COMPUTING SCIENCE
Gauss's Day of Reckoning
A famous story about the boy wonder of mathematics has taken on a life of its own
Brian Hayes
Making History
If Sartorius did not specify a series running from 1 to 100, where
did those numbers come from? Could there be some other document from
Gauss's era that supplies the missing details? Perhaps someone to
whom Gauss told the story "with amusement and relish" left
a record of the occasion. The existence of such a corroborating
document cannot be ruled out, but at present there is no evidence
for it. None of the works I have seen makes any allusion to another
early source. If an account from Gauss's lifetime exists, it remains
so obscure that it can't have had much influence on other tellers of
the tale.
In the literature I have surveyed, the 1-100 series makes its first
appearance in 1938, some 80 years after Sartorius wrote his memoir.
The 1-100 example is introduced in a biography of Gauss by Ludwig
Bieberbach (a mathematician notorious as the principal instrument of
Nazi anti-Semitism in the German mathematical community).
Bieberbach's telling of the story is also the earliest I have seen
to specify Gauss's strategy for calculating the sum—the method
of forming pairs that add to 101. Should Bieberbach therefore be
regarded as the source from whom scores of later authors have
borrowed these "facts"? Or is this a case of multiple
independent invention?
If you think it utterly implausible that two or more authors would
come up with the same example and the same method, then Bieberbach
himself is disqualified as the source. A full millennium before
Gauss and Büttner had their classroom confrontation,
essentially the same problem and solution appeared in an
eighth-century manuscript attributed to Alcuin of York.
Furthermore, in the years since Bieberbach wrote, there is
unmistakable evidence of independent invention. Not all versions
agree that the sequence of numbers was the set of consecutive
integers from 1 through 100. Although that series is the
overwhelming favorite, many others have been proposed. Some are
slight variations: 0-100 or 1-99. Several authors seem to feel that
adding up 100 numbers is too big a job for primary-school students,
and so they trim the scope of the assignment, suggesting 1-80, or
1-50, or 1-40, or 1-20, or 1-10. A few others apparently think that
1-100 is too easy, and so they give 1-1,000 or else a series in
which the difference between successive terms is a constant other
than 1, such as the sequence 3, 7, 11, 15, 19, 23, 27. (The example
series chosen by various authors and other features of the versions
are tabulated in the table at right.)
Perhaps the most influential version of the story after that of
Sartorius is the one told by Eric Temple Bell in Men of
Mathematics, first published in 1937. Bell has a reputation as
a highly inventive writer (a trait not always considered a virtue in
a biographer or historian). He turns the Braunschweig schoolhouse
into a scene of gothic horror: "a squalid relic of the Middle
Ages run by a virile brute, one Büttner, whose idea of teaching
the hundred or so boys in his charge was to thrash them into such a
state of terrified stupidity that they forgot their own names."
Very cinematic! When it comes to the arithmetic, however, Bell is
one of the few writers who scruple to distinguish between fact and
conjecture. He doesn't claim to know the actual numerical series,
but writes: "The problem was of the following sort, 81297 +
81495 + 81693 + ... + 100899, where the step from one number
to the next is the same all along (here 198), and a given number of
terms (here 100) are to be added." (Personally, I'd have a hard
time even writing that problem on a small slate, much less solving it.)
» Post Comment