History of Mathematics II

Math 304, METU, spring semester, 2009/10

Practicalities

Weekly schedule:

• Tuesday, 12:40–14:40 in M-105: this is a change—agreed to by all students in class—from the original schedule (which had the class meeting one hour later)
• Thursday, 14:40–15:30, in M105

Exams:

1. In class, Tuesday, March 30, 12:40–14:40. Solutions with commentary:
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   The exam itself:
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   Scores out of 40: 4, 8, 9, 10, 12, 13, 17, 18, 18, 18, 21, 22, 22, 23, 27, 30, 31, 34, 40. (To see your paper, visit my office!)
2. In class, Tuesday, May 18, 12:40–14:40. Solutions with commentary:
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   The exam itself:
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3. Final: in M-102 on Saturday, May 12, at 13:30 (as scheduled by the university). Solutions:
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   This time the solutions do not include the problems, which are here:
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Text:

No modern textbook, only original sources; see below

Instructor:

David Pierce

Official title of course:

HISTORY OF MATHEMATICAL CONCEPTS II

About the course

This course is a continuation of Math 303, but that course is not a prerequisite for this one. Practices will be as in Math 303:

• attendence is required;
• all students will spend time making presentations at the blackboard;
• there is no “textbook”.

This course will make no attempt to fit the catalogue description. Some phrases in that description are apparently based on chapter titles in Boyer's History of Mathematics. But again, this course will not follow a textbook; we shall read original sources (albeit in translation, from Arabic, Latin, French, …). This approach is slower, but more honest to the title of the course. Why?

• I accept the conclusion of the philosopher R. G. Collingwood that history is the history of thought. This means, in particular, that doing history of mathematics means thinking the mathematical thoughts of past mathematicians.
  • This is difficult work, but nobody else can do it for us.
  • This work can hardly be done without looking directly at what these mathematicians actually wrote.
• Second-hand accounts of past mathematics may give a misleading view, as for example by translating everything into modern algebraic terms.

Anybody who is interested can read a conventional “history of mathematics” on their own. But there is no substitute for working together, as a group, to understand some old piece of original mathematics.

Some students took Math 303 in hope of learning some history in the sense of stories. The words “history” and “story” are indeed cognate, coming through French from the Latin historia, which is from the Greek ιστορια. However, we know almost nothing about the personal lives of ancient mathematicians. About more recent mathematicians, more is known. For example, there is this interesting piece of information:

After his death, Newton's body was discovered to have had massive amounts of mercury in it, probably resulting from his alchemical pursuits. Mercury poisoning could explain Newton's eccentricity in late life. [Wikipedia, accessed 2010.02.17]

This is irrelevant to the understanding of Newton's mathematics (though it might be used as an excuse for not understanding Newton).

Some students in Math 303 were disappointed in the quality of some of their classmates' presentations. However, student presentations are essential to this course. You don't really understand something unless you can stand up and talk about it. Also, in this course, everybody should have read what is being presented at the blackboard, and everybody should be prepared to criticize a faulty presentation, or to raise questions.

Texts

The contents of the course are under development, but the sequence will be something like the following. We shall need to refer to the Ancients, since later writers refer to them:

• Book II of Euclid's Elements (298 KB; for the full text see the Math 303 website)
• Books I and II of the Conics of Apollonius of Perga.

I shall probably just lecture on the foregoing as needed. Everybody should read and be prepared to talk about the following, as announced in class:

1. Muhammad ibn Musa al-Khwarizmi (c.780–850), Compendium on Calculation by Completion [al-jabr] and Reduction [al-muqabala]: our selections, on solving quadratic equations, are taken from The Mathematics of Egypt, Mesopotamia, China, India, and Islam: A Sourcebook (edited by Victor Katz); they are available as a pdf file (1.1 MB) and in the library photocopy shop. Two older translations of Al-Khwarizmi's work are available on the web:

   • Robert of Chester's Latin translation of the Algebra of Al-Khowarizmi, with an introduction, critical notes, and an English version by Louis Charles Karpinski, 1915

   • The Algebra of Mohammed Ben Musa. Edited and translated [from the Arabic] by Frederic Rosen, 1831

2. Thabit ibn Qurra (836–901) on quadratic equations; our selections, from the Katz volume, are available with those of al-Khwarizmi

3. Omar Khayyam (1048–1131), Algebra; our selections, on solving cubic equations by conic sections, are available as a pdf file (608 KB), and in the library photocopy shop, together with the Al-Khwarizmi and the ibn Qurra

4. Gerolamo Cardano (1501–1556), Ars Magna. Available in the library photocopy shop are chapters I, II, VI, XI, XXXVII, and a part of XXXIX of this book. The selections are given both in the original Latin and in the version found in The Great Art, or The Rules of Algebra, by Girolamo Cardano, translated and edited by T. Richard Witmer (1968). One should look at the Latin to see that Cardano did not use any of our current notation. A more faithful translation of small selections is found in A Source Book in Mathematics, 1200–1800, edited by D. J. Struik (1969).

   The Latin text in the library photocopy shop is from an edition of Cardano's complete works, published in Lyon in 1663 (after Cardano's death). It has many mistakes, some noted by Witmer, some not. (It is all I could find on the web. The Ars Magna is Section 4.4 of the complete works.)

5. François Viète (1540–1603), Introduction to the Analytic Art (translated from the Latin by J. Winfree Smith and included as an appendix to Jacob Klein, Greek Mathematical Thought and the Origin of Algebra, Dover Publications, 1992), especially chapters I–III, along with number 5 of the “laws of zetetics” in chapter V; the whole text is in the library photocopy shop

6. René Descartes (1596–1650), Geometry: we read through p. 55 in the translation (from the French and Latin, 1925) by David Eugene Smith and Marcia L. Latham, available as a pdf file (13 MB) taken from the Online Books Page, or in the library photocopy shop

7. Isaac Newton (1643–1726), Principia. In the library photocopy shop are the Definitions and Axioms, along with Book 1, Sections I–III. On April 30, I put a contemporary translation there (to supplement the old Motte translation. The author's preface should be read as well. The text is taken from Motte's 1729 English translation, as transcribed at Wikisource. A facsimile of this edition is also available from Google Books. We read:

   • Definitions
   • Axioms, or Laws of Motion
   • Lemmas 1–11
   • Proposition 1 and Corollaries 1–4
   • Proposition 2
   • Proposition 3 and Corollaries 1–3
   • Proposition 4 and Corollaries 1–9
   • Proposition 6 and Corollaries 1 & 5
   • Proposition 7 and Corollaries 2 & 3
   • Proposition 9
   • Lemma 12
   • Proposition 10 and Corollaries 1 & 2
   • Proposition 11
   • Proposition 12
   • Lemma 13
   • Lemma 14 and Corollary 1
   • Proposition 13 and Corollaries 1 & 2
   • Proposition 14 and corollary
   • Proposition 15 and corollary
   • Proposition 16 and Corollaries 1–3
   • Proposition 17 and Corollary 1

The Princeton Companion to Mathematics (edited by Timothy Gowers, 2008) contains, in part VI, short biographies of 96 mathematicians. The table of contents begins thus:

VI.1 Pythagoras (ca. 569 B.C.E.–ca. 494 B.C.E.) 733
VI.2 Euclid (ca. 325 B.C.E.–ca. 265 B.C.E.) 734
VI.3 Archimedes (ca. 287 B.C.E.–212 B.C.E.) 734
VI.4 Apollonius (ca. 262 B.C.E.–ca. 190 B.C.E.) 735
VI.5 Abu Ja'far Muhammad ibn Musa al-Khwarizmi (800–847) 736
VI.6 Leonardo of Pisa (known as Fibonacci) (ca. 1170–ca. 1250) 737
VI.7 Girolamo Cardano (1501–1576) 737
VI.8 Rafael Bombelli (1526–after 1572) 737
VI.9 François Viète (1540–1603) 737
VI.10 Simon Stevin (1548–1620) 738
VI.11 René Descartes (1596–1650) 739
VI.12 Pierre Fermat (160?–1665) 740
VI.13 Blaise Pascal (1623–1662) 741
VI.14 Isaac Newton (1642–1727) 742

We have no writings of Pythagoras; of the remaining thirteen mathematicians on this list, we have (in 503 and 504) read the works of seven, along with two others.

Grading

I shall take attendance for each hour of class. I shall give at least one examination in term, and one final examination.

There will be an attendence score and an exam score. You are guaranteed only that the lesser of these two scores will determine your grade, by the catalogue scheme. (Grading in Math 303 was more lenient than this; but repetition of this leniency is not guaranteed.)