Why Beauty is Truth Page 3

  Babylonian base–60 numerals.

  The Babylonians were the earliest known culture to use something akin to our positional notation. But there was one significant difference. In the decimal system, every time a digit is moved one place to the left, its numerical value is multiplied by ten. So 20 is ten times 2, and 200 is ten times 20. In the Babylonian system, each move to the left multiplies a number by sixty. So “20” would mean 2 times 60 (120 in our notation) and “200” would mean 2 times 60 times 60 (7200 in our notation). Of course, they didn’t use the same “2” symbol; they wrote the number “two” using two copies of a tall, thin wedge symbol, as shown in the figure above. Numbers from one to nine were written by grouping that many copies of the tall wedge. For numbers greater than nine, they added another symbol, a sideways wedge, which denoted the number ten, and they used groups of these symbols to denote twenty, thirty, forty, and fifty. So, for instance, our “42” was four sideways wedges followed by two tall wedges.

  For reasons we can only guess at, this system stopped at 59. The Babylonians did not group six sideways wedges to make 60. Instead, they reverted to the tall thin wedge previously used to mean “one,” and used it to mean “one times sixty.” Two such wedges meant 120. But they might also mean “two.” Which meaning was intended had to be inferred from context, and from the position of the symbols relative to each other. For example, if there were two tall wedges, a space, and two more tall wedges, then the first group meant “one hundred twenty” and the second “two”—much as the two symbols 2 in our 22 mean twenty and two respectively.

  This method extended to much larger numbers. A tall wedge could mean 1, or 60, or 60 × 60 = 3,600, or 60 × 60 × 60 = 216,000, and so on. The three bottom groups in the figure indicate 60 × 60 + 3 × 60 + 12, which we would write as 3,792. A big problem here is that the notation has some ambiguities. If all you see is two tall wedges, does this mean 2, 60 × 2, or 60 × 60 × 2? Does a sideways wedge followed by two tall ones mean 12 × 60 + 2 or 12 × 60 × 60 + 2, or even 10 × 60 × 60 + 2 × 60? By Alexander the Great’s time, the Babylonians had removed these ambiguities by using a pair of diagonal wedges to indicate that no number occurred in a given slot; in effect, they had invented a symbol for zero.

  Why did the Babylonians use this sexagesimal system rather than the familiar decimal system? They may have been influenced by a useful feature of the number 60: its large variety of divisors. It is divisible exactly by the numbers 2, 3, 4, 5, and 6. It is also divisible by 10, 12, 15, 20, and 30. This feature is rather pleasant when it comes to sharing things, such as grain or land, among several people.

  A final feature may well have been decisive: the Babylonian method of measuring time. It seems that they found it convenient to divide a year into 360 days, although they were excellent astronomers and knew that 365 was closer, and 365¼ closer still. The lure of the arithmetical relationship 360 = 6 × 60 was too strong. Indeed, when referring to time, the Babylonians suspended the rule that moving symbols one slot to the left multiplied their value by sixty, and replaced that by six, so that what should have meant 3,600 was actually interpreted as 360.

  This emphasis on 60 and 360 still lingers today, in our use of 360 degrees in a full circle—one degree per Babylonian day—and in the 60 seconds in a minute and 60 minutes in an hour. Old cultural conventions have incredible staying power. I find it amusing that in this age of spectacular computer graphics, moviemakers still date their creations in Roman numerals.

  Nabu-Shamash would have learned all of this, except the “zero” sign, at an early stage of his education. He would have become adept at impressing thousands of tiny cuneiform wedges into damp clay at speed. And just as today’s students grapple with the transition from whole numbers to fractions and decimals, Nabu-Shamash would eventually have been faced with the Babylonian method for representing numbers like one-half, or one-third, or the more complicated subdivisions of unity dictated by the brutal realities of astronomical observations.

  To avoid spending whole afternoons drawing wedges, scholars represent cuneiform numbers with a mixture of old and new. They write the decimal numbers depicted in the successive groups of wedges, using commas to separate them. So the final group in the figure would be written 1,3,12. This convention saves a lot of expensive typesetting and is easier to read, so we’ll go along with the scholars.

  How would a Babylonian scribe have written the number “one-half”?

  In our own arithmetic, we solve this problem two different ways. We either write the number as a fraction, ½, or introduce the famous “decimal point” and write it as 0.5. The fractional notation is more intuitive and came earlier historically; decimal notation is more difficult to grasp, but it lends itself better to computation because the symbolism is a natural extension of the “place-value” rules for whole numbers. The symbol 5 in 0.5 means “5 divided by 10,” and in 0.05 it means “5 divided by 100.” Moving a symbol one place to the left multiplies it by 10; moving it one place to the right divides it by 10. All very sensible and systematic.

  As a result, decimal arithmetic is just like whole-number arithmetic, except that you have to keep track of where the decimal point goes.

  The Babylonians had the same idea, but in base 60. The fraction ½ should be some number of copies of the fraction 1/60. Clearly the right number is 30/60, so they wrote “one-half” as 0;30, where scholars use the semicolon to denote the “sexagesimal point,” which in cuneiform notation was again a matter of spacing. The Babylonians managed some fairly advanced calculations: for example, their value for the square root of 2 was 1;24,51,10, which differs from the true value by less than one part in a hundred thousand. They used this precision to good advantage in both theoretical mathematics and astronomy.

  The most exciting technique that Nabu-Shamash would have been taught, as far as our central theme of symmetry is concerned, is the solution of quadratic equations. We know quite a lot about Babylonian methods for solving equations. Of the roughly one million Babylonian clay tablets known to exist, about five hundred deal with mathematics. In 1930, the orientalist Otto Neugebauer recognized that one of these tablets demonstrated a complete understanding of what today we call quadratic equations. These are equations that involve an unknown quantity and its square, together with various specific numbers. Without the square, the equation would be called “linear,” and such equations are the easiest to solve. An equation that involves the cube of the unknown (multiply it by itself, then multiply that by the unknown again) is called “cubic.” The Babylonians seem to have possessed a clever method for finding approximate solutions to certain types of cubic equation, based on numerical tables. All that we are certain of, however, are the tables themselves. We can only infer what they were used for, and cubic equations are most likely. But the tablets Neugebauer studied make it plain that the Babylonian scribes had mastered the quadratic.

  A typical one, which dates back about 4000 years, asks, “Find the side of a square if the area minus the side is 14,30.” This problem involves the square of the unknown (the area of the square) as well as the unknown itself. In other words, it asks the reader to solve a quadratic equation. The same tablet rather offhandedly provides the answer: “Take half of 1, which is 0;30. Multiply 0;30 by 0;30, which is 0;15. Add this to 14,30 to get 14,30;15. This is the square of 29;30. Now add 0;30 to 29;30. The result is 30, the side of the square.”

  What’s going on here? Let’s write the steps in modern notation.

  • Take half of 1, which is 0;30.

  ½

  • Multiply 0;30 by 0;30, which is 0;15.

  ¼

  • Add this to 14,30 to get 14,30;15.

  870¼

  • This is the square of 29;30.

  870¼ = (29½) × (29½)

  • Now add 0;30 to 29;30.

  29½ + ½

  • The result is 30, the side of the square.

  30

  The most complicated step is the fourth, which finds a number (it is 29
½) whose square is 870¼. The number 29½ is the square root of 870¼. Square roots are the main tool for solving quadratics, and when mathematicians tried to use similar methods to solve more complicated equations, modern algebra was born.

  Later we will interpret this problem using modern algebraic notation. But it is important to realize that the Babylonians did not employ an algebraic formula as such. Instead, they described a specific procedure, in the form of a typical example, that led to an answer. But they clearly knew that exactly the same procedure would work if the numbers were changed.

  In short, they knew how to solve quadratics, and their method—though not the form in which they expressed it—was the one we use today.

  How did the Babylonians discover their method for solving quadratics? There is no direct evidence, but it seems likely that they came across it by thinking geometrically. Let’s take an easier problem that leads to the same recipe. Suppose we find a tablet that says, “Find the side of a square if the area plus two of the sides is 24.” In more modern terms, the square of the unknown plus twice the unknown equals 24. We can represent this question as a picture:

  Geometric picture of a quadratic equation.

  Here the vertical dimension of the square and rectangle to the left of the equal sign corresponds to the unknown, and the small squares are of unit size. If we split the tall rectangle in half and glue the two pieces onto the square, we get a shape like a square with one corner missing. The picture suggests that we should “complete the square” by adding in the missing corner (shaded square) to both sides of the equation:

  Completing the square.

  Now we have a square on the left and 25 unit squares on the right. Rearrange those into a 5 × 5 square:

  Now the solution is obvious.

  Thus the unknown plus one, squared, equals five squared. Taking square roots, the unknown plus one equals five—and you don’t have to be a genius to deduce that the unknown is four.

  This geometric description corresponds precisely to the Babylonian method for solving quadratics. The more complicated example from the tablet uses exactly the same recipe. The tablet only states the recipe and doesn’t say where it comes from, but the geometric picture fits other circumstantial evidence.

  2

  THE HOUSEHOLD NAME

  Many of the greatest mathematicians of the ancient world lived in the Egyptian city of Alexandria, a city whose origins lie among five substantial oases to the west of the Nile, out in the Western Desert. One of them is Siwa, notable for its salt lakes, which grow during the winter and shrink in the summer heat. The salt contaminates the soil and creates major headaches for archaeologists because it is sucked up into the ancient stone and mud-brick remains and slowly destroys the fabric of the buildings.

  The most popular tourist site in Siwa is Aghurmi, a former temple dedicated to the god Amun. So holy was Amun that his main aspect was entirely abstract, but he became associated with a more physical entity, the provenance of the god Re, the Sun. Constructed during the 26th Dynasty, the temple of Amun at Siwa was the home of a famous oracle that is particularly associated with two major historical events.

  The first is the destruction of the army of Cambyses II, a Persian king who conquered Egypt. It is said that in 523 BCE, planning to use the oracle of Amun to legitimate his rule, Cambyses sent a military force into the Western Desert. The army reached Bahariya Oasis but was destroyed in a sandstorm on its way to Siwa. Many Egyptologists suspect that the “lost army of Cambyses” may be mythical, but in 2000 a team from Helwan University, looking for oil, found bits of cloth, metal, and human remains in the area, and suggested that these might be the remains of the lost army.

  The second event, two centuries later, is historical fact: a fateful visit to Siwa by Alexander the Great, who was after exactly the same thing as Cambyses.

  Alexander was the son of King Philip II of Macedon. Philip’s daughter Cleopatra of Macedon married King Alexander of Epirus, and Philip was assassinated during the proceedings. The killer may have been Philip’s homosexual lover Pausanias, who was upset because the king had not done anything about some complaint or other that Pausanias had made. Or the murder may have been a Persian plot set up by Darius III. If so, it backfired, because the Macedonian army immediately proclaimed Alexander king, and the 20-year-old monarch famously went on to conquer most of the known world. Along the way, in 332 BCE, he conquered Egypt without a fight.

  Intent on cementing this conquest with an endorsement of his credentials as pharaoh, Alexander made a pilgrimage to Siwa to ask the oracle whether he was a god. He visited the oracle alone, and on his return announced its verdict: yes, the oracle had confirmed that he truly was a god. This verdict became the primary source of his authority. Later, rumors claimed that the oracle had revealed him to be the son of Zeus.

  It is not clear whether the Egyptians were convinced by this rather flimsy evidence or whether, given Alexander’s control of a substantial army, they found it prudent to go along with his story. Perhaps they were fed up with the rule of the Persians and considered Alexander the lesser of two evils—he had been welcomed with open arms by the former Egyptian capital of Memphis for precisely that reason. Whatever the truth behind the history, from that time on, the Egyptians venerated Alexander as their king.

  On the way to Siwa, fascinated by an area of the country lying between the Mediterranean Sea and the lake that came to be known as Mareotis, Alexander decided to have a city constructed there. The city, which he modestly named Alexandria, was designed by Donocrates, a Greek architect, after a basic plan sketched by Alexander himself. The city’s birth has been dated by some to 7 April 331 BCE; this date is disputed by others, but it must be close to 334 BCE. Alexander never saw his creation; his next visit to the area was to be buried there.

  So, at least, goes the time-honored legend, but the truth is probably more complex. It now appears that much of what later became Alexandria already existed when Alexander arrived. Egyptologists discovered long ago that many inscriptions are not all that trustworthy. The great Temple at Karnak, for instance, is riddled with cartouches of Ramses II. But much of it was actually constructed by his father, Seti I, and traces—not always faint—of the father’s inscriptions can be seen beneath those carved for Ramses. Such usurpation was commonplace, and was not even considered disrespectful. In contrast, “defacing” a predecessor’s reliefs—hacking out the pharaoh’s face—was most definitely disrespectful, intentionally depriving that predecessor of his place in the afterlife by destroying his very identity.

  Alexander had his name carved all over the buildings of ancient Alexandria. He had his name carved, so to speak, on the city itself. Where other pharaohs usurped the odd building or monument, Alexander usurped an entire city.

  Alexandria became a major seaport, connected by branches of the Nile and a canal to the Red Sea and thence to the Indian Ocean and the Far East. It became a center of learning, with a celebrated library. And it was the birthplace of one of the most influential mathematicians in history: the geometer Euclid.

  We know much more about Alexander than we do about Euclid—even though Euclid’s long-term influence on human civilization was arguably greater. If there can be such a thing as a household name in mathematics, “Euclid” is it. Although we know little about Euclid’s life, we know a lot about his works. For several centuries, mathematics and Euclid were pretty much synonymous throughout the Western world.

  Why did Euclid become so well known? There have been greater mathematicians, and more significant ones. But for close to two thousand years Euclid’s name was known to every student of mathematics across the whole of Western Europe, and to a lesser extent in the Arab world as well. He was the author of one of the most famous mathematics texts ever written: the Elements of Geometry (usually shortened to Elements). When printing was invented, this work was among the first books to appear in printed form. It has been published in over a thousand different editions, a number exceeded only by t
he Bible.

  We know slightly more about Euclid than we do about Homer. He was born in Alexandria around 325 BCE and died in about 265 BCE.

  Having said that, I am uncomfortably aware that I already need to backtrack. That Euclid existed and was sole author of the Elements is only one of three theories. The second is that he existed but did not write the Elements, at least not on his own. He may have been the leader of a team of mathematicians who collectively produced the Elements. The third theory—far more contentious but within the bounds of possibility—is that the team existed, but much like the group of mostly French, mostly young mathematicians who wrote under the name “Nicolas Bourbaki” in the mid-twentieth century, they took “Euclid” as a collective pseudonym. Nevertheless, the most likely story seems to be that Euclid existed, that he was one person, and that he composed the Elements himself.

  This does not mean that Euclid discovered all of the mathematics contained within his book’s pages. What he did was to collect and codify a substantial part of ancient Greek mathematical knowledge. He borrowed from his predecessors and he left a rich legacy for his successors, but he also stamped his own authority on the subject. The Elements is generally described as a geometry book, but it also deals with number theory and a kind of prototypical algebra—all of it presented in geometrical guise.

  Of Euclid’s life we know very little. Later commentators included a few snippets of information in their works, none of which modern scholars can substantiate. They tell us that Euclid taught in Alexandria, and it is usual to infer that he was born in that city, but we don’t actually know that. In 450 AD, in an extensive commentary on Euclid’s mathematics written more than seven centuries after his death, the philosopher Proclus wrote: