Apologies for the tardiness of this blog, but the Spring 2018 semester is ending shortly and final assignments require my attention. But, there is good news: Ms. Rubin informed me that I should be looking for an e-mail from the Human Resources Department at the University of Central Florida and a tentative start date of May 18, 2018 to begin my employment at the archives. Next week will be when Ms. Rubin informs me more about the process and hopefully what I need to do in regards to it. On that note, the my “elective course” in the History of Mathematics with the late Dr. Howard Eves continues with lessons regarding Greek mathematics and mathematicians.
Resuming from last week, Dr. Eves’ lecture continued from Proclus and A Commentary on the First Book of Euclid’s “Elements“ which contained a secondary recital of Eudemus’ History of Geometry and the method now known as the Eudemian Summary. From there, Dr. Eves explained the role Thales of Miletus had in developing Greek mathematics by traveling to Egypt and measuring the dimensions of the pyramids by the shadows that were projected by the sun. Other contributions of Thales were also discussed such as the Thales Puzzle, his work with angles, and the covalence of triangles. Then came Pythagoras.
From this point forward, Dr. Eves’ lectures described a brief account of Pythagoras’ life, his foundation of Pythagoreanism on the colony of Croton, and his eventually murder. Dr. Eves provided a description of the Pythagorean beliefs in sets of natural numbers, Pythagorean influences in tautologies and deductions and the symbols used in tautologies. Also discussed was contributions of Hippocrates of Chios (not Hippocrates of Kos, the physician).
As a bit of a break, Dr. Eves regaled several anecdotes and stories regarding Pythagoras and the Pythagorean Brotherhood. These included Pythagoras and “The Lure of Geometry” legend, the rule of the Pythagorean Brotherhood of giving credit to “the master” than themselves, Pythagorean spiritual beliefs and the story of Pythagoras meeting a dog, the story of Damon and Pythias, and the story of the Pythagorean and the Inn Keeper. In addition, Dr. Eves told some stories regarding Thales as well.
The next lecture was about the “perfect numbers” and how before 1952 there were only 12 “perfect numbers” were all even numbers. Euclid’s formula was used to find five more perfect numbers through a digital computer. The Swedes found more using their own computer. More were found through using IBM computer, leaving total at 30. At this point the discussion shifts to the history of finding amicable numbers.
Aside from Pythagorean contributions, Pierre de Fermat discovered two more amicable numbers in 1636, but Thābit ibn Qurra has been attributed as the originator. René Descartes and Leonhard Euler discovered more pairs and Nicolò I. Paganini (not the violinist) found a smaller pair in 1866. The discussion then shifted to the development of the two categories of arithmetic (number theory) and logistics, figurate numbers and triangular numbers, pentangular numbers and square root numbers. While Dr. Eves ended the section, he went on to discuss the Pythagorean Theorem.
In the beginning of the next recording, Dr. Eves spoke about the contributions toward Greek polygons by and quirks of Johann Carl Friedrich Gauss (had peculiar publishing habits). This discussion then eased into Pythagorean geometry. Dr. Eves believed Euclid, Hippocrates of Chios, and others failed to make a singular mathematical deductive chain due to theory of parallel minds. This lead to the a discussion of the introduction of postulation methods by Pythagoreans. Stemming from this discussion, topics such as Pythagorean triples, Pythagoreans and the diagonal square, and why the discovery destroyed the Pythagorean philosophies regarding whole numbers (hint: irrational numbers). The latter was broken down for more detail.
After making a reference to A Mathematician’s Apology by G.H. Hardy, a geometrical method of showing the square of irrational numbers was discussed before Dr. Eves explained how ratios and integers undermine the Pythagorean beliefs of whole numbers. Dr. Eves then explained how Pythagoreans attempted to find the square through geometrical methods. At this point, the lecture ended.
The next recording began with a review of the material covered on the next test for Dr. Eves’ students. After which, Dr. Eves began a lecture on the Pythagorean transformation of areas and used triangles and pentagons as examples to demonstrate this transformation. This was followed by a discussion of the Postulates of the Discourse and Euclid’s “definitions” of simplistic mathematical terms in The Elements. Unfortunately, the lecture had to be cut short as it was time to leave for the day.
So, this concludes the retelling of this week’s events. Next week there will be more details regarding my employment endeavors and more from the Howard Eves Audio Collection. Until then, enjoy the weekend and stay safe! Bye!