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One day, on the faraway world of not-medieval-Europe, someone discovered that a type of stone could be made to float. A decade or two later, there are flying castles, built out of and/or on a foundation of said floating stone.

Floating stone doesn't actually float everywhere. There are large lines (let's call them Ley lines) crisscrossing continents and oceans, where stone floats best on, separated by voids where it doesn't float much, or even at all. Altitude above ground also weaken the effect, so you won't see floating castles more than a few km up at most.

If a floating castle drifts away from a Line, it will (hopefully) gently float down and land - though "gently" at the scale of a castle can still be potentially fatal to anything both in or under the castle.

What propulsion would be used on a floating castle with medieval technology?

Standard airship techniques may not work, as flying castles are so incredibly denser and more massive. Ideally, it would allow for a fair bit of control, to avoid getting pushed out of a Line or into a mountain.

Available tech can be from late Roman to mid-late medieval era. There is no magic apart from floating rock, and it doesn't really scale down.

On the other hand, there have been a few decades with incentives to solve new problems, so new tech can be used if it could have reasonably been invented in those conditions. Likewise, flying castles themselves can help in engineering projects, for example to (slowly) move very large charges up.

The following numbers and assumptions are given as a baseline, but can be changed if it makes things more interesting:

• The two main things moving a drifting floating castle are wind and natural currents from the Ley line. Generally, the Ley current is weaker though not negligible, and constant on one place. A castle can also very slowly gain and loose altitude. Assuming no ballast drop, a castle can accelerate at about 5 cm/s², with a max ascending speed of 2m/s, in good conditions (descending speed is limited only by lithobraking).




• The largest flying foundations are roughly 30m radius circles, with 20m-thick foundation. The smallest ones are 8m radius and 6m-thick foundation. The foundation itself is at least 9/10 filled in stone, but there are often small chambres and tunnels for access below. There can be light structures hanging from under the foundation, though most of the mass must be over it.




• Floating stone has a density of about 2.5 g/cm³. So the biggest castles have a foundation of about 150,000 tons, and the smallest of 3,000 tons. (If other properties are needed, we can use granite as placeholder)




• Total non-floating-stone carrying capacity of a castle is about 1/3 of the stone mass, for the best-built (and most expensive) models - so 1,000 to 50,000 tons.




• To save on capacity, walls can be partially built with floating stone, up to about 1/3 of the mass of the foundation. Those walls must be at least 1m thick, but don't count on carrying mass and can be used as support for other structures.




• Flying castles typically have a smaller slab inside, in a long vertical shaft, capable of independent vertical movement. It is usable only when there is little to no horizontal acceleration, and with a max vertical speed of 10 cm/s. It can be used to raise about 1/30 of the mass of the foundation (so 100 to 5,000 tons). Some people are experimenting with cisterns at both ends, using hydraulic power from the upper cistern, then raising spent water from the lower to the upper one. Not sure how well this scheme would work - this would be the subject of another question, but consider that there is some power available from this, if it can help.

Brute force. Lots of inertia in a castle, so you'll have problems making an engine sufficiently powerful to drive a propellor to move it along, even if you had substantially more advantaged technology than was generally available in the medieval period.

Therefore, brute force.

One means of brute force is to just get a lot of slaves on the end of ropes. This might be a good reason to get back to the good old loot'n'enslave warfare that was so popular back in the axial age. A slightly less ethically challenged option would be to use lots of draft animals... oxen would probably do, because you don't want to get up too much speed with a castle.

Another possibility is to use anchors and capstans, and use mechanical advantage to haul your castle up towards an anchor, and then (slightly less) brute force to move your anchor points in the direction you wish to go. You might have to use this method to make headway against very strong winds, and you'll definitely want to be anchoring your castle in place when you're not moving it around.

There's a minor additional benefit here in that you could perhaps fire a lightweight anchor via a siege weapon, and therefore be able to haul yourself along without actually putting troops or animals on the ground. This would be useful for navigating hostile or otherwise impassible terrain such as mountains, swamps or shallow lakes and rivers.

You should work on the physics behind the floating rocks. What is the principle that makes them float?

We know that the force of the stones is parallel (but opposite in verse) to the gravity, so it alway points toward the sky.

This means that the rocks can only make the castle stay afloat. In order to move the castle, you need a force that can be addressed, with a parallel component with respect to the surface.

You said that, far from the Line, the floating power of the rocks gradually vanishes, as well as it is weaker the higher you are from the surface, so we can make some assumptions.
Everything sounds as if there is a repulsive force from the ground (1), as well as a material that can dampen the force toward the sky that is generated by such stones (which should be plentiful far from the Line).

Now, let's say that I can shield my floating stones:

^{The mortals tremble in the presence of my superior Paint skills!}

An unshielded stone, receives repulsive force from all directions, so that all the components parallel to surface cancel themselves. A shielded one, receive only the forces from one direction, so that it has a smaller component toward the sky, but also a component parallel to the surface, pointing in the opposite verse with respect to the opening in the shielding.

So, a castle could have some big unshielded stones, mainly used to keep everything afloat, and one or more partially shielded stones, mounted on joints that can be orientated, to propel and steer the castle itself.

(1) But let's be careful: the repulsive force from something that can be approximated as an infinite plane would be the same at every distance from the plane itself (in other words, it wouldn't fade with height). I suppose that the Line is thin enough that the ground under the castle can't be considered an infinite, repulsive, plane.

Tracks.

I wondered for how long people had been using rail-type tracks to move cargo overland. A long time: I found the Diolkos, used by the ancient Greeks to move ships overland.

https://en.wikipedia.org/wiki/Diolkos

The Diolkos was a trackway paved with hard limestone[26] with parallel
grooves running about 1.60 metres (63 in) apart.[31] The roadway was
3.4 to 6 metres (11 to 20 ft) wide.[26] Since ancient sources tell little about how the ships were hauled across,[24] the mode of ship
transport has largely to be reconstructed from the archaeological
evidence. The tracks indicate that transport on the Diolkos was done
with some sort of wheeled vehicle.

Your ley lines would have parallel tracks along them, probably of stone. Stays down within the tracks would prevent upward motion of the castle and having two would prevent lateral movement out of the influence of the ley lines supporting power.

The castle would be moved by sliding the stays along the tracks. A team of oxen could do this, in the same manner an ox team might pull a barge. Depending on the distance between the two stays you might need 2 teams working together.

You could do away with the stays and rails and just use the ox teams but you would need to have a cart heavy enough to oppose the lifting power of the castle which is more work to move.

Ley-Line Canals

https://commons.wikimedia.org/wiki/File:Segovia_Aqueduct.JPG

Your castles may float along dedicated canals which are specifically built to allow for floating castles to navigate them. The Romans famously built aqueducts for transporting water, if you built something similar, it would allow for you to easily transport your castles around.

The canals would be constructed along the paths of the ley-lines, essentially acting as guides to prevent the castle from drifting away from the ley-line. This also means you will not accidentally drift into a “dead zone” where the stones do not float at all, which would cause a castle to crash.

Sails

Canals alone though wont cause you to move forward. Instead, as you are now floating along fixed lines, like a railroad track, you can now safely use sails to push you forwards. Before, without he canal, you risked being pushed out of the ley-line. Now though, as you are more firmly fixed into the line, there is significantly less of a chance for you to be pushed out, most momentum from the wind should push you forwards (assuming you angle your sails correctly and the wind is not blowing in the complete opposite direction you want to go).

Wheels

Alternatively, you might opt to use wheels to move your castles. Wheels would run along grooves in the canal and could be powered through a number of ways, such as by man power, using beasts of burden or even with falling water like a watermill. Don’t expect the castle to move particularly quickly though.

One problem with the concept of using canals though is they would take a very long time to build everywhere. Perhaps you only have canals on major routs, such as between cities (much like how highways only run between large cities and smaller towns build up along those routs). You could also take from the idea of shipping lanes where canals are only built on well-established and often used routs.

Magnetism

Essentially, your castles could work like levitating trains. Your stones would work like magnets, repelling away from the ley-line which causes them to float. If you can invert that signal, causing them to be pulled towards the ley-line rather than being pushed away, you could propel yourself through the power of magnetism (not really magnets per say but they function similarly enough).

The way magnetic trains work is by turning on and off magnets in a certain order, causing the train to be pulled forward. Your stones could be turned on and off, when they are on, they float on the ley-line, when they are off, they fall.

So how can we turn this into forward movement? Well if we start with all of the stones being turned on, the castle will be floating and relatively stationary (barring the flow of the ley-line and the wind pushing it). What we do next is turn off the stones at the front of the castle, only for a fraction of a second, which will cause the castle to tilt slightly forwards. The force of gravity will start pulling the castle downwards on that side, pulling it forward. To prevent us from crashing, we turn the stone back on, floating us back up, only now we are slightly further forward. If we keep repeating this process rapidly, we will soon start to build up significant speed, despite the castles high mass.

This concept is very similar to that of the rail gun which uses electromagnetic force to move a metallic object at high speed. Contrary to its name, it is not just intended for weaponry, NASA proposed to use a rail gun to accelerate small aircraft or spacecraft into orbit..

You should be able to sail the castle, the issue would be sail material, you'd be using bamboo, timber and metal rather than cloth because the structures are going to be too large for traditional fabric. Controlling sails that large will be challenging but not impossible, I'd think that capstan and treadwheel driven winches would be sufficient.