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So in my setting there is a small and exclusive group of people that have access to devices that work like computers (but which are not the same as computers. They do not use semiconductors and electricity, for instance). These people would use these machines to simulate the world and acquire knowledge of physics and science, putting them way ahead of their time. These machines are condemmed by organized religion because they are borderline prophecy and magic for those seeing it from outside, so these people go about their business in secret.

This particular story tells the tale of Julius, a lad who loses his parents for a plague and is sent to live with his grandfather's brother, a bitter old man with a lot of money named Cesare who seem to always be on guard. The boy arrives at the countryside home of Cesare and is treated like an irritating guest.

The story follows the emotional bonding of Cesare and Julius as he comes as a guest and then slowly progresses towards family. Cesare, the bitter man finaly melts his protections and see in Julius the son he never had and Julius finds in Cesare the guardian that he lost. The entire piece resounds with themes of family union and filial love but also growing up as Julius goes from a naive lad to an able scientific apprencite and lab assistent and seem posed to carry on the work of Cesare, the son the man never had.

Although the themes are important, the backdrop on which they are dramatized is one of science and discovery. The emotional journey of Cesare and Julius happens as Julius goes down into Cesare's research. He finds that Cesare is one of those people that control a computer, then he slowly tries to learn what is going on as he tries to come to grips with what he thinks about all this since he is a god fearing lad. Eventually the alure of forbidden knowledge and curiosity take over and Julius start to secretly studying Cesare's work from the few glimpses he can catch.

The problem on which Cesare is working is of central importance because it also serves to illustrate the power of owning a computer and the role of computer simulations in the scientific discovery process, which is something I would like the readers to experience.

The work is framed in the 1800s. A time of religious fervor and intellectual effervescence that will shake the world. In fact, this is also one of the themes of the work, the tension between tradition and innovation. So in order for the piece to have an interesting take I wanted them to deal with a problem that the scientific community of the 1800s could be aware with the physics of the time, but which was too hard to solve by hand, thereby requiring a computer and giving an advantage to those that owned one.

My first idea was the discovery of Neptune, but Urbain LeVerrier did the computations by hand and managed to get it published. So even though it might have been an extensive and difficult computation to perform, it was nonetheless quite possible with the computational resources of the time. I wanted something equally iconic and worldview shattering but harder, way harder. A juicy problem that is right there but is untractable without a computer.

What difficult, important problems could possibly be conceived with the knowledge of 1800s physics but were untractable without the employment of a digital computer's computational power?


Edit

I see a lot of great examples like Wheather forecasting and dynamical Many-body problems. However it ocurred to me that the examples (probably due to my lack of clarity in the question) are too general.

What do I mean by this? Well let's take the example of the N-Body problems for N>2, kindly suggested by @b.Lorenz, which is currently our best-ranked answer.

Although I can appreciate the difficulties to obtain analytical solutions to an N-Body problem and the contributions that having a computer would make to it, I was wondering what specific N-body problem would be relevant?b In the story, Julius and Cesare are posed to make a discovery using their computer. An important discovery. I wondered what that discovery might be. For instance, I mentioned the discovery of Neptune as my initial consideration.

This discovery could be a specific case of solving an N-body problem. The characters could simulate a system with a fictitious mass positioned so as to rid the predictions from inconsistencies with the observations, and then extrapolate the system into the future to make an observation of the hidden planet. That would be awesome. But I also mentioned that Urbain LeVerrier made a ton of approximations, calculated the thing by hand using data collected from astronomical observations and still got it right. Neptune was observed in the sky roughly around where LeVerrier predicted. So this particular problem of an N-body computation is not ideal because it is not a problem that highlights the advantages of computer simulation, it is not out of reach for a dedicated investigator with a few assistants working for, say, a year or a few months.

So what would be a good example then? What particular case of an N-body phenomenon would be relevant and known for 1800s physicists that they just wouldn't be able to investigate even if they managed to have the relevant insight?

Likewise, what specific case of wheather forecast or hurricane prediction would be dramatic enough that it would shatter the scientific community when solved, is knowable from the physics of the day, but is out of reach for the actual solution even to the brilliant minds that happened to stumble upon it? I know that "hurricane prediction" would be an incredibly useful but I'm thinking more on a specific instance.


Edit 2

So again I have noticed that I have a somewhat lack of clarity. AlexP asked what do I mean by "solve". Admitedly, solving a problem might mean a lot of things, particularly in physics.

Although I accept answers that deviate from this norm, if they are good and interesting, by "solving" I mean:

"Numerically integrating difficult systems of differential equations in order to obtain accurate numerical predictions for phenomena that, at the time of the setting, cannot be analytically obtained, that is by algebraically solving the system."

I require that the phenomenon in question be, if not useful or critical for the entirety of the society, at least scientifically interesting. For instance, the discovery of Neptune did not spell doom for the society, nor would it have suffered dire consequences if the planet had not been discovered, nevertheless it was important because it basically rewrote what we knew about our cosmic home, and that was a big deal for the astronomy of that time.

As I said, I will accept things that are not part of the above set, as long as they are interesting and enticing, but I believe that numerical integration of difficult systems of equations is a nice, basic aim.

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  • $\begingroup$ A digital computer ? If you mean an electronic computer of the current type, that requires considerable knowledge of atomic, molecular and quantum theory to build. Science in 1800 Earth had no concept of even the electron or any structure internal to atoms. Do you mean mechanical computers (essentially slightly programmable calculators) ? Have you read The Difference Engine which has much to say on this subject. $\endgroup$ – StephenG Mar 4 at 10:11
  • $\begingroup$ This is an interesting reference, I wasn't aware of it. I am going to look it up, thanks :). And yes, it is an analog computer, not a digital one with semiconductors and electricity. $\endgroup$ – urquiza Mar 4 at 10:16
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    $\begingroup$ Double pendulums... look deceptively simply and elegant until you try to solve ;D $\endgroup$ – user6760 Mar 4 at 11:05
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    $\begingroup$ Hurricane predictions: There's nothing particularly new or special about thermodynamics, air pressure, rotational motion, and water mass. $\endgroup$ – user535733 Mar 4 at 13:42
  • $\begingroup$ Great insight. Wheather forecast may be tedious today, but being a professional meteorologist in the 1800s could have been a friggin' challenge. There is just a few issues with it. The data that wheather forecasting agencies use to feed into the models is collected from devices distributed all around the world. I have no idea whether such infrastructure existed in the 1800s and it would be just a case of working with it. Also, it would help if the problem at hand would lead to an important, impact discovery. "Predicting hurricanes" is useful, but general. Anything more specific in mind? $\endgroup$ – urquiza Mar 4 at 13:51
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Gravitational N-body problems for N>2 are chaotic and in general unsolvable analytically. The laws and the resulting equations were known since Newton.

Using perturbation theory, a lot of work, and a load of creative ideas, great scientists of the 18-19th century, like Lagrange, could approximately solve some cases of interest (long term stability of solar system, Lagrange points, finding planets by their gravitational influence on other planets), but for example predicting the outcome (outbound velocities) of a close encounter between three celestial objects of comparable mass, was beyond them.

Using digital computers, this can be solved to arbitrary precision. (Error will remain, but the more compute you throw at it, the smaller it gets)

In theory, everything that a (non-quantum) computer can calculate, can be calculated by hand, but you would need a great army of calculators employed day and night, and they will make a lot of mistakes. So it is safe to say that only a computer can practically solve such a problem.

It is not hard to imagine, that such a problem could have practical importance too, for example predicting whether a dwarf planet will hit Earth or make a close miss after a close encounter with two other asteroids. (Ceres and Eros were discovered in the early 1800-s so they are detectable with 19th century telescopes)

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  • $\begingroup$ Using digital computers, this can be solved to arbitrary precision. (Error will remain, but the more compute you throw at it, the smaller it gets) Errors are reduced by better algorithms more than raw power. You make much better error margins by designing the algorithm well based on the physics. $\endgroup$ – StephenG Mar 4 at 11:28
  • $\begingroup$ I like this idea a lot. However, your example lacks focus. Could you perhaps suggest a specific instance where the N-Body solution would generate a shattering discovery? For instance, in my question I mentioned that "the discovery of Neptune" as the great leap the characters will make. They could do it by simulating the solar system with a fictitious Neptune to show that the system predicts the observations better than current models. But LeVerrier made a ton of approximations and solved the damn thing by hand and still got it right. Any examples where he couldn't do it? $\endgroup$ – urquiza Mar 4 at 13:58
  • $\begingroup$ @StephenG Sure, they need algoritmic ideas and higher order integration schemes, but they directly benefit from more compute by reducing the timestep and increasing floating proint precision $\endgroup$ – b.Lorenz Mar 4 at 14:33
  • $\begingroup$ @urquiza Arguably I have such an example (close 3-body encounters and planetary defense) It is not a very fundamental scientific discovery on its own, but if they can foretell that a seriously big but not civilization-ending impact is impending and have time to prepare (like stock food for years expecting sudden planetary cooling from dust), I would consider it an impactful result. $\endgroup$ – b.Lorenz Mar 4 at 14:39
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    $\begingroup$ Equally impactful would be proving that it's not going to hit, no matter what the doomsayers in the church and public square are saying. That might fit the themes of the work better. $\endgroup$ – Cadence Mar 5 at 3:12
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The short answer is all of them.

The long one, well, it depends.

See, when i was on my first class on finite elements analysis the teacher, trying to make us undestand how the program we use works, made us do the math by hand. We had a cube with a force aplyed to one of its vertices, and we did a very rough grid on it, with a point in each vertices. And took some good hours to do.

The computer, on the other hand, did a really good grid with thousands of points, doing a much better simulation than i did, and it took it just some seconds.

Whatever we can do, by hand, on classical physics the computer can too, but faster.

When trying to resolve a complicated problem you can break it in very simple parts, but will take you much more time.

There is, of course, some cases in wich you can only use a computer, but i think this is the wrong way to look at this question.

For example, in ww2 artillery boats had mechanical computers they used to know where and when to fire a canon to accuratly shot down an ennemy vessel. The calculus involved are really simple parabolic projectile physics, but the time in wich they needed the answer was really short. So they could see a enemy ship and shoot it down fast.

A human could not compete with a computer in this case.

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In general, I think fluid simulations are a good example of a thing where computers can really help us out. They are nice and intuitive, and fairly practical. They are also a good example of something where we could do the symbolic calculations on simplified problems worked, but doing it computationally lets us handle real-world applications.

If you want a specific example, I suggest:

Steam engines are getting good around the time when you've set the story. This is a great problem for a computer-aided design because it doesn't rely on super high precision observations of natural phenomena outside of the character's control. The goal here is to look at a proposed design and simulate it. So however powerful your computer-like-devices are, your character can come up with a design that is as simple or complex as necessary.

The ability to investigate the simulation in detail at each time-step will translate to increases in efficiency and performance for your character's engines. It is plausible enough that your character has been trained in the art of making these engines by his uncle -- little does the church know, he's actually using science!

He could also simulate propellor blades or maybe even boats generally. This isn't as tied to the timeframe (although maybe he works on steam boats).

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One field were having access to computers completely trumps not having them is cryptography. Essentially the computer owners can brute force most cyphers created and used for hand coding and can easily create cyphers that are impossible to crack by hand. But this is more diplomacy then science and also requires intercepting messages in the first place.

A more sciency area is math, especially number theory. Gauss computed primes by hand up to a million or so just to get an idea of what their distribution looks like. Mersenne primes can also be computed much farther with a computer.

Finally I think the astronomy you mentioned yourself can provide good examples. Discovering Neptune is much easier with computers than without and so could have happened earlier. Presumably there are a few similar examples.

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  • $\begingroup$ Thank you for your input quarangue. Indeed, cryptography and prime numbers are important things that digital computers turned upside down. However, you will notice that the question refers to problems in classical physics, not math in general. Also, let's face it, prime numbers would hardly indispose you with the Inquisition and threaten your very existence in this world, so where's the fun in that ;P ? $\endgroup$ – urquiza Mar 4 at 11:15
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I think the answer might be "none that are particularly useful for the society in question".

If you look at typical uses of computers today:

Meteorology: requires an array of accurate real-time temperature/pressure/wind-speed data in order to predict the weather.

Military: In principle useful for ballistics but without rapid communications a pre-existing hand-calculated table would be more useful on the field.

CAD Engineering: outputs plans/models/designs that are really only useful with good quality raw materials and modern fabrication machinery.

Finance: banking/stock-market/business operations all require fast and reliable communications.

Telecommunications: similarly - needs infrastructure.

PDAs, lap-tops, GPS etc: all only useful in a very different society.

Sciences - Physics, Chemistry, Astronomy, etc: generally the information will only be as useful as existing theories. No point in modelling protein folding or electromagnets if you haven't yet discovered atoms or quantum physics, or Maxwell's laws.

About the only things I could imagine you being able to do would be faster prediction of things like tide tables, eclipses, log tables, and planetary motions (although people were already doing that manually using calculus/algebra etc.). It might also be useful in cryptography, but due to limitations in communications you wouldn't be sending vast amounts of data, so probably no more useful than one-time pads etc. You could, of course, find large primes, solve Diophantic equations etc., but as you have already commented - that isn't physics or technology as such.

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  • $\begingroup$ You're underestimating the usefulness of a ballistic calculator in the field for the military branch that snatched up mechanical means of aiming as fast as they could be developed: naval gunfire. $\endgroup$ – Keith Morrison Mar 5 at 2:14
  • $\begingroup$ As recently as the Great War, the military of many nations managed to successfully bombard each other with long-range projectile weapons, both on land and on sea. They did use mechanical calculators in some situations, but managed quite well without computers. $\endgroup$ – Penguino Mar 5 at 2:47
  • $\begingroup$ Mechanical ballistic calculators were used on warships almost as soon as big-gun warships were in production. The Royal Navy introduced the Dreyer Table prior to World War One and the Mark IV version was on all its capital ships by mid-1916. $\endgroup$ – Keith Morrison Mar 5 at 18:14

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