Editing Exsecant

{{short description|Trigonometric function defined as secant minus one}}
[[File:Exsecant and versine.png|thumb|upright=1.25|The exsecant and versine functions substitute for the expressions {{math|1=exsec&thinsp;''x'' = sec&thinsp;''x''&thinsp;&minus;&thinsp;1}} and {{math|1=vers&thinsp;''x'' = 1&thinsp;&minus;&thinsp;sec&thinsp;''x''}} which appear frequently in certain applications.{{r|cajori}}]]
[[File:Versine, chord, and exsecant as line segments.png|thumb|upright=1.25|The names exsecant, versine, chord, etc. can also be applied to line segments related to a circular arc.{{r|segments}} The length of each segment is the radius times the corresponding trigonometric function of the angle.]]

The '''external secant''' function (abbreviated '''exsecant''', symbolized '''exsec''') is a [[trigonometric function]] defined in terms of the [[secant (trigonometry)|secant]] function:

<math display=block>\operatorname{exsec} \theta = \sec\theta - 1 = \frac{1}{\cos\theta} - 1.</math>

It was introduced in 1855 by American [[civil engineering|civil engineer]] [[Charles Haslett]], who used it in conjunction with the existing [[versine]] function, <math>\operatorname{vers}\theta = 1 - \cos\theta,</math> for designing and measuring [[circular arc|circular]] sections of [[railroad]] track.{{r|haslett}} It was adopted by [[surveying|surveyors]] and civil engineers in the United States for railroad and [[geometric design of roads|road design]], and since the early 20th century has sometimes been briefly mentioned in American trigonometry textbooks and general-purpose engineering manuals.{{r|trigbooks}} For completeness, a few books also defined a '''coexsecant''' or '''excosecant''' function (symbolized '''coexsec''' or '''excsc'''), <math>\operatorname{coexsec} \theta = {}</math><math>\csc\theta - 1,</math> the exsecant of the [[complementary angle]],{{r|bohannan}}{{r|hall}} though it was not used in practice. While the exsecant has occasionally found other applications, today it is obscure and mainly of historical interest.{{r|atlas}}

As a [[line segment]], an external secant of a [[circle]] has one endpoint on the circumference, and then extends radially outward. The length of this segment is the radius of the circle times the trigonometric exsecant of the central angle between the segment's inner endpoint and the [[point of tangency]] for a line through the outer endpoint and [[tangent]] to the circle.

== Etymology ==

The word ''secant'' comes from Latin for "to cut", and a general [[secant line]] "cuts" a circle, intersecting it twice; this concept dates to antiquity and can be found in Book 3 of [[Euclid's Elements|Euclid's ''Elements'']], as used e.g. in the [[intersecting secants theorem]]. 18th century sources in [[Neo-Latin|Latin]] called ''any'' non-[[tangent]]ial line segment external to a circle with one endpoint on the circumference a ''secans exterior''.{{r|Latin}}

The trigonometric [[secant function|''secant'']], named by [[Thomas Fincke]] (1583), is more specifically based on a line segment with one endpoint at the center of a circle and the other endpoint outside the circle; the circle divides this segment into a radius and an external secant. The external secant segment was used by [[Galileo Galilei]] (1632) under the name ''secant''.{{r|galileo}}

== 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.

== Catastrophic cancellation for small angles ==

Naïvely evaluating the expressions <math>1 - \cos \theta</math> (versine) and <math>\sec \theta - 1</math> (exsecant) is problematic for small angles where <math>\sec \theta \approx \cos \theta \approx 1.</math> Computing the difference between two approximately equal quantities results in [[catastrophic cancellation]]: because most of the digits of each quantity are the same, they cancel in the subtraction, yielding a lower-precision result.

For example, the secant of {{math|1°}} is approximately {{math|1={{val|1.000152}}}}, with the leading several digits wasted on zeros, while the [[common logarithm]] of the exsecant of {{math|1°}} is approximately {{math|{{val|-3.817220}}}},{{r|log exsec}} all of whose digits are meaningful. If the logarithm of exsecant is calculated by looking up the secant in a six-place [[trigonometric table]] and then subtracting {{math|1}}, the difference {{math|1=sec&thinsp;1° &minus;&thinsp;1 ≈&nbsp;}}{{wbr}}{{math|1={{val|0.000152}}}} has only 3 [[significant digits]], and after computing the logarithm only three digits are correct, {{math|1=log(sec&thinsp;1° &minus;&thinsp;1) ≈&nbsp;}}{{wbr}}{{math|1=<span class="nowrap">−3.81<span style="color:#a00">8<span style="margin-left:.25em;">156</span></span></span>}}.<ref>The incorrect digits are highlighted in red.</ref> For even smaller angles loss of precision is worse.

If a table or computer implementation of the exsecant function is not available, the exsecant can be accurately computed as <math display=inline>\operatorname{exsec} \theta = \tan \theta \, \tan \tfrac12\theta\vphantom\Big|,</math> or using versine, <math display=inline>\operatorname{exsec} \theta = \operatorname{vers} \theta \, \sec \theta,</math> which can itself be computed as <math display=inline>\operatorname{vers} \theta  = 2 \bigl({\sin \tfrac12\theta}\bigr)\vphantom)^2\vphantom\Big| = {}</math>{{wbr}}{{nobr|<math>\sin \theta \, \tan \tfrac12\theta\,\vphantom\Big|</math>;}} Haslett used these identities to compute his 1855 exsecant and versine tables.{{r|Haslett summary}}{{r|nagle}}

For a sufficiently small angle, a circular arc is approximately shaped like a [[parabola]], and the versine and exsecant are approximately equal to each-other and both proportional to the square of the arclength.{{r|shunk}}

==Mathematical identities==

==={{anchor|arcexsec}}Inverse function===
The [[inverse function|inverse]] of the exsecant function, which might be symbolized {{math|arcexsec}},{{r|hall}} is well defined if its argument <math>y \geq 0</math> or <math>y \leq -2</math> and can be expressed in terms of other [[inverse trigonometric function]]s (using [[radian]]s for the angle):

<math display=block>
\operatorname{arcexsec}y = \arcsec(y+1) = \begin{cases}
{\arctan}\bigl(\!{\textstyle \sqrt{y^2+2y}}\,\bigr) & \text{if}\ \ y \geq 0, \\[6mu]
\text{undefined} & \text{if}\ \ {-2} < y < 0, \\[4mu]
\pi - {\arctan}\bigl(\!{\textstyle \sqrt{y^2+2y}}\,\bigr) & \text{if}\ \ y \leq {-2}; \\
\end{cases}_{\vphantom.}
</math>

the arctangent expression is well behaved for small angles.{{r|scheme inverse}}

===Calculus===

While historical uses of the exsecant did not explicitly involve [[calculus]], its [[derivative (mathematics)|derivative]] and [[antiderivative]] (for {{mvar|x}} in radians) are:{{r|mathworld}}

<math display=block>\begin{align}
\frac{\mathrm{d}}{\mathrm{d}x}\operatorname{exsec} x &= \tan x\,\sec x, \\[10mu]
\int\operatorname{exsec} x\,\mathrm{d}x &= \ln\bigl|\sec x + \tan x\bigr| - x + C,\vphantom{\int_|}
\end{align}</math>

where {{math|ln}} is the [[natural logarithm]]. See also [[Integral of the secant function]].

===Double angle identity===

The exsecant of twice an angle is:{{r|hall}}

<math display=block>\operatorname{exsec} 2\theta = \frac{2 \sin^2 \theta} {1 - 2 \sin^2 \theta}.</math>

==See also==
* [[Chord (geometry)]] – A line segment with endpoints on the circumference of a circle, historically used trigonometrically
* [[Exponential minus 1]] – The function <math>x \mapsto e^x - 1,</math> also used to improve precision for small inputs

== Notes and references ==

{{reflist |30em |refs=

<ref name=allen>{{cite book 
  | last = Allen | first = Calvin Frank
  | year = 1894 | orig-year = 1889 
  | title = Railroad Curves and Earthwork 
  | place = New York 
  | publisher = Spon & Chamberlain 
  | url = https://archive.org/details/railroadcurvesea00allerich/page/20/ 
  | page = 20
  }}</ref>

<ref name=calvert>{{cite web 
  | last = Calvert | first = James B. 
  | year = 2007 | orig-year = 2004
  | title = Trigonometry 
  | url = http://www.du.edu/~jcalvert/math/trig.htm  | access-date = 2015-11-08 | url-status = dead 
  | archive-url = https://web.archive.org/web/20071002214133/http://mysite.du.edu/~jcalvert/math/trig.htm | archive-date = 2007-10-02 
  }}</ref>

<ref name=segments>The original conception of trigonometric functions was as line segments, but this was gradually replaced during the 18th and 19th century by their conception as length ratios between sides of a right triangle or abstract functions; when the exsecant was introduced, in the mid 19th century, both concepts were still common. {{pb}}
{{cite journal
  | last = Bressoud | first = David
  | year = 2010
  | title = Historical Reflections on Teaching Trigonometry
  | journal = Mathematics Teacher
  | volume = 104 | number = 2 | pages = 106–112
  | doi = 10.5951/MT.104.2.0106
  | url = http://patthompson.net/ThompsonCalc/Ch2_Graphics/Bressoud-Trig.pdf
  }} {{pb}}
{{cite journal
  | last = Van Sickle | first = Jenna
  | year = 2011
  | title = The history of one definition: Teaching trigonometry in the US before 1900
  | journal = International Journal for the History of Mathematics Education
  | volume = 6 | number = 2 | pages = 55–70
  | url = https://www.academia.edu/6967946 | url-access = registration
  }}</ref>

<ref name=trigbooks>{{cite book 
  | title = Trigonometry 
  | last1 = Kenyon | first1 = Alfred Monroe 
  | last2 = Ingold  | first2 = Louis 
  | year = 1913 
  | place = New York 
  | publisher = [[The Macmillan Company]] 
  | page = 5
  | url = https://archive.org/details/trigonometry01ingogoog/page/n23
  }} {{pb}}
{{cite book 
  | last1 = Hudson | first1 = Ralph Gorton 
  | last2 = Lipka | first2 = Joseph 
  | year = 1917 
  | title = A Manual of Mathematics 
  | place = New York 
  | publisher = [[John Wiley & Sons]] 
  | page = 68 
  | url = https://books.google.com/books?id=-_0SAQAAMAAJ }} {{pb}}
{{cite book
  | last1 = McNeese | first1 = Donald C.
  | last2 = Hoag | first2 = Albert L.
  | year = 1957
  | title = Engineering and Technical Handbook
  | place = Englewood Cliffs, NJ
  | publisher = Prentice-Hall
  | lccn = 57-6690
  | pages = 147, 315–325 (table 41)
  | url = https://archive.org/details/engineeringtechn0000mcne/page/147/
  | url-access = limited
  }} {{pb}}
{{cite book
  | last = Zucker | first = Ruth
  | year = 1964
  | chapter = 4.3.147: Elementary Transcendental Functions - Circular functions
  | page = 78
  | editor1-last = Abramowitz | editor1-first = Milton | editor1-link = Milton Abramowitz
  | editor2-last = Stegun | editor2-first = Irene A. | editor2-link = Irene Stegun
  | title = [[Abramowitz and Stegun|Handbook of Mathematical Functions]]
  | chapter-url = https://personal.math.ubc.ca/~cbm/aands/page_78.htm
  | place = Washington, D.C.
  | publisher = National Bureau of Standards
  | lccn = 64-60036
  }} </ref>

<ref name=frye>{{cite book 
  | last = Frye | first = Albert I. 
  | year = 1918  | orig-year = 1913 
  | title = Civil engineer's pocket-book: a reference-book for engineers, contractors and students containing rules, data, methods, formulas and tables 
  | edition = 2nd
  | place = New York 
  | publisher = [[D. Van Nostrand Company]] 
  | page = 211 
  | url = https://archive.org/details/civilengineerspo00frye/page/211/
  }}</ref>
  
<ref name=mathworld>{{cite web 
  | last = Weisstein | first = Eric W. | author-link = Eric Wolfgang Weisstein 
  | title = Exsecant 
  | work = [[MathWorld]] 
  | publisher = [[Wolfram Research, Inc.]] 
  | date = 2015 | orig-year = 2005 
  | url = http://mathworld.wolfram.com/Exsecant.html
  | access-date = 2015-11-05
  }}</ref>
  
<ref name=atlas>{{cite book 
  | last1 = Oldham | first1 = Keith B. 
  | last2 = Myland | first2 = Jan C. 
  | last3 = Spanier | first3 = Jerome 
  | date = 2009 | orig-year = 1987 
  | title = An Atlas of Functions 
  | edition = 2nd 
  | publisher = Springer 
  | at = Ch. 33, "The Secant sec(x) and Cosecant csc(x) functions", §33.13, p. 336
  | isbn = 978-0-387-48806-6 
  | doi = 10.1007/978-0-387-48807-3
  | url = https://books.google.com/books?id=UrSnNeJW10YC&dq=exsecant&pg=PA336 
  | quote = Not appearing elsewhere in the ''Atlas'' [...] is the archaic ''exsecant'' function [...].
  }}</ref>

<ref name=galileo>
Galileo used the Italian ''segante''. {{pb}}
{{cite book 
  | last = Galilei | first = Galileo | author-link = Galileo Galilei 
  | year = 1632 
  | title = Dialogo di Galileo Galilei sopra i due massimi sistemi del mondo Tolemaico e Copernicano 
  | title-link = Dialogue Concerning the Two Chief World Systems 
  | trans-title = Dialogue on the Two Chief World Systems, Ptolemaic and Copernican 
  | language = Italian
  }} {{pb}}
{{cite book   
  | editor-last = Finocchiaro | editor-first = Maurice A. 
  | last = Galilei | first = Galileo | author-link = Galileo Galilei 
  | year = 1997 | orig-year = 1632 
  | title = Galileo on the World Systems: A New Abridged Translation and Guide 
  | publisher = [[University of California Press]] 
  | isbn = 9780520918221 
  | pages = 184 (n130), 184 (n135), 192 (n158) 
  | quote = Galileo's word is ''segante'' (meaning secant), but he clearly intends ''exsecant''; an exsecant is defined as the part of a secant external to the circle and thus between the circumference and the tangent.
  }} {{pb}}
{{cite journal
  | last = Finocchiaro | first = Maurice A.
  | year = 2003
  | title = Physical-Mathematical Reasoning: Galileo on the Extruding Power of Terrestrial Rotation
  | journal = Synthese
  | volume = 134
  | number = 1–2, Logic and Mathematical Reasoning
  | pages = 217–244
  | doi = 10.1023/A:1022143816001
  | jstor = 20117331
  }}</ref>

<ref name=nagle>{{cite book 
  | last = Nagle | first = James C. 
  | year = 1897 
  | title = Field Manual for Railroad Engineers 
  | edition = 1st
  | chapter = IV. Transition Curves 
  | place = New York 
  | publisher = [[John Wiley and Sons]] 
  | at = §§ 138–165, {{pgs|110–142}}; [https://archive.org/details/fieldmanualforra00naglrich/page/332/ Table XIII: Natural Versines and Exsecants], {{pgs|332–354}} 
  | chapter-url = https://archive.org/details/fieldmanualforra00naglrich/page/110/
  }} {{pb}}
Review: {{cite journal 
  | date = 1897-12-03 
  | title = ''Field Manual for Railroad Engineers''. By J. C. Nagle 
  | type = Review 
  | journal = [[The Engineer (UK magazine)|The Engineer]] 
  | volume = 84 
  | page = 540 
  | url = https://archive.org/details/sim_engineer_july-2-december-31-1897_84/page/540/mode/1up
  }}</ref>
  
<ref name=libraries>{{cite web 
  | last = Simpson | first = David G. 
  | date = 2001-11-08 
  | title = AUXTRIG
  | type = [[Fortran 90]] source code 
  | publisher = [[NASA Goddard Space Flight Center]] 
  | location = Greenbelt, MD 
  | url = http://www.davidgsimpson.com/software/auxtrig_f90.txt 
  | access-date = 2015-10-26 
  }} {{pb}}
{{cite web 
  | last = van den Doel | first = Kees 
  | date = 2010-01-25 
  | title = jass.utils Class Fmath 
  | work = JASS - Java Audio Synthesis System 
  | version = 1.25 
  | url = http://www.cs.ubc.ca/~kvdoel/jass/doc/jass/utils/Fmath.html#aexsec%28double%29 
  | access-date = 2015-10-26 
  }} {{pb}}
{{cite web
  | title = MIT/GNU Scheme – Scheme Arithmetic
  | publisher = [[Massachusetts Institute of Technology]]
  | type = [[MIT/GNU Scheme]] source code
  | version = v.&nbsp;12.1
  | at = <code>exsec</code> function, <code>arith.scm</code> lines 61–63
  | date = 2023-09-01
  | access-date = 2024-04-01
  | url = https://git.savannah.gnu.org/cgit/mit-scheme.git/tree/src/runtime/arith.scm?h=release-12#n61
  }}</ref>

<ref name="scheme inverse">
{{cite web
  | title = 4.5 Numerical operations
  | website = MIT/GNU Scheme Documentation
  | publisher = [[Massachusetts Institute of Technology]]
  | version = v.&nbsp;12.1
  | at = procedure: aexsec
  | date = 2023-09-01
  | access-date = 2024-04-01
  | url = https://www.gnu.org/software/mit-scheme/documentation/stable/mit-scheme-ref/Numerical-operations.html#index-aexsec
  }} {{pb}}
{{cite web
  | title = MIT/GNU Scheme – Scheme Arithmetic
  | publisher = [[Massachusetts Institute of Technology]]
  | type = [[MIT/GNU Scheme]] source code
  | version = v.&nbsp;12.1
  | at = <code>aexsec</code> function, <code>arith.scm</code> lines 65–71
  | date = 2023-09-01
  | access-date = 2024-04-01
  | url = https://git.savannah.gnu.org/cgit/mit-scheme.git/tree/src/runtime/arith.scm?h=release-12#n65
  }}</ref>

<ref name=Latin>
{{cite book 
  | last1 = Patu<!-- (Patu de Mello)? --> | first1 = Andræâ-Claudio (André Claude)
  | last2 = Le Tort | first2 = Bartholomæus
  | year = 1745 
  | editor-first = Franciscus (Dominique-François) | editor-last = Rivard | editor-link = :fr:Dominique-François Rivard 
  | title = Theses Mathematicæ De Mathesi Generatim 
  | language = Latin 
  | place = Paris 
  | publisher = Ph. N. Lottin 
  | page = 6 
  | url = https://books.google.com/books?id=R7RQAAAAcAAJ
  }} {{pb}}
{{cite book 
  | last = Lemonnier | first = Petro (Pierre) | author-link = Petro Lemonnier 
  | year = 1750 
  | title =  Cursus Philosophicus Ad Scholarum Usum Accomodatus 
  | volume = 3 
  | language = Latin 
  | pages = 303– 
  | editor1-first = Ludovicum (Ludovico) | editor1-last = Genneau 
  | editor2-first = Jacobum (Jacques) | editor2-last = Rollin 
  | location = [[Collegio Harcuriano]] ([[Collège d'Harcourt]]), Paris 
  | url = https://books.google.com/books?id=R49XAAAAcAAJ
  }} {{pb}}
{{cite book 
  | last = Thysbaert | first = Jan-Frans 
  | year = 1774 
  | title = Geometria elementaria et practica 
  | language = Latin 
  | publisher = Lovanii, e typographia academica 
  | chapter = Articulus II: De situ lineæ rectæ ad Circularem; & de mensura angulorum, quorum vertex non est in circuli centro. §1. De situ lineæ rectæ ad Circularem. Definitio II: [102] 
  | page = 30, foldout 
  | url = https://books.google.com/books?id=jmY-AAAAcAAJ
  }} {{pb}}
{{cite book 
  | last = van Haecht | first = Joannes 
  | year = 1784 
  | title = Geometria elementaria et practica: quam in usum auditorum 
  | language = Latin 
  | publisher = Lovanii, e typographia academica 
  | page = 24, foldout 
  | chapter = Articulus III: De secantibus circuli: Corollarium III: [109] 
  | url = https://books.google.com/books?id=4_1AAAAAcAAJ
  }}</ref>

<ref name=bohannan>{{cite book 
  | last = Bohannan | first = Rosser Daniel
  | year = 1904 | orig-year = 1903
  | title = Plane Trigonometry
  | place = Boston
  | publisher = Allyn and Bacon
  | chapter = $131. The Versed Sine, Exsecant and Coexsecant. §132. Exercises 
  | pages = 235–236
  | chapter-url = https://archive.org/details/planetrigonometr00boharich/page/235/
  }}</ref>

<ref name=hall>{{cite book
  | title = Plane Trigonometry 
  | last1 = Hall | first1 = Arthur Graham 
  | last2 = Frink  | first2 = Fred Goodrich 
  | year = 1909 
  | chapter = Review Exercises
  | place = New York 
  | publisher = [[Henry Holt and Company]] 
  | at = §&nbsp;"Secondary Trigonometric Functions", {{pgs|125–127}}
  | chapter-url = https://archive.org/details/planetrigonometr00hallrich/page/125/ 
  }}</ref>

<ref name=searles>{{cite book 
  | last = Searles | first = William Henry 
  | year = 1880 
  | title = Field Engineering. A hand-book of the Theory and Practice of Railway Surveying, Location, and Construction
  | location = New York 
  | publisher = [[John Wiley & Sons]] 
  | url = https://archive.org/details/fieldengineering00sear_0/ }} {{pb}}
{{cite book 
  | last1 = Searles | first1 = William Henry 
  | last2 = Ives | first2 = Howard Chapin
  | year = 1915 | orig-year = 1880 
  | title = Field Engineering: A Handbook of the Theory and Practice of Railway Surveying, Location and Construction
  | edition = 17th
  | location = New York 
  | publisher = [[John Wiley & Sons]] 
  | url = https://archive.org/details/fieldengineering00sear/
  }}</ref>

<ref name=haslett>{{cite book
  | last = Haslett | first = Charles 
  | editor-last = Hackley | editor-first = Charles W.
  | year = 1855
  | contribution = The Engineer's Field Book
  | pages = 371–512
  | title = The Mechanic's, Machinist's, and Engineer's Practical Book of Reference; Together with the Engineer's Field Book
  | place = New York
  | publisher = James G. Gregory
  | url = https://archive.org/details/mechanicsmachini00hasl/
  }} {{pb}}
As the book's editor Charles W. Hackley explains in the preface, "The use of the more common trigonometric functions, to wit, sines, cosines, tangents, and cotangents, which ordinary tables furnish, is not well adapted to the peculiar problems which are presented in the construction of Railroad curves. [...] Still there would be much labor of computation which may be saved by the use of tables of external secants and versed sines, which have been employed with great success recently by the Engineers on the [[Ohio and Mississippi Railroad]], and which, with the formulas and rules necessary for their application to the laying down of curves, drawn up by Mr. Haslett, one of the Engineers of that Road, are now for the first time given to the public." ([https://archive.org/details/mechanicsmachini00hasl/page/n19/ {{pgs|vi–vii}}]) {{pb}}
Charles Haslett continues in his preface to the ''Engineer's Field Book'': "Experience has shown, that versed sines and external secants as frequently enter into calculations on curves as sines and tangents; and by their use, as illustrated in the examples given in this work, it is believed that many of the rules in general use are much simplified, and many calculations concerning curves and running lines made less intricate, and results obtained with more accuracy and far less trouble, than by any methods laid down in works of this kind. [...] In addition to the tables generally found in books of this kind, the author has prepared, with great labor, a Table of Natural and Logarithmic Versed Sines and External Secants, calculated to degrees, for every minute; also, a Table of Radii and their Logarithms, from 1° to 60°." ([https://archive.org/details/mechanicsmachini00hasl/page/373/ {{pgs|373–374}}]) {{pb}}
Review: {{cite journal
  | editor-last = Poor | editor-first = Henry Varnum | editor-link = Henry Varnum Poor
  | date = 1856-03-22
  | title = ''Practical Book of Reference, and Engineer's Field Book''. By Charles Haslett
  | type = Review
  | journal = American Railroad Journal
  | volume = XII | series = Second Quarto Series | number = 12
  | id = Whole No. 1040, Vol. XXIX
  | page = 184
  | url = https://archive.org/details/5088829_29/page/n196/mode/1up
  }}</ref>

<ref name="Haslett summary"> {{harvnb
  | Haslett | 1855
  | loc = [https://archive.org/details/mechanicsmachini00hasl/page/415/ {{pgs|415}}]
  }}</ref>

<ref name="van brummelen">{{cite book
  | last = Van Brummelen | first = Glen | author-link = Glen Van Brummelen
  | year = 2021
  | title = The Doctrine of Triangles
  | publisher = Princeton University Press
  | chapter = 2. Logarithms
  | pages = 62–109
  | isbn = 9780691179414
  }}</ref>

<ref name=cajori>{{cite book
  | last = Cajori | first = Florian | author-link = Florian Cajori
  | year = 1929
  | title = A History of Mathematical Notations |volume = 2
  | location = Chicago | publisher = [[Open Court Publishing Company|Open Court]]
  | at = §527. "Less common trigonometric functions", {{pgs|171–172}} 
  | url = https://archive.org/details/b29980343_0002/page/172
  }} </ref>

<ref name=gillespie>{{cite book
  | last = Gillespie | first = William M.
  | year = 1853
  | title = A Manual of the Principles and Practice of Road-Making  
  | place = New York
  | publisher = A. S. Barnes & Co.
  | pages = 140–141 
  | url = https://archive.org/details/manualofprincipl00gill/page/140/
  }}</ref>
  
<ref name=jordan>{{cite book
  | last = Jordan | first = Leonard C.
  | year = 1913
  | title = The Practical Railway Spiral
  | place = New York
  | publisher = D. Van Nostrand Company
  | page = 28
  | url = https://archive.org/details/practicalrailwa00jordgoog/page/n39/ 
  }}</ref>

<ref name=euler>{{cite journal
  | last = Thornton-Smith | first = G. J.
  | year = 1963
  | title = Almost Exact Closed Expressions for Computing all the Elements of the Clothoid Transition Curve
  | journal = Survey Review
  | volume = 17
  | issue = 127
  | pages = 35–44
  | doi = 10.1179/sre.1963.17.127.35
  }}</ref>

<ref name=canals>{{cite journal
  | last1 = Doolittle | first1 = H. J. 
  | last2 = Shipman | first2 = C. E. 
  | year = 1911
  | journal = Proceedings of the American Society of Civil Engineers
  | volume = 37 |number = 8
  | department = Papers and Discussions
  | title = Economic Canal Location in Uniform Countries 
  | pages = 1161–1164
  | url = https://archive.org/details/proceedings37amer/page/1161/
  }}</ref>

<ref name=roads>For example: {{pb}}
{{cite book 
  | last = Hewes | first = Laurence Ilsley
  | year = 1942
  | title = American Highway Practice
  | place =  New York
  | publisher = John Wiley & Sons
  | page = 114
  | url-access = limited
  | url = https://archive.org/details/americanhighwayp0001laur/page/114/
  }} {{pb}}
{{cite book 
  | last = Ives | first = Howard Chapin
  | year = 1966 | orig-year = 1929
  | title = Highway Curves 
  | edition = 4th 
  | place = New York 
  | publisher = John Wiley & Sons 
  | lccn = 52-9033
  }} {{pb}}
{{cite book
  | last = Meyer | first = Carl F.
  | year = 1969 | orig-year = 1949
  | title = Route Surveying and Design
  | edition = 4th
  | place = Scranton, PA
  | publisher = International Textbook Co.
  | url-access = limited
  | url = https://archive.org/details/routesurveyingde00meye/page/16/
  }}</ref>

<ref name=shunk>{{cite book
  | last = Shunk
  | first = William Findlay
  | year = 1918 | orig-year = 1890
  | title = The Field Engineer: A Handy Book of Practice in the Survey, Location, and Track-Work of Railroads
  | edition = 21st
  | place = New York
  | publisher = D. Van Nostrand Company 
  | url = https://archive.org/details/fieldengineer02shun/page/36/
  | page = 36
  }}</ref>

<ref name="log exsec"> In a table of logarithmic exsecants such as {{harvnb|Haslett|1855|loc=[https://archive.org/details/mechanicsmachini00hasl/page/417/mode/1up p. 417]}} or {{harvnb|Searles|Ives|1915|loc=[https://archive.org/details/fieldengineering00sear/page/135/mode/1up II. p. 135]}}, the number given for {{math|1=log&thinsp;exsec&thinsp;1°}} is {{math|{{val|6.182780}}}}, the correct value plus {{math|10}}, which is added to keep the entries in the table positive.
</ref>

}} <!-- END REFLIST-->

{{Trigonometric and hyperbolic functions}}

[[Category:Trigonometric functions]]