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EDIT: Just to clarify, when I refer to magic, I am referring to the supernatural variety, not the stage variety.

PREFACE: It was recommended to me that I might try and have this answered by the historically inclined by the people over in StackExchange Worldbuilding, so here I am. I would greatly appreciate any real-world insights people might have into this topic.

ORIGINAL QUESTION:
I've long had difficulty trying to put this into words, and trying to find reliable sources that I could look into to inform me on how to move forward with my writing. I'm hoping someone out there more learned than I can point me in the right direction.

I want to create a magic system, but I don't want to create a magic system based on the more cliche offerings you find in video-games or YA fantasy, stuff that hinges on, say, four basic elements, etc. I have two very specific parameters I'm trying to work within to make this system and I hope it helps explaining to you what I will need to move forward:

The magic system is at least somewhat based on the real-world origins of magic, culturally speaking. I am trying my hardest to find some reliable historical accounts that aren't heavily biased or religious in nature that can tell me, perhaps from an anthropological standpoint (or any other scientific one, I'm not picky) that I can base this magic system on. Like, for example, I've found some articles explaining how some formal practices as we know them may have originated in Egypt, and the idea of magic and magicians might have Persian roots, etc., but I don't know how trustworthy they are. This also includes the rationale behind its practice, if possible: like, say, how practitioners used it, why they carried out certain rituals to do it, the context of and understanding behind things like spells, religious connotation, where people actually thought magic came from and what it was, etc. Basically anything that gives me a timeline to work with, and a greater understanding of where our modern understanding of magic came from.

The magic system is not showy, based on visuals, and is almost terrifying. I hesitate to list examples, but I like how some older films used to present the idea of magic, at least visually. I liked, for example, how it was depicted in the movie DRAGONSLAYER, or maybe the better example is in LORD OF THE RINGS, especially in the battle between Saruman and Gandalf: there aren't laser light-shows, it's invisible, it's subtle, it's even scary and transformative, more upsetting and less convenient than stuff you might see in WORLD OF WARCRAFT, etc. I feel like there's more power in the less-is-more depiction here. I don't know that necessarily helps anyone give me an answer, but it is part of my overall process, so...including it here.

I hope I've made it clear enough: if not I'm willing to answer questions to hone in more on a clearer question. Please let me know if you have any other questions that will help you answer the question and I will do my best to answer. Thank you.

Magical practices are either historical or fictionalized. The former are mixed up with religion, as you noted, and will show a lot of regional and epochal variation. The latter have already synthesized and refocused, but don't generally come with footnotes and historical justifications. For example, the magic system in "Monty Python and the Holy Grail" appears to have roots in Anglo-Saxon and Celtic traditions, but saying that doesn't explain why Tim the Enchanter would use explosives or wear ram's horns.

I'm skeptical that any magical practices are or were universal. To create or identify a magic system more strongly linked to historical practices, I think you need to pick a regional/temporal/cultural context as a model and work forward from there. Some of your reading material could include, for example, Magic in the Graeco-Roman world, Magic in the Ancient World, Aztec use of entheogens, and Magical Elements in the Avesta and Nerang Literature. The fact that religion is prominent in magic is an inherent part of historical practices; for your model to be accurate, you might want to incorporate religious elements instead of avoiding them.

You need a copy of Authentic Thaumaturgy, by Isaac Bonewits. He was a modern-day ritual magician, who managed to earn a degree in magic from the University of California.

Authentic Thaumaturgy is his attempt to systematise the underlying rules of historical magic systems: it's written as a tool for creating magic systems for role-playing games. It's available in PDF from its most recent publisher, Steve Jackson Games.

Not to contradict or compete with Aaron Brick's or John Dallman's posted answers, just helping with some additional resources as food for thought.

Here are three sources that explore magic use in ancient Egypt (a potentially rich source for ideas):

1. Ancient History Encyclopedia - Magic in Ancient Egypt
   


2. Ancient Origins - The Magic of Heka
   


3. BBC - Ancient Egyptian Magic
   

Brandon Sanderson (author of The Wheel of Time series) has a blog where he offers Sanderson's Laws of Magic:

1. Sanderson's First Law


2. Sanderson's Second Law


3. Sanderson's Third Law

Also, Mythcreants has a blog with resourceful articles, including creating Rational and Ecclectic magic systems, and Limits on Magic among many others.

Also can recommend Orson Scott Card's book How to Write Science Fiction and Fantasy which includes a chapter on creating magic systems.

Also also :-) you might wish to peruse the topic of Alchemy