This book is not quite what I was hoping for, and I suppose in some way that is why I cannot give it the full 5 stars despite doing an admirable job in its actual intent. The book is less about the science and medicine of Ancient Greece and Ancient China, than on what I would call the Weltanschauung, or world-view, behind science and medicine in Greece and China from approximately 400 BC to 200 AD. Instead of calling it a Weltanschauung, they call the cultural/political/societal milieu of each society a manifold. They then proceed to examine what drove the societies at the times in question, and explain what the typical philosopher, scientist, or skilled artisan would be expected to do and believe.
This is an interesting comparison, and can help explain how the Greeks and Chinese arrived at the conceptual frameworks they used in science. For this, the book is excellent. My main problem is that it does not give me a good sense of how science and philosophy was actually used in each society. I have a lot more experience with Greek science, and so know approximately how their methods worked and how they compare with modern methods. The authors are careful to not judge either society with modern methods, but it would have been helpful to know from a modern perspective how "accurate" certain ideas are. Here I'm thinking primarily of astronomy, where it would have been nice to know how Chinese (and Greek, though, I am aware of the cognitive models) thought of calculations in a physical sense, or if they did. These were the type of insights I was looking for and did not find in the book. I learned a great deal about what drove people in philosophy, but didn't get a good sense of the actual accomplishments made.
Quite an interesting book, that if you have an interest in how/why Chinese/Greek science became what it was, is a good introduction. If you're looking for specific scientific comparisons, then this is not the right book.