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Gauss's Day of Reckoning

A famous story about the boy wonder of mathematics has taken on a life of its own

Brian Hayes


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 the mathematician.

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 fifteenth year.

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.

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