Infinitesimal: How a Dangerous Mathematical Theory Shaped the Modern World (42 page)

He continues: “It is a well-known rule among mathematicians that the sum of an arithmetical progression, or the aggregate of all the terms, equals the sum of its extremes multiplied by half the total number of terms.” This is a simple rule, familiar today to many a high school student. The sum of all the numbers from 1 to 10, for example, is 11 (that is, 1
+
10) times 5 (half the number of terms in the sequence), that is, 55. Designating the infinitesimal magnitude of a single point by the letter “o,” Wallis then uses the rule to sum up all the indivisible lines that compose the triangle:

Therefore, if we consider the smallest term “o” (since we suppose that a point equals “o” in magnitude as well as zero in number), the sum of the two ends is the same as the largest term. I substitute the altitude of the figure for the number of terms in the progression, for this reason, that if we suppose the number of terms to be
∞
, then the sum of their lengths is
×
Base
(since the base is equal to the sum of the two ends).

Wallis is after the total length of all the lines that make up the triangle. Since they are infinite in number, and range from zero (or “o”) to the length of the base, their combined length is
×
Base
. He now multiplies this by the thickness of each line:

But we suppose the thickness or altitude of each (line or parallelogram) to be
×
the Altitude of the Triangle
; by which the sum of the lengths is to be multiplied. Therefore
×
A
multiplied by
×
Base
will give the area of the triangle. That is
.

Figure 9.1. Wallis’s triangles, composed of parallel lines. From
De sectionibus conicis
, prop. 3. (Oxford, Leon Lichfield, 1655)

And that is how Wallis calculated the area of the triangle: He summed up the lengths of all the component lines as an arithmetical progression, and then multiplied the sum by the “thickness” of each line. Arriving in this way at an equation that had
∞
in the numerator and
∞
in the denominator, he canceled them against each other and ended up with the familiar formula
. QED.

Now, it is probably an understatement to say that no modern mathematician would follow Wallis in these wild and woolly calculations. Nor would many of his contemporaries, including all the Jesuits and Fermat, among others. Apart from the problematic assumption that a surface is made up of lines with a certain (very small) thickness, Wallis is also assuming without proof that the rules for summing up a finite series also apply to an infinite one. And if these unsubstantiated assumptions are not questionable enough, Wallis then casually divides infinite by infinity, or, to use his own notation,
∞
by
∞
. In modern mathematics
is undefined, for the simple reason that if
=
a
, then
∞
=
a
×
∞
, and since any number multiplied by
∞
equals
∞
,
a
can be any number. But Wallis treats
as an ordinary algebraic expression, and cancels
∞
by
∞
. When criticized by Fermat and others, Wallis seemed unconcerned with the logical difficulties of his procedures, refusing to concede any of their points. His approach, after all, was not meant to demonstrate his adherence to strict formal rigor. It was, rather, designed to make mathematics acceptable to his fellows at the Royal Society.

What did Wallis accomplish with his unconventional approach? For one, he posited that geometrical objects were objects “out there” in the world, and could be investigated as such, just like any natural object. This is the exact opposite of the traditional view that held that all geometrical objects should be constructed from first principles. It also runs counter to Hobbes’s view that geometrical objects are perfectly known because we construct them. For Wallis, in contrast, the triangle already exists in the world, and the geometer’s job is to decipher its hidden characteristics—just like a scientist trying to understand a geological rock formation or the biological systems of an organism. Drawing on common sense and intuition about the physical world, Wallis concluded that a triangle is composed of parallel lines next to each other, just as a rock formation is made of geological strata, a piece of wood is made of fibers, or (following Cavalieri) a piece of cloth is made of threads.