De Gradibus

De Gradibus (Latin: Concerning degrees) was written by the Father of Arab Philosophy, Al-Kindi (801-873CE). In it, he applies mathematics to medicine, demonstrating a method he invented to determine the proper strength of a drug for a patient. Also, he discusses the degrees of the phases of the moon and how they help a physician to determine the most crucial days of a patient's illness.

When it was translated into Latin, the complex mathematical reasoning made it difficult for Western Europeans to grasp. Roger Bacon appreciated his approach, and endorsed it thusly:

The degree can only be determined by the method taught by Al-Kindi’s De gradibus, one extremely difficult and almost entirely unknown among Latin physicians of these days, as everyone is aware. Whoever wants to become perfect in this philosopher’s art must know the fundamentals of mathematics, because the species of greater and lesser inequality, the species of ratios, and the very difficult rules of fractions are all used by this author.

Plinio Prioreschi, a 20th century expert on the history of medicine, credits Al-Kindi with the earliest attempt to quantify medicine.

Al-Kindi was heavily influenced by noted Greek physician Galen (129-216CE). The stereotype of a Muslim rejecting any non-Muslim source of knowledge is tossed out by Al-Kindi's approach to knowledge. He wrote:

We must not hesitate to recognize the truth and to accept it no matter what is its origin, no matter if it comes to us from the ancients or from foreign people. My purpose is first to write down all that the ancients have left us on a given topic and then, using the Arabic tongue and taking into account the customs of our time and our capacities, to complete what they have not fully expressed.

How did Arabic works come to be available to European scholars. Was it haphazard, or was there a deliberate move to share knowledge. Tomorrow we will learn about Gerard of Cremona, and for a double treat, we will also talk about Gerard of Cremona. (Not a typo.)

The Philosophy of Music

Much of the medieval attitude about music and its forms came from Boethius (c.480-524 CE). In his de institution music [Latin: "On the laws of music"] he distinguished three types of music:

Instrumental music
Human music
Mundane music

Detail from copy of De musica [source]

Although today we use the word "mundane" to refer to something ordinary, it comes from Latin and refers to the world; Boethius uses it to refer to the music made by the world, that is, the so-called "music of the spheres": that sound, inaudible to human ears, that was made by the friction of the spheres surrounding the Earth in which the planets and other heavenly bodies traveled.

Instrumental music referred to music made by one of several different agents. It could come from something under tension (such as with stringed instruments), by wind, by water, or by percussion. It should be noted that Boethius was not referring to "musical instruments" as much as he used the term to mean that some physical agency was causing the sound. That could be a rushing stream, the wind in the trees, and falling rocks as much as manufactured devices in the hands of a musician. Later writers included singing as part of this category.

Human music was therefore not referring to singing by humans. For the Middle Ages, "music" was all about harmony, and the "harmony" between the physical body and the spiritual side was a serious topic. For example, you must nourish the physical body, but you must not eat so much that you fall into the sin of gluttony. Proper proportion was everything.

In fact, music (as opposed to mere noise) was all about harmony and proportion. That is why Music was studied in the medieval university only after mastering Arithmetic. A contemporary of Boethius, Cassiodorus (485-c.585 CE), compared the two by explaining that

Arithmetic is the discipline of absolute numerable quantity. Music is the discipline which treats of numbers in their relation to those things which are found in sound.

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.

Omar Khayyam, Mathematician

First page of "Cubic equation and
intersection of conic sections"

A book of verses underneath the bough

A flask of wine, a loaf of bread and thou

Beside me singing in the wilderness

And wilderness is paradise now.

The Rubaiyat of Omar Khayyam, translated by Edward Fitzgerald in 1859, made Khayyam the most famous Persian poet in the 19th century. Few people realize that Khayyam did not need Fitzgerald to be famous. Centuries earlier, he was one of the most influential thinkers produced by the Middle East.

Born in Nishapur on the 18th of May in 1048, he spent part of his youth in Balkh, which would produce Rumi 80 years after Khayyam's death. He studied under the well-known scholars Mansuri and Nishapuri. He put his education to work: as an adult, he was either teaching algebra and geometry, studying the stars, working on calendar reform, acting as a court advisor, or learning medicine. He taught the works of Avicenna.

The Tomb

He was best known in his lifetime and afterward for his mathematical writing, especially on algebra. Many of the principles of algebra that made their way to Europe came from Khayyam's Treatise on Demonstration of Problems on Algebra (1070).

One of his claims was that the solution of cubic equations cannot be solved by a ruler and compass. He said it required the use of conic sections, and announced his intention to write a paper that lays out the "fourteen forms with all their branches and cases." He never got around to it, and 750 years would have to pass before someone produced the proof of Khayyam's claim.

Omar Khayyam died on 4 December 1131 at the age of 83, and was buried in what is now the Khayyam Garden in Nishapur. A  mausoleum was built in 1963 to house his remains.

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.")

Cross-referencing an Eclipse

Diagram of an eclipse from a modern translation of Hipparchus

It is not always easy to figure out dates from classical or medieval writings. Chroniclers did not necessarily strive for the kind of historical accuracy which 21st-century audiences expect. When they wanted to be precise, they often expressed themselves in ways that do not provide a proper context for the modern scholar.

Consider, for instance, Pappus of Alexandria, whom the Encyclopedia Britannica calls "the most important mathematical author writing in Greek during the later Roman Empire." [source] He wrote many important texts, but we knew little of his life.

I mentioned the other day how Suidas' Lexicon gives us data on works and events otherwise lost to history. The entry for Pappus reads:

Alexandrian, philosopher, born in the time of the elder emperor Theodosius, when the philosopher Theon also flourished, the one who wrote about Ptolemy’s Canon. His books [are] Description of the Inhabited World; Commentary on the 4 Books of the Great Syntaxis of Ptolemy; The Rivers in Libya; Dream-Interpretations. [source]

We know that Theodosius reigned from 372-395 CE, so it gives us a time frame for Pappus. This creates a small head-scratcher, however. Pappus claims to have calculated and observed an eclipse in the month of Tybi (the fifth month of the Coptic calendar). There is a problem with this dating: no eclipse occurred during the month of Tybi during the reign of Theodosius that Pappus could have observed! Could the Suidas be wrong? Certainly. But then... what is right?

There is, as it turns out, a 10th century copy of a work by Theon of Alexandria (the one mentioned in the Suidas entry) that has a marginal note next to an entry on the Emperor Diocletian (who reigned from 284-305 CE), stating "at that time wrote Pappus." Is it possible that the composer of Suidas had access to that work and assumed that it meant Pappus flourished when Theon did? If we look closer to the reign of Diocletian, we discover that there was an eclipse in the month of Tybi which would place it (using the modern method of dating) on 18 October 320 CE. If Pappus observed it himself in 320, it isn't likely that he was flourishing over 50 years later. This places him firmly in the earlier part of the 4th century.

Pappus is far more important than as an example of the care with which modern historians must date historical events. Some of his eight-volume work on mathematics is extant; and deals with many facets of geometry and carefully lays out the mathematical findings of his predecessors and how their work builds on each other over time. He also worked on several problems such as inscribing regular polyhedrons inside a sphere, conic sections, trisecting an angle, and many more. He has a theorem named after him, as well as the Pappus chain, the Pappus configuration, and the Pappus graph.

His commentary on Ptolemy provides us with insight into some lost works of classical astronomy, such as an astronomical work by Hipparchus on eclipses (illustrated in the above figure).

Doctor Profundus

I have written about the Oxford Calculators, four men at Oxford University in the second quarter of the 14th century who made great strides in science and philosophy by treating things like heat and light as if they were quantifiable, even though they did not have ways to measure them. They engaged in "thought experiments" and used mathematics to determine the validity of their points. They were not always right in the end, but they were meticulous in their approach. One of the four was so esteemed that he was called Doctor Profundus, the "Profound Doctor."

Thomas Bradwardine (c.1290-1349) had a reputation as a precocious student at Balliol College. We know he was there by 1321, and later took a doctor of divinity degree. A gifted scholar and theologian, he wrote theories on the Liar Paradox and other logical "insolubles." The Liar Paradox is the statement "I am a liar." For it to be true, the speaker must be a liar; but if it is a true statement then the speaker is not lying. Resolving with logic how such statements can be understood had been tackled for centuries. Bradwardine's work Insolubilia presented complex solutions for puzzles/statements like this.

Like many university men of his day, Bradwardine followed an ecclesiastical career path. After serving as chancellor of the university, he became chancellor of the diocese of London and Dean of St.Paul's. He was also chosen to be chaplain and confessor to Edward III (mentioned in this blog numerous times), celebrating victory masses after campaigns of the Hundred Years War and being entrusted with diplomatic missions. The only time he did not have Edward's support was when John Stratford, Archbishop of Canterbury, died. Bradwardine was elected archbishop by the canons of Canterbury, but Edward opposed the choice, preferring his own chancellor at the time, John de Ufford. When de Ufford died of the Black Death (this was in 1349), Edward allowed Bradwardine to assume the position. Bradwardine had to travel to Pope Clement VI in Avignon for confirmation. but on his return, he succumbed to the Black Death on 26 August. He had been archbishop for 40 days.

That career would not have secured his place in history, however, even with his work attacking the Pelagian heresy. As one of the Oxford Calculators, he developed the "mean speed" theorem and the Law of Falling Bodies before Galileo. He studied "star polygons" (how regular polygons "tile" or fit together in patterns) before Kepler. He developed mnemonic techniques to improve mental abilities, explaining them in De Memoria Artificiali (On Artificial Memory).

One of his theories involved the vacuum of space. Aristotle felt that a vacuum needed a container, because an open space would automatically become filled by matter outside that space flowing into it. Therefore, according to Aristotle, no vacuum could exist above the world, because there was no container beyond the world to maintain the vacuum. Bradwardine was not satisfied with this. The infinity of space was a hot topic in the Middle Ages and Renaissance. His De causa Dei (On the Causes of God) argued that God Himself was infinite, and therefore space beyond our world extended infinitely. (This was different from suggesting that God created separately a space that was infinite.) He also suggested that this infinity could include other worlds that God could create and rule over.

Battle of the Numbers

Among the accomplishments of Hermann of Reichenau, he also provides us with the set of rules for one of Europe's oldest board games, developed by a monk to teach Boethian number theory, called Rithmomachy or Arithmomachia, "Battle of the Numbers."

The pieces on the rectangular 8x16 board, their "ranks" and their allowed moves are determined by mathematical rules based on their geometry (Circles, Squares, Triangles, Pyramids) and the numbers marked on their surfaces. I could not possibly explain the rules in a short post—nor should I be able to, since the intent was to design a game that truly requires a grasp of mathematical functions and the skill to apply them quickly. Feel free to educate yourselves on the rules here and here.

Laser-etched pieces. [link]

It was more than just a game of strategy like chess (to which it has some resemblance). According to a 2001 book, Rithmomachy

combined the pleasures of gaming with mathematical study and moral education. Intellectuals of the medieval and Renaissance periods who played this game were not only seeking to master the principles of Boethian mathematics but were striving to improve their own understanding of the secrets of the cosmos. [The Philosopher's Game, Anne Moyer]

The game became popular as a teaching aid in monasteries in France and Germany, and even reached England where Roger Bacon recommended it to students at Oxford. Over the centuries it spread as an intellectual pastime, and by the Renaissance it had spread enough that instructions were being printed in French, German, Italian and Latin. Sadly (mercifully?), the game fell out of popularity and the public's consciousness after the 1600s until modern historians re-discovered it.