The idea is that the concepts of DNA, inheritance, and micro-organisms were discovered significantly earlier in history. Roughly 50 years after the founding of Alexandria as a city state in 332 B.C., Greek scholars ended up developing the first microscopes. During their studies of human anatomy, they made the startling discovery of cells.
They posited theories that something within these small parts of our body (DNA) strongly affected the traits passed on from parents to their children. This appeared to be true, given what they could observe in plants, domestic animals, and even humans.
Under orders from Lysimachus, they began the ambitious project of selectively breeding humans in an attempt to bring out these characteristics:
Greater strength, stamina, and physical coordination (agility/dexterity).
Assume their work would be continued by the Roman Empire, and later the Byzantine and Holy Roman Empires as time marched on. By 1806, when the Holy Roman Empire was dissolved at last, how much progress could feasibly have been made through a selective breeding program?