In astronomy, geography, and related sciences and contexts, a direction or plane passing by a given point is said to be vertical if it contains the local gravity direction at that point.
Conversely, a direction or plane is said to be horizontal (or leveled) if it is perpendicular to the vertical direction. In general, something that is vertical can be drawn from up to down (or down to up), such as the y-axis in the Cartesian coordinate system.
Historical definition
The word horizontal is derived from the Latin , which derives from the Greek , meaning 'separating' or 'marking a boundary'. The word vertical is derived from the late Latin , which is from the same root as vertex, meaning 'highest point' or more literally the 'turning point' such as in a whirlpool.
Girard Desargues defined the vertical to be perpendicular to the horizon in his 1636 book Perspective.
Geophysical definition
The plumb line and spirit level
In physics, engineering and construction, the direction designated as vertical is usually that along which a plumb-bob hangs. Alternatively, a spirit level that exploits the buoyancy of an air bubble and its tendency to go vertically upwards may be used to test for horizontality. A water level device may also be used to establish horizontality.
Modern rotary laser levels that can level themselves automatically are robust sophisticated instruments and work on the same fundamental principle.
The spherical Earth
When the curvature of the Earth is taken into account, the concepts of vertical and horizontal take on yet another meaning. On the surface of a smoothly spherical, homogenous, non-rotating planet, the plumb bob picks out as vertical the radial direction. Strictly speaking, it is now no longer possible for vertical walls to be parallel: all verticals intersect. This fact has real practical applications in construction and civil engineering, e.g., the tops of the towers of a suspension bridge are further apart than at the bottom.
Also, horizontal planes can intersect when they are tangent planes to separated points on the surface of the Earth. In particular, a plane tangent to a point on the equator intersects the plane tangent to the North Pole at a right angle. (See diagram). Furthermore, the equatorial plane is parallel to the tangent plane at the North Pole and as such has claim to be a horizontal plane. But it is. at the same time, a vertical plane for points on the equator. In this sense, a plane can, arguably, be both horizontal and vertical, horizontal at one place, and vertical at another.
Further complications
For a spinning earth, the plumb line deviates from the radial direction as a function of latitude. Only on the equator and at the North and South Poles does the plumb line align with the local radius. The situation is actually even more complicated because Earth is not a homogeneous smooth sphere. It is a non homogeneous, non spherical, knobby planet in motion, and the vertical not only need not lie along a radial, it may even be curved and be varying with time. On a smaller scale, a mountain to one side may deflect the plumb bob away from the true zenith.
On a larger scale the gravitational field of the Earth, which is at least approximately radial near the Earth, is not radial when it is affected by the Moon at higher altitudes.
Independence of horizontal and vertical motions
Neglecting the curvature of the earth, horizontal and vertical motions of a projectile moving under gravity are independent of each other. Vertical displacement of a projectile is not affected by the horizontal component of the launch velocity, and, conversely, the horizontal displacement is unaffected by the vertical component. The notion dates at least as far back as Galileo.
When the curvature of the Earth is taken into account, the independence of the two motion does not hold. For example, even a projectile fired in a horizontal direction (i.e., with a zero vertical component) may leave the surface of the spherical Earth and indeed escape altogether.