Gradian
The gradian, also known as the gon or grade, is a unit of angular measurement defined as one-hundredth of a right angle, equivalent to 1/400 of a full circle.[1][2] It measures angles in a centesimal system, where a right angle equals 100 gradians and a complete revolution equals 400 gradians.[3] One gradian corresponds to 0.9 degrees or approximately 0.015708 radians (π/200).[4][5] This unit facilitates decimal-based calculations, aligning with the metric system's emphasis on powers of ten, and subdivisions include centigradians (0.01 gradian), often used for precise measurements.[6] The gradian originated in France during the development of the metric system in the late 18th and early 19th centuries, proposed as a rational alternative to the sexagesimal degree system to promote decimal consistency in scientific and technical fields.[7] It gained traction alongside other metric reforms, such as the centigrade temperature scale, but its adoption remained limited outside specific applications.[8] Primarily used in surveying, civil engineering, and navigation—especially in Europe—the gradian simplifies right-angle divisions and decimal arithmetic for land measurement and mapping tasks.[9][10] It appears in some geological and mining contexts, as well as on scientific calculators and software supporting angular computations, though degrees and radians dominate globally.[6][8]Definition and Fundamentals
Definition
The gradian, also known as the gon or grade, is a unit of plane angle defined as one four-hundredth of a full circle.[1] This makes it a centesimal measure, where the entire circumference is partitioned into 400 equal gradian units for angular quantification.[5] In this system, a right angle—or quadrant—corresponds precisely to 100 gradians, emphasizing its alignment with decimal subdivisions.[11] The gradian's structure thus divides the circle into parts that are multiples of 0.01 of a quadrant, promoting ease in decimal arithmetic for geometric computations.[9]Symbol and Notation
The gradian, serving as a decimal-based unit for plane angle measurement, employs specific symbols and notations in technical literature and standards. The international standard designates "gon" as the official name and symbol for the unit.[12] In contemporary usage, particularly in mathematical and engineering contexts, the primary notation for expressing angles in gradians is a superscript "g" placed after the numerical value, analogous to the degree symbol; for example, a right angle is written as $100^\text{g}.[13] This superscript form distinguishes gradian measurements from degrees while maintaining compact readability in formulas and diagrams. The unit symbol "gon" is used for the unit itself, while the superscript "g" denotes angles measured in gradians. Alternative notations include the abbreviations "gr" and "gon", which appear in various international texts and software implementations for compatibility and clarity.[14] Historical variations trace back to early French developments, where the unit was termed "grade" and abbreviated as "grd" in older texts, reflecting its origins in metric system proposals.[14] The International Organization for Standardization (ISO) established "gon" as the preferred symbol in ISO 80000-3:2019 to promote uniformity across languages and avoid ambiguity with other terms like "grad" for gradient.[12]Historical Development
Origins and Etymology
The gradian, also known as the grade or gon, emerged from efforts by the French Academy of Sciences in the 18th and 19th centuries to reform angular measurement as part of the broader metric system overhaul, aiming to replace the cumbersome sexagesimal divisions of the circle (based on 360 degrees) with a purely decimal system for simplified calculations in science and engineering.[15] Early proposals during the French Revolution sought a universal, rational framework tied to natural phenomena, much like the metre's basis in Earth's meridian; the decimal system for angles was introduced by the law of 11 Brumaire Year IV on 1 November 1795, where the right angle equaled 100 grades, dividing the full circle into 400 grades to align with base-10 arithmetic.[15] This work emphasized the practical benefits of decimal subdivisions for fields like astronomy and geodesy, where traditional degrees complicated computations. In 1897, a commission including Henri Poincaré advocated for the system's adoption, highlighting its advantages for calculations without needing two-digit multiplications in conversions.[15] Etymologically, the term "grade" derives from the French "grade," meaning a step or degree, reflecting the unit's conception as incremental divisions akin to steps in a decimal progression. To promote linguistic neutrality and avoid confusion with the English "grade" denoting slope or incline, the name evolved to "gon" in the 20th century, drawn from the Greek "gōnia" (γωνία), signifying corner or angle, paralleling its use in terms like "polygon."[16]Adoption and Decline
The gradian experienced limited adoption primarily in European surveying contexts. It was employed in French surveying practices until the mid-20th century, aligning with the country's metric reforms and facilitating decimal-based angular calculations in land measurement and mapping, as well as in Swiss systems, where the gon appears in official projection formulas for coordinate transformations. The unit's inclusion in ISO standards, such as ISO 80000-1:2009 for general quantities and units and ISO/IEC 13249-3:2016 for information technology data types, recognizes it as a valid plane angle measure but renders it non-mandatory alongside the preferred radian. The gradian's decline stemmed from the dominant tradition of the degree unit in astronomy, navigation, and international scientific literature, where compatibility with historical tables and instruments favored the sexagesimal system. Post-1970s computational developments further entrenched this shift, as early digital surveying software and calculators were predominantly programmed for degrees, creating inertia against adopting the gradian despite its decimal advantages. By the late 20th century, it had become largely obsolete outside niche European applications, supplanted by degrees for broader interoperability.Conversions and Mathematical Relations
Formulas for Conversion
The gradian, also known as the gon, is defined such that a full circle corresponds to 400 gradians, providing a basis for conversions to other angular units. This equivalence stems from the unit's design, where 400 gradians equal 360 degrees and 2π radians.[17] To convert gradians to degrees, the formula is derived by dividing the full-circle values: degrees = gradians × (360/400) = gradians × 0.9. Thus, 1 gradian = 0.9 degrees. The inverse conversion is gradians = degrees × (400/360) = degrees × (10/9).[17] For conversion to radians, the relation follows from the full-circle equivalences: radians = gradians × (2π/400) = gradians × (π/200). Therefore, 1 gradian = π/200 radians, approximately 0.01570796 radians. The bidirectional formula is gradians = radians × (400/(2π)) = radians × (200/π).[18][17] These formulas reflect the gradian's alignment with a decimal structure for angular measurement, facilitating calculations in systems preferring base-10 divisions.[18]Equivalences with Other Units
The gradian, denoted as gon, equates to one-fourth of a right angle, making a full circle 400 gradians, which corresponds exactly to 360 degrees, 2π radians, and 1 turn. Similarly, a right angle measures 100 gradians, equivalent to 90 degrees, π/2 radians, and 0.25 turns. These relations stem from the gradian's centesimal basis, dividing the circle into 400 equal parts for alignment with decimal systems.[1] In comparisons to sexagesimal subdivisions, 1 gradian equals 0.9 degrees and thus 54 arcminutes, while 1 degree approximates 1.111 gradians (precisely 10/9 gradians). One gradian further subdivides to 3240 arcseconds.[19] Such equivalences facilitate interoperability in fields like surveying, where gradians align with metric precision.[1] The following table summarizes breakdowns of a full circle across key units, including percentages for proportional representation:| Description | Gradians (gons) | Degrees (°) | Radians (rad) | Percentage of Circle (%) |
|---|---|---|---|---|
| Full Circle | 400 | 360 | 2π | 100 |
| Right Angle (Quadrant) | 100 | 90 | π/2 | 25 |
| 1 Degree | 1.111... (10/9) | 1 | π/180 | 0.277... (1/360) |
| 1 Turn | 400 | 360 | 2π | 100 |