Infinitesimal: How a Dangerous Mathematical Theory Shaped the Modern World (18 page)

In the parallelogram ABCD in which the side AB is greater than the side BC, a diameter BD is traced with a point E along it, and EF and EG parallel to AB and BC respectively, then EF is greater than EG, and the same is true for all other similar parallels. Therefore all the lines similar to EF in the triangle ABD are greater than all the lines similar to EG in the triangle CDB, and therefore the triangle ABD is greater than the triangle CDB. Which is false, because the diameter BD divides the parallelogram down the middle.

Figure 3.8. Torricelli, the paradox of the parallelogram. Based on E. Torricelli,
Opera omnia
, vol. 1, part 2, p. 417.

The conclusion that the two halves of the rectangle differ in size is absurd, but it seems to follow easily from the concept of indivisibles. What is to be done? Ancient mathematicians, well aware that infinitesimals could lead to such contradictions, simply banned them from mathematics. Cavalieri reintroduced indivisibles but tried to deal with such contradictions by inscribing rules into his procedures to ensure those contradictions would not arise. For example, he insisted that, in order to compare “all the lines” in one figure with “all the lines” in the other, the lines in both figures must all be parallel to a single line he called the “regula.” Since the lines
EF
and
EG
in Torricelli’s paradox are not parallel, then Cavalieri could claim that they should not be compared at all, and the paradox can be averted. In practice, however, Cavalieri’s artificial limitations were ignored both by his followers, who saw them as inconvenient hindrances, and by his critics, who did not believe that they resolved the fundamental problem.

Torricelli took a different approach. Instead of trying to evade the paradox, he made a sustained effort to understand it and what it meant for the structure of the continuum. His conclusion was startling: The reason all the short lines parallel to
EG
produce an area equal to the same number of longer lines parallel to
EF
is that the short lines are “wider” than the long lines. More broadly, according to Torricelli, “that indivisibles are all equal to each other, that is that points are equal to points, lines are equal in width to lines, and surfaces are equal in thickness to surfaces, is an opinion that seems to me not only difficult to prove, but in fact false.” This is a stunning idea. If some indivisible lines are “wider” than others, doesn’t that mean that they can in fact be divided, to reach the width of the “thin” lines? And if indivisible lines have a positive width, doesn’t it follow that an infinite number of them would add up to an infinite magnitude—not to the finite area of the triangles
ADB
and
CDB
? And the very same applies to points with a positive size and surfaces with a “thickness.” The assumption seems absurd, but Torricelli insisted that his paradoxes indicated that there was no other explanation. And not only that: he founded his entire mathematics approach on precisely this idea.

In order to transform this basic insight into a mathematical system, it was not enough to say in principle that indivisibles differed in size from one another; it was necessary to determine by precisely how much they differed from one another. For this, Torricelli turned once again to the paradox of the parallelogram. In the diagram, the same number of long lines
EF
and short lines
EG
produce exactly the same total area. For this to be true, the short lines
EG
need to be “wider” by exactly the same proportion as the lines
EF
are “longer.” That in turn is the ratio of
BC
to
BA
, which is, in other words, the slope of the diagonal
BD
. At a stroke, Torricelli transformed a rather dubious speculation about the composition of the continuum into a quantifiable and usable mathematical magnitude.

Torricelli then showed exactly how to make mathematical use of indivisibles with “width” by calculating the slope of the tangent of a class of curves that we would characterize as
y
m
=
kx
n
, and that he called an “infinite parabola.” In this he went well beyond Cavalieri, who calculated areas and volumes enclosed in geometrical curves, but never their tangents. Indeed, Cavalieri’s insistence on comparing only conglomerations of “all the lines” or “all the planes” left no room for the delicate calculation of tangents, which are slopes calculated at single indivisible points. But Torricelli’s more flexible method, which distinguished between the magnitudes of different indivisibles, made this possible. He first pointed to the figures
ABEF
and
CBEG
in the paradox of the parallelograms. The two figures, known as “semi-gnomons,” are equal in area because they complete the equal triangles
DFE
and
EGD
to the equal triangles
ADB
and
CDB
. This will always be true, furthermore, no matter where the point
E
is positioned on the diagonal
DB
, even as it is moved to the point
B
itself. Accordingly, the line
BC
is equal in area, or “quantity,” to the line
AB
, even though the line
AB
is longer. This is the case because, just like the semi-gnomon
CBEG
, the indivisible line
BC
is “wider” than
AB
by precisely the same ratio that
AB
is longer.

Now, as long as we are dealing with straight lines, such as the diagonal
BD
, the semi-gnomons are always equal, and the “width” of the indivisibles is given by the simple ratio of the slope. But what happens if, instead of a straight line, we are given a generalized parabola, which in modern terms would be given as
y
m
=
kx
n
? In this “infinite parabola,” the semi-gnomons are no longer equal, but they do hold a fixed relationship. As Torricelli proved using the classic method of exhaustion, if the segment on the curve is very small, the ratio of the two semi-gnomons is as
. And if the width of the semi-gnomons is only a single indivisible, then the “size” of the indivisible lines that meet at the curve is as
.

Figure 3.9. Semi-gnomons meeting at a segment of an “infinite parabola.” If the segment is very small, or indivisible, then the ratio of the areas of the semi-gnomons is as
.

This result enabled Torricelli to calculate the slope of the tangent at every point on the “infinite parabola,” shown as the curve
AB
in
figure 3.10.
Torricelli’s key insight is that at the point
B
, where the two indivisible lines
BD
and
BG
meet the curve, they also meet the straight line that is the curve’s tangent at that point. And whereas the “area” of the two indivisibles is as
relative to the curve, it is equal relative to the straight tangent—if the tangent is extended to become the diagonal of a rectangle. Accordingly, in Figure 3.10, the ratio of the “areas” of
BD
and
BG
is
, but the ratio of the “areas” of
BD
and
BF
is 1. This means that the ratio of the “areas” of
BF
and
BG
is
. Now,
BF
and
BG
have the same “width” in Torricelli’s scheme, because they both meet the curve
BF
(or its tangent) at point
B
at precisely the same angle. The difference between the two segments is only in their lengths, and it follows that the length
BF
is to the length
BG
as
. Now,
BF
is equal to
ED
, and
BG
is equal to
AD
, and therefore the ratio of the abscissa
ED
of the tangent to the abscissa
AD
of the curve is
, or, more simply,
ED
=
. Therefore the slope
of the tangent at point
B
is
. In this way, the slope of an “infinite parabola” can be known at any given point on the “infinite parabola,” based on its abscissa and ordinate.