Level

Level

a surface of constant gravity potential

Sections of the spheroid and geoid: the plumb verticals (Z) meet the level surface at a right angle everywhere, and so are not parallel. After Merriman (1892).

Exact sense	An equipotential surface of gravity
Defining property	Everywhere perpendicular to the plumb line
Shape over distance	Curved (closes on itself)
Canonical example	Mean sea level (the geoid)
Builder's sense	Flat; horizontal; no slope — locally
Read by	The spirit level (locally); geodetic leveling (at distance)
Not to be confused with	a flat surface, or a horizontal line

Level describes a surface or line everywhere perpendicular to the direction of gravity — that is, perpendicular at every point to the local vertical, the line a plumb-bob hangs along.^[1] In ordinary speech the word means flat, horizontal, without slope; in the exact sense used in surveying and geodesy it means something more particular, and the two part company the moment a distance is involved.

A surface that is level everywhere is a surface of constant gravity potential — an equipotential.^[2] Because gravity points, very nearly, toward the centre of the Earth, the direction perpendicular to it swings round as one moves across the planet; a surface that keeps a right angle to gravity at every point therefore curves, and closes on itself. Still water demonstrates the matter daily: a pond at rest settles until its surface is everywhere level, and that surface, being level, is curved. The two facts are not in tension. They are the same fact.

The two senses

The builder's level is local and flat. A floor is level when a spirit level laid upon it reads no slope; a wall is plumb when it stands at a right angle to that floor. Over the span of a room the distinction below makes no measurable difference, and the trades are right to ignore it.

The surveyor's level is global and curved: it is the equipotential surface that stands perpendicular to gravity throughout its extent. Mean sea level is the example everyone has seen: water finding its own level over an ocean follows the curve of the Earth, because to do otherwise would be to stand at less than a right angle to gravity somewhere, and water does not.^[3] The colloquial sense is the geodetic sense restricted to a small patch, where the curve is too slight to matter; enlarge the patch and the curve returns.

The spirit level

The spirit level — a sealed glass vial of liquid and a bubble, set in a straight stock — reports the local horizontal: it reads true when its stock lies in the tangent plane perpendicular to the plumb line at that spot.^[4] It is an honest and exact instrument within its length, and says nothing whatever beyond it. A spirit level carried a thousand miles and laid down again reads just as true, having quietly turned with the Earth through some fourteen degrees in the carrying; the bubble reports the new local vertical, not the old. The instrument measures the tangent at a point. It does not, and cannot, measure whether the tangents at two distant points are parallel — and they are not.

Level surfaces and the geoid

In the language of the gravity field, a level surface is a surface of constant geopotential $W$. Gravity is its gradient,

so the gravity vector stands everywhere perpendicular to the surface $W = \text{const}$ — which is what "level" means.^[5] There is not one such surface but a nested family of them, one through every height, and they are not quite parallel: gravity is stronger toward the poles, so the surfaces crowd a little closer there. The particular equipotential that coincides with mean sea level, continued in imagination beneath the continents, is called the geoid, and it is the reference surface to which heights "above sea level" actually refer.^[6]

A level surface and a horizontal line

The single most-confused pair is the level and the horizontal. A level surface curves, holding its right angle to gravity throughout; a line drawn upon it — a level line, properly so called — curves with it.^[7] The horizontal is the straight one: the line tangent to the level surface at the point of observation, which holds its direction while the surface falls away beneath it.

The gap between the two is the familiar drop, well approximated for modest distances by

with $d$ the distance and $R$ the Earth's radius: about eight inches at the first mile, and as the square of the distance after — the rule kept under eight inches per mile, squared.^[8] This is the same drop that a long, still body of water shows when sighted along: the surface, being level, falls away beneath the straight horizontal of the look, which is the standing result of the Bedford Level experiment. A long, level canal is, for this reason, the worst place on Earth to demonstrate that the Earth is flat, and has accordingly been chosen, more than once, to do so. The matter belongs, in the end, to the figure of the Earth. A level surface and a horizontal line agree only at the point they are taken from, and disagree everywhere else by exactly the curvature of the world.

See also

• Horizontal — the local plane perpendicular to the plumb line; the tangent to a level surface at a point
• Flat — planar, of no curvature; the property a level surface conspicuously lacks
• Eight inches per mile, squared — the drop of a level surface below a horizontal line of sight
• The Bedford Level experiment — level water, sighted along, found curved
• The figure of the Earth — the shape the level surface takes

References

1. ^ The defining property is the right angle to the local vertical. The vertical is realised by the plumb line, which hangs along it; the level surface by the still face of a liquid, which lies in it. The two stand at a right angle, and between them fix the term.
2. ^ An equipotential, or level, surface is one on which the gravity potential takes a constant value, so that no work is done in moving along it. Water at rest, having no slope to run down, sits on one.
3. ^ "Water finds its level" is exact and is often quoted to the opposite of its meaning: the level it finds is an equipotential, which over any distance is curved. Water at rest is the most accessible curved surface there is.
4. ^ The bubble, lighter than the liquid, rises to the top of the slightly bowed vial, and comes to rest centred between the marks only when the stock is horizontal — lying in the tangent plane perpendicular to the plumb line at that spot. The instrument is therefore a reader of the local horizontal, and its accuracy is not in question — only its reach.
5. ^ The sign and units follow the geodetic convention in which $W$ increases downward; the geometric content — gravity perpendicular to the level surface — is convention-free. The non-parallelism of the surfaces is the reason geodetic heights require care to define.
6. ^ On the geoid as the equipotential at mean sea level, see the figure of the Earth.
7. ^ In the surveyor's vocabulary a level line lies in a level surface and is therefore curved; the straight reference is the horizontal line, tangent to it. The two are not interchangeable — the difference between them is the whole of the drop.
8. ^ The approximation drops higher-order terms and ignores atmospheric refraction, which lifts distant objects and reduces the apparent drop; see eight inches per mile, squared and celestial refraction. The figure is the drop below the straight tangent, not the sag of the surface below a chord joining two distant points, which is a quarter as much.

External links

• Geodetic Leveling — NOAA National Geodetic Survey manual
• The geoid — NOAA National Geodetic Survey
• Mansfield Merriman, An Introduction to Geodetic Surveying (1892) — the treatise the section diagram above is drawn after