# The convexity of water

I am curious; I understand how all of Earth’s water is convex - that is, it bends along with the curvature of the planet - but I am not sure as to how it retains it’s seemingly flat surface so well, being a liquid. I have read a few books on pseudoscience, and apparently there were a few Biblical ultra-fundamentalists in the 1800’s who did experiments showing that water was indeed flat to prove that Earth itself is flat. It can be quite hard to measure between two horizons out on the ocean, since there are so many miles between two points. Could anyone here tel me a bit about how this water/curvature thing works? Also, if there is any scientific literature you could point me to, it would be appreciated.

“I am not sure as to how it retains it’s seemingly flat surface so well, being a liquid.”

Is this question a joke in some form?

Taking a wild guess at to what the OP is really asking, the ocean is curved as stated. It appears flat for the same reason the land appears flat - because the earth is big enough so that the small portions humans can see to the horizon are curved so slightly that an eye can’t distinguish much curvature.

Why does water do this as well as land? Because water is extremely incompressible for a liquid. The top few tenths of a percent are roiled by winds but the thousands of feet of water will sit as a single body just like a conteinent.

Having a bit of a hair in my a posteriori about the Flat Earth idea, this is of some interest to me. Charles Johnson, of the Flat Earth Society, as late as the 1980’s, used the flatness of water as a proof (sic) of the flatness of the earth.

However… If you actually want to observe the “bulge” of flat water, go to a large lake or bay. Take a small telescope and, ideally, a tripod. Take sightings. Go to the other end of the bay and take more sightings.

(But, of course, it might be waves or agitated water that “heaps up” in the middle, right?)

In the 1800’s, a bet was made on this issue, and was to be settled by placing posts of equal height along the sides of a long canal. The posts’ height was measured up from the water level. A telescope was placed at one end of the row of posts. If there was a “bulge,” then the middle posts should appear higher than the end posts.

The result? The flat earth guy couldn’t see the bulge, and tried to refuse to pay off on the bet. It took lengthy court action to pry the money out of him.

(Cite: “Flat Earth: The History of an Infamous Idea” by Christine Garwood. Fun book!)

If you have the stomach for it, here is a web site about Charles Johnson.

I haven’t heard about anyone trying to prove the earth was flat by measuring points. I suspect that you are misremembering a much weirder 19th century group, Cyrus Teed’s Koreshan cult (ages before the Branch Davidians used the name). Mr. teed believed that the water on the earth wasn’t convex, or even flat. It was concave, and we all lived on the inside of a “bubble” in an infinite expanse of dirt. To prove that (as they put it) “We Live Inside”, they ran a straight line along a long straight drainage canal and proved to their satisfaction that it plunged into the earth, ths provin the concavity of the world.

http://www.weirdus.com/states/florida/fabled_people_and_places/hollow_earth/index.php

Related subject: a US naval guy I knew once told me that the largest US aircraft carriers were so long that they had to take into account the curvature of the earth in their design (only by an inch or so but still, impressive). Is that true or was he just filling me with seamen?

An aircraft carrier must confront waves that are much, much higher than an inch. If they had to accommodate such a small effect from curvature, the waves would kill them.

Along those lines, the towers of the Golden Gate Bridge are not perfectly parallel… they are further apart at the top to account for the Earth’s curvature.

As somebody who’s spent some limited time at sea, I must say I’m surprised that the simplex experiment hasn’t been mentioned yet. Find an opportunity on a fairly clear day to sit in the wake of a large ship (ideally a tall ship, as sails and hull make a nice contrast) and note how the hull disappears under the horizon before the sails do. I’ve seen this effect through regular binoculars plenty of times.

A Nimitz class carrier is 333m long. I get the curvature at that distance to be 9mm, one third of an inch. the keel and deck lines have been drawn straight in any project I’ve been involved with, buoyancy and longitudinal strength calcs included. Maybe the measurements of building block positions on the assembly site? If they need a long horizontal measure, they need to take into account the fraction of degree angle? I’m no expert in this, but their laser equipment is easily capable of fraction of mm accuracy and they often seem to use that too.

I’ve seen this on an ocean beach. You can’t see the lower portion of ships passing in the distance on a clear day.

That’s when they’re going over the edge.

I don’t know anything about ship building, but as far as the curvature goes, my back-of-the-napkin calculations concur with yours. In optical surveying, you have to account for curvature error over long traverses. The formula is:

C = 0.0239 D^2, where D = distance in 1000s of feet

Over 1100 feet (~333m), that works out to about 3 hundredths of a foot, which is about a third of an inch, like you say. Perhaps naively, I wouldn’t think that this requires any specific attention in design for a couple reasons:

I’d have to ask one the engineers around here, but I’m pretty sure that 3 hundredths in 1000 is well within tolerance for elevations for general construction purposes. I.e., not that you would ever pour a continuous 1100 ft. foundation slab, but if you did, you’d consider yourself lucky to not have any more error than that.

On that note, you don’t build something the size of an aircraft carrier as a continuous, totally rigid body for much the same reason you don’t build bridges or (large) foundations that way. The design of a ship of that size is going to have to accommodate a lot of flexing and expansion. For instance, if I haven’t botched the calculation, 1000 feet of steel will expand/contract by 3 inches over a temperature delta of 40F (you can get that much just from day/night cycles). In a bridge or a building, you use expansion joints. I don’t know how it works in ship design, but something has to take up that slack gracefully unless you want the structure to warp or for bolts/rivets/welds to sheer/split under the strain.

Point being, thermal expansion alone induces variations that are easily an order of magnitude greater than the curvature issue. Even if curvature does have some kind of structural implication, I would suspect that the normal expansion relief mechanisms that are required for much larger and less exotic reasons are more than capable of dealing with it. So, again - I don’t really know anything about ship design, but my suspicion is that this isn’t something specifically considered in the design process.

To the OP:

I’m not sure exactly what you want explained. The oceans are curved for the same reason that the rest of the planet is round. Stars and planets are roughly spherical because of gravity. Gravity may be a fairly weak force, but it can act over any distance and it is always attractive. Moreover, a “point mass” will exert a gravitational field of equal strength in every direction. And any non-point mass (i.e. real-world objects with actual volume) will act just like a point mass if you’re far enough away.

The significance is this: anything massive enough to start sucking matter into its gravity well is going accrue that matter more or less uniformly in every direction because that’s how gravity operates. And a three-dimensional object which is exactly symmetrical in every direction is, you guessed it: a sphere. Hence why the earth, including its oceans, is more or less spherical.

If you’re asking why small drops of water are round(ish), that’s a different and somewhat more complicated matter. The short answer is that water has cohesion: molecules of water cling slightly to other water molecules. Although it arises from a totally different process than gravity (the electromagnetic force), this cohesive force also tends to act uniformly in every direction. So a small drop of water floating in deep space will take on a highly spherical shape even without gravity. On earth, if you have a drop of water sitting on a table, it wants to be nice perfect sphere. But gravity is trying to crush it into the table, which winds up flattening it out into an oblong oval-like cross section instead.

Thanks to all of you for your responses. As weird as it may seem, this really is a serious question. In regards to the hull disappearing before the mast idea, I fully understand and accept it. When confronted with this idea, the people who believed the earth was flat explained it by saying that it is an illusion caused by the bending of light as it gets farther away from you. Again, I don’t believe this stuff. Some time ago I was doing a study on pseudoscience for a college course and I came across this group of people called the Universal Zetetic Society, a small group of Biblical fundamentalists in 1800’s England that used the Bible to prove that the Earth was flat, along with bonkers experiments. Not being an expert on geology/oceanography, I posted this question to see if anyone could explain this a bit more to me. In regards to the “bending of light” idea mentioned above, could anyone inform me the reasons as to why that’s impossible? Note: I am pretty sure I understand it, but I’d just like some outside comments.

Without knowing specifically how they were trying to explain it, I can’t specifically debunk it. However, atmospheric refraction is certainly a real thing, and there is an indirect relationship with the curvature of the earth.

When light passes from one medium to another, it refracts (i.e. bends). From a classical point of view, think of light as perfect little sinusoidal wave. Now imagine that the bottom half of that wave is stuck in molasses while the top half is in open air. When the wave dips down into the molasses, it moves slower than when swings up into the air portion. Because of this, it behaves a lot like a car with one side stuck in mud. If the wheel on the left is spinning faster than the wheel on the right, the car is going to toe to right. Similarly, the wave of light is going to bend towards the slow side.

This is refraction. Light travels slower through any medium that isn’t vacuum because it interacts with the electrons in matter, and these interactions take time. Different substances and different densities of the same substance slow down light by different amounts. We quantify this by giving every substance an “index of refraction”. When light passes from one medium to another with a different index of refraction, it bends towards the substance that slows it down more by an amount that depends on the difference between the two indices.

The quantum explanation for refraction is … complicated. And when you get right down to it, it isn’t really an explanation at all. It’s more like: “this is a mathematical model for the behavior of light which is really, really accurate according to every experiment we’ve ever conducted. Take it or leave it.” As Feynman pointed out in his famous lectures, this behavior is really very weird and no one can explain why light behaves in the very weird way that it does instead of some other way. And unless we can someday find a way to peak beyond the veil of the Heisenberg uncertainty principle, that’s really the end of the story; nature just is what it is.

But I digress. Suppose, for the sake of argument, that the earth IS a sphere. Now imagine that you level up a laser so that it’s perfectly horizontal to the surface the earth. Or, I should say, perfectly tangent to it. As that beam of light travels away from the laser, it is gradually getting farther away from the surface of the earth, due to the curvature. And because of that, the atmosphere that it is travelling through is getting gradually thinner. Although this is a gradual, blurry boundary rather than a sharp one, it is still the case that the light is passing between mediums with different indices of refraction (due to the density). It travels slower through denser air than thinner air, so the effect is that it refracts inwards towards the center of the earth by a small amount. Instead of travelling in a straight line, it takes on a slightly curved trajectory. The effect is small, but it’s still another one of those things that you have to account for if you’re doing optical surveying over long distances.

The argument that the mast-before-the-hull effect is just an illusion due to refraction is bogus simply because it’s a self-defeating argument. Suppose now that the earth is actually flat (on average). Local terrain might not be quite flat, but certainly the ocean would be, being a liquid. You have one observer standing on a pier 10 feet above the surface of the ocean, and another observer standing on the deck of a distance ship, also 10 feet above the surface of the water. If the earth is flat, light travelling from the distant ship to the observer on the dock will remain at a constant altitude, and therefore a constant average density of atmosphere (i.e. minor random fluctuations due to heat, etc. not withstanding). Since there isn’t a change in refractive index, the light won’t refract, and there is no bending that could possibly explain why you would see the mast before the hull.

In other words, the admission that distant light at approximately the same elevation is getting bent at all is an admission that the earth isn’t flat. Perhaps not round, but definitely not flat.

What’s really funny is…this is an actual effect… But it works in the opposite way!

The refraction of light allows us to “see” just a little bit over the horizon. For instance, take a typical sunset. The sun gets lower and lower, then just touches the horizon. It looks like it is sitting on the horizon. Well, if the earth’s atmosphere were, at that moment, magically removed, and replaced by a vacuum with no index of refraction – you would see that the sun is actually partially below the horizon.

It is a very small effect, but you can measure it. (Use eye protection!) Sit by the sea or lake shore, and measure the angle of the sun above the horizon (azimuth) every ten minutes. Graph the angle vs. time. It’s a straight line, until very close to sundown, at which the graphed line bends, just a little.

So, we can see the hull of a ship just a little farther out to sea than we would if the air didn’t refract light.

(Another “hockeystick” graph!)

I’ve just stumbled across the folks the OP was clearly talking about – it’s Samuel Birley Rowbotham ( 1816-1884), who wrote under the pseudonym “Parallax”, and claimed to have observed the effect on the Old Bedford Canal in Norfolk, England, in 1838. He published first a pamphlet and then a book on his observations. These were later repeated in 1870 by John Hampden, who believed them, and Alfred Russell Wallace (the co-discoverer of Evolution), who was a surveyor and corrected for errors in the work, getting a result that supported a curved earth.

Since Wikipedia will be out in a few hours: