To Explain the World: The Discovery of Modern Science (41 page)

Thus, in order for the lowest vibrations of both strings each to go through whole numbers of cycles in the same time, the quantity
L
₂
/
L
₁
must be a ratio of whole numbers—that is, a rational number. (In this case, in the same time each overtone of each string will also go through a whole number of cycles.) The sound produced by the two strings will thus repeat itself, just as if a single string had been plucked. This seems to contribute to the pleasantness of the sound.

For instance, if
L
₂
/
L
₁
= 1/2, then the vibration of string 2 of lowest frequency will go through two complete cycles for every complete cycle of the corresponding vibration of string 1. In this case, we say that the notes produced by the two strings are an octave apart. All of the different C keys on the piano keyboard produce frequencies that are octaves apart. If
L
₂
/
L
₁
= 2/3, the two strings make a chord called a fifth. For example, if one string produces middle C, at 261.63 cycles per second, then another string that is 2/3 as long will produce middle G, at a frequency 3/2 × 261.63 = 392.45 cycles per second.
*
If
L
₂
/
L
₁
= 3/4, the chord is called a fourth.

The other reason for the pleasantness of these chords has to do with the overtones. In order for the
N
₁
th overtone of string 1 to have the same frequency as the
N
₂
th overtone of string 2, we must have
vN
₁
/2
L
₁
=
vN
₂
/2
L
₂
, and so

L
₂
/
L
₁
=
N
₂
/
N
₁

Again, the ratio of the lengths is a rational number, though for a different reason. But if this ratio is an irrational number, like
π
or the square root of 2, then the overtones of the two strings can never match, though the frequencies of high overtones will come arbitrarily close. This apparently sounds terrible.

4. The Pythagorean Theorem

The so-called Pythagorean theorem is the most famous result of plane geometry. Although it is believed to be due to a member of the school of Pythagoras, possibly Archytas, the details of its origin are unknown. What follows is the simplest proof, one that makes use of the notion of proportionality commonly used in Greek mathematics.

Consider a triangle with corner points
A
,
B
, and
P
, with the angle at
P
a right angle. The theorem states that the area of a square whose side is
AB
(the hypotenuse of the triangle) equals the sum of the areas of squares whose sides are the other two sides of the triangle,
AP
and
BP.
In modern algebraic terms, we can think of
AB
,
AP
, and
BP
as numerical quantities, equal to the lengths of these sides, and state the theorem as

AB
²
=
AP
²
+
BP
²

The trick in the proof is to draw a line from
P
to the hypotenuse
AB
, which intersects the hypotenuse at a right angle, say at point
C.
(See Figure 2.) This divides triangle
ABP
into two smaller right triangles,
APC
and
BPC.
It is easy to see that both of these smaller triangles are similar to triangle
ABP
—that is, all their corresponding angles are equal. If we take the angles at the corners
A
and
B
to be
α
(alpha) and
β
(beta), then triangle
ABP
has angles
α
,
β
, and 90°, so
α
+
β
+ 90° = 180°. Triangle
APC
has two angles equal to
α
and 90°, so to make the sum of the angles 180° its third angle must be
β
. Likewise, triangle
BPC
has two angles equal to
β
and 90°, so its third angle must be
α
.

Because these triangles are all similar, their corresponding sides are proportional. That is,
AC
must be in the same proportion to the hypotenuse
AP
of triangle
ACP
that
AP
has to the hypotenuse
AB
of the original triangle
ABP
, and
BC
must be in the same proportion to
BP
that
BP
has to
AB.
We can put this in more convenient algebraic terms as a statement about ratios of the lengths
AC
,
AP
, etc.:

It follows immediately that
AP
²
=
AC
×
AB
, and
BP
²
=
BC
×
AB.
Adding these two equations gives

AP
²
+
BP
²
= (
AC
+
BC
) ×
AB

But
AC
+
BC
=
AB
, so this is the result that was to be proved.

Figure 2. Proof of the Pythagorean theorem.
This theorem states that the sum of the areas of two squares whose sides are
AP
and
BP
equals the area of a square whose sides are the hypotenuse
AB
. To prove the theorem, a line is drawn from
P
to a point
C,
which is chosen so that this line is perpendicular to the line from
A
to
B
.

5. Irrational Numbers

The only numbers that were familiar to early Greek mathematicians were rational. These are numbers that are either whole numbers, like 1, 2, 3, etc., or ratios of whole numbers, like 1/2, 2/3, etc. If the ratio of the lengths of two lines is a rational number, the lines were said to be “commensurable”—for instance,
if the ratio is 3/5, then five times one line has the same length as three times the other. It was therefore shocking to learn that not all lines were commensurable. In particular, in a right isosceles triangle, the hypotenuse is incommensurable with either of the two equal sides. In modern terms, since according to the theorem of Pythagoras the square of the hypotenuse of such a triangle equals twice the square of either of the two equal sides, the length of the hypotenuse equals the length of either of the other sides times the square root of 2, so this amounts to the statement that the square root of 2 is not a rational number. The proof given by Euclid in Book X of the
Elements
consists of assuming the opposite, in modern terms that there is a rational number whose square is 2, and then deriving an absurdity.

Suppose that a rational number
p
/
q
(with
p
and
q
whole numbers) has a square equal to 2:

(
p
/
q
)
²
= 2

There will then be an infinity of such pairs of numbers, found by multiplying any given
p
and
q
by any equal whole numbers, but let us take
p
and
q
to be the smallest whole numbers for which (
p
/
q
)
²
= 2. It follows from this equation that

p
²
= 2
q
²

This shows that
p
²
is an even number, but the product of any two odd numbers is odd, so
p
must be even. That is, we can write
p
= 2
pʹ
, where
pʹ
is a whole number. But then

q
²
= 2
p
ʹ
²

so by the same reasoning as before,
q
is even, and can therefore be written as
q
= 2
qʹ
, where
qʹ
is a whole number. But then
p
/
q
=
pʹ
/
qʹ
, so

(
p
ʹ/
q
ʹ)
²
= 2

with
pʹ
and
qʹ
whole numbers that are respectively half
p
and
q
, contradicting the definition of
p
and
q
as the smallest whole numbers for which (
p
/
q
)
²
= 2. Thus the original assumption, that there are whole numbers
p
and
q
for which (
p
/
q
)
²
= 2, leads to a contradiction, and is therefore impossible.

This theorem has an obvious extension: any number like 3, 5, 6, etc. that is not itself the square of a whole number cannot be the square of a rational number. For instance, if 3 = (
p
/
q
)
²
, with
p
and
q
the smallest whole numbers for which this holds, then
p
²
= 3
q
²
, but this is impossible unless
p
= 3
pʹ
for some whole number
pʹ
, but then
q
²
= 3
pʹ
²
, so
q
= 3
qʹ
for some whole number
q
, so 3 = (
pʹ
/
qʹ
)
²
, contradicting the statement that
p
and
q
are the smallest whole numbers for which
p
²
= 3
q
²
. Thus the square roots of 3, 5, 6 . . . are all irrational.

In modern mathematics we accept the existence of irrational numbers, such as the number denoted
whose square is 2. The decimal expansion of such numbers goes on forever, without ending or repeating; for example,
= 1.414215562. . . . The numbers of rational and irrational numbers are both infinite, but in a sense there are far more irrational than rational numbers, for the rational numbers can be listed in an infinite sequence that includes any given rational number: