Maddox, John. "False Calculation of pi by Experiment," Nature 370 (4 August 1994) p. 323.

 

Maddox, John. "False Calculation of pi by Experiment," Nature 370 (4 August 1994) p. 323.

The irrational number pi is inevitably a great source of wonder and a stimulant of speculation.
Sadly, perhaps, the sport of adding a few extra digits to the decimal value of pi has been killed by
computers. But historically, the number embodies the mystery of circularity by relating the
circumference of a circle to its diameter (by simply multiplication of the latter). That pi is
irrational is the origin of the belief that it is not possible to ‘square the circle,' although a little
reflection will show that there is nothing wrong with the square root of pi except that, like pi
itself, it is also irrational. What can be objectionable about a square with sides whose length is an
irrational number?

In any case, the numerical value of pi can be obtained experimentally, so to speak, without ever
drawing a circle and measuring its circumference, but simply by the manipulation of straight
lines. The first claim to that effect appears to be due to Buffon (strictly, Le comte de Buffon),
who pointed out in 1777 that pi appears explicitly in the calculation of the probability that, if a
straight object such as a needle is thrown randomly onto a flat surface rules with parallel lines,
the needle will intersect one of the lines. The simplest case is when the length of the needle, say
l, is less than the separation of the parallel lines, say d, when the probability that the needle will
intersect one of the lines is 2l/pid.

From that point on, it was open to anybody to seek a value for pi simply by dropping a needle
onto a surface rules with parallel lines set further apartment from each other than the length of
the needle. This apparently became one of the great intellectual pastimes of the nineteenth
century. If N is the number of times the needle is dropped and H is an experimental estimate of
pid/2l, giving 2lN/dH as the estimated value of pi. The most celebrated of the estimates obtained
in their way is due to the Italian M. Lazzarini, who announced in 1901 a values of pi =
3.141529... In the true value of pi, the last digit should be a ‘6,' not a ‘9,' so that the result is
accurate to a few parts in 10 million.

Lee Badger, from the Webster State University at Ogden in Utah, evidently shares the view that
this result is too good to be true. Writing in the Mathematical Association of America's
Pedagogical Mathematics Magazine (67, 83; 1994), Badger describes the result as "lucky." That
is a charitable way of putting it. The truth is that if Lazzarini;s result had been published in 1994
and not in 1901, it would be called a barefaced fraud. Indeed, Badger himself, after elegantly
demonstrating that Lazzarini;s good luck must somehow have been contrived, himself uses the
word "hoax" to describe how an even better approximation to pi might be obtained. In short,
Badger's tale should be a warning to all those who pollute the literature that their misdeeds will
follow them to the grave.

The details of Lazzarini's experiment are to the point. His needle was 2.5 cm long (that is l), his
parallel lines were separated by 3.0 cm (d), he dropped his needle onto the marked grip 3,408
times (N) and recorded 1,808 intersections of the needle with a grid-line (H). The exceptional
quality of Lazzarini;s good luck is easily appreciated: one hit more or less, giving 1,809 or 1,807
rather than 1,808 hits, would have produced a variation of 1,2,000 from the value of pi, yielding
a departure from the true value in the third rather than the seventh decimal place.

There are other grounds for worrying about the precision of the result, not the least of which are
the unavoidable imprecisions in the length of the needle (l) and the spacing between the lines of
the grid (d). The obvious difficulty is than an error in either translates directly in a commensurate
error in the estimate of pi obtained by dropping a needle onto a rules grip. Would Lazzarini have
had access to the metrology equipment that would have allowed his measurements of l and d to
be accurate to a few parts in 10 million?

Badger, evidently one in whom the seeds of suspicion arise only with difficulty, puts must of the
same point in yet another way: why, he muses, should Lazzarini have dropped his needle onto his
grid exactly 3,408 times and not, for example, 3,500 times? Is there the possibility, only the
slightest possibility, of course, that Lazzarini was guided by his knowledge that the number
355/113 is a rather approximation to pi first described as such in the fifth century by a Chinese
mathematician?

For as well-mannered a critic of even deceased fellow-beings as Badger, it is evidently distasteful
to face up to the enormity of what Lazzarini may have done. The reported dimensions of his
experimental equipment nevertheless give the show away. For one thing, the ratio 2l/d = 5/3.
Simply multiplying that by the reported ratio of needle-throws to hits (3,408/1,808 = 213/113
after dividing both numerator and denominator by 16) gives the magic ratio 355/113.

But charitable Badger turns the problem around. He allows that Lazzarini may have had the ratio
of 355/113 somewhere in mind (and probably nearer the front of the it than the back), meaning
that the dimensions of his equipment would have enabled him to made a choice every 213 throws
of the needed; how good now is my approximation to pi, he might have asked himself after every
213 throws.

On that view, Lazzarini's reported success would have arisen on the sixteenth attempt, but
elsewhere the errant experimental mathematician claims to have case his 2.5-cm needle 4,000
times. The charitable question is that of the probability that, at some multiple of 213 throws of
the needle is a staggering 0.3, or 30 percent. Lazzarini may not have been a fraud after all!

Sadly for the memory of the dead, Badger then proceeds to put the knife in. Lazzarini was
apparently unwise enough to report not just the number of his hits after 3,408 cases of his needle,
but also at intermediate values from those expected by somebody with a value of pi in mind turns
out to be consistently smaller than would be expected if the events contrived were random.
Indeed, Badger concludes that the chance that the fluctuation from the ideal reported by Lazzarini
would be as small amounts to merely 0.000003, or 3 in a million. Unsurprisingly, he concludes
that "it seems likely that the experiment was not done."

That, of course, is how those who concoct data are most commonly found out. It is easy enough
to ensure that the final result includes a reasonable error, but much more difficult to arrange that
the errors in a series of data bear a reasonable relationship to what random processes would yield.
And those who have concocted the numbers can always say that the case against them rests "only
on probability," as if that were without meaning. Yet, curiously enough, Lazzarini
non-experiment is not without meaning. Indeed, it inspired generations of people to believe that
there is indeed a connection between circularity and rectilinear geometry. If it was merely a
gedanken experiment, it may nevertheless have served what is called a heuristic purpose of some
importance.