Haversine formula

The haversine formula determines the great-circle distance between two points on a sphere given their longitudes and latitudes. Important in navigation, it is a special case of a more general formula in spherical trigonometry, the law of haversines, that relates the sides and angles of spherical triangles.

The first table of haversines in English was published by James Andrew in 1805,[1] but Florian Cajori credits an earlier use by José de Mendoza y Ríos in 1801.[2][3] The term haversine was coined in 1835 by James Inman.[4][5]

These names follow from the fact that they are customarily written in terms of the haversine function, given by hav θ = sin2(⁠θ/2⁠). The formulas could equally be written in terms of any multiple of the haversine, such as the older versine function (twice the haversine). Prior to the advent of computers, the elimination of division and multiplication by factors of two proved convenient enough that tables of haversine values and logarithms were included in 19th- and early 20th-century navigation and trigonometric texts.[6][7][8] These days, the haversine form is also convenient in that it has no coefficient in front of the sin2 function.

A diagram illustrating great-circle distance (drawn in red) between two points on a sphere, P and Q. Two antipodal points, u and v are also shown.
  1. ^ van Brummelen, Glen Robert (2013). Heavenly Mathematics: The Forgotten Art of Spherical Trigonometry. Princeton University Press. ISBN 9780691148922. 0691148929. Retrieved 2015-11-10.
  2. ^ de Mendoza y Ríos, Joseph (1795). Memoria sobre algunos métodos nuevos de calcular la longitud por las distancias lunares: y aplicacion de su teórica á la solucion de otros problemas de navegacion (in Spanish). Madrid, Spain: Imprenta Real.
  3. ^ Cajori, Florian (1952) [1929]. A History of Mathematical Notations. Vol. 2 (2 (3rd corrected printing of 1929 issue) ed.). Chicago: Open court publishing company. p. 172. ISBN 978-1-60206-714-1. 1602067147. Retrieved 2015-11-11. The haversine first appears in the tables of logarithmic versines of José de Mendoza y Rios (Madrid, 1801, also 1805, 1809), and later in a treatise on navigation of James Inman (1821). (NB. ISBN and link for reprint of second edition by Cosimo, Inc., New York, 2013.)
  4. ^ Inman, James (1835) [1821]. Navigation and Nautical Astronomy: For the Use of British Seamen (3 ed.). London, UK: W. Woodward, C. & J. Rivington. Retrieved 2015-11-09. (Fourth edition: [1].)
  5. ^ "haversine". Oxford English Dictionary (2nd ed.). Oxford University Press. 1989.
  6. ^ H. B. Goodwin, The haversine in nautical astronomy, Naval Institute Proceedings, vol. 36, no. 3 (1910), pp. 735–746: Evidently if a Table of Haversines is employed we shall be saved in the first instance the trouble of dividing the sum of the logarithms by two, and in the second place of multiplying the angle taken from the tables by the same number. This is the special advantage of the form of table first introduced by Professor Inman, of the Portsmouth Royal Navy College, nearly a century ago.
  7. ^ W. W. Sheppard and C. C. Soule, Practical navigation (World Technical Institute: Jersey City, 1922).
  8. ^ E. R. Hedrick, Logarithmic and Trigonometric Tables (Macmillan, New York, 1913).