Deciphering Zero

Source

Ah, numbers. We use them every day. We also know that there are different sets of numbers. We have Arabic numerals for everyday use, and we have Roman numerals for special events, like Superbowls and the year a movie came out.

Roman numerals were used exclusively in the Middle Ages for a long time. They were inconvenient for large sums, but Western Europe had no other option. Eventually, however, along came so-called Arabic numerals. They were introduced by Leonard of Pisa, better known today as Fibonacci. Fibonacci's Liber abaci ("Book of calculating"; it wasn't about the abacus) introduced Arabic numerals (which probably came originally from India) and a decimal system, with "places" for ones, tens, hundreds, and so forth. With these new numbers came something very new and strange to them: what we call "zero."

Of course they did not call it "zero" when it was first introduced. The Arabic word was ṣifr, or zephir, which when filtered through Old French became cifre and eventually the English cipher. John Sacrobosco (c.1195 - c.1256; mentioned here) in The Craft of Numbering explained:

A cipher tokens nought, but he makes the figure that comes after to betoken more than he should; thus 10. Here the figure of 1 betokens 10, and if the cipher were away, ..., he should betoken only 1, for then he should stand in the first place. [paraphrased]

The concept of the zero was so mysterious, the new number system so different and difficult to master (the British Exchequer clung to Roman numerals—at least partially—until the mid-17th century), that using them seemed like a secret code. The words encipher and decipher grew from the ability to make and read this code and understand the zero.

Scholar of the Supernatural

[I am on a brief vacation, so here is a post from the past. This post first appeared 23 August 2012.]

In Dante's Inferno, the eighth circle is reserved for sorcerers, astrologers, and false prophets. There the narrator sees Michael Scot. You might think, if someone were so well-known after his death, that we would know more about him. Well, we know a little, but we have some cool stories.

Michael Scot, depicted here tearing up the Scriptures.*

Michael Scot was no doubt born in Scotland, although other locations (like Salerno and Toldeo) have tried to claim him. Dates of 1175-c.1232 seem to work for what little we know of his life. We know that Pope Honorius wrote to Stephen Langton on 16 January 1223, urging him to grant Scot a religious position, and that Honorius himself nominated Scot for Archbishop of Cashel. If Scot was educated sufficiently to be offered these positions, he would not have lived until 1290, which is the date Sir Walter Scott offers for his death. (Scott was confusing Scot with a Sir Michael Scot who lived later.)

Scot turned down the position in Cashel; it looks like he did hold benefices in Italy, however, spending time in Bologna and Palermo before going to Toledo in Spain. It was probably in Spain that he learned Arabic, which helped get him invited to the court of Holy Roman Emperor Frederick II. Besides translating texts for Frederick, he was a court astrologer, saying of the work:

Every astrologer is worthy of praise and honor, since by such a doctrine as astrology he probably knows many secrets of God, and things which few know.

This was not likely to endear him to the Roman Catholic Church.

Although he was known in his lifetime as a brilliant Aristotelian scholar, and Fibonacci's Liber Abaci was dedicated to him, his books on alchemy and astrology and the occult sciences earned him a reputation for magic. A Bronze Age circle of stones in northwest England called "Long Meg and Her Daughters" was supposedly a coven of witches turned to stone by Scot. Other stories have him hosting feasts served by invisible spirits. Boccaccio refers to him in the Decameron as a magician. It is also told (long after the fact) that he predicted he would die from a small stone falling on his head from a great height. He always wore an iron cap to prevent it, but he removed the cap when entering a church one day (more not to stand out than for reverence of God, we are told), and a small stone of the size he predicted fell on his head. He picked up the stone, recognized that his prophecy was coming true, put his affairs in order, and died of the head wound shortly after! His reputation (helped by the dearth of facts) has made him a prime subject for fiction right up to the present day.

*From a fresco painted between 1366 and 1388 by Andrea Bonaiuti in the Cappellone degli Spagnoli of Santa Maria Novella in Florence. St. Dominic preaches to the crowd.

Al-Gebra

Recent stamp commemorating al-Khwārizmī

Algebra—a method for doing computations using non-number symbols (such as "x" and "y") in equations—keeps coming up in conversations around me lately, and I realize I haven't addressed its Medieval origins yet.

Perhaps I should say Classical origins, since the Babylonians developed an arithmetical system for dealing with linear and quadratic equations. The Greeks, Chinese, and Egyptians used a kind of geometric algebra in the centuries BCE. A Greek mathematician in Alexandria in the 3rd century CE, Diophantus, is sometimes called the "Father of Algebra" based on his series of books, Arithmetica, that deal with solving algebraic equations.

Diophantus has a rival for that title, however.

An Arab mathematician named Muhammad ibn-Mūsā al-Khwārizmī (c.780-850) wrote a book in Baghdad in 825 called Kitab al-jabr wal-muqubala ["The book of restoration and balancing"]. Specifically, the process of turning the equation x - 2 = 12 to x = 14 was called jabr because one was "restoring" the x. The process of turning x + y = y + 7 into x = 7 was muqubala because one was "balancing" the two sides.

The word al-jabr, "restoration," eventually became the sole label for this method of mathematics.*

A Latin translation of his work was circulating in Europe in the 12th century. Fibonnaci is believed to have been exposed to Arabic mathematics, which might be why he was able to come close to solving the equation x³ + 2x² + cx = d.

So al-Khwārizmī gets the title "Father of Algebra" because the branch of mathematics is named for his book describing it. He also gets the honor of naming a different mathematical term: his name was Latinized into Algorithmus, from which we derive the term "algorithm."

*Interestingly, this word's non-mathematical definition of "restoration" made it suitable for other uses.  It made its way into European parlance via Arabic, and "algebrista" became a title for a "bone-setter." The term could also apply to barbers, because they did bone-setting as well as blood-letting. (The term for a blood-letter was "sangrador.")

Jacob Anatoli

Daily Medieval has frequently mentioned the importance of Arabic texts in the transmission of knowledge to Western Europe. Arabic, however, was not a commonly known language, and Arabs did not have a strong presence in Western Europe. Arabic culture often brushed up against Latin culture in the southern Mediterranean, as mentioned Salerno, or when a scholar such as Michael Scot made it a point to learn Arabic. Scot probably had help in the form of Jacob Anatoli.

Jacob ben Abba Mari ben Simson Anatoli (c.1194-1256) grew up in southern France, and gained such a reputation for scholarship that he was invited to Naples by Frederick II, who gathered several other academics to his court, such as Scot and Fibonacci. Anatoli became known for his translations of Arabic texts into Hebrew, and he very likely aided Michael Scot in his Arabic translations. Roger Bacon explains that Scot was aided by a Jew named Andreas, and some scholars believe "Andreas" to be a misunderstanding of "Anatoli."

Of his non-translations, the greatest work is the Malmad ha-Talmidim (the title is a pun, being interpreted either "Teacher of the Students" or "Goad to the Students"). The Malmad shows a wide range of knowledge, incorporating the Old Testament and Jewish commentators, but also the New Testament, Aristotle, Plato, and Averroes. His egalitarian approach to Christian and Muslim matters was refreshing, but Judaism still had special status; he wrote "the Greeks had chosen wisdom as their pursuit; the Romans, power; and the Jews, religiousness." He tells us that a non-Jew who seeks religious Truth should be respected by Judaism and not mocked.

Anatoli extended this intellectual courtesy to Frederick II, incorporating remarks by the emperor in his works. He also mentions a Christian whom Anatoli considers a second master (after Anatoli's own mentor, Samuel ibn Tibbon); this "master" has been equated to Michael Scot.

As for his Arabic translations, Anatoli's crucial contribution was exposing the West to the work of Averroes, one of two Arab scholars (the other was the medical expert Avicenna) whose work is considered fundamental to the Middle Ages.We'll look at Averroes tomorrow.

The Abacus

After mentioning Fibonacci's work, the Liber Abacus, it occurred to me that the place of the abacus in history deserved a little attention.

The Salamis Tablet, 300 BCE

Like the etymology for book, the word "abacus" does not start out to "mean" a frame with wires and beads. The word "abacus" first enters print in the English language in 1387. The Latin word from which it is lifted refers to a sandboard, a counting board covered in sand that allows you to draw with your finger. Latin took the word from the Greek abax, abakos, a board covered with sand for the purpose of drawing figures and calculating. At some point, the sand was replaced with counters of wood or stone that were moved from column to column for calculations, and the board itself was designed to facilitate calculations

In 1846, on the island of Salamis, a white marble counting board was discovered. The Salamis Tablet has been studied extensively, and one scholar has made a video of its proper use.

But when did abacus come to refer to the wooden frame with beads on wires? A reconstruction of a 1st century Roman abacus shows a board with grooves to keep the round beads in line. Visually, it resembles the abacus with which we are familiar. Gerbert of Aurillac (c.946-1003), one of the most influential scientific minds of his era, pushed the use of the abacus as a method of calculating much more swiftly than when using Roman numerals. He was able to promote its use even more when he became Pope Sylvester II.

The abacus in the form we think of it seems to come from China in the 2nd century BCE. Called a suanpán ("counting tray"), it was built with rods that held beads, 2 on an upper deck and 5 on a lower. Now called the "2/5 abacus," the two decks allowed the user to use larger numbers without adding 1+1+1+1, etc. Other versions had different numbers of rods, and different numbers of beads on them.

Abacus showing 87,654,321

Visually, it is very much like the Roman abacus mentioned above. Commerce between Rome and China was not unknown, but a direct influence cannot be proven. Still, the wooden-framed Chinese suanpán was so much like the Roman abacus that it was natural that the West would use the same name for the new device. In fact, no one type of the many objects used for calculating universally replaced the others. Counting boards of clay or wax were used well past the Middle Ages. In fact, until just after 2000, some accounting schools in China required proficiency in using the bead abacus.

Fibonacci

While the foundation of the Tower of Pisa was being being laid, a man was born nearby who developed math skills that might have helped the ill-fated architectural wonder.
Leonardo Pisano ("of Pisa") (c.1175-c.1250) was the son of Guglielmo Bonacci. Although known as "Leonardo Pisano" during his lietime, he signed his name "Bonacci" on his writings; an 1838 writer referred to him as "Fibonacci"—short for filius Bonacci ("son of Bonacci")—and the name stuck with his modern fans.

His father was a customs officer in Algeria, and between living there and traveling around the coast of the Mediterranean, Leonardo grew up exposed to education outside of the Greco-Roman/Western European tradition. He recognized the advantages of the Hindu-Arabic system of numbers over using Roman numerals, and worked to popularize it in Europe, starting with his 1202 work Liber Abaci ("Book of the Abacus" or "Book of Calculating"). In it, he presented to Europe the decimal system by which we all learn the four basic mathematical functions in school.

Calculating with the four functions in a decimal system.

The decimal system, with its "places" for ones and tens and hundreds, etc.,  was much "neater" than the system of Roman numerals and included a digit for "zero." Roman numerals had no "zero," and the words null or nihil were used to express a lack of something. The Roman tradition had great difficulty with the concept of "nothing" in math, because it seemed inappropriate to have a "something" that would indicate a "nothing."

If people have heard of Fibonacci in the present day, it is usually because of a particular sequence of numbers associated with him. In Chapter 12 of the Liber Abaci, he presents a math problem: how many rabbits are created in one year starting with one pair? After describing the progression in words, he shows the number progression as 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377. Examples of this series existed prior to Fibonnaci, and it is likely that he was simply repeating something he had learned, but 19th century mathematician Edouard Lucas called this sequence the Fibonacci numbers. They have been found to relate to many phenomena found in nature. (A thorough discussion is impossible here, but look.)

A webpage with some simple representations of the Fibonacci sequence is here.