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I've noticed that most magic systems appear sometime between a civilization's Bronze age and Medieval period, sometimes as early as the stone age.

What properties would a magic system need to have for its discovery to come much later in technological development? Specifically, around the time that electricity is beginning to be harnessed and steam-powered trains exists and railroads connect several major cities, i.e. the late industrial revolution.

Magic can't be limited by genetics. If it were a matter of "having the gift", magic would have been discovered much earlier when the first magical child sneezed fire on one fateful day.

Magic can't be based on a spiritual connection to nature or some divine power. Since spirituality comes much earlier in human development, magic would have been discovered alongside spirituality in this case.

Magic would likely be scientific or mathematical in some sense, but it needs to have certain properties to avoid its discovery alongside alchemy (as in the precursor to chemistry, not magical chemistry), heliocentric astronomy, the printing press, physics, and calculus.

What limiting factors would delay the discovery of magic into the late first industrial revolution?

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    $\begingroup$ "Since spirituality comes much earlier in human development" and yet the rise of the spiritualist movement as we more or less know it today is in the 19th century. It gave us such things as tarot cards and the Ouija board that are still associated with spirits and magic today, yet weren't really that to begin with. It wasn't even some grand discovery that kicked it off, either, one of the biggest contributors was a grand scam by the Fox sisters and others trying to cash in on the spiritualism popularity thus feeding it more. You can have a phenomena start just because. $\endgroup$ – VLAZ Aug 11 at 18:00
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    $\begingroup$ There are numerous examples where the people in the world lost or ignored a piece of knowledge only to rediscover it later. Here is a dumb example I very recently found out about - potatoes were imported from America but not because they were considered food. The Eurapeans only found out this property later. The Aztec did prepare and eat potatoes before the Europeans came in, though, and peoples around America had been consuming them for centuries. I don't think it's THAT unbelievable to have a somewhat critical knowledge not surface up until later, even though people should have figured it. $\endgroup$ – VLAZ Aug 11 at 18:09
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    $\begingroup$ "I've noticed that most magic systems appear sometime between a civilization's Bronze age and Medieval period": for example, Scientology. neoclassical economics, psychoanalysis, chiropractic, Lysenkoism, chromotherapy, theosophy, ... $\endgroup$ – AlexP Aug 11 at 18:55
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    $\begingroup$ Only beings that have entered the Celestial Realm can use magic. As such Yuri Gagarin became the first mage. Neil Armstrong and Buzz Aldrin became the first archmages. The effect was first noticed when Laika seemed to survive much longer that expected. $\endgroup$ – dbc Aug 13 at 16:52
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    $\begingroup$ If you Ask instead what properties of steam, electromagnetism or relativity delayed their discovery, isn't the real Answer, "None at all"? What delayed their discovery was in no way intrinsic; it was human ignorance, even among the greatest scientists. Is that not so? Why should that be different for magic? School-kids today are taught what makes rainbows and to us, that seems quite simple but until 400 years ago, was it not an inexplicable wonder? $\endgroup$ – Robbie Goodwin Aug 14 at 21:39

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Make magic require materials that were not available, or at least not in sufficient quantities, before the industrial age.

Consider, for example, magic wands from the Harry Potter franchise. Despite having a genetic component to magic, wizards nevertheless use wands to focus their power, and wands must be made from specific magical materials. In Harry Potter, these are generally taken from magical creatures, like phoenix feather--but what if the best magical focus were actually pure crystalline titanium? (Or even silicon--after all, trapping lightning in a flattened purified rock to make it think for us is pretty darn magical...)

The serial novel Ra by Sam Hughes features the discovery of magic around 1970, purely by chance when someone accidentally figure out the correct words to utter, but magical machinery involves a lot of modern industrial manufacturing processes and the main character has a steel staff as her main piece of magical kit--even if the basic discovery of magic energy had accidentally happened earlier, no one would've been able to make any significant use of it anyway prior to the industrial age, when specific alloys of steel could be reliably mass produced and machinery could be devised for storing and handling thaumic currents.

If efficient use of magic requires access to pure metallic aluminum, metallic titanium, plastics, or any other products of modern materials science, or if it requires larger power sources that are simply not available in nature and could only be experimented with after the introduction of steam engines and/or electrical generators, then humanity will have no way of discovering magic until they first develop those power sources and materials.

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    $\begingroup$ Indeed, the steam engine itself provides a useful parallel, being the textbook example of a technology delayed by the need for suitable materials $\endgroup$ – Pingcode Aug 12 at 2:52
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    $\begingroup$ Spoilers for Thorgal (en.wikipedia.org/wiki/Thorgal) follows.This comic (as far as I remember, it's been many years since I read it) is set in the Viking Age, but the characters get access to advanced alien technology late in the series. One of the protagonists have supernatural powers but has very limited control (think random bursts of energy) until he gets access to a headband from his alien ancestory which is basically a computer with a neural interface. This enables him to focus his power to regenerate tissue, make objects levitate etc. $\endgroup$ – glaux Aug 12 at 3:37
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    $\begingroup$ Here come the Uranium Wizzards! Uranium was discovered in 1789 (it's the first radioactive element to be discovered) and isolated in 1841. The unique radioactive effects may serve as pseudoscientifical explanation of its influence on magic. If you don't like the name, I'll add it to list of possible band names. $\endgroup$ – Prieforprook Aug 12 at 12:41
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    $\begingroup$ @Prieforprook I would go to a Uranium Wizards concert. Doesn't even matter what kind of music it is. I just want the tour t-shirt. :) $\endgroup$ – T.J.L. Aug 12 at 13:53
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    $\begingroup$ @Prieforprook Reminds me of the GURPS Technomancer setting, in which it all nuclear energy is magical in nature, but it took the first atomic bomb detonation at the Trinity site (and Oppenheimer's words) in 1945 to allow magic to work inside the Earth's atmosphere. $\endgroup$ – notovny Aug 12 at 19:24
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Magic was known all along, but was terribly unsystematic and badly known.

The first industrial revolution changed this both by raising standards of living so that it became feasible to educate many more children rather than have them drudging in the fields, producing more potential magic researchers, and by developing more systematic ways to study things.

Some proportion of the potential researchers did research, and got better knowledge. And the improved communications let them accumulate knowledge even more quickly.

Rather like, oh, science and technology

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    $\begingroup$ “Any sufficiently analysed magic is indistinguishable from science” - Girl Genius $\endgroup$ – Joe Bloggs Aug 12 at 6:47
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    $\begingroup$ While this could be true, I feel it's unlikely to delay things long enough, if it's solely a knowledge requirement. The more likely emergence point would be somewhere in the range of 1400-1600 (Renaissance to early Enlightenment), some 200-300 years early at best. In particular, the primary thing this idea needs to worry about is the introduction of the printing press. That's supposing you didn't have some Roman emperor fund a serious study ~100AD, or any of the earlier empires. $\endgroup$ – Clockwork-Muse Aug 12 at 16:39
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    $\begingroup$ @Clockwork-Muse Look at math for example - in principle it's purely knowledge based, but its development is still linked to technological applications of it. If magic is completely useless without current state of the art math then the fact that magic can be useful would only be discovered once math was developed to this degree for other reasons. $\endgroup$ – Nobody Aug 12 at 21:03
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    $\begingroup$ @Clockwork-Muse, there's nothing about the Theory of Evolution that requires anything more significant than basic observation in order to develop the basics, yet it took until the 19th Century before it was put together. Humans have been breeding plants for millennia, yet Mendelian genetics only arrived at the same time. It was the 17th Century before geology began systematic development, and yet the first steps also depended on simple observation that could have been (and were, in isolated instances) thousands of years earlier. Sometimes things just don't happen even if they could. $\endgroup$ – Keith Morrison Aug 12 at 21:23
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    $\begingroup$ @Clockwork-Muse That's not what I meant to say, what I meant to say is that those theoretical mathematician's research wouldn't be funded if we didn't have a technologically advanced society which can afford to pay for research that doesn't need to have direct practical applications. And a patron or two just doesn't cut it, to get to a modern level of math you need tens of thousands or more full time math researchers, over a longer time frame, communicating with each other, possibly using powerful computers. $\endgroup$ – Nobody Aug 12 at 21:39
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Advanced chemicals.

Traditional witchcraft depends on natural ingredients like cobweb or toadstool. What if magical chemistry needs more advanced ingredients? Many of the chemicals had become available only during industrial age. An alchemist might have luck with synthesizing one or two of them - but not the whole set.

Here are some of the chemicals that don't occur naturally, but very ubiquitous today, together with the year of discovery (synthesis/isolation):

Prussian blue       1703(?)
Nickel              1751
Chromium            1797
Morphine            1803  
Portland cement     1824
Metallic aluminum   1825
Isopropyl alcohol   1920

When combined in certain proportions, novel chemicals would produce effects (sometimes, very powerful) that can not be predicted by chemistry. Some people can call that magic.

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    $\begingroup$ Nickel doesn't occur naturally? $\endgroup$ – T.J.L. Aug 12 at 13:54
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    $\begingroup$ @T.J.L. It only occurs naturally in alloys with iron or copper, and in ancient times it was often wrongly identified as silver and used for the same purposes (e.g. making coins).. $\endgroup$ – alephzero Aug 12 at 15:07
  • $\begingroup$ Portland cement though is inferior to the now lost recipe for opus caementicium discovered ~150 BC. Which might make for an interesting plot point: magic was previously discovered, but the technique or required material was lost shortly after it's discover, or even hidden. $\endgroup$ – Garrett Motzner Aug 13 at 22:38
  • $\begingroup$ I've read that aluminum DOES occur naturally, but is harder to find the natural lumps than to find a similar amount of gold. This led to some French ruler, probably Napoleon, providing aluminum tableware to the most honored people at banquets, but gold tableware instead to those somewhat less honored. $\endgroup$ – user6030 Aug 14 at 1:40
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Magic requires an extraordinary amount of power to function. Before the Industrial Era, there weren't many high-output generators, and almost none of them used electricity directly.

Perhaps your world has a magical material that is inert under most conditions, but is activated when exposed to a high amount of energy. While lightning might activate it before this point, the charge it puts out likely wouldn't last long enough for the material to have any long-term effect. Perhaps it would just gain some magical attributes for a short period of time, such as a constant warmth. But with access to more consistent electrical power, that magical energy can finally be maintained and harnessed by people for various purposes.

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(...) Specifically, around the time that electricity is beginning to be harnessed and steam-powered trains exists and railroads connect several major cities.

Magic is powered by alternating current, or by steam powered engines.

This kinda happened in our own world, if you consider electromagnetic fields to be a form of magic. People in the 19th century surely thought the radio was a work of wizards. The following explanation of the phenomenon is often misattributed to Albert Einstein:

You see, wire telegraph is a kind of a very, very long cat. You pull his tail in New York and his head is meowing in Los Angeles. Do you understand this? And radio operates exactly the same way: you send signals here, they receive them there. The only difference is that there is no cat.

This rhymes well with tropes such as using lightning to imbue life into corpses, a la Frankenstein.

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    $\begingroup$ Oooh. I like the idea of electromagnetism-based magic. Coils are trivial to make as long as there is a consistent electricity source, so it's something that couldn't be done before electricity. AC current in particular is not naturally occuring, which is a nice bonus. $\endgroup$ – Beefster Aug 11 at 20:33
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    $\begingroup$ I expected the "if you consider..." link to be to some sort of tvtropes page about reframing modern technology as spells and amulets. Your choice was so much better $\endgroup$ – QuadmasterXLII Aug 13 at 0:53
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The most obvious would be a magic system that is based on technology or electricity, such as electrokinesis. This type of magic could activate when a character holds a certain object, or when they are affected by electricity. However, that gets thwarted by lightning.

Since the industrial revolution was basically the start of mass pollution, perhaps a magic system based on smog, or when the atmosphere isn't clean, would be appropriate.

However, I'd just introduce an object powered by electricity, or perhaps a radio or some other device, that activates that character's powers, which can only be used when they have that object on their person.

I guess said magic could be scientific, or maybe steam-powered, since the locomotive was invented around this time, or maybe telegraph-based? Maybe it could be based on the Bessemer process (https://interestingengineering.com/35-inventions-that-changed-the-world), which produced better-quality steel. Maybe it could be activated by steel, which became much more common during the Ind. Rev? Other technologies: arc lamps, modern matches, electromagnets, photographs, the Dynamo Generator, the hydrogen fuel cell, dynamite, and incandescent lamps.

So, unless you introduce an invention/object that grants/develops existing powers, or make it based on electricity, or the physics of electricity, I can't really see a way.

P.S. Dynamite-mancy would be cool.

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    $\begingroup$ You likely wouldn't need to worry too much about lightning - too unpredictable, and almost certainly going to destroy whatever it hits. Static electricity, though, would be a more likely avenue (although the voltage/amperage will usually be limited) $\endgroup$ – Clockwork-Muse Aug 12 at 16:43
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The society that knows magic was isolated.

In addition to technological innovations, the late industrial revolution was also the age of exploration. The culture which understands magic is isolated - perhaps pacific islanders, or an Amerind group. Exploring Europeans realize what this group has and they begin studying it in earnest.

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It was fatal in the wild, but researchers can conduct controlled experiments and their successors can learn

(Loosely inspired by Charles Stross's Laundry Files series.)

Magical phenomena have always taken place, but they almost always kill the recipient in various interesting ways. Visible causes of death may include spontaneous human combustion, suffocation or heart attacks for no apparent cause but with a look of great horror on their face, or vanishing completely. You'll notice that this is pretty much a list of what you'll find in any "supernatural and unexplained events" book today. No-one has had a chance to find out why, because no-one generally gets a second chance to say "I wonder why that happened...?"

The Industrial Revolution had a lot of unexpected side effects. The biggest one was the democratisation of learning. People went to science demonstrations by people like Humphrey Davy in the same way we'd go to the cinema. Science was cool. It was still rare for working-class children to get much of an education, but very promising children could get scholarships, and middle-class children definitely could get a good education. Not only that, but with so many scientific advances and so many skilled trades opening up, adult education was huge. It's no coincidence that science and engineering generally followed an accelerating trend from then on, because what gets you progress is not usually one genius but a thousand "merely" talented people. Much of what was discovered then did not particularly need the technological apparatus of the Industrial Revolution (Darwin didn't need steam engines to work out evolution) but it absolutely required that social environment.

So, what might our hypothetical gentleman scientists have investigated?

There are all sorts of strange rituals and beliefs in human society, especially in the old country traditions and sayings. Throwing salt over your shoulder if you spill it, saying hello to magpies, a belief that chewing willow bark will help pregnancy pains, touching wood after a future prediction, red sky at night predicting good weather the next day, women being bad luck on ships, and so on. (You'll notice two of those are broadly correct, of course!) We have plenty of evidence of gentleman scientists deciding to apply the scientific method to folk remedies. So it's almost inevitable that if magic actually happens, someone will run across it.

The difference this time is that they're looking for it, and they're prepared. It's possible that they may have an idea of what could happen and take some precautions to improve their chances - perhaps earthing wires, perhaps a rope tied to their wrist, perhaps a diving suit. The main thing though is that they document what they're going to do before they do it, like all good scientists. And this time, whatever happens to them, a record exists of precisely what they did to kill themselves. All these gentleman scientists were in contact with others working in the same area, of course, so if one hits a phenomenon which kills him then all his friends know exactly how to reproduce it. The state of his body gives them a useful guide on what might have happened and how to stop it happening to them, and a repeat of the experiment can be done with observers behind safety shields.

This is almost exactly how other high-risk science ran at the time. The discovery of fluorine and fluorinated compounds, and then the discovery of what not to do with them in a confined space, was basically a succession of people saying "what if I do this?" followed by a fatal explosion. (If you haven't read the Things I Won't Work With blogs, I would recommend them even if you only have a hazy knowledge of chemistry.) Even in more mundane areas of engineering, variability in the quality of materials led to some serious bridge collapses, and to steam engine boilers exploding catastrophically. The leading edge of science and engineering always has some casualties.

But however dangerous this is, eventually our body of gentleman scientists are going to figure out how to work safely with it. Whether the answer is earthing, rubber suits, sprinkled salt or holy water, a sacrificial goat or whatever, if an outcome can be reproduced then people can figure out mechanisms to deal with it. Many people blew themselves up or poisoned themselves with fluorine, phosphor and sodium (three elements which are just looking for an opportunity to kill you); but now sodium monofluorophosphate is what you brush your teeth with. Magic will go exactly the same way.

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    $\begingroup$ There's a rather good bit in the middle of the first Laundry Files book where 19th century scientists are studying the "Basilisk" effect and perform things like the slit-experiment, firing the basilisk through a cloud chamber and a bunch of other tests, growing more and more curious as they realise that they're dealing with something wildly unlike anything they're accustomed to. $\endgroup$ – Ruadhan Aug 12 at 8:16
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    $\begingroup$ @Ruadhan I do like the historical asides in the earlier Laundry Files books, especially how they tie in actual historical figures like Arthur Ransome. The hallmark of all Stross work though is the quality of worldbuilding. I hope everyone on this SE site has read them as a benchmark for what the good stuff looks like. :) $\endgroup$ – Graham Aug 12 at 10:18
  • $\begingroup$ Ahem, which two of those are broadly correct? ;) $\endgroup$ – N Unnikrishnan Aug 13 at 11:58
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    $\begingroup$ @Ruadhan And then only with modern processing technology are they able to simulate the basilisk in a DSP so it can a be used for practical purposes. $\endgroup$ – user1937198 Aug 13 at 18:35
  • $\begingroup$ @NUnnikrishnam Which two of what? $\endgroup$ – Graham Aug 13 at 19:02
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Magic Is Math

Magic has always existed in the world - but it isn't the requirement of rare materials that restricts it, nor having a certain blood lineage. And its secrets were known as early as the first arithmetic was being invented.

But early magic was slow - very slow, in fact so much so that it was considered little more than a curiosity, because what you could accomplish with it was severely limited by how quickly you could run computations - which on a human level is far, far too sluggish.

Then Along Came Charles Babbage

When Babbage invented the Difference Engine, in our world it was not widely adopted for practical use - but in this world, rapid mathematical calculations would have a practical purpose. They would greatly speed up the rate at which magic could be performed.

This invention was created in 1822 - which is the very late period of the first industrial revolution.

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    $\begingroup$ This was my thought too, although I didn't think about the Babbage engine. I was thinking more along the lines of mathematics progressing in line with engineering, or an understanding of Calculus (17th century) being a necessary precursor for the development of magical symbology. But I like yours better, just think - Charles Babbage being the father of modern magic, Ada Lovelace the mother. $\endgroup$ – Doug R. Aug 14 at 12:40
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    $\begingroup$ PS - on second thought, I seem to remember reading a story about 30 years ago that dealt with computational magic. The premise was that magic ebbed and flowed, and spells had to take this into account (or something along those lines). It was very computationally-intensive to calculate this, until computers were invented and it was possible to crunch the numbers years in advance, like tide charts. $\endgroup$ – Doug R. Aug 14 at 12:43
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Aluminium. Aluminium like element

I will simply use Aluminium and you can substitute that with whatever name you like.

So reading the wiki page for our Aluminium friend says that they were super rare. They were more expensive than gold and people did not know how to make large amounts of them. Sure it dates back to the Greeks but we can ignore that. So until people knew how to make Aluminium in large amounts, they could not use magic. Pretty simple really.

Now this will lead to something like this: after your Bayer and Hall–Héroult process is made then they discover interesting side effects for using large amounts of Aluminium. So those are science men and they mess around with your Aluminium the lo and behold: Aluminium, used properly, is the stuff of magic.

The rest is up to you. Perhaps it needs to be in contact with the person, or it needs to make a certain shape, a pentagram is an obvious answer, or even it can be eaten...etc. Honestly whatever suits your world most. Also the quantity aspect solves why it was never apparent that it was magical. Until we made tons of it it was making so little magic that people did not notice.

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Penicillin. Penicillin like thing. I'm not good with chemistry so bear with me. The same Aluminium idea is the same. Only it's something like a plant or a combination or things. Basically penicillin. Read on the actual penicillin and change whatever you feel like it.

This again means that it possible to create the stuff of magic in a lab. We only needed to mess around with the right stuff and viola. MAGIC.

While at the same time insures that no society before those with your limitation can make them.

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A rare natural element that is only accessible with deep mining capability. Obvious things is obvious. We needed to go so deep underground or underwater to extract Contrivium, terrible writing advice if you don't know or the classic: Phlebotinum So after the technology of the world advanced to that point we started seem this thing. And it did not take long to people to figure out it's magical. And even if people back in the day found an ore or a gem or whatever. They still could not get more so it was more like one time thing.

A simple point

"Magic would have to be scientific or mathematical in some sense"

I see this as a problem. The scientific method as we know it is new. Sure. But can you point to the Roman, Muslims, Chinese, Greeks, Egyptians...etc at their highest point in history and say: they new nothing of how to manipulate the natural stuff into useful things?

Because science is all about that honestly. OK we can argue but point is that a bunch of people wanted to make X happen, so they got together and made X without magic and by only using their brains and a little, or much, labor.

That is the same principle behind sending someone to the moon or making a box of whatever materials to call it your own and to protect you.

Basically the fundamental principles of making a spear is more or the less the same behind making a fighter jet. And mathematics, afaik, are not limited by science I'd say. So it's a particular level of scientific knowledge and practices you are looking for.

Fair enough and there is no problem with that. Though lets not forget that in a 100 years our science might look like that of idiots to them. So it is NOT a matter of a constant thing.

So scientific discoverer and ability are subjective.

Anyway all I'm saying is that confusing a certain level of scientific knowledge with science in general is not a good idea for your setting.

And if that does not work for whatever reason then just tell us what other factors are into this.

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Magic is Environmental

In the Shadowrun setting, mana comes in cycles. The world did end in 2012 as predicted by the Mayans, but that was just the Fifth World, the Sixth World immediately followed it. Even cycles are high magic, odd cycles are low/no magic. Each even cycle ends with a visit from magically apocalyptic creatures called Horrors, who wreak havoc on humanity until the level of magic plummets, but that's neither here nor there.

The idea is that all the mystical and spiritual traditions that are passed down over centuries are real, based on things that worked in the past, but there isn't enough ambient magical energy. Once there is enough energy available in the environment, things that people had been doing for centuries just start working. This doesn't preclude any of the other restrictions on magic, like genetics or skill. Even with the right genes or the right techniques, without a sufficient ambient magic field, none of the rest matters.

In your case, if the industrial revolution coincided with the return of sufficient ambient magic energy, scientists and engineers would begin studying the phenomena. Telegraphs and faster travel methods would allow stories to get to potential investigators much more quickly than in ancient times, and gather data in larger quantities. Your smart folks would have access to empirical evidence and be able break down the myths and legends into reality and truth.

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Running in a different direction from Graham's mention of the Laundry Files series by Charles Stross

In that series, Magic is a branch of applied mathematics. Magic users have existed in various forms for much of human history, but generally have poor results and a low survival rate. It's only with the invention of mechanical and electro-mechanical computers, and the discovery of the underlying equations (Turing's Last Theorem) that magic has become readily accessible.

The Oft-repeated description is that simulating the right/wrong equation sets up resonances in the multiverse, resonances that can be felt and heard by entities further down the mandelbrot set from us. With the right equipment, and the right mathematics, you can talk to them, and they'll talk back and help do things. Essentially demonology.

The upshot is that most of those entities are little more than computer-programs themselves, known in-universe as "feeders in the night". Summoning one of those is a great way to land yourself a zombie. A zombie that can be bound and programmed with a verbal programming language of sorts. The good guys use them as night-watchmen.

But there are bigger ones, vasty intelligences from the nightmares of H.P.Lovecraft, and summoning one of those is a fantastic way to hasten the heat-death of your universe. Fortunately it's extremely difficult to do and requires a lot of blood-sacrifice (death means the destruction of the information in a person's mind, which sets up spacetime ripples and can power the invocations)
The Nazis had a solid go at summoning such an intelligence in the 2nd world war.

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  • $\begingroup$ Nice. I did consider making the Stross reference stronger in my answer and tying it in with Babbage's Difference Engine, which would firmly place it in the late Victorian period. But then you also have to justify why the many thousands of human computers ("computer" as a job description) didn't find it, and that's one weak point which Stross never really filled out. $\endgroup$ – Graham Aug 12 at 10:24
  • $\begingroup$ Someone had to discover it eventually! The existence of electro-mechanical computers simply makes it easier to hit upon (the protagonist of the series discovered it accidentally while writing a graphics renderer) One imagines that some 19th century Computers did find it accidentally, but didn't have the theoretical background to actually understand what they'd found before their brains got filled with holes by daemons. Ritual magic is something that's been a thing in the Laundryverse for much much longer though, it just that ritual doesn't lend itself to scientific study. $\endgroup$ – Ruadhan Aug 12 at 11:19
  • $\begingroup$ I guess my starting point is that the existence of ritual magic does mean it's been discovered. Those rituals are a way to work (relatively) safely with magic, even if it's not brilliantly understood. I'm thinking like the same way engineers put in rules of thumb for how thick bridge girders or boiler walls should be, without perfectly understanding grain structure formation. For sure science could optimise that, but it would not be something new. For the OP's requirements, I read it as more like genuinely new discoveries by Pasteur, Darwin. $\endgroup$ – Graham Aug 12 at 11:40
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If you want to have a sudden late explosion of magic why not have it based in symbiosis with an extremophile microorganism?

Think midichlorians from Star Wars. A cavern found when mining deep underground is glowing and warping spacetime, filled with billions of tiny organisms where they evolved without light or much chemical sustenance or maybe they were discovered when an asteroid hit the planet or some other method of discovery or dispersion throughout the globe.

After the discovery of the # microorganism it began to propagate throughout the natural environment unconstrained by the previous prison it was in. As a hardy organism different to anything else encountered, the immune systems of most species had no defence against it. Some people had a reaction and died, the majority adapted to live with the tiny organism, much like how all of our cells now contain a mitochondria.

Not long after the spacetime warping microorganisms began fusing with natural creatures, unusual effects began to be seen as said creatures began to exert some level of control over space, warping gravity to move objects without touching them, causing electrical and thermal abnormalities. As humans began to experiment with these new abilities they found that by using certain tones of voice, noises, movements of their body hosting the organisms they could produce a vibration/sympathetic response in the organisms in their body or ones nearby to produce more complicated effects etc etc.

It naturally leans more towards sorcery than wizardry but could be flavoured as either. If you want to make it somehow exclusive then have it passed in families who guard their bloodlines carefully and consider their blood sacred and ban transfusions for fear of others gaining the same abilities.

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  • $\begingroup$ A virus, perhaps? $\endgroup$ – Brian Lacy Aug 12 at 18:13
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Some ideas how to spin this:

  • the discovery of radioactivity (around 1896, matching your timeframe) and further research of it enables spells and magic: wikipedia because handwavium

  • deep (industrial) mining released some kind of trapped (angelic/demonic/alien?) entity that grants magic(very advanced technology) to its followers

  • technology advancement (spinnig two elaborately inlayed gold/platinum/copper circles antidirectionally at 250mph while funneling X-ray beams through it) allows opening the long-thought-lost and buried/hidden portals into different faery realms. So far tales about magic existed, but humans had no access to it - now we can conquer faerie and grab all its magic resources (shiny berries/faery liver anyone?) for ourselfs

  • unrelated ambient magic levels rising/falling (Wikipedia - Magic_in_Earthdawn) Magic mana availability raises/falls and only with a certain level of magic ambiency it can be harnessed. High levels of mana correspond to demons roaming earth, killing humanity. Last time round was just before ice-age and all but few humans were eradicated, a long magic draugh period followed and now it rises again by pure happenstance. All magic cookies currently show dark things to come ...

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Magic has always been here in small quantities. So yes witches could use magic but mostly relied on 'headology' and basic herb based medicine.

Magical energy (Mana) slowly leaks into this world (dimension). However it wasn't until the Trinity test (first atomic bomb) that large amounts started flowing in. The atomic bomb blasts weakened the barrier between the magical realms and ours. So now we have lots of mana, ready to be harnessed.


Edit: Oops I got the time period wrong.

Updated version is that even small explosions cause small leaks. So cannon fire causes Mana to leak in for a while. 
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  • $\begingroup$ This would put the timeframe into the 1940s rather than the mid-1800s $\endgroup$ – Beefster Aug 13 at 19:56
  • $\begingroup$ Gunpowder was used in miliary as early as 904 in china - your gunpowder proposal would have put China to dominate the world under the late Tang Dynasty... $\endgroup$ – Trish Oct 12 at 14:05
  • $\begingroup$ The idea of gunpowder magic however was good enough to spark me asking a question about it...starting in 904 AD! $\endgroup$ – Trish Oct 12 at 15:29
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Two ideas that come to mind.

The Stross' Laundry Files world is already referenced in a couple answers. However, Stross, for his own reasons, wanted magic to be ancient, in order to tie its existence in the world and link to the ancient horrors. In your case what can be done, is that

Magic is not inherently present in all of maths, but rather in a specific sub-field of mathematics, which has only been discvered in the last 100 years, and which requires pre-existing mathematical apparatus, not available in earlier times.

If you look at Mathematics timeline, one thing that immediately appeals to me is fractals and chaos theory. Fractals almost look magical on their own. However, if that doesn't appeal to you, take a peek at maths stack exchange, there will be plenty questions present-day maths, which seem damn magical to me, and much of which, I presume, require mathematical knowledge and know-how not available 200 years ago, not to speak of earlier. Anything that requires group theory, for example.

Radioactivity is tied to magic.

Until 19th century, people had no idea that something like radioactivity -- completely undetectable to our senses -- could exist. It existed in the world, of course, but we were completely oblivious of it. It only comes natural that when radioactivity was discovered, its effects might include not only the physical, which we have studied so well, but also magical. Either as a source of magic, enabler, or something, radioactive materials are the only way people get to actually do magic. And only in 19th/20th century we came to be able to produce radioactive materials in significant enough quantities for us to use them magically.

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  • $\begingroup$ You say fractals. @Zibbobz said the Babbage Machine to make calculations fast. And one of Rick Cook's Wizardry books had the Wiz trying to calculate the world's key (which was a fractal) and it required an IBM mainframe. $\endgroup$ – Jerry Jeremiah Aug 19 at 8:00
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Electricity!

You see, magic needs electricity to work. It needs energy, as magic does not allow you to break the energy conservation principle - it simply allows you to use energy to interact with the world in new and unusual ways.

Pre-industrial age, the best you could get was the mage's own muscles with very tiny electric current, or maybe an electric-eel if you lived in a good place. Ancient Egyptian wizards used the acid batteries for their rituals - how do you think they've built pyramids?

So cue the electricity, and some gifted people suddenly discover that they can do more than move a single scrap of paper on a windless day. They can now move whole cabinets when holding a live wire.

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  • $\begingroup$ Welcome to WorldBuilding.SE! There's no need to tell us that your solution is just fantasy, as we're already well aware of that. That's the whole point of the site, after all. $\endgroup$ – F1Krazy Aug 12 at 13:45
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Electricity generates Mana needed for Magic

In order to make magic work a certain amount of Mana is needed, however Mana is only generated near strong electrical currents, and prior to the industrial revolution your only real chance of finding any was being in a thunderstorm.

As the Mana dissipates fairly quickly while there might have been the occasional wizard back in the day, their opportunities to practice and learn the craft were severely limited.

This does mean that if you want to practice magic in the wilderness you need to bring not just a battery, but some sort of device that uses a fair amount of current.

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Hamilton

In 1843, around the William Rowan Hamilton published an idea for using imaginary extra dimensions to perform lower-dimensional operations. These new mechanics were used by James Clerk Maxwell in the 1850s and 1860s to unify and explain two new mysterious forces : electricity (discovered by Charles-Augustin de Coulomb in 1785), and magnetism by way of galvanic induction discovered by Michael Faraday in the 1810s through 1830s.

Hamilton's technique, extending complex (imaginary) numbers into quaternions, would precede octonions which are currently being looked into for application in string and M-theory.

Additionally, Hamilton developed a way of looking at things that departed from time- or location- dependency, looking for alternative ways of viewing chemical or phyical processes (such as their total energy and a range of statistic probabilities for values such as postion). Hamiltonian mechanics is the foundation of statistical interpretations of quantum mechanics, and enhrined this way by the Hamiltonian

If your magic is something that can be codified, but is very intangible, Hamilton's math was probably necessary to give it form.

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  • $\begingroup$ Freaky. I was just reading about complex numbers. $\endgroup$ – Beefster Aug 12 at 16:26
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Aliens cue the hand gesture, you know the one

"Magic" as commonly defined, exists in the universe and is generally thought to be more powerful and efficient than "mere" science by all the various sentient races. The most magically powerful races use special drives that "suck" magical energy from the surrounding area to go faster than light, power warding shields, lightning/lasers, and what have you for their interplanetary ships. Well in the Far Distant Past a ship was damaged in the solar system such that its drive was always on, but didn't power anything. The magical feedback from all this would kill a recovery team, and it really wasn't worth the effort of fixing anyway, so it was left. Meanwhile on earth magic doesn't exist, because it's being constantly drained away by the drive. Anyone who thinks shamanistic mumbo-jumbo from pre-history is actually based on any version of reality is a crackpot.

Roll forward to 1780/whatever year you want, the ship finally crashes into the dark side of the moon, its main drive is hit by an asteroid, it falls into the sun/earth, whatever takes your fancy. Now all the sudden BOOM, magic is HERE. Maybe the people with innate ability start doing magic/blowing themselves up. Maybe just the kids do it because if you don't use magic starting in childhood the ability atrophies and can't be re-ignited. Maybe it's super specific so only the (now ex) crackpots who were determined to "prove" magic beforehand can even fiddle with it now. Maybe people start accidentally stumbling upon words/gestures/symbols of power. It's a pretty flexible origin-of-magic starting point.

This is basically an amalgamation of the Emberverse (where, though it's only mentioned tangentially, alien skulduggery is the reason electronics/modern tech fails and a "soft" form of magic emerges) and a short story I read ages ago where an alien FTL drive makes it seem like the speed of light is constant from earth. This prohibits mankind from figuring out the key math to FTL and assuming it's impossible.

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Let's try a different answer that plays not on what was invented during the period, but the advancements that the period brought to the world: Subtlety and magic's prerogative to keep the user alive are the two main reasons.

Disclaimer: I am going to base this on the premise that "discovery" for the purposes of this question means both the actual discovery of magic as an ability, actual proof that it is magic, and the somewhat widespread dissemination of that information. A painter can have a magical talent, but unless they are overtly using magic to paint, then it's just a man's claim.

Subtlety

Magic, until the First Magical Revolution, is just not about big flashy effects. It is subtle, and easily mistaken for chance, good genetics, or just having the gift.

Magic up until this point is the twice a decade prodigy in a trade, that child that can pick up a tool and just understand how to use it in the way they want to. It is the soldier that has survived a dozen battles through injury and infection to survive to old age despite the odds. It is the cunning thief that uses their "instinct" to avoid trouble from all sorts of people. It's the craftsman that creates something that lasts just a bit longer than it normally should. There is nothing that screams overt magic, but a myriad of coincidence and good fortune that is more than just good genes.

If you want to go that way, there are a blessed few that have figured out the secret of making magic flashy and powerful and truly bend nature to their will. We call those people "gods" and they form the basis of mythology.

Staying Alive

Magic's first prerogative is to keep its user alive. When a person is sick or injured, it is magic that assists in the healing process. Whether it is disease, malnutrition, or injury, magic supplements the body until it is in peak condition, with or without magical help. As above, this is a subtle boost and not visibly fast healing. For those that lack the power, then their self-healing abilities can only go so far or fail outright when put under too much stress.

Since most magic is tied up keeping the body in prime (or just functional) shape, there is not the power to actually be flashy and powerful. And for those that have all their magic tied into their health, well they have the power, just apparently not the ability.

The Industrial Revolution

What does this have to do with the First Industrial Revolution? Technically nothing. In this case it is not any one invention, process, or discovery that will enable the widespread discovery of magic. It the other things that comes from this revolution that will enable the discovery of magic.

Better living conditions can lead to better health -- now magic that might have been tied into keeping people healthy is now free to be used and expressed. As such, there is more of a chance to be either blatantly flashy or actually observed and questioned

The lower infant mortality the revolution brings on allows more children to survive and awaken to the potential to be magical and for that magic to supplement the child's health. Also since they're healthier, they have more of a change to have free magic to use.

More organized science coming from a larger population and a boom in scientific progress can lead to discoveries that, while individuals dismissed single events, together they paint a trend of anomalies. Anomalies like magical phenomenon. New science and math may give symbols and non-instinctual understanding of magic allowing for collaborative discoveries where if there were any before, they were individual ones.

Just a higher population in both number and density can lead to just a higher chance that magic will be discovered and studied. As well as the mage not being killed on principal. While the probability is non-zero earlier, there are more chances now as opposed to then.

The enhanced understandings of the world leads to things that were once magic to be understood. Things that might have been written in antiquity as magic and wizardry are now science tricks. We have removed a critical mass of mundane magic and have definitively found the gems of real magic within.

Admittedly, none of these outright prevent magic from being discovered earlier. The idea is that while discover was a non-zero chance then, it is a much higher chance now.

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Magic Requires Knowledge of Thermodynamics

You can’t break the laws of physics unless you know where to press.

Within limits of their abilities, training, and experience, a wizard can cause any physical process to happen just by willing it, as long as energy and entropy are totally balanced. This is different from non-magical processes where energy is conserved but entropy increases.

Example: A wizard can stare at a bucket of water and will it to freeze, but in order for the spell to work, he/she has to will the surrounding air to be heated by an equivalent amount.

This system requires that the first wizards to reach awareness of their powers had to know at least 19th-century physics before they could accidentally discover their abilities.

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  • $\begingroup$ I'm no chemistry expert, but wouldn't it violate the conservation of entropy to transfer energy arbitrarily? $\endgroup$ – Beefster Aug 13 at 15:29
  • $\begingroup$ Yes, that's the whole point of how this magic system works. It's only when wizards have industrial-age knowledge of thermodynamics, that they can control entropy and break free of non-magical physics. Perhaps the extra entropy is absorbed into their souls, and makes them really exhausted. $\endgroup$ – Alex R Aug 13 at 18:39
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Magic Spells are Computer Code

  • 1819: Charles Babbage
  • 1842: Luigi Menabrea
  • 1843: Ada Lovelace
  • 1936: Alan Turing

These are approximately the years the first humans to fully grasp the concept of computer code documented their thoughts. It's possible some wizards discovered it prior to these dates, but it wasn't until programming was thoroughly studied by mere mortals, that the wizards could practice magic safely. Both Ada Lovelace and Alan Turing died young and under suspicious circumstances. What secrets did they take to their graves?

Imagine all magical spells have to be cast using some kind of programming language, and a serious bug such as an infinite loop or a memory leak can kill the wizard.

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Supernatural abilities have been locked away deep inside us for most of our history and became only then awakened for the first time when the marvels of modern medicine & psychology conincided with the political will generated by the Cold War, experimentation with New Ages concepts in american society as well as the applied ruthlessness of the CIA to enable a U.S. marine to kill a goat just by staring at it.
And the rest is history, as they say...

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  • $\begingroup$ This would date magic much later than I had in mind; this would relate to the second industrial revolution. $\endgroup$ – Beefster Aug 12 at 16:12
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Actually it was genetic but the mutation for it only appeared fairly recently. At first there was only a small family of magic users and they kept it under tight control. Then one of the existing magic users broke away and started having a bunch of children and formed a splinter group. Now there's political tension between the two groups so they decided to marry off their children to their allies (rulers of other countries) to increase their base of support. The allies want as many magical decedents as possible. A few of the grandchildren decided they wanted to just marry for love and start to mix into the normal population. Suddenly magic users are absolutely everywhere and it's impossible to put the genie back in the bottle, being a magic user is such a desirable trait.

Kind of like atomic weapons, but sentient.

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  • $\begingroup$ This would be too slow to propagate for what I had in mind. $\endgroup$ – Beefster Aug 12 at 16:11
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Magic requires making deals with trans-dimensional entities. : The deal making process is complex and requires many things which don't normally happen at the same time (hold a rock over a candle during a lunar eclipse while being on a boat in a river, then...).

The discovery on how to do this was luck, although the distribution of the knowledge of how to do it is a modern thing.


Magic requires a soul artifact to do even basic stuff. It's easy to make a soul artifact if you already have access to one, but it's impossible to do without one.

The first artifact wasn't made by us and sat in a museum for many years before anyone figured out its use.

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Population Density

If magic is based on having a large number of thinking minds in the area (not participating, just being there), then only some really large cities can support magic. Ancient Rome might have qualified, but only in it's peak. Many modern cities would also be enough. (and some New York city apartment complexes might be enough on their own.)

The magic in this case probably starts with active thinking minds disturbing the surface of a "mana-space". The magician then capitalizes on the energy of the collective disturbance.

Just to throw things for a loop, different cities might have different flavors, so that a magician from one city might not be able to work in another city. Or maybe he/she would need to acclimate to the new city. This might get further modified by cultural differences affecting how different the magic is.

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I'd like to point you towards the anime Youjo Senki. In there, magic is a gift you're born with, but in order to use it you really need an operation orb to make it work. From the wiki it says

An Operation Orb is a tool mages use to interfere with reality through magic according to their will. A breakthrough to recreate miraculous phenomena through the combination of both mana and analog calculation.

Not much has been revealed about the inner working of Operation orbs, but the orb's main function is to process the world data and convert mana of user mages.

If you look at the history of magic in that world it then says

In ancient time, magic exists in various forms in many regions, civilizations, cultures, and even traditional customs. However, those who could trigger a miracle by their own are extremely rare. For those who had magic potential, most of them could only increase drug effect a bit, breathe in water temporarily, heal a bit faster, etc. People in some regions decided to research such phenomenons instead of fearing them, finding that some specific stuff could be used as catalysts to make magic usage more efficient. They even found that some heroes in history actually used holy relics to trigger miracles.

As magic was accomplished with various magical devices which were not very efficient and stable, the Industrial Revolution occurred. Magic then went through a revolution following other fields of study that were influenced by war. In their search to optimize spell composition, the analogue computer Differential Analyzer caught their attention. Inspiring engineer to duplicate computerization by magic and invent the modern magical device, Operational Orb.

The design of the magical computer that facilitates mages to compose spells is based on a pocket watch. Making the Operational Orb a device that is compact, sturdy and precise. In Tanya's era, an operation orb could be more expensive than a flagship tank or aircraft. In short, normal orbs cost as much as the most powerful weapons.

This sounds rather close to what you seem to be looking for. Just get rid of the holy relics and such, and until you got computing tech good enough to make the orbs there will be no true usage of magic other than those passive effects (and paybe some parlour tricks).

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This answer assumes that what matters for your purposes is not that magic is completely unknown, just that it's functionally unusable prior to the industrial revolution (which for most stories is what's likely to matter).

It's always been around, but doing useful things with it requires large numbers of skilled workers to be involved in controlling it.

Using modern computing terms, magic has to scale out to be useful. Having just one person casting magic imposes limits in some way that make it either inherently useless, impractical, or downright dangerous. Maybe there's a physical limit to how fast a person can safely form a spell by themselves, perhaps there some limit to how much magical energy a person can channel, etc. However, more than one person can contribute to the same spell at once, so a bunch of skilled mages working in parallel can safely do big, useful spells that could not be cast by a single person alone.

In more technical terms, your total power and total efficiency when spell-casting are both proportionate to the number of people actually casting the spell. To limit things such that large numbers of people are required, either power or total efficiency (or both) has to scale exponentially proportionate to the number of people involved (preferably with an exponent just above 1, that way you need lots of people to be involved), but either logarithmically or linearly (with a very shallow slope) proportionate to the skill of the individual casters. This will result in a situation where the most efficient way to cast a big spell is to gather a very large number of mages to cast it.

That situation is the same one that lead to industrialization happening when it did. For factories and mass production to be possible, you similarly need very large amounts of either unskilled or semi-skilled labor. This, in turn, means that the same things that made industrialization possible (namely significant improvements to agriculture that allowed for far fewer people to be needed to produce food combined with an overall increase in the standard of living for most people) would make practical use of magic beyond stuff like simple party tricks possible as well.

From there, you can then further delay the practical use of magic by constructing the system of magic such that there's a minimum amount of skill required to be able to work safely with other mages. This will in turn require some minimum amount of education to be able to actually work as a mage, which will limit how many people can e trained as mages proportionate and therefore put a practical upper limit on the possible scale of spell-casting based on what proportion of your population is required to support the rest of the population at a basic level.

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    $\begingroup$ This doesn't make the magic not discovered. Not useful =/= not discovered. $\endgroup$ – IT Alex Aug 12 at 14:24
  • $\begingroup$ +1 for first paragraph in particular. $\endgroup$ – Jontia Aug 12 at 16:54

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