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Gradian

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"Grade (angle)" redirects here. For the grade of an inclined surface, see Grade (slope).
"Gon (angle)" redirects here. For n-gon, see Regular polygon.
Gon
Unit ofAngle
Symbolᵍ, gon, grad, grd
Conversions
1 ᵍ in ...... is equal to ...
   turns   ⁠1/400⁠ turn
   radians   ⁠π/200⁠ rad
≈ 0.0157.. rad
   milliradians   5π mrad
≈ 15.71.. mrad
   degrees   ⁠9/10⁠°
   minutes of arc   54′

The gradian is a unit of measurement of an angle, defined as one hundredth of the right angle (in other words, there are 100 gradians in 90 degrees).[1][2][3][4] It is equivalent to ⁠1/400⁠ of a turn, ⁠9/10⁠ of a degree, or ⁠π/200⁠ of a radian. Measuring angles in gradians is said to employ the centesimal system of angular measurement.[5][6][7]:22[Note 1]

The gradian is also known as the gon (from Greek γωνία/gōnía for angle), grad, or grade. In continental Europe, the French term centigrade was in use for one hundredth of a grad. This was one reason for the adoption of the term Celsius to replace centigrade as the name of the temperature scale.[11][12]

Gradians are principally used in surveying (especially in Europe),[13][6][14] and to a lesser extent in mining[15] and geology.[16][17]

As of May 2020, the gon is officially a legal unit of measurement in the European Union[18]: 9  and in Switzerland.[19]

The gradian is not part of the International System of Units (SI).[20][18]: 9–10 

History and name

The unit originated in connection with the French Revolution in France as the 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, ⁠1/360⁠ of a turn), the name gon was later adopted, first in those regions, later as the international standard. In German, the unit was formerly also called Neugrad (new degree), likewise nygrad in Swedish, Danish and Norwegian (also gradian), and 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,[13][6][14] mining[15] and geology.[16][17] The French artillery[who?] has used the grad for decades.[citation needed] Today, the degree, ⁠1/360⁠ of a turn, or the mathematically more convenient radian, ⁠1/2π⁠ of a turn (used in the SI system of units) are generally used instead.

In the 1970s and 1980s most scientific calculators offered the grad as well as radians and degrees for their trigonometric functions.[21] In the 2010s, some scientific calculators lack support for gradians.[22]

Symbol

ᵍ
Gon
In UnicodeU+1D4D ᵍ MODIFIER LETTER SMALL G
Related
See alsoU+00B0 ° DEGREE SIGN

The international standard symbol for this unit today is "gon" (see ISO 31-1). 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°.

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.

0° = 0 gradians
90° = 100 gradians
180° = 200 gradians
270° = 300 gradians
360° = 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 grad, the direction to one's left is 17 grad, to one's right 217 grad and behind one 317 grad. A disadvantage is that the common angles of 30° and 60° in geometry must be expressed in fractions (⁠33+1/3⁠ grad and ⁠66+2/3⁠ grad, respectively). Similarly, in one hour (⁠1/24⁠ day), Earth rotates by 15° or ⁠16+2/3⁠ gon (see also decimal time). These observations are a consequence of the fact that the number 360 has more divisors than the number 400 does; notably, 360 is divisible by 3, while 400 is not. There are eleven factors of 360 less than or equal to its square root: {2, 3, 4, 5, 6, 8, 9, 10, 12, 15, 18}. However, there are only seven for 400: {2, 4, 5, 8, 10, 16, 20}.

In the 18th century, the metre was defined as the forty-millionth part of a meridian. Thus, one grad of arc along the Earth's surface corresponded to 100 kilometres of distance at the equator; 1 centigrad of arc equaled 1 kilometre; 0.1 cc (centi-centigrads) of arc equaled 1 metre.[23]

Conversion

Conversion of common angles
Turns Radians Degrees Gradians
0 turn 0 rad 0° 0g
⁠1/72⁠ turn ⁠π/36⁠ or ⁠𝜏/72⁠ rad 5° ⁠5+5/9⁠g
⁠1/24⁠ turn ⁠π/12⁠ or ⁠𝜏/24⁠ rad 15° ⁠16+2/3⁠g
⁠1/16⁠ turn ⁠π/8⁠ or ⁠𝜏/16⁠ rad 22.5° 25g
⁠1/12⁠ turn ⁠π/6⁠ or ⁠𝜏/12⁠ rad 30° ⁠33+1/3⁠g
⁠1/10⁠ turn ⁠π/5⁠ or ⁠𝜏/10⁠ rad 36° 40g
⁠1/8⁠ turn ⁠π/4⁠ or ⁠𝜏/8⁠ rad 45° 50g
⁠1/2π or 𝜏⁠ turn 1 rad approx. 57.3° approx. 63.7g
⁠1/6⁠ turn ⁠π/3⁠ or ⁠𝜏/6⁠ rad 60° ⁠66+2/3⁠g
⁠1/5⁠ turn ⁠2π or 𝜏/5⁠ rad 72° 80g
⁠1/4⁠ turn ⁠π/2⁠ or ⁠𝜏/4⁠ rad 90° 100g
⁠1/3⁠ turn ⁠2π or 𝜏/3⁠ rad 120° ⁠133+1/3⁠g
⁠2/5⁠ turn ⁠4π or 2𝜏 or α/5⁠ rad 144° 160g
⁠1/2⁠ turn π or ⁠𝜏/2⁠ rad 180° 200g
⁠3/4⁠ turn ⁠3π or ρ/2⁠ or ⁠3𝜏/4⁠ rad 270° 300g
1 turn 𝜏 or 2π rad 360° 400g

Not part of the SI system of units

The gradian is not part of the International System of Units (SI). The EU directive on the units of measurement[18]: 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.[20] The previous edition mentioned it only in a footnote, which said the following:[24]

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 10 000 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

Notes

  1. ^ 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 Ref. [8] The gradations were in one-hundredths of a full revolution.[9][10]

References

  1. ^ Harris, J. W. and Stocker, H. Handbook of Mathematics and Computational Science. New York: Springer-Verlag, p. 63, 1998.
  2. ^ http://mathworld.wolfram.com/Gradian.html
  3. ^ https://www.nist.gov/pml/nist-guide-si-appendix-b9-factors-units-listed-kind-quantity-or-field-science#ANGLE
  4. ^ Patrick Bouron (2005). Cartographie: Lecture de Carte (PDF). Institut Géographique National. p. 12. Archived from the original (PDF) on 2010-04-15. Retrieved 2011-07-07.
  5. ^ Balzer, Fritz (1946). Five Place Natural Sine and Tangent Functions in the Centesimal System. Army Map Service, Corps of Engineers, U.S. Army.
  6. ^ a b c Zimmerman, Edward G. (1995). "6. Angle Measurement: Transits and Theodolites". In Minnick, Roy; Brinker, Russell Charles (eds.). The surveying handbook (2nd ed.). Chapman & Hall. ISBN 041298511X. {{cite book}}: External link in |chapterurl= (help); Unknown parameter |chapterurl= ignored (|chapter-url= suggested) (help)
  7. ^ Gorini, Catherine A. (2003). The Facts on File Geometry Handbook. Infobase Publishing. ISBN 978-1-4381-0957-2.
  8. ^ Cajori, Florian (1899). A History of Physics in Its Elementary Branches: Including the Evolution of Physical Laboratories. Macmillan. The angle through which the torsion-head must be deflected was measured in centesimal divisions of the circle
  9. ^ Ohm, Georg Simon (1826). "Bestimmung des Gesetzes, nach welchem Metalle die Contactelektricität leiten, nebst einem Entwurfe zur Theorie des Voltaischen Apparates und des Schweiggerschen Multiplikators" (PDF). Journal für Chemie und Physik. 46: 137–166. Archived from the original (PDF) on 23 May 2020. Template:Lang-de (p. 147) [the amount of rotation at the top of the torsion balance was read in hundred parts of an entire revolution]
  10. ^ Keithley, Joseph F. (1999). The Story of Electrical and Magnetic Measurements: From 500 BC to the 1940s. John Wiley & Sons. ISBN 978-0-7803-1193-0. It hung on a ribbon torsion element with a knob on top, graduated in 100 parts.
  11. ^ Frasier, E. Lewis (February 1974), "Improving an imperfect metric system", Bulletin of the Atomic Scientists: 9ff. On 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.
  12. ^ Mahaffey, Charles T. (1976), Metrication problems in the construction codes and standards sector, NBS Technical Note 915, U.S. Department of Commerce, National Bureau of Commerce, Institute for Applied Technology, Center for Building Technology, The term "Celsius" was adopted instead of the more familiar "centigrade" because in France the word centigrade has customarily been applied to angles.
  13. ^ a b Kahmen, Heribert; Faig, Wolfgang (2012). Surveying. De Gruyter. ISBN 9783110845716.
  14. ^ a b Schofield, Wilfred (2001). Engineering surveying: theory and examination problems for sudents (5th ed.). Butterworth-Heinemann. ISBN 9780750649872.
  15. ^ a b Sroka, Anton (2006). "Contribution to the prediction of ground surface movements caused by a rising water level in a flooded mine". In Sobczyk, Eugeniusz; Kicki, Jerzy (eds.). International Mining Forum 2006, New Technological Solutions in Underground Mining: Proceedings of the 7th International Mining Forum, Cracow - Szczyrk - Wieliczka, Poland, February 2006. CRC Press. ISBN 9780415889391.
  16. ^ a b Gunzburger, Yann; Merrien-Soukatchoff, Véronique; Senfaute, Gloria; Piguet, Jack-Pierre; Guglielmi, Yves (2004). "Field investigations, monitoring and modeling in the identification of rock fall causes". In Lacerda, W.; Ehrlich, Mauricio; Fontoura, S. A. B.; Sayão, A. S. F. (eds.). 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. Vol. 1. CRC Press. ISBN 978-1-4822-6288-9.
  17. ^ a b Schmidt, Dietmar; Kühn, Friedrich (2007). "3. Remote sensing: 3.1 Aerial Photography". In Knödel, Klaus; Lange, Gerhard; Voigt, Hans-Jürgen (eds.). Environmental Geology: Handbook of Field Methods and Case Studies. Springer Science & Business Media. ISBN 978-3-540-74671-3.
  18. ^ a b c "Directive 80/181/EEC". 27 May 2009. Archived from the original on 22 May 2020. On the approximation of the laws of the Member States relating to units of measurement and on the repeal of Directive 71/354/EEC.
  19. ^ "941.202 Einheitenverordnung". Archived from the original on 22 May 2020.
  20. ^ a b The International System of Units (PDF) (9th ed.), International Bureau of Weights and Measures, Dec 2022, ISBN 978-92-822-2272-0
  21. ^ Maloney, Timothy J. (1992), Electricity: Fundamental Concepts and Applications, Delmar Publishers, p. 453, ISBN 9780827346758, On most scientific calculators, this [the unit for angles] is set by the DRG key
  22. ^ Cooke, Heather (2007), Mathematics for Primary and Early Years: Developing Subject Knowledge, SAGE, p. 53, ISBN 9781847876287, Scientific calculators commonly have two modes for working with angles – degrees and radians
  23. ^ Cartographie – lecture de carte – Partie H Quelques exemples à retenir. Archived 2 March 2012 at the Wayback Machine
  24. ^ International Bureau of Weights and Measures (2006), The International System of Units (SI) (PDF) (8th ed.), ISBN 92-822-2213-6, archived (PDF) from the original on 2021-06-04, retrieved 2021-12-16
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