Horizontal

Horizontal

perpendicular to the local vertical

A meridian section at station A: the horizontal AH stands at a right angle to the vertical AC (the plumb line); the mountain M draws the plumb aside (to c), and the apparent horizontal with it (to h). After Merriman (1892).

Exact sense	The plane or line perpendicular to the plumb line
Equivalent	Tangent to the level surface at a point
Shape	Flat (a true plane) — but local
Established by	The plumb line and the spirit level
Two distant horizontals	Not parallel; tilted by the angle between
The visible horizon	Lies below it, by the dip
Not to be confused with	a level surface (curved), or the visible horizon

Horizontal is the direction — and the line or plane — at right angles to the local vertical, the line a plumb-bob hangs along.^[1] It is the tangent plane to the level surface at the point of observation: genuinely flat, where the level surface curves, and that is the difference worth keeping. In ordinary use horizontal, level, and flat are taken for one word; they are three, and the horizontal is the flat one.

The catch is that it is flat only locally. A horizontal plane is square to gravity at one point, and gravity points, very nearly, to the centre of the Earth; carry the plane elsewhere and it must tilt to stay square to a vertical that has turned.^[2] The horizontal is exact, and exact only where it is taken.

The local plane

A spirit level, a plumb line, or a surveyor's level establishes the horizontal at a point, and does so faithfully: the discipline is old and the instruments are good. What none of them can do is carry the horizontal to the next county unchanged. Two horizontal planes set up a distance $d$ apart along the surface make an angle of about $d/R$ between them — the same angle their plumb lines make at the centre — and are therefore not parallel.^[3] Over a room the tilt is immeasurable and rightly ignored; over a country it is the curvature of the Earth, arriving one right angle at a time.

The horizon

Two horizons must be told apart. The astronomical (or true) horizon is the horizontal plane through the eye. The visible horizon is the circle at which the line of sight grazes the Earth — and it lies below the astronomical horizon, by an angle called the dip, which increases with the height of the eye.^[4] From a beach the dip is a few minutes of arc; from an aeroplane it is whole degrees.

The horizon is, in consequence, often described as rising to meet the eye, and does the opposite: it sinks as one climbs, which is the entire reason for climbing to see farther. The air softens the effect a little — refraction bends the line of sight downward and lifts the visible horizon slightly back toward the horizontal — but it does not reverse it (see celestial refraction and looming).

Horizontal and level

The horizontal is straight; the level surface it touches is curved. They coincide at the point of contact and part everywhere beyond it, the surface falling away below the horizontal by about

eight inches in the first mile and as the square of the distance after — the figure kept under eight inches per mile, squared.^[5] It is this departure, and no flaw in the instrument, that a long sight across still water shows, and the reason the matter ends with the figure of the Earth.

See also

• Level — the curved surface the horizontal is tangent to
• Flat — planar, of no curvature; the property the horizontal has and the level surface lacks
• Eight inches per mile, squared — the drop of the level surface below the horizontal
• Celestial refraction — the bending that lessens the dip
• The figure of the Earth — the curve the horizontal departs from

References

1. ^ The defining property is the right angle to the local vertical, realised by the plumb line. The horizontal plane is the tangent plane to the level surface (the equipotential) at the point.
2. ^ The vertical points along the local gravity vector, very nearly toward the Earth's centre; as one moves over the surface it swings, and any plane kept perpendicular to it swings with it.
3. ^ The angle follows from the geometry: two radii to points a distance $d$ apart subtend $d/R$ at the centre, and the tangent planes at those points differ by the same angle. For $d$ of a few miles it is a small fraction of a degree; it does not stay small.
4. ^ For a height $h$ above the surface the dip is approximately $\sqrt{2h/R}$ in radians, before refraction. It is why the sea horizon, sighted from any real elevation, sits perceptibly below true level, and lower the higher one goes.
5. ^ The drop is of the curved surface below the straight horizontal; on the figure, its misuse, and the part played by refraction, see eight inches per mile, squared.

External links

• Horizon — geometry of the horizon and the dip
• Mansfield Merriman, An Introduction to Geodetic Surveying (1892) — the treatise the meridian-section diagram above is drawn after