Infinitesimal: How a Dangerous Mathematical Theory Shaped the Modern World (45 page)

Wallis could easily have proven this simple result by giving the general formula for the sum of the sequence of natural numbers beginning with 0, and dividing it by the sum of an equal number of the largest term:
divided by
n
(
n
+ 1)
, which immediately gives
. But his goal was not to calculate the ratio, but to demonstrate the usefulness of the method of induction: try one case, then another, and then another. If the theorem holds in all cases, then to Wallis it is proven and true. “
Induction
,” he wrote many years later, is “a very good Method of
Investigation
 … which doth very often lead us to the early discovery of a General Rule.” Most important, “it need not … any further Demonstration.”

Once he had established this first theorem, Wallis moved on to do the same for more complex series: what if, instead of adding up a sequence of the natural numbers and dividing the sum by the same number of the largest term, he added up the squares of the natural numbers and divided the sum by an equal number of the largest square? Using his favored method of induction, he tries it out. Starting with the simplest case, he gets:

He then adds more terms, calculating the sum in each case:

Looking at the different cases, Wallis deduces that the more terms there are in the series, the closer the ratio approaches
. For an infinite series, he concludes, the difference will vanish entirely. He writes it up in a theorem (proposition 21):

If there is proposed an infinite series, of quantities that are as squares of arithmetic proportionals (or as a sequence of square numbers) continually increasing, beginning from a point of 0, it will be to a series of the same number of terms equal to the greatest as 1 to 3.

Wallis’s proof requires only a single sentence: the result, he writes, is “clear from what has gone before.” Induction needs no further support.

Wallis tries out one more series of this type, looking at the cubes rather than the squares of the natural numbers: