A Brief Guide to the Great Equations (3 page)

The term, ‘Pythagorean theorem’, is popularly used to refer to two different things: a rule and a proof. The rule is simply a fact. It states an equality between the lengths of the sides of a right-angled triangle: the length of the hypotenuse squared (
c
²
) is equal to the sum of the squares of the two other sides: (
a
²
+
b
²
). That rule has a practical value: it allows us, for instance, to calculate the length of that hypotenuse if we know the lengths of the two sides. The proof is different. It’s the demonstration of how we know this fact to be true.

It is confusing that this phrase can refer to both. It’s a confusion embedded in the word ‘theorem.’ The word can mean a result that is (or is assumed to be) proven. It comes from the Greek for ‘to look at’ or ‘contemplate’, and has the same root as ‘theatre.’ When people like Hobbes see the Pythagorean theorem, they can pay attention
to two very different things: to the product, rule, or thing proven – the hypotenuse rule – or to the process, the proving, or the way it is known.

The rule is extremely important, crucial to describing the space around us. It is invaluable to carpenters, architects, and surveyors in small and large-scale construction projects. This is one reason Freemasons – the esoteric organization said to have been born in medieval stonemason guilds – adopted the Pythagorean theorem as a symbol. One piece of Masonic literature cites the Pythagorean theorem as ‘containing or representing the truth upon which Masonry is based, and the basis of civilization itself’,
⁵
and a simplified version of the diagram accompanying Euclid’s proof, called the ‘Classic Form’, is often emblazoned on carpets in Masonic lodges. The rule characterizes celestial spaces as well, thus is essential to navigation and astronomy.

This rule was known long before Euclid or even Pythagoras. The fact that sides of specific lengths – 3, 4, and 5 units, say, or 6, 8, and 10 – create a ‘set square’ with a right-angled triangle between the two shorter sides was an empirical discovery known to ancient craftsmen. Such trios of numbers are called ‘Pythagorean triplets’, and their independent discovery in different lands is not surprising given their simplicity and practical importance. Another ancient discovery seems to have been the rule
c
²
=
a
²
+
b
²
for such triplets. A Babylonian cuneiform tablet of about 1800
BC
, known as Plimpton 322 after the collection in which it resides at Columbia University, contains a table of fifteen rows of Pythagorean triplets. The tablet was evidently a trigonometric table or teaching aid for the rule to figure out hypotenuses of right-angled triangles. It contains no variables, but it seems to have been intended to communicate the rule via a list of examples.
⁶

A Babylonian cuneiform tablet of about 1800 BC, known as Plimpton 322 after the collection in which it resides at Columbia University. The tablet, evidently a trigonometric table or teaching aid for the rule to figure out hypotenuses of right-angled triangles, contains a table of fifteen rows of Pythagorean triplets.

The rule was also known in ancient India. Applications of it are found in the
Śulbasūtras
, the texts that accompany the Sutras or ‘sacred teachings’ of the Buddha, which seem to have been written between 500 and 100
BC
but clearly pass on knowledge of much earlier times. In their instructions for constructing ritual areas they display considerable geometrical knowledge, though it is expressed informally and approximately, and without much justification.
⁷

The earliest existing Chinese writing on astronomy and mathematics, the
Zhou Bi Suan Jing
(‘Gnomon of the Zhou’, containing texts dating from the first century
BC
but whose contents are said to be centuries earlier), likewise exhibits knowledge of the rule. One application is in a calculation of how far the sun is from the earth. The reasoning process involves a bamboo tube and its shadow, and assumes that the earth is flat; the
Zhou Bi
is famous among historians of science for being ‘the only rationally based and fully mathematicised account of a flat earth cosmos.’
⁸
The earliest extant version contains an often-reproduced diagram against a chessboardlike background from which one can readily see that the area of the square built on the hypotenuse is the same as the combination of the areas on the other two sides – but this almost certainly dates from a third century ad source, long after Euclid.

Diagram from a late edition of the
Zhou Bi
. The characters refer to the colors of the squares.

The Babylonian tablet, the Indian
Śulbasūtras
, and the Chinese
Zhou Bi
each exhibit knowledge of the rule as part of a body of mathematical knowledge applied to some other purpose: educational in the case of Plimpton 322, religious in the case of the
Śulbasūtras
, astronomical in the case of the
Zhou Bi
. In these and in other ancient texts the rule is presented without explicit justification, mainly as a way of finding distances and checking results, though occasionally with more formality.

Indeed, the Pythagorean theorem is surely unique among mathematical landmarks for the range of colorful practical illustrations, ranging from prosaic to poetic, over its thousands of years of history, involving the dimensions of fields, canals, clotheslines, footpaths, roads, and aqueducts. From an Egyptian manuscript: ‘A ladder of 10 cubits has its foot 6 cubits from a wall; how high does it reach?’ From a medieval Italian manuscript: ‘A spear 20 ft. long leans against a tower. If its end is moved out 12 ft., how far up the tower does the spear reach?’ An Indian text asks readers to compute the depth of a pond, swimming with red geese, if the tip of a lotus bud were about 9 inches above the water, but was blown over by the wind – its stem fixed to the bottom – and vanished beneath the water at a distance of about 40 inches. These kinds of exercises make mathematics fun!

The rule has become a model piece of knowledge, and knowing it is often symbolic of human intelligence itself. At the end of the movie the
Wizard of Oz
, the Scarecrow – to show he truly does have a brain – states a botched version: ‘The sum of the square roots of any two sides of an isosceles triangle is equal to the square root of the remaining side.’ The levity is perfect, for it spares us in the audience from really having to follow it, and keeps what’s happening in the realm of fairy tale.

But proving a rule is much different from just knowing it. A proof demonstrates the general validity of a result based on first principles – for its own sake, not linked with a practical end, and with the focus less on the result than on how one arrives at it; on the process by which we come to trust it. A proof recounts the journey by which we know an equation. To provide the proof of a rule therefore involves a different perspective on mathematics than just stating the rule. For a proof is not an assertion of authority but an acknowledgement of intellectual democracy. It does not simply pass on a piece of wisdom from one’s precursors as a tour de force of intellect, a stroke of genius. It does not say, ‘This is a fact!’ or ‘This is how a genius told us to do it.’ Instead, the proof of a result says that the journey is something
anyone
can take, in principle at least, thanks to the matrix of mathematical definitions and concepts that we already possess. It therefore says in effect, ‘Follow this, and you’ll see that we know all the steps how to get there
already
!’ Giving the proof of a rule therefore establishes a landmark that anyone can get to by following the path indicated, and that one can trust to orient oneself while making further journeys in unexplored territory. Proofs of key equations transform mathematics from a complex terrain into a landscape by erecting landmarks. The rest of mathematics is still present, but in the background.

Although the first proof of the hypotenuse rule is traditionally ascribed to Pythagoras (ca. 569–475
BC
), the claim that his proof was the earliest was first advanced half a millennium later, and is almost certainly untrue.
⁹
The idea of proof seems to have originated in ancient Greece, and took hundreds of years to develop. It culminated in Euclid’s
Elements
, which presents mathematical knowledge entirely in the form of explicit, formal proofs. The proof of the Pythagorean theorem is the next-to-last one of Book I. In a right-angled triangle, the square on the side opposite the right angle is equal to the sum of the squares on the other two sides. Proposition
48, the last proof of Book I, is the converse: if the square on one side of a triangle is equal to the sum of the squares on the other two sides, it is a right-angled triangle. The proof is as follows: Build a square on each side of a right-angled triangle. Draw a line from the vertex of the right angle, perpendicular to the hypotenuse, to that square’s far side. This divides the big square into rectangles. Each rectangle turns out to be the same size as one of the squares: the sum of the smaller squares thus equaling the area of the square on the hypotenuse. Interestingly, Euclid’s proof is associated with the distinctive image created by its lines, and has been called the windmill, peacock, or bridal chair proof after fanciful images that it has been taken to suggest.

A classic diagram illustrating the proof in Euclid’s
Elements
.

Every great discovery seems to generate the irresistible urge to scour through records to see if anyone else discovered it earlier, discovered it but did not write it down, or brushed up against it without discovering it. The Pythagorean theorem, as we seem forever fated to call it, was no exception. For historians, showing how close a people came to proving the Pythagorean theorem appears to be a way to try to show how advanced that civilization was – and claims have been made for the Babylonian, Indian, and Chinese discovery of the Pythagorean theorem based on Plimpton 322, the
Śulbasūtras
, the
Zhou Bi
, and other texts.
¹⁰
But in the process, it is easy and tempting to confuse or ignore the difference between the Pythagorean theorem, the empirically determined rule, and the Pythagorean theorem, the proof of the equation.

Occasionally, humans have taken the journey on their own, discovering the Pythagorean theorem without the aid of teachers. One is the French mathematician and philosopher Blaise Pascal, whose father forbade any discussion of mathematics around the household, afraid that the subject might distract his child from the all-important studies of Greek and Latin. But the young Pascal began to explore geometry with the aid of a piece of charcoal, in the process discovering many of the proofs codified in Euclid’s
Elements
, including the Pythagorean theorem.
¹¹

It is also possible to discover new proofs of the theorem. For if the Pythagorean theorem is unique among mathematical landmarks for the range of its applications and examples, it is also unique for the range of ways that it has been proven. Most proofs are based on the same axioms, but follow different paths to the climax. Many – especially the earliest proofs such as Socrates’, Euclid’s, and in the later Chinese manuscript
Zhou Bi
– are geometrical, where
a
,
b
, and
c
refer to lengths of various sides of shapes, and the proof proceeds by manipulating the shapes and showing something about their areas. Other proofs are algebraic, or based on more complex mathematics where the numbers refer to abstract things, and can even refer to vectors. Some so-called proofs, though, assume results that are proven by the Pythagorean theorem, and so are really circular arguments. The algebraic approach – which the Babylonians understood – is what produced the
c
²
=
a
²
+
b
²
version of the rule.