Can we prove that plumb line is vertical to ground?

Question

Using a plumb line to make sure a wall is vertical for instance, is as far as I know one of the most primary tools in the sense that the very-very ancient builders used it as an instrument.
I was wondering how could one assume at that time that the weight is always vertical to the ground?
Is it a case of experimenting and observing and assuming that it seems vertical and later on we confirmed it?
Or is there actual a way to prove it in some kind of geometric approach? (Like when they measured heights using tricks with triangles and the sun's shadow)

Answer 1

I was wondering how could one assume at that time that the [string] is always vertical to the ground?

Because the weighted string was the best definition for vertical they had.

Before urbanization, Homo Sap. was an astoundingly perceptive observer of their surroundings. The interesting bit about plumb lines as a tool is not the observation that it is vertical, but utilizing it as a tool!

The concepts of "vertical" (as defined by something hanging down, without swinging, without being pushed by wind and so on) and "horizontal" (as defined by still water, for example) are quite probably older than the species; you can show this by constructing a clearly non-vertical plumb line, or non-horizontal still water, and observe how primates and many other mammals find those confusing and/or intriguing. I suspect that this is part of the vestibular system in mammals.

Things like "ground" and "trees" are just visual cues. I'd wager the overwhelming majority of Homo specimens have known that not all ground is horizontal, and that not all trees are vertical, because they have seen counterexamples.

It is only when both trees and ground are tilted the same way, that human perception is fooled. If the tilt is hard to perceive, so that our sense of balance seems to agree with the trees and the ground, humans may think that things roll and liquids run in the "wrong direction". But, very, very few Homo Sap. Urbanus are smart enough to grab a piece of string, a small weight, and compare whether their sense of vertical agrees with what they can test.

Do not assume that pre-Sapiens humans were all similarly stupid. The amazing jump into tool constructors is realizing how to use the phenomenon to create new tools. That is truly a quantum jump -- there are many tool-using species, but very few species that can construct completely new tools. Even some Corvidae, which are not even mammals, are smart enough to use small pebbles to raise the water level in a transparent container, to get access to a tidbit to eat. However, it is a big step from that into realizing it can be used to make waterways passable by constructing weirs and locks. Or, similarly, from knowing that a weighted string is (in suitable conditions) vertical, to using it as a vertical standard to compare other things to.

Is it a case of experimenting and observing and assuming that it seems vertical and later on we confirmed it?

No. They used is as the definition of vertical.

Or is there actual a way to prove it in some kind of geometric approach?

I suspect that most people would not see any need to prove it, especially if they have never seen a non-vertical plumb line (except in high wind or similar conditions).

Answer 2

The plumb line technically isn't always pointing vertically down. When you first let go of it, it'll swing around a little bit and finally come to rest in a vertical position. So the real question isn't "why it's always pointing vertically?," because that's not true, but rather: "why does the plumb line have a vertical stable equilibrium position?" Note: equilibrium position is the position at which an object will stay indefinitely unless acted on by a net external force, and a stable equilibrium position is an equilibrium position that the object will return to if perturbed slightly from it (for example, the top of a hill is an unstable equilibrium because an object will shoot down the hill if perturbed slightly from the top of it, but the bottom of a hill is a stable equilibrium because if an object testing at the bottom of a hill is perturbed slightly, it will return to the bottom. In this case the vertical position of the plumb line is a stable equilibrium position, because if perturbed slightly the plumb line will swing back and forth a bit and then return to its vertical position).

Now that we've defined the question in a more rigorous, if not slightly too formal way, let's think about it. We need to somehow figure out why something happens. There are two methods to do this: either prove that it must happen, or find a contradiction in reality if it were not to happen. Because if there's a contradiction in reality if this physical phenomena were not to occur, then it must occur, or we have a bad model of reality. Let's assume that Newton's laws are a relatively good model of reality for now and see if a contradiction in reality would occur were the plumb line not to always come to a stable vertical equilibrium.

So let's look at what happens to the plumb line when it's not vertical and show that it must return to a vertical position. This is nowhere near an actual proof using physical laws but rather just a logical walkthrough based on the results of Newton's laws. If the plumb line is displaced from its vertical position, gravity pulls down on it. And, while I won't get into rotational motion and torque here, I'm just going to state that that gravitational force combined with the tension force that keeps moving in a circle (for without that tension force, the plumb bob would just fall straight down - as you can imagine would happen if you cut the string) add together in such a way as to fight any displacement from vertical position. Namely, they fight to push the angle it's displaced from the vertical back to 0. And this will make the plumb bob go from being displaced by some angle to being displaced by a lesser and lesser angle until it passes through vertical and then due to the velocity it's gained, will be displaced to the other side. Here, once again, tension + gravity will push it back towards vertical and it will oscillate, or swing, back to the other side. And this process would continue forever, but friction robs the system of energy until finally it rests in a vertical position.

Answer 3

Yes! Beside the plumb line for vertical reference, they also dug a small trench and filled it with water. The water level served as true horizontal axis. This ensure a 90 degree vertical axis since the ground may be sloped.

Answer 4

This question is about the definitions of "vertical" and "horizontal". The ancient builders had no doubts that (i) these directions are assumed by plumbs hanging down, resp., surfaces of fluids, and (ii) that these two directions are orthogonal to each other.

At each point ${\bf p}$ on the surface of the earth we can feel the gravitational field ${\bf F}$. For all practical purposes this field is homogeneous near ${\bf p}$, i.e., ${\bf F}({\bf x})={\bf F}({\bf p})$ for all ${\bf x}$ in some neighborhood of ${\bf p}$. The direction of ${\bf F}({\bf p})$ is called the downwards vertical at ${\bf p}$, resp., in the neighborhood of ${\bf p}$. Letting an "ideal mass point" hang down on an "ideal string" near ${\bf p}$ it is an easy exercise to show that the potential energy of this mass is minimal if the string has the direction of ${\bf F}({\bf p})$. On the other hand, and this is more difficult to show, the potential energy of an "ideal cup of tea" near ${\bf p}$ is minimal if the upper surface of the tea is orthogonal to ${\bf F}({\bf p})$.

Answer 5

Ancient people presumably assumed (subconsciously, for the most part) the following axioms:

1. When an object is dropped, it falls in a straight line.
2. All such straight lines are parallel to a particular, constant straight line, which we can call "the vertical".

Axiom 2 is of course false, but locally it's approximately true. In fact certain ancient people such as the Greeks believed that the Earth was a sphere and that everything fell towards its center, so they would have had no problem ditching axiom 2 and instead defining "vertical" as being the direction towards the Earth's center. But even without knowledge of the shape of the Earth, axiom 2 is approximately true and quite intuitive.

Anyway, once these two axioms are assumed, it's reasonably obvious that a weight on a string will pull the string in a straight line parallel to "the vertical". After all, if the weight were in any position other than directly below the point where the string is being held, it would be possible for it to fall down a little bit more without breaking the string. Only when the string is parallel to the vertical will the weight not be able to continue falling.

Being able to measure this "vertical" direction is useful for a lot reasons. In architecture, for instance, if the top and the bottom of a building aren't on a line parallel to the vertical, then the top will "want" to fall through the air, whereas if the top - bottom line is vertical, then the top "wants" to fall through the brick, so it's secure.