Gauss's Day of Reckoning
A famous story about the boy wonder of mathematics has taken on a life of its own
I started my survey with five modern biographies of Gauss: books by
G. Waldo Dunnington (1955), Tord Hall (1970), Karin Reich (1977), W.
K. Bühler (1981) and a just-issued biography by M. B. W. Tent
(2006). The schoolroom incident is related by all of these authors
except Bühler. The versions differ in a few details, such as
Gauss's age, but they agree on the major points. They all mention
the summation of the same series, namely the integers from 1 to 100,
and they all describe Gauss's method in terms of forming pairs that
sum to 101.
None of these writers express much skepticism about the anecdote
(unless Bühler's silence can be interpreted as doubt). There is
no extended discussion of the story's origin or the evidence
supporting it. On the other hand, references in some of the
biographies did lead me to the key document on which all subsequent
accounts seem to depend.
This locus classicus of the Gauss schoolroom story is a
memorial volume published in 1856, just a year after Gauss's death.
The author was Wolfgang Sartorius, Baron von Waltershausen,
professor of mineralogy and geology at the University of
Göttingen, where Gauss spent his entire academic career. As
befits a funerary tribute, it is affectionate and laudatory throughout.
In the portrait Sartorius gives us, Gauss was a wunderkind.
He taught himself to read, and by age three he was correcting an
error in his father's arithmetic. Here is the passage where
Sartorius describes Gauss's early schooling in the town of
Braunschweig, near Hanover. The translation, except for two phrases
in brackets, is by Helen Worthington Gauss, a great-granddaughter of
In 1784 after his seventh birthday the little
fellow entered the public school where elementary subjects were
taught and which was then under a man named Büttner. It was
a drab, low school-room with a worn, uneven floor.... Here among
some hundred pupils Büttner went back and forth, in his
hand the switch which was then accepted by everyone as the final
argument of the teacher. As occasion warranted he used it. In
this school—which seems to have followed very much the
pattern of the Middle Ages—the young Gauss remained two
years without special incident. By that time he had reached the
arithmetic class in which most boys remained up to their
Here occurred an incident which he
often related in old age with amusement and relish. In this
class the pupil who first finished his example in arithmetic was
to place his slate in the middle of a large table. On top of
this the second placed his slate and so on. The young Gauss had
just entered the class when Büttner gave out for a problem
[the summing of an arithmetic series]. The problem was barely
stated before Gauss threw his slate on the table with the words
(in the low Braunschweig dialect): "There it lies."
While the other pupils continued [counting, multiplying and
adding], Büttner, with conscious dignity, walked back and
forth, occasionally throwing an ironical, pitying glance toward
this the youngest of the pupils. The boy sat quietly with his
task ended, as fully aware as he always was on finishing a task
that the problem had been correctly solved and that there could
be no other result.
At the end of the hour the slates
were turned bottom up. That of the young Gauss with one solitary
figure lay on top. When Büttner read out the answer, to the
surprise of all present that of young Gauss was found to be
correct, whereas many of the others were wrong.
Incidental details from this account reappear over and over in later
tellings of the story. The ritual of piling up the slates is one
such feature. (It must have been quite a teetering heap by the time
the hundredth slate was added!) Büttner's switch (or cane, or
whip) also made frequent appearances until the 1970s but is less
common now; we have grown squeamish about mentioning such barbarities.
What's most remarkable about the Sartorius telling of the
story is not what's there but what's absent. There is no mention of
the numbers from 1 to 100, or any other specific arithmetic
progression. And there is no hint of the trick or technique that
Gauss invented to solve the problem; the idea of combining the
numbers in pairs is not discussed, nor is the formula for summing a
series. Perhaps Sartorius thought the procedure was so obvious it
needed no explanation.
A word about the bracketed phrases: Strange to report, the
Worthington Gauss translation does mention the first 100
integers. Where Sartorius writes simply "eine arithmetischen
Reihe," Worthington Gauss inserts "a series of numbers
from 1 to 100." I cannot account for this interpolation. I can
only guess that Worthington Gauss, under the influence of later
works that discuss the 1-to-100 example, was trying to help out
Sartorius by filling in an omission. The second bracketed passage
marks an elision in the translation: Where Sartorius has the pupils
"rechnen, multiplizieren und addieren," Worthington Gauss
writes just "adding." I'll have more to say on this point below.