Versine

From Wikipedia, the free encyclopedia
Jump to: navigation, search

The versine or versed sine is a trigonometric function already appearing in some of the earliest trigonometric tables. The versine of an angle equals 1 minus its cosine.

There are several related functions, most notably the coversine and haversine. The latter, half a versine, is of particular importance in the haversine formula of navigation.

Overview[edit]

The versine[1][2][3][4][5] or versed sine[3][6][7][8][9] is a trigonometric function already appearing in some of the earliest trigonometric tables. It is written as versin(θ),[3][7][8] sinver(θ),[10][11] vers(θ),[1][2][3][4][5][6] ver(θ)[12] or siv(θ).[13][14] In Latin, it is known as the sinus versus[13][14] (flipped sine), versinus, versus or the sagitta (arrow).

Expressed in terms of the meanwhile more commonly used "vertical" sines (sinus rectus) and cosines (cosinus rectus) functions, the versine is equal to 1 − cos(θ), or 2 sin2(θ/2).

There are several related functions corresponding to the versine:

In full analogy to the above-mentioned four functions another set of four "half-value" functions exists as well:

History and applications[edit]

Versine and coversine[edit]

Sine, cosine, and versine of angle θ in terms of a unit circle with radius 1, centered at O. This figure also illustrates the reason why the versine was sometimes called the sagitta, Latin for arrow.[32] If the arc ADB of the double-angle Δ = 2θ is viewed as a "bow" and the chord AB as its "string", then the versine CD is clearly the "arrow shaft".

The ordinary sine function (see note on etymology) was sometimes historically called the sinus rectus ("vertical sine"), to contrast it with the versed sine (sinus versus).[33] The meaning of these terms is apparent if one looks at the functions in the original context for their definition, a unit circle:

For a vertical chord AB of the unit circle, the sine of the angle θ (representing half of the subtended angle Δ) is the distance AC (half of the chord). On the other hand, the versed sine of θ is the distance CD from the center of the chord to the center of the arc. Thus, the sum of cos(θ) (equal to the length of line OC) and versin(θ) (equal to the length of line CD) is the radius OD (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 from C to the circle.

This figure also illustrates the reason why the versine was sometimes called the sagitta, Latin for arrow,[32] from the Arabic usage sahem[34] of the same meaning. This itself comes from the Indian word 'sara' (arrow) that was commonly used to refer to "utkrama-jya". If the arc ADB of the double-angle Δ = 2θ is viewed as a "bow" and the chord AB as its "string", then the versine CD 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 the abscissa (the horizontal axis of a graph).[32]

In 1821, Cauchy used the terms sinus versus (siv) for the versine and cosinus versus (cosiv) for the coversine.[13][14][nb 1]

The trigonometric functions can be constructed geometrically in terms of a unit circle centered at O.

Historically, the versed sine was considered one of the most important trigonometric functions.[9][33][34]

As θ goes to zero, versin(θ) is the difference between two nearly equal quantities, so a user of a trigonometric table for the cosine alone would need a very high accuracy to obtain the versine in order to avoid catastrophic cancellation, making separate tables for the latter convenient.[9] 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 its logarithm is defined everywhere except for the single angle (θ = 0, 2π,...) where it is zero—thus, one could use logarithmic tables for multiplications in formulas involving versines.

In fact, the earliest surviving table of sine (half-chord) values (as opposed to the chords tabulated by Ptolemy and other Greek authors), calculated from the Surya 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°).[33]

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

Haversine[edit]

The haversine, in particular, was important in navigation because it appears in the haversine formula, which is used to accurately compute distances within reason on an astronomic spheroid (see issues with the earth's radius vs. sphere) given angular positions (e.g., longitude and latitude). One could also use sin2(θ/2) directly, but having a table of the haversine removed the need to compute squares and square roots.[9]

An early utilization by José de Mendoza y Ríos of what later would be called haversines is documented in 1801.[11][35]

The first known English equivalent to a table of haversines was published by James Andrew in 1805.[36][37][38]

In 1835, the term haversine (notated as log. haversine, log. havers. and hav.) was coined[39] by James Inman[11][40][41] in the third edition of his work Navigation and Nautical Astronomy: For the Use of British Seamen to simplify the calculation of distances between two points on the surface of the earth using spherical trigonometry for applications in navigation.[1][39] Inman also used the terms nat. versine and nat. vers. for versines.[1]

Other high-regarded tables of haversines were those of Richard Farley in 1856[36][42] and John Caulfield Hannyngton in 1876.[36][43]

The haversine continues to be used in navigation and has even found new applications in recent decades, like in Bruce D. Stark's method for clearing lunar distances utilizing Gaussian logarithms since 1995[44][45] or in a more compact method for sight reduction since 2014.[28]

Modern uses[edit]

Whilst the usage of the versine, coversine and haversine as well as their inverse functions can be traced back centuries, the names for the other five cofunctions appear to be of much younger origin.

One period (0 < θ < π/2) of a versine or, more commonly, a haversine (or havercosine) waveform is also commonly used in signal processing and control theory as the shape of a pulse or a window function (including Hann, Hann–Poisson and Tukey windows), because it smoothly (continuous in value and slope) "turns on" from zero to one (for haversine) and back to zero.[nb 2] In these applications, it is named Hann function or raised-cosine filter. Likewise, the havercosine is used in raised-cosine distributions in probability theory and statistics.

In the form of sin2(θ) the haversine of the double-angle Δ describes the relation between spreads and angles in rational trigonometry, a proposed reformulation of metrical planar and solid geometries by Norman John Wildberger since 2005.[46]

As sagitta and cosagitta, double-angle Δ variants of the haversine and havercosine have also found new uses in describing the correlation and anti-correlation of correlated photons in quantum mechanics.[47]

Mathematical identities[edit]

Definitions[edit]

[2][3][4][5][6][7][8][13][14] Versin plot.png
[15] Vercosin plot.png
[2][4][5][6][8][11][13][14][16][18][19][20][21] Coversin plot.png
[22] Covercosin plot.png
[2][4][5][6][23] Haversin plot.png
[29] Havercosin plot.png
[17] Hacoversin plot.png
[31] Hacovercosin plot.png

Derivatives and integrals[edit]

[3] [2][3]
[16] [16]
[23] [23]

Inverse functions[edit]

Inverse functions like arcversine[30] (arcversin, arcvers,[30] avers,[48][49] aver), arcvercosine (arcvercosin, arcvercos, avercos, avcs), arccoversine[30] (arccoversin, arccovers,[30] acovers,[48][49] acvs), arccovercosine (arccovercosin, arccovercos, acovercos, acvc), archaversine (archaversin, archav,[30] haversin−1,[50] invhav,[30][51][52][53] ahav,[30][48][49] ahvs, ahv, hav−1[54][55]), archavercosine (archavercosin, archavercos, ahvc), archacoversine (archacoversin, ahcv) or archacovercosine (archacovercosin, archacovercos, ahcc) exist as well:

[30][48][49] (for 0 ≤ y ≤ 2)[30]
[30][48][49] (for 0 ≤ y ≤ 2)[30]
[30][48][49][50][51][52][54][55] (for 0 ≤ y ≤ 1)[30]

Other properties[edit]

These functions can be extended into the complex plane.[3][16][23]

Maclaurin series:

[23]

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 to π/2

When the versine v is small in comparison to the radius r, it may be approximated from the half-chord length L (the distance AC shown above) by the formula

.[56]

Alternatively, if the versine is small and the versine, radius, and half-chord length are known, they may be used to estimate the arc length s (AD in the figure above) by the formula

This formula was known to the Chinese mathematician Shen Kuo, and a more accurate formula also involving the sagitta was developed two centuries later by Guo Shoujing.[57]

A more accurate approximation used in engineering[58] is

Arbitrary curves and chords[edit]

The term versine 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 distance v from the chord to the curve (usually at the chord midpoint) is called a versine measurement. For a straight line, the versine of any chord is zero, so this measurement characterizes the straightness of the curve. In the limit as the chord length L goes to zero, the ratio 8v/L2 goes to the instantaneous curvature. This usage is especially common in rail transport, where it describes measurements of the straightness of the rail tracks[59] and it is the basis of the Hallade method for rail surveying.

The term sagitta (often abbreviated sag) is used similarly in optics, for describing the surfaces of lenses and mirrors.

See also[edit]

Notes[edit]

  1. ^ a b c d e f Some English sources confuse the versed cosine with the coversed sine. Historically (f.e. in Cauchy, 1821), the sinus versus (versine) was defined as siv(θ) = 1−cos(θ) and the cosinus versus (what is now also known as coversine) as cosiv(θ) = 1−sin(θ). However, in their 2009 English translation of Cauchy's work, Bradley and Sandifer associate the cosinus versus (and cosiv) with the versed cosine (what is now also known as vercosine) rather than the coversed sine. Similarly, in their 1968/2000 work, Korn and Korn associate the covers(θ) function with the versed cosine instead of the coversed sine.
  2. ^ a b The abbreviation hvs sometimes used for the haversine function in signal processing and filtering is also sometimes used for the unrelated Heaviside step function.

References[edit]

  1. ^ a b c d e f g Inman, James (1835) [1821]. Navigation and Nautical Astronomy: For the Use of British Seamen (3 ed.). London, UK: W. Woodward, C. & J. Rivington. Retrieved 2015-11-09.  (Fourth edition: [1].)
  2. ^ a b c d e f g h i j Zucker, Ruth (1983) [June 1964]. "Chapter 4.3.147: Elementary Transcendental Functions - Circular functions". In Abramowitz, Milton; Stegun, Irene Ann. Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables. Applied Mathematics Series. 55 (Ninth reprint with additional corrections of tenth original printing with corrections (December 1972); first ed.). Washington D.C., USA; New York, USA: United States Department of Commerce, National Bureau of Standards; Dover Publications. p. 78. ISBN 0-486-61272-4. LCCN 64-60036. MR 0167642. ISBN 978-0-486-61272-0. LCCN 65-12253. 
  3. ^ a b c d e f g h Weisstein, Eric W. "Versine". MathWorld. Wolfram Research, Inc. Archived from the original on 2010-03-31. Retrieved 2015-11-05. 
  4. ^ a b c d e f g h i Tapson, Frank (2004). "Background Notes on Measures: Angles". 1.4. Cleave Books. Archived from the original on 2007-02-09. Retrieved 2015-11-12. 
  5. ^ a b c d e f g h i Oldham, Keith B.; Myland, Jan C.; Spanier, Jerome (2009) [1987]. "32.13. The Cosine cos(x) and Sine sin(x) functions - Cognate functions". An Atlas of Functions: with Equator, the Atlas Function Calculator (2 ed.). Springer Science+Business Media, LLC. p. 322. doi:10.1007/978-0-387-48807-3. ISBN 978-0-387-48806-6. LCCN 2008937525. 
  6. ^ a b c d e f g h 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.  (See errata.)
  7. ^ a b c d e Boyer, Carl Benjamin (1969) [1959]. "5: Commentary on the Paper of E. J. Dijksterhuis (The Origins of Classical Mechanics from Aristotle to Newton)". In Clagett, Marshall. Critical Problems in the History of Science (3 ed.). Madison, Milwaukee, and London: University of Wisconsin Press, Ltd. pp. 185–190. ISBN 0-299-01874-1. LCCN 59-5304. 9780299018740. Retrieved 2015-11-16. 
  8. ^ a b c d e f Swanson, Todd; Andersen, Janet; Keeley, Robert (1999). "5 (Trigonometric Functions)". Precalculus: A Study of Functions and Their Applications (PDF). Harcourt Brace & Company. p. 344. Archived (PDF) from the original on 2003-06-17. Retrieved 2015-11-12. 
  9. ^ a b c d Calvert, James B. (2007-09-14) [2004]. "Trigonometry". Archived from the original on 2007-10-02. Retrieved 2015-11-08. 
  10. ^ 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). 2. Leipzig, Germany: B. G. Teubner. p. 231. Retrieved 2015-12-09. 
  11. ^ a b c d e f g Cajori, Florian (1952) [1929]. A History of Mathematical Notations. 2 (2 (3rd corrected printing of 1929 issue) ed.). Chicago, USA: Open court publishing company. p. 172. ISBN 978-1-60206-714-1. 1602067147. Retrieved 2015-11-11. The haversine first appears in the tables of logarithmic versines of José de Mendoza y Rios (Madrid, 1801, also 1805, 1809), and later in a treatise on navigation of James Inman (1821). See J. D. White in Nautical Magazine (February and July 1926).  (NB. ISBN and link for reprint of 2nd edition by Cosimo, Inc., New York, USA, 2013.)
  12. ^ a b c d e f g h Shaneyfelt, Ted V. "德博士的 Notes About Circles, ज्य, & कोज्य: What in the world is a hacovercosine?". Hilo, Hawaii: University of Hawaii. Archived from the original on 2015-09-19. Retrieved 2015-11-08. 
  13. ^ a b c d e f g Cauchy, Augustin-Louis (1821). "Analyse Algébrique". Cours d'Analyse de l'Ecole royale polytechnique (in French). 1. L'Imprimerie Royale, Debure frères, Libraires du Roi et de la Bibliothèque du Roi. [2] (reissued by Cambridge University Press, 2009; ISBN 978-1-108-00208-0)
  14. ^ a b c d e f g Bradley, Robert E.; Sandifer, Charles Edward (2010-01-14) [2009]. Buchwald, J. Z., ed. Cauchy’s Cours d’analyse: An Annotated Translation. Sources and Studies in the History of Mathematics and Physical Sciences. Cauchy, Augustin-Louis. Springer Science+Business Media, LLC. pp. 10, 285. doi:10.1007/978-1-4419-0549-9. ISBN 978-1-4419-0548-2. LCCN 2009932254. 1441905499, 978-1-4419-0549-9. Retrieved 2015-11-09.  (See errata.)
  15. ^ a b c d Weisstein, Eric W. "Vercosine". MathWorld. Wolfram Research, Inc. Archived from the original on 2014-03-24. Retrieved 2015-11-06. 
  16. ^ a b c d e f Weisstein, Eric W. "Coversine". MathWorld. Wolfram Research, Inc. Archived from the original on 2005-11-27. Retrieved 2015-11-06. 
  17. ^ a b c d e f Weisstein, Eric W. "Hacoversine". MathWorld. Wolfram Research, Inc. Archived from the original on 2014-03-29. Retrieved 2015-11-06. 
  18. ^ a b Ludlow, Henry Hunt; Bass, Edgar Wales (1891). Elements of Trigonometry with Logarithmic and Other Tables (3 ed.). Boston, USA: John Wiley & Sons. p. 33. Retrieved 2015-12-08. 
  19. ^ a b Wentworth, George Albert (1903) [1887]. Plane Trigonometry (2 ed.). Boston, USA: Ginn and Company. p. 5. 
  20. ^ a b Kenyon, Alfred Monroe; Ingold, Louis (1913). Trigonometry. New York, USA: The Macmillan Company. pp. 8–9. Retrieved 2015-12-08. 
  21. ^ a b Anderegg, Frederick; Roe, Edward Drake (1896). Trigonometry: For Schools and Colleges. Boston, USA: Ginn and Company. p. 10. Retrieved 2015-12-08. 
  22. ^ a b c d Weisstein, Eric W. "Covercosine". MathWorld. Wolfram Research, Inc. Archived from the original on 2014-03-28. Retrieved 2015-11-06. 
  23. ^ a b c d e f g h Weisstein, Eric W. "Haversine". MathWorld. Wolfram Research, Inc. Archived from the original on 2005-03-10. Retrieved 2015-11-06. 
  24. ^ Fulst, Otto (1972). "17, 18". In Lütjen, Johannes; Stein, Walter; Zwiebler, Gerhard. Nautische Tafeln (in German) (24 ed.). Bremen, Germany: Arthur Geist Verlag. 
  25. ^ a b Sauer, Frank (2015) [2004]. "Semiversus-Verfahren: Logarithmische Berechnung der Höhe" (in German). Hotheim am Taunus, Germany: Astrosail. Archived from the original on 2013-09-17. Retrieved 2015-11-12. 
  26. ^ Rider, Paul Reece; Davis, Alfred (1923). Plane Trigonometry. New York, USA: D. Van Nostrand Company. p. 42. Retrieved 2015-12-08. 
  27. ^ "Haversine". Wolfram Language & System: Documentation Center. 7.0. 2008. Archived from the original on 2014-09-01. Retrieved 2015-11-06. 
  28. ^ a b Rudzinski, Greg (July 2015). Ix, Hanno. "Ultra compact sight reduction". Ocean Navigator. Portland, ME, USA: Navigator Publishing LLC (227): 42–43. ISSN 0886-0149. Retrieved 2015-11-07. 
  29. ^ a b c d Weisstein, Eric W. "Havercosine". MathWorld. Wolfram Research, Inc. Archived from the original on 2014-03-29. Retrieved 2015-11-06. 
  30. ^ a b c d e f g h i j k l m n van Vlijmen, Oscar (2005-12-28) [2003]. "Goniology". Eenheden, constanten en conversies. Archived from the original on 2009-10-28. Retrieved 2015-11-28. 
  31. ^ a b c d e Weisstein, Eric W. "Hacovercosine". MathWorld. Wolfram Research, Inc. Archived from the original on 2014-03-29. Retrieved 2015-11-06. 
  32. ^ a b c "sagitta". Oxford English Dictionary (3rd ed.). Oxford University Press. September 2005.  (Subscription or UK public library membership required.)
  33. ^ a b c Boyer, Carl Benjamin; Merzbach, Uta C. (1991-03-06) [1968]. A History of Mathematics (2 ed.). New York, USA: John Wiley & Sons. ISBN 978-0471543978. 0471543977. 
  34. ^ a b Miller, Jeff (2007-09-10). "Earliest Known Uses of Some of the Words of Mathematics (V)". New Port Richey, Florida, USA. Archived from the original on 2015-09-05. Retrieved 2015-11-10. 
  35. ^ de Mendoza y Ríos, Joseph (1795). Memoria sobre algunos métodos nuevos de calcular la longitud por las distancias lunares: y aplication de su teórica á la solucion de otros problemas de navegacion (in Spanish). Madrid, Spain: Imprenta Real. 
  36. ^ a b c Archibald, Raymond Clare (1945-07-11). "197: Natural and Logarithmic Haversines". Recent Mathematical Tables (PDF). Mathematical Tables and other Aids to Computation (MTAC) (Review). 1. The National Research Council, Division of Physical Sciences, Committee on Mathematical Tables and Other Aids to Computation; American Mathematical Society. pp. 421–422. doi:10.1090/S0025-5718-45-99080-6. Archived (PDF) from the original on 2015-11-19. Retrieved 2015-11-19.  [3]
  37. ^ Andrew, James (1805). Astronomical and Nautical Tables with Precepts for finding the Latitude and Longitude of Places. T. XIII. London. pp. 29–148.  (A 7-place haversine table from 0° to 120° in intervals of 10".)
  38. ^ van Brummelen, Glen Robert (2013). Heavenly Mathematics: The Forgotten Art of Spherical Trigonometry. Princeton University Press. ISBN 9780691148922. 0691148929. Retrieved 2015-11-10. 
  39. ^ a b "haversine". Oxford English Dictionary (2nd ed.). Oxford University Press. 1989. 
  40. ^ White, J. D. (February 1926). "(unknown title)". Nautical Magazine.  (NB. According to Cajori, 1929, this journal has a discussion on the origin of haversines.)
  41. ^ White, J. D. (July 1926). "(unknown title)". Nautical Magazine.  (NB. According to Cajori, 1929, this journal has a discussion on the origin of haversines.)
  42. ^ Farley, Richard (1856). Natural Versed Sines from 0 to 125°, and Logarithmic Versed Sines from 0 to 135°. London.  (A haversine table from 0° to 125°/135°.)
  43. ^ Hannyngton, John Caulfield (1876). Haversines, Natural and Logarithmic, used in Computing Lunar Distances for the Nautical Almanac. London.  (A 7-place haversine table from 0° to 180°, log. haversines at intervals of 15", nat. haversines at intervals of 10".)
  44. ^ 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. Retrieved 2015-12-02. 
  45. ^ 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. Retrieved 2015-12-02. [4][5]
  46. ^ Wildberger, Norman John (2005). Divine Proportions: Rational Trigonometry to Universal Geometry (1 ed.). Australia: Wild Egg Pty Ltd. ISBN 0-9757492-0-X. Retrieved 2015-12-01. 
  47. ^ Stávek, Jiří (2013-10-18). "On the Trigonometric Loophole". Applied Physics Research. Prague, CZ: Canadian Center of Science and Education. 5 (6). doi:10.5539/apr.v5n6p48. eISSN 1916-9647. ISSN 1916-9639. Archived from the original on 2015-11-19. Retrieved 2015-11-19. 
  48. ^ a b c d e f Simpson, 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. Retrieved 2015-10-26. 
  49. ^ a b c d e f van 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. Retrieved 2015-10-26. 
  50. ^ a b mf344 (2014-07-04). "Lost but lovely: The haversine". Plus magazine. maths.org. Archived from the original on 2014-07-18. Retrieved 2015-11-05. 
  51. ^ a b Skvarc, 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. Retrieved 2015-11-28. 
  52. ^ a b Skvarc, 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. Retrieved 2015-11-28. 
  53. ^ Ballew, Pat (2007-02-08) [2003]. "Versine". Math Words, page 4. Versine. Archived from the original on 2007-02-08. Retrieved 2015-11-28. 
  54. ^ a b Weisstein, Eric W. "Inverse Haversine". MathWorld. Wolfram Research, Inc. Archived from the original on 2008-06-08. Retrieved 2015-10-05. 
  55. ^ a b "InverseHaversine". Wolfram Language & System: Documentation Center. 7.0. 2008. Retrieved 2015-11-05. 
  56. ^ 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. 
  57. ^ Needham, Noel Joseph Terence Montgomery (1959). Science and Civilisation in China: Mathematics and the Sciences of the Heavens and the Earth. 3. Cambridge University Press. p. 39. ISBN 9780521058018. 
  58. ^ Boardman, Harry (1930). Table For Use in Computing Arcs, Chords and Versines. Chicago Bridge and Iron Company. p. 32. 
  59. ^ Nair, P. N. Bhaskaran (1972). "Track measurement systems—concepts and techniques". Rail International. International Railway Congress Association, International Union of Railways. 3 (3): 159–166. ISSN 0020-8442. OCLC 751627806. 

External links[edit]