"with coefficients in the field generated by these numbers. Therefore, any number that is constructible by a sequence of steps is a root of a minimal polynomial whose degree is a power of two. Note also that radians (60 degrees, written 60°) is constructible. The argument below shows that it is impossible to construct a 20° angle. This implies that a 60° angle cannot be trisected, and thus that an arbitrary angle cannot be trisected. Denote the set of rational numbers by . If 60° could be trisected, the degree of a minimal polynomial of over would be a power"
"Minute The Minute is a unit of time or angle. As a unit of time, the Minute is most of times equal to (the first sexagesimal fraction) of an hour, or 60 seconds. In the UTC time standard, a minute on rare occasions has 61 seconds, a consequence of leap seconds (there is a provision to insert a negative leap second, which would result in a 59-second minute, but this has never happened in more than 40 years under this system). As a unit of angle, the minute of arc is equal to of a degree, or 60 seconds (of"
"arcminutes and arcseconds to measure angles when the object being measured is too small for direct visual inspection. For instance, a toolmaker's optical comparator will often include an option to measure in ""minutes and seconds"". Minute and second of arc A minute of arc, arcminute (arcmin), arc minute, or minute arc 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 – it is for this reason that the Earth's circumference is almost exactly 21,600 nautical miles. A minute of arc"
"Minute and second of arc A minute of arc, arcminute (arcmin), arc minute, or minute arc 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 – it is for this reason that the Earth's circumference is almost exactly 21,600 nautical miles. A minute of arc is of a radian. A second of arc, arcsecond (arcsec), or arc second is of an arcminute, of a degree, of a turn, and (about ) of a radian. These units originated in Babylonian astronomy"
"the location of a place on Earth east or west of the Prime Meridian. Each degree of longitude is sub-divided into 60 minutes, each of which is divided into 60 seconds. A longitude is thus specified in sexagesimal notation as 23° 27′ 30″ E. For higher precision, the seconds are specified with a decimal fraction. An alternative representation uses degrees and minutes, where parts of a minute are expressed in decimal notation with a fraction, thus: 23° 27.5′ E. Degrees may also be expressed as a decimal fraction: 23.45833° E. For calculations, the angular measure may be converted to radians,"
"Degree (angle) A degree (in full, a degree of arc, arc degree, or arcdegree), usually denoted by ° (the degree symbol), is a measurement of a plane angle, defined so that a full rotation is 360 degrees. It is not an SI unit, as the SI unit of angular measure is the radian, but it is mentioned in the SI brochure as an accepted unit. Because a full rotation equals 2 radians, one degree is equivalent to radians. The original motivation for choosing the degree as a unit of rotations and angles is unknown. One theory states that it is"
"geometry, it is the number of seconds in a minute, and the number of minutes in a degree. In normal space, the three interior angles of an equilateral triangle each measure 60 degrees, adding up to 180 degrees. Because it is divisible by the sum of its digits in base 10, it is a Harshad number. A number system with base 60 is called sexagesimal (the original meaning of ""sexagesimal"" is sixtieth). It is the smallest positive integer that is written with only the smallest and the largest digit of base 2 (binary), base 3 (ternary) and base 4 (quaternary)."
"Clock angle problem Clock angle problems are a type of mathematical problem which involve finding the angles between the hands of an analog clock. Clock angle problems relate two different measurements: angles and time. The angle is typically measured in degrees from the mark of number 12 clockwise. The time is usually based on 12-hour clock. A method to solve such problems is to consider the rate of change of the angle in degrees per minute. The hour hand of a normal 12-hour analogue clock turns 360° in 12 hours (720 minutes) or 0.5° per minute. The minute hand rotates"
"the angle is measured as long as each angle is measured in the same unit. One ""radian"" is the angle subtended by an arc of a circle that has the same length as the circle's radius. The radian is the derived quantity of angular measurement in the SI system. By definition, it is dimensionless, though it may be specified as ""rad"" to avoid ambiguity. Angles measured in degrees, are shown with the symbol °. Subdivisions of the degree are minute (symbol ', 1' = 1/60°) and second {symbol "", 1"" = 1/3600°}. An angle of 360° corresponds to the angle"
". It is approximately equal to one milliradian ( ). A mil measuring of a revolution originated in the imperial Russian army, where an equilateral chord was divided into tenths to give a circle of 600 units. This may be seen on a lining plane (an early device for aiming indirect fire artillery) dating from about 1900 in the St. Petersburg Museum of Artillery. Degree (angle) A degree (in full, a degree of arc, arc degree, or arcdegree), usually denoted by ° (the degree symbol), is a measurement of a plane angle, defined so that a full rotation is 360"
"arcsecond on a 10 m class telescope. Minutes (′) and seconds (″) of arc are also used in cartography and navigation. At sea level one minute of arc along the equator or a meridian (indeed, any great circle) equals exactly one geographical mile along the Earth's equator or approximately one nautical mile (1852 meters, or ≈1.15078 statute miles). A second of arc, one sixtieth of this amount, is roughly 30 meters or 100 feet. The exact distance varies along meridian arcs because the figure of the Earth is slightly oblate (bulges a third of a percent at the equator). Positions"
"consults the home port clock, every hour's discrepancy between them translates into another fifteen degrees of longitude. One degree of longitude equals four minutes of time the world over, although in terms of distance, one degree shrinks from 60.15 nautical miles or 111 km [Earth's circumference being: 21,653.521 nautical miles, or 24,901.55 statute miles at the Equator], to virtually nothing at the poles. Precise knowledge of the hour in two different places at once - a longitude prerequisite so easily accessible today from any pair of cheap wristwatches - was utterly unattainable up to and including the era of pendulum"
"11 o'clock are equated to half value, and winds at 12 and 6 o'clock are equated to no value. The minute of angle correction (full value) is then commonly estimated as ((Range [meters] / 100) × Wind [mph]) / ""C"", where ""C"" is a constant. The constant ""C"" equals 15 for ranges from 100 to 500 meters, 14 for 600 meters, 13 for 700–800 meters, 12 for 900 meters, and 11 for 1,000 meters. For full-value winds, this full windage correction is used. For half-value winds, the minute of correction in windage given by this formula is halved; for no-value"
"can be interpreted as a sexagesimal number, meaning seconds. As with the ancient Babylonian sexagesimal system, however, each of the three sexagesimal digits in this number (3, 23, and 17) is written using the decimal system. Similarly, the practical unit of angular measure is the degree, of which there are 360 (six sixties) in a circle. There are 60 minutes of arc in a degree, and 60 arcseconds in a minute. In some usage systems, each position past the sexagesimal point was numbered, using Latin or French roots: ""prime"" or ""primus"", ""seconde"" or ""secundus"", ""tierce"", ""quatre"", ""quinte"", etc. To this"
"on the main scale; multiply this number by 5 and the product will be the number of minutes to be added to the whole number of degrees. For example: Zero on the vernier scale has moved 28 whole degrees to the right of the 0 on the main scale and the 3rd line on the vernier scale coincides with a line upon the main scale as indicated. Multiplying 3 by 5, the product, 15, is the number of minutes to be added to the whole number of degrees, thus indicating a setting of 28 degrees and 15 minutes. Protractors have"
"latitude could be used as the basis for a unit of measurement for distance and proposed the nautical mile as one minute or one-sixtieth () of one degree of latitude. As one degree is of a circle, one minute of arc is of a circle (or, in radians, ). These sexagesimal (base 60) units originated in Babylonian astronomy. Gunter used Snell's circumference to define a nautical mile as 6,080 feet, the length of one minute of arc at 48 degrees latitude. Since the earth is not a perfect sphere but is an oblate spheroid with slightly flattened poles, a minute"
"degree is ""110.6 kilometres"". The circles of longitude, meridians, meet at the geographical poles, with the west-east width of a second naturally decreasing as latitude increases. On the Equator at sea level, one longitudinal second measures ""30.92 metres"", a longitudinal minute is ""1855 metres"" and a longitudinal degree is ""111.3 kilometres"". At 30° a longitudinal second is ""26.76 metres"", at Greenwich (51°28′38″N) ""19.22 metres"", and at 60° it is ""15.42 metres"". On the WGS84 spheroid, the length in meters of a degree of latitude at latitude φ (that is, the distance along a north–south line from latitude (φ − 0.5)"
"are traditionally given using degrees, minutes, and seconds of arcs for latitude, the arc north or south of the equator, and for longitude, the arc east or west of the Prime Meridian. Any position on or above the Earth's reference ellipsoid can be precisely given with this method. However, when it is inconvenient to use base-60 for minutes and seconds, positions are frequently expressed as decimal fractional degrees to an equal amount of precision. Degrees given to three decimal places ( of a degree) have about the precision of degrees-minutes-seconds ( of a degree) and specify locations within about 120"
"The length of 1 minute of latitude is (1.00 nautical miles), while the length of 1 second of latitude is (see nautical mile). In 1687 Isaac Newton published the ""Philosophiæ Naturalis Principia Mathematica"", in which he proved that a rotating self-gravitating fluid body in equilibrium takes the form of an oblate ellipsoid. (This article uses the term ""ellipsoid"" in preference to the older term ""spheroid"".) Newton's result was confirmed by geodetic measurements in the 18th century. (See Meridian arc.) An oblate ellipsoid is the three-dimensional surface generated by the rotation of an ellipse about its shorter axis (minor axis). ""Oblate"
"is a further method of visualising fractions of 60, since we are very used to expressing fractions of an hour (60 minutes) when telling the time. Thus: The angle is the angle of the wind or tide as it presents itself relative to the ship or aircraft, so if the wind is coming from the left at a relative angle of 30°, we use a sine of 0.5. Once the sine has been estimated, the drift due to wind or tide can be estimated accordingly by resolving the velocity of the wind or tide into a forward component and a"
"fractional portions of Babylonian numerals and other mesopotamian systems, by Hellenistic astronomers using Greek numerals for the fractional portion only, and is still used for modern time and angles, but only for minutes and seconds. However, not all of these uses were positional. Modern time separates each position by a colon or point. For example, the time might be 10:25:59 (10 hours 25 minutes 59 seconds). Angles use similar notation. For example, an angle might be 10°25'59"" (10 degrees 25 minutes 59 seconds). In both cases, only minutes and seconds use sexagesimal notation—angular degrees can be larger than 59 (one"
"those points (unless on the equator, where these are equal); the difference is less than . A geographical mile is defined to be the length of one minute of arc along the equator (one equatorial minute of longitude), therefore a degree of longitude along the equator is exactly 60 geographical miles or 111.3 kilometers, as there are 60 minutes in a degree. The length of 1 minute of longitude along the equator is 1 geographical mile or , while the length of 1 second of it is 0.016 geographical mile or . Planetary co-ordinate systems are defined relative to their"
"determine latitude at sea. He reasoned that the lines of latitude could be used as the basis for a unit of measurement for distance and proposed the nautical mile as one minute or one-sixtieth () of one degree of latitude. As one degree is of a circle, one minute of arc is of a circle – such that the polar circumference of the Earth would be exactly 21,600 miles. Gunter used Snell's circumference to define a nautical mile as 6,080 feet, the length of one minute of arc at 48 degrees latitude. In 1791, the French Academy of Sciences selected"
"is required, the numbers work out with the same rule: 1% over 1 NM ≈ 60' It is also useful to find out the lateral deviation from a given VOR course or radial: Each dot on a VOR indicator represents 2° of deviation, or 200' per dot per DME. There are other applications to this rule. One such application is time drift. An hour is equal to 60 minutes, and a minute is equal to 60 seconds, so some other relationships between angle and time can be observed. If a pilot is flying a leg of 120 miles and finds"
"the 10 minute entry on the 75 degree page, shown above-right, which is 0.9666746. However, this answer is only accurate to four decimal places. If one wanted greater accuracy, one could interpolate linearly as follows: From the Bernegger table: The difference between these values is 0.0000745. Since there are 60 seconds in a minute of arc, we multiply the difference by 50/60 to get a correction of (50/60)*0.0000745 ≈ 0.0000621; and then add that correction to sin (75° 9′) to get : A modern calculator gives sin (75° 9′ 50″) = 0.96666219991, so our interpolated answer is accurate to the"
"The angle may be measured in degrees or in time, with 24 = 360° exactly. In astronomy, hour angle is defined as the angular distance on the celestial sphere measured westward along the celestial equator from the meridian to the hour circle passing through a point. It may be given in degrees, time, or rotations depending on the application. In celestial navigation, the convention is to measure in degrees westward from the prime meridian (Greenwich hour angle, GHA), from the local meridian (local hour angle, LHA) or from the first point of Aries (sidereal hour angle, SHA). The hour angle"
"of longitude, to correlate with the established 24-hour day convention. Finally, it may be the case that more than one of these factors has come into play. According to that theory, the number is approximately 365 because of the apparent movement of the sun against the celestial sphere and that it was rounded to 360 for some of the mathematical reasons cited above. For many practical purposes, a degree is a small enough angle that whole degrees provide sufficient precision. When this is not the case, as in astronomy or for geographic coordinates (latitude and longitude), degree measurements may be"
"Gradian The gradian is a unit of measurement of an angle, equivalent to formula_1 of a turn, formula_2 of a degree, or formula_3 of a radian. The gradian is defined as formula_4 of the right angle (in other words, there are 100 gradians in the right angle), which implies a full turn being 400 gradians. It is also known as gon (from Greek /""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"
"a trigonometric table, which he needed when computing the eccentricity of the orbits of the Moon and Sun. He tabulated values for the chord function, which gives the length of the chord for each angle. He did this for a circle with a circumference of 21,600 and a radius (rounded) of 3438 units: this circle has a unit length of 1 arc minute along its perimeter. He tabulated the chords for angles with increments of 7.5°. In modern terms, the chord of an angle equals the radius times twice the sine of half of the angle, i.e.: He described the"
"one minute of arc along a meridian of a spherical earth. An ellipsoid model leads to a variation of the nautical mile with latitude. This was resolved by defining the nautical mile to be exactly 1,852 metres. However for all practical purposes distances are measured from the latitude scale of charts. As the Royal Yachting Association says in its manual for day skippers: ""1 (minute) of Latitude = 1 sea mile"", followed by ""For most practical purposes distance is measured from the latitude scale, assuming that one minute of latitude equals one nautical mile"". There are six auxiliary latitudes that"
"this system. In most mathematical work beyond practical geometry, angles are typically measured in radians rather than degrees. This is for a variety of reasons; for example, the trigonometric functions have simpler and more ""natural"" properties when their arguments are expressed in radians. These considerations outweigh the convenient divisibility of the number 360. One complete turn (360°) is equal to 2"""" radians, so 180° is equal to radians, or equivalently, the degree is a mathematical constant: 1° = . The turn (or revolution, full circle, full rotation, cycle) is used in technology and science. One turn is equal to 360°."
"particular knot can be tied into the string if travelling to a known latitude. The knots were typically tied to measure angles of one finger-width. When held at arm's length, the width of a finger measures an angle that remains fairly similar from person to person. This was widely used (and still is today) for rough angle measurements, an angle known as ""issabah إصبع"" for 'finger' in Arabic, or a ""chih"" in Chinese. By modern measure, this is about 1 degree, 36 minutes, and 25 seconds, or just over 1.5 degrees. It is equal to the arcsine of the ratio"
"degrees to (φ + 0.5) degrees) is about Similarly, the length in meters of a degree of longitude can be calculated as An alternative method to estimate the length of a longitudinal degree at latitude formula_5 is to assume a spherical Earth (to get the width per minute and second, divide by 60 and 3600, respectively): where Earth's average meridional radius formula_7 is . Since the Earth is not spherical that result can be off by several tenths of a percent; a better approximation of a longitudinal degree at latitude formula_5 is where Earth's equatorial radius formula_10 equals ""6,378,137 m"""