Even though the greeks did not rely on a systematic study which method did they use to begin to know the world

The answer is observation. Observation is the gaining of information from a chief source. In living beings, observation works through the senses. In science, observation can also consist of the recording of data via the use of instruments. Greek philosophers move toward the big questions of life sometimes in an unpretentious scientific way, sometimes in mystic ways, but always in an imaginative fashion using observation.