"Lecture" for Week 6

Renaissance mathematics

To make things come out more evenly, some of the reading (not homework) for this week extends into the "transition period". Viete really belongs with the Renaissance cubic-equation people, anyway.

Here are some things that I thought just might be giving you trouble:
• Stillwell p. 93 refers to Euclid's proposition VI.28 without quoting it. In Heath's translation it reads:

"To a given straight line to apply a parallelogram equal to a given rectilinear figure and deficient by a a parallelogrammic figure similar to a given one: thus the given rectilineal figure must not be greater than the paralllelogram described on the half of the straight line and similar to the defect."

That is Euclid's prealgebraic Greek filtered through Heath's Downton-Abbey era English. Perhaps you would have preferred to remain ignorant.

• Stillwell p. 96 refers to constructible numbers without clearly defining them. Greenberg's text (see "Lecture" 4), p. 141, explains that the field K of constructible numbers is the closure of the rational numbers with respect to the square-root operation (applied arbitrarily often together with the more elementary arithmetic operations), or, equivalently, the smallest subfield of the real numbers with the property that every positive element has a square root within the field. These are precisely the numbers that are constructible by straightedge and compass in the Euclidean sense: given a segment, you can construct another segment for which the ratio of the two lengths is the number in question. Thus the square root of 2 is constructible (Pythagorean theorem), but the cube root of 2 is not (duplication of the cube is impossible).
• Allen (pp. 8 and 14) and Stillwell (p. 105) refer briefly to translations of the Elements (by Pacioli into Latin and by Tartaglia into Italian). The story of editions and translations of Euclid is really quite tortured. Greenberg gives the best short summary I have seen:

"... the Elements, written around 300 B.C., has survived, though not in an original manuscript written by Euclid himself. The version we use today has been reconstructed from a tenth-century Greek copy found around 1800 in the Vatican Library and from Arabic translations of other lost Greek copies and revisions.... The first printed version ... appeared in Venice in 1482 (Campanus' translation from the Arabic [into Latin]).... A new Greek text was compiled in the 1880s by Heiberg, and that was translated into English in 1908 by Sir Thomas Heath; it is the version to which English speakers mainly refer."

Heath goes into the history at great length in his History of Greek Mathematics and even greater length in his commentary to his translation of Euclid. Some main points: The book of Campanus was followed quickly by a rival translation by Zamberti, who worked from a Greek source, but one that was far from authentic. Contrary to Allen, Pacioli came next, attempting to correct Campanus and refute Zamberti. In 1533 a Greek edition was published by Grynaeus, which became the standard source until 1800. Tartaglia's translation was probably made from the Latin versions of Campanus and Zamberti, not directly from Greek (or Arabic). "All our Greek texts of the Elements up to a century [now two centuries!] ago depended upon manuscripts containing Theon's recension of the work" -- and Heath does not have a good opinion of Theon (Hypatia's father).

The point of all this is that early modern Europe's knowledge of what Euclid actually wrote suffered from, first, repeated translation through at least one intermediate language, and, second, the dubious quality of the surviving ancient Greek manuscripts. Theon and other ancient commentators did not distinguish between making an improved edition of Euclid's book and writing a new book for pedagogical purposes -- it is as if all freshman calculus and physics textbooks today were, or purported to be, new editions of Newton's Principia. Add the facts that inevitably some of the "improvements" were wrong and that there was no reliable way of archiving old versions (through many centuries of war and social decay), and it's clear that one had a mess.

• In Allen's discussion (p. 8) of the notation in Triparty, it was not immediately obvious to me that the second and fourth lines are translations of the first and third lines into modern notation. Thus .6.2 means 6x2 and .9.2.m means 9x-2, etc.

Cubic equations

Several years ago I actually needed to understand the exact solutions of a cubic equation for a research purpose. Here is what I learned. Since writing that I've learned still more, but I'll just point out a paper that interprets the constants that show up in the [delFerro-Tartaglia-]Cardano solution in terms of geometrical properties of the graph of the cubic polynomial, at least in the case where it is not monotonic:

R. W. D. Nickalls, Math. Gazette 77 (1993) 354-359.

Announcements and procedures

• The book review will be graded by me, not Ngoc Do, so be sure to mail it to me (March 2). I will be out of town late March 3 and all day March 4, so please do not expect an immediate response.
• The instructions for the "position paper" due April 13 are the same as in previous semesters. See the "announcements" part of the course web page.
• About the term paper due May 5, see items on the "Some useful links" page.