- Due today, 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! I'll get those back next time.
- Sorry, I haven't finished grading your problems.
- It's time for course evaluations. Please take the time to visit eval.nku.edu, and give me your feedback. Thanks!
- Lessons from 7.1/7.2
- Commercial from Adam; last time, Debra!
- Lessons from 7.3
- 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, replaced 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 (1735), which helped lead to the creation of 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...."
- This one reads like a soap opera! It's an interesting twist to have so much information, when our problem up til now has been too little....
- Late 1400s: The University of Bologna is credited with holding the line on mathematics (one of the only places in Europe), although Regiomontanus (at the University of Vienna) discovered the power of the press, and began to print his own and other (particularly Greek) works.
(From the previous section we learned that, as of 1452, students at Paris merely had to take an oath that they'd read the first six books of Euclid! p. 312) - "Mathematics benefited immensely from ... the discovery, translation, and circulation of ancient Greek texts." Nonetheless, "The mathematician ... often tried little more than to comprehend what the ancients had done." (p. 315) This changed when the Italians showed how to solve the general cubic equation....
- Some began publishing in "the vernacular" (not in Latin anymore). In English, it was Robert Recorde (1510-1558) who produced popular textbooks in English (he is credited with introducing the "=" sign). Look at p. 318, and try to read a page from The Whetstone of Whitte (1557)....
Here's the title page (rough), and a little clearer.
- The history of the cubic is presented in brief in this section, with recollections of
- Duplicating the cube,
- Diophantus's cubic,
- Omar Khayyam's work on "the 13 cubics",
- Fibonacci's challenge problems,
- The Bologna school gets credit for the solution during the 1500s (P. 319).
- The story begins with Fra Luca Pacioli (c. 1445-1514), who wrote the Summa de arithmetica, Geometria, proportioni, et Proportionalita (a summary of contemporary mathematics).
In this work he essentially issued a challenge:
"...that the solution of the cubic equation was as impossible as the quadrature of the circle." (p. 319)
and, like any good challenge, challengers rose up to attempt the impossible.
- Pacioli's student was non other than Leonardo da Vinci (April 15, 1452 - May 2, 1519),
who was also the illustrator of Pacioli's De divina proportione:
- In the image of Pacioli there is a water-filled Rhombicuboctahedron), as well as other interesting mathematical objects).
- Pacioli's student was non other than Leonardo da Vinci (April 15, 1452 - May 2, 1519),
who was also the illustrator of Pacioli's De divina proportione:
- del Ferro (February 6, 1465 -- November 5, 1526) "shattered" his prediction by solving for the solutions of the special cubic
Since at the time the methods by which one advanced were different (e.g. public competitions), it did not behoove one to share one's methods -- so del Ferro stayed mum about the solution, sharing it only with a few, including his student Antonio Fiore.
- Tartaglia
(1499/1500 -- December 13, 1557) -- the "stammerer".
- Tartaglia (tahr 'tal yuh) had been traumatized by nearly dying in a French massacre of those in his town of Brescia
- Poor, insecure, he nonetheless made good, and became a first rate mathematician (writing on tombstones, for lack of better writing surfaces!).
- He applied mathematics to the "art" of artillary fire: "Anticipating Galileo, Tartaglia taught that falling bodies of different weights traverse equal distances in equal times." (p. 321)
- "Had a habit of claiming other people's discoveries as his own...." (such as Pascal's triangle -- everybody wants that one!)
- 1580: a friend sends some challenging cubic problems
(p. 321). Finally manages to solve not only the cubic of del
Ferro, but also all cubics of the form
- Fiore challenges Tartaglia to a public contest, and Tartaglia humiliates him, by solving all his challenge problems (quickly), and also by giving Fiore cubics that reduce only to the latter form (which Fiore couldn't solve). Tartaglia scores!
- Girolamo
Cardano (Cardan) (1501--1576)
- Along comes the "scoundral" Cardan:
- Publishes the horoscope of Christ
- Loses jobs right and left, in scandal
- competent -- maybe even great -- physician
- lousy astrologer (misreads King Edward VI's horoscope, promising long life, great deeds, etc.). King dies young.
- But, a man of honor, it is said that when the day came that he'd predicted would be his last, he committed suicide rather than be a liar! (p. 322).
- Cardan begs Tartaglia for the solution methods; Tartaglia refuses; more begging, and the oath that it would be kept secret. Tartaglia acquieses, and then -- upon rumors that del Ferro had actually had the solution -- Cardan goes to Bologna to verify the rumors, claims that del Ferro had it, and then publishes the method in his own volume. The method is forever after known as "Cardan's formula."
- Along comes the "scoundral" Cardan:
- in Cardan's Liber de Ludo Aleae (Book on Games of Chance) he "broke the ground for a theory of probability more than 50 years before Fermat and Pascal." (p. 323)
- Ars_Magna --
The Great Art
- on algebraic equations. "Cardan was no mere plagiarist, but one who combined a measure of honest toil with his piracy." (p. 323)
- We want to examine how to derive the "cubic formula".
- In the process of using "his" formula, Cardan encountered
complex roots ("ghosts of real numbers", as Napier called
them). "Putting aside the mental tortures involved, multiply
by
....": sum 10, product 40.
- "So progresses arithmetic subtlety the end of which, as is said, is as refined as it is useless." (p. 323)
- Let's take a look at Bombelli's cubic,
- Rafael Bombelli (1526-1572)
- First to write the rules of negative numbers
- First to write complex numbers as sums of real and imaginary parts. "From this time on, imaginary numbers lost some of their mystical character...."