Wednesday, 26 October 2022

#26 Versine... from old Math Terms notes


The versine of an angle, A, is an almost extinct expression for the quantity 1-cos(A). Up to the 1600's this was probably the second most common trigonometric value used. The Latin word versed relates to turning, and the "versed sine" was, in essence, the sine turned 90 degrees.

In 1835, James Inman introduced the term haversine to describe a value of 1/2 of the versine, "half-versine". The haversine was an important formula in spherical geometry and navigation, since it gave a simple way to find the approximate distance between two points on the earth using the Longitude and Longitudes. If we consider two points on the unit sphere, with positions given as (lat1, long1) and (lat2, long2) in radians, then the distance between them is given by  where dLat and dLong are the differences in the latitiudes and longitudes. Tables for Navigation contained both Hav(x) and its inverse invHav(x) and the logs of these values to assist in prosthaphaeresis . To find the distances on the earth, the answer would be multiplied by the radius of the earth. According to Jeff Miller's web site, the word first appeared in the third edition of Navigation and Nautical Astronomy for the use of British Seamen.

The mathematical terms converse and inverse are both from the same root. Many other words come less directly from this root. A plow turns dirt up and over and creates a furrow, a straight line of dirt along the ground. Things laid out along a straight line were sometimes said to resemble the furrow and called verses, and thus words in a line of a poem became a verse. To reverse is to turn back, and the obverse side is the side you see when you turn something over, and your vertebra are the joints that allow you to turn.

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