Can we prove that plumb line is vertical to ground?

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Solution 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).

Solution 2

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.

Solution 3

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 ${\bf F}({\bf p})$.

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Updated on August 01, 2022

Comments

  • Jim
    Jim over 1 year

    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)

    • bof
      bof over 7 years
      How was "vertical" defined, if not by the direction of the plumb line? And what does vertical "to ground" mean, what does the ground have to do with it?
    • Jim
      Jim over 7 years
      @bof:I would assume that "vertical" would be in their approach something that is in the same direction as a standing man.When someone makes a pillar I would guess that he would understand that it should be pointed upwards to be solid. But if the plumb line would actually indicate what they need, I am not clear how they concluded they could count on that tool
    • Jack M
      Jack M over 7 years
      It's not always vertical to the ground, assuming by vertical you mean perpendicular. Try standing on a slope and holding a plumbline..
    • franz lemmermeyer
      franz lemmermeyer over 7 years
      If you ask whether "we" can prove it, then the answer is no. The earth is approximately a rotating ellipsoid, so except at the poles and the equator a plumb line will neither point to the center nor in a direction perpendicular to the tangent plane.
    • Jim
      Jim over 7 years
      @franzlemmermeyer:My question is how could they assume that a hanging object is perpedicular to the ground so that they can use it as reference for building a pillar or wall
    • Jim
      Jim over 7 years
      @JackM:Yes I mean perpedicular. How would you erect a pillar in a slope? I am not sure if my question is clear or confusing
    • Jack M
      Jack M over 7 years
      @Jim I don't know how I would erect a pillar on a slope, but what I can tell you is that a plumbline certainly isn't going to be perpendicular to the ground.
    • Jim
      Jim over 7 years
      @JackM:It is not and that is my point. They needed some kind of reference to something that is always a right angle. So I am curious how/if they somehow could prove that this tool is dependable for that. If that was something that "seemed" good approximation why didn't they used their own bodies instead of this tool as a point of reference? Do I make sense? May be I am not expressing my question properly
    • Jack M
      Jack M over 7 years
      @Jim I think I understand now. I think the question you're trying to ask is "Without using any modern physics, how can we explain what's so special about the direction in which a plumbob pulls its string?".
    • Jim
      Jim over 7 years
      @JackM:Yes that is it. I used the terms vertical/perpendicular as these are fundamental in geometry and the plumb line seemed like a tool being used as we use a compass and ruler
  • Jim
    Jim over 7 years
    My question is about ancient builders. Why are you mentioning Newton's laws?
  • Jim
    Jim over 7 years
    I am sorry but I am confused. My question is about builders in our remote past and the fact that they depended on the plumb line as a basic tool. How can their thinking process been around "the gravitational field F"?
  • Jim
    Jim over 7 years
    This is helpful.I have the following question: What we define as vertical/perpedicular is something that is 90 degrees with another surface. Now I assume that the vertical that they intuitively understood was that also. I.e. the erect/straight body. Also I assume that when trying to erect e.g. a pillar in order to remain standing they would also build it vertical to the ground. So my question is, why didn't they just use their straight body as a reference when building and considered using such a tool? Am I clear on my question?
  • Nominal Animal
    Nominal Animal over 7 years
    "Why didn't they just use their straight body as a reference?" - That is at the very heart of the matter: A tool user might use a level, or a plumb line if shown -- similar to puzzle tests done on primates and corvids --, but only a tool creator will think of using a plumb line as a vertical standard, instead of being satisfied with their "feels". The deceptively simple answer to this is that Homo Sapiens used to be very, very adept at that. In fact, Homo neanderthalensis was already an adept tool creator almost a hundred thousand years ago, creating extremely sharp flint blades etc.
  • Nominal Animal
    Nominal Animal over 7 years
    The deep answer is anthropology. Why weren't they satisfied with what they could sense directly, and sought tools instead? Why did they observe their surroundings with such a keen eye -- consider prehistoric astronomy, megaliths, etc. I suspect the answer is very much cultural; that that "state" is the niche Homo evolved for, and competed in better than any other species. A key point I am trying to make is that urban Homo sapiens has (culturally) lost most of that drive, and therefore views those old achievements as "miraculous".
  • Nominal Animal
    Nominal Animal over 7 years
    Homo sapiens is still evolving, however. It has been said that the rate of changes is now faster than ever before. Personally, I think Homo sap. urbanus is developing towards eusociality, based on how much effort they put into social structures and "social intelligence", and how poor they are at observing their non-human surroundings. That theory also well explains why current humans find old ideas, mythologies, and techniques so "weird".
  • Jim
    Jim over 7 years
    The very primitive man so, inherently "felt" something as erect/perpendicular without a need to prove it. How is it possible that builders in our more recent past (after 1000BC) depended on this without trying to "formalize" why they use it? Seems strange to me that they used the tool standing on a slope without considering to what it is perpendicular with and only after 17th century AD we formulated gravity theory (so the plumb line is perpendicular to the center of the earth). May be I am mixing things?
  • Nominal Animal
    Nominal Animal over 7 years
    Did they? I do not think so. When they created tools, they certainly kept refining their tool-creation methods. For example, flint knives are much sharper than any metal; flint shards -- using the same method we believe neanderthalensis already used tens of thousands of years ago -- are still used as the sharp tips for atomic force microscopy and other atomic-scale tools, as their edge and point is only a few atoms thick. Same for vertical and horizontal: it is not about formalization, but creating tools to overcome limitations. Here, limitations in relying on human senses alone.
  • Jim
    Jim over 7 years
    With your last point I agree in the sense that the principles remain the same, our implementation of them improves. But I don't have a reason to think that they did not use it on a slope. This tool is still in use by workers in constructions.
  • Jim
    Jim over 7 years
    These axioms are sensible to me. But believed that the Earth was a sphere and that everything fell towards its center seems weird. Do you have any reference example? Egyptians used the plumb line and did not have any idea of the earth being round as far as I know
  • Jack M
    Jack M over 7 years
    The Greeks believed the Earth was round, see for example Eratosthenes - I believe it's also mentioned in Aristotle's work somewhere (IIRC he argued the Earth was round on the basis of its round shadow during a lunar eclipse). When I say they knew things fall towards the center I'm mainly thinking of an argument of Archimedes in his work on fluid dynamics where I think he assumes this, but again I think Aristotle mentions it at some point too.
  • Jim
    Jim over 7 years
    Even in that case, believing to proving is not the same. The concept of proof was introduced by that time. And believing that earth is round is not an axiom. But what you posted as axioms in your answer are indeed helpful for me
  • Nominal Animal
    Nominal Animal over 7 years
    @Jim: I know I have. Water levels are similarly used for finding horizontal levels, even when there is an obstruction so that a laser level cannot be used. (Very nice for laying concrete.) I'd say those who build stuff to last never rely on their senses, as our senses are far from precise, but use tools to overcome the limitations. Again, the quantum leap is not the use of the tool, but the realization that you can construct a tool to get better results than without.
  • Jim
    Jim over 7 years
    Using a plumb line is a fundamental way to intuitively understand that the radius of a circle is at right angle with the tangent. And right angles is fundamental in geometry. It is very puzzling to me that the early scientists i.e. after 1500 AD "played" with things like converting a circle to square or measuring the distance the sun but such a basic tool was left to intuition without a need to prove of its validity. Overall your answer is not only very helpful but very educational and interesting