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This question pursues the history of this noun and its etymons; compare with ELU and FSE.

physic (n.)
[1.] c. 1300, fysike, "art of healing, medical science," also "natural science" (c. 1300),
[2.] from Old French fisike "natural science, art of healing" (12c.)
[3.] and directly from Latin physica (fem. singular of physicus) "study of nature,"

[4.] from Greek physike (episteme) "(knowledge) of nature,"
from fem. of physikos "pertaining to nature," from physis "nature,"
from phyein "to bring forth, produce, make to grow" (related to phyton "growth, plant," phyle "tribe, race," phyma "a growth, tumor")
from PIE root *bheue- "to be exist, grow" (see be).
[…]

Historically, how was 4 connected to 1-3?

I do not understand the underlying notions because I (in 2016) cannot empathise with the perspective of the Ancient Greeks, Romans, or Old French of the equivalence of (knowledge of) nature, natural science, healing, medical science, and modern physics.

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As the knowledge grows various disciplines separate from each other. In the beginning there was only one "science" and it was called philosophy. Mathematics and astronomy separated very early. All the rest was "physics", the study of nature. Medicine also separated early, but not completely, because very little specifically medical knowledge was available. Until the modern era many scientists also practiced medicine, and such things as astrology were considered indispensable for medical education. We see a lot of traces of this situation in the language: even in our time a mathematician or a physicist is called "Doctor of Philosophy (PhD)". But if you hear just "Doctor" you may assume that the person probably practices medicine.

As you noticed the word "physics" in Greek means just the knowledge of the nature. But nature includes not only the nature of stars and the laws of motion, but also the human and animal and plant body.

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