A Brief Guide to the Great Equations (4 page)

In the fourth century ad, the Greek geometer Pappus of Alexandria discovered a theorem that extended Euclid’s. A few centuries later, Arabic mathematician Thābit Ibn Qurra (836–901), working in Baghdad, provided several new proofs in revising an earlier Arabic translation of the
Elements
. Two and a half centuries later, the Hindu mathematician Bhaskara (b. 1114) was so enamored of the visual simplicity of the
Zhou Bi
proof that he redid it in the form of
a simple diagram, and instead of an explanation wrote a single word of instruction: ‘See.’

Later, Italian artist Leonardo da Vinci, Dutch scientist Christiaan Huygens, and German philosopher Gottfried Leibniz (1646–1716), all contributed new proofs. So did U.S. Congressman James Garfield, in 1876, before he became the twentieth U.S. president. Indeed, over a dozen
collections
of proofs of the Pythagorean theorem have appeared: in 1778, a list of thirty-eight was published in Paris, in 1880 a monograph appeared in Germany with forty-six proofs, while in 1914 a list of ninety-six proofs was published in Holland. The
American Mathematical Monthly
, the first general-interest mathematical magazine in the U.S., began publishing proofs in its first issues, starting in 1894. With some condescension, it stated that problem solving ‘is one of the lowest forms of mathematical research’, being applied and without scientific merit. Nevertheless, the magazine promised to devote ‘a due portion of its space to the solution of problems’ such as the Pythagorean theorem, to serve an educational purpose. ‘It [problem solving] is the ladder by which the mind ascends into the higher fields of original research and investigation. Many dormant minds have been aroused into activity through the mastery of a single problem.’
¹²
In 1901, after publishing about a hundred proofs, its editor abandoned the effort, announcing that ‘there is no limit to the number of proofs – we just had to quit.’

Diagram on the basis of which U.S. President James A. Garfield invented a proof.

One who refused to quit was a schoolteacher and subscriber from Ohio named Elisha S. Loomis – a mason – who had contributed some of the proofs. Loomis continued to collect them, many passed on by teachers of bright youngsters who knew of his interest. In 1927 (by then a college professor) Loomis published
The Pythagorean Proposition
, a book containing 230 proofs; in 1940, the 87-year-old Loomis published a second edition containing 370 proofs.
¹³
He dedicated both books to his Masonic lodge. Loomis divided the contents into geometric, algebraic, dynamic, and quaternion proofs. Most were geometric: number 31 was Huygens’s; 33, Euclid’s; 46, da Vinci’s; 225, Bhaskara’s; 231, Garfield’s; and the
Zhou Bi
’s was 243. Of the algebraic proofs, Leibniz’s was number 53. Loomis prized the way that the challenge of coming up with a new proof tested the mettle of students and, evidently fascinated by the process of proof, liked to signal interesting proofs, interesting people who had contributed proofs, or to commend youthful contributors.
¹⁴
He was disapproving of those who, he thought, disrespected the subject. He chastised some American geometry textbooks that omitted Euclid’s proof – possibly to show ‘originality or independence’ – remarking wryly that ‘the leaving out of Euclid’s proof is like the play of Hamlet with Hamlet left out.’
¹⁵
His final sentence of the second edition: ‘And the end is not yet.’
¹⁶

Loomis was correct; it wasn’t. The
Guinness Book of World Records
Web site, under ‘Most Proofs of Pythagoras’s Theorem’, recently named a Greek who, it is claimed, has discovered 520 proofs. By the time you read this, more have surely appeared.

All these proofs provoke two questions. The first is: Why isn’t one proof enough? We know why one application is not enough: the point of a rule is that it applies to many different circumstances. But proofs? A small number of proofs of the Pythagorean theorem generalize the theorem that Euclid proved, and thus extend what he did.
Most in Loomis’s collections, however, are not of that type. Nor do they make the result more certain than it already is. Their fascination lies in the scientific desire not merely to discover, but to view a discovery from as many angles as possible – to convert implicit possibilities, or merely hypothesized or assumed results, into actualities. Science aims to enrich the world, to increase the variety of its forms, to let the reality of the things in the world show themselves. As science progresses, the landscape of the world develops with it.

The second question is: Why all the attention to
this
particular theorem, which has fascinated amateurs and professionals for thousands of years? A part of the answer is surely personal biography: the Pythagorean theorem tends to be the first deep proof that each of us encounters, the first proof where – as Hobbes’s experience shows – it is not obvious what it is we are setting out to prove. It is the first journey of mathematical discovery where we find something genuinely new at the other end. But that must be only a small part of the answer, for we also learn other beautiful proofs early on, such as of the irrationality of the square root of two or of the infinity of primes. We also learn proofs that are similar to the Pythagorean theorem (for instance, Proposition 31 of Book VI of Euclid’s
Elements
), or much more powerful and useful than the Pythagorean theorem, without these attracting anywhere near the same degree of attention. A striking example of the latter is the law of cosines –
c
²
=
a
²
+
b
²
– 2
ab
cos θ – which covers all triangles, not just right-angled triangles, and relates the lengths of the sides to the cosine of one of the angles; the Pythagorean theorem is but a special case of the law of cosines. Yet this law communicates no special magic – partly because one has to know trigonometry to prove it – and one can hardly imagine a Hobbes becoming as transformed.

The full answer as to why the Pythagorean theorem seems magical is threefold: the visibility of the hypotenuse rule’s applications, the accessibility of the proof, and the way that actually proving the theorem seems to elevate us to contemplate higher truths and thus acquaint us with the joy of knowing.

First, the theorem characterizes the space around us, and we thus
encounter it not only in carpentry and architecture, physics and astronomy, but in nearly every application and profession. The Pythagorean rule for a distance in three-dimensional space – the diagonal of a shoebox, say – is the square root of
x
²
+
y
²
+
z
²
; for four-dimensional Euclidean space it’s the square root of
x
²
+
y
²
+
z
²
+
w
²
; in Minkowski’s interpretation of Einstein’s special theory of relativity, the four-dimensional space-time version is
x
²
+
y
²
+
z
²
1 (
ct
)
²
, where
c
is the speed of light. Suitably adapted, this formula enters into the equations of thermodynamics, in describing the three-dimensional motions of masses of molecules. It also enters into both the special and the general theory of relativity. (In the former, it is used to describe the path of light moving in one reference frame from the point of view of another, while in the latter, in a still more complex extension, it is used to describe the motion of light in curved, four-dimensional space-time.) And it is generalized still further in higher mathematics. In
The Pythagorean Theorem: A 4,000-Year History
, Eli Maor calls the Pythagorean theorem ‘the most frequently used theorem in all of mathematics.’
¹⁷
This is not only because of its direct use but also due to what Maor calls ‘ghosts of the Pythagorean theorem’ – the host of other expressions that derive, directly or indirectly, from it. An example is Fermat’s famous ‘last theorem’, finally proven in 1994, which asserts that no integers satisfy the equation
a
ⁿ
=
b
ⁿ
+
c
ⁿ
(all variables stand for positive integers) for any
n
greater than two. Though, being the denial rather than the assertion of an equality, Fermat’s last theorem cannot be put in the form of an equation.

Second, as Hobbes’s experience indicates, even though the Pythagorean theorem involves a bit of knowledge whose proof seems implausible at the beginning, it can be proved simply and convincingly even without mathematical training. This is one reason why philosophers and scientists from Plato onward use it as an emblematic demonstration of reasoning itself. In
On the World Systems
, Galileo cited Pythagoras’s experience proving the theorem to illustrate the distinction between certainty and proof – what we now call the context of discovery and the context of justification.
¹⁸
In
The Rules
for the Direction of the Human Mind
(Rule XVI), French philosopher and scientist René Descartes used the Pythagorean theorem to show the virtues of symbolic notation, which he was introducing into mathematics. G.W.F. Hegel viewed the proof as ‘superior to all others’ in the way it illustrates what it means for geometry to proceed scientifically, which for him meant showing how an identity contains differences.
¹⁹
German philosopher Arthur Schopenhauer, one of the few critics of the way Euclid proved the Pythagorean theorem, viewed that proof as emblematic for another reason. Mocking it as a ‘mousetrap’ proof that lures readers in and then ‘springs’ a trap on them, Schopenhauer thought it logically true but overtly complicated, did not like the fact that not all its steps were intuitive (he much preferred proofs that appealed to intuition), and maintained that Euclid’s proof is a classical illustration of a misleading demonstration. Indeed, he saw Euclid’s proof of the Pythagorean theorem as emblematic of all that was
wrong
with the philosophy of his day, for it emphasized the triumph of sheer logic over insight and educated intuition. Hegel’s philosophical system, for Schopenhauer, was in effect no more than one huge conceptual mousetrap.
²⁰

Third, the Pythagorean theorem makes accessible the visceral thrill of discovery. Whenever we prove it, we can hardly be said to be ‘learning’ anything, for we learned the hypotenuse rule as schoolchildren. But as the proof proceeds – as we set the problem in a bigger context, and as the little pieces begin to snap together with an awesome inevitability – we seem to be taken out of the here and now to someplace else, a realm of truths far more ancient than we, a place we can reach with a little bit of effort no matter where we are. In that place, this particular right-angled triangle is nothing special; all are the same and we do not have to start the proof all over again to be certain of it. Something lies behind this particular triangle, of which it is but an instance. The experience is comforting, even thrilling, and you do not forget it. The proof arrives as the answer to a puzzle in a language that you did not have beforehand, a language that arrives in that instant yet which, paradoxically, you sense
you already possessed. Without that moment of insight, the Pythagorean theorem remains a rule handed down authoritatively, rather than a proof gained insightfully.

All three components of the magic of the Pythagorean theorem are evident in the earliest known, most celebrated, and most complexly described story of a journey to the Pythagorean theorem. That occurs in Plato’s dialogue
Meno
, written about 385
BC
, or somewhat more than a century after Pythagoras and almost a century before Euclid’s
Elements
. It is the first extended illustration of the mathematical knowledge of ancient Greece that exists. In the
Meno
, Socrates coaxes a slave boy, ignorant of mathematics, to prove a particular instance of the theorem, one involving an isosceles right-angled triangle.

The principal participants are Socrates and Meno, a handsome youth from Thessaly. Meno is impatient, balks at difficult ideas, and likes impressive-sounding answers – a teacher’s nightmare. He’s been pestering Socrates about how it is possible to learn virtue. Socrates finds it difficult to get Meno’s mind going; his name, appropriately, means ‘stand fast’ or ‘stay put.’ The word ‘education’ means literally ‘to lead out.’ Socrates cannot lead Meno much of anywhere.

At one point, Meno throws up his hands and asks Socrates – in a famous query known as Meno’s paradox – how it is possible to learn anything at all. If you don’t know what you are looking for, you won’t be able to recognize it when you come across it – while if you do know you don’t bother to go looking for it. Meno is implying that it is fruitless even to try.