I'm the alchemy guy... but I wasn't going to answer this, because the question has a ton of misconceptions to get through before giving you an answer.

And then, I've actually already answered this before on this subreddit (many times). ...but no one else answered, so here I am:

Many factors changed the world from supernatural answers to scientific ones with the question: "what is matter?"

In the middle ages, all matter was made of four elements. And then all knowledge and theories followed that one foundation.

We do see empirical research in the middle ages, and even peer review, and even natural theories, and not just supernatural ones. So there's already one misconception. But you do have a point to a degree.

So what changed? The Reformation and the Counter-Reformation is really the crux of your answer. When the Reformation happened, protestants ranted and raved against the Aristotelean dogma and the Old Ways of the catholic church, which was steeped in ritual and superstition.

With the Counter-Reformation the Priests quickly changed their tune (the laymen were much slower) superstition was out!

So when did the last supernatural explanation of something happen that was peer reviewed? Well... the 20th century. Think of the psychic experiments of the CIA! Or the more serious investigation of ghosts, etc in the 19th century and again in the 1950's (let's say).

So there's another misconception: it didn't stop until way later than you think (and it's still around, the scientists that are curious in those ways are ridiculed now, though) And the other misconception: we had empirical experiments in the 13th century (before the renaissance even) We had peer review in many ways: All sorts of books on alchemy and magic and astrology were sent to other naturalists, astronomers, mathematicians, physicians, etc (scientists) for them to write a commentary on (read: peer review)

Okay, so to summarize: scientific peer review happened MUCH earlier than the Royal Society. It just wasn't the norm everyone had to follow. Superstition and supernatural explanations still make it into academia TODAY (if not mainstream) see: Creationists. Some of those guys are tenured and have PhD's in biology, etc. And they still believe the world was created by supernatural means. They are scientists (though ridiculed) and it's 2015 the last time I checked.

So there's no division point. It's a slooow transition that took 400 years and is still going on. Maybe not a smooth transition, maybe some steps, but a slow transition nonetheless.

But that's not the answer you wanted to hear. So here it is: In the 18th century strides were made. Alchemists had discovered several actual elements (like phosphorous by Hennig Brand, using an alchemical recipe that had aspects of sympathetic magic and involved him distilling his own urine. We start to see ideas of conservation of mass, etc in chemical reactions. We see the beginnings of the periodic table, and we've already been seeing people disagreeing with Aristotle and the four elements for centuries before the 17th century (but they were in the minority) so FOR ME, the end of the old and then beginning of the new is 1805 with the publication of John Dalton's Atomic Theory ..again, it was a slow transition, but if you want a date where the old was dead, and modern science was definitely here to stay: 1805 is a date as good as any :)

If you want to read WAY, WAY more about all of this, I've answered this same question many different ways: https://www.reddit.com/r/AskHistorians/wiki/profiles/bemonk