Wednesday, September 23, 2009, 3-4pm
130 ECC

Trigonometric Tables, their Utility and Making in Late Imperial China

Dr. Jeff Chen
Dept of Mathematics
St Cloud State University

Integration of Jesuit and Chinese methods in astronomy was one of the phases Xu Guangqi (1579-1659) prescribed for the Calendar Reform in 1630s. One of the obstacles the integration of the two methods was the incommensurability of trigonometric tables the Jesuits introduced and the measuring unit for arcs in the traditional Chinese system. As an important computing instrument, trigonometric tables in China as well as its utility and making warrant close examination.

The Jesuits utilized trigonometric tables to simplify the computations in astronomy while Chinese astronomers, before the arrival of the Jesuits, employed the method of interpolation to serve the same computational needs. Although the Jesuits provided the basic principles of making trigonometric tables, there were technical details left unexplained, which made the reconstruction of a complete trigonometric table impossible. Some Chinese scholars in the 18th century tried to remedy this situation by changing the measuring unit of arcs from 360 degree for the full circle to some other units. Such changes rarely had any following. In the 19th century, after the publication of Geyuan milü jiefa (Quick Methods for Circle-Division and Determining the Precise ratio of the circle) by Ming Antu (1692?-1765?), a trigonometric treatise which discussed the power series approach of finding the length of an arc from its sine value and related properties provided an easy and fast way to complete the construction of trigonometric tables. In this presentation, I compare the Chinese indigenous computation methods, trigonometric tables introduced by Jesuit introduced by Jesuits, and trigonometric tables constructed by Chinese scholars in the 17th-19th century utilizing various computation methods to investigate how Chinese scholars at different time viewed and received trigonometric tables.

Wednesday, October 7, 2008, 3-4pm
ECC 130

Depressing the Cubic

Dr. William Branson
Dept of Mathematics
St Cloud State University

A cubic equation x³ = ax² + b can be depressed into y3 = Ay + B. Algebraically, this is done by setting y = x - (a/3). Girolamo Cardano, in his 1542 classic Ars Magna , gave a geometric demonstration of this, followed by a statement of the algebraic rule. His geometric construction and his algebraic rule, however, have little connection--it's almost as if they were excerpts from different books, spliced together. And so they were--medieval Italian mathematics had two distinct strains that rarely met: algebraic and geometric. Cardano's text is one of the first that tried to tie the two together. He was not, from our point of view, entirely successful, for his geometric arguments are rarely proofs of his algebraic claims. Yet his peers acclaimed his book. And so, in this talk, we will address two things. First, what was his geometric argument for depressing the cubic? This is a nice bit of solid geometry that is rarely seen in modern mathematics courses. Second, what is a proof? Cardano's proofs are not convincing to us, yet they were to his peers. Similarly, our proofs are found convincing to us, but researchers in automated proof checking would argue that, in the future, any proof that doesn't start with the ZFC axioms will be found unconvincing.

Monday, November 16, 2009, 3-4pm
ECC 130

On Constructibility Results for a Class of Non-Selfadjoint Analytic Perturbations of Matrices with Degenerate Eigenvalues

Mr. Aaron Welters
Dept of Mathematics
University of California, Irvine

In this talk, I consider the problem of finding explicit recursive formulas to compute the perturbed eigenvalues and eigenvectors of non-selfadjoint analytic perturbations of matrices with degenerate eigenvalues. Based on some math-physics problems arising from the study of slow light in photonic crystals, we single out a class of perturbations that satisfy what I call the generic condition. It will be shown that for this class of perturbations, the problem mentioned above of finding explicit recursive formulas can be solved. Using these recursive formulas, I will list the first and second order terms for the perturbed eigenvalues and eigenvectors of perturbations belonging to this class.