Measuring the World . . .

     Yesterday afternoon, at the Goethe Institut in Washington DC, I saw a wonderful film, "Measuring the World." Based on a popular 2005 novel by Daniel Kehlmann, the story of a friendship between preeminent German mathematician Carl Friedrich Gauss (1777-1855) and Prussian naturalist Alexander von Humboldt (1769-1859).  The film offers a delightful interplay of personalities and ideas as it darts between the explorations of these two men -- one digging inside his head for mathematics and the other traveling over mountains, through jungles, across oceans. 

     Early in the film Gauss' talent is revealed through a well-known story in which he quickly finds a tedious sum.  His teacher, expecting to have some idle time while students calculate, asks for the sum of the counting numbers from 1 to 100.  That is, the students should find S, where:

     S = 1 + 2 + 3 + 4 + 5 + . . . + 99 + 100

Rearranging the numbers, Gauss considered the sum this way:

     S = (1+100) + (2+99) + (3+98) + (4+97) + . . . + (49+52) + (50+51)

And this sum of 50 pairs is   50 x 101.       So    S = 5050.

Sherman Stein's Mathematics, The Man-Made Universe (Freeman, 1976) speaks of Gauss often -- and the closing words of its final Chapter 19 offer these words from a letter written by Gauss to Janos Bolyai (1802-1860).
    
     Surely it is not the knowing
     but the learning, not the possessing
     but the acquiring, not the being-there
     but the getting-there, that affords
     the greatest satisfaction.  If I have
     clarified and exhausted something, I leave it
     to go again into the dark.