, ...
and a decreasing sequence
a1, a2, a3, ...
such that both sequences have
limit pi.
an = K
tan(pi/K), bn = K sin(pi/K),
where
K = 3.2 n-1. Equally, we have
a n+1 =
2K tan(pi/2K), b n+1 = 2K sin(pi/2K),
and it is not a difficult exercise in trigonometry to show that
(1) . . .
(1/an + 1/bn) = 2/a^(n+1)
Archimedes,
starting from a1 = 3 tan(pi/3) = 3 sqrt 3 and b1 = 3 sin(pi/3) = 3sqrt3/2,
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
< pi < 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
Except for Tsu Ch'ung Chi, 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.
Tsu Ch'ung Chi (430-501 AD) 355/113
al'Khwarizmi (c. 800 ) 3.1416
Al'Kashi (c. 1430) 14 places
Vičte (1540-1603) 9 places
Romanus (1561-1615) 17 places
Van Ceulen (c. 1600) 35 places
Al'Khwarizmi lived in Bagdad, 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 Tsu Ch'ung Chi, one need hardly add, lived in China.
The European Renaissance
brought about in due course a whole new mathematical world. Among the first
effects of this reawakening was the emergence of mathematical formulae for pi. One of the earliest was that of Wallis (1616-1703)
These are both dramatic
and astonishing formulae, for the expressions on the right are completely
arithmetical in character, while pi arises in the first instance from geometry. They show the surprising results that infinite processes can achieve and point the way to the wonderful richness of modern mathematics.
From the point of view
of the calculation of pi, however, neither is of any use at all. In Gregory's series, for example, to get 4 decimal places correct we require the error to be less than 0.00005 = 1/20000, and so we need about 10000 terms of the series. However, Gregory also showed the more general result
An even better idea is to take the formula
Clearly we shall get very rapid convergence indeed if we can find a formula something like
pi
/4 =
tan^-1 (1/a) + tan^-1(1/b)
with a and b large. In 1706 Machin found such a formula:
With a formula like
this available the only difficulty in computing pi is the sheer boredom of
continuing the calculation. Needless to say, a few people were silly enough to
devote vast amounts of time and effort to this tedious and wholly useless
pursuit. One of them. an Englishman named Shanks, used Machin's formula to
calculate pi to 707 places, publishing the results of many years of labour in 1873. Shanks has achieved immortality for a very curious reason which we shall explain in a moment.
Here is a summary of how the improvement went:
Very soon after Shanks' calculation a curious statistical freak was noticed by De Morgan, who found that in the last of 707 digits there was a suspicious shortage of 7's. He mentions this in his Budget of Paradoxes of 1872 and a
curiosity it remained until 1945 when Ferguson discovered that Shanks had made
an error in the 528th place, after which all his digits were wrong. In 1949 a
computer was used to calculate pi to 2000 places. In this and all subsequent computer expansions the number of 7's does not differ significantly from its expectation, and indeed the sequence of digits has so far passed all statistical tests for randomness.
We should say a little
of how the notation pi arose. Oughtred in 1647 used the symbol d/pi for the
ratio of the diameter of a circle to its circumference. David Gregory (1697)
used pi/r for the ratio of the circumference of a circle to its radius. The
first to use pi with its present meaning was an Welsh mathematician William Jones in 1706 when he states 3.14159 andc. = pi
. Euler adopted the symbol in 1737 and it quickly became a standard notation.
We conclude with one
further statistical curiosity about the calculation of pi, namely Buffon's
needle experiment. If we have a uniform grid of parallel lines, unit distance
apart and if we drop a needle of length k < 1 on the grid, the probability
that the needle falls across a line is 2k/pi. Various people have tried to
calculate pi by throwing needles. The most remarkable result was that of Lazzerini (1901), who made 34080 tosses and got
Still on the theme of
phoney experiments, Gridgeman, in a paper which pours scorn on Lazzerini and
others, created some amusement by using a needle of carefully chosen length k =
0.7857, throwing it twice, and hitting a line once. His estimate for pi was thus given by
It is almost
unbelievable that a definition of pi was used, at least as an excuse, for a
racial attack on the eminent mathematician Edmund Landau in 1934. Landau had
defined pi in this textbook published in Göttingen in that year by the, now
fairly usual, method of saying that pi/2 is the value of x between 1 and 2 for which cos x vanishes. This unleashed an academic dispute which was to end in Landau's dismissal from his chair at Göttingen. Bieberbach, an eminent number theorist who disgraced himself by his racist views, explains the reasons for Landau's dismissal:-
As a postscript, here
is a mnemonic for the decimal expansion of pi. Each successive digit is the number of letters in the corresponding word.
2/pi =
(3.3.5.5.7.7....)/(2.2.4.4.6.6....)
and one of the best-known is
pi
/4 = 1
- 1/3 + 1/5 - 1/7 + ....
This formula is sometimes attributed to Leibniz (1646-1716) but is seems to have been first discovered by James Gregory (1638- 1675).
(3) . . .
tan^-1 x = x - x^3 /3 + x^5 /5 - ... (-1 <= x <= 1)
from which the first series results if we put x = 1. So using the fact that
tan^-1
(1/sqrt 3) = pi/6 we get
pi
/6 = (1/sqrt
3)(1 - 1/(3.3) + 1/(5.3.3) - 1/(7.3.3.3) + ...
(4) . . .
pi/4 = tan^-1 (1/2) + tan^-1 (1/3)
and then calculate the two series obtained by putting first 1/2 and the 1/3 into (3).
(5) . . .
pi/4 = 4 tan^-1 (1/5) - tan^-1 (1/239)
Actually this is not at all hard to prove, if you know how to prove (4) then there is no real extra difficulty about (5), except that the arithmetic is worse. Thinking it up in the first place is, of course, quite another matter.
1699: Sharp used Gregory's result to get 71 correct digits
Shanks knew
that pi was irrational since this had been proved in 1761 by Lambert.
Shortly after Shanks' calculation it was shown by Lindemann that pi is
transcendental, that is, pi is not the solution of any polynomial equation
with integer coefficients. In fact this result of Lindemann showed that
'squaring the circle' is impossible. The transcendentality of pi means that there is no ruler and compass construction to construct a square equal in area to a given circle.
1701: Machin used an improvement to get 100 digits and the following used his methods:
1719: de Lagny found 112 correct digits
1789: Vega got 126 places and in 1794 got 136
1841: Rutherford calculated 152 digits and in 1853 got 440
1873: Shanks calculated 707 places of which 527 were correct
You can see 2000 places of pi.
pi=
355/113 = 3.1415929
which,
incidentally, is the value found by Tsu Ch'ung Chi. This outcome is suspiciously
good, and the game is given away by the strange number 34080 of tosses. Kendall
and Moran comment that a good value can be obtained by stopping the experiment
at an optimal moment. If you set in advance how many throws there are to be then
this is a very inaccurate way of computing pi. Kendall and Moran comment that you would do better to cut out a large circle of wood and use a tape measure to find its circumference and diameter.
2 x 0.7857
/ pi = 1/2
from which he got
the highly creditable value of pi = 3.1428. He was not being serious!
Thus the valiant rejection by the Göttingen student body which a great mathematician, Edmund Landau, has experienced is due in the final analysis to the fact that the un-German style of this man in his research and teaching is unbearable to German feelings. A people who have perceived how members of another race are working to impose ideas foreign to its own must refuse teachers of an alien culture.
G.H. Hardy replied immediately to Bieberbach in a
published note about the consequences of this un-German definition of pi
There are many of us, many Englishmen and many Germans, who said things during the War which we scarcely meant and are sorry to remember now. Anxiety for one's own position, dread of falling behind the rising torrent of folly, determination at all cost not to be outdone, may be natural if not particularly heroic excuses. Prof. Bieberbach's reputation excludes such explanations of his utterances, and I find myself driven to the more uncharitable conclusion that he really believes them true.
Not only in Germany
did pi present problems. In the USA the value of pi gave rise to heated political debate. In the State of Indiana in 1897 the House of Representatives unanimously passed a Bill introducing a new mathematical truth.
Be it enacted by the General Assembly of the State of Indiana: It has been found that a circular area is to the square on a line equal to the quadrant of the circumference, as the area of an equilateral rectangle is to the square of one side.
The Senate of Indiana showed a little more sense and postponed indefinitely the adoption of the Act!
(Section I, House Bill No. 246, 1897)
How I want a drink, alcoholic of course, after the heavy lectures involving quantum mechanics. All of thy geometry, Herr Planck, is fairly hard...
3.14159265358979323846264...