A Brief Guide to the Great Equations (13 page)

In 1771, a fire destroyed much of St. Petersburg. With his house burning, Euler himself – weak and blind – was carried to safety on the shoulders of a friend. He calculated on. On September 18, 1783, he tutored one of his grandchildren in maths, worked out some problems regarding the paths of hot-air balloons, and considered possible orbits of the recently discovered planet Uranus, when his pipe suddenly dropped from his mouth. In the same breath, ‘he ceased to calculate and live.’
⁷

Today, the modern metropolis of mathematics is much larger still than it was in Euler’s time. It is now laid out in huge boroughs, including analysis, algebra, and topology. Euler helped advance all three. His textbook on algebra,
Vollständige Anleitung zur Algebra
(Complete Instruction in Algebra, published in English as
Elements of Algebra
), presents the field essentially in the form it is today. He also made some of the first forays into topology, a field that did not exist yet, thanks to his famous solution to the Konigsberg bridge problem, involving the question of whether you could cross the seven bridges spanning the banks and islands of that city in a single walk without crossing any bridge twice – although topology would not be recognized as a borough for a hundred years or so.

But Euler was known as the master architect of analysis: scholars often called him ‘analysis incarnate.’ His most important single work in this field is a two-volume textbook, written during his Berlin years, entitled
Introductio ad analysin infinitorum
(
Introduction to Infinite Analysis
, 1748). In it, Euler unveiled numerous discoveries about functions involving infinite series, supplied proofs of theorems that others had left missing or incomplete, simplified many mathematical techniques, and proposed definitions and symbols that have since become standard, including π and
e
. ‘The
Introductio
did for analysis what Euclid’s
Elements
had done for geometry and al-Khowârizmî’s
Hisâb al-jabr wa’l muquâbalah
for algebra. It was a classic text from which whole generations were inspired to learn their analysis, especially their knowledge of infinite series.’
⁸

But the
Introductio
did much more than reorganize analysis. By translating many mathematical terms and expressions into the language of infinite series, it transformed analysis from a newly developing area of mathematics, alongside the existing fields of geometry and algebra, into its principal region. It all but made analysis the centre city of mathematics.

In the
Introductio
, Euler announced the dramatic discovery of a deep connection between exponential functions, trigonometric functions, and imaginary numbers. The proof grew out of his studies of exponential functions. In simplest terms, an exponential function involves a number called the base and another number set to the upper right of the base, called the exponent, with the exponent indicating how many times the base is multiplied by itself to produce the value of the function (this notation was invented by Descartes). A simple example of an exponential function is
y
= 2
^x
, where 2 is the base and
x
the exponent. For any integer
x
, this gives rise to a finite series of terms and an integer product. For instance, 2
²
= 2 × 2 = 4, 2
³
= 2 × 2 × 2 = 8, 2
⁴
= 2 × 2 × 2 × 2 = 16, and so forth.

These integer pairs of numbers can be treated as belonging to a curve. On the infinite number of points on such a curve, only a few pairs are integers; the values on the dotted curve in between include decimals like 3.81 and even irrational numbers like
and π. What does it mean to multiply a number such as 2 by itself 2.31 or
or π times? For
rational numbers expressible in the form
p
/
q
, this had to mean the
p
th root of 2 to the
q
. For example, 2 to the power 3.81 (3 381/100) would be the 100th root of 2 to the power 381. For irrational numbers, it would mean filling in the missing point on that curve, which can be calculated as the limit of an infinite sequence. Thus 2
^π
is the limit of 2
³
, 2
^3.1
, 2
^3.14
, ..., 2
^3.1415926
, ...as we take more and more decimal values of π.

In
Chapter VII
of the
Introductio
, Euler showed that, in choosing the base for an exponential function, there were numerous mathematical advantages to selecting the number created by adding up the following infinite series:

The sum of these terms, Euler noted, is the irrational number 2.718281828459 ..., which ‘for the sake of brevity’ he will represent as
e
. This number is the base of natural logarithms and one of the most important mathematical constants. Euler then noted that if we use
e
as the base of our exponents, then the function
e
^x
can be calculated for any
x
by an infinite series:

This is known as the exponential function, an example of what is called the Taylor series.
⁹

In
Chapter VIII
, Euler turned to trigonometric functions. He began by reviewing the fact that if the diameter of a circle is 1 its circumference is an irrational number, 3.14159265...which ‘for the sake of brevity’ he says he will call π. He then described properties of the trigonometric functions, which associate to the measure of an angle in a right-angled triangle the numbers created by various ratios of the sides. The sine function, for instance, associates to the measure of one of the acute angles in a right-angled triangle the ratio of the length of the side opposite that angle to the length of the hypotenuse. The sine function can be generalized from acute angles
to arbitrary angles as follows: Draw a right-angled triangle
ABC
, with hypotenuse
BC
of length 1, in the (
x
,
y
)-plane so that vertex
B
lies at the origin (0,0), vertex
A
lies on the positive
x
-axis, and vertex
C
lies above the
x
-axis. Let
a
be the measure of the angle ∠
ABC
, measured counterclockwise from the positive
x
-axis. Then sin
a
is the ratio of the lengths
AC
/
BC
, but since
BC
= 1, sin
a
= length
AC
= the
y
-coordinate of the point
C
. If we take ‘
y
-coordinate of
C
’ as the definition for sin
a
(
a
is the measure of angle ∠
ABC
), then we have a definition that works for any angle: rotate
BC
through an angle
a
(starting from the positive
x
-axis and measured counterclockwise) and record the
y
-coordinate of
C
. Then the sine goes from 0 to 1 (at 90 degrees) back to 0 (180 degrees), thence to –1 (270 degrees), again to 0 (360 degrees), and repeats that pattern through successive cycles of 360 degrees, producing a pattern called the ‘sine wave’ familiar from oscilloscopes. The general cosine function is defined in the same way, except taking the
x
-coordinate of
C
as the value. As the angle varies, the cosine goes from 1 to 0 to –1 to 0 to 1, repeating just as does the sine function, making the same pattern as the sine function, but out of phase.

Euler then runs through various more or less obvious properties of sines and cosines, including the fact that, from a simple application of the Pythagorean theorem, (sin
x
)
²
+ (cos
x
)
²
= 1.

Continuing to summarize things that Newton and other predecessors knew, Euler next showed how trigonometric functions involving sines and cosines could also be expressed in terms of infinite series. For instance, the function sin
x
can be expressed as the following infinite sum of terms, which we’ll put in
this font
so that we can simply and easily follow the terms: