Infinitesimal: How a Dangerous Mathematical Theory Shaped the Modern World (46 page)

The method of induction proves itself once again. As the number of terms increases, the ratio approaches ever more closely to
, which leads to proposition 41:

If there is proposed an infinite series of quantities that are as cubes of arithmetic proportionals (or as a sequence of pure numbers) continually increasing, beginning from a point or 0, it will be to a series of the same number of terms equal to the greatest as 1 to 4.

Like the theorems that preceded it, this, too, requires no proof beyond self-evident induction.

In modern notation, Wallis’s three theorems would look like this:

Wallis considers these ratios important steps on the road to calculating the area of a circle, because each algebraic ratio corresponded for him to a particular geometrical case. The first one shows the ratio between a triangle and its enclosing rectangle, just as Wallis shows in his proof of the area of the triangle in
De sectionibus conicis
. The series 0, 1, 2, 3,…,
n
represents the lengths of the parallel lines that make up the triangle, and the series
n
,
n
,
n
,
n
,…
,
n
represents an equal number of parallel lines composing the enclosing rectangle. The ratio between them,
, is indeed the ratio between the areas of the triangle and rectangle (see
Figure 9.1
). The second case corresponds to the ratio between a half-parabola and its enclosing rectangle or, more precisely, the ratio between the area outside the half-parabola and the rectangle. The parallel lines composing this area increase as squares, that is, 0, 1, 4, 9,…,
n
2
, whereas the rectangle is represented by
n
2
,
n
2
,
n
2
,…,
n
2
. Wallis, in effect, shows that the ratio between the area outside a parabola and the area of its enclosing rectangle is
. The third ratio (
Figure 9.3
) corresponds to a steeper “cubic” parabola, showing that the ratio here is
. While Wallis still has a long way to go before calculating the more difficult ratio between the quarter-circle and its enclosing square, his strategy for arriving there is clearly taking shape.

With these results established, Wallis now makes use of induction once again to arrive at an even more general theorem: what is true for natural numbers, their squares, and their cubes, must be true for all powers
m
of natural numbers:

Wallis does not quite write the results in this form. Lacking our modern notation, he uses a table, where he assigns a ratio,
, to the “first power,” another ratio,
, to the “second power,” another ratio,
, to the “third power,” and so on. The table is open-ended, and the rule is manifest: for any power
m
, the ratio will be
.