Editing Exsecant (section)

== 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]]