Infinitesimal: How a Dangerous Mathematical Theory Shaped the Modern World (44 page)

The more lines we have in the quadrant and in the square—or, as we would say today, as
n
approaches infinity—the closer this number draws to the ratio between the area of the quadrant and the area of the enclosing square.

Figure 9.2. Parallel lines compose the surface of the quadrant. After John Wallis,
Arithmetica infinitorum
(Oxford: Leon Lichfield, 1656), p. 108, prop. 135.

Now, the precise length of each of the parallel lines
r
that compose the quadrant is dependent on its distance from the first and longest line,
R
. If we divide the distance
R
into
n
equal parts, and consider each equal part a unit, then the length of the line closest to
R
is
; the line next to that will be
, the one after that
, and so on, until we reach the circumference, where the last line is
,
which is zero. The ratio between the lines dividing the quadrant and the same number of lines dividing the enclosing square will therefore be

Wallis’s goal in the
Arithmetica infinitorum
is to calculate this ratio as
n
increases to infinity, and this proves to be no easy task. He approaches the result through a succession of approximations of similar series, that draw ever closer to the desired ratio. But far more significant than Wallis’s calculation of the area of a circle is his method for summing infinite series that ultimately lead to his final result.

Suppose, he suggests at the start of
Arithmetica infinitorum
, that we have a “series of quantities in arithmetic proportion, continually increasing, beginning from a point or 0 … thus as 0, 1, 2, 3, 4, etc.” What, he asks, is the ratio of the sum of the terms of the series to the sum of an equal number of the largest term? Wallis decides to try it out. He begins with the simplest case, of the two-term series 0, 1. The ratio is, accordingly,

He tests other cases:

Every case yields the same result, and Wallis draws a definite conclusion: “If there is taken a series of quantities in arithmetic proportion (or as the natural sequence of numbers) continually increasing, beginning from a point or 0, either finite or infinite in number (for there will be no reason to distinguish), it will be to a series of the same number of terms equal to the greatest, as 1 to 2.”