Gradian
{{short description|Unit of measurement of an angle, equal to 1/400th of a circle}}
{{redirect|Grade (angle)|the grade of an inclined surface|Grade (slope)}}
{{Redirect|Gon (angle)|n-gon|Regular polygon}}
{{distinguish|Gradient}}
{{Infobox unit
| name = gon
| othernames = gradian, grade, grad
| image = Boussole en grades table conversion (cropped).jpg
| caption = Compass graded with 400 gon
| standard =
| quantity = Angle
| symbol = gon
| symbol2 = ᵍ
| symbol3 = grad
| units1 = turns
| inunits1 = {{sfrac|1|400}} turn
| units2 = radians
| inunits2 = {{sfrac|{{pi}}|200}} rad
≈ 0.0157... rad
| units3 = milliradians
| inunits3 = 5{{pi}} mrad
≈ 15.71... mrad
| units4 = degrees
| inunits4 = 0.9°
| units5 = minutes of arc
| inunits5 = 54′
}}
In trigonometry, the gradian{{snd}}also known as the gon ({{etymology|grc|{{Wikt-lang|grc|γωνία}} ({{grc-transl|γωνία}})|angle}}), grad, or grade{{cite web |last=Weisstein|first=Eric W. |title=Gradian |url=https://mathworld.wolfram.com/Gradian.html |access-date=2020-08-31|website=mathworld.wolfram.com |language=en}}{{snd}}is a unit of measurement of an angle, defined as one-hundredth of the right angle; in other words, 100 gradians is equal to 90 degrees.{{cite book |last1=Harris|first1=J. W. |last2=Stocker|first2=H. |title=Handbook of Mathematics and Computational Science |location=New York |publisher=Springer-Verlag |page=63 |year=1998}}{{cite web |url=https://nist.gov/pml/nist-guide-si-appendix-b9-factors-units-listed-kind-quantity-or-field-science#ANGLE |title=NIST Guide to the SI, Appendix B.9: Factors for units listed by kind of quantity or field of science |publisher=NIST |website=nist.gov |url-status=dead |archive-url=https://web.archive.org/web/20170417071705/https://www.nist.gov/pml/nist-guide-si-appendix-b9-factors-units-listed-kind-quantity-or-field-science |archive-date=2017-04-17}}{{cite book|url=http://webdav-noauth.unit-c.fr/files/perso/pbarbier/cours_unit/Elements_de_base_de_la_cartographie.pdf|page=12|date=2005|publisher=Institut Géographique National|author=Patrick Bouron|access-date=2011-07-07|title=Cartographie: Lecture de Carte|url-status=dead|archive-url=https://web.archive.org/web/20100415034329/http://webdav-noauth.unit-c.fr/files/perso/pbarbier/cours_unit/Elements_de_base_de_la_cartographie.pdf|archive-date=2010-04-15}} It is equivalent to {{sfrac|1|400}} of a turn,{{Cite web|title=Gradian|url=https://artofproblemsolving.com/wiki/index.php/Gradian|access-date=2020-08-31|website=Art of Problem Solving}} {{sfrac|9|10}} of a degree, or {{sfrac|{{pi}}|200}} of a radian. Measuring angles in gradians (gons) is said to employ the centesimal system of angular measurement, initiated as part of metrication and decimalisation efforts.{{cite book |last1=Balzer |first1=Fritz |title=Five Place Natural Sine and Tangent Functions in the Centesimal System |date=1946 |publisher=Army Map Service, Corps of Engineers, U.S. Army. |url=https://books.google.com/books?id=2IxG7M9Yw0UC&q=centesimal&pg=PP9 |language=en}}{{cite book |last1=Zimmerman |first1=Edward G.|title=The surveying handbook |publisher=Chapman & Hall|editor-first1=Roy |editor-last1=Minnick| editor-first2=Russell Charles |editor-last2=Brinker | chapter= 6. Angle Measurement: Transits and Theodolites |isbn=041298511X |edition=2nd|year=1995|chapter-url=https://books.google.com/books?id=2gB7w9XlNJAC&q=centesimal&pg=PA80}}{{cite book |last1=Gorini |first1=Catherine A. |title=The Facts on File Geometry Handbook |date=2003 |publisher=Infobase Publishing |isbn=978-1-4381-0957-2 |url=https://books.google.com/books?id=PlYCcvgLJxYC |language=en |page=[https://books.google.com/books?id=PlYCcvgLJxYC&q=centesimal&pg=PA22 22]}}{{efn|On rare occasions, centesimal refers to the division of the full angle (360°) into hundred parts. One example is the description of the gradations on Georg Ohm's torsion balance in a history of physics book from 1899.{{cite book |last1=Cajori |first1=Florian |title=A History of Physics in Its Elementary Branches: Including the Evolution of Physical Laboratories |date=1899 |publisher=Macmillan |isbn=9781548494957 |url=https://books.google.com/books?id=7Y5KAAAAMAAJ&dq=%22history%22%20centesimal%20angle&pg=PA229 |language=en|quote="The angle through which the torsion-head must be deflected was measured in centesimal divisions of the circle"}} The gradations were in one-hundredths of a full revolution.{{cite journal|first=Georg Simon|last=Ohm|title=Bestimmung des Gesetzes, nach welchem Metalle die Contactelektricität leiten, nebst einem Entwurfe zur Theorie des Voltaischen Apparates und des Schweiggerschen Multiplikators|journal=Journal für Chemie und Physik|year=1826|volume=46|pages=137–166|url=http://www2.ohm-hochschule.de/bib/textarchiv/Ohm.Bestimmung_des_Gesetzes.pdf|archive-url=https://web.archive.org/web/20200523161946/http://www2.ohm-hochschule.de/bib/textarchiv/Ohm.Bestimmung_des_Gesetzes.pdf|archive-date=23 May 2020|quote={{langx|de|wurde die Größe der Drehung oben an der Drehwage in Hunderttheilen einer ganzen Umdrehung abgelesen}} (p. 147) [the amount of rotation at the top of the torsion balance was read in hundred parts of an entire revolution]}}{{cite book |last1=Keithley |first1=Joseph F. |title=The Story of Electrical and Magnetic Measurements: From 500 BC to the 1940s |date=1999 |publisher=John Wiley & Sons |isbn=978-0-7803-1193-0 |url=https://www.google.com/search?tbm=bks&q=%22It+hung+on+a+ribbon+torsion+element+with+a+knob+on+top%2C+graduated+in+100+parts.%22 |language=en|quote=It hung on a ribbon torsion element with a knob on top, graduated in 100 parts.}}}}
In continental Europe, the French word centigrade, also known as centesimal minute of arc, was in use for one hundredth of a grade; similarly, the centesimal second of arc was defined as one hundredth of a centesimal arc-minute, analogous to decimal time and the sexagesimal minutes and seconds of arc.{{cite book | last=Klein | first=H.A. | title=The Science of Measurement: A Historical Survey | publisher=Dover Publications | series=Dover Books on Mathematics | year=2012 | isbn=978-0-486-14497-9 | url=https://books.google.com/books?id=CrmuSiCFyikC&pg=PA114 | access-date=2022-01-02 | page=114}} The chance of confusion was one reason for the adoption of the term Celsius to replace centigrade as the name of the temperature scale.{{citation|title=Improving an imperfect metric system|first=E. Lewis|last=Frasier|journal=Bulletin of the Atomic Scientists|date=February 1974|volume=30|issue=2|pages=9–44|doi=10.1080/00963402.1974.11458078|bibcode=1974BuAtS..30b...9F}}. On [https://books.google.com/books?id=IwwAAAAAMBAJ&pg=PA42 p. 42] Frasier argues for using grads instead of radians as a standard unit of angle, but for renaming grads to "radials" instead of renaming the temperature scale.{{citation|title=Metrication problems in the construction codes and standards sector|series=NBS Technical Note 915|first=Charles T.|last=Mahaffey|journal=Final Report National Bureau of Standards|publisher=U.S. Department of Commerce, National Bureau of Commerce, Institute for Applied Technology, Center for Building Technology|year=1976|bibcode=1976nbs..reptU....M|url=https://archive.org/stream/metricationprobl915maha|quote=The term "Celsius" was adopted instead of the more familiar "centigrade" because in France the word centigrade has customarily been applied to angles.}}
Gradians (gons) are principally used in surveying (especially in Europe),{{cite book|first1= Heribert |last1=Kahmen|first2=Wolfgang|last2= Faig |title=Surveying |isbn=9783110845716|publisher=De Gruyter|year=2012|url=https://books.google.com/books?id=7nMiAAAAQBAJ&dq=%22gon%22%20%20%22surveying%22%20%22Europe%22&pg=PA58}}{{cite book |first=Wilfred |last=Schofield|title=Engineering surveying: theory and examination problems for students |publisher=Butterworth-Heinemann |isbn=9780750649872 |edition=5th|year=2001|url=https://books.google.com/books?id=1wYBMUYV0b0C&q=gon&pg=PA23}}
and to a lesser extent in mining{{cite book |title=International Mining Forum 2006, New Technological Solutions in Underground Mining: Proceedings of the 7th International Mining Forum, Cracow - Szczyrk - Wieliczka, Poland, February 2006 |isbn=9780415889391|year=2006|editor-first1=Eugeniusz |editor-last1=Sobczyk|editor-first2= Jerzy|editor-last2=Kicki|publisher=CRC Press|chapter=Contribution to the prediction of ground surface movements caused by a rising water level in a flooded mine|author-first=Anton|author-last=Sroka|chapter-url=https://books.google.com/books?id=-smnDwAAQBAJ&q=gon&pg=PA68}} and geology.{{cite book |editor-last1=Lacerda |editor-first1=W. |editor-last2=Ehrlich |editor-first2=Mauricio |editor-last3=Fontoura |editor-first3=S. A. B. |editor-last4=Sayão |editor-first4=A. S. F. |title=Landslides: Evaluation & Stabilization/Glissement de Terrain: Evaluation et Stabilisation, Set of 2 Volumes: Proceedings of the Ninth International Symposium on Landslides, June 28 -July 2, 2004 Rio de Janeiro, Brazil |date=2004 |volume=1|publisher=CRC Press |isbn=978-1-4822-6288-9 |author-last1=Gunzburger|author-first1=Yann|author-last2=Merrien-Soukatchoff|author-first2=Véronique|author-last3=Senfaute|author-first3=Gloria|author-last4=Piguet|author-first4=Jack-Pierre|author-last5=Guglielmi|author-first5=Yves|chapter=Field investigations, monitoring and modeling in the identification of rock fall causes|chapter-url=https://books.google.com/books?id=DLPNBQAAQBAJ&dq=mining%20measurement%20%22gon%22%20-%22data%20mining%22&pg=PA561}}{{cite book |editor-last1=Knödel |editor-first1=Klaus |editor-last2=Lange |editor-first2=Gerhard |editor-last3=Voigt |editor-first3=Hans-Jürgen |title=Environmental Geology: Handbook of Field Methods and Case Studies |date=2007 |publisher=Springer Science & Business Media |isbn=978-3-540-74671-3 |author-first1= Dietmar|author-last1=Schmidt |author-first2=Friedrich |author-last2=Kühn|chapter=3. Remote sensing: 3.1 Aerial Photography|chapter-url=https://books.google.com/books?id=QF5_Xvhm8KQC&dq=geology%20%22gon%22&pg=PA28}}
The gon (gradian) is a legally recognised unit of measurement in the European Union{{Cite web |url=https://eur-lex.europa.eu/legal-content/EN/TXT/PDF/?uri=CELEX:01980L0181-20090527&from=EN |title=Directive 80/181/EEC |date=27 May 2009 |quote=On the approximation of the laws of the Member States relating to units of measurement and on the repeal of Directive 71/354/EEC. |archive-url=https://web.archive.org/web/20200522202337/https://eur-lex.europa.eu/legal-content/EN/TXT/PDF/?uri=CELEX%3A01980L0181-20090527&from=EN |archive-date=22 May 2020}}{{rp|9}} and in Switzerland.{{cite web |url=https://www.admin.ch/opc/de/classified-compilation/19940345/ |title=941.202 Einheitenverordnung |language=de |archive-url=https://web.archive.org/web/20200522202750/https://www.admin.ch/opc/de/classified-compilation/19940345/ |archive-date=22 May 2020}} However, this unit is not part of the International System of Units (SI).{{rp|9–10}}
History and name
The unit originated in France in connection with the French Revolution as the {{lang|fr|grade}}, along with the metric system, hence it is occasionally referred to as a metric degree. Due to confusion with the existing term grad(e) in some northern European countries (meaning a standard degree, {{sfrac|1|360}} of a turn), the name gon was later adopted, first in those regions, and later as the international standard.{{which|date=December 2024}} In France, it was also called {{lang|fr|grade nouveau}}. In German, the unit was formerly also called {{lang|de|Neugrad}} (new degree) (whereas the standard degree was referred to as {{lang|de|Altgrad}} (old degree)), likewise {{lang|da|nygrad}} in Danish, Swedish and Norwegian (also gradian), and {{lang|is|nýgráða}} in Icelandic.
Although attempts at a general introduction were made, the unit was only adopted in some countries, and for specialised areas such as surveying, mining and geology. Today, the degree, {{sfrac|1|360}} of a turn, or the mathematically more convenient radian, {{sfrac|1|2{{pi}}}} of a turn (used in the SI system of units) is generally used instead.
In the {{nowrap|1970s{{hsp}}{{ndash}}}}1990s, most scientific calculators offered the gon (gradian), as well as radians and degrees, for their trigonometric functions.{{citation|title=Electricity: Fundamental Concepts and Applications|first=Timothy J.|last=Maloney|publisher=Delmar Publishers|year=1992|isbn=9780827346758|page=[https://archive.org/details/electricityfunda0000malo/page/453 453]|quote=On most scientific calculators, this [the unit for angles] is set by the DRG key|url=https://archive.org/details/electricityfunda0000malo/page/453}} In the 2010s, some scientific calculators lack support for gradians.{{citation|title=Mathematics for Primary and Early Years: Developing Subject Knowledge|first=Heather|last=Cooke|publisher=SAGE|year=2007|isbn=9781847876287|page=53|url=https://books.google.com/books?id=C3t4B3stB0UC&pg=PA53|quote=Scientific calculators commonly have two modes for working with angles – degrees and radians}}
= Symbol =
{{infobox symbol
|name= Gon
|mark = ◌ᵍ
|unicode={{unichar|1D4D|modifier letter small g|nlink=modifier letter}}
|see also={{unichar|00B0|nlink=Degree sign}}
}}
The international standard symbol for this unit is "gon" (see ISO 31-1, Annex B).{{needs update|reason=ISO 31-1 has been superseded twice, so this information may be obsolete.|date=December 2024}} Other symbols used in the past include "gr", "grd", and "g", the last sometimes written as a superscript, similarly to a degree sign: 50g = 45°.
A metric prefix is sometimes used, as in "dgon", "cgon", "mgon", denoting respectively 0.1 gon, 0.01 gon, 0.001 gon.
Centesimal arc-minutes and centesimal arc-seconds were also denoted with superscripts c and cc, respectively.
{{SI multiples
| unit=gon
| symbol=gon
}}
Advantages and disadvantages
Each quadrant is assigned a range of 100 gon, which eases recognition of the four quadrants, as well as arithmetic involving perpendicular or opposite angles.
:
| align="right"| 0° | align="center"| = | align="right"| 0 gradians |
| align="right"| 90° | align="center"| = | align="right"| 100 gradians |
| align="right"| 180° | align="center"| = | align="right"| 200 gradians |
| align="right"| 270° | align="center"| = | align="right"| 300 gradians |
| align="right"| 360° | align="center"| = | align="right"| 400 gradians |
One advantage of this unit is that right angles to a given angle are easily determined. If one is sighting down a compass course of 117 gon, the direction to one's left is 17 gon, to one's right 217 gon, and behind one 317 gon. A disadvantage is that the common angles of 30° and 60° in geometry must be expressed in fractions (as {{sfrac|33|1|3}} gon and {{sfrac|66|2|3}} gon respectively).
Conversion
{{Table of angles}}
Relation to the metre <span class="anchor" id="Metre"></span>
{{further|History of the metre#Meridional definition}}
{{see also|Arc length#Great circles on Earth|Nautical mile#Similar definitions}}
File:Kilometre definition.svg was one ten-millionth of the distance from the North Pole to the equator, measured along a meridian through Paris.]]
In the 18th century, the metre was defined as the 10-millionth part of a quarter meridian.
Thus, 1 gon corresponds to an arc length along the Earth's surface of approximately 100 kilometres; 1 centigon to 1 kilometre; 10 microgons to 1 metre.[http://webdav-noauth.unit-c.fr/files/perso/pbarbier/cours_unit/Elements_de_base_de_la_cartographie.pdf Cartographie – lecture de carte – Partie H Quelques exemples à retenir]. {{webarchive|url=https://web.archive.org/web/20120302111251/http://webdav-noauth.unit-c.fr/files/perso/pbarbier/cours_unit/Elements_de_base_de_la_cartographie.pdf|date=2 March 2012}}. (The metre has been redefined with increasing precision since then.)
Relation to the SI system of units <span class="anchor" id="SI"></span>
The gradian is not part of the International System of Units (SI). The EU directive on the units of measurement{{rp|9–10}} notes that the gradian "does not appear in the lists drawn up by the CGPM, CIPM or BIPM." The most recent, 9th edition of the SI Brochure does not mention the gradian at all.{{SIbrochure9th}} The previous edition mentioned it only in the following footnote:{{SIbrochure8th}} {{blockquote|The gon (or grad, where grad is an alternative name for the gon) is an alternative unit of plane angle to the degree, defined as (π/200) rad. Thus there are 100 gon in a right angle. The potential value of the gon in navigation is that because the distance from the pole to the equator of the Earth is approximately {{val|10000|u=km}}, 1 km on the surface of the Earth subtends an angle of one centigon at the centre of the Earth. However the gon is rarely used.}}
See also
- {{Annotated link |Angular frequency}}
- {{Annotated link |Milliradian}} (primarily military use)
- {{Annotated link |Harmonic analysis}}
- {{Annotated link |Jean-Charles de Borda}}
- {{Annotated link |Repeating circle}}
- {{Annotated link |Spread (rational trigonometry)}}
- {{Annotated link |Steradian}} (the "square radian")
Notes
{{notelist}}
References
{{Reflist}}
External links
- [http://mathforum.org/library/drmath/view/55451.html Ask Dr Math]
- Definitions of [http://www.sizes.com/units/grade.htm grade], [http://www.sizes.com/units/gon.htm gon] and [http://www.sizes.com/units/temperature_centigrade.htm centigrade] on sizes.com
- [http://www.cleavebooks.co.uk/dictunit/notesa.htm Dictionary of Units]