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29-623: Three variants were studied, one with three F-1 engines in the first stage, one with four, and one with five. Without the S-II stage, which made up a large fraction of the mass of the Saturn V, a version of the INT-20 using an unmodified five-engine version of the S-IC booster would be greatly overpowered and accelerate substantially faster than the Saturn V. This would create excessive aerodynamic stress in

58-588: A quarter meridian . So ⁠ 10,000,000 m / 90 × 60 ⁠ = 1,851.85 m ≈ 1,852 m became the metric length for a nautical mile. France made it legal for the French Navy in 1906, and many metric countries voted to sanction it for international use at the 1929 International Hydrographic Conference. Both the United States and the United Kingdom used an average arcminute—specifically,

87-406: A degree (5866 ⁠ 2 / 3 ⁠ feet per arcminute ). In 1633, William Oughtred suggested 349,800 feet to a degree (5830 feet per arcminute). Both Gunter and Oughtred put forward the notion of dividing a degree into 100 parts, but their proposal was generally ignored by navigators. The ratio of 60 miles, or 20 leagues, to a degree of latitude remained fixed while the length of the mile

116-423: A degree) of latitude at the equator, so that Earth's polar circumference is very near to 21,600 nautical miles (that is 60 minutes × 360 degrees). Today the international nautical mile is defined as 1,852 metres (about 6,076 ft; 1.151 mi). The derived unit of speed is the knot , one nautical mile per hour. There is no single internationally agreed symbol, with several symbols in use. The word mile

145-579: A minute of arc of a great circle of a sphere having the same surface area as the Clarke 1866 ellipsoid . The authalic (equal area) radius of the Clarke 1866 ellipsoid is 6,370,997.2 metres (20,902,222 ft). The resulting arcminute is 1,853.2480 metres (6,080.210 ft). The United States chose five significant digits for its nautical mile, 6,080.2 feet , whereas the United Kingdom chose four significant digits for its Admiralty mile, 6,080 feet. In 1929

174-496: Is allowed to take on arbitrary real values. The infinitesimal arc length in these coordinates is So the length of a curve γ {\displaystyle \gamma } from p {\displaystyle p} to q {\displaystyle q} is a functional of the curve given by According to the Euler–Lagrange equation , S [ γ ] {\displaystyle S[\gamma ]}

203-498: Is from the Latin phrase for a thousand paces: mille passus . Navigation at sea was done by eye until around 1500 when navigational instruments were developed and cartographers began using a coordinate system with parallels of latitude and meridians of longitude . The earliest reference of 60 miles to a degree is a map by Nicolaus Germanus in a 1482 edition of Ptolemy 's Geography indicating one degree of longitude at

232-645: Is minimized if and only if where C {\displaystyle C} is a t {\displaystyle t} -independent constant, and From the first equation of these two, it can be obtained that Integrating both sides and considering the boundary condition, the real solution of C {\displaystyle C} is zero. Thus, ϕ ′ = 0 {\displaystyle \phi '=0} and θ {\displaystyle \theta } can be any value between 0 and θ 0 {\displaystyle \theta _{0}} , indicating that

261-450: Is not a perfect sphere ), as well as on spheroidal celestial bodies . The equator of the idealized earth is a great circle and any meridian and its opposite meridian form a great circle. Another great circle is the one that divides the land and water hemispheres . A great circle divides the earth into two hemispheres and if a great circle passes through a point it must pass through its antipodal point . The Funk transform integrates

290-469: The Equator contains " milaria 60 ". An earlier manuscript map by Nicolaus Germanus in a previous edition of Geography states " unul gradul log. et latitud sub equinortiali formet stadia 500 que fanut miliaria 62 ⁠ 1 / 2 ⁠ " ("one degree longitude and latitude under the equator forms 500 stadia , which make 62 ⁠ 1 / 2 ⁠ miles"). Whether a correction or convenience,

319-519: The Saturn IB and Saturn V, and re-using Saturn V components would reduce costs and simplify ground operations compared to building an entirely new launcher in that payload range. Nautical mile A nautical mile is a unit of length used in air, marine, and space navigation , and for the definition of territorial waters . Historically, it was defined as the meridian arc length corresponding to one minute ( ⁠ 1 / 60 ⁠ of

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348-451: The class of all regular paths from a point p {\displaystyle p} to another point q {\displaystyle q} . Introduce spherical coordinates so that p {\displaystyle p} coincides with the north pole. Any curve on the sphere that does not intersect either pole, except possibly at the endpoints, can be parametrized by provided ϕ {\displaystyle \phi }

377-493: The curve must lie on a meridian of the sphere. In a Cartesian coordinate system , this is which is a plane through the origin, i.e., the center of the sphere. Some examples of great circles on the celestial sphere include the celestial horizon , the celestial equator , and the ecliptic . Great circles are also used as rather accurate approximations of geodesics on the Earth 's surface for air or sea navigation (although it

406-453: The distance along a great circle was 60 miles per degree. However, these referred to the old English mile of 5000 feet and league of 15,000 feet, relying upon Ptolemy's underestimate of the Earth's circumference . In the early seventeenth century, English geographers started to acknowledge the discrepancy between the angular measurement of a degree of latitude and the linear measurement of miles. In 1624 Edmund Gunter suggested 352,000 feet to

435-458: The extra cost and complexity of the fifth engine was unjustified. A four-engine variant would launch with four engines firing and shut down two engines 146 seconds after launch. The remaining two engines would burn until first-stage shutdown 212 seconds after launch. This variant could put approximately 132,000 pounds (60,000 kg) into a 100 nautical mile (185 km or 115 statute mile) orbit, versus around 250,000 pounds (110,000 kg) for

464-496: The great circles on the n -sphere are the intersection of the n -sphere with 2-planes that pass through the origin in the Euclidean space R . Half of a great circle may be called a great semicircle (e.g., as in parts of a meridian in astronomy ). To prove that the minor arc of a great circle is the shortest path connecting two points on the surface of a sphere, one can apply calculus of variations to it. Consider

493-572: The international nautical mile was defined by the First International Extraordinary Hydrographic Conference in Monaco as exactly 1,852 metres (which is 6,076.12 ft). The United States did not adopt the international nautical mile until 1954. Britain adopted it in 1970, but legal references to the obsolete unit are now converted to 1,853 metres (which is 6,079.40 ft). The metre

522-424: The location. Great circle In mathematics , a great circle or orthodrome is the circular intersection of a sphere and a plane passing through the sphere's center point . Any arc of a great circle is a geodesic of the sphere, so that great circles in spherical geometry are the natural analog of straight lines in Euclidean space . For any pair of distinct non- antipodal points on

551-470: The low atmosphere. Several solutions to this problem were considered. Using the original five-engine S-IC would require three engines to be shut down 88 seconds after launch, with the remainder of the first-stage flight flown on only two engines. This meant that a considerable amount of the firing time would be carrying three engines of "dead weight". As a consequence the extra payload over a four-engine variant would only have been about one thousand pounds, and

580-430: The measurement based on this ( ⁠ 40,075.017 km / 360 × 60 ⁠ = 1,855.3 metres) is known as the geographical mile . Using the definition ⁠ 1 / 60 ⁠ of a degree of latitude on Mars , a Martian nautical mile equals to 983 m (1,075 yd). This is potentially useful for celestial navigation on a human mission to the planet , both as a shorthand and a quick way to roughly determine

609-435: The points (the intrinsic distance on a sphere), and is proportional to the measure of the central angle formed by the two points and the center of the sphere. A great circle is the largest circle that can be drawn on any given sphere. Any diameter of any great circle coincides with a diameter of the sphere, and therefore every great circle is concentric with the sphere and shares the same radius . Any other circle of

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638-403: The poles and 1,843 metres at the Equator. France and other metric countries state that in principle a nautical mile is an arcminute of a meridian at a latitude of 45°, but that is a modern justification for a more mundane calculation that was developed a century earlier. By the mid-19th century, France had defined a nautical mile via the original 1791 definition of the metre , one ten-millionth of

667-467: The reason for the change from 62 ⁠ 1 / 2 ⁠ to 60 miles to a degree is not explained. Eventually, the ratio of 60 miles to a degree appeared in English in a 1555 translation of Pietro Martire d'Anghiera 's Decades: "[Ptolemy] assigned likewise to every degree three score miles." By the late 16th century English geographers and navigators knew that the ratio of distances at sea to degrees

696-497: The sphere is called a small circle , and is the intersection of the sphere with a plane not passing through its center. Small circles are the spherical-geometry analog of circles in Euclidean space. Every circle in Euclidean 3-space is a great circle of exactly one sphere. The disk bounded by a great circle is called a great disk : it is the intersection of a ball and a plane passing through its center. In higher dimensions,

725-430: The sphere, there is a unique great circle passing through both. (Every great circle through any point also passes through its antipodal point, so there are infinitely many great circles through two antipodal points.) The shorter of the two great-circle arcs between two distinct points on the sphere is called the minor arc , and is the shortest surface-path between them. Its arc length is the great-circle distance between

754-447: The three-stage Saturn V. The three-engine variant would burn all three engines up to first-stage shutdown at 146 seconds after launch. This variant could put approximately 78,000 pounds (35,000 kg) of payload into a 100 nautical mile (185 km) orbit, around 2.5 times the useful payload of the Saturn IB. Both three- and four-engine variants would therefore have provided useful payload capacities ( Saturn C-3 ) intermediate between

783-571: Was constant along any great circle (such as the equator , or any meridian), assuming that Earth was a sphere. In 1574, William Bourne stated in A Regiment for the Sea the "rule to raise a degree" practised by navigators: "But as I take it, we in England should allowe 60 myles to one degrée: that is, after 3 miles to one of our Englishe leagues, wherefore 20 of oure English leagues shoulde answere to one degrée." Likewise, Robert Hues wrote in 1594 that

812-491: Was originally defined as 1 ⁄ 10,000,000 of the length of the meridian arc from the North pole to the equator (1% of a centesimal degree of latitude), thus one kilometre of distance corresponds to one centigrad (also known as centesimal arc minute) of latitude. The Earth's circumference is therefore approximately 40,000 km. The equatorial circumference is slightly longer than the polar circumference –

841-453: Was revised with better estimates of the earth’s circumference. In 1637, Robert Norwood proposed a new measurement of 6120 feet for an arcminute of latitude, which was within 44 feet of the currently accepted value for a nautical mile. Since the Earth is not a perfect sphere but is an oblate spheroid with slightly flattened poles, a minute of latitude is not constant, but about 1,862 metres at

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