- Returned today, 11/30: a one-page, typed proposal for your paper/project. It should give me a good idea of what your focus is.
- Due Today: the remaining problems from chapter 4 (4.4 and 4.5) and the first problem set (5.3) from chapter 5. This is the last problem set I'll collect. Remaining problems will make up part of your final exam.
- Due Monday, 12/7: please provide a revised
one-page, typed proposal for your paper/project. It
should let me know that you're making good progress!
Use complete sentences, self-edit, peer-edit, use the writing center, etc.
- Lessons from 6.3
- Commercial from Jason; next Adam, then Debra
- Lessons from 7.1
- "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). Here is the solution in the Liber Abaci
- A model for recursive definition:
with
and
.
- 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, which is based on the identity
- 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).
- Things are bleak, as the Four Horsemen of the Apolcolypse can attest.
- The 14th century starts off with something thoroughly modern: climate change!
Interesting, isn't it, that a cold snap (the "Little Ice Age") could result in famine that kills ten percent of many towns.
- 1347: the Black Death arrives in the Mediterranean, probably from
the Crimea, via caravans from China:
Calculus, the theory of colors, and the law of universal gravitation. That's all. What will you do/have you done by the time you reach(ed) 24?
Towns in France are so heavily depopulated that they send to Italy for new citizens. Those who perish most are those who stayed at the helm, natural leaders. Deprived of leadership, society was destabilized.
- War: The Hundred_Year_War rages between France and England (e.g. Joan of Arc).
- Feudalism reigns. Brigands (veterans of the wars) roam the countryside, stirring up trouble. Peasants' Revolt of 1381 helps bring feudalism to an end.
- Despair in the 14th century was illustrated by the Danse_Macabre:
- Meanwhile, growth in population drives a Renaissance, or "rebirth", which
is traditionally said to have begun in Italy in the 14th century. Italy
was spared some of the hits taken by the northern countries of Europe.
The Renaissance saw the transition from the feudal, ecclesiastical society to a more secular, urban society. A part of that was the transformation of the so-called "Cathedral Universities." These had been founded in the 12th and 13th centuries (many during the lifetime of Fibonacci, e.g. Paris, AD 1200, or Oxford -- 1214), as alternatives to the monestaries, for the teaching of more secular subjects (e.g. medicine); but they were undergoing changes.
It's interesting to see how the role of student has changed. Students formerly paid for their lessons, and they wanted them just so: see the second paragraph of p. 312. I showed this section to a colleague, and he and I both agreed that we would be delighed to have such students! The control of the wedding day, and leaving town without permission aside....
Ultimately, the State takes over the universities, and the students lose their power. Foiled again!
The focus away from mathematics was changing. It's interesting that mathematics was not encouraged because it was perceived as a means for "practical application". How little scholars in the Middle Ages remembered of Plato's love of pure mathematics as a mark of an educated person; or Archimedes preference for the pure over the applied.
- Also about this time paper and movable type spur on the
development of literacy: books become available. Paper replaces
parchment, which had replaced papyrus. The Gutenberg Bible is one of
the last documents printed on parchment (requiring the skins of 300
sheep, p. 306).
Paper is credited to the Chinese as of AD 105; passing through the hands of the Arabs, only to be relinquished to the Christian conquerers of the Arabic Moors in Spain.
- In 1453 Constantinople fell to the Turks, much to the surprise of
Christiandom: the positive side is that scholars who'd retreated to the
capital of the Byzantine empire now moved on to Italy with their Greek
texts in tow, and this may have helped fuel the Renaissance. It
certainly improved the mathematical textbooks, that had been translated
by the process of "telephone" we discussed earlier, based on the Arabic.
- Folks turn to strong monarchies to save them from the chaos:
- Louis XI in France (1461)
- Ferdinand and Isabella in Spain (1477)
- Henry VII in England (1485, patriarch of the Tudor dynasty)
- 1478: the first popular math textbook (Treviso Arithmetic)
is published in Treviso, Italy. It is written for "young people
preparing to enter commercial careers", so could be called a Business
Math textbook.
Euclid's Elements follow (poor Arabic translation) in 1482, replace by a good Greek-translated version in 1505.
- Johannes Muller (1436-1476)
- Known as Regiomontanus, the Latin name of Konigsberg (famous for Euler's Bridges of Konigsberg problem, which helped lead to graph theory).
- The crater Regiomontanus on the Moon is named after him.
- His De Triangulis Omnimodis (On Triangles of all Kinds) represented the birth of trigonometry as a field of study separate from astronomy.
- His astonomical work brought him to the attention of Pope Sextus IV to help coordinate the ecclesiastical calendar with astronomical events, but he died before he could.
- Columbus carried Regiomontanus's almanac, the Ephemerides Astronomicae on his trips to the New World, and used a total eclipse of the moon predicted by Regiomontanus to frighten the "Indians" into provisioning his ships (p. 308). This is another example of the importance of applied math!;)
- The Gregorian calendar was ultimately adopted in Catholic
countries in 1582, and by England in 1752. This led to a curious
situation -- illustrated by the calendar of September, 1752:
September 1752 Su Mo Tu We Th Fr Sa 1 2 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30'The countryside erupted in riots as people demanded the return of their "lost days."' (p. 308) - The section ends on a discussion of the "cult of the classics"
which developed which the reintroduction of the Greek works.
- "Antiquity, both Latin and Greek, offered a model of perfection by which to judge all civilizations...." (p. 313).
- This did have a positive effect: "scholars rummaged through old libraries... collecting, copying..., and diffus[ing the] treasures they had unearthed." (p. 313)
- "Renaissance men venerated manuscripts."
- Focused on reviving, rather than creating. "They slipped into the habit of pretending to be Greeks or Romans, even going through the motions of reviving pagan religious rituals." (p. 313)
- Burton ends on this sad note: "Renaissance educators subordinated, and sometimes even impugned, learning from experience and direct observation. The effect was to impede the study of the physical sciences and mathematics.... The Renaissance... nevertheless indirectly opened the way to the Scientific Revolution of the 1600s...."