- I finally finished grading the homework.
- Section 3.3
- Section 3.4
- Section 3.5
- Homeworks 4.1/2 and 4.3 are due today.
- 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 5.3/5.4
- Commercial from Jennifer; Nathan's next; then Tom.
- Lessons from 5.5
- 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 of Alexandria (c. 290 - c. 350) 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.
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)
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
Muslems (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)
- Arabic Mathematics
- A new empire, ruled by Caliphs, and based on the Islamic faith (founded by Muhammad (c. 570- 632)), swept the old civilizations away:
- 635: Damascus fell
- 637: Jerusalem
- c. 640: Egypt falls
- 711: cross the straits of Gibraltar
- A map of the Islamic empire (circa 750)
- Capital moved to Baghdad, circa 762 (population 800,000 in the 9th century -- larger than Constantinople)
- Caliph al-Ma'mun (786-833) sets up the "House of Wisdom", where extant bodies of knowledge were translated into Arabic ("...practically the whole extant body of Greek scientific ... writings had been recorded in the Arabic language." -- p. 240).
- Mathematicians:
- "most illustrious": Mohammed ibn Musa al-Khowarizmi (c. 780-850) (al khwahr 'iz mee)
- Friend of Caliph al-Ma'mun, associated with the House of Wisdom
- Through his two works (on arithmetic and algebra) Europe became aware of Hindu numerals and algebraic approach. (p. 240)
- Book of addition and subtraction according to the Hindu Calculation explained the Hindu decimal system of numerals, including making use of zero (p. 241).
- Interestingly, no copies of the Arabic remain, but it came to Europe in a Latin version from the 12th century.
- As the translation was from Arabic, the numerals aquired the name "Arabic" numerals. And so it goes....
- Europeans also became acquainted with algebra through al-Khowarizmi's work, Hisab al-jabr w'al muqabalah -- "The Science of reunion and reduction" -- translated as Liber Algebrae et Almucabola.
- reunion: moving negative terms to one side or the other, to make them positive
- reduction: cancellation of like terms
- al-Khowarizmi lives on in the word "algorithm", a Latin corruption of his name. Algorism "...meant the art of computing with Hindu-Arabic numerals." (p. 241)
- al-Khowarizmi is credited by our author with preserving the methods of an old Persian school -- he was not the inventor.
- His algebra suffered from a "complete lack of mathematical symbols" (p. 241) -- still using "flowery prose" instead of notation such as Diophantus attempted.
- "...the Arabs, inspired by Euclid's Elements, seemed to believe that an argument had to be geometric to be convincing." (p. 242).
- Three fundamental types of quadratic equations:
(NB: Which equation is missing?!)
One way of thinking about this is that Arabic mathematicians are reflecting more of the Greek tradition than the Indian.
-
- Example of completing the square: p. 243
- Moving from the geometric, to the algorithmic (even if the notation has not arrived): read the passage on p. 245, beginning "If you wish to subtract the root of 4 from the root of 9...."
"The old, ingenious Babylonian tricks and devices for solving individual problems are finally seen as part of al-Khowarizmi's systematic reduction of the quadratics to their standard types...."
- According to our author, irrational roots of quadratics were acceptable, but they did not perceive the reality of negative solutions to an equation.
- Abu Kamil (c. 850-930) -- "The Reckoner from Egypt"
- Commented and elaborated on al-Khowarizmi's work in his Book of Algebra, which Fibonacci would draw heavily on for his Liber Abaci.
- Expands on al-Khowarizmi's work, nonetheless. Adds methods of his own. Let's look at his problem on p. 244:
- Abu Kamil's major advance: irrational coefficients in indeterminate equations
- He'd developed a "calculus of radicals", based on the identity
- Thabit ibn Qurra (c. 836-901)
- Excellent translator of Greek works
- "Thabit's thorough revision of [the Elements] is the most faithful rendition into Arabic..., which in turn came to the West through Gerard of Ceremona's Latin adaptation in the twelfth century." (p. 246)
- His Book on the Determination of Amicable Numbers is considered "...the first [known] completely original mathematics written in Arabic." (p. 246)
- Nicomachus had mentioned them, but didn't give any theory.
- Thabit's result: if
,
, and
are all prime, then
and
are "amicable numbers".
- Try
:
- Generalized the Pythagorean theorem, and devised novel proofs of the Pythagorean theorem.
- An astronomer in Baghdad, he drew attention to errors in Ptolemy's Almagest.
- Abu Bakr al-Karaji (d. 1029)
- "The monomials are infinite", he declared.
- Developed an algebra of polynomials, as well as
- Several results on binomial expansions, including discovering Pascal's triangle a few years before Pascal (1623-1662).
- al-Samaw'al (c. 1180) took his formula for the sum of cubees in an interesting geometrical direction: let's look at p. 249.
- "most illustrious": Mohammed ibn Musa al-Khowarizmi (c. 780-850) (al khwahr 'iz mee)
- A new empire, ruled by Caliphs, and based on the Islamic faith (founded by Muhammad (c. 570- 632)), swept the old civilizations away:
- Chinese Mathematics -- to come!
