Next up on the list are The Works of Archimedes. This Dover edition is basically your only option these days. Sir Thomas Heath's translation is the only game in town and Dover's edition is the only edition still in print. If you are desperate for a better quality binding, Cambridge University Press put out Heath's original translation in hardcover in 1897. I opted to track down a copy of that. The downside is that Archimedes' The Method would not be rediscovered for nearly another decade after the CUP edition came out. They released a pamphlet supplement of The Method also by Heath. Unfortunately, I only discovered this after reading all of the 1897 edition. The supplement is quite rare now but I found exactly one copy in the hands of a Latvian book dealer. It was surprisingly cheap. Unfortunately, it will probably take several weeks to arrive so I may need make an addendum to this post if anything really interesting crops up in there. So, while I have not seen the Dover edition in person, it is probably the best of bet for any sane reader who does not want to spend a lot of time and cash with antiquarian book dealer listings.
Archimedes' work greatly resembles that of Euclid, from a modern perspective. So, a lot of what I said about the eye-opening experience of Euclid could also easily apply to Archimedes, if Archimedes is the reader's first introduction to Greek mathematics. However, I think Euclid would be easier on the first time reader. Euclid's Elements describes a system of geometry with enough detail that one can just start on the first page with no prior knowledge and make it all the way through to the end. Archimedes is not so forgiving. While Euclid's work can and did (even still does, in certain corners) serve as a textbook, most of Archimedes' works were simply problems that he wanted to tackle. His intended audience was fellow mathematicians, such as Dositheus of Pelusium, Conon of Samos, and other members of the illustrious Alexandrian crowd. Further, Archimedes' surviving works make reference to and depend on other works that do not survive. Since we have a full-blown system of geometry and mathematics of our own to rely on, filling in the missing piecesis not too difficult. However, these things together mean that Euclid is going to provide a smoother first hit for a prospective student of Greek mathematics.
Archimedes has a lot of very Euclid-like proofs on spheres, cylinders, circles, conoids, and spheroids. But Archimedes is at his best when he tries to tackle specific and concrete problems. This is why, despite his great purely mathematical achievements, he is considered the father of engineering; while he was a great mathematician, he was the greatest engineer. I will give a few examples of my favorites.
Archimedes, like many mathematicians of the centuries, sought to figure out the ratio of a circle's circumference to its diameter, i.e. pi. There is an age-old philosophical question about whether or not there is a difference between a circle and a polygon with an infinite (or simply extremely high) number of sides. An ideal circle certainly has no sides. But maybe in the real universe of Planck measurements there is no difference. Ultimately, for Archimedes' purposes, and most purposes involving pi, the answer does not really matter. The length of the sides of a polygon are very easy to measure. And a polygon drawn hugging the inside or outside of a circle will more closely approximate the circle as the number of sides goes up. Therefore, a polygon with a high number of sides can give a good approximation of pi. And the more sides one adds, the closer one gets. Archimedes chooses to go up to 96 sides, one polygon hugging the outside and one hugging the inside of a circle. This gives a lower bound for pi at 3 10/71 and an upper bound at 3 1/7. If these two values are averaged, it gives pi accurate up to 3.141. Further sides could be used to produce infinitely more values. Christoph Grienberger, an Austrian mathematician, pushed Archimedes' approach all the way to 39 digits of accuracy. Interestingly, Wikipedia is very misleading on this topic. Archimedes stopped at his 96-sided polygon proof because it was more than sufficient and smashed the accuracy of all existing estimates. Extending it further was an obvious practice left up to the reader. He had solved pi.
Archimedes also has a fascinating work where he describes the process of calculating how many grains of sand it would take to fill up the universe. He does this in response to the notion that there is no number big enough to count the grains of sand on Earth. His numbers for the size of the Earth and the universe are wildly off. His estimation for the size of the Earth is off by an order of magnitude and he dismisses the approximations of his own day which were actually fairly accurate. Still, while his methodology is bogus, he comes up with the same basic idea of our modern scientific notation of large numbers to represent the number of grains of sand it would take to fill the universe. In a sense, he fumbles the theoretical but nails the practical here.
I was always fascinated by the mechanics of see saws as a child, particularly the "trick" of moving along the seesaw to allow children of different sizes to still play with one another. He describes the math behind this process, along with a few methods for figuring out the center of gravity of common shapes, in his On the Equilibrium of Planes. The math here was surprisingly simple and a real personal joy for me.
Archimedes also describes the mechanics of hydrostatics, i.e. how things float, sink, and/or displace liquid. In other words, Archimedes knows what floats your boat. Given that we have already established that he also knows what tilts your seesaw, I think you are in for a good time. The math to solve the famous problem of how much gold is really in a crown is all laid out here, though he does not actually recount the story. It is possible that he came up with the method and wrote it down before discovering that particular practical use. It is also possible that the story of Archimedes and the gold crown was merely an illustrative story that later came to be treated as history by Vitruvius.
Heath, in his introduction, argues that Archimedes was fairly close to giving us Calculus. And I think he is right. Archimedes' section on spirals describes them both in geometric terms, i.e. moving along a circle and a line at the same time, and in terms of a change in magnitude over time. He also tackles the problem of calculating the area of these spirals. Elsewhere, he deals with calculating the area under many rather complicated curves. Combined with his fascination with mechanics, who knows what could have been, if only more interest had been given to Archimedes, more of his works preserved, or the scholarly community in Alexandria not hobbled over the years?