In the mid 1600s Europeans began to get serious about natural philosophy, or, as we would say, science. They founded academies to study it, set up correspondence committees to share result across the continent, began publishing the first journals to report on their findings. Which raised a very important question: what was natural philosophy, and what was not?
The new discipline first of all required belief in a natural world that was in some way separate from things one might call divine or supernatural. Many theologians simply refused to accept this division, arguing that the whole universe from the throne of God to London to South America to the pit of Satan was entirely of the Almighty, and therefore all knowledge was theological knowledge. But the idea rapidly became entrenched that one could advance knowledge by setting that question aside and considering things like magnets or balls rolling down inclined planes from a strictly material perspective; God of course might decide to change things around, but since in practice he rarely did so, the assumption that he would not held sway.
But once people had divided the world into natural and supernatural it remained to map out the boundary between them. It was no easy task. For example one much debated question was whether life could be explained in entirely material terms or required something supernatural, some divine spark or elan vital, to separate it from inanimate matter. When you consider how far they were from being able to explain anything about how living organisms worked, it took a certain amount of hubris to assert that this was in fact an entirely natural question and therefore under the purview of the new societies. But many did assert this, and others held that while the ultimate mystery of life might be beyond science, science could still learn at vast amount about it. Like the question of how God operates in the world, the question of life's origin was set aside, and work pushed forward on understanding how seeds developed and what the organs of the body did.
One very interesting debate concerned alchemy. Alchemists considered what they did scientific in our sense of the word, meaning they were doing experiments aimed at understanding, and gaining power over, brute matter. Many of Europe's top scientists were alchemists, including Isaac Newton and Robert Boyle. Nonetheless, alchemy was very quickly written out of Natural Philosophy. I think this was ultimately a matter of style. Alchemists wrote in a deliberately obscure way, using ancient, opaque terminology and exposition that amounted to misdirection, and to outsiders the whole business looked like mysticism. Plus, alchemists had been wrestling with the same problem (turning lead into gold) for 1500 years without making any discernible progress. So they were booted out of the scientific academies, and by 1750 alchemy had slid out of elite culture and into the realm of cranks.
Likewise witchcraft. Should scientists investigate the claim that certain fey old women could curse sheep to death? The English Royal Society was asked to do so, but refused. Organized science never took a stand about witchcraft in the early period but made their opinion fairly clear by resolutely refusing to get involved.
On the other hand, the notion that bodies could influence each other from a distance was brought back in. To Galileo this was the very definition of magic, so he rejected the idea that the moon could influence the tides. But experimental study of magnets showed that this absolutely does happen, and when Newton formulated his theory of gravity he simply asserted that while he could not explain how the force could cross vast distances, he knew that it did.
These questions are still with us; somewhere in the world right now two scientists are arguing about whether the many worlds interpretation of quantum mechanics is science, or whether it is scientific to speculate that DNA arrived on earth from space. But they were especially acute in science's early days.