Next up on the list is Hippocrates, Volume I: Ancient Medicine (Loeb Classical Library, No.147). Well, that's just the first volume. The complete Hippocrati ccorpus comes in ten volumes. This is almost certainly more than is in the corresponding Great Books volume. Someone less crazy than myself may wish to pick up something cheaper and more compact like Hippocratic Writings (Penguin Classics).
That said, you can never really go wrong with Loeb editions. The translations are never worse than serviceable. And on top of the translation, each Loeb volume has the original text on the page opposite the translation. Each work comes with a brief introduction, usually spending more time on the manuscript tradition than the layman is likely to care for. They are hardcover editions with excellent binding and a compact size. I have also never seen a library copy of a Loeb edition that wasn't still in great shape. In the academic world, Loeb editions are often the standard version for the original text. And if you can read the original language at all, it is great boon. Sadly, I know only a few words of Greek and largely picked up the Loeb version because it's the only complete translation of the Hippocratic corpus available. But I will almost certainly stick to Loeb editions when I reach Latin texts since that's one dead language I can read.
Hippocrates was one of the earliest great medical practitioners that we know of. Works believed to have been written by him are abundant. Works falsely ascribed to him are likewise abundant. The confusion is partly due to the fact that an entire school of thought, the Hippocratic school, emerged from his teachings and writings. His writings fell out of use in the Latin West during the medieval period but were used widely in the Arab world during the same time period. His work is largely the reason for the superiority of so-called "Arab" medicine during the medieval period. He regained popularity in the Latin West during the Renaissance and managed to fade out of use only when he had finally been surpassed.
The corpus consists of a great deal of medical advice that is still valid. For example, many of his surgeries and methods for dealing with broken bones are still more or less the same as one would find in a modern hospital. His work with diseases, however, is much more limited. He had no understanding of germ theory, for example. And he cannot really be faulted for that. Germ theory only became a dominant idea when we could literally see the damn things under a microscope. If Hippocrates could see bacteria, he may have modified some of his views. So, really, it was a failure in the field of optics, not the field of medicine. A similar argument can be made for the field of astronomy. It's all just theorizing until someone whips out the optics and starts measuring things.
Hippocrates had some understanding of the limits of his knowledge, however. This can be seen in his treatments. While much theorizing is done about wet versus dry, hot versus cold, and other such out-moded ways of thinking about the body, his actual treatments hint at his level of faith in this system of thinking. His treatments for diseases are usually all about keeping the patient comfortable and nourished. Nourishment is more important than it may first seem. Many diseases either dampen the appetite or outright make a person unable to digest normal food. These days we can just feed a person intravenously no matter the situation. He proposes various gruel-like foods for most situations. If things get worse, there are things like hydromel (honey diluted in water). Not all of them are bad ideas if you find yourself with the flu or food poisoning.
Hippocrates was also careful to record his cases. Even if he failed to save a patient, that knowledge could help others recognize the same illness and potentially save lives down the road. And even continued failure had the potential to teach something new. He would always have clear notes about what didn't work so that he could try something new. For this reason Hippocrates is often credited with being the first real empiricist. This is often in the context of contrasting him to Aristotle. I do not believe that that is entirely fair to Aristotle. Aristotle, especially in his biological text that no-one reads, relies heavily on observation. It is true that most of his works relies largely on reasoning from first principles but that is not his only method. Likewise, Hippocrates sometimes attempts to develop theories based on widely-held ideas of his own time. Therefore, I would argue that the difference is merely one of degree, not one of kind. Greek thought was not as monolithic as commonly portrayed. Who knows how much more varied the picture would look if the works of more authors had survived?