Tales of Statisticians
Carl Friederich Gauss
30 Apr 1777 - 23 Feb 1855Gauss is generally recognized as the greatest mathematician of modern times; he is ranked with Archimedes and Newton for his profound understanding of number. Beyond that, his work draws a line that separates the old stuff from what can be called modern mathematics. The difference is not a matter of insight as such, but rather of the rigorous clarification of the insight.
As a child, Gauss displayed a phenomenal talent for mental computation, and also for its shortcuts.
When he was ten, he and his classmates were given the problem of correctly adding the hundred numbers 81297 + 81495 + 81693 . . . + 100899. No sooner was the problem stated than Gauss immediately wrote down the answer, and, as was the custom, flung his slate on the table by way of announcing that he was done. The rest of the class labored on for an hour, until finally all the slates were in. Gauss's was the only correct answer.
Of course it's simple. The hundred numbers differ by 198 at each step, and it is a trivial problem to add them in one's head:
Take first an easier case: the sum of the n = 4 numbers 2 + 6 + 8 + 12. They form pairs, from the outermost two to the innermost two. We are only interested in the sum of the series. It then suffices to note that the outermost pair sum to 14, and each may be replaced by the average of that sum, namely 7. So also for the next pair. Then we can regard all these numbers as simply being 7, and multiplying by the total number of numbers (4), we at once reach a grand total of 28. What we have done is to replace the problem of adding a long string of numbers with the easier problem of multiplying two numbers together. In Gauss's problem, the arithmetic is even easier, since we will multiply the average size by 100, the number of numbers to be added. Multiplying by 100 is approximately the easiest arithmetical operation in the world. We need only average the first and last numbers: (81297 + 100899) / 2 = 182196/2 = 91098. Then each of the 100 numbers has an average size of 91098. Add two zeroes at the end (to multiply by 100) and we have the answer: 9109800. Bingo!
It's easy when you know how.
For a boy of ten to spontaneously figure out how; well, that is not quite so easy. Gauss's phenomenal ability attracted the attention of the Duke of Brunswick, who agreed in 1791, Gauss being then fourteen, to pay for Gauss's later education.
Among the things that Gauss encountered in his early years was the binomial theorem, which gives a series for the expansion of (1 + x)*n, where n is any number. The series as usually given is the following:
1 + (n)x/1 + (n)(n - 1)x*2 /1 x 2 +(n)(n - 1)(n - 2)x*3 / 1 x 2 x 3 . . .
If n = 2, the series terminates at the third term, since the fourth and all higher terms include a multiplier (n - 2), which would equal zero. Similarly for any positive whole number value of n. But when n is not a positive whole number, the series does not terminate; it continues indefinitely. Most of the time, that "infinite" series converges, meaning that successive terms are smaller, and add less and less to the total, which thus approaches closer and closer to some specific value. That value can be calculated, to any desired precision. So far so good, but mathematicians calculating with the formula would sometimes get ridiculous answers. This occurred when the series happened not to converge, but expanded to give an infinite total value. The mathematicians shrugged and moved on. Gauss was the first to ask, under what conditions do such series converge? The answer defines the zone within which a given formula is safe to use. In this additional rigor lies the possibility of working safely with infinite processes, which is what the calculus routinely does. Newton (and/or Leibniz) had provided the new tool. Gauss and his contemporaries began to write its user manual.
Working with numbers in this way, Gauss not only rediscovered known theorems (such as the law of biquadratic reciprocity, here left undefined), but proved their correctness in a way that still stands as valid. This and other remarkable things were written up in his first published work, the Disquisitiones Arithmeticae, completed in 1798 but delayed in publication until 1801. It was in seven sections, an eighth having been omitted to keep down the cost of publication. Gauss's lost eighth chapter has caused as much gnashing of mathematical teeth, over the centuries, as has the niggardly margin of Fermat's Diophantus, which would not permit him to write out the "remarkable" proof of his Last Theorem, but only the fact that he had discovered it. Even the seven section Disquisitiones soon became rare, owing to the failure of the publisher. Gauss's favorite pupil Eisenstein, for instance, never possessed a copy.
On the first day of the new century, in 1801, a new planet, Ceres, had been discovered in a position which made it extremely difficult to observe. Only a few observations existed from which to calculate its orbit, and predict when it would again return to view. Such calculations (as Newton observed) are among the most arduous in astronomy. But the philosophical interest in planets was then intense. Hegel had proved, from the perfection of the number of seven, that the astronomers' search for planets beyond the then known seven was vain and futile. Hegel was a fool, but even the nonfools wanted to find Ceres again. Gauss bent himself to the task of computing its orbit, and succeeded. The heavens made his fortune, and from that moment on, Gauss was a man with a reputation. Humboldt was the great talent broker of his day. "Who is the greatest mathematician in Germany," said Humboldt to Laplace. "Pfaff" answered Laplace. "What about Gauss," asked Humboldt. "Oh," said Laplace, "Gauss is the greatest mathematician in the world."
Gauss's patron the Duke of Bruswick was fatally wounded in the Battle of Austerlitz, which was fought against Napoleon's forces on 2 December 1806. His death left Gauss without support. Humboldt then secured for Gauss the Directorship of the Göttingen Observatory. Göttingen University is a fine place, perhaps a little remote from the centers of power. To its mathematical eminence, and its tradition of political and social liberality, Gauss contributed much. In this situation, precarious and isolated yet conducive, Gauss remained all the rest of his life.
In 1809 appeared his second masterpiece. Appropriately to his new duties, it was titled Theoria Motus Corporum Coelestium. It analyzed planetary motions, and discussed the problems of determining orbits in general. Gauss's interest in the difficulties of observation, and his improvements in instruments of observation, dated back to before 1800; these interests and his calculations of the law of observational error now found a place in print. Among the details was the method of least squares, which (as his notebooks show) he was indeed using before 1806, the date when Legendre had published a version of it. Gauss was also the first to develop the utility of the normal distribution curve, which had been discovered earlier by de Moivre. Discovery is important. Making previously discovered things available for use is important also, and it is not entirely inappropriate that this distribution is now often called "Gaussian."
On 22 August 1811 there appeared what was called the Great Comet. Gauss again calculated its orbit, and his results were again confirmed. The following year, those who saw in the comet a warning of the fall of kingdoms saw their hypothesis confirmed too, when the shreds of Napoleon's defeated army marched back from Moscow. At this period Gauss more quietly conquered a mathematical kingdom for himself, by investigating the analytic functions of the complex numbers which he had described in the Disquisitiones. Gauss found the fundamental theorem in 1811, communicated it to his friend Wolfgang Bolyai, and then did nothing further. It was later rediscovered by Cauchy, and yet again, still later, by Weierstrass. Gauss did publish in 1812 an important work on the hypergeometric series, which arises in statistics in problems of "draw without replacement" from a finite stock of indistinguishable objects. Again, he found the conditions under which the series converges, and is mathematically tractable.
So it went with other problems; Gauss not only made discoveries, he determined the conditions under which those discoveries could be responsibly handled. Sometimes he had to invent new numbers as the terms in which to report those solutions, such as the Gaussian complex integers (those complex numbers, of the form a + bi, in which a and b are rational integers, and i is the square root of -1).
Mathematics had never seen anything like this. Gauss himself was modest about it. His remark on his own work was like that of Bach (who had said, "ich habe fleissig sein müssen; wer es gleichfalls ist, wird eben so weit kommen"). Bell translates Gauss's remark thus: "If others would but reflect on mathematical truths as deeply and as continuously as I have, they would make my discoveries." Those discoveries went beyond number, and included such practical devices as the electric telegraph, which he developed with his colleague Wilhelm Weber, the heliotrope (for sending messages with light), and the bifilar magnetometer (used in his researches on magnetism).
Statistically, Gauss was interested in the theory of insurance (so Weber tells us in 1841; Gauss himself never published a word of it), and in determining the optimum number of jurors and witnesses, a question which also occupied his contemporary Poisson and other social philosophers. He studied the laws of infant mortality, and for years directed the widows' fund at the University of Göttingen. In 1823, Gauss put forward a new argument for preferring his principle of least squares, accepting the principle that minimizing variance was preferable to maximizing likelihood, but holding to his previous (normal) law of error. (This position turns out to be precarious; as Bertrand has remarked, for small values of x, any number of functions will give values approximately equal to the error curve).
Modest he was, about his own discoveries, but Gauss was not always correspondingly generous to others who had put in the necessary effort and had made discoveries of their own. His indifference to publishing his own results was the hallmark of his whole career. Beyond that, he was often surly toward the young, and at the most unfortunate possible times. He should have been on better terms with Jacobi, who spent his short life largely repeating results that Gauss had earlier reached, but never revealed. Gauss's treatment of Abel, the needy and brilliant Norwegian mathematician whose reputation would have been made by a word from Gauss, except that the word never came, is one of the low points of mathematical history. Gauss was silent rather than commendatory about Cauchy's work on the theory of functions of a complex variable, and about Hamilton's discovery of quaternions (the algebra in which b x a does not equal a x b). Both results had been anticipated in Gauss's earlier, but unpublished, work.
Pauca sed matura — such was Gauss's motto. Few but ripe. It will seem to posterity that he applied it too rigorously, both to himself and to others. All in all, the effect of that motto on mathematical history, whether that history was made by Gauss or by others, has not been a happy one. The nineteenth century partly spent itself in repeating itself, ignorant of what it had already accomplished. Gauss's legacy is tremendous, but two thirds of it is shadow: it did not really happen until later.
The visible third puts him firmly among the immortals. And yet, and yet . . .
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