Editing Exsecant (section)

== History and applications ==
In the 19th century, most [[railroad]] tracks were constructed out of [[circular arc|arcs of circles]], called ''simple curves''.{{r|allen}} [[Surveying|Surveyors]] and [[civil engineering|civil engineers]] working for the railroad needed to make many repetitive trigonometrical calculations to measure and plan circular sections of track. In surveying, and more generally in practical geometry, tables of both "natural" trigonometric functions and their [[common logarithm]]s were used, depending on the specific calculation. Using logarithms converts expensive multiplication of multi-digit numbers to cheaper addition, and logarithmic versions of trigonometric tables further saved labor by reducing the number of necessary table lookups.{{r|van brummelen}}

The ''external secant'' or ''external distance'' of a curved track section is the shortest distance between the track and the intersection of the tangent lines from the ends of the arc, which equals the radius times the trigonometric exsecant of half the [[central angle]] subtended by the arc, <math>R\operatorname{exsec}\tfrac12\Delta.</math>{{r|frye}} By comparison, the ''versed sine'' of a curved track section is the furthest distance from the ''long [[chord (geometry)|chord]]'' (the line segment between endpoints) to the track{{r|gillespie}} – cf. [[Sagitta (geometry)|Sagitta]] – which equals the radius times the trigonometric versine of half the central angle, <math>R\operatorname{vers}\tfrac12\Delta.</math> These are both natural quantities to measure or calculate when surveying circular arcs, which must subsequently be multiplied or divided by other quantities. Charles Haslett (1855) found that directly looking up the logarithm of the exsecant and versine saved significant effort and produced more accurate results compared to calculating the same quantity from values found in previously available trigonometric tables.{{r|haslett}} The same idea was adopted by other authors, such as Searles (1880).{{r|searles}} By 1913 Haslett's approach was so widely adopted in the American railroad industry that, in that context, "tables of external secants and versed sines [were] more common than [were] tables of secants".{{r|jordan}}

In the late-19th and 20th century, railroads began using arcs of an [[Euler spiral]] as a [[track transition curve]] between straight or circular sections of differing curvature. These spiral curves can be approximately calculated using exsecants and versines.{{r|jordan}}{{r|euler}}

Solving the same types of problems is required when surveying circular sections of [[canal]]s{{r|canals}} and roads, and the exsecant was still used in mid-20th century books about road surveying.{{r|roads}}

The exsecant has sometimes been used for other applications, such as [[Euler–Bernoulli beam theory|beam theory]]<ref>{{cite journal | last = Wilson | first = T. R. C. | title = A Graphical Method for the Solution of Certain Types of Equations | department = Questions and Discussions | journal = The American Mathematical Monthly | volume = 36 | number = 10 | year = 1929 | pages = 526–528 | jstor = 2299964 }}</ref> and [[depth sounding]] with a wire.<ref>{{cite journal | last = Johnson | first = Harry F. | year = 1933 | title = Correction for inclination of sounding wire | journal = The International Hydrographic Review | volume = 10 | number = 2 | pages = 176–179 | url = https://journals.lib.unb.ca/index.php/ihr/article/view/28265 }}</ref>

In recent years, the availability of [[calculator]]s and [[computer]]s has removed the need for trigonometric tables of specialized functions such as this one.{{r|calvert}} Exsecant is generally not directly built into calculators or computing environments (though it has sometimes been included in [[software libraries]]),{{r|libraries}} and calculations in general are much cheaper than in the past, no longer requiring tedious manual labor.