History of Mathematics 80402, Spring 2005

217 Szprinzak, 14:00 - 15:45 Thursdays.

Office hours: Monday 16:00, Mathematics 104.
e-mail: ehud AT math.huji.ac.il

Some final papers.

The papers are mostly in Hebrew; the English titles below are inexact. All papers, with comments, can be found in Orna's office.
Mayan mathematics and the notion of zero
Euclid's construction of the regular pentagon
Talmudic mathematics
Origins of the three body problem in Newton and Lagrange
Hilbert, and Kantian puzzles
Origins of the calculus of variations: the brachistochrone.
Decidability of geometry from Descartes to Tarski
Well-ordering of the natural numbers in Euclid VII
Newton's Opticks
Georg Cantor
Mishnat Hamidot
History of zero
Gauss's proofs of the fundamental theorem of Algebra
Galois's Galois theory
Euler and the origin of the theory of graphs
Fibonacci and angle trisection

Basic texts

Secondary references

Reading Assignments

  1. Plato, Meno, dialog of Socrates with the boy. (p. 431 of v.1 in the Hebrew translation (Menon) by Y. Liebes, v. 1.
    English translation by B. Jowett can be found in the MIT Internet Classics Archive
    Extract from Plato's seventh letter
    Definitions and Basic Notions of Book I.
  2. Read the statements of the Postulates and all Propositions of Book I. When the statement is not clear, try to find the meaning in the proof. Understand proofs of 1-4, 9,10,27-32,43,44.
  3. Book II. Read and translate propositions into algebraic notation, without looking up the commentary. Book III: read statements. Proof of III 16.
  4. Construction of pentagon. Book IV, Proposition 11 and backtrack (including IV 2, III 32, III 37.)
  5. III 16. Book V. X 1.
  6. (for May 19) Principia The three laws (axioms), Book I Section 1, Book I Section 2 Prop. 1.
  7. Descartes, from The Geometry (selection from Struik, distributed.)

Rough description of content of classes

  1. February 24.
    General background of Greek mathematics. Geography. Egyptian and Babylonian predecessors; reception by Moslem and European cultures. . Pythagoreans. Alexandria. Euclid.
  2. March 3.
    A greek view of mathematics: Plato: 7th letter. Menon. Logical arguments.
    Euclid, Book I: Definitions, Common notions.
  3. March 10.
    Euclid: Axioms. Propositions of Book I.
    Rigor of proofs.
    Construction with straightedge and compass.
    Delian problems: Trisection of angle. Duplication of cube. Squaring of circle. Pappus' classification of solutions: plane (= straightedge and compass), solid (=conic sections), "linear" = with other curves.
  4. March 17
    Zeno's paradoxes (from Heath.)
    Euclid: Books II; geometric algebra. Book III. Book IV: construction of pentagon.
  5. March 18.
    Modern view of equivalence between geometry and algebra. Impossibility of ``plane solution'' of the Delian problems. Symmetries (automorphisms of fields) Fields generated by square roots have no symmetries of order three. But the field generated by a primitive $9$ root of unity has an automorphism of order $3$. (A complete proof with these ideas requires Galois's notion of normal field extensions, that we will not go into.)
  6. March 24.
    Continuation; see Geometry from Algebra. Proof using the notion of the dimension of a field over the rational field, rather than by consideration of symmetries. This may be closer to Wantzel's original proof (1837). We did we briefly mention one of the main principles of Galois theory: A (algebraic-) geometric construction has a unique solution even with complex coordinates if and only if, algebraically, it can be performed by rational operations : +,x,/,-
    As an example, we saw the construction of pythagorean triples by intersection of circle with line through a point on a circle, and another rational point. Excercise: work this out.
    Diophantus was mentioned.
    We also squared one of the Lunes of Hippocrates (source: Heath. See also the St. Andrews' entry on Hippocrates of Chios )
  7. March 31. Constructions: the regular solids. Classification theorem: XIII-18. A modern viewpoint, symmetry.
    Book V: comparable magnitudes. By contrast, the horn angle, III 16.
    Book X: commensurables and incommensurables. Propositions 1,2,3. The geometric Euclidean algorithm. Conjectural description in modern terms of the crisis of incommensurability and its effect and resolution. (1) Possible early Pythagorean correspondence of geometry with algebra: attempt to identify geometric lengths with ratios of numbers. (2) incommensurability of the diagonal and side of a square. Euclid, Proposition VIII 8, and implications. (3) Algebra from geometry: addition and subtraction of equivalence classes of lines. Multiplication and division. (4) Book V, Eudoxus: identification of the field of lengths with a subfield of the real numbers.
  8. May 5. Background on the mathematics of the scientific revolution, de Revolutionibus to Principia
  9. May 12, May 19, May 20, May 26, June 2. Texts by Leibniz, Newton, Galileo, Descartes (photocopied). Geometric and algebraic foundations for infinitesimal analysis. Newton's proofs of equivalence of global elliptic orbits (or more generally orbits along conic sections) with an infinitesimal law: accelaration towards a fixed center, with magnitude inversely proportional to the distance squared. Newton's proof of the transcendence of trigonometric and related functions (Lemma 28.) Flashforwards: Hausdorff metric on closed sets in the plane, Bezout's theorem.
  10. June 9, Descartes. (Flashforward to Tarski.)


The papers are due by August 31. Work in pairs is encouraged.
The paper should be typed and submitted online, as well as in person.
The expected length is about five pages for an individual work, seven for two authors. Please clear the suggested topic with me in advance:
When sending e-mail related to this class, please write the words: History of Mathematics in the subject line.

Examples of student projects from other classes.

  • Three sets of student papers from Gregory Cherlin's History of Mathematics classes
  • See also some wider projects from St. Andrews .
  • Some ideas related to material studied so far:

    The reception of extension of Euclid in Islamic mathematics; the connection of algebra and geometry in Thabit ibn Qurra; in Al Biruni and Omar Khayyam. You can begin with the St. Andrew's site or Hogend's bibliography (the books by Beggren and by Kennedy can be found in the national library.) There is also Heath's chapter "Euclid in Arabia."

    Euclid in the renaissance. For Durer and the connection of geometry and art in the 16th century, Martin Kemp's book The science of art has been recommended to me.

    Isaac Barrow

    Some of you may prefer projects with a more mathematical focus. Ideally, you can describe a mathematical idea both historically and from a modern viewpoint. Some examples:

    Choose one of the Delian problems. Beginning with some of the references on this page, or otherwise, study the history of solutions and attempted solutions, and describe one or two solutions mathematically in detail.

    Euclid's fifth axiom of Book I, and attempted proofs and replacements.

    Construction of one of the regular solids.

    Construction of the regular pentagon.

    Areas and volumes.

    The number theory books of Euclid. E.g. proof and statement of VIII 8. Today this would be deduced from the unique decomposition to primes. How does Euclid's proof compare?

    The geometric algebra thesis. What statements would be needed in order to interpret a field in a plane, together with a notion of line segments of equal length? Which of these statements are present in Euclid, and to what extent do they help understand the contents of the first books?