Hello, everyone!

I am pleased to report that I have reached the twenty-fourth disc of the Howard Eves Audio Collection. This means there are four more discs left of my “elective, non-credit course” in the History of Mathematics with the late Professor Howard Eves. Certainly, the summary contents of the recording will posted here and there were no interruptions aside from my mandatory breaks. So, without further ado, here are the day’s events.

After placing my backpack in my own cubby locker, I wanted to check if anyone had broken down the remaining boxes after I left yesterday. Unfortunately, that was not the case. So, I finished breaking the boxes down and stacking them in the shelves as I had done before. When I was done, I gathered the materials for the audio collection and went to the first floor.

Resuming where I left off on last Friday, the recording continued Dr. Eves’ lecture as he was teaching about drawing tangents with straight edges and how Pascal’s hexagon theorem was used in tangents. He then moved on to Book III of Apollonius’ *Conics*, where he began explaining terminologies and subjects in Projective Geometry: Harmonic Properties, Theorems Concerning Products of Segments of Intersecting Chords, and Focal Properties among others. The professor explained that the focal directrix property was not mentioned in any of the books, but Dr. Eves theorized that the concept might have been in the lost Book VIII. He then began to demonstrate the theorems in Book III.

After his demonstrations, Dr. Eves began explaining about the other books: Book IV of *Conics** *dealt with Converses of the theorems in Book III, Book V of *Conics *had Apollonius using normals in his systems, and Dr. Eves made comparisons of how Apollonius conducted his analysis to Cartesian analysis. He noted Book VI of *Conics* was devoted to equal and similar conics and the professor used the example of the George Washington Bridge and the Palisades to demonstrate parabolas in those concepts. Book VII featured Conjugate Diameters, so Dr. Eves demonstrated what conjugate diameters are. The professor then remarked Book VIII being lost then ended the class.

The next disc was organized in bizarre fashion: Part Two of the lecture was Track One, while Part One of the lecture was Track Two. I made note of this and switched my notes to reflect this case. After beginning Track Two, Dr. Eves gave an overview of the previous class on Apollonius before he began his lecture with Apollonius’ other written works.

The other written works by Apollonius are all currently lost, they were mentioned in the works of Pappus of Alexandria. After giving a summary of Apollonius’ *On Proportional Section*, *On Spatial Section, On Determinate Section*, and *Tangencies*, Dr. Eves noted that the knowledge from this works were preserved after these books were translated in Arabic. He also discussed The Problem of Apollonius, the use of a Straight Line or Circle (a Stircle, as he called it), and demonstrated a Nine Point Circle. It was during his discussion of the Nine Point Circle where the professor detoured the class into a brief lecture on how the theorem was discovered by Karl Wilhelm von Feuerbach. Dr. Eves told of the mathematician’s odd life and loss of sanity before resuming the lecture on Apollonius’ written works.

The next works were Apollonius’ *On Verging Constructions, *followed by Apollonius’ *Plane Loci. *At this point*, *Dr. Eves demonstrated a problem that lead to a discussion to the Circle of Apollonius. He then name prominent geometers who had given great solutions to the problem such as Princess Elisabeth of Bohemia (Dr. Eves erroneously referred to her as the daughter of Frederick the Great, when she was the daughter of Frederick V). He then demonstrated a problem in Book VII before commenting on efforts to restore the lost books. At this point, Dr. Eves commented that he believed Jakob Steiner had surpassed Apollonius’ skills as a geometer.

Dr. Eves began a new section with a lecture on Hipparchus, Menelaus, and Claudius Ptolemy, which he clarified Claudius Ptolemy did not belong to the same royal family who ruled Egypt after Alexander the Great. His lecture began with an Early History of Trigonometry from Egyptian sources, Babylonian sources, and how Greek interest in astronomy lead to spherical trigonometry. He then focused on Hipparchus.

Dr. Eves listed the accomplishments of Hipparchus of Nicaea: The Lunar Month (the mean, specifically), discovered the procession of the equinoxes, computed the lunar parallax, determined the perigee of the Moon to the Earth, cataloged the positions of 850 fixed stars, introduced the degree, minute, and the second to the Greeks, and advocated a prototype of latitude and longitude. Unfortunately, the audio was a little bit choppy for two minutes. Thankfully, the choppiness ended as Dr. Eves commented on Theon of Alexandria mentioning a 12 book treatise by Hipparchus in his own works. The professor discussed Hipparchus’ Table of Chords before switching to the next scholar, Menelaus of Alexandria.

The major work discussed was* Spirica* by Menelaus of Alexandria, which held Greek theories of Spherical Trigonometry. The concepts Spherical Triangles and Congruent Cases of Spherical Triangles were detailed by the professor. Dr. Eves mentioned how Menelaus discovered the sum of the three angles of a spherical triangle is greater than 180 degrees. He noted how while Book II was devoted to astronomy, Book III developed Spherical Trigonometry further. At this point, Dr. Eves mentioned how he had learned in Spherical Trigonometry in grade school as well as how it is part of pilot training. Unfortunately, Claudius Ptolemy was not mentioned as the recording ended as Dr. Eves demonstrated Menelaus’ theorems.

The next recording began with a brief overview of the previous class by Dr. Eves, which apparently ended as they were going to discuss Hero of Alexandria (also known as Heron of Alexandria). The professor resumed this lecture by describing the inventions of Heron and his works in mechanics and geometry. The written works of Heron was the next topic.

Heron’s *Metrica *revealed how the ancient Greek solved the area of triangle in terms of its three sides. Dr. Eves also mentioned that Heron was the one who developed the approximation of square roots. In his summary of Heron’s *Pneumatica,* which described approximately 100 machines within, Dr. Eves told of a colloquium on Heron’s machines that was taught in the Math Department at the time. After mentioning Heron’s *On the Dioptra *and Heron’s *Catoptrica,* the professor changed the subject to Ancient Greek Algebra.

Dr. Eves mentioned how the characterization of three stages of the development of algebraic notation was first formed in 1842 to give a better context for the next subject. The first of the three stages was the Rhetorical Stage. The professor mentioned that some origins of this form of symbolic algebra can be found in Metrodorus the Grammarian’s *Greek Anthology*. Unfortunately, my time to leave came as Dr. Eves was beginning to work through the related problem studies.

Well, I should be able to finish the twenty-fourth disc tomorrow as well as twenty-five. At this pace, I might finish the Collection by the end of the week. That is if no further interruptions occur. If such things come to pass, then the events will be recorded. Enjoy the rest of the afternoon and stay safe. Bye!