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It is the 10th Century and the hero, Ibn Muadhe has discovered trigonometry about six centuries before Cartes, and has received a wonderful gift.

A very spiritual Avicenna had rushed by camel and horse from Persia to bring two special stones each the size of a person's hand, which can communicate with each other much like present-day telephones, with exceptions. They can only communicate by a straight line and only through atmosphere. They can work in rain and fog, not that I imagine that's a huge issue in Al-Andalus.

The Caliph of Cordova is fascinated and has demanded Ibn Muadhe establish a communication system as far as possible for strategic purposes.

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The problem lies in the fact that this is 10th Century Islamic tech in what is now southern Spain, and, as well, Ibn Muadhe knows the Earth is of course spherical and not flat.

Using this tech and the resources of the Caliph's coffers, how can he get these two magical stones as far a distance as possible? And how far would that be?

I'm thinking tall structures with the stones atop, but hoping for a creative solution using the properties that are allowed and the technology level involved. I'm willing to bend the rules of the properties of the stones just a bit if it can get me a really great distance in a clever way.

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This is a common nautical problem

One of the first things you learn as a sailor is how to find navigation guides. For a lighthouse, for example, you need to know how far away you can see it, so you can estimate your distance when you first see it at night.

For each of two elevated objects, you multiple 1.17 times the square-root of your height of eye (or height of object) to get distance to the horizon. Then add the distances for two objects together to get total visible distance.

How big were structures in the 10th century?

The tallest building in the 10th century (and, infact every century until the 13th) was the Great Pyramid at Giza. Taking a look at the other buildings I could find, not many (if any) were over 100m.

How far can we see with such structures?

So using 100m (328 feet for our thumb rule), we get a distance to the horizon of about 21 nautical miles. So with two such towers, we could expect to see around 42 nautical miles, or 80 km. Therefore, we can assume our straight line comms will work for about 80km.

For Cordoba related references, Jaen is about 80km away, Sevilla is 120, Malaga is 135, and Toledo is 230.

But wait!

There are mountains in Spain, and those are going to give you a much more significant height advantage than towers. Grenada is about 740m, while Cordoba is at 120m, for a total height difference of 620m. If we put a tower on top of a 620m hill, we get about 57nm to the horizon, for a total of 145 km. Conveniently, Grenada is 130 km away from Cordoba, so just about perfect for this communication method.

Best alternative?

Another method would be to put one message stone on the top of a tower at a high point of a mountain. For example, Pico Veleta is about 3398 m tall, and only 25km from Grenada. The mountain-to-tower range from the peak of that would be about 265 km. Thus, if you make Grenada your capital city, you could communicate with a ~6 hour delay for a fast rider to take a message up the mountain, over most of Andalusia and to the Tangiers and Nador on the African coast.

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If the stones use EM spectrum, just bounce off the ionosphere. If they use lasers, mount primitive prisms on poles every twenty miles. If they are message passers, fire them up on ballista. If they....

The issue you are facing is that any sufficiently advanced magic is indistinguishable from technology. You could use any method, up to and including mounting a hermit every fifty miles to swing a live cat in circles thus powering the stones to bend.

Poor cat.

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I think that terrain will get in the way more than the curvature of the earth. So do what you would do to get over local obstacle.

Place them on mountain tops with a semaphore system from mountain to base.

Invent hot air balloons and use those as your communications links.

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From eyesight to horizon on a "Flat" surface, you can see about three miles. For an observer at the top of a Battleship (About 180 feet), you can see around 16 miles.

The tallest mountain in the area you've specified is Mulhacén, which is about 11,400 feet tall. Establishing an outpost on the top of that would give you a distance of about 130 miles theoretical distance. Of course, things like other mountains might get in the way, so it's unlikely that it will get that far. If you had more than two stones, this would be a lot more useful (Climbing mountains isn't exactly the easiest thing), since you could relay a message to the top of the mountain, and then have the mountain relay it to someone else. BUT you specified two stones.

The usefulness of placing one of the stones on the mountain is probably not that great, since you'd still have to send messages up and down the mountain.

Honestly, a system of semaphore stations space out on mountaintops, etc would work better for quicker communications. Instant communication stones sound like a really great idea for long-distance communications, but where they could really shine is in short-range communications. Like in combat. You can often see what your other section is doing, but communicating to them can be hard. Instant communications with two sections allows for much faster response time. No need for runners, no need for flags, or trumpets, etc.

EDIT Oh, and a tool that is useful for figuring out how far you can theoretically go can be found here.

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If they work through rain and fog, I'm assuming water doesn't stop your magic stones communications, so the answer from there seems obvious use the stones across oceans.

As has been pointed out in other answers, mountain tops give you additional line of sight over the horizon, if they can communicate through ocean water this would increase the apparent elevation by a lot. The Mediterranean is ~1500 m deep on average and the Atlantic is ~3300 m deep on average, essentially communicating across water is between mountain tops. For comparison the highest point in the Iberian Peninsula is Mulhacen at 3,478 m.

This could potentially allow communications between the Balearic Islands in the Mediterranean and the mainland (~100-200 km), across the Mediterranean to North Africa, or longer distances between ships at sea.

The ships at sea in the Atlantic would likely be the longest distance available up to 400+ km apart (like communicating between two 3,000 m tall mountain tops), but isn't likely to be the most useful strategically.

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