![Boussole en grades table conversion (cropped)](https://upload.wikimedia.org/wikipedia/commons/c/c6/Boussole_en_grades_table_conversion_%28cropped%29.jpg)
In
trigonometry
Trigonometry () is a branch of mathematics that studies relationships between side lengths and angles of triangles. The field emerged in the Hellenistic world during the 3rd century BC from applications of geometry to astronomical studies. T ...
, the gradian, also known as the gon (from grc, γωνία, gōnía, angle), grad, or grade, is a
unit of measurement
A unit of measurement is a definite magnitude (mathematics), magnitude of a quantity, defined and adopted by convention or by law, that is used as a standard for measurement of the same kind of quantity. Any other quantity of that kind can ...
of an
angle
In Euclidean geometry, an angle is the figure formed by two Ray (geometry), rays, called the ''Side (plane geometry), sides'' of the angle, sharing a common endpoint, called the ''vertex (geometry), vertex'' of the angle.
Angles formed by two ...
, defined as one hundredth of the
right angle
In geometry and trigonometry, a right angle is an angle of exactly 90 Degree (angle), degrees or radians corresponding to a quarter turn (geometry), turn. If a Line (mathematics)#Ray, ray is placed so that its endpoint is on a line and the ad ...
; in other words, there are 100 gradians in 90 degrees.
[Harris, J. W. and Stocker, H. Handbook of Mathematics and Computational Science. New York: Springer-Verlag, p. 63, 1998.] It is equivalent to of a
turn, of a
degree
Degree may refer to:
As a unit of measurement
* Degree (angle), a unit of angle measurement
** Degree of geographical latitude
** Degree of geographical longitude
* Degree symbol (°), a notation used in science, engineering, and mathematics
...
, or of a
radian
The radian, denoted by the symbol rad, is the unit of angle in the International System of Units (SI) and is the standard unit of angular measure used in many areas of mathematics. The unit was formerly an SI supplementary unit (before that c ...
. Measuring angles in gradians is said to employ the ''centesimal'' system of angular measurement, initiated as part of
metrication
Metrication or metrification is the act or process of converting to the metric system of measurement. All over the world, countries have transitioned from local and traditional units of measurement to the metric system. This process began in F ...
and
decimalisation
Decimalisation or decimalization (see spelling differences) is the conversion of a system of currency or of weights and measures to units related by powers of 10.
Most countries have decimalised their currencies, converting them from non-decimal ...
efforts.
In continental
Europe
Europe is a large peninsula conventionally considered a continent in its own right because of its great physical size and the weight of its history and traditions. Europe is also considered a Continent#Subcontinents, subcontinent of Eurasia ...
, 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
Decimal time is the representation of the time of day using units which are decimally related. This term is often used specifically to refer to the time system used in France for a few years beginning in 1792 during the French Revolution, whic ...
and the
sexagesimal
Sexagesimal, also known as base 60 or sexagenary, is a numeral system with sixty as its base. It originated with the ancient Sumerians in the 3rd millennium BC, was passed down to the ancient Babylonians, and is still used—in a modified form ...
minutes and seconds of arc
A minute of arc, arcminute (arcmin), arc minute, or minute arc, denoted by the symbol , is a unit of angular measurement equal to of one degree. Since one degree is of a turn (or complete rotation), one minute of arc is of a turn. The ...
.
The chance of confusion was one reason for the adoption of the term ''
Celsius
The degree Celsius is the unit of temperature on the Celsius scale (originally known as the centigrade scale outside Sweden), one of two temperature scales used in the International System of Units (SI), the other being the Kelvin scale. The ...
'' to replace ''centigrade'' as the name of the temperature scale.
Gradians are principally used in
surveying
Surveying or land surveying is the technique, profession, art, and science of determining the terrestrial two-dimensional or three-dimensional positions of points and the distances and angles between them. A land surveying professional is ca ...
(especially in Europe),
and to a lesser extent in
mining
Mining is the extraction of valuable minerals or other geological materials from the Earth, usually from an ore body, lode, vein, seam, reef, or placer deposit. The exploitation of these deposits for raw material is based on the economic via ...
and
geology
Geology () is a branch of natural science concerned with Earth and other astronomical objects, the features or rocks of which it is composed, and the processes by which they change over time. Modern geology significantly overlaps all other Ear ...
.
the gon is officially a legal unit of measurement in the
European Union
The European Union (EU) is a supranational political and economic union of member states that are located primarily in Europe. The union has a total area of and an estimated total population of about 447million. The EU has often been des ...
and in
Switzerland
). Swiss law does not designate a ''capital'' as such, but the federal parliament and government are installed in Bern, while other federal institutions, such as the federal courts, are in other cities (Bellinzona, Lausanne, Luzern, Neuchâtel ...
.
The gradian is not part of the
International System of Units
The International System of Units, known by the international abbreviation SI in all languages and sometimes pleonastically as the SI system, is the modern form of the metric system and the world's most widely used system of measurement. E ...
(SI).
History and name
The unit originated in connection with the
French Revolution
The French Revolution ( ) was a period of radical political and societal change in France that began with the Estates General of 1789 and ended with the formation of the French Consulate in November 1799. Many of its ideas are considere ...
in
France
France (), officially the French Republic ( ), is a country primarily located in Western Europe. It also comprises of Overseas France, overseas regions and territories in the Americas and the Atlantic Ocean, Atlantic, Pacific Ocean, Pac ...
as the , along with the
metric system
The metric system is a system of measurement that succeeded the Decimal, decimalised system based on the metre that had been introduced in French Revolution, France in the 1790s. The historical development of these systems culminated in the d ...
, 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, of a turn), the name ''gon'' was later adopted, first in those regions, and later as the international standard. In France, it was also called . In
German
German(s) may refer to:
* Germany (of or related to)
**Germania (historical use)
* Germans, citizens of Germany, people of German ancestry, or native speakers of the German language
** For citizens of Germany, see also German nationality law
**Ger ...
, the unit was formerly also called (new degree) (whereas the standard degree was referred to as (old degree)), likewise in
Danish
Danish may refer to:
* Something of, from, or related to the country of Denmark
People
* A national or citizen of Denmark, also called a "Dane," see Demographics of Denmark
* Culture of Denmark
* Danish people or Danes, people with a Danish a ...
,
Swedish
Swedish or ' may refer to:
Anything from or related to Sweden, a country in Northern Europe. Or, specifically:
* Swedish language, a North Germanic language spoken primarily in Sweden and Finland
** Swedish alphabet, the official alphabet used by ...
and
Norwegian
Norwegian, Norwayan, or Norsk may refer to:
*Something of, from, or related to Norway, a country in northwestern Europe
*Norwegians, both a nation and an ethnic group native to Norway
*Demographics of Norway
*The Norwegian language, including the ...
(also ''gradian''), and 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
Surveying or land surveying is the technique, profession, art, and science of determining the terrestrial two-dimensional or three-dimensional positions of points and the distances and angles between them. A land surveying professional is ca ...
,
mining
Mining is the extraction of valuable minerals or other geological materials from the Earth, usually from an ore body, lode, vein, seam, reef, or placer deposit. The exploitation of these deposits for raw material is based on the economic via ...
and
geology
Geology () is a branch of natural science concerned with Earth and other astronomical objects, the features or rocks of which it is composed, and the processes by which they change over time. Modern geology significantly overlaps all other Ear ...
.
The French armed forces' artillery units have used the gon for decades. Today, the degree, of a
turn, or the mathematically more convenient radian, of a turn (used in the
SI system of units) is generally used instead.
In the 1990s, most
scientific calculator
A scientific calculator is an electronic calculator, either desktop or handheld, designed to perform mathematical operations. They have completely replaced slide rules and are used in both educational and professional settings.
In some areas ...
s offered the gon, as well as radians and degrees, for their
trigonometric functions
In mathematics, the trigonometric functions (also called circular functions, angle functions or goniometric functions) are real functions which relate an angle of a right-angled triangle to ratios of two side lengths. They are widely used in all ...
. In the 2010s, some scientific calculators lack support for gradians.
Symbol
The international standard symbol for this unit today is "gon" (see
ISO 31-1
ISO 31-1 is the part of international standard ISO 31 that defines names and symbols for quantities and units related to ''space and time''. It was superseded in 2006 by ISO 80000-3.
Definitions
Its definitions include:
Annex A
Annex A ...
). Other symbols used in the past include "gr", "grd", and "g", the last sometimes written as a superscript, similarly to a degree sign: 50
g = 45°.
A
metric prefix
A metric prefix is a unit prefix that precedes a basic unit of measure to indicate a multiple or submultiple of the unit. All metric prefixes used today are decadic. Each prefix has a unique symbol that is prepended to any unit symbol. The pre ...
sometimes is used, as in "dgon", "cgon", "mgon", 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.
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.
:
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 gon and gon, respectively).
Similarly, in one hour ( day), Earth rotates by 15° or gon (see also
decimal time
Decimal time is the representation of the time of day using units which are decimally related. This term is often used specifically to refer to the time system used in France for a few years beginning in 1792 during the French Revolution, whic ...
). 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 twelve factors of 360 less than or equal to its square root: . However, there are only eight for 400: .
Conversion
Relation to the metre
![Kilometre definition](https://upload.wikimedia.org/wikipedia/commons/1/14/Kilometre_definition.svg)
In the 18th century, the
metre
The metre (British spelling) or meter (American spelling; see spelling differences) (from the French unit , from the Greek noun , "measure"), symbol m, is the primary unit of length in the International System of Units (SI), though its pref ...
was defined as the 10-millionth part of a
quarter meridian
In geodesy and navigation, a meridian arc is the curve between two points on the Earth's surface having the same longitude. The term may refer either to a segment of the meridian, or to its length.
The purpose of measuring meridian arcs is to d ...
.
Thus, one gon corresponds to an
arc length
ARC may refer to:
Business
* Aircraft Radio Corporation, a major avionics manufacturer from the 1920s to the '50s
* Airlines Reporting Corporation, an airline-owned company that provides ticket distribution, reporting, and settlement services
* ...
along the Earth's surface of approximately 100 kilometres; 1 centigon to 1 kilometre; 10 microgon to 1 metre.
Cartographie – lecture de carte – Partie H Quelques exemples à retenir
Relation to the SI system of units
The gradian is ''not'' part of the International System of Units
The International System of Units, known by the international abbreviation SI in all languages and sometimes pleonastically as the SI system, is the modern form of the metric system and the world's most widely used system of measurement. E ...
(SI). The EU directive on the units of measurement notes that the gradian ''does not appear in the lists drawn up by the CGPM
The General Conference on Weights and Measures (GCWM; french: Conférence générale des poids et mesures, CGPM) is the supreme authority of the International Bureau of Weights and Measures (BIPM), the intergovernmental organization established i ...
, CIPM
The General Conference on Weights and Measures (GCWM; french: Conférence générale des poids et mesures, CGPM) is the supreme authority of the International Bureau of Weights and Measures (BIPM), the intergovernmental organization established i ...
or BIPM
The International Bureau of Weights and Measures (french: Bureau international des poids et mesures, BIPM) is an intergovernmental organisation, through which its 59 member-states act together on measurement standards in four areas: chemistry, ...
.'' The most recent, 9th edition of the SI Brochure
The International System of Units, known by the international abbreviation SI in all languages and sometimes Pleonasm#Acronyms and initialisms, pleonastically as the SI system, is the modern form of the metric system and the world's most wid ...
does not mention the gradian at all. The previous edition mentioned it only in a footnote, which said the following:
See also
* Angular frequency
In physics, angular frequency "''ω''" (also referred to by the terms angular speed, circular frequency, orbital frequency, radian frequency, and pulsatance) is a scalar measure of rotation rate. It refers to the angular displacement per unit tim ...
* Angular mil
A milliradian ( SI-symbol mrad, sometimes also abbreviated mil) is an SI derived unit for angular measurement which is defined as a thousandth of a radian (0.001 radian). Milliradians are used in adjustment of firearm sights by adjusting ...
(military measurement)
* Harmonic analysis
Harmonic analysis is a branch of mathematics concerned with the representation of Function (mathematics), functions or signals as the Superposition principle, superposition of basic waves, and the study of and generalization of the notions of Fo ...
* Jean-Charles de Borda
Jean-Charles, chevalier de Borda (4 May 1733 – 19 February 1799) was a French mathematician, physicist, and Navy officer.
Biography
Borda was born in the city of Dax to Jean‐Antoine de Borda and Jeanne‐Marie Thérèse de Lacroix.
In 175 ...
* Repeating circle
Developed from the reflecting circle, the repeating circle is an instrument for geodetic surveying, invented by Etienne Lenoir in 1784, while an assistant of Jean-Charles de Borda, who later improved the instrument. It was notable as being the ...
* Spread (rational trigonometry)
''Divine Proportions: Rational Trigonometry to Universal Geometry'' is a 2005 book by the mathematician Norman J. Wildberger on a proposed alternative approach to Euclidean geometry and trigonometry, called rational trigonometry. The book advocat ...
* Steradian
The steradian (symbol: sr) or square radian is the unit of solid angle in the International System of Units (SI). It is used in three-dimensional geometry, and is analogous to the radian, which quantifies planar angles. Whereas an angle in radian ...
(the "square radian")
Notes
References
{{Reflist
External links
Ask Dr Math
* Definitions o
an
on sizes.com
Units of plane angle
Decimalisation
Metrication
Non-SI metric units