XVlll
Introduction
We
continue with a study
of
Liu Hui's third century A.D. commentary on the Chinese
classic
Nine Chapters
in
the Mathematical
Art
and
of
the lost work
of
the fifth century
astronomer Zu Chongzhi. Marshall Clagett's translation
of
Verba Filiorum, the Latin
version
of
the 9th century Arabic
Book
oj
Knowledge
oj
the Measurement
oj
Plane and
Spherical Figures
completes our first millenium extracts.
The next selection jumps forward
500
years and discusses the tombstone
of
Ludolph
van Ceulen which recorded the culminating computation
of
pi by purely Archimedian
techniques to
35
places as performed by Ludolph, using 2
62
_gons,
before
1615.
We
com-
plete this period with excerpts from three great transitional thinkers:
Fran~ois
Viete
(1540-1603) whose work greatly influenced that
of
Fermat; John Wallis (1616-1703), to
whom Newton indicated great indebtedness; and the Dutch polymath Christian Huygens
(1629-1695), who correctly formalized Willebrord Snell's acceleration
of
Archimedes'
method and
was
thus able to recapture Van Ceulen's computation with only 2
30
-gons. In
a part
of
this work, not reproduced here, Huygens vigorously attacks the validity
of
Gregory's argument for the transcendence
of
pi.
From Newton to Hilbert (Papers
[16]
to
[24])
These comprise many
of
the most significant papers on pi. After visiting Newton's contri-
bution
we
record a discussion
of
the arctangent series for pi variously credited to the
Scot James Gregory, the German Leibniz, and to the earlier Indian MMhava. In this
period
we
move from the initial investigations
of
irrationality,
by
Euler and Lambert, to
one
of
the landmarks
of
nineteenth century mathematics, the proof
of
the transcendence
of
pi.
The first paper
is
a selection from Euler and it demonstrates Euler's almost unparal-
leled-save
for
Ramanujan-ability
to formally manipulate series, particularly series for
pi.
It
is
followed by an excerpt from Lambert and a discussion
by
Struik
of
Lambert's
proof
of
the irrationality
of
pi, which
is
generally credited as the first proof
of
its
irrationality. Euler had previously proved the irrationality
of
e.
Lambert's proof
of
the
irrationality
of
pi
is
based on a complicated continued fraction expansion. Much simpler
proofs are to be found in
[33], [48].
There
is
a selection from Shank's self-financed publication that records
his
hand calcu-
lation
of
607
digits
of
pi.
(It
is
in fact correct only to
527
places, but this went unnoticed
for almost a century.) The selection
is
included to illustrate the excesses that this side
of
the story has evoked. With a modern understanding
of
accelerating calculations this
computation, even done
by
hand, could be considerably simplified. Neither Shanks's
obsession with the computation
of
digits nor
his
error are
in
any
way
unique. Some
of
this
is
further discussed in
[64].
The next paper
is
Hermite's
1873
proof
of
the transcendence
of
e.
It
is
followed
by
Lindemann's
1882
proof
of
the transcendence
of
pi. These are, arguably, the most impor-
tant papers in the collection. The proof
of
the transcendence
of
pi laid to rest the possibil-
ity
of
"squaring the circle," a problem that had been explicit since the late 5th
c.
B.C.
Hermite's seminal paper on
e in many ways anticipates Lindemann, and it
is
perhaps
surprising that Hermite did not himself prove the transcendence
of
pi. The themes
of
Hermite's paper are explored and expanded in a number
of
later papers in this volume.
See
in particular Mahler
[42].
The last two papers offer simplified proofs
of
the transcen-
dence. One
is
due to Weierstrass in
1885
and the other to Hilbert in
1893.
Hilbert's elegant
proof
is
still probably the simplest proof
we
have.
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