Seeing beyond the Horizon

From a hilltop, the horizon lies below the horizontal level at an angle called the “dip”. Around AD 1020, the brilliant Persian scholar al-Biruni used a measurement of the dip, from a mountain of known height, to get an accurate estimate of the size of the Earth. It is claimed that his estimate was within 1% of the true value but, since he was not aware of atmospheric refraction and made no allowance for it, this high precision must have been fortuitous  [TM213 or search for “thatsmaths” at].

Snowdonia photographed from the Ben of Howth, 12 January 2021. Photo: Niall O’Carroll (Instagram).

From a standpoint by the ocean shore, the horizon appears to be about 5 km away. From an elevation of 100 metres, it is some 36 km and from a mountain peak at 500 metres, about 80 km. There is a simple formula for the distance (in kilometres),

d = \sqrt{2RH/1000} ,

where R is the Earth’s radius (6,370 km) and H is the elevation in metres. This formula gives the distance to the geometric horizon, but the visible horizon may be closer or farther away: the transparency of the atmosphere, which allows light to pass through it without scattering, depends upon prevailing meteorological conditions, and variations in density cause light to deviate from a straight path.


In extremely clean, clear air, the visibility can be over 200 km, but this is rare. Visibility is normally limited by turbulence, humidity and pollution and is usually no greater than about 80 km. The ideal conditions are a crisp, cold day with still air, low humidity and an absence of pollution. The colder the air, the less humidity it can hold.

January 2021 was unusually cold, with air temperatures across the country well below average. On 12th January, photographer Niall O’Carroll took a remarkable photograph from Howth Head, showing the mountains of North Wales, with Snowdon prominent in centre-shot.

Snowdonia photographed from the Ben of Howth, 12 January 2021. Photo: Niall O’Carroll (Instagram).

Using the above formula, the horizon from the top of Snowdon, at 1085 m, is 118 km. The horizon from the Ben of Howth (171 m) is at 47 km. The sum of these distances (165 km) exceeds the great circle distance of 140 km between the two peaks. Therefore, each is above the geometric horizon of the other. However, the photograph shows several lesser peaks in Wales, and extensive lower ground. How can this be?


Atmospheric refraction is the bending of light from a straight line as it passes through the atmosphere. There are several factors influencing refraction – temperature, pressure and humidity amongst them – but the most important is the decrease of density with height. Normally, the air temperature drops with increasing altitude – mountain tops are colder than lower ground. . But in an inversion, cold air underlies warmer air.

Air is compressible and its density decreases exponentially with height. Light travels more easily through air of lower density. The route taken between two points by a light ray is the path that can be traversed in the least time. It is easier for light to travel along a route that passes upward through thinner air. As a result, the transition from cold dense air to thinner warmer air bends the light, and objects below the (geometric) horizon can become visible: in suitable weather conditions, we can actually see “beyond the horizon”

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That’s Maths II: A Ton of Wonders

by Peter Lynch now available.
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>>  Review in The Irish Times  <<

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