May 10, 2018 – “The Three Unsolved Problems of Higher Geometry from Antiquity”

Hello, everyone!

I sincerely wish everyone has been enjoying their summer break (however large or small it may be), but this does not end my volunteer work at the University of Central Florida Library Special Collections and University Archives. Actually, my status as a volunteer ends next week and such it will be for next week’s post. Instead, expect two posts next week as I become an official employee on Friday, May 18. So far there are no hiccups in the system and everything seems to be on track. But, that is next week and this week continues with Dr. Eves’ History of Mathematics non-credit course.

Resuming from last week, the recording from Dr. Eves spoke of the Three Unsolved Problems of Higher Geometry from Antiquity: the Duplication of a Cube, the Trisection of an Angle, and the Quadrature of a Circle (Square of a Circle). Before explaining the contents of this lecture, I have to note there was a bit of a mix-up as I was listening to the recording. If by going in chronological order, the disc that should have in the fifteenth case was instead in the sixteenth case and vice versa. That was confusing when I switched discs, but was eventually sorted out.

To get back on topic, Dr. Eves began by lecturing on the Duplication of a Cube. He explained the contributions of Hippocrates of Chios’ reduction to the insertion of two mean proportionals and how the concept was used to solving edges of a cube. From this point, Dr. Eves made sure students understood what Higher Geometry was and the role spheres had its study. The next contribution  was the lost solution of Eudoxus of Cnidus (lost as in the actual solution has been lost to the annals of time and only references to it have been passed down through secondary sources) and the influences of Archytas’ teachings on Eudoxus’ other contributions.

The next major development was solutions provided by Menaechmus’ invention of the conic sections. After explaining what conic sections were, Dr. Eves highlighted the debate between mathematicians regarding the origins of analytical geometry (was it discovered or invented) before explaining his own personal findings in the debate. This lead to discussions of other major developments such as the Conchoids of Nicomedes, Diocles’ Cissoid, and the contributions of Apollonius of Perga. It was during this section that Dr. Eves explained the lack of awareness by practitioners until late nineteenth and/or early twentieth centuries that these equations could not be solved by restricted tools (straight edge and the compass).

Dr. Eves moved on to the next topic of Mechanical Solutions and the ill-attribution of them to Plato (Plato apparently despised them). He demonstrated how Plato supposedly invented a mechanical way of drawing a certain figure. Dr. Eves concluded the topic with the Life, Contributions, and Tragic Demise of Eratosthenes. Eratosthenes was one of those humans gifted with mental and physical prowess that unfortunately was stricken blind as he entered advanced age. Unable to read nor observe nature, the Greek became depressed and resorted to the extreme measure of starving himself to death to end his suffering.

The next topic was the Trisection of an Angle and theories of its two possible origins: multi-section of lines or constructing regular polygons (nine-sided, specifically). The first Greek mathematician discussed was Archimedes and his different strategies developed to solve trisection of triangles. From here, Dr. Eves explained in greater detail the Conchoid of Nicomedes and the “tomahawk” trisection that was developed as an alternate method. He also explained the meaning of “QED” before demonstrating how two tomahawks could be used to quintet-sect an angle.

The topic of Geometric Progression was discussed and how Elementary Algebra can be used in the equations. The Sum of the Terms to Infinity for the Convergent Geometric Series was discussed next followed by a discussion of the flaws of approximating an angle (the larger the angle, the more inaccurate the approximation). Unfortunately, the recording ended as Dr. Eves was discussing the contributions that were later made by Albrecht Dürer.

On the next disc, Dr. Eves lectured on the Quadrature of a Circle after reminding his students of materials that would be covered on the next test. Egyptian approximations were briefly discussed before stressing the importance of Hippocrates of Chios’ contributions, especially the Hippocratic lunes. After demonstrating the Hippocratic lunes, the professor discussed Christian Goldbach’s Conjecture and the similar status the two methods shared. From here, Archimedes’ life was revisited and his contribution of the Spiral of Archimedes. Dr. Eves commented that history was the story of  The Story of Thinkers versus Thugs (Thugs win, but Thinkers’ legacy outlasts Thugs).

After explaining the Quadratix of Hippias, Dr. Eves gave directions for his students regarding the deadline for paper assignments as well as Final Exam schedule (no classes). Unfortunately, my time had run out and was forced to end my session as Dr. Eves resumed his lecture. The good news is that by the end of next Wednesday, I will have only ten more discs to listen to before finalizing the processing of the collection.

That is it for this week. Next week begins my summer semester, ends my work as a volunteer, and starts my duties as an employee. Very exciting developments indeed! So, everyone enjoy their weekend and stay safe. Bye!

Author: 57r3l574d

I am currently a Graduate Student at the University of Central Florida and simultaneously employed by the university library's Special Collections and University Archives as a Other Personnel Service (OPS) Student. Expected to graduate in 2019 with a History MA - Public History Track.

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