- I'm embarrassed to say that I still haven't graded your
homework. Many apologies.
- Section 3.3
- Section 3.4
- Section 3.5
- Homeworks 4.1/2 and 4.3 due Wednesday.
- Remove problem #1 from the homework assignment for 4.4. It's too much (at least the way I ended up doing it!).
- Lessons from 5.3
- Commercial from Nathan; Who's next (I forgot!)?
- Lessons from 5.4
- The Cattle Problem of Archimedes
- Sent by Archimedes to Eratosthenes
- A system of equations, which, when reduced, gives rise to
a Pell equation:
- Fancifully derived from Homer's Odyssey.
- A numerical problem that required 22 centuries to solve.
- Early Mathematics in India
- Alexander's invasion of India stimulate communication of ideas. "It seems likely that Indian mathematics was directly influenced and inspired by the Greeks...." (p. 227)
- mathematics as an outgrowth of astronomy/astrology
- writers lacked algebraic symbolism, so they expressed problems in flowery verse; they liked "Behold!" proofs.
- Indians were using zero by 458 AD.
- 5th Century AD: Aryabhata
- born about 476
- Speculation that he worked at the University in Nalanda
- investigated arithmetic and geometric series
- tables of sines of angles in the first quadrant
- worked on linear and quadratic indeterminate equations
- Gave
as an approximation for
- 6th Century: Brahmagupta,
of Bhinmal
- Introduced negative numbers
- Two roots of a quadratic equation
- Worked on cyclic quadrilaterals (one of your problems in section 4.4 is his)
- "Where Diophantus sought to solve equations in the rational numbers, the Indian mathematicians admitted only positive integers as solutions." (p 228 -- Now called Diophantine equations).
- "Althought Aryabhata knew of a method for finding
a solution of
- So let's look at the general solution of the linear problem above, then look at a 12th century example (p. 230 -- Say quickly, mathematician!)
- 12th century: Bhaskara
(1114-1185), of Bijapur_district, India
- He addressed his "flowery prose" to his unwed daughter, whose marriage was fated (by the stars) to be unhappy so she never wed. She got a book named after her, as a consolation prize (Lilavati -- The Beautiful)
- His major work ("Siddhanta Siromani") became known in Europe via Arabic translation, 1587. (If you look at the Early Modern dates in the back of our book, they start with Mersenne -- 1588!).
- "Swarm of bees" problem, p. 229
- Now we turn to China, where some interesting Diophantine problems
were being considered:
- 6th Century Chang Ch'iu-chien, whose Mathematical Classic contained the famous "Hundred Fowls" problem, which we will consider (p. 231).
- Sun-Tsu (first century AD) poses the first in a long line
of so-called "Chinese Remainder" problems (p. 231)
He was not aware that there are an infinite number of solutions (although only a finite number of sensible ones).
- The section closes with an interesting advance by the Indians in solving those Pell equations (e.g. the Cattle Problem): Brahmagupta knew that given two sets of solutions of a Pell equation, a third solution could be obtained (p. 232).
- Pappus was not known as a great mathematician, but his
Mathematical Collection was a consolidation of the geometric
knowledge of his time. In it he summarizes or references many great
works, some of which are now unknown. Our author cites especially his
summary of Eudoxus's
He discovered that "...the quadratrix of Hippias could be obtained as the intersection of a cone of revolution with a right cylinder whose base was the spiral of Archimedes." (p. 234) Cool!
Speculated that the three classic problems of antiquity -- squaring the circle, trisecting the angle, and doubling the cube -- could not be solved using the classical criteria (p. 235).
- Hypatia (370-415)
- daughter and pupil of Theon of Alexandria (it was his copy of the Elements of Euclid that was the most ancient until the Vatican copy swiped by Napolean appeared).
- The "First Woman Mathematician" (p. 235) (wait! what about Theano, p. 92!)
- Reported to have made commentaries on Diophantus's and Apollonius's works.
- Lectured at the Museum on mathematics and philosophy.
- "Waylaid by a mob of religious zealots", she was flayed by oyster shells, torn limb-from-limb, and her remains burned.
- Proclus
(410-485)
- Trained in Alexandria, but returned to Athens to head Plato's Academy.
- He had great "access to numerous historical and critical works that have since vanished completely." We know of them through him. His Eudemian Summary of the History of Geometry by Eudemus "is of incomparable value as our main source of information on Greek geometry from Thales to Euclid." (p. 236).
- The Museum was plundered by the Christians, and then by the
Moslems (who burned what was left). Interesting quote: "Either the
manuscripts contain what is in the Koran, in which case we do not have
to read them, or they contain what is contrary to the Koran, in which
case we must not read them." (p. 236) So the manuscripts were burned to
fire the public baths.... Argh.
- Boethius (475-524)
- "Last of the Romans, first of the scholastics" -- a bridge between antiquity and the Middle Ages
- Wrote The Consolation of Philosophy while on "death row".
- Provided scholars with textbooks on the quadrivium (arithmetic, geometry, music, and astronomy). It wasn't much, but it helped keep the spark alive.
- In the Byzantine empire they, too, were working to preserve the Greek works.
- Cassiodorus
(circa 480-575)
- Young friend of Boethius
- "Founded a large monastery with the conscious aim of making it a center of Christian learning and scholarship... involving the creation of a scriptorium for the translation into Latin of the classical texts...." (p. 238)
- His Introduction to Divine and Human Writings gave "a brief discussion of each of the seven liberal arts. This primitive textbook served as the basis of the curriculum of the church schools in the early Middle Ages." (p. 239)