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Versine

From Wikipedia, the free encyclopedia
1 minus the cosine of an angle
"versin" redirects here. For the Polish village, seeVersin (village).
Trigonometry
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Look up versine or versed sine in Wiktionary, the free dictionary.

Theversine orversed sine is atrigonometric function found in some of the earliest (SanskritAryabhatiya,[1]Section I)trigonometric tables. The versine of an angle is 1 minus itscosine.

There are several related functions, most notably thecoversine andhaversine. The latter, half a versine, is of particular importance in thehaversine formula of navigation.

Aunit circle withtrigonometric functions.[2]

Overview

[edit]

Theversine[3][4][5][6][7] orversed sine[8][9][10][11][12] is atrigonometric function already appearing in some of the earliest trigonometric tables. It is symbolized in formulas using the abbreviationsversin,sinver,[13][14]vers, orsiv.[15][16] InLatin, it is known as thesinus versus (flipped sine),versinus,versus, orsagitta (arrow).[17]

Expressed in terms of commontrigonometric functions sine, cosine, and tangent, the versine is equal toversin⁡θ=1−cos⁡θ=2sin2⁡θ2=sin⁡θtan⁡θ2{\displaystyle \operatorname {versin} \theta =1-\cos \theta =2\sin ^{2}{\frac {\theta }{2}}=\sin \theta \,\tan {\frac {\theta }{2}}}

There are several related functions corresponding to the versine:

  • Theversed cosine,[18][nb 1] orvercosine, abbreviatedvercosin,vercos, orvcs
  • Thecoversed sine orcoversine[19] (in Latin,cosinus versus orcoversinus), abbreviatedcoversin,covers,[20][21][22]cosiv, orcvs[23]

Special tables were also made of half of the versed sine, because of its particular use in thehaversine formula used historically innavigation.

  • Thehaversed sine[24] orhaversine (Latinsemiversus),[25][26] abbreviatedhaversin,semiversin,semiversinus,havers,hav,[27][28]hvs,[nb 2]sem, orhv.[29] It is defined as

hav θ=sin2⁡(θ2)=1−cos⁡θ2{\displaystyle {\text{hav}}\ \theta =\sin ^{2}\left({\frac {\theta }{2}}\right)={\frac {1-\cos \theta }{2}}}

History and applications

[edit]

Versine and coversine

[edit]
Sine, cosine, and versine of angleθ in terms of aunit circle with radius 1, centered atO. This figure also illustrates the reason why the versine was sometimes called thesagitta, Latin forarrow.[17][30] If the arcADB of the double-angleΔ = 2θ is viewed as a "bow" and thechordAB as its "string", then the versineCD is clearly the "arrow shaft".
Graphs of historical trigonometric functions compared with sin and cos – inthe SVG file, hover over or click a graph to highlight it

The ordinarysine function (see note on etymology) was sometimes historically called thesinus rectus ("straight sine"), to contrast it with the versed sine (sinus versus).[31] The meaning of these terms is apparent if one looks at the functions in the original context for their definition, aunit circle:

For a verticalchordAB of the unit circle, the sine of the angleθ (representing half of the subtended angleΔ) is the distanceAC (half of the chord). On the other hand, the versed sine ofθ is the distanceCD from the center of the chord to the center of the arc. Thus, the sum of cos(θ) (equal to the length of lineOC) and versin(θ) (equal to the length of lineCD) is the radiusOD (with length 1). Illustrated this way, the sine is vertical (rectus, literally "straight") while the versine is horizontal (versus, literally "turned against, out-of-place"); both are distances fromC to the circle.

This figure also illustrates the reason why the versine was sometimes called thesagitta, Latin forarrow.[17][30] If the arcADB of the double-angleΔ = 2θ is viewed as a "bow" and the chordAB as its "string", then the versineCD is clearly the "arrow shaft".

In further keeping with the interpretation of the sine as "vertical" and the versed sine as "horizontal",sagitta is also an obsolete synonym for theabscissa (the horizontal axis of a graph).[30]

In 1821,Cauchy used the termssinus versus (siv) for the versine andcosinus versus (cosiv) for the coversine.[15][16][nb 1]

The trigonometric functions can be constructed geometrically in terms of aunit circle centered atO.

Asθ goes to zero, versin(θ) is the difference between two nearly equal quantities, so a user of atrigonometric table for the cosine alone would need a very high accuracy to obtain the versine in order to avoidcatastrophic cancellation, making separate tables for the latter convenient.[12] Even with a calculator or computer,round-off errors make it advisable to use the sin2 formula for small θ.

Another historical advantage of the versine is that it is always non-negative, so itslogarithm is defined everywhere except for the single angle (θ = 0, 2π, …) where it is zero—thus, one could uselogarithmic tables for multiplications in formulas involving versines.

In fact, the earliest surviving table of sine (half-chord) values (as opposed to thechords tabulated by Ptolemy and other Greek authors), calculated from theSurya Siddhantha of India dated back to the 3rd century BC, was a table of values for the sine and versed sine (in 3.75° increments from 0 to 90°).[31]

The versine appears as an intermediate step in the application of thehalf-angle formula sin2(⁠θ/2⁠) =⁠1/2⁠versin(θ), derived byPtolemy, that was used to construct such tables.

Haversine

[edit]

The haversine, in particular, was important innavigation because it appears in thehaversine formula, which is used to reasonably accurately compute distances on an astronomicspheroid (see issues with theEarth's radius vs. sphere) given angular positions (e.g.,longitude andlatitude). One could also use sin2(⁠θ/2⁠) directly, but having a table of the haversine removed the need to compute squares and square roots.[12]

An early utilization byJosé de Mendoza y Ríos of what later would be called haversines is documented in 1801.[14][32]

The first known English equivalent to atable of haversines was published by James Andrew in 1805, under the name "Squares of Natural Semi-Chords".[33][34][17]

In 1835, the termhaversine (notated naturally ashav. orbase-10 logarithmically aslog. haversine orlog. havers.) was coined[35] byJames Inman[14][36][37] in the third edition of his workNavigation and Nautical Astronomy: For the Use of British Seamen to simplify the calculation of distances between two points on the surface of the Earth usingspherical trigonometry for applications in navigation.[3][35] Inman also used the termsnat. versine andnat. vers. for versines.[3]

Other high-regarded tables of haversines were those of Richard Farley in 1856[33][38] and John Caulfield Hannyngton in 1876.[33][39]

The haversine continues to be used in navigation and has found new applications in recent decades, as in Bruce D. Stark's method for clearinglunar distances utilizingGaussian logarithms since 1995[40][41] or in a more compact method forsight reduction since 2014.[29]

Modern uses

[edit]

While the usage of the versine, coversine and haversine as well as theirinverse functions can be traced back centuries, the names for the other fivecofunctions appear to be of much younger origin.

One period (0 <θ < 2π) of a versine or, more commonly, a haversine waveform is also commonly used insignal processing andcontrol theory as the shape of apulse or awindow function (includingHann,Hann–Poisson andTukey windows), because it smoothly (continuous in value andslope) "turns on" fromzero toone (for haversine) and back to zero.[nb 2] In these applications, it is namedHann function orraised-cosine filter.

Mathematical identities

[edit]

Definitions

[edit]
versin(θ):=2sin2(θ2)=1−cos⁡(θ){\displaystyle {\textrm {versin}}(\theta ):=2\sin ^{2}\!\left({\frac {\theta }{2}}\right)=1-\cos(\theta )\,}[4]
coversin(θ):=versin(π2−θ)=1−sin⁡(θ){\displaystyle {\textrm {coversin}}(\theta ):={\textrm {versin}}\!\left({\frac {\pi }{2}}-\theta \right)=1-\sin(\theta )\,}[4]
vercosin(θ):=2cos2(θ2)=1+cos⁡(θ){\displaystyle {\textrm {vercosin}}(\theta ):=2\cos ^{2}\!\left({\frac {\theta }{2}}\right)=1+\cos(\theta )\,}[18]
haversin(θ):=versin(θ)2=sin2(θ2)=1−cos⁡(θ)2{\displaystyle {\textrm {haversin}}(\theta ):={\frac {{\textrm {versin}}(\theta )}{2}}=\sin ^{2}\!\left({\frac {\theta }{2}}\right)={\frac {1-\cos(\theta )}{2}}\,}[4]

Circular rotations

[edit]

The functions are circular rotations of each other.

versin(θ)=coversin(θ+π2)=vercosin(θ+π){\displaystyle {\begin{aligned}\mathrm {versin} (\theta )&=\mathrm {coversin} \left(\theta +{\frac {\pi }{2}}\right)=\mathrm {vercosin} \left(\theta +\pi \right)\end{aligned}}}

Derivatives and integrals

[edit]
ddxversin(x)=sin⁡x{\displaystyle {\frac {\mathrm {d} }{\mathrm {d} x}}\mathrm {versin} (x)=\sin {x}}[42]∫versin(x)dx=x−sin⁡x+C{\displaystyle \int \mathrm {versin} (x)\,\mathrm {d} x=x-\sin {x}+C}[4][42]
ddxvercosin(x)=−sin⁡x{\displaystyle {\frac {\mathrm {d} }{\mathrm {d} x}}\mathrm {vercosin} (x)=-\sin {x}}∫vercosin(x)dx=x+sin⁡x+C{\displaystyle \int \mathrm {vercosin} (x)\,\mathrm {d} x=x+\sin {x}+C}
ddxcoversin(x)=−cos⁡x{\displaystyle {\frac {\mathrm {d} }{\mathrm {d} x}}\mathrm {coversin} (x)=-\cos {x}}[19]∫coversin(x)dx=x+cos⁡x+C{\displaystyle \int \mathrm {coversin} (x)\,\mathrm {d} x=x+\cos {x}+C}[19]
ddxhaversin(x)=sin⁡x2{\displaystyle {\frac {\mathrm {d} }{\mathrm {d} x}}\mathrm {haversin} (x)={\frac {\sin {x}}{2}}}[24]∫haversin(x)dx=x−sin⁡x2+C{\displaystyle \int \mathrm {haversin} (x)\,\mathrm {d} x={\frac {x-\sin {x}}{2}}+C}[24]

Inverse functions

[edit]

Inverse functions likearcversine (arcversin, arcvers,[8] avers,[43][44] aver),arcvercosine (arcvercosin, arcvercos, avercos, avcs),arccoversine (arccoversin, arccovers,[8] acovers,[43][44] acvs),arccovercosine (arccovercosin, arccovercos, acovercos, acvc),archaversine (archaversin, archav, haversin−1,[45] invhav,[46][47][48] ahav,[43][44] ahvs, ahv, hav−1[49][50]),archavercosine (archavercosin, archavercos, ahvc),archacoversine (archacoversin, ahcv) orarchacovercosine (archacovercosin, archacovercos, ahcc) exist as well:

arcversin⁡(y)=arccos⁡(1−y){\displaystyle \operatorname {arcversin} (y)=\arccos \left(1-y\right)\,}[43][44]
arcvercos⁡(y)=arccos⁡(y−1){\displaystyle \operatorname {arcvercos} (y)=\arccos \left(y-1\right)\,}
arccoversin⁡(y)=arcsin⁡(1−y){\displaystyle \operatorname {arccoversin} (y)=\arcsin \left(1-y\right)\,}[43][44]
arccovercos⁡(y)=arcsin⁡(y−1){\displaystyle \operatorname {arccovercos} (y)=\arcsin \left(y-1\right)\,}
archaversin⁡(y)=2arcsin⁡(y)=arccos⁡(1−2y){\displaystyle \operatorname {archaversin} (y)=2\arcsin \left({\sqrt {y}}\right)=\arccos \left(1-2y\right)\,}[43][44][45][46][47][49][50]
archavercos⁡(y)=2arccos⁡(y)=arccos⁡(2y−1){\displaystyle \operatorname {archavercos} (y)=2\arccos \left({\sqrt {y}}\right)=\arccos \left(2y-1\right)}
archacoversin⁡(y)=arcsin⁡(1−2y){\displaystyle \operatorname {archacoversin} (y)=\arcsin \left(1-2y\right)\,}
archacovercos⁡(y)=arcsin⁡(2y−1){\displaystyle \operatorname {archacovercos} (y)=\arcsin \left(2y-1\right)\,}

Other properties

[edit]

These functions can be extended into thecomplex plane.[42][19][24]

Maclaurin series:[24]

versin⁡(z)=∑k=1∞(−1)k−1z2k(2k)!haversin⁡(z)=∑k=1∞(−1)k−1z2k2(2k)!{\displaystyle {\begin{aligned}\operatorname {versin} (z)&=\sum _{k=1}^{\infty }{\frac {(-1)^{k-1}z^{2k}}{(2k)!}}\\\operatorname {haversin} (z)&=\sum _{k=1}^{\infty }{\frac {(-1)^{k-1}z^{2k}}{2(2k)!}}\end{aligned}}}
limθ→0versin⁡(θ)θ=0{\displaystyle \lim _{\theta \to 0}{\frac {\operatorname {versin} (\theta )}{\theta }}=0}[8]
versin⁡(θ)+coversin⁡(θ)versin⁡(θ)−coversin⁡(θ)−exsec⁡(θ)+excsc⁡(θ)exsec⁡(θ)−excsc⁡(θ)=2versin⁡(θ)coversin⁡(θ)versin⁡(θ)−coversin⁡(θ)[versin⁡(θ)+exsec⁡(θ)][coversin⁡(θ)+excsc⁡(θ)]=sin⁡(θ)cos⁡(θ){\displaystyle {\begin{aligned}{\frac {\operatorname {versin} (\theta )+\operatorname {coversin} (\theta )}{\operatorname {versin} (\theta )-\operatorname {coversin} (\theta )}}-{\frac {\operatorname {exsec} (\theta )+\operatorname {excsc} (\theta )}{\operatorname {exsec} (\theta )-\operatorname {excsc} (\theta )}}&={\frac {2\operatorname {versin} (\theta )\operatorname {coversin} (\theta )}{\operatorname {versin} (\theta )-\operatorname {coversin} (\theta )}}\\[3pt][\operatorname {versin} (\theta )+\operatorname {exsec} (\theta )]\,[\operatorname {coversin} (\theta )+\operatorname {excsc} (\theta )]&=\sin(\theta )\cos(\theta )\end{aligned}}}[8]

Approximations

[edit]
Comparison of the versine function with three approximations to the versine functions, for angles ranging from 0 to 2π
Comparison of the versine function with three approximations to the versine functions, for angles ranging from 0 toπ/2

When the versinev is small in comparison to the radiusr, it may be approximated from the half-chord lengthL (the distanceAC shown above) by the formula[51]v≈L22r.{\displaystyle v\approx {\frac {L^{2}}{2r}}.}

Alternatively, if the versine is small and the versine, radius, and half-chord length are known, they may be used to estimate the arc lengths (AD in the figure above) by the formulas≈L+v2r{\displaystyle s\approx L+{\frac {v^{2}}{r}}}This formula was known to the Chinese mathematicianShen Kuo, and a more accurate formula also involving the sagitta was developed two centuries later byGuo Shoujing.[52]

A more accurate approximation used in engineering[53] isv≈s32L128r{\displaystyle v\approx {\frac {s^{\frac {3}{2}}L^{\frac {1}{2}}}{8r}}}

Arbitrary curves and chords

[edit]

The termversine is also sometimes used to describe deviations from straightness in an arbitrary planar curve, of which the above circle is a special case. Given a chord between two points in a curve, the perpendicular distancev from the chord to the curve (usually at the chord midpoint) is called aversine measurement. For a straight line, the versine of any chord is zero, so this measurement characterizes the straightness of the curve. In thelimit as the chord lengthL goes to zero, the ratio⁠8v/L2⁠ goes to the instantaneouscurvature. This usage is especially common inrail transport, where it describes measurements of the straightness of therail tracks[54] and it is the basis of theHallade method forrail surveying.

The termsagitta (often abbreviatedsag) is used similarly inoptics, for describing the surfaces oflenses andmirrors.

See also

[edit]

Notes

[edit]
  1. ^abSome English sources confuse the versed cosine with the coversed sine. Historically (f.e. inCauchy, 1821), thesinus versus (versine) was defined as siv(θ) = 1−cos(θ), thecosinus versus (what is now also known as coversine) as cosiv(θ) = 1−sin(θ), and the vercosine as vcsθ = 1+cos(θ). However, in their 2009 English translation of Cauchy's work,Bradley and Sandifer associate thecosinus versus (and cosiv) with theversed cosine (what is now also known as vercosine) rather than thecoversed sine. Similarly, in their 1968/2000 work,Korn and Korn associate the covers(θ) function with theversed cosine instead of thecoversed sine.
  2. ^abThe abbreviationhvs sometimes used for the haversine function in signal processing and filtering is also sometimes used for the unrelatedHeaviside step function.

References

[edit]
  1. ^The Āryabhaṭīya by Āryabhaṭa
  2. ^Haslett, Charles (September 1855). Hackley, Charles W. (ed.).The Mechanic's, Machinist's, Engineer's Practical Book of Reference: Containing tables and formulæ for use in superficial and solid mensuration; strength and weight of materials; mechanics; machinery; hydraulics, hydrodynamics; marine engines, chemistry; and miscellaneous recipes. Adapted to and for the use of all classes of practical mechanics. Together with the Engineer's Field Book: Containing formulæ for the various of running and changing lines, locating side tracks and switches, &c., &c. Tables of radii and their logarithms, natural and logarithmic versed sines and external secants, natural sines and tangents to every degree and minute of the quadrant, and logarithms from the natural numbers from 1 to 10,000. New York, USA: James G. Gregory, successor of W. A. Townsend & Co. (Stringer & Townsend). Retrieved2017-08-13.[…] Still there would be much labor of computation which may be saved by the use of tables ofexternal secants andversed sines, which have been employed with great success recently by the Engineers on theOhio 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. […] In presenting this work to the public, the Author claims for it the adaptation of a new principle in trigonometrical analysis of the formulas generally used in field calculations. 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. The examples given have all been suggested by actual practice, and will explain themselves. […] As a book for practical use in field work, it is confidently believed that this is more direct in the application of rules and facility of calculation than any work now in use. 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°. […]1856 edition
  3. ^abcInman, James (1835) [1821].Navigation and Nautical Astronomy: For the Use of British Seamen (3 ed.). London, UK: W. Woodward, C. & J. Rivington. Retrieved2015-11-09. (Fourth edition:[1].)
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  11. ^Korn, Grandino Arthur;Korn, Theresa M. (2000) [1961]. "Appendix B: B9. Plane and Spherical Trigonometry: Formulas Expressed in Terms of the Haversine Function".Mathematical handbook for scientists and engineers: Definitions, theorems, and formulars for reference and review (3 ed.). Mineola, New York, USA:Dover Publications, Inc. pp. 892–893.ISBN 978-0-486-41147-7. (Seeerrata.)
  12. ^abcCalvert, James B. (2007-09-14) [2004-01-10]."Trigonometry".Archived from the original on 2007-10-02. Retrieved2015-11-08.
  13. ^Edler von Braunmühl, Anton (1903).Vorlesungen über Geschichte der Trigonometrie - Von der Erfindung der Logarithmen bis auf die Gegenwart [Lectures on history of trigonometry - from the invention of logarithms up to the present] (in German). Vol. 2. Leipzig, Germany:B. G. Teubner. p. 231. Retrieved2015-12-09.
  14. ^abcCajori, Florian (1952) [March 1929].A History of Mathematical Notations. Vol. 2 (2 (3rd corrected printing of 1929 issue) ed.). Chicago, USA:Open court publishing company. p. 172.ISBN 978-1-60206-714-1. 1602067147. Retrieved2015-11-11.The haversine first appears in the tables of logarithmic versines ofJosé de Mendoza y Rios (Madrid, 1801, also 1805, 1809), and later in a treatise on navigation ofJames Inman (1821). See J. D. White inNautical Magazine (February andJuly 1926).{{cite book}}:ISBN / Date incompatibility (help) (NB. ISBN and link for reprint of 2nd edition by Cosimo, Inc., New York, USA, 2013.)
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  32. ^de Mendoza y Ríos, Joseph (1795).Memoria sobre algunos métodos nuevos de calcular la longitud por las distancias lunares: y aplicación de su teórica á la solucion de otros problemas de navegacion (in Spanish). Madrid, Spain: Imprenta Real.
  33. ^abcArchibald, Raymond Clare (1945)."Recent Mathematical Tables : 197[C, D].—Natural and Logarithmic Haversines..."Mathematical Tables and Other Aids to Computation.1 (11):421–422.doi:10.1090/S0025-5718-45-99080-6.
  34. ^Andrew, James (1805).Astronomical and Nautical Tables with Precepts for finding the Latitude and Longitude of Places. Vol. T. XIII. London. pp. 29–148. (A 7-placehaversine table from 0° to 120° in intervals of 10".)
  35. ^ab"haversine".Oxford English Dictionary (2nd ed.).Oxford University Press. 1989.
  36. ^White, J. D. (February 1926). "(unknown title)".Nautical Magazine. (NB. According toCajori, 1929, this journal has a discussion on the origin of haversines.)
  37. ^White, J. D. (July 1926). "(unknown title)".Nautical Magazine. (NB. According toCajori, 1929, this journal has a discussion on the origin of haversines.)
  38. ^Farley, Richard (1856).Natural Versed Sines from 0 to 125°, and Logarithmic Versed Sines from 0 to 135°. London.{{cite book}}: CS1 maint: location missing publisher (link) (Ahaversine table from 0° to 125°/135°.)
  39. ^Hannyngton, John Caulfield (1876).Haversines, Natural and Logarithmic, used in Computing Lunar Distances for the Nautical Almanac. London.{{cite book}}: CS1 maint: location missing publisher (link) (A 7-placehaversine table from 0° to 180°,log. haversines at intervals of 15",nat. haversines at intervals of 10".)
  40. ^Stark, Bruce D. (1997) [1995].Stark Tables for Clearing the Lunar Distance and Finding Universal Time by Sextant Observation Including a Convenient Way to Sharpen Celestial Navigation Skills While On Land (2 ed.). Starpath Publications.ISBN 978-0914025214. 091402521X. Retrieved2015-12-02. (NB. Contains a table ofGaussian logarithmslg(1+10−x).)
  41. ^Kalivoda, Jan (2003-07-30)."Bruce Stark - Tables for Clearing the Lunar Distance and Finding G.M.T. by Sextant Observation (1995, 1997)" (Review). Prague, Czech Republic.Archived from the original on 2004-01-12. Retrieved2015-12-02.[2][3]
  42. ^abcWeisstein, Eric Wolfgang."Versine".MathWorld.Wolfram Research, Inc.Archived from the original on 2010-03-31. Retrieved2015-11-05.
  43. ^abcdefSimpson, David G. (2001-11-08)."AUXTRIG" (Fortran 90 source code). Greenbelt, Maryland, USA:NASA Goddard Space Flight Center.Archived from the original on 2008-06-16. Retrieved2015-10-26.
  44. ^abcdefvan den Doel, Kees (2010-01-25)."jass.utils Class Fmath".JASS - Java Audio Synthesis System. 1.25.Archived from the original on 2007-09-02. Retrieved2015-10-26.
  45. ^abmf344 (2014-07-04)."Lost but lovely: The haversine".Plus magazine. maths.org.Archived from the original on 2014-07-18. Retrieved2015-11-05.{{cite news}}: CS1 maint: numeric names: authors list (link)
  46. ^abSkvarc, Jure (1999-03-01)."identify.py: An asteroid_server client which identifies measurements in MPC format".Fitsblink (Python source code).Archived from the original on 2008-11-20. Retrieved2015-11-28.
  47. ^abSkvarc, Jure (2014-10-27)."astrotrig.py: Astronomical trigonometry related functions" (Python source code). Ljubljana, Slovenia: Telescope Vega,University of Ljubljana.Archived from the original on 2015-11-28. Retrieved2015-11-28.
  48. ^Ballew, Pat (2007-02-08) [2003]."Versine".Math Words, page 4. Versine. Archived from the original on 2007-02-08. Retrieved2015-11-28.
  49. ^abWeisstein, Eric Wolfgang."Inverse Haversine".MathWorld.Wolfram Research, Inc.Archived from the original on 2008-06-08. Retrieved2015-10-05.
  50. ^ab"InverseHaversine".Wolfram Language & System: Documentation Center. 7.0. 2008. Retrieved2015-11-05.
  51. ^Woodward, Ernest (December 1978).Geometry - Plane, Solid & Analytic Problem Solver. Problem Solvers Solution Guides.Research & Education Association (REA). p. 359.ISBN 978-0-87891-510-1.
  52. ^Needham, Noel Joseph Terence Montgomery (1959).Science and Civilisation in China: Mathematics and the Sciences of the Heavens and the Earth. Vol. 3.Cambridge University Press. p. 39.ISBN 9780521058018.{{cite book}}:ISBN / Date incompatibility (help)
  53. ^Boardman, Harry (1930).Table For Use in Computing Arcs, Chords and Versines.Chicago Bridge and Iron Company. p. 32.
  54. ^Nair, P. N. Bhaskaran (1972). "Track measurement systems—concepts and techniques".Rail International.3 (3). International Railway Congress Association,International Union of Railways:159–166.ISSN 0020-8442.OCLC 751627806.

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