- Homework 4.1/2 is graded:
- You needed to use the common notions, axioms, and previous propositions to prove your results. You weren't allowed your own "common notions" of things!
- #6: many proved the converse
- #10: could have done it much more easily using #8 (prop. 35)
- #11: invoke #8 (but can't use it directly)
- #11: if you haven't drawn the picture, you're cheating!;)
- #12: you need the result that the inscribed triangle in a semi-circle is right.
- Homework 4.3 isn't graded yet.
- For Monday, 11/30: please provide a one-page, typed proposal for your paper/project. It should give me a good idea of what your focus is.
- For Monday, 12/7: please provide a one-page, typed revised proposal for
your paper/project. It should let me know that you're making
good progress!
- Problems from 6.1/2 and 6.3 have also been assigned.
- For the week after Thanksgiving, the remaining problems from chapter 4 and the first problem set (5.3) from chapter 5 are due. This is the last problem set I'll collect. Remaining problems will make up part of your final exam.
- Lessons from 6.1/6.2
- Commercial from Paul; next (in order): Chris, Jason, Adam, Debra
- Lessons from 6.3
- Square roots were obtained by Liu Hui in an interesting way, digit
by digit (p. 258): find the
.
- This is exactly the same as Kim's method.
- Section 6.1: The Decline and Revival (and decline!) of Learning
- The section begins with a recap of the
destruction of Roman culture, and the Dark Ages (the
depth reached in the 7th century), while the Islamic
culture was arising, and Brahmagupta was working
industriously away in India.
- The Battle
of Tours (732) saw the Muslim army turned
back to Spain by Charles Martel, grandfather
of Charlemagne.
- The Christian Church and the monastic schools are keeping the lights on, but just barely: their aim was producing ecclesiastical leaders, not scholars. There was suspicion of the pagans, like Archimedes....
- Our author admirably gives a shout-out to the scribes, who
are diligently copying the texts (and keeping "the ancient
culture from extinction" (p. 272), although other civilizations
aren't going to let that happen -- I get a little heavier whiff
of Euro-centrism from our author in this section!).
- Charlemagne
(742-814) crowned Holy Roman Emperor on Christmas Day, 800.
- His empire
- Alcuin of York
(c. 740-804) Charlemagne's mathematician
- Englishman, renowned scholar
- Wrote textbooks for all the liberal arts
- Searcher after books
- Introduced the Carolingian_minuscule script, "the source of our printed alphabet today" (p. 272).
- Unfortunately, the principal sort of problem studied was how to calculate the ecclesiastical calendar....
- His
Propositiones
ad Acuendos Juvenes
(Propositions for Sharpening
Youthful Minds) presented 53 puzzles,
including some "river crossing problems":
- the three men and their sisters crossing
- The Farmer, wolf, goat, bag of beans crossing
- Gave an "impulse" to education, which endured after his death.
- Brief Renaissance, followed by invasions from the north (Norsemen -- Go Norse! Norse pride!), Magyars from the east, Saracens from the south, ....
- The good news was that no one culture makes a permanent takeover of Christian Europe, and the monasteries continued to function. Charlemagne's empire broke up into what would become France, Germany, and Italy today.
- Eventually the monastic schools were displaced in importance by the "cathedral schools" -- Cologne, Tours, Liege, Chartres, Reims, and Paris, which became the great universities of the 13th and 14th centuries (more in chapter 7).
- Greek wisdom would be reintroduced to Europe via Arabic,
and through Spain.
- Cordoba, in Spain, in the 7th century, had
- 600 mosques
- 600,000 volumes in their library
- and was an intellectual center of the western caliphate
- Burton credits the Arabic translations of the classic Greek works as "Islam's great and enduring contribution to the advancement of knowledge" (p. 275).
- Christians "recovered" Toledo, Spain, in 1085. It too had been an important center. It became important to translate these Arabic works.
- Cordoba, in Spain, in the 7th century, had
- The crusadors of the 11th century, began to bring back Greek knowledge to Europe. Translators like Adelard_of_Bath (1080-1152) and Gerard_of_Cremona (1114-1187) translated the classics from Arabic into Latin (the lingua franca of Europe). The work could be dangerous: there's the interesting tale of Gerard in the disguise of an Islamic student taking in lectures in Cordoba (he would have been killed if discovered).
- The first printed copy of Campanus's Euclid's Elements comes out in 1482. Translated directly from Greek, it was an improvement on the translations derived from Arabic, which came out via a process of "telephone" (p. 276).
- "By the middle of the 1200s... Western scholars stood once again on the solid foundation of Hellenistic thought." (p. 279)
- The section begins with a recap of the
destruction of Roman culture, and the Dark Ages (the
depth reached in the 7th century), while the Islamic
culture was arising, and Brahmagupta was working
industriously away in India.
- Section 6.2: The Liber_Abaci and Liber Quadratorum
- "The greatest mathematician of the Middle Ages was Leonardo of Pisa, better known as Fibonacci.... [T]he mathematical renaissance of the West dates from him." (c. 1170-1250)
- The Leaning_Tower_of_Pisa and he are just about twins.
- Read the opening sentences of his Liber Abaci (p. 280):
- These are the nine figures of the Indians:
- 9 8 7 6 5 4 3 2 1
What a start! Arabic numerals introduced, via his "Book of Counting" (his father ran a customshouse). The arabic numbers were first introduced in the Codex Vigilanus around 976 -- p. 282). By the beginning of printed books, c. 1450, our arabic numerals had taken the form in which we know them today.
- "The Liber Abaci embodied virtually all the arithmetical knowledge of Fibonacci's time...." (p. 280)
- Even so, it was not translated into English until 2002, on its 800th anniversary!
- Let's do one problem from the Liber Abaci: #3, p. 287 (we'll come back for more in a bit!).
- The Liber_Quadratorum
- Devoted to Diophantine problems of second degree, including the tournament problems from the court of Emperor Frederick II (proposed by John_of_Palermo
- Notice Fibonacci's notation
- (arabic in origin) for writing fractions (see p. 285)
- right to left....
- Remember that Fibonacci borrowed from Abu Kamil commentaries on al-Khowarizmi's work.
- Jordanus_de_Nemore (c. 1225).
- Another high point of the 13th century mathematical scene in Europe.
- May have been at the University of Toulouse (founded 1229).
- Known almost exclusively through his works:
- De Triangulus -- medieval geometry at its best ("rigorous -- and frequently new -- proofs of Euclidean Theorems" -- p. 286)
- De Numeris Datis -- on advanced algebra
- Notation, again: Jordanus used letters to represent numerical quantities throughout, advancing the evolution of algebraic notation
- Forgotten for awhile, but resurrected by Francois Vieta 350 years later!
- Numerical illustrations used Roman numerals
- There followed a period without any particular luminous successors (until the opening salvos of the Renaissance).
- "Fibonacci is remembered today mainly because a nineteenth century French number theorist, Edouard Lucas, attached his name to a sequence that appeared in a trivial problem in the Liber Abaci." (p. 289)
- The Problem: bunnies (p. 289)
- A model for recursive definition
- We've already talked about them a little (ratio of successive is golden, see p. 294)
- Let's look at the famous Fibonacci puzzle, p. 293
- Fibonacci
Nim (we'll adjust the result of #1, p. 295)
- The Fibonacci numbers provide a unique decomposition of the natural numbers, analogous to the primes.
- Every natural number is either Fibonacci, or can be written as the sum of non-consecutive Fibonacci numbers in a unique way (up to order).